sucky3

European Journal of Operational Research 171 (2006) 516–535

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A bargaining model with asymmetric informationfor a single supplier–single buyer problem

Eric Sucky *

Department of Supply Chain Management, School of Business and Economics, Goethe-University Frankfurt,

Mertonstr. 17, 60054 Frankfurt, Germany

Received 7 December 2001; accepted 23 August 2004Available online 30 October 2004

Abstract

Banerjee�s joint economic lot size (JELS) model represents one approach to minimizing the joint total relevant costof a buyer and a supplier by using a joint optimal order and production policy. The implementation of a jointly optimalpolicy requires coordination and cooperation. Should the buyer have the market power to implement his own optimalpolicy as that one to be used in the exchange process no incentive exists for him to choose a joint optimal policy. Ajoint policy can therefore only be the result of a bargaining process between the parties involved. The supplier maymake some sort of concession such as a price discount or a side payment in order to influence the buyer�s order policy.A critical assumption made throughout in supply chain literature is that the supplier has complete knowledge about thebuyer�s cost structure. Clearly, this assumption will seldom be fulfilled in practice. The research presented in this paperprovides a bargaining model with asymmetric information about the buyer�s cost structure assuming that the buyer hasthe power to impose its individual optimal policy.� 2004 Elsevier B.V. All rights reserved.

Keywords: Production; Supply chain management; Coordination; Asymmetric information

1. Introduction

A supply chain can be considered as a network of different geographically dispersed facilities where rawmaterials, intermediate products, or finished products are transformed and transportation links that con-nect the facilities ([10, p. 15], [37, p. 5]). One of the major tasks of supply chain management is to coordinatethe processes in the supply chain in such a way, that a given set of objectives is achieved [39, pp. 8–9]. Most

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E. Sucky / European Journal of Operational Research 171 (2006) 516–535 517

commonly, the relevant objectives, pursued by supply chain management, are minimizing system-wide costswhile satisfying a predetermined service level [30, p. 835]. The complexity in coordinating the processes insupply chains is introduced by the organizational structure within the network. In general, a supply chain iscomposed of independent firms with individual preferences [7, p. 113]. Therefore, in contrast to the man-agement of multi-echelon systems, that coordinates inventory, production and distribution decisions atmultiple locations of one firm, supply chain management involves coordination of such decisions amongmultiple and independent firms [27, pp. 794–795]. Bhatnagar et al. [5] identify the issue of coordina-tion—at the most general level, which they call general coordination—in integrating decisions of differentfunctions. Within this problem of functional coordination Bhatnagar et al. [5] as well as Thomas and Grif-fin [41] distinguish three categories: (1) supply–production coordination, (2) production–distribution coor-dination, and (3) inventory–distribution coordination. In this paper we will focus on the first category,which is also called buyer–supplier coordination [41, p. 2]. For each set of nodes in a supply chain, e.g.a location of a manufacturer and a site of an assembler, a supplier–buyer relationship can be identified[2, p. 199]. Material flows from a supplier to a buyer while information and financial flows are bi-direc-tional. Both, in the scientific discussion and in practice considerable attention is paid to the importanceof a coordinated relationship between suppliers and buyers in supply chains. As Goyal and Gupta [21] note,coordination between the supplier and the buyer can be mutually beneficial to both. Studies on buyer–sup-plier coordination have focused on determining the order and production policy which is jointly optimal forboth. Using such a joint optimal order and production policy—as opposed to independently derived pol-icies—leads to a significant total cost reduction. However, there is an additional set of problems involved inimplementing joint policies.

In this paper we will analyze the typical case of a buyer–supplier relationship with a strong buyer. Shouldthe buyer have the market power to implement his own optimal policy as that one to be used in the ex-change process no incentive exists for him to choose a cooperative policy. To provide an incentive to orderin quantities suitable to the supplier, there exists the opportunity for the supplier to offer a price discount ora side payment to the buyer. One critical assumption made throughout the literature dealing with incentiveschemes to influence buyer�s ordering policy is that the supplier has complete knowledge about the buyer�scost structure. The research presented in this paper offers a bargaining model with asymmetric informationabout buyer�s cost structure assuming that the buyer has the market power to impose his individual optimalpolicy. The remainder of the paper is organized as follows. Section 2 presents the review of the literaturedealing with the problem of buyer–supplier coordination. In Section 3—on the basis of the well-knownjoint economic lot size model suggested by Banerjee [4]—we will show that from an individual point of viewneither the buyer nor the supplier has an incentive to choose a cooperative policy. We describe the quan-titative losses due to a cooperative policy, both from the buyer�s and the supplier�s perspective. After that,we analyze the bargaining process between the parties involved. In the first approach (Section 4), it is pre-sumed that the supplier has complete knowledge about the buyer�s cost structure. Afterwards, in the secondmodel—presented in Section 5—we analyze a more realistic scenario. The supplier has incomplete informa-tion about the buyer�s cost structure, i.e. the buyer can have different functions of total cost, assumed by thesupplier. We will present a self-selection model for the determination of the optimal set of offers which arespecifically designed for the buyer�s different cost structures.

2. Literature review

There is a large and growing stream of literature on buyer–supplier coordination. Most of the literatureon buyer–supplier coordination focuses either on deriving a coordinated order and production policy in thecontext of a given supply contract, or on determination optimal contract parameters given the functionalform of that contract. In the last category, the primary focus is in design and evaluation of supply contracts

518 E. Sucky / European Journal of Operational Research 171 (2006) 516–535

between independent buyers and suppliers in supply chains. Cachon [8] and Tsay et al. [42] provide com-prehensive literature reviews of designing supply contracts. Corbett and Tang [12] study the design of sup-ply contracts in the presence of asymmetric information. In this paper, however, we will focus on the firstcategory: Coordinating order and production policies under a given supply contract. From the contractualperspective, we assume that a fixed unit-price contract, often so called wholesale-price contract, exists be-tween the buyer and the supplier initially. Therefore, in the context of a given supply contract, the coordi-nation of the buyer�s order policy and the supplier�s production policy does not affect the total volumeprovided to the buyer.

When the buyer and the supplier treat inventory problems singly under deterministic conditions, it is wellknown that the economic order quantity (EOQ) formula or the economic lot size (ELS) formula gives anindividual optimal solution. However, in general, an order policy based on the EOQ solution is unaccept-able to the supplier, and likewise, a production and delivery policy based on the ELS solution is unaccept-able to the buyer [32, p. 312]. Each party has the lowest total relevant cost when their individual optimalorder or production and delivery policy is realized, but the other party experiences a considerable loss, com-pared to its own individual optimal policy [33, p. 26]. The problem of buyer–supplier coordination has beenemphasized in numerous publications. Detailed reviews of available approaches for coordinating singlebuyer–single supplier systems are given by Bhatnagar et al. [5], Goyal and Gupta [21], Sharafali and Co[38] and Thomas and Griffin [41].

In general, inventory decisions in a buyer–supplier system are made independently. That is, the sup-plier determines its own optimal policy to produce and to supply a product and the buyer may also cal-culate its own optimal policy to order from the supplier. As a result, the supplier�s optimal deliveryquantity may not necessarily be the same as the buyer�s optimal order quantity. In that case, the adop-tion of the buyer�s optimal ordering policy places the supplier at a cost disadvantage. Otherwise, theadoption of the supplier�s optimal delivery policy is disadvantageous from the buyer�s perspective. Asa remedy, Goyal [15,18], Banerjee [4], and Landeros and Lyth [29], present some models for coordinatingbuyer–supplier systems with centralized control. The researchers suggest approaches for determining anintegrated order and delivery policy, minimizing the joint total relevant cost incurred by both parties. Itis shown that in integrated models one party�s gain resulting from the integrated policy in comparison ofthe other party�s optimal policy exceeds the other party�s loss. Thus, the net benefit can be shared byboth parties (the supplier and the buyer) in some equitable fashion [21, p. 262]. While all models usethe accepted EOQ formula to determine buyer�s individual optimal order policy, the distinguishing fea-ture is the assumed production and delivery policy of the supplier. Goyal [15] was the first who intro-duced an integrated policy for a single supplier-single buyer system. He assumes that the buyer�sdemand is uniform with respect to time, there is no lead time for the supplier and the buyer, and thesupplier and the buyer cooperate with the objective of minimizing the joint total relevant cost. The jointrelevant costs are the costs of holding and ordering at the buyer�s side and the set-up cost at the sup-plier�s side. However, the effect of production rate on calculating average inventory is ignored, by assum-ing an infinite production rate. Banerjee [4] generalizes Goyal�s model by integrating a finite productionrate. Therefore, the holding cost of the supplier will be considered by determining the integrated orderand delivery policy. Assuming a production rate greater than the demand rate, the supplier makes theproduction set-up every time the buyer places an order and supplies on a lot-for-lot basis. Goyal [18]relaxes the lot-for-lot assumption by considering that each production batch is dispatched to the buyerin an integer number of equal sized sub-batches. In contrast to Banerjee [4] who assumes that the entirelot is conveyed as a whole from the supplier to the buyer (sequential movement), Goyal [18] allows trans-portation of sub-batches whose size may be larger than one unit and smaller than the entire lot (com-bined movement). However, the entire lot has to be completed before any shipments from that lotcan take place. The possibility to produce more than one shipment in a given production run does alterthe supplier�s average inventory as well as the number of production set-ups. It is shown, that the joint


total relevant cost is further reduced. Landeros and Lyth note that Goyal�s model overlooked theprocessing cost per shipment [29, p. 154]. Therefore, Landeros and Lyth [29] consider fixed delivery costassociated with each shipment to the buyer. However, this approach does not generalize Goyal�s model.In the presence of centralized control one simply has to increase the fixed order processing costs of thebuyer by the fixed transportation cost per shipment which does not alter the solution methodology. How-ever, all models mentioned above assume that the whole production batch must be finished before anyshipments from the batch can take place.

Another class of models for coordinating buyer–supplier systems with centralized control allows theshipment of sub-batches to the next stage before the whole lot is finished at the preceding one. Based onan idea of Szendrovits [40] in the context of a serial multi-stage production system, this control policywas first applied in the area of supplier–buyer coordination by Joglekar [25]. Agrawal and Raju [1] considerthis case of a supplier being able to ship a number of sub-batches before the whole production batch is fin-ished. Analogous to Szendrovits [40], they provide sub-batches of uniform size with a constant time allow-ance between any two consecutive supplies. Based on a much earlier idea set out by Goyal [16], Goyal [19]provides an alternative shipment policy, in which all sub-batches are in general of unequal size. This pro-duction and delivery policy involves successive shipments within a production batch so that the size of thesub-batches increases according to a geometric series. Chatterjee and Ravi [9] present an equivalent modelconsidering fixed delivery costs associated with each shipment on the supplier�s side contrary to fixed trans-portation cost per shipment as a component of the buyer�s fixed order processing costs. Viswanathan [43]shows, based on a simulation study, that neither a policy with equal sized sub-batches nor a policy withunequal sized sub-batches dominates the other. However, Hill [23] derives the structure of the true optimalsolution for the sizes of the sub-batches for a single supplier–single buyer system acting in a static determin-istic environment. For the determination of the optimal sizes of the sub-batches for the buyer–supplier sys-tem with centralized control see also Hill [22] and Goyal [20].

Nevertheless, neither the models with equal sized sub-batches nor the models with unequal sized sub-batches consider the power structure in buyer–supplier relationships. Three typical cases can be applicable:(1) The buyer dominates the supplier. (2) The supplier dominates the buyer. (3) The buyer and the supplierhave equal power. In this paper we analyze the first case. If the buyer is strong, the supplier will be forced toaccept the buyer�s individual optimal order policy. In this situation, the supplier may make some sort ofconcession such as a price discount or a side payment in order to influence the buyer�s ordering behaviour.The supplier�s problem to influence the buyer�s order policy has been analyzed by several authors, includingMonahan [34,35], Banerjee [3], Goyal [17], Lee and Rosenblatt [31] and Joglekar [25]. The authors computea price discount which compensates the loss of the buyer, if he changes his order policy to a cooperativepolicy. Miller and Kelle [33] and Kelle et al. [28] also suggest some quantitative models to support nego-tiations in Just-In-Time supply systems. However, one critical assumption made throughout in the litera-ture dealing with supplier–buyer coordination is that the supplier has complete knowledge about thebuyer�s cost structure [11, p. 444]. Solely Corbett and de Groote [11] address the bargaining process betweena buyer and a supplier under asymmetric information. Other than the existing joint economic lot-sizing lit-erature assuming that the supplier has full information about the buyer�s cost structure, the authors con-sider the situation that the buyer�s holding cost is unobservable to the supplier. Corbett and de Groote[11] present a game theoretical analysis and determine an optimal discount scheme in order to influencethe buyer�s ordering behaviour. Nevertheless, the buyer�s ordering costs are assumed as common knowl-edge. In this case, the problem of information asymmetry is eliminated. The buyer�s holding cost can bespecified as an implicit function of his ordering cost and his EOQ. Thus, if the supplier knows the buyer�sordering cost and his EOQ, the supplier can also derive the buyer�s holding cost. The research presented inthis paper provides a bargaining model with incomplete information about the buyer�s ordering cost andholding cost, assuming that the buyer has the market power to implement his EOQ in case of a break-downin negotiations.


3. Ordering and production policies in the JELS model

3.1. Individual optimal policies

Following Banerjee [4], the discussion and analysis in this paper is restricted to the case of a single sup-plier and single buyer of a specific product. The buyer and the supplier are separate and independent organ-izations. The demand for the product is assumed constant and deterministic. Shortages are not permitted atthe buyer�s end, and the time horizon over which the product is ordered by the buyer and supplied by thesupplier is infinite. The supplier fabricates the regarded product at a finite production rate. The transpor-tation time between the supplier and the buyer is assumed to be zero. The optimality criterion for the buyerand the supplier is the minimization of their own total relevant cost per period. The buyer�s total relevantcost per period consists of the order processing cost and the inventory holding cost. The supplier�s costfunction includes the production set-up cost and the inventory holding cost. Both parties determine the eco-nomic order quantity or lot size using the well known economic order quantity (EOQ) formula or economiclot size (ELS) formula [4, pp. 293–294].

3.1.1. Individual optimal ordering policy

The objective for the buyer is to determine the optimal ordering policy minimizing his total relevant costper period without shortages. We refer to him as party (A), and hence generally use subscript ‘‘A’’ to des-ignate his set of parameters or decisions. It is easy to see that for an optimal ordering policy every order,with quantity xA [unit], is received precisely when the inventory level drops to zero [6, pp. 145–147]. Thedemand rate at the buyer�s end is d [unit/period]. The inventory holding cost of (A) is hA [$/(unit and per-iod)] and the order processing cost amounts to B [$]. The total relevant cost per period of (A) is given by

KAðxAÞ ¼ B � dxA

þ xA2� hA ½$=period�: ð1Þ

The economic order quantity (EOQ), x�A [unit], and the minimum total relevant cost per period, KAðx�AÞ[$/period], for the buyer (A) are given by

x�A ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � B � d

hA

s; KAðx�AÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � B � d � hA

p: ð2Þ

3.1.2. Individual optimal production policy

The objective for the supplier is to find the optimal production policy minimizing his total relevant costper period. We refer to him as party (P) and use subscript ‘‘P’’ to designate the set of parameters or deci-sions of the supplier. The production rate amounts to p [unit/period], whereas the relation p > d will alwayshold. The supplier follows a lot-for-lot policy, i.e. (P) produces the ordered quantity of the product and, oncompletion of the batch, ships the entire lot to (A). The average inventory is given by xP

2� dp [unit] [4, p. 294].

The inventory holding cost of (P) is hP [$/(unit and period)] and the production set-up cost is R [$]. Thetotal relevant cost per period of (P) is given by

KP ðxP Þ ¼ R � dxP

þ xP2� dp� hP ½$=period�: ð3Þ

The supplier�s economic lot size (ELS), x�P [unit], and his minimum total relevant cost per period, KP ðx�P Þ[$/period], are given by

x�P ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � R � p

hP

s; KP ðx�P Þ ¼ d �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � R � hP

p

s: ð4Þ


3.2. Integrated production and ordering policy

If (P) follows a lot-for-lot strategy, the lot size corresponds to the quantity delivered. For (A), the orderquantity corresponds to the quantity delivered. Since we have excluded shortages by definition, an equal lotsize xG = xA = xP for (A) and (P) is the logical consequence. This leads to the inventory cycles for (A) and(P) as represented in Fig. 1.

Banerjee [4] determines the joint lot size xG = xA = xP minimizing the joint total relevant cost for boththe buyer and the supplier. The joint total relevant cost of (A) and (P) for a joint lot size xG = xA = xP canbe derived from Eqs. (1) and (3) as follows [4, p. 299]:

KGðxGÞ ¼ B � dxG

þ xG2� hA þ R � d

xGþ xG

2� dp� hP ½$=period�: ð5Þ

The joint optimal lot size, x�G [unit], and the minimum joint total relevant cost per period, KGðx�GÞ [$/per-iod], are given by

x�G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � d � ðBþ RÞhA þ d

p � hP

s; KGðx�GÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � d � ðBþ RÞ � hA þ

dp� hP

� �s: ð6Þ

3.3. Comparison of the individual and integrated policies

The relation of set-up cost of (P) to the ordering cost of (A) and the relation of the holding cost of (P) tothe holdings cost of (A) for any joint lot size xG can be expressed with the parameters a and b [4, p. 295]:

a ¼R � d

xG

B � dxG

¼ RB

and b ¼xG2� dp � hP

xG2� hA

¼ d � hPp � hA

: ð7Þ

The following relations can be derived:

x�A ¼ffiffiffiba

r� x�P ; x�P ¼

ffiffiffiab

r� x�A; x�G ¼

ffiffiffiffiffiffiffiffiffiffiffi1þ a1þ b

s� x�A ¼

ffiffiffiffiffiffiffiffiffiffi1þ 1

a

1þ 1b

s� x�P : ð8Þ

Inventory Buyer (A)

Time

xA

Inventory Supplier (P)

Time

xP

Fig. 1. Inventory cycles for (A) and (P).

Fig. 2. Individual and integrated policies.


A uniform lot size x�A ¼ x�P ¼ x�G only applies for a = b. For a 5 b follows x�A 6¼ x�P , and the joint optimallot size x�G is located within the interval between the individual optimal solutions of (A) and (P), i.e.x�G 2 �x�A; x�P ½ if x�A > x�P and x�G 2 �x�P ; x�A½ if x�A > x�P . In case of a 5 b ðx�A 6¼ x�P Þ a solution with an uniformlot size xG = xA = xP only exists if (P) adapts to (A) ðxP ¼ x�A ¼ xGÞ, (A) adapts to (P) ðxA ¼ x�P ¼ xGÞ orif (A) and (P) select another joint lot size, for example x�G ðxA ¼ xP ¼ x�GÞ. Fig. 2 exemplary shows the costfunctions of (A) and (P), the function of the joint total relevant cost and the individual and integrated pol-icies in case of x�A < x�P .

Deviations from the individual optimal policy always lead to an increase of relevant cost. From the sup-plier�s point of view KP ðxGÞ > KP ðx�P Þ applies for any joint policy xG 6¼ x�P . In the same way,KAðxGÞ > KAðx�AÞ applies for any joint policy xG 6¼ x�A from the buyer�s perspective. Using the relations (8)the consequences of the alternative policies x�A, x

�P and x�G for (A) and (P) can be calculated (see Table 1).

The adoption of the buyer�s optimal ordering policy as the joint policy ðxP ¼ x�A ¼ xGÞ places the supplier

at a cost disadvantage: The relevant costs of (P) increase by the factor 12�

ffiffiab

qþ

ffiffiba

q� �in relation to his indi-

vidual minimum KP ðx�P Þ. Otherwise, the adoption of the supplier�s optimal delivery policy is disadvanta-geous from the buyer�s perspective. In case of xA ¼ x�P ¼ xG, the relevant costs of (A) also increase by the

factor 12�

ffiffiab

qþ

ffiffiba

q� �in relation to his individual minimum KAðx�AÞ. The realization of the joint optimal

policy ðxA ¼ xP ¼ x�GÞ leads to an increase of the relevant costs of (A) by the factor 12�

ffiffiffiffiffiffi1þa1þb

qþ

ffiffiffiffiffiffi1þb1þa

q� �in

relation to his individual minimum KAðx�AÞ whereas the relevant costs of (P) increase by the factor

12�

ffiffiffiffiffiffi1þ1

a

1þ1b

rþ


b

1þ1a

r� �in relation to his individual minimum KP ðx�P Þ. With a 5 b and a,b > 0, the following

relations can be derived:

1

2�

ffiffiffiab

rþ

ffiffiffiba

r !>

1

2�


sþ

ffiffiffiffiffiffiffiffiffiffiffi1þ b1þ a

r !> 1; ð9Þ

1

2�

ffiffiffiab

rþ

ffiffiffiba

r !>

1

2�


a

1þ 1b

sþ


b

1þ 1a

s0@

1A > 1: ð10Þ

Table 1Individual economic consequences of alternative policies

Order quantity/lot size Total relevant cost for (A) Total relevant cost for (P)

xA ¼ x�A, xP ¼ x�A ¼ffiffiba

q� x�P KAðxAÞ ¼ KAðx�AÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � B � d � hA

pKP ðxP Þ ¼ KP ðx�AÞ ¼ 1

2 �ffiffiab

qþ

ffiffiba

q� �� KP ðx�P Þ

xA ¼ x�P ¼ffiffiab

q� x�A, xP ¼ x�P KAðxAÞ ¼ KAðx�P Þ ¼ 1

2 �ffiffiab

qþ

ffiffiba

q� �� KAðx�AÞ KP ðxP Þ ¼ KP ðx�P Þ ¼ d �

ffiffiffiffiffiffiffiffiffiffi2�R�hP

p

qxA ¼ x�G ¼

ffiffiffiffiffiffiffi1þa1þb

q� x�A,

xP ¼ x�G ¼ffiffiffiffiffiffi1þ1

a

1þ1b

r� x�P

KAðxAÞ ¼ KAðx�GÞ ¼ 12 �

ffiffiffiffiffiffiffi1þa1þb

qþ

ffiffiffiffiffiffiffi1þb1þa

q� �� KAðx�AÞ KP ðxP Þ ¼ KP ðx�GÞ ¼ 1

2 �ffiffiffiffiffiffi1þ1

a

1þ1b

rþ


b

1þ1a

r� �� KP ðx�P Þ


The inequalities (9) and (10) show that the deviation from the individual optimal policy always leads toan increase in cost by a factor greater than one. In addition to that, it can be observed that the increase incost resulting of the other party�s individual optimal policy is greater than the increase initiated by theimplementation of the joint optimal policy. Table 1 summarizes the individual economic consequencesof alternative joint policies.

The analysis of the individual economic consequences of alternative joint policies shows that neither thebuyer nor the supplier have an incentive to choose a cooperative policy deviating from their individual opti-mal policies. The buyer and the supplier are separate and independent firms. In this context, it is reasonableto assume that each party will act in their own best interests. If (A) and (P) behave individually and ration-ally, they want to select their individual optimal policies x�A and x�P in any case. Therefore, for a 5 b a jointpolicy xG = xA = xP can only result from negotiations between the parties involved.

4. A bargaining game with complete information

4.1. The bargaining model

If (A) and (P) behave individually rational, they select their individual optimal policies x�A and x�P . If (A)or (P) has market power to impose its individual optimal policy on (P) or (A), respectively, then no incen-tive exists for either (A) or (P) to deviate from their individual optimal policy x�A or x�P . The weaker playermust deviate from his individual optimal policy and adapt to the stronger player�s policy. However if nego-tiations are possible, the weaker player may try to persuade the stronger player to select a policy other thanits individual optimal policy by a side payment. A side payment is defined as an additional monetary trans-fer between supplier (buyer) and buyer (supplier) that is used as an incentive for deviating from the indi-vidual optimal policy [36, p. 22]. It is to be examined, which joint policy xG = xA = xP is implemented if(A) and (P) negotiate on the joint order quantity and lot size. The bargaining game (N,U,u*) is describedby the set of players N = {(A), (P)}; the set U of feasible payoff combinations (in this case cost combina-tions) u 2 U and the threat point u* 2 U realized in the case of a break-down in negotiations [13, p.238]. The bargaining problem arises from the fact that at least one cost combination u 0 2 U with the fol-lowing property exists: u0i 6 u�i for all players i 2 N; u0i < u�i for at least one player i 2 N. The bargaininggame to be examined can be described as follows:

• (A) has the market power to implement his optimal policy, i.e. xG ¼ x�A results in case of a break-down innegotiations. The threat point is given by u� ¼ KAðx�AÞ;KP ðx�AÞ

� �.

• For (A) to select a policy other than his individual optimal policy x�A the increase in total relevant costmust be at least compensated by a side payment from (P). Thus, (P) offers (A) a joint policy xG with anassociated side payment at a value of z [$/period]. The feasible cost combinations are given byu = (KA(xG) � z,KP(xG) + z), for xG > 0, z P 0.


• (P) makes a take-it-or-leave-it-offer and the game is immediately terminated after acceptance or refusalof the offer by (A) [14, pp. 57–61].

• No transaction cost arise and (P) has complete knowledge about the cost function of (A).

4.2. Bargaining solution with complete information

In this section will be analyzed, which joint policy xG and associated side payment z are offered to (A) by(P). (A) has the market power to implement his individual optimal policy x�A. If (A) behaves individually ra-tional, no incentive exists for deviating from this individual optimal policy. Without negotiations x�A will be

implemented, withKAðx�AÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � B � d � hA

pandKP ðx�AÞ ¼ 1

2�

ffiffiab

qþ

ffiffiba

q� �� KP ðx�P Þ. If (P) offers (A) a joint pol-

icy xG against a side payment at a value of z, the following non-linear minimization problem is to be solved:

min KP ðxG; zÞ ¼ KP ðxGÞ þ z ð11Þs.t. KAðxGÞ � z 6 KAðx�AÞ; ð12Þ

z P 0; ð13ÞxG > 0. ð14Þ

The non-linear optimization problem given in (11)–(14) can be solved by using the Lagrangian approach.Nevertheless, the solution can be determined as follows: Regarding the minimization problem (11), for eachgiven policy xG > 0 the transfer payment z should be selected as small as possible while satisfying restriction(12). Therefore, for each given xG the corresponding optimal value of z is exactly given at the point where(12) is satisfied as an equation. Replacing z in the objective function (11) by KAðxGÞ � KAðx�AÞ and minimiz-ing the resulting cost expression, the optimal offered policy and the corresponding side payment can easilycalculated. The supplier offers the joint optimal policy (6) and a side payment, which compensates exactlythe increase in cost of (A), induced by deviating from the individual optimal policy x�A:

x�G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � d � ðBþ RÞhA þ d

p � hP

s; z ¼ KAðx�GÞ � KAðx�AÞ ¼

1

2�


sþ


r !� 1

!�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � B � d � hA

p:

ð15Þ
If (A) behaves individually rationally, he accepts this offer, since condition (12) is fulfilled. (A) realizes his
minimal total cost KAðx�GÞ � z ¼ KAðx�GÞ � KAðx�GÞ þ KAðx�AÞ ¼ KAðx�AÞ. The offer is attractive for (P) only ifthe total relevant cost plus the side payment are lower than with implementation of x�A, i.e. the followingmust apply:

KP ðx�GÞ þ KAðx�GÞ � KAðx�AÞ < KP ðx�AÞ: ð16Þ
Condition (16) can be transformed:
1

2�


a

1þ 1b

sþ


b

1þ 1a

s0@

1A �KP ðx�P Þþ

1

2�


sþ


r !� 1

!�KAðx�AÞ<

1

2�

ffiffiffiab

rþ

ffiffiffiba

r !�KP ðx�P Þ:

ð17Þ
It can be shown that condition (17) is satisfied for all a 5 b and a,b > 0, i.e. (P) realizes a bargaining
surplus V P ðx�GÞ at a value of:

V P ðx�GÞ ¼ KP ðx�AÞ � KP ðx�GÞ � z ¼ KP ðx�AÞ � KP ðx�GÞ � KAðx�GÞ þ KAðx�AÞ > 0: ð18Þ

Table 2Example 1

a = 1.35, b = 0.6 Total relevant cost for (A)(d = 10,000, B = 100, hA = 50)

Total relevant cost for (P)(p = 15,000, R = 135, hP = 45)

Joint total relevant costfor (A) and (P)

x�A ¼ 200 KAðx�AÞ = 10,000 KP ðx�AÞ ¼ 9750 KGðx�AÞ = 19,750x�P ¼ 300 KAðx�P Þ = 10,833.33 KP ðx�P Þ ¼ 9000 KGðx�P Þ = 19,833.33x�G ¼ 242:38 KAðx�GÞ = 10,185.25 KP ðx�GÞ ¼ 9205:46 KGðx�GÞ = 19,390.72


(P) behaves like a master planner and selects the joint optimal order quantity and lot size x�G. With completeinformation negotiations based on the assumption made, lead to the joint optimal order quantity and lotsize. It can be shown that this result is independent of who possesses market power to implement its indi-vidual optimal policy. Likewise it can be shown that this result is also independent of who makes the take-it-or-leave-it-offer. A fundamental principle illustrated here appears to apply to most, if not all, singlesource purchases: if the two parties negotiate cooperatively, with complete information, a side payment willlead to a contract from which both parties will benefit [36, p. 25]. With respect to the work of Banerjee [3], ithas to be noted that a solution identical to the presented one can be obtained—as was shown by Banerjee[3]—if the supplier grants an appropriate all-unit quantity discount to the buyer, provided that the buyer�sholding cost hA does not depend on the unit price which the buyer has to pay to the supplier.

4.3. A numerical example (1)

Consider the information for the buyer (A) and the supplier (P) given in Table 2. The buyer (A) selectshis optimal policy x�A ¼ 200 and realizes his minimum total relevant cost KAðx�AÞ = 10,000. Without bargain-ing, the supplier�s total relevant cost are KP ðx�AÞ ¼ 9750. Therefore, the supplier (P) offers the joint optimalpolicy x�G ¼ 242:38 and a side payment z ¼ KAðx�GÞ � KAðx�AÞ ¼ 185:25 which exactly compensates the in-crease in cost of (A) induced by deviating from x�A ¼ 200 to x�G ¼ 242:38. It is individually rational forthe buyer (A) to accept this offer: KAðx�GÞ � z = 10,000. Then, (P) realizes KP ðx�GÞ þ z ¼ 9390:71 and a bar-gaining surplus of V P ðx�GÞ ¼ KP ðx�AÞ � KP ðx�GÞ � z ¼ 359:29.

5. A bargaining game with asymmetric information

5.1. The bargaining situation with asymmetric information

In the preceding paragraph it is shown that a joint optimal policy x�G results from a bargaining processbetween the buyer and the supplier. This results however from the assumption that both players involvedhave complete information, and can design the side payment scheme accordingly. The results change, assoon as the players have private information about their cost functions. In the following it is analyzed whichbargaining solutions will be implemented if the players have incomplete information. Furthermore, we as-sume that the buyer has market power to impose his optimal policy x�A on the supplier. However, it willbe also assumed, that the supplier has asymmetric information about the buyer�s cost structure, i.e. (P) pos-sesses incomplete information about the type i 2 I of the buyer. The buyer (A) will only select a policy otherthan the individual optimal policy x�A, as was shown, if the increase in total relevant cost resulting from thispolicy is compensated by (P). Therefore the knowledge of the buyer�s cost function is of the highest interestfor the supplier (P). However, (A) has no incentive to report truthfully on his cost structure [26, p. 493].Moreover, the buyer has an incentive to overstate his own total relevant cost in order to obtain a side pay-ment z as high as possible. In order to bring (A) to a truthful report, (P) can employ screening: the supplier(P) determines several offers which are specifically designed for the different assumed types of (A). These


offers must be designed to be incentive-compatible, so that they are attractive for each assumed type of (A) tochoose the offer which is designed for that specific type of (A) [24, pp. 561–562]. By accepting one of theseoffers (A) indirectly reports truthfully about his cost situation. In the following, from the point of view of (P),it will be assumed that two alternative cost functions of (A) are possible: KA

1 ðxÞ and KA2 ðxÞ, i.e. from the view

of (P) two different types of buyer (A) exist: (A1) and (A2). Type of buyer (A1) is characterized by the inven-tory holding cost hA,1, the order processing cost B1, and the individual optimal order policy x�A;1. Accordingly,for the type of buyer (A2), we use hA,2, B2, and x�A;2. In the following section, for calculating the optimal set ofoffers, we take the reasonable assumption that the inequation hA,1 5 hA,2 will always hold.

5.2. The screening model

The following bargaining game has two stages: in the first stage, (P) makes an offer, and then, in the sec-ond stage, (A) can either accept or reject. After that, the game ends. Both players (A) and (P) are assumed tobe risk-neutral. From the supplier�s point of view two different types of buyer (A) exist: With a probability ofx1 > 0 the buyer is type (A1) and with a probability of x2 = 1 � x1 > 0 type (A2). With complete knowledgeabout the buyer�s cost structure, (P) would offer the joint optimal policy x�G and a side payment z which com-pensates for the increase of cost. Not being informed about the buyer�s true cost structure it will be optimalfor (P) to offer a set of offers (x1,z1,x2,z2), i.e. two separate joint policies x1 and x2 with corresponding sidepayments for the assumed buyer types (A1) and (A2). If (P) does not do this and only offers the joint optimalpolicy for type (A1) while (A) is the type of buyer (A2), it is possible that there is an incentive for type (A2) todisguise itself as type (A1) to realize a profit KA

1 ðx1Þ � KA2 ðx1Þ. To avoid that, (P) has to offer joint policies x1,

x2 and side payments z1, z2 which guarantee that each type of (A) accepts the adequate offer. The non-linearoptimization problem of (P) is to minimize the expected value of his total cost:

min E½KP ðx1; z1; x2; z2Þ� ¼ x1 � ðKP ðx1Þ þ z1Þ þ x2 � ðKP ðx2Þ þ z2Þ ð19Þs.t. KA

1 ðx1Þ � z1 6 KA1 ðx�A;1Þ; ð20Þ

KA2 ðx2Þ � z2 6 KA

2 ðx�A;2Þ; ð21ÞKA

1 ðx1Þ � z1 6 KA1 ðx2Þ � z2; ð22Þ

KA2 ðx2Þ � z2 6 KA

2 ðx1Þ � z1; ð23Þz1; z2 P 0; ð24Þx1; x2 > 0. ð25Þ

Conditions (20) and (21) ensure individual rationality: for both assumed types of (A), it must be moreattractive to accept the offer than the threat point. Conditions (22) and (23) ensure incentive-compatibility,so that it is attractive for each assumed type of (A) to choose the offer which is designed for that specifictype of (A). The cost functions KP(x1) and KA(x1) are strictly convex in x1 and the cost functions KP(x2) andKA(x2) are strictly convex in x2. Therefore, the Karush–Kuhn–Tucker-conditions (see Appendix A) are suf-ficient for an optimal solution. Using the Karush–Kuhn–Tucker-conditions the optimal set of offers can bedetermined. The analysis of the Karush–Kuhn–Tucker-conditions leads to six possible sets of offers (seeAppendix A). The set of offers which satisfies all Karush–Kuhn–Tucker-conditions is a feasible solutionsatisfying the constraints (20)–(24) and is also the optimal solution of the minimization problem (19).

Set of offers 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis
x1 ¼
2 � d � ðB1 þ RÞhA;1 þ d

p � hPwith z1 ¼ KA

1 ðx1Þ � KA1 ðx�A;1Þ; ð26Þ


x2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � d � ðB2 þ RÞhA;2 þ d

p � hP

swith z2 ¼ KA

2 ðx2Þ � KA2 ðx�A;2Þ: ð27Þ

Offered joint policies are equal to the joint optimal policy, which can be derived from the JELS model orthe presented bargaining game with complete information. The offered payments z1 and z2 compensate ex-actly the increases of cost, induced by the transition from x�A;1 to x1 or from x�A;2 to x2. The optimality test ofthe set of offers 1 is described in Appendix A. The set of offers 1 is not the optimal one if the buyer has anincentive to choose a contract which is not designed for his true cost structure. Thereby we can distinguishtwo cases: First, the true (or nearly true) cost function of (A) is KA

1 ðxÞ, but the buyer has an incentive toaccept the offer which is designed on the basis of the cost function KA

2 ðxÞ, i.e. (A1) has an incentive to imitate(A2). Second, the true (or nearly true) cost function of the buyer is KA

2 ðxÞ, but the buyer has an incentive toaccept the offer which is designed on the basis of the cost function KA

1 ðxÞ. In both cases the buyer woulddisguise itself as the other type in order to gain a profit of imitation KA

1 ðx�1Þ � KA1 ðx2Þ þ z2 or

KA2 ðx�2Þ � KA

2 ðx1Þ þ z1. To avoid that, the sets of offers number 2 to 6 are designed.

Set of offers 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis
x1 ¼
ðx1 � Rþ x1 � B2 þ B1 � B2Þ � dx1

2� dp � hP þ

x1

2� 1

2

� �� hA;2 þ 1

2� hA;1

with z1 ¼ KA1 ðx1Þ � KA

1 ðx�A;1Þ; ð28Þ


p � hP

swith z2 ¼ KA

2 ðx2Þ � KA2 ðx1Þ þ KA


Imitation of (A1) by (A2) will be avoided by offering an—from the point of view of KA2 ðxÞ—unattractive

policy x1, which deviates from the joint optimal policy determined by the JELS model. Additionally, the of-fered payment z2 consists of the compensation for the increase of cost, induced by the transition from x�A;2 tox2 and a bonus for not imitating (A1). The optimality test of the set of offers 2 is described in Appendix A.

x1 ¼




x2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2 � Rþ x2 � B1 þ B2 � B1Þ � d

x2

2� dp � hP þ

x2

2� 1

2

� �� hA;1 þ 1

2� hA;2

swith z2 ¼ KA


The set of offers 3 is designed for the case that the buyer has an incentive to accept the offer which isdesigned on the basis of the cost function KA

2 ðxÞ while the buyer�s true cost function is KA1 ðxÞ. Set of offers

3 avoids the imitation of (A2) by (A1) on the one hand by offering an—from the point of view of KA1 ðxÞ—

unattractive policy x2. On the other hand, the side payment z1 exceeds the compensating of the increases ofcost at the buyer�s side, induced by deviating from the individual optimal policy x�1 to x1. The optimality testof the set of offers 3 is described in Appendix A.

x1 ¼




x21;2 ¼�KA

2 ðx�A;2Þ �KA1 ðx�A;1Þ

hA;1 � hA;2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKA


hA;1 � hA;2

!2

� 2 � ðB1 �B2Þ � dhA;1 � hA;2

vuut with z2 ¼ KA2 ðx2Þ �KA

2 ðx�A;2Þ:

ð33Þ
In this case the imitation of (A2) by (A1) is prevented only by a unattractive policy x2. For x21 ; x22 > 0
result two sets of offers 4, which have to be proofed for optimality. The optimality test of the set of offers4 is described in Appendix A. Regarding equation (33), note that we have taken the reasonable assumptionfor the screening model that is, to presume that the inequation hA,1 5 hA,2 will always hold.

Set of offers 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !v
x11;2 ¼�
KA1 ðx�A;1Þ �KA

2 ðx�A;2ÞhA;2 � hA;1

�KA


hA;2 � hA;1

2

� 2 � ðB2 �B1Þ � dhA;2 � hA;1

uut with z1 ¼ KA1 ðx1Þ �KA

1 ðx�A;1Þ;

ð34Þ


p � hP

swith z2 ¼ KA


In this case the imitation of (A1) by (A2) is prevented only by an—from the point of view of KA2 ðxÞ—unat-

tractive changed quantity x1. For x11 , x12 > 0 two sets of offers 5 have to be proofed for optimality (seeAppendix A).

Set of offers 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv
x11;2 ¼�
KA1 ðx�A;1Þ �KA

2 ðx�A;2ÞhA;2 � hA;1

�KA


hA;2 � hA;1

!2

� 2 � ðB2 �B1Þ � dhA;2 � hA;1

uut with z1 ¼ KA1 ðx1Þ �KA

1 ðx�A;1Þ;

ð36Þ

x21;2 ¼�KA


hA;1 � hA;2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKA


hA;1 � hA;2

!2

� 2 � ðB1 �B2Þ � dhA;1 � hA;2

vuut with z2 ¼ KA2 ðx2Þ �KA

2 ðx�A;2Þ:

ð37Þ
Set of offers 6 is designed for the special case that the buyer has an incentive to accept the offer designed
on the basis of KA2 ðxÞ when his true cost function is KA

1 ðxÞ and, simultaneously, the buyer has an incentive toaccept the offer designed on the basis of KA

1 ðxÞ if his true cost function is KA2 ðxÞ. The supplier (P) can deter-

mine an optimal set of offers for the assumed cost functions KA1 ðxÞ and KA

2 ðxÞ of (A): the set of offers whichsatisfies the Karush–Kuhn–Tucker-conditions is the optimal set of offers. If the buyer behaves rationally, itis attractive for each type of (A) to choose the offer which is designed for that specific type of (A). The ex-pected value of the supplier�s bargaining surplus is given by the difference between the expected value of histotal relevant cost with and without bargaining.

5.3. A numerical example (2)

Consider the following information for both types of a buyer (A) and a supplier (P) in Table 3, withx1 = x2 = 0.5, d = 10,000 and p = 15,000. The buyer has the market power to implement his optimal pol-

Table 3Example 2

Supplier (P) Type of buyer (A1) Type of buyer (A2)

R = 135, hP = 45 B1 = 100, hA,1 = 50 B2 = 50, hA,2 = 100x�P ¼ 300 x�A;1 ¼ 200 x�A;2 ¼ 100KP ðx�P Þ ¼ 9000 KA

1 ðx�A;1Þ = 10,000 KA2 ðx�A;2Þ = 10,000

Fig. 3. Joint optimal policies for different assumed types of (A).


icy. Without bargaining, the total relevant cost of the supplier (P) are KP ðx�A;1Þ ¼ 9750 if the buyer is type(A1) and KP ðx�A;2Þ = 15,000 if the buyer is type (A2). The expected value of the supplier�s total cost withoutbargaining amounts to E[KP(x1 = 200,z1 = 0,x2 = 100,z2 = 0)] = 12,375. In this example, the set of offers 4is optimal for the supplier: (x1 = 242.38,z1 = 185.25,x2 = 141.42,z2 = 606.57). The expected value of sup-plier�s total cost amounts to: E[KP(x1,z1,x2,z2)] = 10,832.31, i.e. the supplier (P) realizes a bargaining sur-plus of E[VP(x1,z1,x2, z2)] = 1542.69.

Fig. 3 shows the cost functions for type (A1), type (A2) and (P), the individual optimal policies x�A;1, x�A;2,

x�P and the joint policies x1, x2 in this example.In summary it follows that the screening between the possible buyer types (A2) and (A1) succeeds, if the

supplier can estimate

(1) the possible cost structure of the buyer and(2) the probabilities x1 and x2 sufficiently exact.

Regarding the influence of the probabilities x1 and x2 on the optimal set of offers we know that the set ofoffers 2 and 3 are directly influenced by one of these values. For example, set of offers 2—as well as set ofoffers 5—is designed for the case that (A2) has an incentive to imitate (A1). Imitation of (A1) by (A2) will beavoided only by offering an unattractive policy x1 (set of offers 5) or—in addition to that—by a side pay-ment z2 which contains a bonus for not imitating (A1) (set of offers 2). If the supplier assumes a relativelylow probability x2 for (A2) and therefore a high probability x1 for (A1) it is advantageous for the supplierto offer set of offers 2, because x1 in this set of offer is closer to the joint optimal policy as x1 in the set ofoffers 5. Knowing this circumstance the supplier can experiment with different combinations of theprobabilities.


6. Conclusion

In this paper we propose a joint order quantity and lot size in a single buyer-single supplier system as abargaining solution on the basis of Banerjee�s JELS model. It is well known that the joint optimal policywill always result in savings in total system cost. However, should one of the parties involved, the buyer orthe supplier, have the more powerful position to enforce its individual optimal policy no incentive exists forthis party to choose a cooperative policy. We consider a typical case: The buyer dominates the supplier. Ifthe buyer has the market power to impose his individual optimal policy on the supplier then no incentiveexists for the buyer to deviate from his individual optimal policy. A joint optimal policy can therefore onlyresult from a bargaining process. The research presented in this paper offers a bargaining model with asym-metric information on the buyer�s cost structure assuming that the buyer has the market power to imple-ment his optimal policy. It was shown that the optimal set of offers will be determined if the supplier canestimate the possible cost structure of the buyer as well as the associated probabilities sufficiently well. Thesupplier can employ the screening model developed above as a quantitative tool supporting contractnegotiations. Experiments with different combinations of assumed cost structures and associated probabil-ities enables the supplier to derive a ‘‘good’’ set of offers. Thus, we conclude that complete information ofboth players, the buyer and the supplier, does not represent a necessary prerequisite for a bargainingsolution.

Appendix A

The optimization problem with constraints (19)–(24) can be transformed to the following unconstrainedminimization problem:

Lðx1; z1; x2; z2Þ ¼ x1 � ðKP ðx1Þ þ z1Þ þ x2 � ðKP ðx2Þ þ z2Þ � k1 � ðKA1 ðx�A;1Þ � KA

1 ðx1Þ þ z1Þ � k2 � ðKA2 ðx�A;2Þ

� KA2 ðx2Þ þ z2Þ � l1 � ðKA

1 ðx2Þ � z2 � KA1 ðx1Þ þ z1Þ � l2 � ðKA

2 ðx1Þ � z1 � KA2 ðx2Þ þ z2Þ:

ðA:1Þ

With x1,x2 > 0 the Karush–Kuhn–Tucker- (KKT-) conditions can be deduced as follows:

x1 �oKP ðx1Þox1

þ k1 �oKA

1 ðx1Þox1

þ l1 �oKA

1 ðx1Þox1

� l2 �oKA

2 ðx1Þox1

¼ 0; ðA:2Þ

x2 �oKP ðx2Þox2

þ k2 �oKA

2 ðx2Þox2

� l1 �oKA

1 ðx2Þox2

þ l2 �oKA

2 ðx2Þox2

¼ 0; ðA:3Þ

x1 � k1 � l1 þ l2 P 0 and z1 � ðx1 � k1 � l1 þ l2Þ ¼ 0; ðA:4Þ

x2 � k2 þ l1 � l2 P 0 and z2 � ðx2 � k2 þ l1 � l2Þ ¼ 0; ðA:5Þ

KA1 ðx�A;1Þ � KA

1 ðx1Þ þ z1 P 0 and k1 � ðKA1 ðx�A;1Þ � KA

1 ðx1Þ þ z1Þ ¼ 0; ðA:6Þ

KA2 ðx�A;2Þ � KA

2 ðx2Þ þ z2 P 0 and k2 � ðKA2 ðx�A;2Þ � KA

2 ðx2Þ þ z2Þ ¼ 0; ðA:7Þ

KA1 ðx2Þ � z2 � KA

1 ðx1Þ þ z1 P 0 and l1 � ðKA1 ðx2Þ � z2 � KA

1 ðx1Þ þ z1Þ ¼ 0; ðA:8Þ

KA2 ðx1Þ � z1 � KA

2 ðx2Þ þ z2 P 0 and l2 � ðKA2 ðx1Þ � z1 � KA

2 ðx2Þ þ z2Þ ¼ 0: ðA:9Þ


The values of all the Lagrange�s multipliers k1, k2, l1 and l2 can either be zero or greater than zero.Therefore, 16 possible combinations exist for the Lagrange�s multipliers k1, k2, l1 and l2 (see Table 4).

In case of x�A 6¼ x�P the side payments z1 and z2 have to be greater than zero. With z1,z2 > 0 and the KKT-conditions (A.4) and (A.5) follows:

TablePossib

Case

k1k2l1l2

TableReleva

Case

12345678

x1 � k1 � l1 þ l2 ¼ 0 and x2 � k2 þ l1 � l2 ¼ 0: ðA:10Þ
It applies x1,x2 > 0, with x2 = 1 � x1 > 0. Conditions (A.10) show that all combinations for the La-
grange�s multipliers with k1 = l1 = 0 and k2 = l2 = 0 are infeasible regarding the KKT-conditions. Fur-thermore, from (A.10) results:

x2 � k2 þ x1 � k1 ¼ 0 () x1 þ x2 ¼ k1 þ k2 () 1 ¼ k1 þ k2: ðA:11Þ
Conditions (A.11) shown that all combinations with k1 = k2 = 0 are infeasible. This analysis leads to
eight feasible combinations of the Lagrange�s multipliers (see Table 5). For these eight feasible combina-tions six possible sets of offers can be derived for the supplier.

A.1. Determination of the set of offers 1

For the first case (k1 > 0, k2 > 0, l1 = 0, l2 = 0) the set of offers can be derived as follows. From condi-tions (A.10) and l1 = l2 = 0 follows k1 = x1 and k2 = x2. With x1,x2 > 0, l1 = l2 = 0, k1 = x1 and k2 = x2

the KKT-conditions (A.2) and (A.3) reduce to:

x1 �oKP ðx1Þox1

þ x1 �oKA

1 ðx1Þox1

¼ 0 () oKP ðx1Þox1

¼ � oKA1 ðx1Þox1

; ðA:12Þ

x2 �oKP ðx2Þox2

þ x2 �oKA

2 ðx2Þox2



: ðA:13Þ

4le combinations for the Lagrange�s multipliers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

=0 >0 =0 =0 =0 >0 >0 >0 =0 =0 =0 >0 =0 >0 >0 >0=0 =0 >0 =0 =0 >0 =0 =0 >0 >0 =0 >0 >0 =0 >0 >0=0 =0 =0 >0 =0 =0 >0 =0 >0 =0 >0 >0 >0 >0 =0 >0=0 =0 =0 =0 >0 =0 =0 >0 =0 >0 >0 =0 >0 >0 >0 >0

5nt combinations of the Lagrange�s multipliers

k1 k2 l1 l2

k1 > 0 k2 > 0 l1 = 0 l2 = 0k1 > 0 k2 = 0 l1 = 0 l2 > 0k1 = 0 k2 > 0 l1 > 0 l2 = 0k1 > 0 k2 > 0 l1 > 0 l2 = 0k1 = 0 k2 > 0 l1 > 0 l2 > 0k1 > 0 k2 = 0 l1 > 0 l2 > 0k1 > 0 k2 > 0 l1 = 0 l2 > 0k1 > 0 k2 > 0 l1 > 0 l2 > 0


Conditions (A.12) and (A.13) represent the first-order conditions for the joint economic order quantityand lot size as presented in Section 3.2. The offered joint policies are given by (26) and (27). They are equalto the joint optimal policy, which can be derived from the JELS model. With k1 > 0, k2 > 0 and the KKT-conditions (A.6) and (A.7), the offered side-payments can be derived: z1 ¼ KA

1 ðx1Þ � KA1 ðx�A;1Þ,

z2 ¼ KA2 ðx2Þ � KA

2 ðx�A;2Þ. The set of offers 1 (see (26) and (27) in Section 5.2) satisfies the KKT-conditions(A.2)–(A.7). The set of offers 1 is feasible and optimal if the KKT-conditions (A.8) and (A.9) are satisfied.Considering the side payments according to (26) and (27) the optimality criteria for set of offers 1 are givenby:

KA1 ðx2Þ � KA

2 ðx2Þ þ KA2 ðx�2Þ � KA

1 ðx�1Þ P 0; ðA:14Þ

KA2 ðx1Þ � KA

1 ðx1Þ þ KA1 ðx�1Þ � KA

2 ðx�2Þ P 0: ðA:15Þ

A.2. Determination of the sets of offers 2 and 3

For the second case (k1 > 0, k2 = 0, l1 = 0, l2 > 0) the set of offers can be derived as follows. From con-ditions (A.10) and k2 = l1 = 0 follows l2 = k1 � x1, l2 = x2 and x2 = k1 � x1. With x1 + x2 = 1 it followsthat k1 = 1 and l2 = 1 � x1. For x1,x2 > 0, k2 = l1 = 0, k1 = 1, l2 = x2 and l2 = 1 � x1 the KKT-condi-tions (A.2) and (A.3) reduce to:

x1 �oKP ðx1Þox1

þ oKA1 ðx1Þox1

� ð1� x1Þ �oKA

2 ðx1Þox1

¼ 0; ðA:16Þ

x2 �oKP ðx2Þox2

þ x2 �oKA

2 ðx2Þox2



: ðA:17Þ

Condition (A.17) represents the first-order condition for the joint economic order quantity and lot size.The offered joint policy x2 and the joint optimal policy are identical. With (A.16),

oKP ðx1Þox1

¼ � R � dðx1Þ2

þ 1

2� dp� hP ;

oKA1 ðx1Þox1

¼ �B1 � dðx1Þ2

þ 1

2� hA;1;

oKA2 ðx1Þox1

¼ �B2 � dðx1Þ2

þ 1

2� hA;2

follows x1 according to (28). The side payments z1 and z2 in accordance with (28) and (29) can be deter-mined using the KKT-conditions (A.6) and (A.9). The set of offers 2 satisfies the KKT-conditions(A.2)–(A.6) as well as (A.9). The set of offers 2 is feasible and optimal if the KKT-conditions (A.7) and(A.8) are also satisfied. Therefore, with (A.7) and (A.8) the optimality criteria for the set of offers 2 are givenby:

KA1 ðx1Þ � KA

2 ðx1Þ � KA1 ðx�1Þ þ KA

2 ðx�2Þ P 0; ðA:18Þ

KA1 ðx2Þ � KA

2 ðx2Þ þ KA2 ðx1Þ � KA

1 ðx1Þ P 0: ðA:19Þ

The set of offers for the third case (k1 = 0, k2 > 0, l1 > 0, l2 = 0), i.e. set of offers 3, can be derived in asimilar manner. The set of offers 3 is feasible and optimal if the KKT-conditions (A.6) and (A.9) aresatisfied.

Now, we will analyze case 4, case 7 and case 8 of the feasible combinations for the Lagrange�s multipliers(see Table 5). In the 4th, 7th and 8th case k1 > 0, k2 > 0 is given. Therefore, the side payments z1 and z2 forthese cases can be determined using the KKT-conditions (A.6) and (A.7), i.e. z1 and z2 are exactly at thepoints where the conditions (20) and (21) are satisfied as equations: z1 ¼ KA

1 ðx1Þ � KA1 ðx�A;1Þ, z2 ¼ KA

2 ðx2Þ�KA

2 ðx�A;2Þ. The optimization problem (19)–(25) reduces to:


min E½KP ðx1; z1; x2; z2Þ� ¼ x1 � KP ðx1Þ þ KA1 ðx1Þ � KA

1 ðx�A;1Þ� �

þ x2 � KP ðx2Þ þ KA2 ðx2Þ � KA

2 ðx�A;2Þ� �

ðA:20Þs.t. KA


2 ðx�A;2Þ � KA1 ðx�A;1Þ P 0; ðA:21Þ

KA2 ðx1Þ � KA

1 ðx1Þ þ KA1 ðx�A;1Þ � KA

2 ðx�A;2Þ P 0. ðA:22Þ

For the non-linear minimization problem (A.20)–(A.22) the KKT-conditions are given by:

x1 �oKP ðx1Þox1

þ x1 �oKA

1 ðx1Þox1

� l2 �oKA

2 ðx1Þox1

þ l2 �oKA

1 ðx1Þox1

¼ 0; ðA:23Þ

x2 �oKP ðx2Þox2

þ x2 �oKA

2 ðx2Þox2

� l1 �oKA

1 ðx2Þox2

þ l1 �oKA

2 ðx2Þox2

¼ 0; ðA:24Þ

KA1 ðx2Þ � KA


1 ðx�A;1Þ P 0; ðA:25Þ

l1 � KA1 ðx2Þ � KA


1 ðx�A;1Þ� �

¼ 0; ðA:26Þ

KA2 ðx1Þ � KA


2 ðx�A;2Þ P 0; ðA:27Þ

l2 � KA2 ðx1Þ � KA


2 ðx�A;2Þ� �

¼ 0: ðA:28Þ

A.3. Determination of the sets of offers 4, 5 and 6

In the 4th case (k1 > 0, k2 > 0, l1 > 0, l2 = 0) the KKT-conditions (A.23)–(A.28) reduce to:

x1 �oKP ðx1Þox1

þ x1 �oKA

1 ðx1Þox1

¼ 0; ðA:29Þ

x2 �oKP ðx2Þox2

þ x2 �oKA

2 ðx2Þox2

� l1 �oKA

1 ðx2Þox2

þ l1 �oKA

2 ðx2Þox2

¼ 0; ðA:30Þ

KA1 ðx2Þ � KA


1 ðx�A;1Þ ¼ 0; ðA:31Þ

KA2 ðx1Þ � KA


2 ðx�A;2Þ P 0: ðA:32Þ

The set of offers 4 (see (32) and (33)) can be determined using (A.29) and (A.31). The set of offers 4 isfeasible and optimal, if conditions (A.32) and (A.30) and l1 > 0 are satisfied. With (A.30) and l1 > 0 thefollowing optimality criterion results:

x2 � 12� d

p � hP þ hA;2� �

� ðRþ B2Þ � dx22

� �12� ðhA;1 � hA;2Þ þ ðB2 � B1Þ � d

x22

> 0: ðA:33Þ

The set of offers for the 7th case (k1 > 0, k2 > 0, l1 = 0, l2 > 0) can be derived in a similar manner to theset of offers 4. The resulting set of offers 5 (see (34) and (35)) is the optimal set of offers if the KKT-con-ditions (A.23) and (A.25) and l2 > 0 are satisfied. For (A.23) and l2 > 0 the following condition must besatisfied:


x1 � 12� d

p � hP þ hA;1� �

� ðRþ B1Þ � dx21

� �12� ðhA;2 � hA;1Þ þ ðB1 � B2Þ � d

x21

> 0: ðA:34Þ

For the 8th case (k1 > 0, k2 > 0, l1 > 0, l2 > 0) set of offers 6 follows (see (36) and (37)). The set of offers 6is feasible and optimal, if the conditions (A.33) and (A.34) are satisfied. Finally, it can be shown that in the6th case the resulting set of offers is identical to the set of offers 2 and in the 5th case the set of offers isidentical to the set of offers 3.

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