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

Production, Manufacturing and Logistics

EUROPEANJOURNAL

OF OPERATION

ALwww.elsevier.com/locate/ejor

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 cost of a buyer and a supplier by using a joint optimal order and production policy. The implementation of a jointly optimal policy requires coordination and cooperation. Should the buyer have the market power to implement his own optimal policy as that one to be used in the exchange process no incentive exists for him to choose a joint optimal policy. A joint policy can therefore only be the result of a bargaining process between the parties involved. The supplier may make 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 the buyer’s cost structure. Clearly, this assumption will seldom be fulfilled in practice. The research presented in this paper provides a bargaining model with asymmetric information about the buyer’s cost structure assuming that the buyer has the power to impose its individual optimal policy.© 2004 Elsevier B.V. All rights reserved.

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

1. Introduction

that connect the facilities ([10, p. 15], [37, p. 5] ) . One of the major tasks of supply chain management is to coordinate the processes in the supply chain in such a way, that a given set of objectives is achieved [39, pp. 8-9] . Most

Tel.: +49 69 7982 8201; fax: +49 69 7982 8414.E-mail address: esucky@wiwi.uni-frankfurt.de

0377-2217/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.08.039

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 costs while satisfying a predetermined service level [30, p. 835]. The complexity in coordinating the processes in supply chains is introduced by the organizational structure within the network. In general, a supply chain is composed of independent firms with individual preferences [7, p. 113] . Therefore, in contrast to the management of multi-echelon systems, that coordinates inventory, production and distribution decisions at multiple locations of one firm, supply chain management involves coordination of such decisions among multiple and independent firms [27, pp. 794-795] . Bhatnagar et al. [5] identify the issue of coordination—at the most general level, which they call general coordination—in integrating decisions of different functions. Within this problem of functional coordination Bhatnagar et al. [5] as well as Thomas and Griffin [41] distinguish three categories: (1) supply-production coordination, (2) production-distribution coordination, 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-directional. Both, in the scientific discussion and in practice considerable attention is paid to the importance of 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-supplier coordination have focused on determining the order and production policy which is jointly optimal for both. Using such a joint optimal order and production policy—as opposed to independently derived policies—leads to a significant total cost reduction. However, there is an additional set of problems involved in implementing joint policies.

supplier, there exists the opportunity for the supplier to offer a price discount or a side payment to the buyer. One critical assumption made throughout the literature dealing with incentive schemes to influence buyer’s ordering policy is that the supplier has complete knowledge about the buyer’s cost structure. The research presented in this paper offers a bargaining model with asymmetric information about buyer’s cost structure assuming that the buyer has the market power to impose his individual optimal policy. The remainder of the paper is organized as follows. Section 2 presents the review of the literature dealing with the problem of buyer-supplier coordination. In Section 3—on the basis of the well-known joint economic lot size model suggested by Banerjee [4]—we will show that from an individual point of view neither the buyer nor the supplier has an

2. Literature review

There is a large and growing stream of literature on buyer-supplier coordination. Most of the literature on buyer-supplier coordination focuses either on deriving a coordinated order and production policy in the context of a given supply contract, or on determination optimal contract parameters given the functional form of that contract. In the last category,

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 comprehensive literature reviews of designing supply contracts. Corbett and Tang [12] study the design of supply contracts in the presence of asymmetric information. In this paper, however, we will focus on the first category: Coordinating order and production policies under a given supply contract. From the contractual perspective, 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 coordination of the buyer’s order policy and the supplier’s production policy does not affect the total volume provided to the buyer.

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

edy, Goyal [15,18] , Banerjee [4] , and Landeros and Lyth [29], present some models for coordinating buyer-supplier systems with centralized control. The researchers suggest approaches for determining an integrated order and delivery policy, minimizing the joint total relevant cost incurred by both parties. It is shown that in integrated models one party’s gain resulting from the integrated policy in comparison of the other party’s optimal policy exceeds the other party’s loss. Thus, the net benefit can be shared by both parties (the supplier and the buyer) in some equitable fashion [21, p. 262] . While all models use the 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 introduced an integrated policy for a single supplier-single buyer system. He assumes that the buyer’s demand is uniform with respect to time, there is no lead time for the supplier and the buyer, and the supplier and the buyer cooperate with the objective of minimizing the joint total relevant cost. The joint relevant costs are the costs of holding and ordering at the buyer’s side and the set-up cost at the supplier’s side. However, the effect of production rate on calculating average inventory is ignored, by assuming an infinite production rate. Banerjee [4] generalizes Goyal’s model by integrating a finite production rate. Therefore, the holding cost of the supplier will be considered by determining the integrated order and delivery policy. Assuming a production rate greater than the demand rate, the supplier makes the production 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

E. Sucky / European Journal of Operational Research 171 (2006) 516-535 519

total relevant cost is further reduced. Landeros and Lyth note that Goyal’s model overlooked the processing cost per shipment [29, p. 154]. Therefore, Landeros and Lyth [29] consider fixed delivery cost associated 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 the buyer by the fixed transportation cost per shipment which does not alter the solution methodology. However, all models mentioned above assume that the whole production batch must be finished before any shipments from the batch can take place.

Another class of models for coordinating buyer-supplier systems with centralized control allows the shipment of sub-batches to the next stage before the whole lot is finished at the preceding one. Based on an idea of Szendrovits [40] in the context of a serial multi-stage production system, this control policy was first applied in the area of supplier-buyer coordination by Joglekar [25] . Agrawal and Raju [1] consider this 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 allowance 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 production and delivery policy involves successive shipments within a production batch so that the size of the sub-batches increases according to a geometric series. Chatterjee and Ravi [9] present an equivalent model considering fixed delivery costs associated with each shipment on the supplier’s side contrary to fixed transportation 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 with unequal sized sub-batches dominates the other. However, Hill [23] derives the structure of the true optimal solution for the sizes of the sub-batches for a single supplier-single buyer system acting in a static deterministic environment. For the determination of the optimal sizes of the sub-batches for the buyer-supplier system 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 subbatches consider the power structure in buyer-supplier relationships. Three typical cases can be applicable:(I) The buyer dominates the supplier. (2) The supplier dominates the buyer. (3) The buyer and the supplier have equal power. In this paper we analyze the first case. If the buyer is strong, the supplier will be forced to accept the buyer’s individual optimal order policy. In this situation, the supplier may make some sort of concession 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, including Monahan [34,35], Banerjee [3] , Goyal [17] , Lee and Rosenblatt [31] and Joglekar [25] . The authors compute a price discount which compensates the loss of the buyer, if he changes his order policy to a cooperative policy. Miller and Kelle [33] and Kelle et al. [28] also suggest some quantitative models to support negotiations 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

520 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

3. Ordering and production policies in the JELS model

3.1. Individual optimal policies

time horizon over which the product is ordered by the buyer and supplied by the supplier is infinite. The supplier fabricates the regarded product at a finite production rate. The transportation time between the supplier and the buyer is assumed to be zero. The optimality criterion for the buyer and the supplier is the minimization of their own total relevant cost per period. The buyer’s total relevant cost per period consists of the order processing cost and the inventory holding cost. The supplier’s cost function includes the production set-up cost and the inventory holding cost. Both parties determine the economic order quantity or lot size using the well known economic order quantity (EOQ) formula or economic lot size (ELS) formula [4, pp. 293-294] .

3.1.1. Individual optimal ordering policyThe objective for the buyer is to determine the optimal ordering policy

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

KA { x a )= B • — + f • Ha [$/period]./2 • B • d

KA(XA) = \ J2 • B • d • h A . (2)

3.1.2. Individual optimal production policyThe objective for the supplier is to find the optimal production policy

minimizing his total relevant cost per period. We refer to him as party (P) and use subscript ‘‘P’’ to designate the set of parameters or decisions of the supplier. The production rate amounts to p [unit/period], whereas the relation p > d will always hold. The supplier follows a lot-for-lot policy, i.e. (P) produces the ordered quantity of the product and, on completion of the batch, ships the entire lot to (A). The average inventory is given by X- • p [unit] [4, p. 294] . The inventory holding cost of (P ) is h P [$/(unit and period)] and the production set-up cost is R [$]. The total relevant cost per period of (P ) is given by

Kp (xp )=R • — + “ •d • hP [$/period].

(3)XP 2 p

The supplier’s economic lot size (ELS), xp [unit], and his minimum total

E. Sucky / European Journal of Operational Research 171 (2006) 516-535 521

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 order quantity corresponds to the quantity delivered. Since we have excluded shortages by definition, an equal lot size x G = x A = x P 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 x G = x A = x P minimizing the joint total relevant cost for both the buyer and the supplier. The joint total relevant cost of (A) and (P) for a joint lot size x G = x A = x P can be derived from Eqs. (1) and (3) as follows [4, p. 299] :

KG(xG)=B • — + (( • hA + R • — + Tr • - • hp [$/period]. (5)xG 2 xG 2 p

The joint optimal lot size, XG [unit], and the minimum joint total relevant cost per period, KG(x*G) [$/period], are given by

xG =]j2 h- +^ .;R); KG(xG) 2 • - -(B + R) • ^hA + P ■ hfj. (6)

3.3. Comparison of the individual and integrated policies

relation of the holding cost of (P) to the holdings cost of (A) for any joint lot size x G can be expressed with the parameters a and b [4, p. 295]:

a =R •—

___XG .B-xf

R and b B f • P • hP

d • hP

f • hA p • hAThe following relations can be derived:

* r b * * *XA = a •Xp; xp = b • XA ; X -A=M

(7)

(8)X

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

522 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

( A ) adapts to (P) (xA = xp = x G) or

A uniform lot size x *A = xP = xG only applies for A = ft. For A 5 B follows x *A = xP, and the joint optimallot size xG is located within the interval between the individual optimal solutions of (A) and (P), i.e.xG 2 ]x_A, xP [ if xA > xP and xG 2 ]xP, x A [ if xA > xP. In case of A 5 B (xA = xP) a solution with an uniformlot size x G = x A = xP only exists if (P) adapts to (A) (xP = xA = xG)if (A) and (P) select another joint lot size, for example xG (xA = xP = x(

functions of (A ) and (P ), the function of the joint total relevant cost and the individual and integrated pol-icies in case of xA < xP.

Deviations from the individual optimal policy always lead to an increase of relevant cost. From the sup-plier’s point of view KP(xG) > KP(xP) applies for any joint policy xG = xP. In the same way,KA (xG) > KA (x_A) applies for any joint policy xG = xA from the buyer’s perspective. Using the relations (8)

2 • + ^relations can be derived:2 • rb+ra +

/T+P1 + a ,

2 • (rb+f ) > 2 • (Sr+b+sm

> 1,

> 1.

(9)

(10)

O R D E R Q U A N T I T Y / L O T

T O T A L R E L E V A N T C O S T F O R ( A)

T O T A L R E L E V A N T C O S T F O R ( P)

*P

II

II

£II

K A ( X A ) = K A ( X A ) = P 2 ' B ' d ' hA

K P ( X P ) = K P ( X A ) = 1 ' ( qA + qA ' K P

%a — X P — ' X _ A , Ka(xa) = KA(xp) = 1 ' ( 0 + qfj ' K P ( X P ) = K P ( X P ) = D '

2 ' R' H PV PX A = X G = Y ' R P F '

X A > K A ( X A ) = K A ( X G ) = 1 '

qI P F + ' K A ( X A )

K P ( X P ) = K P ( X * G ) = 1 ' ( q^ + q^ ) ' K P

( X * )

X P = X G = ' X *

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

The analysis of the individual economic consequences of alternative joint policies shows that neither the buyer nor the supplier have an incentive to choose a cooperative policy deviating from their individual optimal policies. The buyer and the supplier are separate and independent firms. In this context, it is reasonable to assume that each party will act in their own best interests. If (A ) and (P ) behave individually and rationally, they want to select their individual optimal policies x “ A and xp in any case. Therefore, for a 5 b a joint policy x G = x A = x P 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 xA and x * p . If (A) or (P ) has market power to impose its individual optimal policy on (P ) or (A ), respectively, then no incentive exists for either (A) or (P) to deviate from their individual optimal policy xA or xp. The weaker player must deviate from his individual optimal policy and adapt to the stronger player’s policy. However if negotiations are possible, the weaker player may try to persuade the stronger player to select a policy other than its individual optimal policy by a side payment. A side payment is defined as an additional monetary transfer between supplier (buyer) and buyer (supplier) that is used as an incentive for deviating from the individual optimal policy [36, p. 22] . It is to be examined, which joint policy x G = x A = x P is implemented if (A ) and (P ) negotiate on the joint order quantity and lot size. The bargaining game (N , U , u *) is described by the set of players N = {(A),(P)}; the set U of feasible payoff combinations (in this case cost combinations) 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 ' 2 U with the fol-lowing property exists: u 6 u* for all players i 2 N ; u < u* for at least one

min K p (x G, z) = K p (x G) + z (11)

s.t. KA(XG) — z 6 K A (xA), (12)

z P 0; (13)

xG > 0. (14)524 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

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

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

Bargaining solution with complete information

In this section will be analyzed, which joint policy x G 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 rational, no incentive exists for deviating from this individual optimal policy. Without negotiations xA will be

implemented, with KA (xA) = p2 • B • d • h A and K p (xA) = 2 • ^[ + [^ ' KP

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 each given policy x G >0 the transfer payment z should be selected as small as possible while satisfying restriction (12). Therefore, for each given x G 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 K A ( x G ) — KA(xA) and minimizing the resulting cost expression, the optimal offered policy and the corresponding side payment can easily calculated. The supplier offers the joint optimal policy (6) and a

2 • d • (B( B + R) z = KA(xG) — KA(xA) =1 1 + a I 1 +

y2 ^ 1 + f + 1 + a— 1 2 • B • d • hA.

(15)

x

If (A) behaves individually rationally, he accepts this offer, since condition (12) is fulfilled. (A) realizes his minimal total cost KA(xG) — z = KA(xG) — KA(xG) + KA(xA) = KA(xA). The offer is attractive for (P) only if the total relevant cost plus the side payment are lower than with implementation of xA, i.e. the following must apply:KP(xG) + KA(xG) — KA(xA) < Kp(xA).

Condition (16) can be transformed:

(16)

2 - (S++S+lKP W)+(2 - (S+b+rm•KA(xA) < 2 • (r+ra)KP«).

(17)

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 Vp (xG) at a value of:

vp (xG)KP (xA) — Kp (xG) — z = Kp (xA) — Kp (xG)KA(xG)+KA(xA) > 0. (18)

a = 1 . 3 5 , B = 0 . 6

T O T A L R E L E V A N T C O S T F O R ( A )

( D = 1 0 , 0 0 0 , B = 1 0 0 ,

T O T A L R E L E V A N T C O S T F O R ( P)

( P = 1 5 , 0 0 0 , R = 1 3 5 ,

J O I N T T O T A L

R E L E V A N T C O S T

F O R ( A) A N D ( P)X A = 2 0 0 K A ( X A ) = 1 0 , 0 0 0 K P ( X A ) = 9 7 5 0 K G ( X A ) = 1 9 , 7 5 0X P = 3 0 0 K A ( X P ) = 1 0 , 8 3 3 . 3 3 K P ( X P ) = 9 0 0 0 K G ( X P ) =

1 9 , 8 3 3 . 3 3X G = 2 4 2 . 3 8 K A ( X G ) = 1 0 , 1 8 5 . 2 5 K P ( X G ) = 9 2 0 5 . 4 6 K G ( X G ) = 1 9 , 3 9 0 . 7 2

( P ) behaves like a master planner and selects the joint optimal order quantity and lot size x “ G . With complete information negotiations based on the assumption made, lead to the joint optimal order quantity and lot size. It can be shown that this result is independent of who possesses market power to implement its individual 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, single source purchases: if the two parties negotiate cooperatively, with complete information, a side payment will lead to a contract from which both parties will benefit [36, p. 25]. With respect to the work of Banerjee [3], it has 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’s holding cost h A does not depend on the unit price which the buyer has to pay to the supplier.

4.2. A numerical example (1)

Consider the information for the buyer (A) and the supplier (P) given in Table 2. The buyer (A) selects his optimal policy x * A = 200 and realizes his minimum total relevant cost KA(xA) = 10,000. Without bargaining, the

5. A bargaining game with asymmetric information

5.1. The bargaining situation with asymmetric information

have complete information, and can design the side payment scheme accordingly. The results change, as soon as the players have private information about their cost functions. In the following it is analyzed which bargaining solutions will be implemented if the players have incomplete information. Furthermore, we assume that the buyer has market power to impose his optimal policy xA on the supplier. However, it will be also assumed, that the supplier has asymmetric information about the buyer’s cost structure, i.e. (P) possesses incomplete information about the type i 2 I of the buyer. The buyer (A) will only select a policy other than the individual optimal policy xA, as was shown, if the increase in total relevant cost resulting from this policy is compensated by (P ). Therefore the knowledge of the buyer’s cost function is of the highest interest for 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 payment z as high as

526 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

offers must be designed to be incentive-compatible, so that they are attractive for each assumed type of ( A ) to choose the offer which is designed for that specific type of (A) [24, pp. 561-562] . By accepting one of these offers (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: Kf (x) and Kf (x), 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 inventory holding cost hA1, the order processing cost B1, and the individual optimal order policy xf 1. Accordingly, for the type of buyer (A2), we use hA 2, B2, and xf 2. In the following section, for calculating the optimal set of offers, we take the reasonable assumption that the inequation hA1 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 second stage, (A ) can either accept or reject. After that, the game ends. Both players (A ) and (P ) are assumed to be risk-neutral. From the supplier’s point of view two different types of buyer (A) exist: With a probability of ro1 > 0 the buyer is type (A1) and with a probability of m 2 = 1 — m1 > 0 type (A2). With complete knowledge about the buyer’s cost structure, (P) would offer the joint optimal policy x * G and a side payment z which compensates for the increase of cost. Not being informed about the buyer’s true cost structure it will be optimal for (P) to offer a set of offers (x1, z1, x2, z2), i.e. two separate joint policies x1 and x2

min E[KP(x1, zb X2, Z2)] = ®1 • (KP(X1) +Z1) + m2 • (KP(X2) +Z2) (19)s.t. Kf(x1) — Z1 6 Kf (xf 1),

(20)Kf(x2)— Z2 6 Kf(xf;2),

(21)Kf (x1) — Z1 6 Kf (x2) — Z2,

(22)Kf (x2) Z2 6 Kf(x1)—Z1, (23)Conditions (20) and (21) ensure individual rationality: for both assumed types

of (A), it must be more attractive 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 specific type of (A). The cost functions KP(x1) and KA(x1) are strictly convex in x1 and the cost functions KP(x2) and KA(x2) are strictly convex in x2. Therefore, the Karush-Kuhn-Tucker-conditions (see Appendix A) are sufficient for an optimal solution. Using the Karush-Kuhn-Tucker-conditions the optimal set of offers can be determined. The analysis of the Karush-Kuhn-Tucker-conditions leads to six possible sets of offers (see Appendix A). The set of offers which satisfies all Karush-Kuhn-Tucker-conditions is a feasible solution satisfying the constraints (20)-(24) and is also the optimal solution of

with Z1 = Kf (x1) — Kf (xf ;1), (26)x1 =2 • d • (BR)

E. Sucky / European Journal of Operational Research 171 (2006) 516-535 527

x2 = \ ^ h d ' + d +h R With Z2 = K f ^ ) _ K f (XA,2)- (27)

Offered joint policies are equal to the joint optimal policy, which can be derived from the JELS model or the presented bargaining game with complete information. The offered payments z1 and z2 compensate exactly the increases of cost, induced by the transition from xf ^ to x1 or from xf ,2 to x 2 . The optimality test of the set of offers 1 is described in Appendix A. The set of offers 1 is not the optimal one if the buyer has an incentive to choose a contract which is not designed for his true cost structure. Thereby we can distinguish two cases: First, the true (or nearly true) cost function of (A) is Kf (x), but the buyer has an incentive to accept the offer which is designed on the basis of the cost function Kf (x), i.e. (A1) has an incentive to imitate (A2). Second, the true (or nearly true) cost function of the buyer is Kf (x), but the buyer has an incentive to accept the offer which is designed on the basis of the cost function Kf (x). In both cases the buyer would disguise Set of offers 2

x1

x2

(

s(x1 • R -K X1 • B2B1 — B2) • d

0J2 • p • hp + 72 — 2 • hf ,2 + 2

with z1 Kf (x1) — Kf (xf ,1),

2 • d • (B2 + R)hf ,2 + p •

with z2 Kf(x2) — Kf (x1)+Kf(x1)—Kf(xf,1).

(28)

(2

Imitation of (A1) by (A2) will be avoided by offering an—from the point of view of Kf (x)—unattractive policy x1, which deviates from the joint optimal policy determined by the JELS model. Additionally, the offered payment z2 consists of the compensation for the increase of cost, induced by the transition from xf 2 to x2 and a bonus for not imitating (A1). The optimality Set of offers 3

x1

x2

s2 • d • (B1 + R)hf ,1 + p •

with z1

(x2 • R + a > 2 • B1 + B2 -

x2 d h x2 1 \ hT • p • hp + 2" —

= Kf (x1) — Kf (x2)+ Kf (x2) — Kf (xf ,2),

JB 1 ) , d with z2 = Kf (x2) — Kf (xf ,2)

.+ 2 • hf ,2 ,

(30)

(31)

The set of offers 3 is designed for the case that the buyer has an incentive to accept the offer which is designed on the basis of the cost function Kf (x) while the buyer’s true cost function is Kf (x). Set of offers 3 avoids the imitation of (A2) by (A1) on the one hand by offering an—from the point of view of Kf (x)— unattractive policy x 2. On the other hand, the side payment z 1 exceeds the compensating of the increases of cost at the buyer’s side, induced by deviating from the individual optimal policy xj to Set of offers 4

x12 • d • (B1 + R)hf ,1 + p •

with z1 Kf(x1) — Kf (xf,1), (32)

528 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

X\2 =

K 2(x 2,2 )-K f(XA,l) 1 ^ hA,2 ±\

' K2 (xA,2) K1 (xA,1 A 2 ' (B1 — B2)' d

with z2 = K

2 (x2 ) K2 (xA,2

)

(33)In this case the imitation of (A2) by (A1) is prevented only by a

unattractive policy x2. For x2l, x22 > 0 result two sets of offers 4, which have to be proofed for optimality. The optimality test of the set of offers 4 is described in Appendix A. Regarding equation (33), note that we have taken the reasonable assumption for the screening model that is, to presume that Set of offers 5

x11,2

K 2(x 2,1 )-KA(x A,2 ) hA,2 ^ hA,1 ±\

' K1 (xA,1 ) K2 (xA,2 A 2 (B2 — B1 )' d

hA,2 - hA,1 hA,2 - hA,1with Z1 = K2 (x1)- K2

(xA,1), (34)

x2 = 2 ' ̂ , ̂ R) with Z2 = KA(x2)-KA(xA,2)- (35)2,2 ^ p P

In this case the imitation of (A 1) by (A2) is prevented only by an—from the point of view of KA (x)—unattractive changed quantity x1. For x11, x12 > 0 two sets of offers 5 have to be proofed for optimality (see Appendix A).

Set of offers 6

x11,2 =K 1 (x A,1 ) K 2 (x A,2 )

hA,2 - hA,1 ±\' K1 (xA,1 ) K2 (xA,2 A 2 (B2 — B1 )' d

with Z1 = K2 (x1)- K2

(xA,1), (36)

x21,2

K2(x 2,2 ) K ~ 2 (x .2,1) ^2,1 — h2,2 ±\

' K2 (x2,2) K1 (x2,1 A 2 '(B1 — B2)' d

h2,1 —h2,2 h2,1 — h2,2with z2 = K2(x2) K2(x2,2)-

(37)

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 K2(x) when his true cost function is K2(x) and, simultaneously, the buyer has an incentive to accept the offer designed on the basis of K2(x) if his true cost function is K2(x). The supplier (P) can determine an optimal set of offers for the assumed cost functions K2(x) and K2(x) of (A): the set of offers which satisfies the Karush-Kuhn-Tucker-conditions is the optimal set of offers. If the buyer behaves rationally, it is attractive for each type of (A ) to choose the offer which is designed for that specific type of (A ). The expected value 5.3. A numerical example (2)

Consider the following information for both types of a buyer (A) and a supplier (P) in Table 3, with x1 = x2 = 0.5, d = 10,000 and p = 15,000. The buyer has the market power to implement his optimal pol-

S U P P L I E R ( P ) T Y P E O F B U Y E R ( A1) T Y P E O F B U Y E R ( A2)

R = 1 3 5 , hP = 4 5 B , = 1 0 0 , hA1 = 5 0 B 2 = 5 0 , H A , 2 = 1 0 0

ii O O A II O O

OOII

K P ( X P ) = 9 0 0 0 K A ( X A , 1 ) = 1 0 , 0 0 0 K A ( X A , 2) = 1 0 , 0 0 0

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

icy. Without bargaining, the total relevant cost of the supplier (P) are Kp( x * A

x) = 9750 if the buyer is type (Ax) and Kp(x_A 2) = 15,000 if the buyer is type (A2). The expected value of the supplier’s total cost without bargaining amounts to E[KP(x1 = 200,z1 = 0,x2 = 100,z2 = 0)] = 12,375. In this example, the set of offers 4 is optimal for the supplier: ( x 1 = 242.38,z1 = 185.25, x2 = 141.42,z2 = 606.57). The expected value of supplier’s total cost amounts to: E[KP(x1,z1, x2,z2)] = 10,832.31, i.e. the supplier (P) realizes a bargaining surplus 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 xA t, xA 2, xp 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 of offers 2 and 3 are directly influenced by one of these values. For example, set of offers 2—as well as set of offers 5—is designed for the case that (A2) has an incentive to imitate (A1).

Xi — ki - 1i +12 P 0 and Z1 • (xi — ki — li + 1 2 ) = 0, (A. 4)

X 2 - k2 + ii - i2 P 0 and Z2 • (x 2 - k2 + li - 1 2 ) = 0, (A.5)

KA (xf,i) - K f (x i)+ Z1 P 0 and k i • (KA(xA,i)

- Kf(xi ) + Z 1 ) = 0, (A.6)

K 2 (x A,2 ) - K f (x 2)+z2 P 0 and

k 2 • (Kf(xA, 2 ) - Kf(x2 ) + Z 2 ) =

0,(A.7)

Kf (X 2 ) - -Z2 - K f (x i)+zi P 0 and li • (Kf(x 2 ) -Z2 - - K A

(xi) + Z i ) = 0; (A.8)

Kf (xi) - - Z1 - K f (x 2) + Z2 P 0 and l2 • (Kf(x 1 ) -Zi - K f (X 2 ) + Z 2 ) = 0 . (A.9)

0

0

C A SE 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6K 1 = 0 > 0 = 0 = 0 = 0 > 0 > 0 > 0 = 0 = 0 = 0 > 0 = 0 > 0 > 0 > 0K 2 = 0 = 0 > 0 = 0 = 0 > 0 = 0 = 0 > 0 > 0 = 0 > 0 > 0 = 0 > 0 > 0

11 = 0 = 0 = 0 > 0 = 0 = 0 > 0 = 0 > 0 = 0 > 0 > 0 > 0 > 0 = 0 > 0

1 2 = 0 = 0 = 0 = 0 > 0 = 0 = 0 > 0 = 0 > 0 > 0 = 0 > 0 > 0 > 0 > 0T A B L E 5R E L E V A N T C O M B I N A T I O N S O F T H E M U L T I P L I E R S

C A S E K 1 K 2 1 1 1 2

1 K > 0K 2 > 0 1 1 = 0 1 2 = 0

2 K > 0 K 2 = 0 1 1 = 0

OA

3 K = 0 K 2 > 0 T R V O

1 2 = 0

4 K > 0 K 2 > 0 T S:

V O 1 2 = 0

5 K [ = 0 K 2 > 0 T K V O

OA

6 K > 0 K 2 = 0 T S:

V O

OA

7 K > 0 K 2 > 0 1 1 = 0

OA

8 K > 0 K 2 > 0

T S:

V O

OA

532 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

economic order quantity and lot size as presented in Section 3.2. The offered joint policies are given by (26) and (27) . They are equal to 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 = Kf(x1) — Kf(xf;1), z2 = Kf(x2) — Kf(xf.2). 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 given by:

Kf (x2) — Kf (X2) + Kf (x2) — Kf (x*) p 0,

(A.14)A.2. Determination of the sets of offers 2 and 3

derived as follows. From conditions (A.10) and k2 = l1 = 0 follows i2 = k1 — ro1, i2 = m 2 and m 2 = k1 — x1. With x1 + x2 = 1 it follows that k1 = 1 and i2 = 1 — x1. For x1, x2 >0, k2 = i1 = 0, k1 = 1, i2 = m 2 and i2 = 1 — x 1 the KKT-conditions (A.2) and (A.3) reduce to:

-KP (x1) -Kf(x0

0Kf(x1)0, (A.16)

-KP (x2) -Kf (x2)- — + ®2 • - “

OX2 OX2

0- K P (x2)

0X

-K f (x 2)-X2 (A.17)

order quantity and lot size. The offered joint policy x2 and the joint optimal policy are identical. With (A.16) ,

-KP

(x1)-x1

R • d 1 d -Kf (x1)(x1)2 2 p -X1

B 1 • d(x1)2

+ ^ • hf ,1;-Kf (x1) B2 • d 1—5-----= — 7—vT + • hf .2

o x 1 ( x 1 ) 2 2follows x 1 according to (28). The side payments z 1 and z 2 in accordance with (28) and (29) can be determined 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 given by:

Kf (X1) — Kf (X1) — Kf (x*)+Kf (x2) P 0,Kf (X2) — Kf (X2)+Kf (X1) — Kf (X1) P 0.

(A.19)The set of offers for the third case (k1 = 0, k2 > 0, i1 >0, i2 = 0), i.e. set of

offers 3, can be derived in a similar manner. The set of offers 3 is feasible and optimal if the KKT-conditions (A.6) and (A.9) are satisfied.

iers (see Table 5). In the 4th, 7th and 8th case k1 > 0, k2 > 0 is given. Therefore, the side payments z1 and z2 for these cases can be determined using the KKT-conditions (A.6) and (A.7) , i.e. z1 and z2 are exactly at the points where the conditions (20) and (21) are satisfied as equations: z1 = Kf (x1) — Kf (xf 1), z2 = Kf (x2) — Kf (xf 2). The optimization problem (19)—(25) reduces to:

E. Sucky / European Journal of Operational Research 171 (2006) 516-535 533

min E [ K P (xi,zi,x 2 , z 2 ) \ = X • [ k p (xi) +Kf(xi) -K+ ®2 • (K( x 2 ) + -K^xA^))(A.20)

s.t. Kf(x2) - K A2 ( x 2) + K A

2 ( x * A a ) -Kf(xAi) p 0,

(A.21)

KA(xi) - Kf (xi)+ Kf (xf,i) - KA (xf,2) P 0.x 1

X 2

0KP fo) | _ 0KA (x1) 0KA (x1) 0KA (x1)0x1

0KP (x2)

+ Xi

+ X

0x1

0KA (x2)

1

0x2 ' 2 0x211 0x2

KA(x2) - KA(x2)+KA(xA,2) - KA(xA,i)

P 0,

11 • (KA(X2) -KA(x2)+KA(xA,2) -KA

(xA,i)) = 0, KA(xi) - KA(xi)+KA(xA,i)

- KA(xA,2) P 0,

0x1

0KA(x2)

+1'

+ 1

0x1

0KA(x2)0x2

(A.23)

(A.24

)

(A.25

)

(A.26

0

0

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

In the 4th case (k1 >0, k2 > 0, i1 >0, i2 = 0) the KKT-conditions (A.23)-(A.28) reduce to:

x 1 •0KP(xi) ,

0KA(xi)0, (A.29)

X •0KP fo) , 0KA (x2)

+ X • 2

0 x2 0 x2

1i •0KA(x2)

0x2+ 1 •8KA(x2)

0x2 0, (A:30)

KA(x2) - KA(x2)+KA(xA,2) - KA(xA,i)

= 0,

KA(xi) - KA(xi)+KA(xA,i) - KA(xA,2) P

(A.3

1)

(A.3The set of offers 4 (see (32) and (33) ) can be determined using (A.29) and (A.31) . The set of offers 4 is feasible and optimal, if conditions (A.32) and (A.30) and 11 > 0 are satisfied. With (A.30) and 11 > 0 the following optimality criterion results:

x2 • J • - • hP + hA,2 - (R + B2

) • “2> 0.

(A.33)2 • (hA,1

- hA,2) + (B2

- B1) • Jr

rs 5 (see (34) and (35)) is the optimal set of offers if the KKT-con- ditions (A.23) and (A.25) and 12 > 0 are satisfied. For (A.23) and 12 > 0 the following condition

534 E. Sucky / European Journal of Operational Research 171 (2006) 516-535

x i • 1 • p • h p + h A ;\ — {R + B i) - X

2 •{hA,2 — hA,1) + {B1 — B2)-X2 > •{ • J

For the 8th case (k1 >0, k2 > 0, i1 >0, i2 > 0) set of offers 6 follows (see (36) and (37)). The set of offers 6 is feasible and optimal, if the conditions (A.33) and (A.34) are satisfied. Finally, it can be shown that in the 6th case the resulting set of offers is identical to the set of offers 2 and in the 5th case the set of offers is identical to the set of offers 3.

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