Research ArticleA Multiproduct Single-Period Inventory Management Problemunder Variable Possibility Distributions
Zhaozhuang Guo12 Shengnan Tian1 and Yankui Liu1
1Risk Management amp Financial Engineering Laboratory College of Management Hebei University Baoding Hebei 071002 China2Fundamental Science Department North China Institute of Aerospace Engineering Langfang Hebei 065000 China
Correspondence should be addressed to Zhaozhuang Guo zhaozhuang2004163com
Received 2 August 2017 Revised 21 September 2017 Accepted 4 December 2017 Published 20 December 2017
Academic Editor Jean-Pierre Kenne
Copyright copy 2017 Zhaozhuang Guo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In multiproduct single-period inventory management problem (MSIMP) the optimal order quantity often depends on thedistributions of uncertain parameters However the distribution information about uncertain parameters is usually partiallyavailable To model this situation a MSIMP is studied by credibilistic optimization method where the uncertain demand andcarbon emission are characterized by variable possibility distributions First the uncertain demand and carbon emission arecharacterized by generalized parametric interval-valued (PIV) fuzzy variables and the analytical expressions about themean valuesand second-ordermoments of selection variables are established Taking second-ordermoment as a riskmeasure a new credibilisticmultiproduct single-period inventorymanagement model is developed undermean-moment optimization criterion Furthermorethe proposed model is converted to its equivalent deterministic model Taking advantage of the structural characteristics of thedeterministicmodel a domain decompositionmethod is designed to find the optimal order quantities Finally a numerical exampleis provided to illustrate the efficiency of the proposed mean-moment credibilistic optimization methodThe computational resultsdemonstrate that a small perturbation of the possibility distribution can make the nominal optimal solution infeasible In this casethe decision makers should employ the proposed credibilistic optimization method to find the optimal order quantities
1 Introduction
The MSIMP is a classical inventory management problemIn order to maximize (minimize) the total expected profit(cost) the decision makers have to make the optimal orderquantities at the beginning of the period At the end ofthe selling period either stock-out or excess inventory willoccur The two possibilities should be considered duringthe decision-making process The popularity of the MSIMPis due to its applicability in retailing and manufacturingindustries Hadley and Whitin [1] first considered a MSIMPwith storage capacity or budget constraints and proposeda dynamic programming solution procedure to find theoptimal order quantities Since then many researchers havedeveloped stochastic MSIMP For instance Nahmias andSchmidt [2] discussed the MSIMP under the linear anddeterministic constraints on budget or space H-S Lau and
A H L Lau [3] extended the MSIMP to handle multicon-straint and presented a Lagrangian-based numerical solutionprocedure for the MSIMP When the conditions of closed-form expressions did not hold Erlebacher [4] proposed aneffective heuristic solution Moon and Silver [5] dealt withthe MSIMP subject to not only a budget constraint on thetotal value of the replenishment quantities but also fixedcosts for nonzero replenishment Furthermore Abdel-Maleket al [6] considered a MSIMP under a budget constraintwith probabilistic demand and random yield Zhang [7]considered theMSIMPwith both supplier quantity discountsand a budget constraint and formulated it as a mixedinteger nonlinear programming model In order to dealwith the possible shortage of limited capacity Zhang andDu [8] discussed zero lead time outsourcing strategy andnonzero lead time outsourcing strategy They also developedthe structural properties and solution procedures for their
HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2159281 14 pageshttpsdoiorg10115520172159281
2 Mathematical Problems in Engineering
profit-maximization models Abdel-Malek and Montanari[9] proposed a methodology for studying the dual of thesolution space of the MSIMP with two constraints and intro-duced an approach to obtain the optimal order quantitiesof each product In addition Huang et al [10] studied acompetitive MSIMP with shortage penalty cost and partialproduct substitution In view of risk preference Ozler et al[11] proposed the MSIMP under a Value at Risk constraintVan Ryzin and Mahajan [12] reviewed the contributions tomultiproduct inventory problem with demand substitutionUnder mean-variance and utility function approaches VanMieghem [13] studied multiproduct single-period networksrsquoproblems in probabilistic framework
When the exact probability distribution of demand isunavailable probabilistic robust optimization method [14] isa tool to deal with the corresponding uncertainty in inventorymanagement problem Based on the assumption that demandwas described by discrete or interval scenarios Vairaktarakis[15] discussed several minimax regret formulations for theMSIMP with a budget constraint When the distributionof demand had known support mean and variance Kam-burowski [16] presented the theoretical foundations for ana-lyzing the inventory management problem They derived theclosed-form formulas for the worst-case and best-case orderquantities Shu et al [17] considered the distribution-freesingle-period inventory management problem by borrowingan economic theory from transportation disciplines Moonet al [18] found the differences between normal distributionapproaches and distribution-free approaches in four scenar-ios with mean and variance Under interval demand uncer-tainty Solyali et al [19] proposed a new robust formulationwhich could solve the intractability issue for large probleminstances As for recent development in stochastic inventorymanagement problems the interested reader may furtherrefer to [20ndash24]
Most of the extensions of inventorymanagement problemhave been made in the probabilistic framework whereuncertain parameters are characterized by random variablesHowever in some cases there are not enough data to deter-mine the exact probability distribution of random variablebecause of economic reason or technical difficulty In sucha case the variable is approximately specified based on theexperiences and subjective judgments of the experts in relatedfields so fuzzy inventory management problem is also anactive research area Fuzzy set theory was applied in the earlyinventorymanagement literature [25 26] In the area of fuzzyMSIMP Mandal and Roy [27] considered a multiproductdisplayed inventory model under shelf-space constraint infuzzy environment where the demand rate of a product wasconsidered as a function of the displayed inventory levelUnder fuzzy demand environment Ji and Shao [28] studiedthe MSIMP and formulated three kinds of models Dutta[29] formulated a fuzzy MSIMP model whose objective wasto maximize the total profit by considering fuzzy demandsIn fuzzy-stochastic environment Saha et al [30] developedmultiproduct multiobjective supply chain models with bud-get and risk constraints where themanufacturing costs of theitems were fuzzy variables and the demands for the productswere random variables Based on credibility measure Guo
[31] proposed two single-period inventory models wherethe uncertain demands were characterized by discrete andcontinuous possibility distributions respectively Tian andGuo [32] formulated a credibilistic optimization model fora single-product single-period inventory problem with twosuppliers
The work mentioned above studied inventory manage-ment problem under the assumption that the exact possibilitydistribution of fuzzy variable was available which motivatesus to study the MSIMP from a new perspective The motiva-tion of this paper is based on the following considerationsFirst shorter product life cycles and growing innovationrates make the market demand extremely variable In thiscase the distribution information about market demand isonly partially available It is reasonable to assume that theexact possibility distribution is embodied in a zonal areafor a practical MSIMP so the interval-valued fuzzy variableis introduced to characterize uncertain market demandSecond the optimal order quantities for different productsare heavily influenced by the carbon emission constraint Insome practical inventorymanagement problems it is difficultto determine the exact carbon emission during logisticactivities Under credibilistic carbon emission constraint aparametric credibilistic optimization model is developed forMSIMP To the best of our knowledge this issue has not beenaddressed in the literature
This paper studies MSIMP by parametric credibilisticoptimization method where uncertain market demand anduncertain carbon emission are characterized by generalizedPIV possibility distributions Decision makers can makeinformed decisions based on a tradeoff model between themean total profit and the second-ordermoment of total profitunder budget constraint and uncertain carbon emissionconstraint The strength of the proposed method is that thedistributions of market demand and carbon emission can betailored to the partial information at hand That is whenthe distribution information about uncertain parameters ispartially available the proposed method is more convenientfor modeling uncertain demand and carbon emission in apractical MSIMP The proposed credibilistic optimizationmethod differs from the existing MSIMP literature in thefollowing several aspects (i) A novel method is introducedto model the perturbation distributions of uncertain demandand carbon emission which is different from the existingliterature (ii) For PIV fuzzy variable its lambda selectionvariable is introduced as its representative the possibilitydistribution of lambda selection can traverse the entiresupport of the PIV fuzzy variable as the lambda parametervaries its values (iii) On the basis of L-Smultiple integral twonew optimization indexes mean and second-order momentabout the total profit are defined to build a parametric cred-ibilistic optimization model under credibilistic constraintof carbon emission (iv) A domain decomposition methodis designed to divide the original credibilistic optimizationmodel into several equivalent parametric programming sub-models which can be solved by conventional optimizationsoftware
The remainder of this paper is organized as followsAfter introducing some basic concepts in fuzzy possibility
Mathematical Problems in Engineering 3
theory Section 2 discusses the properties about generalizedPIV fuzzy variable and its selection variable In Section 3a new parametric credibilistic optimization model is firstdeveloped for MSIMP where uncertain demand and uncer-tain carbon emission are characterized by variable possibilitydistributions Then the equivalent deterministic model ofthe proposed parametric credibilistic optimization modelis discussed in this section A new domain decompositionmethod is also designed in this section to find the optimalorder quantities In Section 4 some numerical experimentsare conducted to demonstrate the validity of the proposedcredibilistic optimizationmethod Section 5 gives the conclu-sion of the paper
2 Generalized PIV Fuzzy Variables
First in this section some basic concepts in fuzzy possibilitytheory are recalled [33ndash36]
Let Γ be the universe of discourseP(Γ) the power set ofΓ and Pos P(Γ) 997891rarr R([0 1]) a fuzzy possibility measureThe triplet (ΓP(Γ) Pos) is called a fuzzy possibility space
Let 120585 be a type 2 fuzzy variable defined on the space(ΓP(Γ) Pos) If for any 119903 isin R the secondary possibilitydistribution function 120583120585(119903) = Pos120585 = 119903 is a subinterval[120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] of [0 1] then 120585 is called a PIV fuzzy vari-able where 120579119897 120579119903 isin [0 1] are two parameters characterizingthe degree of uncertainty that 120585 takes the value 119903
A type 2 fuzzy variable 120585 is called a generalized PIVnormal fuzzy variable [36] if its secondary possibility distri-bution is the subinterval
[(1 minus 120579119897) 119890minus(119903minus120583)221205902 119890minus(119903minus120583)221205902 + (1 minus 119890minus(119903minus120583)221205902) 120579119903] (1)
of [0 1] for 119903 isin R where 120583 isin R 120590 gt 0 and 120579119897 120579119903 isin [0 1] aretwo parameters characterizing the degree of uncertainty that120585 takes on the value 119903 When 120579119897 = 120579119903 = 0 the correspondingfuzzy variable is denoted by 120585119899 whose possibility distributionis called the nominal possibility distribution of 120585 In thefollowing 120585 sim 119899(120583 1205902 120579119897 120579119903) means that 120585 is a generalizedPIV normal fuzzy variable
A type 2 fuzzy variable 120578 is called a generalized PIVtriangular fuzzy variable [36] if its secondary possibilitydistribution is the subinterval [(119903 minus 1199031)(1199032 minus 1199031) minus 120579119897((119903 minus1199031)(1199032 minus 1199031)) (119903 minus 1199031)(1199032 minus 1199031) + 120579119903((1199032 minus 119903)(1199032 minus 1199031))] of[0 1] for 119903 isin [1199031 1199032] and the subinterval [(1199033 minus 119903)(1199033 minus1199032) minus 120579119897((1199033 minus 119903)(1199033 minus 1199032)) (1199033 minus 119903)(1199033 minus 1199032) + 120579119903((119903 minus1199032)(1199033 minus 1199032))] of [0 1] for 119903 isin [1199032 1199033] where 1199031 lt 1199032 lt1199033 are real numbers and 120579119897 120579119903 isin [0 1] are two parameterscharacterizing the degree of uncertainty that 120578 takes on thevalue 119903 When 120579119897 = 120579119903 = 0 the corresponding fuzzy variableis denoted by 120578119899 whose possibility distribution is calledthe nominal possibility distribution of 120578 In the following120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) means that 120578 is a generalized PIVtriangular fuzzy variable
For a PIV fuzzy variable its lambda selection is defined in[34] Assume that 120585 is a PIV fuzzy variable with the secondarypossibility distribution 120583120585(119903) = [120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] For any120582 isin [0 1] a fuzzy variable 120585120582 is called a lambda selection
of 120585 if 120585120582 has the following generalized parametric possibilitydistribution
Obviously the possibility distribution of lambda selectionvariable depends on the parameter 120582 That is the possibilitydistribution of lambda selection variable can traverse theentire support of PIV fuzzy variable as the lambda parametervaries its value in the interval [0 1]
Based on L-S integral [37] the mean value of a fuzzyvariable 120585 is defined as
119864 (120585) = int(minusinfin+infin)
119903 dCr 120585 le 119903 (3)
where the credibility Cr120585 le 119903 is computed by
In addition the second-order moment of a fuzzy variable120585 is defined as
119872(120585) = int(minusinfin+infin)
[119903 minus 119864 (120585)]2 dCr 120585 le 119903 (5)
where 119864(120585) is the mean value of 120585 defined by (3)For lambda selection variable its mean value and second-
order moment are important optimization indices in theMSIMP The following theorems establish their analyticalexpressions which will be used in the rest of the paper Forthe sake of presentation the proofs of the following theoremsare provided in the appendix
Theorem1 Let 120585120582 be a lambda selection of the generalized PIVnormal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the mean value ofthe lambda selection 120585120582 is
119864 (120585120582) = 120596120583 (6)
where 120596 = 1 minus (1 minus 120582)120579119897 minus 120582120579119903Theorem 2 Let 120578120582 be a lambda selection of the generalizedPIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then the meanvalue of the lambda selection 120578120582 is
119864 (120578120582) = [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334+ 14120582120579119903 (1199031 minus 21199032 + 1199033)
(7)
Theorem 3 Let 120585120582 be a lambda selection of the generalizedPIV normal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the second-order moment of the lambda selection 120585120582 is
119872(120585120582) = 120596 [21205902 + (120596 minus 1)2 1205832] (8)
4 Mathematical Problems in Engineering
Theorem 4 Let 120578120582 be a lambda selection of the general-ized PIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then thesecond-order moment of the lambda selection 120578120582 is
119872(120578120582)= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2]
+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(9)
where119898 = 119864(120578120582)In the next section the distribution information about
uncertain demand and uncertain carbon emission is partiallyavailable and characterized by generalized PIV normal fuzzyvariable and triangular fuzzy variable respectively
3 Credibilistic OptimizationModel for MSIMP
In order to model MSIMP some necessary notations areprovided in the following subsection
31 Notations
Fixed Parameters
119899 number of products119894 product index 119894 = 1 2 119899119888119894 procurement cost for unit product 119894119892119894 goodwill cost for unit unmet demand of product 119894119901119894 retailerrsquos sales price for unit product 119894119904119894 salvage value for unit residual product 119894120579119897119894 downward perturbation degree of nominal possi-bility distribution for product 119894120579119903119894 upward perturbation degree of nominal possibil-ity distribution for product 119894120582119894 lambda selection parameter of demand distribu-tion for product 119894120583120582119894 mean value of the lambda selection variable forproduct 119894119863119894 largest market demand for product 119894119861 total investment amount119870 total carbon emission allowance from government120573 predetermined confidence level119873+ the set of nonnegative integers
Decision Variables
119876119894 retailerrsquos order quantity for product 119894
Uncertain Parameters
120585119894 uncertain market demand with variable possibilitydistribution
120578119894 uncertain carbon emission due to logistic activitiesfor product 119894120587 uncertain profit for retailer
32 Credibilistic OptimizationModel and Its Equivalent Deter-ministic Form In this subsection aMSIMP is studied wherethe uncertain demand and uncertain carbon emission arecharacterized by generalized PIV fuzzy variables At thebeginning of selling season the retailer is interested in deter-mining the order quantity119876119894 for product 119894 to satisfy customerdemand for each product For product 119894 the distributioninformation of uncertain demand is only partially knownbased on the expertsrsquo experiences or subjective judgmentsAssume that the uncertain demand for product 119894 (119894 =1 2 119899) is characterized by generalized PIV normal fuzzyvariable 120585119894 = 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 119899 and the largestmarket demand for product 119894 is no more than 119863119894 At the endof the period if 119876119894 ge 120585119894 then 119876119894 minus 120585119894 units are salvaged for aper-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894minus119876119894 units representlost sales cost for a per-unit cost 119892119894
The profit for the retailer stemming from the sales ofproduct 119894 is represented as
120587 (119876119894 120585119894) = (119901119894 minus 119888119894) 120585119894 minus (119888119894 minus 119904119894) (119876119894 minus 120585119894)+minus (119901119894 minus 119888119894 + 119892119894) (120585119894 minus 119876119894)+ (10)
for 119894 = 1 2 119899 respectivelyThe profit function for product 119894 cannot be directly
maximized because it is a fuzzy variable In order to transformthe fuzzy objective into a crisp one the mean profit of120587(119876119894 120585119894) is computed by
120587 (119876119894 119903) dCr 120585119894 le 119903 (11)
Since 120585119894 has an interval-valued possibility distributionrobust optimization method (see [38ndash41]) can be used tomodel the MSIMP
In this paper the lambda selection variable 120585120582119894 is employedto represent the generalized PIV fuzzy variable 120585119894 In this casethe mean value of profit 120587(119876119894 120585120582119894 ) is computed by
120587 (119876119894 119903)minus 119864 [120587 (119876119894 119903)]2 dCr 120585120582119894 le 119903= ℎ [(119888119894 minus 119901119894 minus 119892119894)2 1198762119894 + 2119892119894119863119894 (119888119894 minus 119901119894 minus 119892119894) 119876119894+ 11989221198941198631198942] + 2 [119888119894 (119901119894 minus 119904119894) + 1199042119894 ]119876119894 minus 119901119894119904119894
sdot int1198761198940
Cr 120585120582119894 le 119903 d119903 + 2119892119894119876119894 (119901119894 + 119892119894 minus 119888119894)sdot int119863119894119876119894
Cr 120585120582119894 le 119903 d119903 minus 2 [(119901119894 minus 119904119894)2 + 1198921198942]sdot int1198761198940
119903Cr 120585120582119894 le 119903 d119903 minus 21198922119894 int119863119894119876119894
119903Cr 120585120582119894 le 119903 d119903+ 1198982119894 (ℎ119894 minus 2)
(14)
where119898119894 = 119864[120587(119876119894 120585120582119894 )]As a result the total profit of the retailer in MSIMP is
Π(119876 120585120582) = 119899sum119894=1
120587 (119876119894 120585120582119894 ) (15)
Based on L-Smultiple integral themean total profit of theretailer is computed by
119864 [Π (119876 120585120582)] = int sdot sdot sdot intR119899
Π(119876 120585120582) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (16)
while the second-order moment of the total profit iscomputed by
119872[Π(119876 120585120582)] = int sdot sdot sdot intR119899
Π (119876 120585120582) minus 119864 [Π (119876 120585120582)]2 d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (17)
In order to find the optimal order quantity119876119894 the retailershould take into account the allocation of emission allowance119870 which will be received before the selling season It is well-known that transportation mode has a significant influenceon carbon emission per ton-mile For product 119894 it is usuallydifficult to determine the exact carbon emission duringlogistic activities Based on the retailerrsquo experience assumethat the carbon emission for product 119894 (119894 = 1 2 119899) ischaracterized by generalized PIV triangular fuzzy variable120578119894 sim Tri(1199031119894 1199032119894 1199033119894 120579119897119894 120579r119894)
According to [36] sum119899119894=1 119876119894120578119894 is also a generalized PIVtriangular fuzzy variable its lambda selection variable isdenoted as (sum119899119894=1 119876119894120578119894)120582
Undermean-moment optimization criterion a new para-metric credibilistic optimization model for the MSIMP isformally built as
Objective function (18) in model (119864-119872 1) is to maximizethe tradeoff between the mean total profit and the standardsecond-order moment of the total profit where 120574 is somenonnegative constant that reflects the decisionmakerrsquos degreeof risk aversion Constraint (19) means that the carbonemission due to logistic activities is less than the totalcarbon emission 119870 with a predetermined confidence level120573 Constraint (20) represents the fact that the investmentamount on total production cost has an upper limit on themaximum investment Constraints (21) and (22) ensure thatdecision variables119876119894 (119894 = 1 2 119899) are nonnegative integersin a reasonable range
In order to solve model (119864-119872 1) its equivalent determin-isticmodel is discussed in the following theorem For the sakeof presentation the proof of the following theorem is alsoprovided in the appendix
6 Mathematical Problems in Engineering
Theorem 5 Let 120578119894 be mutually independent fuzzy variablesThenmodel (E-M 1) is equivalent to the following deterministicprogramming model
max119876
Π119890 (119876) minus 120574radicΠ119898 (119876) (E-M 2) (23)
st ( 119899sum119894=1
119876119894120578119894)120582
inf
(120573) le 119870 (24)
119899sum119894=1
119888119894119876119894 le 119861 (25)
0 le 119876119894 le 119863119894 119894 = 1 2 119899 (26)
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
profit-maximization models Abdel-Malek and Montanari[9] proposed a methodology for studying the dual of thesolution space of the MSIMP with two constraints and intro-duced an approach to obtain the optimal order quantitiesof each product In addition Huang et al [10] studied acompetitive MSIMP with shortage penalty cost and partialproduct substitution In view of risk preference Ozler et al[11] proposed the MSIMP under a Value at Risk constraintVan Ryzin and Mahajan [12] reviewed the contributions tomultiproduct inventory problem with demand substitutionUnder mean-variance and utility function approaches VanMieghem [13] studied multiproduct single-period networksrsquoproblems in probabilistic framework
When the exact probability distribution of demand isunavailable probabilistic robust optimization method [14] isa tool to deal with the corresponding uncertainty in inventorymanagement problem Based on the assumption that demandwas described by discrete or interval scenarios Vairaktarakis[15] discussed several minimax regret formulations for theMSIMP with a budget constraint When the distributionof demand had known support mean and variance Kam-burowski [16] presented the theoretical foundations for ana-lyzing the inventory management problem They derived theclosed-form formulas for the worst-case and best-case orderquantities Shu et al [17] considered the distribution-freesingle-period inventory management problem by borrowingan economic theory from transportation disciplines Moonet al [18] found the differences between normal distributionapproaches and distribution-free approaches in four scenar-ios with mean and variance Under interval demand uncer-tainty Solyali et al [19] proposed a new robust formulationwhich could solve the intractability issue for large probleminstances As for recent development in stochastic inventorymanagement problems the interested reader may furtherrefer to [20ndash24]
Most of the extensions of inventorymanagement problemhave been made in the probabilistic framework whereuncertain parameters are characterized by random variablesHowever in some cases there are not enough data to deter-mine the exact probability distribution of random variablebecause of economic reason or technical difficulty In sucha case the variable is approximately specified based on theexperiences and subjective judgments of the experts in relatedfields so fuzzy inventory management problem is also anactive research area Fuzzy set theory was applied in the earlyinventorymanagement literature [25 26] In the area of fuzzyMSIMP Mandal and Roy [27] considered a multiproductdisplayed inventory model under shelf-space constraint infuzzy environment where the demand rate of a product wasconsidered as a function of the displayed inventory levelUnder fuzzy demand environment Ji and Shao [28] studiedthe MSIMP and formulated three kinds of models Dutta[29] formulated a fuzzy MSIMP model whose objective wasto maximize the total profit by considering fuzzy demandsIn fuzzy-stochastic environment Saha et al [30] developedmultiproduct multiobjective supply chain models with bud-get and risk constraints where themanufacturing costs of theitems were fuzzy variables and the demands for the productswere random variables Based on credibility measure Guo
[31] proposed two single-period inventory models wherethe uncertain demands were characterized by discrete andcontinuous possibility distributions respectively Tian andGuo [32] formulated a credibilistic optimization model fora single-product single-period inventory problem with twosuppliers
The work mentioned above studied inventory manage-ment problem under the assumption that the exact possibilitydistribution of fuzzy variable was available which motivatesus to study the MSIMP from a new perspective The motiva-tion of this paper is based on the following considerationsFirst shorter product life cycles and growing innovationrates make the market demand extremely variable In thiscase the distribution information about market demand isonly partially available It is reasonable to assume that theexact possibility distribution is embodied in a zonal areafor a practical MSIMP so the interval-valued fuzzy variableis introduced to characterize uncertain market demandSecond the optimal order quantities for different productsare heavily influenced by the carbon emission constraint Insome practical inventorymanagement problems it is difficultto determine the exact carbon emission during logisticactivities Under credibilistic carbon emission constraint aparametric credibilistic optimization model is developed forMSIMP To the best of our knowledge this issue has not beenaddressed in the literature
This paper studies MSIMP by parametric credibilisticoptimization method where uncertain market demand anduncertain carbon emission are characterized by generalizedPIV possibility distributions Decision makers can makeinformed decisions based on a tradeoff model between themean total profit and the second-ordermoment of total profitunder budget constraint and uncertain carbon emissionconstraint The strength of the proposed method is that thedistributions of market demand and carbon emission can betailored to the partial information at hand That is whenthe distribution information about uncertain parameters ispartially available the proposed method is more convenientfor modeling uncertain demand and carbon emission in apractical MSIMP The proposed credibilistic optimizationmethod differs from the existing MSIMP literature in thefollowing several aspects (i) A novel method is introducedto model the perturbation distributions of uncertain demandand carbon emission which is different from the existingliterature (ii) For PIV fuzzy variable its lambda selectionvariable is introduced as its representative the possibilitydistribution of lambda selection can traverse the entiresupport of the PIV fuzzy variable as the lambda parametervaries its values (iii) On the basis of L-Smultiple integral twonew optimization indexes mean and second-order momentabout the total profit are defined to build a parametric cred-ibilistic optimization model under credibilistic constraintof carbon emission (iv) A domain decomposition methodis designed to divide the original credibilistic optimizationmodel into several equivalent parametric programming sub-models which can be solved by conventional optimizationsoftware
The remainder of this paper is organized as followsAfter introducing some basic concepts in fuzzy possibility
Mathematical Problems in Engineering 3
theory Section 2 discusses the properties about generalizedPIV fuzzy variable and its selection variable In Section 3a new parametric credibilistic optimization model is firstdeveloped for MSIMP where uncertain demand and uncer-tain carbon emission are characterized by variable possibilitydistributions Then the equivalent deterministic model ofthe proposed parametric credibilistic optimization modelis discussed in this section A new domain decompositionmethod is also designed in this section to find the optimalorder quantities In Section 4 some numerical experimentsare conducted to demonstrate the validity of the proposedcredibilistic optimizationmethod Section 5 gives the conclu-sion of the paper
2 Generalized PIV Fuzzy Variables
First in this section some basic concepts in fuzzy possibilitytheory are recalled [33ndash36]
Let Γ be the universe of discourseP(Γ) the power set ofΓ and Pos P(Γ) 997891rarr R([0 1]) a fuzzy possibility measureThe triplet (ΓP(Γ) Pos) is called a fuzzy possibility space
Let 120585 be a type 2 fuzzy variable defined on the space(ΓP(Γ) Pos) If for any 119903 isin R the secondary possibilitydistribution function 120583120585(119903) = Pos120585 = 119903 is a subinterval[120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] of [0 1] then 120585 is called a PIV fuzzy vari-able where 120579119897 120579119903 isin [0 1] are two parameters characterizingthe degree of uncertainty that 120585 takes the value 119903
A type 2 fuzzy variable 120585 is called a generalized PIVnormal fuzzy variable [36] if its secondary possibility distri-bution is the subinterval
[(1 minus 120579119897) 119890minus(119903minus120583)221205902 119890minus(119903minus120583)221205902 + (1 minus 119890minus(119903minus120583)221205902) 120579119903] (1)
of [0 1] for 119903 isin R where 120583 isin R 120590 gt 0 and 120579119897 120579119903 isin [0 1] aretwo parameters characterizing the degree of uncertainty that120585 takes on the value 119903 When 120579119897 = 120579119903 = 0 the correspondingfuzzy variable is denoted by 120585119899 whose possibility distributionis called the nominal possibility distribution of 120585 In thefollowing 120585 sim 119899(120583 1205902 120579119897 120579119903) means that 120585 is a generalizedPIV normal fuzzy variable
A type 2 fuzzy variable 120578 is called a generalized PIVtriangular fuzzy variable [36] if its secondary possibilitydistribution is the subinterval [(119903 minus 1199031)(1199032 minus 1199031) minus 120579119897((119903 minus1199031)(1199032 minus 1199031)) (119903 minus 1199031)(1199032 minus 1199031) + 120579119903((1199032 minus 119903)(1199032 minus 1199031))] of[0 1] for 119903 isin [1199031 1199032] and the subinterval [(1199033 minus 119903)(1199033 minus1199032) minus 120579119897((1199033 minus 119903)(1199033 minus 1199032)) (1199033 minus 119903)(1199033 minus 1199032) + 120579119903((119903 minus1199032)(1199033 minus 1199032))] of [0 1] for 119903 isin [1199032 1199033] where 1199031 lt 1199032 lt1199033 are real numbers and 120579119897 120579119903 isin [0 1] are two parameterscharacterizing the degree of uncertainty that 120578 takes on thevalue 119903 When 120579119897 = 120579119903 = 0 the corresponding fuzzy variableis denoted by 120578119899 whose possibility distribution is calledthe nominal possibility distribution of 120578 In the following120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) means that 120578 is a generalized PIVtriangular fuzzy variable
For a PIV fuzzy variable its lambda selection is defined in[34] Assume that 120585 is a PIV fuzzy variable with the secondarypossibility distribution 120583120585(119903) = [120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] For any120582 isin [0 1] a fuzzy variable 120585120582 is called a lambda selection
of 120585 if 120585120582 has the following generalized parametric possibilitydistribution
Obviously the possibility distribution of lambda selectionvariable depends on the parameter 120582 That is the possibilitydistribution of lambda selection variable can traverse theentire support of PIV fuzzy variable as the lambda parametervaries its value in the interval [0 1]
Based on L-S integral [37] the mean value of a fuzzyvariable 120585 is defined as
119864 (120585) = int(minusinfin+infin)
119903 dCr 120585 le 119903 (3)
where the credibility Cr120585 le 119903 is computed by
In addition the second-order moment of a fuzzy variable120585 is defined as
119872(120585) = int(minusinfin+infin)
[119903 minus 119864 (120585)]2 dCr 120585 le 119903 (5)
where 119864(120585) is the mean value of 120585 defined by (3)For lambda selection variable its mean value and second-
order moment are important optimization indices in theMSIMP The following theorems establish their analyticalexpressions which will be used in the rest of the paper Forthe sake of presentation the proofs of the following theoremsare provided in the appendix
Theorem1 Let 120585120582 be a lambda selection of the generalized PIVnormal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the mean value ofthe lambda selection 120585120582 is
119864 (120585120582) = 120596120583 (6)
where 120596 = 1 minus (1 minus 120582)120579119897 minus 120582120579119903Theorem 2 Let 120578120582 be a lambda selection of the generalizedPIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then the meanvalue of the lambda selection 120578120582 is
119864 (120578120582) = [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334+ 14120582120579119903 (1199031 minus 21199032 + 1199033)
(7)
Theorem 3 Let 120585120582 be a lambda selection of the generalizedPIV normal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the second-order moment of the lambda selection 120585120582 is
119872(120585120582) = 120596 [21205902 + (120596 minus 1)2 1205832] (8)
4 Mathematical Problems in Engineering
Theorem 4 Let 120578120582 be a lambda selection of the general-ized PIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then thesecond-order moment of the lambda selection 120578120582 is
119872(120578120582)= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2]
+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(9)
where119898 = 119864(120578120582)In the next section the distribution information about
uncertain demand and uncertain carbon emission is partiallyavailable and characterized by generalized PIV normal fuzzyvariable and triangular fuzzy variable respectively
3 Credibilistic OptimizationModel for MSIMP
In order to model MSIMP some necessary notations areprovided in the following subsection
31 Notations
Fixed Parameters
119899 number of products119894 product index 119894 = 1 2 119899119888119894 procurement cost for unit product 119894119892119894 goodwill cost for unit unmet demand of product 119894119901119894 retailerrsquos sales price for unit product 119894119904119894 salvage value for unit residual product 119894120579119897119894 downward perturbation degree of nominal possi-bility distribution for product 119894120579119903119894 upward perturbation degree of nominal possibil-ity distribution for product 119894120582119894 lambda selection parameter of demand distribu-tion for product 119894120583120582119894 mean value of the lambda selection variable forproduct 119894119863119894 largest market demand for product 119894119861 total investment amount119870 total carbon emission allowance from government120573 predetermined confidence level119873+ the set of nonnegative integers
Decision Variables
119876119894 retailerrsquos order quantity for product 119894
Uncertain Parameters
120585119894 uncertain market demand with variable possibilitydistribution
120578119894 uncertain carbon emission due to logistic activitiesfor product 119894120587 uncertain profit for retailer
32 Credibilistic OptimizationModel and Its Equivalent Deter-ministic Form In this subsection aMSIMP is studied wherethe uncertain demand and uncertain carbon emission arecharacterized by generalized PIV fuzzy variables At thebeginning of selling season the retailer is interested in deter-mining the order quantity119876119894 for product 119894 to satisfy customerdemand for each product For product 119894 the distributioninformation of uncertain demand is only partially knownbased on the expertsrsquo experiences or subjective judgmentsAssume that the uncertain demand for product 119894 (119894 =1 2 119899) is characterized by generalized PIV normal fuzzyvariable 120585119894 = 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 119899 and the largestmarket demand for product 119894 is no more than 119863119894 At the endof the period if 119876119894 ge 120585119894 then 119876119894 minus 120585119894 units are salvaged for aper-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894minus119876119894 units representlost sales cost for a per-unit cost 119892119894
The profit for the retailer stemming from the sales ofproduct 119894 is represented as
120587 (119876119894 120585119894) = (119901119894 minus 119888119894) 120585119894 minus (119888119894 minus 119904119894) (119876119894 minus 120585119894)+minus (119901119894 minus 119888119894 + 119892119894) (120585119894 minus 119876119894)+ (10)
for 119894 = 1 2 119899 respectivelyThe profit function for product 119894 cannot be directly
maximized because it is a fuzzy variable In order to transformthe fuzzy objective into a crisp one the mean profit of120587(119876119894 120585119894) is computed by
120587 (119876119894 119903) dCr 120585119894 le 119903 (11)
Since 120585119894 has an interval-valued possibility distributionrobust optimization method (see [38ndash41]) can be used tomodel the MSIMP
In this paper the lambda selection variable 120585120582119894 is employedto represent the generalized PIV fuzzy variable 120585119894 In this casethe mean value of profit 120587(119876119894 120585120582119894 ) is computed by
120587 (119876119894 119903)minus 119864 [120587 (119876119894 119903)]2 dCr 120585120582119894 le 119903= ℎ [(119888119894 minus 119901119894 minus 119892119894)2 1198762119894 + 2119892119894119863119894 (119888119894 minus 119901119894 minus 119892119894) 119876119894+ 11989221198941198631198942] + 2 [119888119894 (119901119894 minus 119904119894) + 1199042119894 ]119876119894 minus 119901119894119904119894
sdot int1198761198940
Cr 120585120582119894 le 119903 d119903 + 2119892119894119876119894 (119901119894 + 119892119894 minus 119888119894)sdot int119863119894119876119894
Cr 120585120582119894 le 119903 d119903 minus 2 [(119901119894 minus 119904119894)2 + 1198921198942]sdot int1198761198940
119903Cr 120585120582119894 le 119903 d119903 minus 21198922119894 int119863119894119876119894
119903Cr 120585120582119894 le 119903 d119903+ 1198982119894 (ℎ119894 minus 2)
(14)
where119898119894 = 119864[120587(119876119894 120585120582119894 )]As a result the total profit of the retailer in MSIMP is
Π(119876 120585120582) = 119899sum119894=1
120587 (119876119894 120585120582119894 ) (15)
Based on L-Smultiple integral themean total profit of theretailer is computed by
119864 [Π (119876 120585120582)] = int sdot sdot sdot intR119899
Π(119876 120585120582) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (16)
while the second-order moment of the total profit iscomputed by
119872[Π(119876 120585120582)] = int sdot sdot sdot intR119899
Π (119876 120585120582) minus 119864 [Π (119876 120585120582)]2 d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (17)
In order to find the optimal order quantity119876119894 the retailershould take into account the allocation of emission allowance119870 which will be received before the selling season It is well-known that transportation mode has a significant influenceon carbon emission per ton-mile For product 119894 it is usuallydifficult to determine the exact carbon emission duringlogistic activities Based on the retailerrsquo experience assumethat the carbon emission for product 119894 (119894 = 1 2 119899) ischaracterized by generalized PIV triangular fuzzy variable120578119894 sim Tri(1199031119894 1199032119894 1199033119894 120579119897119894 120579r119894)
According to [36] sum119899119894=1 119876119894120578119894 is also a generalized PIVtriangular fuzzy variable its lambda selection variable isdenoted as (sum119899119894=1 119876119894120578119894)120582
Undermean-moment optimization criterion a new para-metric credibilistic optimization model for the MSIMP isformally built as
Objective function (18) in model (119864-119872 1) is to maximizethe tradeoff between the mean total profit and the standardsecond-order moment of the total profit where 120574 is somenonnegative constant that reflects the decisionmakerrsquos degreeof risk aversion Constraint (19) means that the carbonemission due to logistic activities is less than the totalcarbon emission 119870 with a predetermined confidence level120573 Constraint (20) represents the fact that the investmentamount on total production cost has an upper limit on themaximum investment Constraints (21) and (22) ensure thatdecision variables119876119894 (119894 = 1 2 119899) are nonnegative integersin a reasonable range
In order to solve model (119864-119872 1) its equivalent determin-isticmodel is discussed in the following theorem For the sakeof presentation the proof of the following theorem is alsoprovided in the appendix
6 Mathematical Problems in Engineering
Theorem 5 Let 120578119894 be mutually independent fuzzy variablesThenmodel (E-M 1) is equivalent to the following deterministicprogramming model
max119876
Π119890 (119876) minus 120574radicΠ119898 (119876) (E-M 2) (23)
st ( 119899sum119894=1
119876119894120578119894)120582
inf
(120573) le 119870 (24)
119899sum119894=1
119888119894119876119894 le 119861 (25)
0 le 119876119894 le 119863119894 119894 = 1 2 119899 (26)
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
theory Section 2 discusses the properties about generalizedPIV fuzzy variable and its selection variable In Section 3a new parametric credibilistic optimization model is firstdeveloped for MSIMP where uncertain demand and uncer-tain carbon emission are characterized by variable possibilitydistributions Then the equivalent deterministic model ofthe proposed parametric credibilistic optimization modelis discussed in this section A new domain decompositionmethod is also designed in this section to find the optimalorder quantities In Section 4 some numerical experimentsare conducted to demonstrate the validity of the proposedcredibilistic optimizationmethod Section 5 gives the conclu-sion of the paper
2 Generalized PIV Fuzzy Variables
First in this section some basic concepts in fuzzy possibilitytheory are recalled [33ndash36]
Let Γ be the universe of discourseP(Γ) the power set ofΓ and Pos P(Γ) 997891rarr R([0 1]) a fuzzy possibility measureThe triplet (ΓP(Γ) Pos) is called a fuzzy possibility space
Let 120585 be a type 2 fuzzy variable defined on the space(ΓP(Γ) Pos) If for any 119903 isin R the secondary possibilitydistribution function 120583120585(119903) = Pos120585 = 119903 is a subinterval[120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] of [0 1] then 120585 is called a PIV fuzzy vari-able where 120579119897 120579119903 isin [0 1] are two parameters characterizingthe degree of uncertainty that 120585 takes the value 119903
A type 2 fuzzy variable 120585 is called a generalized PIVnormal fuzzy variable [36] if its secondary possibility distri-bution is the subinterval
[(1 minus 120579119897) 119890minus(119903minus120583)221205902 119890minus(119903minus120583)221205902 + (1 minus 119890minus(119903minus120583)221205902) 120579119903] (1)
of [0 1] for 119903 isin R where 120583 isin R 120590 gt 0 and 120579119897 120579119903 isin [0 1] aretwo parameters characterizing the degree of uncertainty that120585 takes on the value 119903 When 120579119897 = 120579119903 = 0 the correspondingfuzzy variable is denoted by 120585119899 whose possibility distributionis called the nominal possibility distribution of 120585 In thefollowing 120585 sim 119899(120583 1205902 120579119897 120579119903) means that 120585 is a generalizedPIV normal fuzzy variable
A type 2 fuzzy variable 120578 is called a generalized PIVtriangular fuzzy variable [36] if its secondary possibilitydistribution is the subinterval [(119903 minus 1199031)(1199032 minus 1199031) minus 120579119897((119903 minus1199031)(1199032 minus 1199031)) (119903 minus 1199031)(1199032 minus 1199031) + 120579119903((1199032 minus 119903)(1199032 minus 1199031))] of[0 1] for 119903 isin [1199031 1199032] and the subinterval [(1199033 minus 119903)(1199033 minus1199032) minus 120579119897((1199033 minus 119903)(1199033 minus 1199032)) (1199033 minus 119903)(1199033 minus 1199032) + 120579119903((119903 minus1199032)(1199033 minus 1199032))] of [0 1] for 119903 isin [1199032 1199033] where 1199031 lt 1199032 lt1199033 are real numbers and 120579119897 120579119903 isin [0 1] are two parameterscharacterizing the degree of uncertainty that 120578 takes on thevalue 119903 When 120579119897 = 120579119903 = 0 the corresponding fuzzy variableis denoted by 120578119899 whose possibility distribution is calledthe nominal possibility distribution of 120578 In the following120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) means that 120578 is a generalized PIVtriangular fuzzy variable
For a PIV fuzzy variable its lambda selection is defined in[34] Assume that 120585 is a PIV fuzzy variable with the secondarypossibility distribution 120583120585(119903) = [120583120585119871(119903 120579119897) 120583120585119880(119903 120579119903)] For any120582 isin [0 1] a fuzzy variable 120585120582 is called a lambda selection
of 120585 if 120585120582 has the following generalized parametric possibilitydistribution
Obviously the possibility distribution of lambda selectionvariable depends on the parameter 120582 That is the possibilitydistribution of lambda selection variable can traverse theentire support of PIV fuzzy variable as the lambda parametervaries its value in the interval [0 1]
Based on L-S integral [37] the mean value of a fuzzyvariable 120585 is defined as
119864 (120585) = int(minusinfin+infin)
119903 dCr 120585 le 119903 (3)
where the credibility Cr120585 le 119903 is computed by
In addition the second-order moment of a fuzzy variable120585 is defined as
119872(120585) = int(minusinfin+infin)
[119903 minus 119864 (120585)]2 dCr 120585 le 119903 (5)
where 119864(120585) is the mean value of 120585 defined by (3)For lambda selection variable its mean value and second-
order moment are important optimization indices in theMSIMP The following theorems establish their analyticalexpressions which will be used in the rest of the paper Forthe sake of presentation the proofs of the following theoremsare provided in the appendix
Theorem1 Let 120585120582 be a lambda selection of the generalized PIVnormal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the mean value ofthe lambda selection 120585120582 is
119864 (120585120582) = 120596120583 (6)
where 120596 = 1 minus (1 minus 120582)120579119897 minus 120582120579119903Theorem 2 Let 120578120582 be a lambda selection of the generalizedPIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then the meanvalue of the lambda selection 120578120582 is
119864 (120578120582) = [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334+ 14120582120579119903 (1199031 minus 21199032 + 1199033)
(7)
Theorem 3 Let 120585120582 be a lambda selection of the generalizedPIV normal fuzzy variable 119899(120583 1205902 120579119897 120579119903) Then the second-order moment of the lambda selection 120585120582 is
119872(120585120582) = 120596 [21205902 + (120596 minus 1)2 1205832] (8)
4 Mathematical Problems in Engineering
Theorem 4 Let 120578120582 be a lambda selection of the general-ized PIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then thesecond-order moment of the lambda selection 120578120582 is
119872(120578120582)= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2]
+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(9)
where119898 = 119864(120578120582)In the next section the distribution information about
uncertain demand and uncertain carbon emission is partiallyavailable and characterized by generalized PIV normal fuzzyvariable and triangular fuzzy variable respectively
3 Credibilistic OptimizationModel for MSIMP
In order to model MSIMP some necessary notations areprovided in the following subsection
31 Notations
Fixed Parameters
119899 number of products119894 product index 119894 = 1 2 119899119888119894 procurement cost for unit product 119894119892119894 goodwill cost for unit unmet demand of product 119894119901119894 retailerrsquos sales price for unit product 119894119904119894 salvage value for unit residual product 119894120579119897119894 downward perturbation degree of nominal possi-bility distribution for product 119894120579119903119894 upward perturbation degree of nominal possibil-ity distribution for product 119894120582119894 lambda selection parameter of demand distribu-tion for product 119894120583120582119894 mean value of the lambda selection variable forproduct 119894119863119894 largest market demand for product 119894119861 total investment amount119870 total carbon emission allowance from government120573 predetermined confidence level119873+ the set of nonnegative integers
Decision Variables
119876119894 retailerrsquos order quantity for product 119894
Uncertain Parameters
120585119894 uncertain market demand with variable possibilitydistribution
120578119894 uncertain carbon emission due to logistic activitiesfor product 119894120587 uncertain profit for retailer
32 Credibilistic OptimizationModel and Its Equivalent Deter-ministic Form In this subsection aMSIMP is studied wherethe uncertain demand and uncertain carbon emission arecharacterized by generalized PIV fuzzy variables At thebeginning of selling season the retailer is interested in deter-mining the order quantity119876119894 for product 119894 to satisfy customerdemand for each product For product 119894 the distributioninformation of uncertain demand is only partially knownbased on the expertsrsquo experiences or subjective judgmentsAssume that the uncertain demand for product 119894 (119894 =1 2 119899) is characterized by generalized PIV normal fuzzyvariable 120585119894 = 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 119899 and the largestmarket demand for product 119894 is no more than 119863119894 At the endof the period if 119876119894 ge 120585119894 then 119876119894 minus 120585119894 units are salvaged for aper-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894minus119876119894 units representlost sales cost for a per-unit cost 119892119894
The profit for the retailer stemming from the sales ofproduct 119894 is represented as
120587 (119876119894 120585119894) = (119901119894 minus 119888119894) 120585119894 minus (119888119894 minus 119904119894) (119876119894 minus 120585119894)+minus (119901119894 minus 119888119894 + 119892119894) (120585119894 minus 119876119894)+ (10)
for 119894 = 1 2 119899 respectivelyThe profit function for product 119894 cannot be directly
maximized because it is a fuzzy variable In order to transformthe fuzzy objective into a crisp one the mean profit of120587(119876119894 120585119894) is computed by
120587 (119876119894 119903) dCr 120585119894 le 119903 (11)
Since 120585119894 has an interval-valued possibility distributionrobust optimization method (see [38ndash41]) can be used tomodel the MSIMP
In this paper the lambda selection variable 120585120582119894 is employedto represent the generalized PIV fuzzy variable 120585119894 In this casethe mean value of profit 120587(119876119894 120585120582119894 ) is computed by
120587 (119876119894 119903)minus 119864 [120587 (119876119894 119903)]2 dCr 120585120582119894 le 119903= ℎ [(119888119894 minus 119901119894 minus 119892119894)2 1198762119894 + 2119892119894119863119894 (119888119894 minus 119901119894 minus 119892119894) 119876119894+ 11989221198941198631198942] + 2 [119888119894 (119901119894 minus 119904119894) + 1199042119894 ]119876119894 minus 119901119894119904119894
sdot int1198761198940
Cr 120585120582119894 le 119903 d119903 + 2119892119894119876119894 (119901119894 + 119892119894 minus 119888119894)sdot int119863119894119876119894
Cr 120585120582119894 le 119903 d119903 minus 2 [(119901119894 minus 119904119894)2 + 1198921198942]sdot int1198761198940
119903Cr 120585120582119894 le 119903 d119903 minus 21198922119894 int119863119894119876119894
119903Cr 120585120582119894 le 119903 d119903+ 1198982119894 (ℎ119894 minus 2)
(14)
where119898119894 = 119864[120587(119876119894 120585120582119894 )]As a result the total profit of the retailer in MSIMP is
Π(119876 120585120582) = 119899sum119894=1
120587 (119876119894 120585120582119894 ) (15)
Based on L-Smultiple integral themean total profit of theretailer is computed by
119864 [Π (119876 120585120582)] = int sdot sdot sdot intR119899
Π(119876 120585120582) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (16)
while the second-order moment of the total profit iscomputed by
119872[Π(119876 120585120582)] = int sdot sdot sdot intR119899
Π (119876 120585120582) minus 119864 [Π (119876 120585120582)]2 d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (17)
In order to find the optimal order quantity119876119894 the retailershould take into account the allocation of emission allowance119870 which will be received before the selling season It is well-known that transportation mode has a significant influenceon carbon emission per ton-mile For product 119894 it is usuallydifficult to determine the exact carbon emission duringlogistic activities Based on the retailerrsquo experience assumethat the carbon emission for product 119894 (119894 = 1 2 119899) ischaracterized by generalized PIV triangular fuzzy variable120578119894 sim Tri(1199031119894 1199032119894 1199033119894 120579119897119894 120579r119894)
According to [36] sum119899119894=1 119876119894120578119894 is also a generalized PIVtriangular fuzzy variable its lambda selection variable isdenoted as (sum119899119894=1 119876119894120578119894)120582
Undermean-moment optimization criterion a new para-metric credibilistic optimization model for the MSIMP isformally built as
Objective function (18) in model (119864-119872 1) is to maximizethe tradeoff between the mean total profit and the standardsecond-order moment of the total profit where 120574 is somenonnegative constant that reflects the decisionmakerrsquos degreeof risk aversion Constraint (19) means that the carbonemission due to logistic activities is less than the totalcarbon emission 119870 with a predetermined confidence level120573 Constraint (20) represents the fact that the investmentamount on total production cost has an upper limit on themaximum investment Constraints (21) and (22) ensure thatdecision variables119876119894 (119894 = 1 2 119899) are nonnegative integersin a reasonable range
In order to solve model (119864-119872 1) its equivalent determin-isticmodel is discussed in the following theorem For the sakeof presentation the proof of the following theorem is alsoprovided in the appendix
6 Mathematical Problems in Engineering
Theorem 5 Let 120578119894 be mutually independent fuzzy variablesThenmodel (E-M 1) is equivalent to the following deterministicprogramming model
max119876
Π119890 (119876) minus 120574radicΠ119898 (119876) (E-M 2) (23)
st ( 119899sum119894=1
119876119894120578119894)120582
inf
(120573) le 119870 (24)
119899sum119894=1
119888119894119876119894 le 119861 (25)
0 le 119876119894 le 119863119894 119894 = 1 2 119899 (26)
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
Theorem 4 Let 120578120582 be a lambda selection of the general-ized PIV triangular fuzzy variable (1199031 1199032 1199033 120579119897 120579119903) Then thesecond-order moment of the lambda selection 120578120582 is
119872(120578120582)= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2]
+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(9)
where119898 = 119864(120578120582)In the next section the distribution information about
uncertain demand and uncertain carbon emission is partiallyavailable and characterized by generalized PIV normal fuzzyvariable and triangular fuzzy variable respectively
3 Credibilistic OptimizationModel for MSIMP
In order to model MSIMP some necessary notations areprovided in the following subsection
31 Notations
Fixed Parameters
119899 number of products119894 product index 119894 = 1 2 119899119888119894 procurement cost for unit product 119894119892119894 goodwill cost for unit unmet demand of product 119894119901119894 retailerrsquos sales price for unit product 119894119904119894 salvage value for unit residual product 119894120579119897119894 downward perturbation degree of nominal possi-bility distribution for product 119894120579119903119894 upward perturbation degree of nominal possibil-ity distribution for product 119894120582119894 lambda selection parameter of demand distribu-tion for product 119894120583120582119894 mean value of the lambda selection variable forproduct 119894119863119894 largest market demand for product 119894119861 total investment amount119870 total carbon emission allowance from government120573 predetermined confidence level119873+ the set of nonnegative integers
Decision Variables
119876119894 retailerrsquos order quantity for product 119894
Uncertain Parameters
120585119894 uncertain market demand with variable possibilitydistribution
120578119894 uncertain carbon emission due to logistic activitiesfor product 119894120587 uncertain profit for retailer
32 Credibilistic OptimizationModel and Its Equivalent Deter-ministic Form In this subsection aMSIMP is studied wherethe uncertain demand and uncertain carbon emission arecharacterized by generalized PIV fuzzy variables At thebeginning of selling season the retailer is interested in deter-mining the order quantity119876119894 for product 119894 to satisfy customerdemand for each product For product 119894 the distributioninformation of uncertain demand is only partially knownbased on the expertsrsquo experiences or subjective judgmentsAssume that the uncertain demand for product 119894 (119894 =1 2 119899) is characterized by generalized PIV normal fuzzyvariable 120585119894 = 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 119899 and the largestmarket demand for product 119894 is no more than 119863119894 At the endof the period if 119876119894 ge 120585119894 then 119876119894 minus 120585119894 units are salvaged for aper-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894minus119876119894 units representlost sales cost for a per-unit cost 119892119894
The profit for the retailer stemming from the sales ofproduct 119894 is represented as
120587 (119876119894 120585119894) = (119901119894 minus 119888119894) 120585119894 minus (119888119894 minus 119904119894) (119876119894 minus 120585119894)+minus (119901119894 minus 119888119894 + 119892119894) (120585119894 minus 119876119894)+ (10)
for 119894 = 1 2 119899 respectivelyThe profit function for product 119894 cannot be directly
maximized because it is a fuzzy variable In order to transformthe fuzzy objective into a crisp one the mean profit of120587(119876119894 120585119894) is computed by
120587 (119876119894 119903) dCr 120585119894 le 119903 (11)
Since 120585119894 has an interval-valued possibility distributionrobust optimization method (see [38ndash41]) can be used tomodel the MSIMP
In this paper the lambda selection variable 120585120582119894 is employedto represent the generalized PIV fuzzy variable 120585119894 In this casethe mean value of profit 120587(119876119894 120585120582119894 ) is computed by
120587 (119876119894 119903)minus 119864 [120587 (119876119894 119903)]2 dCr 120585120582119894 le 119903= ℎ [(119888119894 minus 119901119894 minus 119892119894)2 1198762119894 + 2119892119894119863119894 (119888119894 minus 119901119894 minus 119892119894) 119876119894+ 11989221198941198631198942] + 2 [119888119894 (119901119894 minus 119904119894) + 1199042119894 ]119876119894 minus 119901119894119904119894
sdot int1198761198940
Cr 120585120582119894 le 119903 d119903 + 2119892119894119876119894 (119901119894 + 119892119894 minus 119888119894)sdot int119863119894119876119894
Cr 120585120582119894 le 119903 d119903 minus 2 [(119901119894 minus 119904119894)2 + 1198921198942]sdot int1198761198940
119903Cr 120585120582119894 le 119903 d119903 minus 21198922119894 int119863119894119876119894
119903Cr 120585120582119894 le 119903 d119903+ 1198982119894 (ℎ119894 minus 2)
(14)
where119898119894 = 119864[120587(119876119894 120585120582119894 )]As a result the total profit of the retailer in MSIMP is
Π(119876 120585120582) = 119899sum119894=1
120587 (119876119894 120585120582119894 ) (15)
Based on L-Smultiple integral themean total profit of theretailer is computed by
119864 [Π (119876 120585120582)] = int sdot sdot sdot intR119899
Π(119876 120585120582) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (16)
while the second-order moment of the total profit iscomputed by
119872[Π(119876 120585120582)] = int sdot sdot sdot intR119899
Π (119876 120585120582) minus 119864 [Π (119876 120585120582)]2 d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (17)
In order to find the optimal order quantity119876119894 the retailershould take into account the allocation of emission allowance119870 which will be received before the selling season It is well-known that transportation mode has a significant influenceon carbon emission per ton-mile For product 119894 it is usuallydifficult to determine the exact carbon emission duringlogistic activities Based on the retailerrsquo experience assumethat the carbon emission for product 119894 (119894 = 1 2 119899) ischaracterized by generalized PIV triangular fuzzy variable120578119894 sim Tri(1199031119894 1199032119894 1199033119894 120579119897119894 120579r119894)
According to [36] sum119899119894=1 119876119894120578119894 is also a generalized PIVtriangular fuzzy variable its lambda selection variable isdenoted as (sum119899119894=1 119876119894120578119894)120582
Undermean-moment optimization criterion a new para-metric credibilistic optimization model for the MSIMP isformally built as
Objective function (18) in model (119864-119872 1) is to maximizethe tradeoff between the mean total profit and the standardsecond-order moment of the total profit where 120574 is somenonnegative constant that reflects the decisionmakerrsquos degreeof risk aversion Constraint (19) means that the carbonemission due to logistic activities is less than the totalcarbon emission 119870 with a predetermined confidence level120573 Constraint (20) represents the fact that the investmentamount on total production cost has an upper limit on themaximum investment Constraints (21) and (22) ensure thatdecision variables119876119894 (119894 = 1 2 119899) are nonnegative integersin a reasonable range
In order to solve model (119864-119872 1) its equivalent determin-isticmodel is discussed in the following theorem For the sakeof presentation the proof of the following theorem is alsoprovided in the appendix
6 Mathematical Problems in Engineering
Theorem 5 Let 120578119894 be mutually independent fuzzy variablesThenmodel (E-M 1) is equivalent to the following deterministicprogramming model
max119876
Π119890 (119876) minus 120574radicΠ119898 (119876) (E-M 2) (23)
st ( 119899sum119894=1
119876119894120578119894)120582
inf
(120573) le 119870 (24)
119899sum119894=1
119888119894119876119894 le 119861 (25)
0 le 119876119894 le 119863119894 119894 = 1 2 119899 (26)
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
120587 (119876119894 119903)minus 119864 [120587 (119876119894 119903)]2 dCr 120585120582119894 le 119903= ℎ [(119888119894 minus 119901119894 minus 119892119894)2 1198762119894 + 2119892119894119863119894 (119888119894 minus 119901119894 minus 119892119894) 119876119894+ 11989221198941198631198942] + 2 [119888119894 (119901119894 minus 119904119894) + 1199042119894 ]119876119894 minus 119901119894119904119894
sdot int1198761198940
Cr 120585120582119894 le 119903 d119903 + 2119892119894119876119894 (119901119894 + 119892119894 minus 119888119894)sdot int119863119894119876119894
Cr 120585120582119894 le 119903 d119903 minus 2 [(119901119894 minus 119904119894)2 + 1198921198942]sdot int1198761198940
119903Cr 120585120582119894 le 119903 d119903 minus 21198922119894 int119863119894119876119894
119903Cr 120585120582119894 le 119903 d119903+ 1198982119894 (ℎ119894 minus 2)
(14)
where119898119894 = 119864[120587(119876119894 120585120582119894 )]As a result the total profit of the retailer in MSIMP is
Π(119876 120585120582) = 119899sum119894=1
120587 (119876119894 120585120582119894 ) (15)
Based on L-Smultiple integral themean total profit of theretailer is computed by
119864 [Π (119876 120585120582)] = int sdot sdot sdot intR119899
Π(119876 120585120582) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (16)
while the second-order moment of the total profit iscomputed by
119872[Π(119876 120585120582)] = int sdot sdot sdot intR119899
Π (119876 120585120582) minus 119864 [Π (119876 120585120582)]2 d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903) (17)
In order to find the optimal order quantity119876119894 the retailershould take into account the allocation of emission allowance119870 which will be received before the selling season It is well-known that transportation mode has a significant influenceon carbon emission per ton-mile For product 119894 it is usuallydifficult to determine the exact carbon emission duringlogistic activities Based on the retailerrsquo experience assumethat the carbon emission for product 119894 (119894 = 1 2 119899) ischaracterized by generalized PIV triangular fuzzy variable120578119894 sim Tri(1199031119894 1199032119894 1199033119894 120579119897119894 120579r119894)
According to [36] sum119899119894=1 119876119894120578119894 is also a generalized PIVtriangular fuzzy variable its lambda selection variable isdenoted as (sum119899119894=1 119876119894120578119894)120582
Undermean-moment optimization criterion a new para-metric credibilistic optimization model for the MSIMP isformally built as
Objective function (18) in model (119864-119872 1) is to maximizethe tradeoff between the mean total profit and the standardsecond-order moment of the total profit where 120574 is somenonnegative constant that reflects the decisionmakerrsquos degreeof risk aversion Constraint (19) means that the carbonemission due to logistic activities is less than the totalcarbon emission 119870 with a predetermined confidence level120573 Constraint (20) represents the fact that the investmentamount on total production cost has an upper limit on themaximum investment Constraints (21) and (22) ensure thatdecision variables119876119894 (119894 = 1 2 119899) are nonnegative integersin a reasonable range
In order to solve model (119864-119872 1) its equivalent determin-isticmodel is discussed in the following theorem For the sakeof presentation the proof of the following theorem is alsoprovided in the appendix
6 Mathematical Problems in Engineering
Theorem 5 Let 120578119894 be mutually independent fuzzy variablesThenmodel (E-M 1) is equivalent to the following deterministicprogramming model
max119876
Π119890 (119876) minus 120574radicΠ119898 (119876) (E-M 2) (23)
st ( 119899sum119894=1
119876119894120578119894)120582
inf
(120573) le 119870 (24)
119899sum119894=1
119888119894119876119894 le 119861 (25)
0 le 119876119894 le 119863119894 119894 = 1 2 119899 (26)
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903
+ (prod119896 =119894
ℎ119896119898119894)2( 119899prod119896=1
ℎ119896 minus 2)]]
+ prod119896 =119894 =119895
ℎ119896 [[sum119895 =119894
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(28)
In Theorem 5 model (119864-119872 2) is a parametric pro-gramming model with respect to parameter 120582 The valueof parameter lambda determines the location and shape ofthe possibility distribution of selection variable Accordingto the definition of lambda selection variable parameter120582 may change its value from 0 to 1 It is highlighted thatthe possibility distribution of lambda selection variable cantraverse the entire support of PIV fuzzy variables as thelambda parameter changes its value in the interval [0 1] Forany given 120582 isin [0 1] the corresponding integer programmingmodel (119864-119872 2) can be solved by conventional optimizationsoftware
33 Domain Decomposition Method Note that the analyticalexpressions of Π119890(119876) and radicΠ119898(119876) include the integralint1198761198940
Cr120585120582119894 le 119903d119903 According to the definition of Cr120585 le119903 the integral int119876119894
0Cr120585120582119894 le 119903d119903 is a piecewise function
with respect to 119876119894 Since decision makers do not know inadvance which subregion the global optimal solution locatesin to solve 2119899 submodels by optimization software to obtain2119899 local optimal solutions is required By comparing theobjective values of the obtained local optimal solutions theglobal optimal solutions 119876lowast119894 119894 = 1 2 119899 can be found
Given the values of distribution parameters 120579119897119894 120579119903119894 and 120582119894the process of domain decompositionmethod is summarizedas follows
Step 1 Solve parametric programming submodels of model(E-M 2) by software Matlab 71 Let 0 le 1198761119894 le 120583119894 120583119894 le1198762119894 le 119872119894 119894 = 1 2 119899 Given a set of values 119876119896119894119905 119896 = 1 or2 119894 = 1 2 119899 119905 = 1 2 2119899 denote the correspondinglocal optimal solutions as 119876lowast119894119905 119894 = 1 2 119899Step 2 Compare the local objective values V119905 = 119864[120587(119876 120585120582)]at local optimal solution 119876lowast119894119905 and find the global maximumprofit by the following formula
V119897 = max1le119905le2119899
V119905 (29)
where 119864[120587(119876 120585120582)] is the mean profit of 120587(119876 120585120582)Step 3 Return119876lowast119894119897 as the global optimal solution tomodel (119864-119872 2) with the global optimal value 119864[120587(119876lowast119894119897 120585120582)]
In the next section the effectiveness of the proposeddomain decomposition method is demonstrated by a practi-cal multiproduct single-period inventory management prob-lem
4 Numerical Experiments
41 Problem Statement In order to illustrate the proposedcredibilistic optimization model (E-M 2) a two-productsingle-period inventory problem is providedwith generalizedPIV normal demand variables The retailerrsquos optimal strategywill be obtained by the proposed credibilistic optimizationmethod Before a hot summer the retailer needs to ordertwo kinds of products air-conditioning (Product 1) andevaporative air cooler (Product 2) The retailer is interestedin determining the order quantity of air-conditioning1198761 andthe order quantity of evaporative air cooler 1198762 to satisfycustomer demand For product 119894 (119894 = 1 2) the distributioninformation of uncertain demand is partially available basedon the expertsrsquo experiences Suppose that the uncertaindemand 120585119894 for product 119894 follows generalized PIV normalpossibility distribution 119899(120583119894 1205902119894 120579119897119894 120579119903119894) 119894 = 1 2 Based onthe practical background of inventory problem the largestmarket demand for product 119894 is nomore than119863119894 At the end ofthe period if119876119894 ge 120585119894 then119876119894 minus120585119894 units are salvaged for a per-unit revenue 119904119894 and if119876119894 lt 120585119894 then 120585119894 minus119876119894 units represent lostsales cost for a per-unit cost119892119894 In view of the carbon emissionconstraint the retailer receives the allocation of emissionallowance119870 = 251000 grams before the summer For product119894 (119894 = 1 2) the distribution information about the unitcarbon emission during logistic activities is partially availablebased on the expertsrsquo experiences Assume that the unitcarbon emissions for two products follow generalized PIV tri-angular possibility distributions Tri(85 100 110 025 015)and Tri(40 50 65 025 015) respectively Due to logisticactivities the sum of emissions is less than the predeterminedtotal emission 119870 with confidence level 120573 = 09 Additionally
Mathematical Problems in Engineering 7
Table 1 Parameters for a two-product single-period inventory problem
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
Table 2 The local optimal solutions with different domains of 1198761 and 1198762The range of 1198761 The range of 1198762 119876lowast1 119876lowast2 119864lowast0 le 1198761 le 800 0 le 1198762 le 2400 800 2378 115885790 le 1198761 le 800 2400 le 1198762 le 6000 800 2400 11655862800 le 1198761 le 3000 0 le 1198762 le 2400 814 2400 11724053800 le 1198761 le 3000 2400 le 1198762 le 6000 813 2410 11749183
800
802
804
806
808
810
812
814
816
818
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 1
02 03 04 05 06 07 08 09 1011
2400
2405
2410
2415
2420
2425
2430
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1011
Figure 1 The optimal order quantities of product 1 and product 2 with 1205822 = 06
the available maximum investment for the retailer is 119861 =$432000 The other pertinent data for the products are givenin Table 1
42 Computational Results In numerical experiments it isassumed that 1205821 = 06 1205822 = 08 and 120574 = 03 Accordingto the proposed domain decomposition method the feasibleregion of the above inventory management problem canbe decomposed into four disjoint subregions of decisionvariables 119876119894 119894 = 1 2 Matlab 71 optimization software isemployed to solve the corresponding parametric program-ming submodels The numerical experiments are conductedon a personal computer (Lenovowith Intel Pentium(R)Dual-Core E5700 300GHz CPU and RAM 400GB) by usingthe Microsoft Windows 10 operating system The compu-tational results are reported in Table 2 By comparing theobtained local optimal solutions the global optimal solution(119876lowast1 119876lowast2 ) = (813 2410) is found with the maximum meantotal profit 1174918343 Sensitivity Analysis for Parameter Lambda By the mean-ings of parameters 1205821 and 1205822 the two parameters determinethe location and shape about the possibility distribution ofselection variable in the support of uncertain demand and
uncertain carbon emission A decision maker may prescribethe values of parameters 1205821 and 1205822 based on his experienceor knowledge If the decision maker cannot identify thevalues of parameters 1205821 and 1205822 he may generate randomlytheir values from some prescribed subintervals of [0 1] Inour experiments to identify the influence of perturbationdistribution on solution results the optimal solutions arefirst computed by adjusting the selection parameter 1205821 in theoptimization problemwith fixed 1205822 = 06When 1205821 increasesits value from 01 to 1 with step 01 the computationalresults about the optimal order quantities of product 1 andproduct 2 are plotted in Figure 1 and the correspondingmean total profits are plotted in Figure 2 From Figures1 and 2 it is found that the optimal order quantities andmean total profit vary while the selection parameter 1205821varies Specifically the optimal order quantity of product 1 ismonotone decreasing with respect to parameter 1205821 while theoptimal order quantity of product 2 is monotone increasingwith respect to parameter 1205821 As a result themean total profitis monotone increasing with respect to parameter 1205821
In the following the optimal solutions are computedby adjusting the selection parameter 1205822 in the optimizationproblem with fixed 1205821 = 06 When 1205822 increases its valuefrom 01 to 1 with step 01 the computational results of the
8 Mathematical Problems in Engineering
times105
108
11
112
114
116
118
12
122
124
126
128
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1011
Figure 2 The mean total profit
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
optimal order quantities of product 1 and product 2 areplotted in Figure 3 and the corresponding mean total profitsare plotted in Figure 4 From Figures 3 and 4 it is concludedthat the optimal order quantity of product 1 is monotoneincreasing with respect to parameter 1205822 while the optimalorder quantity of product 2 is monotone decreasing withrespect to parameter 1205822 As a result the mean total profit ismonotone decreasing with respect to parameter 1205822
The above computational results demonstrate that theoptimal order quantities of product 1 and product 2 dependheavily on the location parameters 1205821 and 1205822 That isthe optimal order quantities of our multiproduct single-period inventory problem depend heavily on the possibilitydistribution of uncertain demand
44 Comparison Study
441 Comparing with Stochastic Optimization Method Inthis subsection the credibilistic optimization method iscompared with stochastic optimization method where thestochastic demands of product 1 and product 2 follow normalprobability distributions N(800 552) and N(2400 752)respectively According to the stochastic optimizationmethod for MSIMP the optimal order quantities for product1 and product 2 are 815 and 2407 with the maximum meanprofit 189530 The solution result is totally different fromour credibilistic optimal solutions reported in Figures 1and 3 Compared with our credibilistic optimal solutionsthe optimal solutions 815 and 2407 to stochastic model arenot feasible solutions to the deterministic programmingmodel in Theorem 5 That is the stochastic optimal solutiondoes not satisfy carbon emission constraint (24) and theinvestment amount constraint (25)
442 Comparing with Fuzzy Optimization Method underFixed Possibility Distribution In this subsection the credi-bilistic optimization method is compared with fuzzy opti-mization method where the uncertain demands of product1 and product 2 follow fixed possibility distributions For
the sake of comparison the fixed possibility distributions aretaken as the nominal possibility distributions of uncertaindemands corresponding to 120579119897119894 = 120579119903119894 = 0 119894 = 1 2 Bysolving the fuzzy optimization model the obtained nom-inal optimal order quantities are 800 and 2400 with thenominal maximum mean total profit 18374885 Obviouslythe nominal maximum mean total profit is larger than theoptimal mean total profits obtained in Figures 2 and 4 Thecomputational results imply that a small perturbation of thenominal possibility distributionmay heavily affect the qualityof optimal solution
To further analyze the influence of the perturbationparameters some additional experiments are conductedwith different values of perturbation parameters 120579119897119894 and 120579119903119894The computational results are reported in Tables 3ndash6 inwhich the robust value is defined as the reduction from thenominal optimal profit to the optimal profits with differentvalues of perturbation parametersThe computational resultsimply that the robust value is increasing with respect toperturbation parameters 120579119897 or 120579119903 that is the larger theperturbation parameter the larger the uncertainty degreeembedded in the generalized PIV possibility distributionof uncertain demand The decision makers can adjust thevalues of perturbation parameters according to their obtaineddistribution information As a consequence the consideredMSIMP depends heavily on the location parameter 120582 andperturbation parameter 120579 For practical inventory man-agement problems if decision makers cannot identify thevalues of parameters 120582 and 120579 they may generate randomlytheir values from some prescribed subintervals of [0 1] Thecomputational results demonstrate the advantages of variablepossibility distributions over fixed possibility distributions
The comparison studies described in Sections 441 and442 lead to the following observations
Firstly stochastic optimization method for MSIMP isbased on the assumption that the market demands areof stochastic nature and the probability distributions ofuncertain parameters are available When the probabilitydistributions of uncertain market demands cannot be deter-mined the stochastic optimizationmethod cannot be used todetermine the optimal order quantities
Secondly in fuzzy MSIMP it is usually assumed that thenominal possibility distributions of uncertain parameters canbe determined exactly and a small perturbation of nominalpossibility distribution will not affect significantly the solu-tion quality The comparison study shows that the robustvalue is increasing with respect to perturbation parametersThe decision makers should adjust the values of perturbationparameters according to their obtained distribution informa-tion
Finally it should be highlighted that under given per-turbation parameters the optimal order quantities dependheavily on the values of location parameter lambda Theproposed parametric credibilistic optimization method iscapable of detecting cases when perturbation distributionscan heavily affect the quality of the nominal solution Inthese cases the decision makers should employ the proposedcredibilistic optimization method to find the optimal order
Mathematical Problems in Engineering 9
800
802
804
806
808
810
812
814
816
818
e opt
imal
ord
er q
uant
ity o
f pro
duct
1
02 03 04 05 06 07 08 09 1012
2400
2402
2404
2406
2408
2410
2412
2414
2416
2418
2420
e o
ptim
al o
rder
qua
ntity
of p
rodu
ct 2
02 03 04 05 06 07 08 09 1012
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
Figure 3 The optimal order quantities of product 1 and product 2 with 1205821 = 06
Table 3 The influence of perturbation parameter 1205791198971 with 1205791199031 = 025 1205791198972 = 015 and 1205791199032 = 021205791198971 119876lowast1 119876lowast2 Themean total profit The robust value005 810 2415 13400340 4974545010 809 2416 13091855 5283030015 809 2417 12788671 5586214020 808 2417 12477149 5897736030 806 2417 11854870 6520015
Table 4 The influence of perturbation parameter 1205791199031 with 1205791198971 = 03 1205791198972 = 015 and 1205791199032 = 021205791199031 119876lowast1 119876lowast2 Themean total profit The robust value005 818 2400 13155432 5219453015 816 2406 12516766 5858119018 813 2409 12316952 6057933025 806 2417 11854870 6520015035 800 2421 11195316 7179569
Table 5 The influence of perturbation parameter 1205791198972 with 1205791198971 = 03 1205791199031 = 025 and 1205791199032 = 021205791198972 119876lowast1 119876lowast2 Themean total profit The robust value010 807 2419 12153199 6221686015 806 2417 11854870 6520015020 804 2414 11549336 6825549025 800 2411 11235260 7139625030 800 2406 10934989 7439896
Table 6 The influence of perturbation parameter 1205791199032 with 1205791198971 = 03 1205791199031 = 025 and 1205791198972 = 0151205791199032 119876lowast1 119876lowast2 Themean total profit The robust value010 800 2422 12665659 5709226015 800 2420 12246912 6127973020 806 2417 11854870 6520015025 812 2412 11456342 6918543030 816 2400 11028701 7346184
10 Mathematical Problems in Engineering
times105
116
117
118
119
12
121
122
e m
ean
tota
l pro
t
02 03 04 05 06 07 08 09 1012
Figure 4 The mean total profit
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
quantities which may immunize against the effect of pertur-bation distribution
5 Conclusions
In this paper the MSIMP has been studied from a newperspective The major new results include the followingseveral aspects
(i) When the distribution information about uncertaindemand and uncertain carbon emission was partially avail-able these uncertain parameters were characterized by gen-eralized PIV fuzzy variables For their selection variables theanalytical expressions of themean and second-ordermomenthave been established
(ii) Two new indexes mean and second-order momentabout the total profit were defined based on L-Smultiple inte-gral and their analytical expressions have been establishedFurthermore a new parametric credibilistic optimizationmodel was developed for MSIMP
(iii) The equivalent deterministic model of the proposedcredibilistic MSIMP has been established According tothe structural characteristics of the equivalent deterministicmodel a domain decompositionmethodwas designed to findthe optimal order quantities
(iv) In numerical experiments the proposed optimizationmethod was compared with stochastic optimization methodand fuzzy optimization method under fixed possibility dis-tribution The computational results demonstrated that asmall perturbation of the demand distribution could makethe nominal optimal solution infeasible and thus practicallymeaningless In this case the decision makers should employthe proposed credibilistic optimization method to find theoptimal order quantities which may immunize against theeffect of perturbation distribution
The developed parametric credibilistic optimizationmodel for MSIMP addressed the effect of perturbation possi-bility distributions In themodel process the generalized PIVfuzzy variables were represented by their lambda selectionsFor a practical MSIMP based on the uncertain distributionsets of generalized PIV fuzzy variables distributionally robustoptimization method will be studied in our future researchExtension to considering decision makersrsquo risk tolerancefuzziness [42] for the MSIMP is another interesting researchdirection In addition hybrid uncertainty and their solutionmethod [43] can be introduced to tackle the MSIMP
Appendix
Proofs of Main Theorems
Proof of Theorem 1 Since 120585 sim 119899(120583 1205902 120579119897 120579119903) the generalizedpossibility distribution of 120585120582 is
120583120585120582 (119903 120579119897 120579119903) = 119890minus(119903minus120583)221205902 [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]+ 120582120579119903 (A1)
According to the definition of credibility measure [44] thecredibility Cr120585120582 le 119903 is computed by
which can generate a measure using the method discussed in[45] By calculation one has
Cr 120585120582 le 119903 =
1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 119903 isin (minusinfin 120583)1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 119903 isin [120583 +infin)
(A3)
According to (3) and (A3) one has
119864 (120585120582) = int(minusinfin+infin)
119903 dCr 120585120582 le 119903= int(minusinfin120583)
119903 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902
+ 1205821205791199032 + int[120583+infin)
119903 d1 minus (1 minus 120582) 120579119897
Mathematical Problems in Engineering 11
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = [1minus (1 minus 120582) 120579119897 minus 120582120579119903] 120583 = 120596120583
(A4)
The proof of theorem is complete
Proof ofTheorem2 Since 120578 sim Tri(1199031 1199032 1199033 120579119897 120579119903) the general-ized possibility distribution of the lambda selection variable120578120582 is
120583120578120582 (119903 120579)
=
120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199032 minus 1199031 119903 isin [1199031 1199032]120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033]
(A5)
According to the definition of credibilitymeasure one has
Cr 120578120582 le 119903 =
0 119903 isin (minusinfin 1199031)12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031) 119903 isin [1199031 1199032)1 minus (1 minus 120582) 120579119897 minus 12 120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 119903 isin [1199032 1199033)1 minus (1 minus 120582) 120579119897 119903 isin [1199033 +infin)
(A6)
By calculation one has
119864 (120578120582) = int[1199031 1199033]
119903 dCr 120578120582 le 119903= int[1199031 1199031]
119903 dCr 120578120582 le 119903+ int(1199031 1199032]
119903 dCr 120578120582 le 119903+ int(1199032 1199033)
119903 dCr 120578120582 le 119903
+ int[1199033 1199033]
119903 dCr 120578120582 le 119903= [1 minus (1 minus 120582) 120579119897] 1199031 + 21199032 + 11990334
+ 14120582120579119903 (1199031 minus 21199032 + 1199033) (A7)
The proof of theorem is complete
Proof of Theorem 3 According to (5) and (6) the second-order moment of 120585120582 is computed as follows
119872(120585120582) = int(minusinfin+infin)
[119903 minus 120596120583]2 dCr 120585120582 le 119903= int(minusinfin120583)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 + 1205821205791199032 + int[120583+infin)
[119903 minus 120596120583]2 d1 minus (1 minus 120582) 120579119897 minus 1 minus (1 minus 120582) 120579119897 minus 1205821205791199032 119890minus(119903minus120583)221205902 minus 1205821205791199032 = 1205962 int
(minusinfin120583)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 minus int
[120583+infin)[119903 minus 120596120583]2 d 119890minus(119903minus120583)221205902 = 120596 [21205902 + (120596 minus 1)2 1205832]
(A8)
The proof of theorem is complete
Proof of Theorem 4 In the following the mean value of 120578120582 isdenoted as119898 that is119898 = 119864(120578120582) According to (5) and (7) the
second-order moment of the lambda selection variable 120578120582 iscomputed by
119872(120578120582) = int[1199031 1199034]
[119903 minus 119864 (120578120582)]2 dCr 120578120582 le 119903= int(1199031 1199032)
[119903 minus 119898]2 d12120582120579119903 + (119903 minus 1199031) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]2 (1199032 minus 1199031)
12 Mathematical Problems in Engineering
+ int(1199032 1199033)
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
[119903 minus 119898]2 d1 minus (1 minus 120582) 120579119897 minus 12 [120582120579119903 + (1199033 minus 119903) [1 minus (1 minus 120582) 120579119897 minus 120582120579119903]1199033 minus 1199032 ] + 12120582120579119903 (1199031 minus 119898)2+ 12120582120579119903 (1199033 minus 119898)2
= 12120582120579119903 [(1199031 minus 119898)2 + (1199033 minus 119898)2] + 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199032 minus 1199031) [(1199032 minus 119898)3 minus (1199031 minus 119898)3]+ 1 minus (1 minus 120582) 120579119897 minus 1205821205791199036 (1199033 minus 1199032) [(1199033 minus 119898)3 minus (1199032 minus 119898)3]
(A9)
The proof of theorem is complete
Proof ofTheorem 5 First objective function (18) is equivalenttomaximizingΠ119890(119876)minus120574radicΠ119898(119876) By calculation themultipleL-S integrals one has
Π119890 (119876) = 119864 [Π (119876 120585120582)]= int sdot sdot sdot int
R119899
119899sum119894=1
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
int sdot sdot sdot intR119899
120587 (119876119894 119903) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
(119898119894prod119896 =119894
ℎ119896) (A10)
Similarly the second-order moment of the total profit iscomputed by
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)= 119899sum119894=1
int sdot sdot sdot intR119899
[120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896]2
d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)+ sum119894 =119895
int sdot sdot sdot intR119899
(120587 (119876119894 119903) minus 119898119894prod119896 =119894
ℎ119896)(120587(119876119895 119903) minus 119898119895prod119896 =119895
ℎ119896) d (Cr 1205851205821 le 119903 times sdot sdot sdot times Cr 120585120582119899 le 119903)
= 119899sum119894=1
[[prod119896 =119894
ℎ119896 int[0119863119894]
120587 (119876119894 119903)2 dCr 120585120582119894 le 119903 + (119898119894prod119896 =119894
ℎ119896)2( 119899prod119896=1
ℎ119896 minus 2)]] + prod119896 =119894 =119895
ℎ119896 [[sum119894 =119895
119898119894119898119895(1 minus 119899prod119896=1
ℎ119896)2]]
(A11)
Next the equivalence between constraint (19) and con-straint (24) is discussed Since 120578119894 (1 le 119894 le 119899) are mutuallyindependent generalized PIV fuzzy variables according to[36] one has119899sum119894=1
where 120579119897 = max1le119894le119899120579119897119894 and 120579119903 = max1le119894le119899120579119903119894Since the credibility Cr120585 le 119903 is a monotone increasing
function Cr(sum119899119894=1 119876119894120578119894)120582 le 119870 ge 120573 is equivalent to
119870 ge inf119909 | Cr( 119899sum
119894=1
119876119894120578119894)120582 le 119909 ge 120573 (A13)
that is 119870 ge (sum119899119894=1 119876119894120578119894)120582inf (120573) The proof of theorem iscomplete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (no 61773150)
References
[1] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall Englewood Cliffs NJ USA 1963
[2] S Nahmias and C P Schmidt ldquoAn efficient heuristic for themulti-item newsboy problem with a single constraintrdquo NavalResearch Logistics Quarterly vol 31 no 3 pp 463ndash474 1984
Mathematical Problems in Engineering 13
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
[3] H-S Lau andAH L Lau ldquoThemulti-productmulti-constraintnewsboy problem Applications formulation and solutionrdquoJournal of Operations Management vol 13 no 2 pp 153ndash1621995
[4] S J Erlebacher ldquoOptimal and heuristic solutions for the multi-item newsvendor problem with a single capacity constraintrdquoProduction Engineering Research and Development vol 9 no 3pp 303ndash318 2000
[5] I Moon and E A Silver ldquoThe multi-item newsvendor problemwith a budget constraint and fixed ordering costsrdquo Journal of theOperational Research Society vol 51 no 5 pp 602ndash608 2000
[6] L Abdel-Malek R Montanari and D Meneghetti ldquoThe capac-itated newsboy problem with random yield The GardenerProblemrdquo International Journal of Production Economics vol115 no 1 pp 113ndash127 2008
[7] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010
[8] B Zhang and SDu ldquoMulti-product newsboy problemwith lim-ited capacity and outsourcingrdquo European Journal of OperationalResearch vol 202 no 1 pp 107ndash113 2010
[9] L L Abdel-Malek and R Montanari ldquoOn the multi-productnewsboy problem with two constraintsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2095ndash2116 2005
[10] D Huang H Zhou and Q-H Zhao ldquoA competitive multiple-product newsboy problem with partial product substitutionrdquoOmega vol 39 no 3 pp 302ndash312 2011
[11] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009
[12] G Van Ryzin and S Mahajan ldquoOn the relationship betweeninventory costs and variety benefits in retail assortmentsrdquoManagement Science vol 45 no 11 pp 1496ndash1509 1999
[13] J A Van Mieghem ldquoRisk mitigation in newsvendor networksResource diversification flexibility sharing and hedgingrdquoManagement Science vol 53 no 8 pp 1269ndash1288 2007
[14] S Wang and W Pedrycz ldquoData-driven adaptive probabilis-tic robust optimization using information granulationrdquo IEEETransactions on Cybernetics 2017
[15] G L Vairaktarakis ldquoRobust multi-item newsboy models witha budget constraintrdquo International Journal of Production Eco-nomics vol 66 no 3 pp 213ndash226 2000
[16] J Kamburowski ldquoThe distribution-free newsboy problemunder theworst-case and best-case scenariosrdquoEuropean Journalof Operational Research vol 237 no 1 pp 106ndash112 2014
[17] M-H Shu C-W Yeh and Y-C Fu ldquoImpacts of transportationcost on distribution-free newsboy problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 307935 2014
[18] I Moon D K Yoo and S Saha ldquoThe distribution-free newsboyproblem with multiple discounts and upgradesrdquo MathematicalProblems in Engineering vol 2016 Article ID 2017253 2016
[19] O Solyali J F Cordeau and G Laporte ldquoThe impact ofmodeling on robust inventory management under demanduncertaintyrdquoManagement Science vol 62 no 4 pp 1188ndash12012016
[20] R Qiu and J Shang ldquoRobust optimisation for risk-aversemulti-period inventory decision with partial demand distributioninformationrdquo International Journal of Production Research vol52 no 24 pp 7472ndash7495 2014
[21] S Fay and J Xie ldquoTiming of product allocation Using proba-bilistic selling to enhance inventorymanagementrdquoManagementScience vol 61 no 2 pp 474ndash484 2015
[22] W Luo and K Shang ldquoJoint inventory and cash managementfor multidivisional supply chainsrdquo Operations Research vol 63no 5 pp 1098ndash1116 2015
[23] X Chen P Hu S Shum and Y Zhang ldquoDynamic stochasticinventory management with reference price effectsrdquoOperationsResearch vol 64 no 6 pp 1529ndash1536 2016
[24] H Mamani S Nassiri and M R Wagner ldquoClosed-FormSolutions for Robust Inventory Managementrdquo ManagementScience vol 63 no 5 pp 1625ndash1643 2017
[25] J Kacprzyk and P Stanieski ldquoLong-term inventory policy-making through fuzzy decision-makingmodelsrdquo Fuzzy Sets andSystems vol 8 no 2 pp 117ndash132 1982
[26] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy modelsfor the newsboy problemrdquo International Journal of ProductionEconomics vol 45 no 1ndash3 pp 435ndash441 1996
[27] N K Mandal and T K Roy ldquoA displayed inventory model withL-R fuzzy numberrdquo Fuzzy Optimization and DecisionMaking AJournal of Modeling and Computation Under Uncertainty vol 5no 3 pp 227ndash243 2006
[28] X Y Ji and Z Shao ldquoFuzzy multi-product constraint newsboyproblemrdquo Applied Mathematics and Computation vol 180 no1 pp 7ndash15 2006
[29] P Dutta ldquoA multi-product newsboy problem with fuzzy cus-tomer demand and a storage space constraintrdquo InternationalJournal of Operational Research vol 8 no 2 pp 230ndash246 2010
[30] A Saha S Kar and M Maiti ldquoMulti-item fuzzy-stochasticsupply chain models for long-term contracts with a profitsharing schemerdquo Applied Mathematical Modelling vol 39 no10-11 pp 2815ndash2828 2015
[31] Z-Z Guo ldquoOptimal inventory policy for single-period inven-tory management problem under equivalent value criterionrdquoJournal of Uncertain Systems vol 10 no 4 pp 302ndash311 2016
[32] S-N Tian andZ-ZGuo ldquoA credibilistic optimization approachto single-product single-period inventory problem with twosuppliersrdquo Journal of Uncertain Systems vol 10 no 3 pp 223ndash240 2016
[33] Z-Q Liu and Y-K Liu ldquoType-2 fuzzy variables and theirarithmeticrdquo Soft Computing vol 14 no 7 pp 729ndash747 2010
[34] Y Liu and Y-K Liu ldquoThe lambda selections of parametricinterval-valued fuzzy variables and their numerical character-isticsrdquo Fuzzy Optimization and Decision Making A Journal ofModeling and Computation Under Uncertainty vol 15 no 3 pp255ndash279 2016
[35] Y Chen and S Shen ldquoThe cross selections of parametricInterval-Valued fuzzy variablesrdquo Journal of Uncertain Systemsvol 9 no 2 pp 156ndash160 2015
[36] Y-K Liu and Z-Z Guo ldquoArithmetic about linear combinationsof GPIV fuzzy variablesrdquo Journal of Uncertain Systems vol 11no 2 pp 154ndash160 2017
[37] M Carter and B van Brunt The Lebesgue-Stieltjes IntegralSpinger-Verlag New York NY USA 2000
[38] Z Guo Y Liu and Y Liu ldquoCoordinating a three level supplychain under generalized parametric interval-valued distribu-tion of uncertain demandrdquo Journal of Ambient Intelligence andHumanized Computing vol 8 no 5 pp 677ndash694 2017
[39] SWang T SNg andMWong ldquoExpansion planning forwaste-to-energy systems using waste forecast prediction setsrdquo NavalResearch Logistics vol 63 no 1 pp 47ndash70 2016
[40] X Bai and Y K Liu ldquoRobust optimization of supply chainnetwork design in fuzzy decision systemrdquo Journal of IntelligentManufacturing vol 27 no 6 pp 1131ndash1149 2016
14 Mathematical Problems in Engineering
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
[41] Y Liu and Y-K Liu ldquoDistributionally robust fuzzy project port-folio optimization problem with interactive returnsrdquo AppliedSoft Computing vol 56 pp 655ndash668 2017
[42] S Wang B Wang and J Watada ldquoAdaptive Budget-PortfolioInvestment Optimization Under Risk Tolerance AmbiguityrdquoIEEE Transactions on Fuzzy Systems vol 25 no 2 pp 363ndash3762017
[43] S Wang and W Pedrycz ldquoRobust granular optimization astructured approach for optimization under integrated uncer-taintyrdquo IEEE Transactions on Fuzzy Systems vol 23 no 5 pp1372ndash1386 2015
[44] B Liu and Y Liu ldquoExpected value of fuzzy variable and fuzzyexpected value modelsrdquo IEEE Transactions on Fuzzy Systemsvol 10 no 4 pp 445ndash450 2002
[45] Y-K Liu and Y Liu ldquoMeasure generated by joint credibilitydistribution functionrdquo Journal of Uncertain Systems vol 8 no9 pp 239-240 2014
