Research ArticlePricing Strategies in Dual-Channel Supply Chain witha Fair Caring Retailer
Lufeng Dai Xifu Wang Xiaoguang Liu and Lai Wei
School of Traffic and Transportation Beijing Jiaotong University Beijing China
Correspondence should be addressed to Lufeng Dai 15114213bjtueducn
Received 11 September 2018 Revised 2 December 2018 Accepted 28 January 2019 Published 18 April 2019
Guest Editor Ahmet Sensoy
Copyright copy 2019 Lufeng Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Manufacturers add online direct channels that inevitably engage in channel competition with offline retail channels Since priceis an important factor in consumers choice of purchasing channel pricing strategy has become a popular topic for research ondual-channel competition and coordination In contrast to previous research on pricing strategies based on the full rationality ofmembers we focus on the impact of retailers fairness concerns on pricing strategies In this study the hybrid dual-channel supplychain consists of one manufacturer with a direct channel who acts as the leader and a retailer who acts as the follower First we usethe Stackelberg game approach to determine the equilibrium pricing strategy for a fair caring retailer Simultaneously we consider acentralized dual-channel supply chain as the benchmark for a comparative analysis of the efficiency of a decentralized supply chainFurthermore we study pricing strategies when the retailer has fairness concerns and determine the complete equilibrium solutionsfor different ranges of the parameters representing cross-price sensitivity and fairness Finally through numerical experimentsthe pricing strategies the profit and utility of the manufacturer and retailer and the channel efficiency of the supply chain arecompared and analysed for two scenarios We find that fairness concerns reduce the manufacturers profits while for the most partthe retailersrsquo profit can be improved however the supply chain cannot achieve complete coordination
1 Introduction
In recent years the number of people who shop online hasgrown Increasingly manufacturers such as Lenovo Dell andNike have added direct channels to increase their profitabilityWhen a manufacturer sells through a traditional retailerand has a direct channel to consumers it is called a dual-channel distribution system In a dual-channel distributionsystem the manufacturer and its retailers sell essentially thesame products Compared with traditional retail channelsonline direct sales channels have lower operating costs andconsumers are given choices that provide them with moreconvenience and price discount (Takahashi K [1]) Supply-demand and competitive relationships coexist between themanufacturer and retailer after an online direct market-ing channel opens Competition may also lead to conflictsbetween the two channels in terms of cross-channel price andoperations
With the popularity of online shopping the gaps betweenchannels in some dual supply chains are narrowing as
consumers have gradually adapted to the heterogeneityof channels and shopped more rationally According toAccentures Chinese consumer research report for 2018[2]preference of consumers for shopping online or going tostores is almost equal particularly smart digital consumerspaying more attention to price comparison China-ASEANMobile Internet Industry Alliance released a survey reporton comparison of online and offline shopping behaviors ofconsumers for 2018 [3]The report shows that the consideredfactors for consumers to choose online or offline shopping arebasically the same Especially when it comes to the productsthat consumers often buy online such as cosmetics clothingelectronic products FMCG etc people tend to pay moreattention to the price of products The narrowing of channeldifferences means that different channels will cover the samecustomer groups and make the problem of channel pricecompetition become more prominent
A large number of investigations and studies have con-firmed that firms similar to individuals have fairness prefer-ences when they are treated unfairly in business (RabinM [4]
HindawiComplexityVolume 2019 Article ID 1484372 23 pageshttpsdoiorg10115520191484372
2 Complexity
Fehr E [5] Kahneman D [6] Kumar N [7]) and have shownthat fairness concerns have a significant impact on decision-making Compared with the traditional supply chain thedual-channel supply chain easily occur unfair cooperationdue to channel competition and other factors which willlead to conflicts of interest and even the breakdown ofcooperation In order to increase sales high-end liquorcompanies such as Yibin Wuliangye Group Company Ltdhave added online direct channels and offered discountedproduct on it (Dong Zhao [8])The goodmarket prospects ofChinese liquor make the liquor enterprises raise the whole-sale price Meanwhile the liquor enterprises were tryingto stabilize the offline retail price and limit the minimumselling price to maintain the brand image (Dong Zhao [8]Chen Xing [9]) The impact of online direct selling and thepassive situation of offline retail made some retailers feelthat they have been treated unfairly and privately took theway of price-off promotions which destroyed the offline pricesystem and damaged the interest ofWuliangye Similar unfairphenomenon also happened in the cooperation betweenGome andGreeElectricAppliances (Xuefei Zhong [10])Howshould manufacturers coordinate channel conflicts by pricestrategies considering retailersrsquo fairness concern
This paper try to apply a fairness preference and thecross-price effect to the dual-channel supply chain decision-making model to obtain results and insights more in linewith actual management practice In this study we consider adual-channel supply chain where a manufacturer acts as theleader and a fair caring retailer is the follower in Stackelberggame and approach the complete equilibrium solutions of thesupply chain members
The remainder of this paper is organized as follows InSection 2 we review dual-channel supply chain literaturesrelated to pricing strategy and fairness In Section 3 wedevelop the structure and demand functions of a dual-channel supply chain and discuss the model assumptionsand describe the parameters In Section 4 we discuss theequilibrium pricing strategies of the members when theretailer has no fairness concerns In Section 5 we analysethe equilibrium pricing strategies of the manufacturer and afair caring retailer In Section 6 a numerical experiment isused to analyse the influence of fairness concerns and cross-price sensitivity on member decisions and supply chain per-formance We conclude by presenting the equilibrium resultsand suggesting directions for future research in Section 7 Allof the proofs are provided in Appendix
2 Literature Review
Our paper is related to two streams of research pricingstrategy and fairness concern in the dual-channel supplychain
21 Pricing Strategy Researches inDual-Channel Supply ChainChiang et al [11] showed that a vertically integrated onlinechannel allows the manufacturer to constrain its retailerrsquospricing behavior in a dual-channel supply chain Guo Yajun[12] showed that adding an online direct channel can expand
the market but it may also exacerbate channel conflictsand depress the retail price leading to a loss in retailersrsquoprofit Cattani et al [13] studied the situation in whichmanufacturers adopt different pricing strategies in order toalleviate channel conflicts and found that reducing wholesaleprices could alleviate the double margin effect and improvesupply chain performance (Qing Fang [14]) Considering thedual-channel supply chain led by retailers and manufactur-ers respectively and making a comparative analysis of theoptimal price decision the dominant party will make thewholesale price beneficial to maximize its profit Yan R [15]constructed the channel demand model based on consumerutility and studied the pricing strategy considering con-sumers have same price sensitivity of different channels Thispaper found that channel conflict can be alleviated by pricestrategy after manufacturers encourage retailers to improveretail services Based on the same channel price sensitivity ofconsumers Tian J F [16] studied the pricing strategy whenmanufacturers develop retail service With the developmentof dual-channel supply chain some researchers have foundthat consumers have gradually adapted to the heterogeneityof channels especially the products that consumers oftenbuy online Therefore price comparison has become thefocus of consumer attention Xu et al [17] analysed theprice comparison behavior of consumers and its impact ondecision-making and profit of supply chain members andfound that the retailers and supplier are all more willing toavoid the existence of price comparison with the objectiveof profit maximization Shen et al [18] researched on thepricing strategy considering price comparison behavior anddesigned the corresponding coordination mechanism Toreduce channel conflict Bo Li [19] considered a consistentpricing strategy in the two channels which means that theprice in the direct channel is equal to that in the retail channelZhang F [20 21] established the dynamic price game modeland analysed the impact of price adjustment on the profit ofsupply chain members Excessive price adjustment is oftendetrimental to their own interests but will make the otherside to get more profits Many scholars (eg [22ndash25]) havestudied the price strategies for different channel structuresand different product strategies
The papers mentioned above have shown that pricestrategies play an important role in allocating channel profitsand coordinating channel conflict In dual-channel supplychain price competition and channel conflicts may makemembers paymore attention to the distribution of profits thispaper make attempts to integrate members fair preferencesinto pricing studies
22 Fairness Concern Researches in Dual-Channel SupplyChain Apart from the single-channel supply chain field fewscholars are engaged in applying fairness preferences to thedual-channel supply chain in their research (YouQ et al [26]Guangxing Wei et al [27]) among which Tengfei Nie andShaofu Du [28] Qinghua Li and Bo Li[29] and Fang Z etal [20] are the most representative Reference [28] studiedthe application of a quantity discount contract in a dyadicsupply chain consisting of one supplier and two retailerswith no cross-price influence between channels Retailers also
Complexity 3
focus on both horizontal and vertical fairness This articledetermines the pricing strategies of the members of a supplychain when the fairness parameter differs Further it alsointroduces other coordination mechanisms to prove that thequantity discount contract cannot fully coordinate the supplychain Reference [29] considered that the retailer providesvalue-added services and they study the pricing decisions ofthe supply chain members for two scenarios one in whichthe retailer has fairness concerns and a second in which itdoes not The partial equilibrium solution of the channelquota is given in this article but the complete equilibriumsolution is not discussed Reference [20] investigated twononcooperative dynamic game models a Stackelberg gamemodel and a vertical Nash game model The paper usednumerical experiments to analyse the influence of the retailerfairness preference on the dynamic behavior of supply chainmembers The FS fairness model is simplified from using apiecewise function to using a continuous function to discusshow the retailers behavior related to its fairness concernsinfluencesmember decisions and utility (Fujing Xu et al [30]Lei Wang et al [31] Bo Li et al [32])
In the aforementioned articles there is a lack of attentionto fairness concerns and cross-price sensitivity Consideringthe influence of fairness concerns on strategy only the localequilibrium solution of the members is inferred and thecomplete process of the memberrsquos game cannot be fullyunderstood based on this analysis In practice when themarket environment and the level of fairness concerns ofthe members change there are multiple equilibriums whichmeans that the members will adopt different strategiesTherefore the complete equilibrium solutions and the cor-responding management significance will become a focus ofthis paper
3 Problem Statement
31 Model Assumption and Notation 119908 per unit wholesaleprice of the manufacturer119901119903 per unit offline retail price of the retailer119901119890 per unit online direct price of the manufacturer119908119899 the superscript n takes the values of 119889lowast and 119891 lowast lowastwhich denote the optimal wholesale pricing strategies withand without fairness concerns119901119899119894 the superscript n takes the values of 119888lowast 119889lowast and 119891lowastlowastwhich denote the optimal strategies under the centralized anddecentralized supply chain without fairness and the strategiesconsidering fairness respectively The subscript 119894 takes thevalues of 119903 and 119890119886 the potential market demand of the channel119888 the manufacturerrsquos marginal cost per unit120579 the cross-price sensitivity between channels120572 a parameter reflecting the per unit difference in thepayoffs of the manufacturer and the retailer when the retailerencounters disadvantageous unfairness119889119903 the demand function of the offline retail channel119889119890 the demand function of the online direct channel120587119894 the subscript 119894 takes the values of 119888 119889 119898 and 119903which denote the total profit of both the centralized and
ManufacturerManufacturer
RetailerRetailer
ConsumerConsumer
w
Pe
Pr
Figure 1 The dual-channel supply chain structure
decentralized supply chains the manufacturerrsquos profit andthe retailerrsquos profits without fairness concerns respectively120587119891119894 the superscript 119891 denotes the scenario with fairnessconcerns the subscript 119894 takes the values of 119888119898 and 119903 whichdenote the total profit of the supply chain the manufacturerrsquosprofit and the retailerrsquos profit respectively
The following assumptions are made in our model (1)The manufacturer and the retailer only sell one kind ofproduct (2)Thepotential share of both the online and offlinechannels is the same (3) Information is symmetrical betweenthe manufacture and the retailer (4)The retailer is in a weakposition
32 The Structure of the Dual-Channel Chain and Chan-nel Demand Function We consider a representative dual-channel structure in our study (see Figure 1)The supply chainconsists of manufacturer M and retailer RThe manufacturercreates infinitely divisible and homogeneous products andsells through both traditional offline retailers and onlinedirect sales channels We establish a Stackelberg game modelto describe the problem between a rational manufacturer anda retailer with a fairness concern The process of the gameis as follows the manufacturer as the initiator of the gamefirst determines the online direct selling price 119901119890 and thewholesale price 119908 the retailer acts as the follower and thensets the offline retail price 119901119903
Linear demand functions are used to characterize channeldemand and have been adopted in studies (Yue X [33] HuangS [34]) and the corresponding demand functions to themanufacturer and the retailer are described as follows
119889119903 = 119886119903 minus 119887119903119901119903 + 120579119901119890 (1)
119889119890 = 119886119890 minus 119887119890119901119890 + 120579119901119903 (2)
The differences in channel characteristics are mainlyreflected in channel price elasticity 119887 and basic marketdemand of channel 119886 Subscript 119903 and 119890 represent the offlineand online channels respectively Channel price elasticitydepicts consumers sensitivity to channel price Basic marketdemand of channel reflects consumers loyalty to the channelWhen channel differences decrease market characteristicscorresponding to the original differentiated channels aregradually converging Some scholars (Yan R [15] GuangyeXu [35]) consider consumers have the same price sensitivityto different channels but the channel loyalty is differentChannel loyalty of consumers is changing with the shoppinghabit Research on Chinarsquos digital consumers released byMcKinsey Greater China in 2017 states that over 90 ofconsumers compare online and offline channels when buyingconsumer electronics (Wei Wang [36]) According to a white
4 Complexity
paper on big data and online cosmetics consumption in 2016although the growth rate of e-commerce has been slowingsince 2012 the proportion of online and offline consumptionis expected to be evenly divided into 2018 (Niuli [37])Therefore this paper considers a situationwhere the potentialshare of the channels is the same Without loss of generalitywe set parameter 119887119903 and 119887119890 to 1 as were done in the studies([23] LiuM [38]) 120579 is the coefficient of cross-price sensitivityThe equation 0 lt 120579 lt 1 reflects the own-price effects whichare greater than the cross-price effects
Obviously it is necessary to impose additional inequityconstraints on the parameters to guarantee the operation ofthe dual channels (i) 119901119903 ge 119908 119901119890 ge 119908 If 119901119890 lt 119908 thenthe retailer will find a less expensive source from the directchannel (ii) 119889119890 ge 0 and 119889119903 ge 0 which ensures that everychannel has sales (iii) 119908 ge 1198884 Pricing Strategy of Members When theRetailer Has No Fairness Concerns
41 Equilibrium Analysis of the Centralized Supply ChainTo examine the efficiency of the decentralized supply chainboth with and without fairness concerns we consider acentralized dual-channel supply chain as a benchmark wherethe manufacturer and the retailer are regarded as a verticallyintegrated supply chain systemThemembersmake decisionsto maximize the overall profit of the supply chain and thewholesale price 119908 is no longer the decision variable in thecentralized supply chain The problem of the supply chainmembers is given as follows
max119901119903119901119890120587119888 = 119889119903 (119901119903 minus 119888) + 119889119890 (119901119890 minus 119888) (3)
119904119905 119901119903 ge 119888119901119890 ge 119888119889119890 ge 0119889119903 ge 0
(4)
By simultaneously solving the first-order conditions ofthe equations above for 119901119903 119901119890 it can be shown that theHessian matrix 119867 is negative definite The optional channelprice 119901119903 119901119890 and the total profit of the supply chain areobtained
119901119888lowast119890 = 119901119888lowast119903 = 1198862 (1 minus 120579) minus 1198882 120587119888 = (120579119888 + 119886 minus 119888)22 (1 minus 120579)
(5)
In an integrated supply chain the decision maker setsa uniform retail price for both online and offline sales toavoid channel competition Obviously with an increase in thecross-price sensitivity coefficient the channel price increasesand the overall profit of the supply chain increases In realityenterprises often adopt the same price for the dual-channelsuch as Suning but a higher level of channel management isrequired in this situation (Chun Yuan et al [39])
42 Equilibrium Analysis of the Decentralized Supply ChainIn this section we consider a decentralized dual-channelsupply chain based on the assumption that neither party inthe supply chain has fairness concerns and that both makedecisions to maximize their individual profits
421 The Retailerrsquos Problem Given the manufacturers net-work direct selling price 119901119890 and wholesale price 119908 accordingto the previous game the profit of the retailer is maximizedas follows
max119901119903120587119903 = 119889119903 (119901119903 minus 119908) (6)
The response function for the retailer can be described asfollows
422 The Manufacturerrsquos Problem The manufacturerrsquos deci-sion problem can be described as follows
max119901119890119908120587119898 = 119889119903 (119908 minus 119888) + 119889119890 (119901119890 minus 119888) (8)
119904119905 119901119903 = 119886 + 119908 + 1205791199011198902 119901119890 ge 119908119889119890 ge 0119889119903 ge 0119908 gt 119888
(9)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction
119867120587119898 [119901119890119908] = [1205792 minus 2 120579120579 minus1] (10)
is negatively definite The manufacturerrsquos profit functionis a concave function of 119901119890 and119908 and the decision problem isa convex optimization problem Thus a unique equilibriumsolution existsTherefore we can deduce the optimal decisionas follows
119901119889lowast119890 = 119908119889lowast = 1198862 (1 minus 120579) minus 1198882 (11)
It is easy to prove (11) and satisfy (9) The optimal retailprice is given by bringing (11) into (7)
119901lowast119903 = (3 minus 120579) 1198864 (1 minus 120579) + (1 + 120579) 1198884 (12)
Complexity 5
By bringing (11) and (12) into (6) and (8) we can obtainthe profit of the manufacturer the profit of the retailer andthe total profit of the supply chain as follows
120587119898 = (120579 + 3) (120579119888 + 119886 minus 119888)28 (1 minus 120579) 120587119903 = (120579119888 + 119886 minus 119888)216 120587119889 = (120579119888 + 119886 minus 119888)2 (120579 + 7)16 (1 minus 120579)
(13)
By analysing the inferred strategies and profits we findthe following the channel efficiency (120587119889120587119888 = (120579 + 7)8) ofthe decentralized supply chain is an increasing function ofthe cross-price coefficient 120579 which indicates that the doublemarginalizationwill beweakenedwhen 120579 increases Similarlywe can draw the same conclusion from the equilibriumpricing strategy The cross-price sensitivity coefficient has apositive impact on the channel price As 120579 increases (see (11)and (12)) both the manufacturer and the retailer will use ahigher pricing strategy to improve its profits which increasesthe overall profit of the supply chain
5 Pricing Strategy of Members When theRetailer Has Fairness Concerns
The supply chain cannot be coordinated when the retailerdoes not have a fairness concern However it is necessary todetermine how the strategy changes when we consider theeffect of fairness concerns and whether channel efficiencycould be improvedThese issues are discussed in the followingsection
51 The Retailerrsquos Problem Because we consider the fairnessconcerns of the retailer in the distribution of profits wemust establish a model for fairness concerns Some mayargue that a more general model that includes both aversionto disadvantageous inequality and aversion to advantageousinequality (for example Fehr and Schmidt [5] and Charness
amp Rabin [40]) is more desirable However a preference foradvantageous inequality is much less prominent (Loewen-stein Thompson and Bazerman [41] did not find it in theirexperiment Ho amp Su [42]) Furthermore we assume that theretailerrsquos fairness reference is the manufacturerrsquos profit 120587119904119903instead of 120574120587119904119903 (120574 gt 0) because the general setting will notproduce substantially different ormore insightful results thana simple setting with 120574 = 1 (Pavlov amp Katok [43] Tengfei Nieand Shaofu Du [28]) The utility function of the retailer canbe written as follows
119880119903 = 120587119903 minus 120572 (120587119898119903 minus 120587119903)+ (14)
where 120587119903 = 119889119903(119901119903 minus 119908) denotes the retailerrsquos monetarypayoff 120587119898119903 = 119889119903(119908 minus 119888) denotes the profits of themanufacturerrsquos offline channel and 120572 denotes the level ofthe retailerrsquos fairness concern about the distribution of offlineprofits as retailers pay more attention to the profits madeby manufacturers from offline channels and compare themto the profits made from the online channel If the retailerrsquosmonetary profit is lower than the equitable profit120587119898119903minus120587119903 ge 0disadvantageous inequality occurs By contrast if 120587119898119903 minus120587119903 lt0 then 119880119903 = 120587119903 which indicates that the retailerrsquos utilityfunction is equal to the profit function
When the retailer faces disadvantage inequality its deci-sion problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (15)
119904119905 119901119903 le 2119908 minus 119888 (16)
The decision of the retailer is
119901119891119903=
119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 11986312119908 minus 119888 119894119891 119901119890 ge 1198701119908 minus 1198631
(17)
where1198701 = (2120572+3)120579(120572+1)1198631 = (120572+2)119888120579(120572+1)+119886120579The corresponding utility of the retailer is
119880119903 = (119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 ge 1198701119908 minus 1198631
(18)
When the retailer is faced with advantageous inequalityits decision problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (19)
119904119905 119901119903 gt 2119908 minus 119888 (20)
The decision of the retailer is
119901119891119903 = 120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 119901119890 le 1198702119908 minus 1198632 (21)
where 1198702 = 31205791198632 = (119886 + 2119888)120579The corresponding utility of the retailer is
119880119903
6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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2 Complexity
Fehr E [5] Kahneman D [6] Kumar N [7]) and have shownthat fairness concerns have a significant impact on decision-making Compared with the traditional supply chain thedual-channel supply chain easily occur unfair cooperationdue to channel competition and other factors which willlead to conflicts of interest and even the breakdown ofcooperation In order to increase sales high-end liquorcompanies such as Yibin Wuliangye Group Company Ltdhave added online direct channels and offered discountedproduct on it (Dong Zhao [8])The goodmarket prospects ofChinese liquor make the liquor enterprises raise the whole-sale price Meanwhile the liquor enterprises were tryingto stabilize the offline retail price and limit the minimumselling price to maintain the brand image (Dong Zhao [8]Chen Xing [9]) The impact of online direct selling and thepassive situation of offline retail made some retailers feelthat they have been treated unfairly and privately took theway of price-off promotions which destroyed the offline pricesystem and damaged the interest ofWuliangye Similar unfairphenomenon also happened in the cooperation betweenGome andGreeElectricAppliances (Xuefei Zhong [10])Howshould manufacturers coordinate channel conflicts by pricestrategies considering retailersrsquo fairness concern
This paper try to apply a fairness preference and thecross-price effect to the dual-channel supply chain decision-making model to obtain results and insights more in linewith actual management practice In this study we consider adual-channel supply chain where a manufacturer acts as theleader and a fair caring retailer is the follower in Stackelberggame and approach the complete equilibrium solutions of thesupply chain members
The remainder of this paper is organized as follows InSection 2 we review dual-channel supply chain literaturesrelated to pricing strategy and fairness In Section 3 wedevelop the structure and demand functions of a dual-channel supply chain and discuss the model assumptionsand describe the parameters In Section 4 we discuss theequilibrium pricing strategies of the members when theretailer has no fairness concerns In Section 5 we analysethe equilibrium pricing strategies of the manufacturer and afair caring retailer In Section 6 a numerical experiment isused to analyse the influence of fairness concerns and cross-price sensitivity on member decisions and supply chain per-formance We conclude by presenting the equilibrium resultsand suggesting directions for future research in Section 7 Allof the proofs are provided in Appendix
2 Literature Review
Our paper is related to two streams of research pricingstrategy and fairness concern in the dual-channel supplychain
21 Pricing Strategy Researches inDual-Channel Supply ChainChiang et al [11] showed that a vertically integrated onlinechannel allows the manufacturer to constrain its retailerrsquospricing behavior in a dual-channel supply chain Guo Yajun[12] showed that adding an online direct channel can expand
the market but it may also exacerbate channel conflictsand depress the retail price leading to a loss in retailersrsquoprofit Cattani et al [13] studied the situation in whichmanufacturers adopt different pricing strategies in order toalleviate channel conflicts and found that reducing wholesaleprices could alleviate the double margin effect and improvesupply chain performance (Qing Fang [14]) Considering thedual-channel supply chain led by retailers and manufactur-ers respectively and making a comparative analysis of theoptimal price decision the dominant party will make thewholesale price beneficial to maximize its profit Yan R [15]constructed the channel demand model based on consumerutility and studied the pricing strategy considering con-sumers have same price sensitivity of different channels Thispaper found that channel conflict can be alleviated by pricestrategy after manufacturers encourage retailers to improveretail services Based on the same channel price sensitivity ofconsumers Tian J F [16] studied the pricing strategy whenmanufacturers develop retail service With the developmentof dual-channel supply chain some researchers have foundthat consumers have gradually adapted to the heterogeneityof channels especially the products that consumers oftenbuy online Therefore price comparison has become thefocus of consumer attention Xu et al [17] analysed theprice comparison behavior of consumers and its impact ondecision-making and profit of supply chain members andfound that the retailers and supplier are all more willing toavoid the existence of price comparison with the objectiveof profit maximization Shen et al [18] researched on thepricing strategy considering price comparison behavior anddesigned the corresponding coordination mechanism Toreduce channel conflict Bo Li [19] considered a consistentpricing strategy in the two channels which means that theprice in the direct channel is equal to that in the retail channelZhang F [20 21] established the dynamic price game modeland analysed the impact of price adjustment on the profit ofsupply chain members Excessive price adjustment is oftendetrimental to their own interests but will make the otherside to get more profits Many scholars (eg [22ndash25]) havestudied the price strategies for different channel structuresand different product strategies
The papers mentioned above have shown that pricestrategies play an important role in allocating channel profitsand coordinating channel conflict In dual-channel supplychain price competition and channel conflicts may makemembers paymore attention to the distribution of profits thispaper make attempts to integrate members fair preferencesinto pricing studies
22 Fairness Concern Researches in Dual-Channel SupplyChain Apart from the single-channel supply chain field fewscholars are engaged in applying fairness preferences to thedual-channel supply chain in their research (YouQ et al [26]Guangxing Wei et al [27]) among which Tengfei Nie andShaofu Du [28] Qinghua Li and Bo Li[29] and Fang Z etal [20] are the most representative Reference [28] studiedthe application of a quantity discount contract in a dyadicsupply chain consisting of one supplier and two retailerswith no cross-price influence between channels Retailers also
Complexity 3
focus on both horizontal and vertical fairness This articledetermines the pricing strategies of the members of a supplychain when the fairness parameter differs Further it alsointroduces other coordination mechanisms to prove that thequantity discount contract cannot fully coordinate the supplychain Reference [29] considered that the retailer providesvalue-added services and they study the pricing decisions ofthe supply chain members for two scenarios one in whichthe retailer has fairness concerns and a second in which itdoes not The partial equilibrium solution of the channelquota is given in this article but the complete equilibriumsolution is not discussed Reference [20] investigated twononcooperative dynamic game models a Stackelberg gamemodel and a vertical Nash game model The paper usednumerical experiments to analyse the influence of the retailerfairness preference on the dynamic behavior of supply chainmembers The FS fairness model is simplified from using apiecewise function to using a continuous function to discusshow the retailers behavior related to its fairness concernsinfluencesmember decisions and utility (Fujing Xu et al [30]Lei Wang et al [31] Bo Li et al [32])
In the aforementioned articles there is a lack of attentionto fairness concerns and cross-price sensitivity Consideringthe influence of fairness concerns on strategy only the localequilibrium solution of the members is inferred and thecomplete process of the memberrsquos game cannot be fullyunderstood based on this analysis In practice when themarket environment and the level of fairness concerns ofthe members change there are multiple equilibriums whichmeans that the members will adopt different strategiesTherefore the complete equilibrium solutions and the cor-responding management significance will become a focus ofthis paper
3 Problem Statement
31 Model Assumption and Notation 119908 per unit wholesaleprice of the manufacturer119901119903 per unit offline retail price of the retailer119901119890 per unit online direct price of the manufacturer119908119899 the superscript n takes the values of 119889lowast and 119891 lowast lowastwhich denote the optimal wholesale pricing strategies withand without fairness concerns119901119899119894 the superscript n takes the values of 119888lowast 119889lowast and 119891lowastlowastwhich denote the optimal strategies under the centralized anddecentralized supply chain without fairness and the strategiesconsidering fairness respectively The subscript 119894 takes thevalues of 119903 and 119890119886 the potential market demand of the channel119888 the manufacturerrsquos marginal cost per unit120579 the cross-price sensitivity between channels120572 a parameter reflecting the per unit difference in thepayoffs of the manufacturer and the retailer when the retailerencounters disadvantageous unfairness119889119903 the demand function of the offline retail channel119889119890 the demand function of the online direct channel120587119894 the subscript 119894 takes the values of 119888 119889 119898 and 119903which denote the total profit of both the centralized and
ManufacturerManufacturer
RetailerRetailer
ConsumerConsumer
w
Pe
Pr
Figure 1 The dual-channel supply chain structure
decentralized supply chains the manufacturerrsquos profit andthe retailerrsquos profits without fairness concerns respectively120587119891119894 the superscript 119891 denotes the scenario with fairnessconcerns the subscript 119894 takes the values of 119888119898 and 119903 whichdenote the total profit of the supply chain the manufacturerrsquosprofit and the retailerrsquos profit respectively
The following assumptions are made in our model (1)The manufacturer and the retailer only sell one kind ofproduct (2)Thepotential share of both the online and offlinechannels is the same (3) Information is symmetrical betweenthe manufacture and the retailer (4)The retailer is in a weakposition
32 The Structure of the Dual-Channel Chain and Chan-nel Demand Function We consider a representative dual-channel structure in our study (see Figure 1)The supply chainconsists of manufacturer M and retailer RThe manufacturercreates infinitely divisible and homogeneous products andsells through both traditional offline retailers and onlinedirect sales channels We establish a Stackelberg game modelto describe the problem between a rational manufacturer anda retailer with a fairness concern The process of the gameis as follows the manufacturer as the initiator of the gamefirst determines the online direct selling price 119901119890 and thewholesale price 119908 the retailer acts as the follower and thensets the offline retail price 119901119903
Linear demand functions are used to characterize channeldemand and have been adopted in studies (Yue X [33] HuangS [34]) and the corresponding demand functions to themanufacturer and the retailer are described as follows
119889119903 = 119886119903 minus 119887119903119901119903 + 120579119901119890 (1)
119889119890 = 119886119890 minus 119887119890119901119890 + 120579119901119903 (2)
The differences in channel characteristics are mainlyreflected in channel price elasticity 119887 and basic marketdemand of channel 119886 Subscript 119903 and 119890 represent the offlineand online channels respectively Channel price elasticitydepicts consumers sensitivity to channel price Basic marketdemand of channel reflects consumers loyalty to the channelWhen channel differences decrease market characteristicscorresponding to the original differentiated channels aregradually converging Some scholars (Yan R [15] GuangyeXu [35]) consider consumers have the same price sensitivityto different channels but the channel loyalty is differentChannel loyalty of consumers is changing with the shoppinghabit Research on Chinarsquos digital consumers released byMcKinsey Greater China in 2017 states that over 90 ofconsumers compare online and offline channels when buyingconsumer electronics (Wei Wang [36]) According to a white
4 Complexity
paper on big data and online cosmetics consumption in 2016although the growth rate of e-commerce has been slowingsince 2012 the proportion of online and offline consumptionis expected to be evenly divided into 2018 (Niuli [37])Therefore this paper considers a situationwhere the potentialshare of the channels is the same Without loss of generalitywe set parameter 119887119903 and 119887119890 to 1 as were done in the studies([23] LiuM [38]) 120579 is the coefficient of cross-price sensitivityThe equation 0 lt 120579 lt 1 reflects the own-price effects whichare greater than the cross-price effects
Obviously it is necessary to impose additional inequityconstraints on the parameters to guarantee the operation ofthe dual channels (i) 119901119903 ge 119908 119901119890 ge 119908 If 119901119890 lt 119908 thenthe retailer will find a less expensive source from the directchannel (ii) 119889119890 ge 0 and 119889119903 ge 0 which ensures that everychannel has sales (iii) 119908 ge 1198884 Pricing Strategy of Members When theRetailer Has No Fairness Concerns
41 Equilibrium Analysis of the Centralized Supply ChainTo examine the efficiency of the decentralized supply chainboth with and without fairness concerns we consider acentralized dual-channel supply chain as a benchmark wherethe manufacturer and the retailer are regarded as a verticallyintegrated supply chain systemThemembersmake decisionsto maximize the overall profit of the supply chain and thewholesale price 119908 is no longer the decision variable in thecentralized supply chain The problem of the supply chainmembers is given as follows
max119901119903119901119890120587119888 = 119889119903 (119901119903 minus 119888) + 119889119890 (119901119890 minus 119888) (3)
119904119905 119901119903 ge 119888119901119890 ge 119888119889119890 ge 0119889119903 ge 0
(4)
By simultaneously solving the first-order conditions ofthe equations above for 119901119903 119901119890 it can be shown that theHessian matrix 119867 is negative definite The optional channelprice 119901119903 119901119890 and the total profit of the supply chain areobtained
119901119888lowast119890 = 119901119888lowast119903 = 1198862 (1 minus 120579) minus 1198882 120587119888 = (120579119888 + 119886 minus 119888)22 (1 minus 120579)
(5)
In an integrated supply chain the decision maker setsa uniform retail price for both online and offline sales toavoid channel competition Obviously with an increase in thecross-price sensitivity coefficient the channel price increasesand the overall profit of the supply chain increases In realityenterprises often adopt the same price for the dual-channelsuch as Suning but a higher level of channel management isrequired in this situation (Chun Yuan et al [39])
42 Equilibrium Analysis of the Decentralized Supply ChainIn this section we consider a decentralized dual-channelsupply chain based on the assumption that neither party inthe supply chain has fairness concerns and that both makedecisions to maximize their individual profits
421 The Retailerrsquos Problem Given the manufacturers net-work direct selling price 119901119890 and wholesale price 119908 accordingto the previous game the profit of the retailer is maximizedas follows
max119901119903120587119903 = 119889119903 (119901119903 minus 119908) (6)
The response function for the retailer can be described asfollows
422 The Manufacturerrsquos Problem The manufacturerrsquos deci-sion problem can be described as follows
max119901119890119908120587119898 = 119889119903 (119908 minus 119888) + 119889119890 (119901119890 minus 119888) (8)
119904119905 119901119903 = 119886 + 119908 + 1205791199011198902 119901119890 ge 119908119889119890 ge 0119889119903 ge 0119908 gt 119888
(9)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction
119867120587119898 [119901119890119908] = [1205792 minus 2 120579120579 minus1] (10)
is negatively definite The manufacturerrsquos profit functionis a concave function of 119901119890 and119908 and the decision problem isa convex optimization problem Thus a unique equilibriumsolution existsTherefore we can deduce the optimal decisionas follows
119901119889lowast119890 = 119908119889lowast = 1198862 (1 minus 120579) minus 1198882 (11)
It is easy to prove (11) and satisfy (9) The optimal retailprice is given by bringing (11) into (7)
119901lowast119903 = (3 minus 120579) 1198864 (1 minus 120579) + (1 + 120579) 1198884 (12)
Complexity 5
By bringing (11) and (12) into (6) and (8) we can obtainthe profit of the manufacturer the profit of the retailer andthe total profit of the supply chain as follows
120587119898 = (120579 + 3) (120579119888 + 119886 minus 119888)28 (1 minus 120579) 120587119903 = (120579119888 + 119886 minus 119888)216 120587119889 = (120579119888 + 119886 minus 119888)2 (120579 + 7)16 (1 minus 120579)
(13)
By analysing the inferred strategies and profits we findthe following the channel efficiency (120587119889120587119888 = (120579 + 7)8) ofthe decentralized supply chain is an increasing function ofthe cross-price coefficient 120579 which indicates that the doublemarginalizationwill beweakenedwhen 120579 increases Similarlywe can draw the same conclusion from the equilibriumpricing strategy The cross-price sensitivity coefficient has apositive impact on the channel price As 120579 increases (see (11)and (12)) both the manufacturer and the retailer will use ahigher pricing strategy to improve its profits which increasesthe overall profit of the supply chain
5 Pricing Strategy of Members When theRetailer Has Fairness Concerns
The supply chain cannot be coordinated when the retailerdoes not have a fairness concern However it is necessary todetermine how the strategy changes when we consider theeffect of fairness concerns and whether channel efficiencycould be improvedThese issues are discussed in the followingsection
51 The Retailerrsquos Problem Because we consider the fairnessconcerns of the retailer in the distribution of profits wemust establish a model for fairness concerns Some mayargue that a more general model that includes both aversionto disadvantageous inequality and aversion to advantageousinequality (for example Fehr and Schmidt [5] and Charness
amp Rabin [40]) is more desirable However a preference foradvantageous inequality is much less prominent (Loewen-stein Thompson and Bazerman [41] did not find it in theirexperiment Ho amp Su [42]) Furthermore we assume that theretailerrsquos fairness reference is the manufacturerrsquos profit 120587119904119903instead of 120574120587119904119903 (120574 gt 0) because the general setting will notproduce substantially different ormore insightful results thana simple setting with 120574 = 1 (Pavlov amp Katok [43] Tengfei Nieand Shaofu Du [28]) The utility function of the retailer canbe written as follows
119880119903 = 120587119903 minus 120572 (120587119898119903 minus 120587119903)+ (14)
where 120587119903 = 119889119903(119901119903 minus 119908) denotes the retailerrsquos monetarypayoff 120587119898119903 = 119889119903(119908 minus 119888) denotes the profits of themanufacturerrsquos offline channel and 120572 denotes the level ofthe retailerrsquos fairness concern about the distribution of offlineprofits as retailers pay more attention to the profits madeby manufacturers from offline channels and compare themto the profits made from the online channel If the retailerrsquosmonetary profit is lower than the equitable profit120587119898119903minus120587119903 ge 0disadvantageous inequality occurs By contrast if 120587119898119903 minus120587119903 lt0 then 119880119903 = 120587119903 which indicates that the retailerrsquos utilityfunction is equal to the profit function
When the retailer faces disadvantage inequality its deci-sion problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (15)
119904119905 119901119903 le 2119908 minus 119888 (16)
The decision of the retailer is
119901119891119903=
119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 11986312119908 minus 119888 119894119891 119901119890 ge 1198701119908 minus 1198631
(17)
where1198701 = (2120572+3)120579(120572+1)1198631 = (120572+2)119888120579(120572+1)+119886120579The corresponding utility of the retailer is
119880119903 = (119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 ge 1198701119908 minus 1198631
(18)
When the retailer is faced with advantageous inequalityits decision problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (19)
119904119905 119901119903 gt 2119908 minus 119888 (20)
The decision of the retailer is
119901119891119903 = 120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 119901119890 le 1198702119908 minus 1198632 (21)
where 1198702 = 31205791198632 = (119886 + 2119888)120579The corresponding utility of the retailer is
119880119903
6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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Complexity 3
focus on both horizontal and vertical fairness This articledetermines the pricing strategies of the members of a supplychain when the fairness parameter differs Further it alsointroduces other coordination mechanisms to prove that thequantity discount contract cannot fully coordinate the supplychain Reference [29] considered that the retailer providesvalue-added services and they study the pricing decisions ofthe supply chain members for two scenarios one in whichthe retailer has fairness concerns and a second in which itdoes not The partial equilibrium solution of the channelquota is given in this article but the complete equilibriumsolution is not discussed Reference [20] investigated twononcooperative dynamic game models a Stackelberg gamemodel and a vertical Nash game model The paper usednumerical experiments to analyse the influence of the retailerfairness preference on the dynamic behavior of supply chainmembers The FS fairness model is simplified from using apiecewise function to using a continuous function to discusshow the retailers behavior related to its fairness concernsinfluencesmember decisions and utility (Fujing Xu et al [30]Lei Wang et al [31] Bo Li et al [32])
In the aforementioned articles there is a lack of attentionto fairness concerns and cross-price sensitivity Consideringthe influence of fairness concerns on strategy only the localequilibrium solution of the members is inferred and thecomplete process of the memberrsquos game cannot be fullyunderstood based on this analysis In practice when themarket environment and the level of fairness concerns ofthe members change there are multiple equilibriums whichmeans that the members will adopt different strategiesTherefore the complete equilibrium solutions and the cor-responding management significance will become a focus ofthis paper
3 Problem Statement
31 Model Assumption and Notation 119908 per unit wholesaleprice of the manufacturer119901119903 per unit offline retail price of the retailer119901119890 per unit online direct price of the manufacturer119908119899 the superscript n takes the values of 119889lowast and 119891 lowast lowastwhich denote the optimal wholesale pricing strategies withand without fairness concerns119901119899119894 the superscript n takes the values of 119888lowast 119889lowast and 119891lowastlowastwhich denote the optimal strategies under the centralized anddecentralized supply chain without fairness and the strategiesconsidering fairness respectively The subscript 119894 takes thevalues of 119903 and 119890119886 the potential market demand of the channel119888 the manufacturerrsquos marginal cost per unit120579 the cross-price sensitivity between channels120572 a parameter reflecting the per unit difference in thepayoffs of the manufacturer and the retailer when the retailerencounters disadvantageous unfairness119889119903 the demand function of the offline retail channel119889119890 the demand function of the online direct channel120587119894 the subscript 119894 takes the values of 119888 119889 119898 and 119903which denote the total profit of both the centralized and
ManufacturerManufacturer
RetailerRetailer
ConsumerConsumer
w
Pe
Pr
Figure 1 The dual-channel supply chain structure
decentralized supply chains the manufacturerrsquos profit andthe retailerrsquos profits without fairness concerns respectively120587119891119894 the superscript 119891 denotes the scenario with fairnessconcerns the subscript 119894 takes the values of 119888119898 and 119903 whichdenote the total profit of the supply chain the manufacturerrsquosprofit and the retailerrsquos profit respectively
The following assumptions are made in our model (1)The manufacturer and the retailer only sell one kind ofproduct (2)Thepotential share of both the online and offlinechannels is the same (3) Information is symmetrical betweenthe manufacture and the retailer (4)The retailer is in a weakposition
32 The Structure of the Dual-Channel Chain and Chan-nel Demand Function We consider a representative dual-channel structure in our study (see Figure 1)The supply chainconsists of manufacturer M and retailer RThe manufacturercreates infinitely divisible and homogeneous products andsells through both traditional offline retailers and onlinedirect sales channels We establish a Stackelberg game modelto describe the problem between a rational manufacturer anda retailer with a fairness concern The process of the gameis as follows the manufacturer as the initiator of the gamefirst determines the online direct selling price 119901119890 and thewholesale price 119908 the retailer acts as the follower and thensets the offline retail price 119901119903
Linear demand functions are used to characterize channeldemand and have been adopted in studies (Yue X [33] HuangS [34]) and the corresponding demand functions to themanufacturer and the retailer are described as follows
119889119903 = 119886119903 minus 119887119903119901119903 + 120579119901119890 (1)
119889119890 = 119886119890 minus 119887119890119901119890 + 120579119901119903 (2)
The differences in channel characteristics are mainlyreflected in channel price elasticity 119887 and basic marketdemand of channel 119886 Subscript 119903 and 119890 represent the offlineand online channels respectively Channel price elasticitydepicts consumers sensitivity to channel price Basic marketdemand of channel reflects consumers loyalty to the channelWhen channel differences decrease market characteristicscorresponding to the original differentiated channels aregradually converging Some scholars (Yan R [15] GuangyeXu [35]) consider consumers have the same price sensitivityto different channels but the channel loyalty is differentChannel loyalty of consumers is changing with the shoppinghabit Research on Chinarsquos digital consumers released byMcKinsey Greater China in 2017 states that over 90 ofconsumers compare online and offline channels when buyingconsumer electronics (Wei Wang [36]) According to a white
4 Complexity
paper on big data and online cosmetics consumption in 2016although the growth rate of e-commerce has been slowingsince 2012 the proportion of online and offline consumptionis expected to be evenly divided into 2018 (Niuli [37])Therefore this paper considers a situationwhere the potentialshare of the channels is the same Without loss of generalitywe set parameter 119887119903 and 119887119890 to 1 as were done in the studies([23] LiuM [38]) 120579 is the coefficient of cross-price sensitivityThe equation 0 lt 120579 lt 1 reflects the own-price effects whichare greater than the cross-price effects
Obviously it is necessary to impose additional inequityconstraints on the parameters to guarantee the operation ofthe dual channels (i) 119901119903 ge 119908 119901119890 ge 119908 If 119901119890 lt 119908 thenthe retailer will find a less expensive source from the directchannel (ii) 119889119890 ge 0 and 119889119903 ge 0 which ensures that everychannel has sales (iii) 119908 ge 1198884 Pricing Strategy of Members When theRetailer Has No Fairness Concerns
41 Equilibrium Analysis of the Centralized Supply ChainTo examine the efficiency of the decentralized supply chainboth with and without fairness concerns we consider acentralized dual-channel supply chain as a benchmark wherethe manufacturer and the retailer are regarded as a verticallyintegrated supply chain systemThemembersmake decisionsto maximize the overall profit of the supply chain and thewholesale price 119908 is no longer the decision variable in thecentralized supply chain The problem of the supply chainmembers is given as follows
max119901119903119901119890120587119888 = 119889119903 (119901119903 minus 119888) + 119889119890 (119901119890 minus 119888) (3)
119904119905 119901119903 ge 119888119901119890 ge 119888119889119890 ge 0119889119903 ge 0
(4)
By simultaneously solving the first-order conditions ofthe equations above for 119901119903 119901119890 it can be shown that theHessian matrix 119867 is negative definite The optional channelprice 119901119903 119901119890 and the total profit of the supply chain areobtained
119901119888lowast119890 = 119901119888lowast119903 = 1198862 (1 minus 120579) minus 1198882 120587119888 = (120579119888 + 119886 minus 119888)22 (1 minus 120579)
(5)
In an integrated supply chain the decision maker setsa uniform retail price for both online and offline sales toavoid channel competition Obviously with an increase in thecross-price sensitivity coefficient the channel price increasesand the overall profit of the supply chain increases In realityenterprises often adopt the same price for the dual-channelsuch as Suning but a higher level of channel management isrequired in this situation (Chun Yuan et al [39])
42 Equilibrium Analysis of the Decentralized Supply ChainIn this section we consider a decentralized dual-channelsupply chain based on the assumption that neither party inthe supply chain has fairness concerns and that both makedecisions to maximize their individual profits
421 The Retailerrsquos Problem Given the manufacturers net-work direct selling price 119901119890 and wholesale price 119908 accordingto the previous game the profit of the retailer is maximizedas follows
max119901119903120587119903 = 119889119903 (119901119903 minus 119908) (6)
The response function for the retailer can be described asfollows
422 The Manufacturerrsquos Problem The manufacturerrsquos deci-sion problem can be described as follows
max119901119890119908120587119898 = 119889119903 (119908 minus 119888) + 119889119890 (119901119890 minus 119888) (8)
119904119905 119901119903 = 119886 + 119908 + 1205791199011198902 119901119890 ge 119908119889119890 ge 0119889119903 ge 0119908 gt 119888
(9)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction
119867120587119898 [119901119890119908] = [1205792 minus 2 120579120579 minus1] (10)
is negatively definite The manufacturerrsquos profit functionis a concave function of 119901119890 and119908 and the decision problem isa convex optimization problem Thus a unique equilibriumsolution existsTherefore we can deduce the optimal decisionas follows
119901119889lowast119890 = 119908119889lowast = 1198862 (1 minus 120579) minus 1198882 (11)
It is easy to prove (11) and satisfy (9) The optimal retailprice is given by bringing (11) into (7)
119901lowast119903 = (3 minus 120579) 1198864 (1 minus 120579) + (1 + 120579) 1198884 (12)
Complexity 5
By bringing (11) and (12) into (6) and (8) we can obtainthe profit of the manufacturer the profit of the retailer andthe total profit of the supply chain as follows
120587119898 = (120579 + 3) (120579119888 + 119886 minus 119888)28 (1 minus 120579) 120587119903 = (120579119888 + 119886 minus 119888)216 120587119889 = (120579119888 + 119886 minus 119888)2 (120579 + 7)16 (1 minus 120579)
(13)
By analysing the inferred strategies and profits we findthe following the channel efficiency (120587119889120587119888 = (120579 + 7)8) ofthe decentralized supply chain is an increasing function ofthe cross-price coefficient 120579 which indicates that the doublemarginalizationwill beweakenedwhen 120579 increases Similarlywe can draw the same conclusion from the equilibriumpricing strategy The cross-price sensitivity coefficient has apositive impact on the channel price As 120579 increases (see (11)and (12)) both the manufacturer and the retailer will use ahigher pricing strategy to improve its profits which increasesthe overall profit of the supply chain
5 Pricing Strategy of Members When theRetailer Has Fairness Concerns
The supply chain cannot be coordinated when the retailerdoes not have a fairness concern However it is necessary todetermine how the strategy changes when we consider theeffect of fairness concerns and whether channel efficiencycould be improvedThese issues are discussed in the followingsection
51 The Retailerrsquos Problem Because we consider the fairnessconcerns of the retailer in the distribution of profits wemust establish a model for fairness concerns Some mayargue that a more general model that includes both aversionto disadvantageous inequality and aversion to advantageousinequality (for example Fehr and Schmidt [5] and Charness
amp Rabin [40]) is more desirable However a preference foradvantageous inequality is much less prominent (Loewen-stein Thompson and Bazerman [41] did not find it in theirexperiment Ho amp Su [42]) Furthermore we assume that theretailerrsquos fairness reference is the manufacturerrsquos profit 120587119904119903instead of 120574120587119904119903 (120574 gt 0) because the general setting will notproduce substantially different ormore insightful results thana simple setting with 120574 = 1 (Pavlov amp Katok [43] Tengfei Nieand Shaofu Du [28]) The utility function of the retailer canbe written as follows
119880119903 = 120587119903 minus 120572 (120587119898119903 minus 120587119903)+ (14)
where 120587119903 = 119889119903(119901119903 minus 119908) denotes the retailerrsquos monetarypayoff 120587119898119903 = 119889119903(119908 minus 119888) denotes the profits of themanufacturerrsquos offline channel and 120572 denotes the level ofthe retailerrsquos fairness concern about the distribution of offlineprofits as retailers pay more attention to the profits madeby manufacturers from offline channels and compare themto the profits made from the online channel If the retailerrsquosmonetary profit is lower than the equitable profit120587119898119903minus120587119903 ge 0disadvantageous inequality occurs By contrast if 120587119898119903 minus120587119903 lt0 then 119880119903 = 120587119903 which indicates that the retailerrsquos utilityfunction is equal to the profit function
When the retailer faces disadvantage inequality its deci-sion problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (15)
119904119905 119901119903 le 2119908 minus 119888 (16)
The decision of the retailer is
119901119891119903=
119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 11986312119908 minus 119888 119894119891 119901119890 ge 1198701119908 minus 1198631
(17)
where1198701 = (2120572+3)120579(120572+1)1198631 = (120572+2)119888120579(120572+1)+119886120579The corresponding utility of the retailer is
119880119903 = (119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 ge 1198701119908 minus 1198631
(18)
When the retailer is faced with advantageous inequalityits decision problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (19)
119904119905 119901119903 gt 2119908 minus 119888 (20)
The decision of the retailer is
119901119891119903 = 120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 119901119890 le 1198702119908 minus 1198632 (21)
where 1198702 = 31205791198632 = (119886 + 2119888)120579The corresponding utility of the retailer is
119880119903
6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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4 Complexity
paper on big data and online cosmetics consumption in 2016although the growth rate of e-commerce has been slowingsince 2012 the proportion of online and offline consumptionis expected to be evenly divided into 2018 (Niuli [37])Therefore this paper considers a situationwhere the potentialshare of the channels is the same Without loss of generalitywe set parameter 119887119903 and 119887119890 to 1 as were done in the studies([23] LiuM [38]) 120579 is the coefficient of cross-price sensitivityThe equation 0 lt 120579 lt 1 reflects the own-price effects whichare greater than the cross-price effects
Obviously it is necessary to impose additional inequityconstraints on the parameters to guarantee the operation ofthe dual channels (i) 119901119903 ge 119908 119901119890 ge 119908 If 119901119890 lt 119908 thenthe retailer will find a less expensive source from the directchannel (ii) 119889119890 ge 0 and 119889119903 ge 0 which ensures that everychannel has sales (iii) 119908 ge 1198884 Pricing Strategy of Members When theRetailer Has No Fairness Concerns
41 Equilibrium Analysis of the Centralized Supply ChainTo examine the efficiency of the decentralized supply chainboth with and without fairness concerns we consider acentralized dual-channel supply chain as a benchmark wherethe manufacturer and the retailer are regarded as a verticallyintegrated supply chain systemThemembersmake decisionsto maximize the overall profit of the supply chain and thewholesale price 119908 is no longer the decision variable in thecentralized supply chain The problem of the supply chainmembers is given as follows
max119901119903119901119890120587119888 = 119889119903 (119901119903 minus 119888) + 119889119890 (119901119890 minus 119888) (3)
119904119905 119901119903 ge 119888119901119890 ge 119888119889119890 ge 0119889119903 ge 0
(4)
By simultaneously solving the first-order conditions ofthe equations above for 119901119903 119901119890 it can be shown that theHessian matrix 119867 is negative definite The optional channelprice 119901119903 119901119890 and the total profit of the supply chain areobtained
119901119888lowast119890 = 119901119888lowast119903 = 1198862 (1 minus 120579) minus 1198882 120587119888 = (120579119888 + 119886 minus 119888)22 (1 minus 120579)
(5)
In an integrated supply chain the decision maker setsa uniform retail price for both online and offline sales toavoid channel competition Obviously with an increase in thecross-price sensitivity coefficient the channel price increasesand the overall profit of the supply chain increases In realityenterprises often adopt the same price for the dual-channelsuch as Suning but a higher level of channel management isrequired in this situation (Chun Yuan et al [39])
42 Equilibrium Analysis of the Decentralized Supply ChainIn this section we consider a decentralized dual-channelsupply chain based on the assumption that neither party inthe supply chain has fairness concerns and that both makedecisions to maximize their individual profits
421 The Retailerrsquos Problem Given the manufacturers net-work direct selling price 119901119890 and wholesale price 119908 accordingto the previous game the profit of the retailer is maximizedas follows
max119901119903120587119903 = 119889119903 (119901119903 minus 119908) (6)
The response function for the retailer can be described asfollows
422 The Manufacturerrsquos Problem The manufacturerrsquos deci-sion problem can be described as follows
max119901119890119908120587119898 = 119889119903 (119908 minus 119888) + 119889119890 (119901119890 minus 119888) (8)
119904119905 119901119903 = 119886 + 119908 + 1205791199011198902 119901119890 ge 119908119889119890 ge 0119889119903 ge 0119908 gt 119888
(9)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction
119867120587119898 [119901119890119908] = [1205792 minus 2 120579120579 minus1] (10)
is negatively definite The manufacturerrsquos profit functionis a concave function of 119901119890 and119908 and the decision problem isa convex optimization problem Thus a unique equilibriumsolution existsTherefore we can deduce the optimal decisionas follows
119901119889lowast119890 = 119908119889lowast = 1198862 (1 minus 120579) minus 1198882 (11)
It is easy to prove (11) and satisfy (9) The optimal retailprice is given by bringing (11) into (7)
119901lowast119903 = (3 minus 120579) 1198864 (1 minus 120579) + (1 + 120579) 1198884 (12)
Complexity 5
By bringing (11) and (12) into (6) and (8) we can obtainthe profit of the manufacturer the profit of the retailer andthe total profit of the supply chain as follows
120587119898 = (120579 + 3) (120579119888 + 119886 minus 119888)28 (1 minus 120579) 120587119903 = (120579119888 + 119886 minus 119888)216 120587119889 = (120579119888 + 119886 minus 119888)2 (120579 + 7)16 (1 minus 120579)
(13)
By analysing the inferred strategies and profits we findthe following the channel efficiency (120587119889120587119888 = (120579 + 7)8) ofthe decentralized supply chain is an increasing function ofthe cross-price coefficient 120579 which indicates that the doublemarginalizationwill beweakenedwhen 120579 increases Similarlywe can draw the same conclusion from the equilibriumpricing strategy The cross-price sensitivity coefficient has apositive impact on the channel price As 120579 increases (see (11)and (12)) both the manufacturer and the retailer will use ahigher pricing strategy to improve its profits which increasesthe overall profit of the supply chain
5 Pricing Strategy of Members When theRetailer Has Fairness Concerns
The supply chain cannot be coordinated when the retailerdoes not have a fairness concern However it is necessary todetermine how the strategy changes when we consider theeffect of fairness concerns and whether channel efficiencycould be improvedThese issues are discussed in the followingsection
51 The Retailerrsquos Problem Because we consider the fairnessconcerns of the retailer in the distribution of profits wemust establish a model for fairness concerns Some mayargue that a more general model that includes both aversionto disadvantageous inequality and aversion to advantageousinequality (for example Fehr and Schmidt [5] and Charness
amp Rabin [40]) is more desirable However a preference foradvantageous inequality is much less prominent (Loewen-stein Thompson and Bazerman [41] did not find it in theirexperiment Ho amp Su [42]) Furthermore we assume that theretailerrsquos fairness reference is the manufacturerrsquos profit 120587119904119903instead of 120574120587119904119903 (120574 gt 0) because the general setting will notproduce substantially different ormore insightful results thana simple setting with 120574 = 1 (Pavlov amp Katok [43] Tengfei Nieand Shaofu Du [28]) The utility function of the retailer canbe written as follows
119880119903 = 120587119903 minus 120572 (120587119898119903 minus 120587119903)+ (14)
where 120587119903 = 119889119903(119901119903 minus 119908) denotes the retailerrsquos monetarypayoff 120587119898119903 = 119889119903(119908 minus 119888) denotes the profits of themanufacturerrsquos offline channel and 120572 denotes the level ofthe retailerrsquos fairness concern about the distribution of offlineprofits as retailers pay more attention to the profits madeby manufacturers from offline channels and compare themto the profits made from the online channel If the retailerrsquosmonetary profit is lower than the equitable profit120587119898119903minus120587119903 ge 0disadvantageous inequality occurs By contrast if 120587119898119903 minus120587119903 lt0 then 119880119903 = 120587119903 which indicates that the retailerrsquos utilityfunction is equal to the profit function
When the retailer faces disadvantage inequality its deci-sion problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (15)
119904119905 119901119903 le 2119908 minus 119888 (16)
The decision of the retailer is
119901119891119903=
119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 11986312119908 minus 119888 119894119891 119901119890 ge 1198701119908 minus 1198631
(17)
where1198701 = (2120572+3)120579(120572+1)1198631 = (120572+2)119888120579(120572+1)+119886120579The corresponding utility of the retailer is
119880119903 = (119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 ge 1198701119908 minus 1198631
(18)
When the retailer is faced with advantageous inequalityits decision problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (19)
119904119905 119901119903 gt 2119908 minus 119888 (20)
The decision of the retailer is
119901119891119903 = 120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 119901119890 le 1198702119908 minus 1198632 (21)
where 1198702 = 31205791198632 = (119886 + 2119888)120579The corresponding utility of the retailer is
119880119903
6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
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[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
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[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
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Complexity 5
By bringing (11) and (12) into (6) and (8) we can obtainthe profit of the manufacturer the profit of the retailer andthe total profit of the supply chain as follows
120587119898 = (120579 + 3) (120579119888 + 119886 minus 119888)28 (1 minus 120579) 120587119903 = (120579119888 + 119886 minus 119888)216 120587119889 = (120579119888 + 119886 minus 119888)2 (120579 + 7)16 (1 minus 120579)
(13)
By analysing the inferred strategies and profits we findthe following the channel efficiency (120587119889120587119888 = (120579 + 7)8) ofthe decentralized supply chain is an increasing function ofthe cross-price coefficient 120579 which indicates that the doublemarginalizationwill beweakenedwhen 120579 increases Similarlywe can draw the same conclusion from the equilibriumpricing strategy The cross-price sensitivity coefficient has apositive impact on the channel price As 120579 increases (see (11)and (12)) both the manufacturer and the retailer will use ahigher pricing strategy to improve its profits which increasesthe overall profit of the supply chain
5 Pricing Strategy of Members When theRetailer Has Fairness Concerns
The supply chain cannot be coordinated when the retailerdoes not have a fairness concern However it is necessary todetermine how the strategy changes when we consider theeffect of fairness concerns and whether channel efficiencycould be improvedThese issues are discussed in the followingsection
51 The Retailerrsquos Problem Because we consider the fairnessconcerns of the retailer in the distribution of profits wemust establish a model for fairness concerns Some mayargue that a more general model that includes both aversionto disadvantageous inequality and aversion to advantageousinequality (for example Fehr and Schmidt [5] and Charness
amp Rabin [40]) is more desirable However a preference foradvantageous inequality is much less prominent (Loewen-stein Thompson and Bazerman [41] did not find it in theirexperiment Ho amp Su [42]) Furthermore we assume that theretailerrsquos fairness reference is the manufacturerrsquos profit 120587119904119903instead of 120574120587119904119903 (120574 gt 0) because the general setting will notproduce substantially different ormore insightful results thana simple setting with 120574 = 1 (Pavlov amp Katok [43] Tengfei Nieand Shaofu Du [28]) The utility function of the retailer canbe written as follows
119880119903 = 120587119903 minus 120572 (120587119898119903 minus 120587119903)+ (14)
where 120587119903 = 119889119903(119901119903 minus 119908) denotes the retailerrsquos monetarypayoff 120587119898119903 = 119889119903(119908 minus 119888) denotes the profits of themanufacturerrsquos offline channel and 120572 denotes the level ofthe retailerrsquos fairness concern about the distribution of offlineprofits as retailers pay more attention to the profits madeby manufacturers from offline channels and compare themto the profits made from the online channel If the retailerrsquosmonetary profit is lower than the equitable profit120587119898119903minus120587119903 ge 0disadvantageous inequality occurs By contrast if 120587119898119903 minus120587119903 lt0 then 119880119903 = 120587119903 which indicates that the retailerrsquos utilityfunction is equal to the profit function
When the retailer faces disadvantage inequality its deci-sion problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (15)
119904119905 119901119903 le 2119908 minus 119888 (16)
The decision of the retailer is
119901119891119903=
119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 11986312119908 minus 119888 119894119891 119901119890 ge 1198701119908 minus 1198631
(17)
where1198701 = (2120572+3)120579(120572+1)1198631 = (120572+2)119888120579(120572+1)+119886120579The corresponding utility of the retailer is
119880119903 = (119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 ge 1198701119908 minus 1198631
(18)
When the retailer is faced with advantageous inequalityits decision problem is
max119901119903119880119903 = (1 + 120572) 119889119903 (119901119903 minus 119908) minus 119889119903 (119908 minus 119888) (19)
119904119905 119901119903 gt 2119908 minus 119888 (20)
The decision of the retailer is
119901119891119903 = 120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 119901119890 le 1198702119908 minus 1198632 (21)
where 1198702 = 31205791198632 = (119886 + 2119888)120579The corresponding utility of the retailer is
119880119903
6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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6 Complexity
= 14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 119901119890 le 1198702119908 minus 1198631
(22)
Proposition 1 The decision and corresponding utility of theretailer can be summarized as follows
119901119891119903 =
120579119901119890 + 119908 + 1198862 119894119891 119901119890 gt 1198702119908 minus 11986322119908 minus 119888 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198632119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 minus 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(23)
119880119903 =
14 (119886 minus 119908 + 120579119901119890)2 119894119891 119901119890 gt 1198702119908 minus 1198631(119908 minus 119888) (120579119901119890 minus 2119908 + 119886 + 119888) 119894119891 1198701119908 minus 1198631 le 119901119890 le 1198702119908 minus 1198631(119887 (120572 + 1) 119901119890 minus (2120572 + 1)119908 + (120572 + 1) 119886 + 120572119888)24 (120572 + 1) 119894119891 119901119890 lt 1198701119908 minus 1198631
(24)
52 The Manufacturerrsquos Problem We divide the feasibleregion of the manufacturerrsquos strategies to obtain the equilib-rium solutions By substituting (12) into 119889119890 ge 0 119889119903 ge 0and summarizing other conditions (119901119890 ge 119908 119908 gt 119888 andthe fairness boundary condition) we can confirm the feasibleregion as shown in Figure 2
The feasible region consists of R1 R2 and R3 R1 andR3 denote the feasible region of the manufacturerrsquos strategywhen the retailer faces both disadvantageous inequality andadvantageous inequality Therefore R2 denotes the feasibleregion when the retailer obtains a fair distribution of theprofits The expressions for the boundary conditions aresummarized in Table 1
It is easy to prove that R2 and R3 satisfy the constraint(119889119903 gt 0) Considering the response functions of the differentregions we can solve the partial equilibrium strategies of themanufacturer accordingly Then we can obtain the optionalsolutions (119908119891lowastlowast 119901119891lowastlowast119890 ) by comparing the partial equilibriumstrategies of the different regions
In 1198771 we denote (1199081198911 1199011198911198901) as the partial equilibriumstrategies and 1205871198911198981 as the optimal profit of the manufacturerThe optimal model is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (25)
119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1) 119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(26)
In R2 we denote (1199081198912 1199011198911198902) as the partial equilibriumstrategies and 1205871198911198982 as their optimal profit The optimal modelis as follows
max1199011198901199081205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (27)
119904119905 119901119903 = 2119908 minus 119888119901119890 ge 1198701119908 minus 1198631119901119890 le 1198702119908 minus 1198632119901119890 gt 1198703119908 minus 1198633119901119890 le 1198706119908 minus 1198636
(28)
In R3 we denote (1199081198913 1199011198911198903) as the partial equilibriumstrategies and 1205871198911198983 denotes the optimal profit for the man-ufacturer as follows
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888) (29)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(30)
Therefore the optimal profit from the decision-making problem for the manufacturer is max120587119891lowast119898 =max1205871198911198981 1205871198911198982 1205871198911198983 Thus we denote (119908119891lowastlowast 119901119891lowastlowast119890 ) as theglobal optimal solution which is referred to as an equilibriumstrategy in the following By substituting (119908119891lowastlowast 119901119891lowastlowast119890 ) into(23) we can deduce the optimal retail price 119901119891119903 Furthermore
Complexity 7
Table 1 Boundary conditions of the feasible region
Expressions for boundary conditions Meaning
L1 119901119890 = (2120572 + 3)120579 (120572 + 1)119908 minus (120572 + 2) 119888120579 (120572 + 1) minus 119886120579 Fairnesscondition
L2 119901119890 = 3120579 minus 119886 + 2119888120579 Fairnesscondition
L3 119901119890 = 119908 Foundationboundary
L4 119901119890 = (2120572 + 1)119908120579 (120572 + 1) minus 119886120579 minus 120572119888120579 (120572 + 1) R1 119889119903 = 0L5 119901119890 = (2120572 + 1) 120579119908(2 minus 1205792) (120572 + 1) + (2 + 120579) 1198862 minus 1205792 minus 120572120579119888(2 minus 1205792) (120572 + 1) R1 119889119890 = 0L6 119901119890 = (3120572 + 2) 1205791199082 (2 minus 1205792) (120572 + 1) +
(120572 + 1) (120579 + 2) 119886 minus ((120572 + 1) (1205792 + 120579 minus 2) + 120572120579) 1198882 (2 minus 1205792) (120572 + 1) R2 119889119890 = 0L7 119901119890 = 1205791199082 minus 1205792 + (120579 + 2) 1198862 minus 1205792 R3 119889119890 = 0
L3
w=c
w
Pe
E
L2
Q
L7
L1
A
C
L6
J
M
R1R2R3
L5
L4
Figure 2 The feasible region of the manufacturerrsquos strategies
Table 2 The partial equilibrium strategies of the manufacturer inR1
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
we can obtain the optimal profit and utility of the retailer aswell as the profit of the manufacturer
First we discuss the pricing strategies of R1 R2 and R3(hereinafter referred to as ldquopartial equilibrium strategiesrdquo)(119908119891lowast119894 119901119891lowast119890119894 ) 119894 = 1 2 3 denotes the extreme point in region
Table 3 The partial equilibrium strategies of the manufacturer inR2
120579 (0 1205796] (1205796 1205795] (1205795 1)1199081198912 119908119891lowast22 119908119891lowast2 119908119891lowast211199011198911198902 119901119891lowast11989022 119901119891lowast1198902 119901119891lowast11989021119894 (119908119891lowast119894119895 119901119891lowast119890119894119895) denotes the maximum point on boundary 119895 inregion 119894Lemma 2 In R1 the partial equilibrium strategies of themanufacturer are shown in Table 2
AppendixA provides the proof of Lemma 2 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Similarly we provide the optional solutions for R2 and R3in Lemmas 3 and 4
Lemma 3 In R2 the partial equilibrium strategies of themanufacturer are shown in Table 3
Appendix B provides the proof of Lemma 3 the analyticalexpressions of all partial equilibrium strategies and thethresholds of the parameters
Lemma 4 In R3 the partial equilibrium strategy of themanufacturer is shown as follows
Thepartial equilibrium strategy of themanufacturer inR3(119908119891lowast32 119901119891lowast11989032) is equal to the partial (0 lt 120579 le 1205796) solution of R2(119908119891lowast22 119901119891lowast11989022) which proves that there is no partial equilibriumstrategy in R3 We can also obtain the same conclusionthrough the following analysis The retailer will decide 119901119891119903 =2119908 minus 119888 rather than 119901119891119903 = (120579119901119890 + 119908 + 119886)2 because of thefairness concern when the manufacturer adopts the optimalpricing strategy (119908119891lowast3 119901119891lowast1198903 ) which reduces the profit of themanufacturer Thus the manufacturer will adopt strategy
8 Complexity
Table 4 The complete equilibrium strategy of the manufacturer
Parameter range Equilibrium strategy120579 120572 120579 120572 119908119891lowastlowast 119901119891lowastlowast119890
0 lt 120579 le 12057960 lt 120572 le 1205721 0 lt 120579 le 1205797 0 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 le 1205721 119908119891lowast22 119901119891lowast119890221205796 lt 120579 le 1205797 0 lt 120572 le 1205721 119908119891lowast1 119901119891lowast11989011205721 lt 120572 lt 1
0 lt 120579 le 1205792 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205792 lt 120579 le 1205797 1205721 lt 120572 lt 1 119908119891lowast22 119901119891lowast119890221205797 lt 120579 le 1205796 1205721 lt 120572 le 1205722 119908119891lowast1 119901119891lowast11989011205722 lt 120572 lt 1 119908119891lowast22 119901119891lowast11989022
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(119908119891lowast22 119901119891lowast11989022) rather than (119908119891lowast3 119901119891lowast1198903 ) after considering thesituation This phenomenon also reflects the diversity ofthe impact of fairness concerns on the memberrsquos decision-making
Appendix C provides the proof of Lemma 4
Proposition 5 By comparing the partial equilibrium strate-gies of R1-R3 we can determine the complete equilibriumstrategies as shown in Table 4
Appendix D provides the proof of Proposition 5 theanalytical expressions of all partial equilibrium strategies andthe thresholds of the parameters
Note that 120579 and 120572 codivide the manufacturers finaldecision space in Table 4 The complete equilibrium solutionis composed of four pricing strategies of the manufacturerand the retailer Additionally there are some remarkablephenomena (1)Themanufacturer does not adopt the pricingstrategy in R3 which is illustrated in Lemma 4 (2)When thecross-price sensitivity coefficient exceeds a threshold 1205793 thepricing strategy of the manufacturer makes the offline salesvolume too small and themanufacturermay choose to cancelthe offline retail channel at this time
The expressions of the optimal solutions are complextherefore the analysis of Proposition 5 is supported bynumerical examples in the next section In addition all of thethresholds for 120579 and 120572 are analytic expressions therefore theinfluence of the parameters on the membersrsquo decisions andsupply chain efficiency is analysed by selecting parametersthat are representative of a real-life situation
6 Numerical Analysis
In this section using numerical experiments we provideadditional management implications to prove the proposi-tions discussed above The analysis is conducted as followswe analyse the impacts of the retailerrsquos fairness concernsand cross-price sensitivity on the pricing strategies and theprofits and utility of the two members in different settings In
particular we focus on the impact of the influencing factoron the variations in channel efficiency when the memberschange their strategies
As this paper mainly studies the influence of fairnessconcerns on dual-channel decision-making several referencevalues are set for the cross-price sensitivity coefficient andan interval simulation is conducted for the fairness concernwhere 120572 varies from 0 to 1 We employ data based on acomparison of previous studies ([23 29]) The cross-pricesensitivity coefficient is set to 03 and 05 which means thatthe coefficient is normal and high respectively The otherbasic parameters in the experiments are set as follows a=1and c=03 The constraint problem is no longer consideredas the complete equilibrium solution satisfies the constraintconditions in the proof The experimental results are shownin Figures 3ndash7
Observation 6 (change in equilibrium strategy)
(1) Effects of 120579 and 120572 on Pricing Strategy As 120572 changesthe manufacturer develops two strategies considering a faircaring retailer as shown in Figures 3 and 4 For convenience(119901119891lowast1199031 119908119891lowast1 119901119891lowast1198901 ) is referred to as equilibrium strategy 1 and(119901119891lowast1199031 119908lowast2 119901lowast1198902) is referred to as equilibrium strategy 2The twoscenarios used for the supply chain are simple in scenario 1the retailer does not have fairness concerns and in scenario2 the retailer does
The decision analyses of the retailer and manufacturerare shown as follows (i) In scenario 2 when the cross-price sensitivity coefficient is normal 120579 = 03 and the levelof the retailerrsquos fairness concern is less The manufacturerwill set a high wholesale price and direct channel price toreduce the profits of an ambitious retailer in a traditionalretail channel which results in a disadvantageous inequalityAs 120572 increases meaning that the retailerrsquos sense of fairnessgrows stronger the manufacturer sets a lower 119908lowast1 and 119901lowast1198901and the retailer will set a higher retail price 119901119891lowast1199031 When theparameters exceed the threshold 120572 (057) the manufacturerwill adopt equilibrium strategy 2 which consists of a lower119908
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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Submit your manuscripts atwwwhindawicom
Complexity 9
pric
e
02 03 04 05 06 07 08 09 101
08
09
1
11
12
13
14
15
p1r (w
flowast1 p
flowaste1 ) = 03
p2r (w
flowast2 p
flowaste2 ) = 03
pflowaste1 = 03
pflowaste2 = 03
plowastr = 03
plowaste = 03
p1r (w
flowast1 p
flowaste1 ) = 05
pflowaste1 = 05
plowastr = 05
plowaste = 05
Figure 3 Impact of 120579 and 120572 on online and offline prices
and 119901119890 to achieve channel fairness (ii) In scenario 2 whenthe cross-price sensitivity coefficient is high 120579 = 05 theprice strategies of the retailer andmanufacturer are all higherthan before (120579 = 05) Another interesting phenomenonis observed As the cross-sensitivity coefficient increasesthe manufacturer becomes more likely to always adopt one
strategy without considering the disadvantageous inequalitywhich leads the retailer to care more about fairness Thisphenomenon reflects the diversity of strategies used bymembers which conflicts with the conclusions of QinghuaLi [29] and proves Proposition 5 (iii)Thewholesale price andonline direct price in scenario 2 are always lower than those
10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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Stochastic AnalysisInternational Journal of
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10 Complexity
w
wflowast1 = 03
wflowast2 = 03
wlowast = 03
1090807060504030201
15
14
13
12
11
1
09
08
07
06
05
wflowast1 = 05
wlowast = 05
Figure 4 Impact of 120579 and 120572 on 119908
for scenario 1 and the gaps slowly increase as 120572 increasesCompared to the manufacturer the offline retail price set bythe retailer in scenario 2 is lower than that in scenario 1 onlywhen the fairness channel exists (iv) Retailers with a strongerfairness concern are more likely to enter a neutral state orobtain a fairness result
Observation 7 (changes in the profits and utility of themembers)
(1) Effects of 120579 and 120572 on the Manufacturerrsquos Profit Wecan deduce some information by observing Figure 5 (i) InFigure 5 120579 = 03 and the manufacturer adopts equilibriumstrategy 1 as the profit of this strategy is higher than thatof equilibrium strategy 1 As 120572 increases the gap betweenthe two strategies decreasesTherefore the manufacturer willchange its strategy if the level of the fairness concern of theretailer exceeds a threshold 120572 A comparison of the profitfor the manufacturerrsquos two strategies is proof of the previous
analysis in Observation 6 While the cross-price sensitivitycoefficient is high when 120579=05 the manufacturer will alwaysadopt equilibrium strategy 1 even though the correspondingprofit decreases as 120572 increases This phenomenon occursbecause the manufacturer would rather have a disadvanta-geous inequality existing than preserve a fairness channel thatcould hurt his interest (ii) Due to retailerrsquos behavior relatedto his fairness concern themanufacturers profit is always lessthan in scenario 1 (iii) Cross-price sensitivity has a positiveeffect on the manufacturer
(2) Effects of 120579 and 120572 on the Retailerrsquos Profit and UtilityFigure 6 shows the utility and profit of the retailer Similarlythe analysis has some implications (i) Compared to scenario1 the profit and utility of the retailer increase because of hisfairness concern which differs from that of themanufacturer(ii) The retailerrsquos utility (120579 = 03) is more than that for (120579 =05) as 120572 increases If 120572 gt 120572 the fairness channel exists andboth the utility and profit increase considerably It can be seen
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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Submit your manuscripts atwwwhindawicom
Complexity 11
profi
ts
Πfm (p
1r (w
lowast1 p
lowaste1)) = 05
Πm(wlowast plowaste ) = 05
Πfm (p
1r (w
lowast1 p
lowaste1)) = 03
Πfm (p
2r (w
lowast2 p
lowaste2)) = 03
Πm(wlowast plowaste ) = 03
02 03 04 05 06 07 08 09 101
03
035
04
045
05
055
06
065
Figure 5 Impact of 120579 and 120572 on 120587119891119898 for the two scenarios
that fairness concerns have a greater effect on the retailersthan cross-price sensitivity
Observation 8 (changes in channel efficiency) The fairnessconcern of the retailer will affect the decisions and profits ofthe members To further illustrate the impact of the retailerrsquosfairness concerns on double marginalization we use (120587119891lowast119903 +120587119891lowast119898 )120587119888lowast119888 to present the channel efficiencies as done in [29]where 120587119888lowast119888 represents the total profit of the centralized supplychain Figure 7 illustrates how the channel efficiencies changeas 120572 increases and 120579 changes
Figure 7 provides valuable information (i) An increasein 120572 widens the gap between the two scenarios when theretailersrsquo fairness concerns do not reach a certain level 120572(120572 lt 120572) The intuitive explanations for this result are asfollows As 120572 increases the manufacturer reduces both thewholesale prices and the network direct prices while theretailers raise retail prices to boost their profits which leadsto double marginalization (ii) We discuss the situation inwhich the level of the retailersrsquo fairness concerns exceeds thethreshold (120572 gt 120572) In this case cross-price sensitivity is
normal The manufacturer adopts a lower pricing strategybecause he is focused on the retailerrsquos fairness concerns Thisadjustment results in a fairness channel Then the retailerobtains a greater profit even after setting a lower priceThe performance of the supply chain is obviously enhancedand tends towards Pareto optimality Simultaneously cross-price sensitivity is high Channel efficiency decreases as 120572increases This change can be explained by the fact that themanufacturerrsquos strategy considers his own interests and hedid not provide a fair distribution for the retailer (iii) Thesupply chain cannot be coordinated by a constant wholesaleprice in scenarios 1 and 2
7 Concluding Remarks
In this paper we considered a dual-channel supply chainthat includes one manufacturer and one retailer A supply-demand relationship and a competitive relationship willcoexist between the manufacturer and retailer after an onlinedirect marketing channel opens Based on this complexrelationship we investigated the considered model with two
12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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12 Complexity
Πfr (p1
r (wflowast1 p
flowaste1 )) = 03
Πfr (p2
r (wflowast2 p
flowaste2 )) = 03
ufr (p1
r (wflowast1 p
flowaste1 )) = 03
ufr (p2
r (wflowast2 p
flowaste2 ) = 03
Πr(plowastr ) = 03
Πfr (p1
r (wflowast1 p
flowaste1 )) = 05
ufr (p1
r (wflowast1 p
flowaste1 )) = 05
Πr(plowastr ) = 05
Πfr (p2
r (wflowast2 p
flowaste2 ) = u
fr (p2
r (wflowast2 p
flowaste2 )
1090807060504030201
profi
ts an
d ut
ilitie
s011
01
009
008
007
006
005
004
003
Figure 6 Impact of 120579 and 120572 on 120587119891119903 and 119906119891119903 for the two scenarios
scenarios to represent whether the retailer has fairness con-cerns Simultaneously the cross-price sensitivity coefficientwhich affects the price competition between dual channels isemphasized in the analysis of pricing decisions By theoreticalderivationwe get the following conclusions andmanagementenlightenments as follows
When the manufacturer faces a rational retailer theoptimal pricing strategy of the decentralized supply chainmembers are increasing functions of the cross-price coeffi-cient 120579Therefore the enterprise managers can set a relativelyhigh retail price and its negative effect on channel demandwill be weakened when all channel consumers pay moreattention to price comparison between channels The resultis more conducive to improving the profits of supply chain
The channel efficiency of the decentralized supply chain isan increasing function of the cross-price coefficient 120579 whichindicates that the double marginalization will be weakenedwhen 120579 increases
Retailers fair concern behavior has an impact on pricestrategy We integrate fair concern into the study of pricestrategy Different from previous studies this paper deducea complete set of pricing decisions when manufacturers bal-ance channel fairness and self-interest 120579 and 120572 codeterminethe manufacturerrsquos final decision which reflect that rationalmanufacturers should take into account both the cross-price impact between channels and retailersrsquo fair behaviorwhen formulating pricing strategies The decision set reflectsthat when the cross-price influence coefficient is fixed the
Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
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[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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Complexity 13
01
chan
nel e
ffici
ency
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 03
(Πlowastr + Πlowast
m)Πclowastc ) = 03
(Πflowastr + Π
flowastm )Πclowast
c ) = 05
(Πlowastr + Πlowast
m)Πclowastc ) = 05
(p2r w
flowast2 p
flowaste2 ) = 03
(p1r w
flowast1 p
flowaste1 ) = 05
(p1r w
flowast1 p
flowaste1 ) = 03
099
098
097
096
095
094
093
092
091
09
08910908070605040302
Figure 7 Impact of 120579 and 120572 on channel efficiencies for the two scenarios
retailers with stronger fair concern are more likely to impelthe manufacturer to make a relatively fair price strategy
Furthermore by numerical experiments the thesis andpropositions are verified Some findings and the correspond-ing management implications are given as follows
When the cross-price influence between channels islower retailers that are more concern about fairness impelmanufacturers set a lower wholesale price and online directprice which is conducive to channel coordination Thisphenomenon is consistent with previous inference Whenthe cross-price influence between channels is higher man-ufacturers establish higher price strategies which are morebeneficial for their own interests Considering the retailersfairness concerns the cross-price effect is not always con-ducive to supply chain coordination which is differentfrom the situation facing rational retailers Therefore whenconfronting a retailer with strong fairness concerns it ismore suitable for the manufacturer to cooperate when the
cross-price impact between channels is relatively low Incontrast among those retailers less concerned about fairnessrational retailers would be better partners
In order to further explore the articles conclusionson the application value of supply chain management Anatural extension of our research would be to test ourmodel predictions For example our analysis predicts that themanufacturer will obtain less profit when he provides a fairdistribution for the retailer in the offline retail channel Inthe home furnishings industry of China the offline retailersfeel that they have been unfairly treated based on the lossof profits due to direct channels Some brands such asKUKA propose giving 10-15 of their profits to dealers [44]To enhance offline marketing Vivo surrenders a greaterportion of the profits to offline mobile phones retailerswhile some brands do not give enough profits whichresults in low sales enthusiasm on the part of retailers [4546]
14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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14 Complexity
The demand functions are constructed for the dual-channel supply chain with homogeneous channel The impli-cation of this paper has limitations to some extent Notall product consumers are primarily concerned with pricewhen choosing shopping channel Furthermore some man-ufacturers adopt a heterogeneous product strategy to effec-tively alleviate the channel conflict Therefore consideringthe heterogeneity of channels and conducting a systematicstatistical analysis of ourmodel is a significant topic for futureresearch
Appendix
A Proof of Lemma 2
In region 1 the manufacturerrsquos problem is as follows
max 1205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)
119901119890 lt 1198701119908 minus 1198631119901119890 ge 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119901119890 lt 1198705119908 minus 1198635
(A1)
Therefore the Hessianmatrix of themanufacturerrsquos profitfunction can be calculated
There is a threshold for the cross-price sensitivity coef-ficient 1205791 = 2radic2(21205722 + 3120572 + 1)(171205722 + 24120572 + 8) When120579 isin (0 1205791) the Hessian matrix is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists If the extremepoint is in the feasible region then the extreme point isthe partial equilibrium solution for the manufacturer Incontrast the partial equilibrium solution is on the boundaryof R1 If 1205791 le 120579 lt 1 the Hessian matrix is neithernegative definite nor positive which can be easily provedIn this case there is no extreme point in the feasible region
which suggests that the optimal solution in R1 is on theboundary
The process of deduction is as follows(1) For inferring the complete partial equilibrium solu-tions in R1 some basic work should be solved includingthe extreme point the optimal value of the boundary andthe corresponding parameter range This part is given inAppendix A1(2) By comparing the results of Appendix A1 we coulddeduce the partial equilibrium solution and the correspond-ing parameter range when the Hessian matrix is negativedefinite (0 lt 120579 lt 1205791) This part is given in Appendix A2(3)When 1205791 le 120579 lt 1 we should determine the optimalvalue of each boundary of R1 and the corresponding param-eter range By comparing the results of Appendix A1 we canobtain the corresponding partial equilibrium solutions Thispart is given in Appendix A3
A1 BasicWork (1)Theoptimal solution on boundary L1 andthe corresponding parameter range are as follows
The manufacturerrsquos problem is as follows
max1199011198901199081205871198911198981 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119891119903 = 119908 (2120572 + 1)2 (120572 + 1) + 1205791199011198902 + 119886 + 120572 (119886 minus 119888)2 (120572 + 1)119901119890 = 1198701119908 minus 1198631119908119862 le 119908 le 119908119860
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(1199081311990111989013) denotes the decision of the manufacturer onthis boundary and the remainder of the boundaries followthis method By solving the problem we can obtain thefollowing
If 0 lt 120579 le 1205799
11990813 = 119908119891lowast13 = 119888 (120572 + 1) 1205792 + ((4119888 minus 119886) 120572 minus 119886 + 2119888) 120579 minus (3119886 + 5119888) 120572 minus 3 (119886 + 119888)2 ((120572 + 1) 120579 + 4120572 + 3) (120579 minus 1) 11990111989013 = 119901119891lowast11989013 = 119908119891lowast13
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
1198831 + ((101205792 minus 120579 minus 8) 1205722 + (131205792 minus 120579 minus 12) 120572 + 41205792 minus 4) 1198881198831
(A16)
where1198831 = 1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8A2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205791We substitute 119908119891lowast1 and 119901119891lowast1198901 into all the boundaries ldquo+rdquodenotes the extreme point that satisfies the correspondingconstraint similarly ldquo-rdquo indicates that the constraint is notsatisfied The results of all the constraints are summarized inTable 5
Analysis 1 (if 0 lt 120572 le 1205721 cap 0 lt 120579 le 1205793 cup 1205721 lt 120572 lt 1cap 1205792 lt 120579 le 1205793) The extreme point satisfies all theconstraints Thus the corresponding partial equilibriumsolution in this situation is
Analysis 2 (if 1205721 lt 120572 lt 1 cap 0 lt 120579 le 1205792) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for L3 of Appendix A1 we can
prove that 1205792 lt 1205799 Therefore we can infer the correspondingpartial equilibrium solution in this situation
Analysis 3 (if 0 lt 120572 lt 1 cap 1205793 lt 120579 lt 1205791) Based onTable 5 only constraint (L3) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of L3Combinedwith the result of L3 of AppendixA1 we can provethat 1(120572+1) lt 1205793Therefore we can infer the correspondingpartial equilibrium solution in this situation
A3 Inferring the Partial Equilibrium Solution When 120579 isin[12057911)Analysis 4 In this case there are no extreme points in thefeasible region which suggests that the optimal solution inR1 is on the boundary We need to infer and compare thesolutions of L1 L3 L4 and L5(1) For all 120572 isin (0 1) it is easy prove that 1205791 gt (minus5120572 minus 2 +radic331205722 + 40120572 + 16)2(120572 + 1) 1205791 gt 1(120572 + 1) and 1 minus 12120572 lt(1205791)2 Then we can infer the solution on each boundarywhich is shown in (2)-(5)(2)
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
By summarizing Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R1 which is shown inTable 2
B Proof of Lemma 3
The problem of the manufacturer is
max 1205871198911198982 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)119904119905 119901119891119903 = 2119908 minus 119888
119901119890 gt 1198701119908 minus 1198631119901119890 lt 1198703119908 minus 1198633119901119890 gt 1198704119908 minus 1198634119908 gt 119888
(B1)
There is a threshold for the cross-price sensitivity coeffi-cient 1205794 = 2radic23 When 120579 isin (0 1205794) the Hessian matrix isnegatively definite If 1205794 le 120579 lt 1 then the Hessian matrixis neither negative definite nor positive definite Similarly weuse a method such as Lemma 2 to deduce the conclusion
B1 Basic Work (1)The result is the same as Appendix A1(2)The optimal solution on boundary L2 and the corre-sponding parameter range are as follows
119908119891lowast22 = (21205792 minus 3120579 minus 6) 119886141205792 minus 18 + (21205793 + 91205792 minus 3120579 minus 12) 119888141205792 minus 18
119901119891lowast11989022 = minus (8120579 + 9) 119886141205792 minus 18 +(61205792 minus 120579 minus 9) 119888141205792 minus 18
(B2)
For all 120579 isin (0 1) 120572 isin (0 1) 119908119876 lt 119908119891lowast22 lt 119908119864 120572 isin (0 1)(3)The optimal solution on boundary L3 and the corre-sponding parameter range are as follows
119908119891lowast26 = 1198864 minus 4120579 + 34119888119901119891lowast11989026 = (2 minus 120579) 1198862 minus 2120579 + 1205792119888
(B4)
(5)The extreme point of the manufacturerrsquos profit func-tion is as follows
119908119891lowast2 = (3120579 + 2) 1198868 minus 91205792 minus(61205792 + 120579 minus 6) 1198888 minus 91205792
119901119891lowast1198902 = (3120579 + 4) 1198868 minus 91205792 minus(61205792 minus 120579 minus 4) 1198888 minus 91205792
(B5)
B2 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794We substitute (119908119891lowast2 119901119891lowast1198902 ) into each boundary ldquo+rdquo indicatesthat the extreme point satisfies the corresponding constraintsimilarly ldquo-rdquo indicates that the constraint is not satisfied Theresults of all the constraints are summarized in Table 6
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
We analyse the equilibrium solution of different param-eter ranges and use (1199081198912 1199011198911198902) to denote the correspondingpartial equilibrium solutions in R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L2) is not satisfied which impliesthat the partial equilibrium solution is on the boundary ofL3 Combined with the results for (2) of Appendix B1 wecan infer the corresponding partial equilibrium solution inthis situation
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) Based on Table 5all the constraints are satisfied which implies that the extremepoint is the partial equilibrium solution
Analysis 3 (if 1205795 lt 120579 lt 1205794 cap 0 lt 120572 lt 1) Based onTable 6 only constraint (L1) is not satisfied which impliesthat the partial equilibrium solution is on the boundary of
L1 Combined with the result of Appendix B1 (1) we caninfer the corresponding partial equilibrium solution in thissituation
B3 Inferring the Partial Equilibrium Solution When 0lt120579lt1205794Analysis 4 (if 1205794 le 120579 lt 1 cap 0 lt 120572 lt 1) Similarly with R1we should determine and compare the optimal solutions forL1 L2 L3 and L4(1)
1205871198982 (119908lowast21 119901lowast11989021) minus 1205871198982 (119908lowast22 119901lowast11989022) = (120579119888 + 119886 minus 119888)2 120572 ((120572 + 1) (121205794 minus 81205793 minus 441205792 minus 8120579 + 16) minus 111205792 minus 16120579 minus 4)
112 ((1205722 + (114) 120572 + 74) 1205792 minus (120572 + 32)2) (97 minus 1205792) gt 0 (B12)
Combined with (1)-(4) the partial equilibrium solutionis
To summarize Analyses 1ndash4 we can obtain the partialequilibrium solution set in region R2 which are shown inTable 3
C Proof of Lemma 4
Themanufacturerrsquos problem in R3 is
max1199011198901199081205871198911198983 = 119889119890 (119901119890 minus 119888) + 119889119903 (119908 minus 119888)
119904119905 119901119903 = 120579119901119890 + 119908 + 1198862 119901119890 ge 1198702119908 minus 1198632
119901119890 ge 1198703119908 minus 1198633119901119890 le 1198707119908 minus 1198637119908 gt 119888
(C1)
The Hessian matrix 119867120587119898 [ 119901119890119908 ] is negative definite Themanufacturerrsquos profit function is a concave function of 119901119890and 119908 and the decision problem is a convex optimizationproblem Thus a unique extreme point exists as follows
Analysis 1 If (119908119891lowast3 119901119891lowast1198903 ) is brought into each boundary thenonly L2 is not satisfied Thus we can deduce that the partialequilibrium solution of R3 is on L2The optimal point for the
Complexity 19
L2 of R2 and R3 is the same Combined with Appendix B1the partial equilibrium solution is
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
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Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
We have obtained the partial equilibrium solution set of R1R2 and R3 The manufacturer would compare the solutionsfor the different regions that are in the same parameter rangeand choose the optimum solution as the equilibrium pricingstrategy The process of inference is as follows(1)By comparing the different thresholds in Lemmas 2ndash4we can divide the parameter ranges that the solution needs tocompare This part is given in Appendix D1(2) Based on the results of Appendix D1 we determinethe complete equilibrium solution This part is given inAppendices D2 and D3
D1 Divide the Parameter Ranges That the Solution Needs toCompare Combined with Lemmas 2 3 and 4 the relevantthresholds are summarized as follows
(3) if 0 lt 120572 le (3 minus 2radic2)3(radic2 minus 1) 997904rArr 0 lt 1205796 lt 1205795 lt1205794 lt 1205793 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 7(4) if (3 minus 2radic2)3(radic2 minus 1) lt 120572 le 1205721 997904rArr 0 lt 1205796 lt 1205795 lt1205793 lt 1205794 lt 1205791 lt 1The problem of the manufacturerrsquos decision is shown in
Table 8(5) if 1205721 lt 120572 lt 1 997904rArr 0 lt 1205792 lt 1205796 lt 1205795 lt 1205793 lt 1205794 lt 1205791 lt1The problem of the manufacturerrsquos decision is shown in
Table 9
D2 R3 and R2 Comparison of the Partial Equilibrium Solu-tions of R3 and R2
Analysis 1 (if 0 lt 120579 le 1205796 cap 0 lt 120572 lt 1) As 119908119891lowast32 = 119908119891lowast22 119901119891lowast11989032 = 119901119891lowast11989022 the partial equilibrium solutions of R3 and R2are the same
Analysis 2 (if 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1) It is easy to provethat 1205871198982(119908119891lowast2 119901119891lowast1198902 ) minus 1205871198983(119908119891lowast32 119901119891lowast11989032) gt 0 Thus the partialequilibrium solutions of R2 are better than those for R3
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
For all 1205795 lt 120579 lt 1 and 0 lt 120572 lt 1 we can prove that(1205722+(114)120572+74)1205792minus(120572+32)2 lt 01198841120572+1198842 lt 0Thereforewe conclude that 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 1205871198982(119908119891lowast32 119901119891lowast11989032) Themanufacturer would choose the solution of R2
Analysis 4 By combining Analysis 1ndash3 we find that thepartial equilibrium solution of R2 is always more profitablethan R3 Combined with Appendix D1 (4)-(6) the problemof the manufacturerrsquos decision can be simplified and is shownin Tables 10 and 11
(1) If 0 lt 120572 le 1205721 See Table 10(2) If 1205721 lt 120572 lt 1 See Table 11
D3 Comparison of the Equilibrium Solutions of R2 andR1 By comparing the solutions of R2 and R3 the moreprofitable solutions are the final equilibrium solutions for themanufacturer and the retailer We use (119908119891lowastlowast 119901119891lowastlowast119890 ) to denotethem Based on Tables 10 and 11 the results are as follows
Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 12 cup 1205792 lt 120579 le1205796 cap 12 lt 120572 lt 1 compare (119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast22 119901119891lowast11989022)
1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast22 119901119891lowast11989022)= (120579119888 + 119886 minus 119888)2 120572 (11988471205722 + 1198848120572 + 1198849)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (71205792 minus 9)
For all 0 lt 120579 le 1205796 1712057921205722+241205792120572+81205792minus161205722minus24120572minus8 gt 01198841 lt 0By making 11988471205722 + 1198848120572 + 1198849 = 0 we can obtain
120572+ = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 + radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28
120572minus = minus2 (120579 + 1) (31205793 + 161205792 + 9120579 minus 6 minus radicminus31205796 minus 41205795 + 571205794 + 21205793 minus 1641205792 minus 48120579 + 64)121205794 + 761205793 + 891205792 minus 4120579 minus 28 (D6)
(1) If 0 lt 120572 le 120572+ 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) gt 0if 120572+ lt 120572 lt 1 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0We make 1205722 = 120572+ We need to compare the size of 1205722 and1205721 (2)
(1205722 minus 1205721) is a monotone increasing function of 120579 Thereexists a threshold 1205797(1205797 isin (1205792 1205796)) If 120579 isin (0 1205797) 1205722 lt 1205721 if120579 isin [1205797 1205796) 1205721 le 1205722 lt 1Solution 1 of Analysis 1 119868119891 0 lt 120579 le 1205796 cap 0 lt 120572 le 1205721Simultaneously 0 lt 120579 le 1205797 0 lt 120572 le 1205722 and 1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium solution is
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 21
Simultaneously 0 lt 120579 le 1205797 1205722 lt 120572 le 1205721and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) le 0 the equilibriumsolution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E2)Simultaneously 1205797 lt 120579 le 1205796 0 lt 120572 le 1205722 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
Solution 2 of Analysis 1 119868119891 1205792 lt 120579 le 1205796 cap 1205721 lt 120572 lt 1Simultaneously 1205792 lt 120579 le 12057971205721 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 )minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E4)Simultaneously 1205797 lt 120579 le 1205796 1205721 lt 120572 le 12057221205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) ge 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E5)Simultaneously 1205797 lt 120579 le 1205796 1205722 lt 120572 lt 1 and1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium
solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E6)Analysis 2 119868119891 0 lt 120579 le 1205792 cap 1205721 lt 120572 lt 1 then compare(119908119891lowast11 119901119891lowast11989011) with (119908119891lowast22 119901119891lowast11989022)Solution of Analysis 2 It is easy to prove that1205871198981(119908119891lowast11 119901119891lowast11989011)minus1205871198982(119908119891lowast22 119901119891lowast11989022) lt 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast22 119901119891lowast11989022) (E7)Analysis 3 119868119891 1205796 lt 120579 le 1205795 cap 0 lt 120572 lt 1 compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast2 119901119891lowast1198902 )1205871198981 (119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982 (119908119891lowast2 119901119891lowast1198902 )= (120579119888 + 119886 minus 119888)2 (120579 + 1) (119884101205722 + 11988411120572 + 11988412)(1712057921205722 + 241205792120572 + 81205792 minus 161205722 minus 24120572 minus 8) (8 minus 91205792)
(D8)
where
11988410 = 181205793 minus 61205792 minus 16120579 + 811988411 = 271205793 minus 24120579 + 811988412 = 91205793 + 31205792 minus 8120579
(D9)
It is easy to prove that 11988410 gt 0 under this condition Bymaking 119884101205722 + 11988411120572 + 11988412 equal to zero we can obtain
120572+= minus91205792 minus 6120579 + 4 + radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2) 120572minus= minus91205792 minus 6120579 + 4 minus radic91205794 + 121205793 + 41205792 + 16120579 + 16(120579 + 1) (3120579 minus 2)
(D10)
Make 1205723 = 120572+ for all 1205796 lt 120579 le 1205795 1205723 gt 0(1) If 0 lt 120572 le 1205723 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) le0If 1205723 lt 120572 then 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus 1205871198982(119908119891lowast2 119901119891lowast1198902 ) gt 0(2)There exists a threshold 1205798 isin (1205796 1205795) if 1205796 lt 120579 le 12057981205723 lt 1 if 1205798 lt 120579 le 1205795 1205723 gt 1
Solution of Analysis 3 1205796 lt 120579 le 1205795 0 lt 120572 lt 1 Simultaneously1205796 lt 120579 le 1205798 0 lt 120572 le 1205723 and 1205871198981(119908119891lowast1 119901119891lowast1198901 ) minus1205871198982(119908119891lowast2 119901119891lowast1198902 ) ge 0 the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E10)Analysis 4 119868119891 1205795 lt 120579 le 1205793 cap 0 lt 120572 lt 1 then compare(119908119891lowast1 119901119891lowast1198901 ) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 4 1205795 lt 120579 le 1205793 0 lt 120572 lt 11205871198981(119908119891lowast1 119901119891lowast1198901 ) gt 1205871198982(119908119891lowast21 119901119891lowast11989021) the equilibrium solutionis
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast1 119901119891lowast1198901 ) (E11)Analysis 5 119868119891 1205793 lt 120579 lt 1 cap 0 lt 120572 lt 1 then compare(119908119891lowast14 119901119891lowast11989014) with (119908119891lowast21 119901119891lowast11989021)Solution of Analysis 5 1205871198981(119908119891lowast14 119901119891lowast11989014) minus 1205871198982(119908119891lowast21 119901119891lowast11989021) gt 0the equilibrium solution is
(119908119891lowastlowast 119901119891lowastlowast119890 ) = (119908119891lowast14 119901119891lowast11989014) (E12)We have summarized (E1)-(E12) and the complete equilib-rium solutions are shown in Table 4
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
22 Complexity
Data Availability
The numerical experiment in this paper is mainly based onthe inferred conclusion in this paper and the correspond-ing parameter assignment is explained at the beginning ofSection 5 Parameter setting mainly refers to the parametersetting of other similar studies and references have been citedin this paper
Disclosure
We declare that the work presented in this manuscriptrepresents our original research that has not been publishedpreviously and is not under consideration for publicationelsewhere in whole or in part
Conflicts of Interest
No conflicts of interest exit in the submission of thismanuscript which is approved by all authors for publication
References
[1] K Takahashi T Aoi D Hirotani and K Morikawa ldquoInventorycontrol in a two-echelon dual-channel supply chain with setupof production and deliveryrdquo International Journal of ProductionEconomics vol 133 no 2 pp 403ndash415 2011
[3] httpwwwcamiacncontent775html[4] M Rabin ldquoIncorporating fairness into game theory and eco-
nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993
[5] E Fehr and K M Schmidt ldquoA theory of fairness competitionand co-operationrdquoThe Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999
[6] D Kahneman J L Knetsch and R HThaler ldquoFairness and theassumptions of economicsrdquo Journal of Business vol 59 no 4pp S285ndashS300 1986
[7] N Kumar and L K Scheer ldquoThe effects of supplier fairness onvulnerable resellersrdquo Journal of Marketing Research vol 32 no1 pp 54ndash65 1995
[8] D Zhao ldquoGame analysis on vertical monopoly based on fixedresale minimum pricemdashTake the case ofMoutai andWuliangyrdquoTimes Finance no 3 pp 88-89 2016
[9] C Xing ldquoDirect stores game chips between Moutai anddealersrdquoWine World no 4 pp 26-27 2012
[10] X Zhong ldquoThe delicate strategic partnership in supply chainmanagement - inspiration of Gome and Gree marketing warrdquoChina Collective Economy no 05 p 50 2007
[11] W K Chiang D Chhajed and J D Hess ldquoDirect marketingindirect profits a strategic analysis of dual channel designrdquoManagement Science vol 49 no 1 pp 1ndash20 2003
[12] G Ya-jun and Z Li-qiang ldquoThe conflict and coordination indual channel based on e-marketrdquo System Engineering Theoryand Practice vol 28 no 9 pp 59ndash66 2008
[13] K Cattani W Gilland H S Heese and J SwaminathanldquoBoiling frogs pricing strategies for a manufacturer addinga direct channel that competes with the traditional channelrdquo
Production Engineering Research and Development vol 15 no1 pp 40ndash56 2006
[14] Q Fang L Ren and Y Wang ldquoPricing strategy of retailer dual-channel supply chain considering predominant powerrdquo Journalof Wuhan University of Science and Technology vol 40 no 4pp 302ndash306 2017
[15] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009
[16] J F Tian T J Fan and J L Hu ldquoPricing policies in a dual-channel supply chainwithmanufacture servicesrdquo inProceedingsof the International Conference on Computer Information Sys-tems amp Industrial Applications 2015
[17] Q Xu Z Liu and B Shen ldquoThe impact of price comparisonservice on pricing strategy in a dual-channel supply chainrdquoMathematical Problems in Engineering vol 2013 Article ID613528 13 pages 2013
[18] C Shen Z Xiong and W Yan ldquoResearch on dual channelpricing and coordination strategy under the network pricecomparisonrdquo Chinese Journal of Management Science vol 22no 1 pp 84ndash93 2014
[19] B Li M Zhu Y Jiang and Z Li ldquoPricing policies of a com-petitive dual-channel green supply chainrdquo Journal of CleanerProduction vol 112 Part 3 pp 2029ndash2042 2016
[20] F Zhang and J Ma ldquoResearch on the complex features about adual-channel supply chain with a fair caring retailerrdquo Commu-nications in Nonlinear Science and Numerical Simulation vol30 no 1-3 pp 151ndash167 2016
[21] F Zhang and C Wang ldquoDynamic pricing strategy and coor-dination in a dual-channel supply chain considering servicevaluerdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 54 pp722ndash742 2018
[22] B Dan G Y Xu and C Liu ldquoPricing policies in a dual-channel supply chain with retail servicesrdquo International Journalof Production Economics vol 139 no 1 pp 312ndash320 2012
[23] Z Ding Research on The Pricing Strategy and CoordinationContracts of Dual Channel Supply Chain Hefei University ofTechnology 2015
[24] J Zhao X Hou Y Guo and J Wei ldquoPricing policies forcomplementary products in a dual-channel supply chainrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 49 pp 437ndash4512017
[25] J Lin and JWang ldquoResearch ofmanufacturersrsquo channel strategyunder dual-channel supply chain based on differentiated prod-uctrdquo Chinese Journal of Management Science vol 26 no 6 pp72ndash84 2018
[26] Y Qu Z Guan R Qu and T Ye ldquoImpact of membersrsquo fairnesspreference and loss-averse on order strategy in hybrid dualchannel supply chainrdquo Chinese Journal of Management vol 14no 01 pp 129ndash138 2017
[27] X Wei Q Lin and Y Qin ldquoOptimal pricing strategies ofdual-channel supply chain under risk aversion and fairnesspreferencerdquo Journal of Chongqing University of Technology(Social Science) vol 30 no 12 pp 50ndash58 2016
[28] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Researchvol 258 no 2 pp 491ndash500 2017
[29] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling 2016
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 23
[30] F Xu R Chuge and W Fan ldquoImpact of horizontal fairnessand vertical fairness on strategies in dual-channel supply chainrdquoJournal of Engineering System vol 29 no 04 pp 527ndash536 2014
[31] L Wang K Cheng and W Shiwei ldquoStudy on pricing strategiesof dual-channel supply chain under fairness preferencerdquo Chi-nese Journal ofManagement Scinece vol 20 no S2 pp 563ndash5682012
[32] B Li Y Li L Hou and P Hou ldquoImpact of fair-minded retaileron decision of supply chain in dual- channelrdquo Control andDecision vol 30 no 05 pp 955ndash960 2015
[33] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Research vol174 no 1 pp 646ndash667 2006
[34] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chainwhen production costsare disruptedrdquo Economic Modelling vol 30 no 1 pp 521ndash5382013
[35] G Y Xu B Dan X M Zhang and C Liu ldquoCoordinating adual-channel supply chain with risk-averse under a two-wayrevenue sharing contractrdquo International Journal of ProductionEconomics vol 147 no 1 pp 171ndash179 2014
[36] W Wang L Bo N Liao and L Xu ldquoRedefining customerexperience in the new retail era- McKinsey China digitalconsumer researchrdquo Science and Technology of China vol 2017no 09 pp 24ndash28 2017
[37] Niuli Trends in Cosmetics Sales Online and Offline OccupyEqual Shares 2017 httpwwwChinachinaChinairncomhyzx20170109140751848shtml
[38] M Liu E Cao C K Salifou et al ldquoPricing strategies of adual-channel supply chain with risk aversionrdquo TransportationResearch Part E Logistics and Transportation Review vol 90pp 108ndash120 2016
[39] C Yuan L Yan and G Chai ldquoA dual-channel cournot gamemodel with remarks on the policy of equal prices on the twochannels of suningrdquo Forecasting vol 33 no 05 pp 65ndash70 2014
[40] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquo The Quarterly Journal of Economics vol 117no 3 pp 817ndash869 2002
[41] G F Loewenstein L Thompson and M H Bazerman ldquoSocialutility and decision making in interpersonal contextsrdquo Journalof Personality and Social Psychology vol 57 no 3 pp 426ndash4411989
[42] T-H Ho and X Su ldquoPeer-induced fairness in gamesrdquoAmericanEconomic Review vol 99 no 5 pp 2022ndash2049 2009
[43] V Pavlov and E Katok Fairness and Coordination Failures inSupply Chain ContractsWorking Paper University of Texas atDalls 2009 httpwwwutdallasedusimekatokfair theorypdf
[44] X Kong ldquoKUKA Online and offline integrationrdquo China ChainStore no 10 pp 52-53 2015
[45] J Guo ldquoOPPO and Vivo Mobile phone marketing strategyanalysisrdquoModern SOE Research no 02 p 96 2018
[46] httpwwwsohucoma128912758 121344
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