Supply chain coordination; A Game Theory approach

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1 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah Jean-Claude Hennet x and Yasemin Arda xx x LI CNR-UMR 668 Université Paul Cézanne Faulté de aint Jerome Avenue Esadrille Normandie Niemen 3397 Marseille Cedex 20 Frane orresponding author phone xx LAA-CNR 7 Avenue du Colonel Rohe 3077 Toulouse Cedex 4 Frane Abstrat In a supply hain organized as a network of autonomous enterprises the main objetive of eah partner is to optimize his prodution and supply poliy with respet to his own eonomi riterion. Conflits of interests and the distributed nature of the deision struture may indue a global loss of effiieny. Contrats an then be used to improve global performane and derease risks. The purpose of the paper is to evaluate the effiieny of different types of ontrats between the industrial partners of a supply hain. uh an evaluation is made on the basis of the relationship between a produer faing a random demand and a supplier with a random lead time. The model ombines queuing theory for evaluation aspets and game theory for deisional purposes. Introdution The supply hains onsidered over different enterprises sharing ommon information and logisti networks. Due to the distributed nature of the system and the deisional autonomy of heterogeneous deision enters organization of tasks and ativities raises some speifi problems of oordination and integration.

2 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 Enterprises an be seen as players in a game defined by a ommon goal but separate onstraints and onfliting objetives. Taking into onsideration that the entities of a supply hain need to ooperate in order to ahieve the global goal a problem appears: how to ooperate without knowing the internal models of the other entities involved? In order to obtain aeptable trade onditions a ertain form of negotiation turns out to be neessary. Game theory provides a mathematial bakground for modeling the system and generating solutions in ompetitive or onfliting situations. The basi rationality priniple of game theory states that eah player ats to optimally aomplish his/her individual goal taking into aount that the others play in the same manner. However if the individual goal of eah player is uniquely to maximize his gain or to minimize his loss the agreements obtained by negotiation may be fragile and will not generally guarantee global optimality for the whole supply hain partiularly when external demand is stohasti. For these reasons muh effort has been reently devoted to oneiving ontrats strengthening the ommitments of partners through risk profit or ost sharing and/or moving the equilibrium state of the game toward a better global performane. Game theoretial appliations in supply hain management are reviewed by Cahon and Netessine 2004 and Leng and Parlar Cahon 2003 reviews the literature on supply hain oordination with ontrats. Examples of ontrat parameters that an be used to ahieve oordination are quantity disounts returns buy baks quantity flexibility and the use of subsidies/penalties. Gupta and Weerawat [7]. onsider the interations between an end-produt manufaturer and an intermediateprodut supplier assuming that the manufaturer has no inventory for end-produts. They desribe revenue sharing ontrats able to oordinate this supply hain.. Gupta et. al. [8] generalize this study to a system in whih the manufaturer has an inventory for end-produts. They propose revenue and ost sharing ontrats with a delay penalty for the supplier in ase of a late delivery by the manufaturer. They also onstrut a searh tehnique in a losed interval to ompute the optimal base stok value for the manufaturer. This paper presents several types of ontrats between a produer and a supplier: prie-only ontrats bakorder osts sharing through transfer payments apaity reservation to assure supply. Eah ontrat is haraterized by several parameters whose values are to be determined through optimization and negotiation using a simple analytial model desribed in the following setion.. Due to the use of suh a general but somewhat simplisti model appliation of the results to a real system

3 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 would require omplementary studies by simulation tests and measurements. However it is believed that the deisional levers studied in the paper are relevant in most applied ases. It is mainly the optimal values of deision variables whih remain to be alulated in pratial appliations. 2 Model and analysis The basi supply hain element onsidered in the paper onsists of one produer and one supplier. uh an element is onsidered generi in terms of trade agreements and produt flows within a supply hain. The produer manufatures and delivers goods to the ustomers using raw produts delivered by his supplier. 2. Basi assumptions Manufaturing and delivery times of the produer are supposed negligible relatively to delivery time from the supplier. Classially the bill of materials applies to determine the quantity of omponents neessary per unit of final produt. But for simpliity of notations a one-to-one orrespondene in tehnial oeffiients is assumed between the onsidered omponent and the final produt. Both demand and delivery proesses are supposed random with respetive average rates and. Parameter is exogenous. Demand is supposed stationary with an average rate supposed known by the produer. Parameter is the main operational deision variable for the supplier. It measures the prodution apaity devoted to this produer. It is also one of the basi negotiation parameters between the supplier and the produer. An additional deision variable for the produer is the referene inventory level of finite produts. The produer is supposed to use an order-driven base stok poliy. When an order omes to her it is immediately satisfied if its amount is available in the stok. If not it has to wait until the inventory has been suffiiently replenished by the arrival of produts from the supplier. In both ases an order is plaed from the produer to the supplier whenever a demand omes and has the same amount in the

4 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 unitary ase. uh a base stok ontrol poliy an also be interpreted as a Kanban mehanism [2]. After an initial inventory replenishment stage the inventory position is held onstant with value. At time t the two random variables; the urrent inventory level It and the number of unompleted orders ut are linked by the relation: I t + u t It an be noted that under the onsidered inventory poliy ut also represents the urrent queue length of orders for the supplier. Consider the following notations: I is the random variable representing the produer inventory level in stationary onditions u is the random variable representing the number of unompleted orders from the produer not yet delivered by the supplier in stationary onditions h is the unit holding ost b is the unit bakorder ost p is the retail prie s is the supplier prodution ost per unit p is the produer prodution ost per unit. Let T be the expeted transfer payment rate from the produer to the supplier. That funtion may depend on a number of quantities aording to the ontrat onsidered. 2.2 The M/M/ model Consider now the model of a unitary demand ourring aording to a Poisson proess with rate. The supplier delay of delivery is modeled as an exponentially distributed servie time with mean value / satisfying the stability ondition / <. Under the - base stok poliy the inventory position is a onstant with value and the number of unompleted orders ut represents the

5 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 queue length of orders for the supplier []. It is a simple M/M/ system with birth-death oeffiients. Aordingly due to equation the inventory level transition graph of Fig.. It is a Markov hain desribed by the Fig.. Transition diagram of the Markov hain of inventory level In stationary onditions Prob u w Prob I w. P w The expeted value of u is defined by 2: w E[ u] wp w w 0 Z. 2 It orresponds to the expeted number of ustomers in the M/M/ queuing system from Little s law the expeted delivery time for the produer is: τ Z 3. Then It is assumed that demands arriving when the stok is empty are bakordered and a bakorder ost b is assoiated to eah unit rate of bakordered sales. Let F be the disrete distribution funtion of u and F u - Fu. The probability of bakorder is: Prob I 0 Prob u F. 4 The expeted amount of bakorders is: + L w P w. 5 w

6 The expeted inventory level I is given by: 0 w P w w I. 6 The produer and the supplier are both supposed risk neutral i.e. their utility funtions are the expeted values of their profits. Prodution osts are supposed proportional to the quantity produed. The expeted profit rate of the produer is given by:. T h b h p T bl hi p π p p p The supplier prodution osts involve a fixed part related to the prodution apaity haraterized by the average prodution rate and a variable part proportional to the expeted sales rate. The expeted profit rate of the supplier is given by:. T π v s 8 As long as the stability ondition > is satisfied the global supply hain expeted profit takes the form: h b h p π π Π v p s p 9 For given values of p and the optimal inventory quantity for the global supply hain is unique and satisfies the lassial disrete newsvendor formula for the M/M/ ase [2] [8]: G * + * Log b h h Log G 0 where stands for the integral part of a real quantity. aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008

7 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 It an be noted that if the transfer funtion T does not depend on as it is the ase in prie-only ontrats the optimal value also maximizes π p sine the inventory is G * ompletely owned by the produer. On the ontrary the value of beomes irrelevant to the supplier. In the opposite ase when T does depend on deisional autonomy of both the supplier and the produer may shift the total inventory quantity to a suboptimal value. Another question of interest in seleting a transfer payment ontrat is the share of the global profit obtained by eah partner. A game-theoreti approah may help determining an appropriate transfer funtion T suh that the global optimal profit ould be obtained with a fair distribution of profit among the partners. 3 A game-theory approah to supply hain oordination 3. A game theory model of a deision entre The enterprises of a supply hain share a ommon goal: to hold a share in a market. They join their resoures to produe and sell partiular goods. o their main fator of integration is the prodution proess for these goods but they have different onstraints and beyond their ommon goal their objetives say profit maximization may be onfliting and even antagonisti. Cooperative game theory an be of great help to design a supply hain or a virtual enterprise by seleting an optimal oalition of partners. But a non-ooperative also alled strategi approah is ertainly more appropriate to determine the set of equilibrium points that an be reahed in trade onditions. A ase of partiular interest is when there exist deisional states from whih neither player has interest to depart. uh ases alled Nash equilibria have reeived a onsiderable attention both in theory and appliations of Game Theory see e.g. [9]. A partiular property of a Nash equilibrium point is that if the internal models of the players are known it is immediately reahed at the first iteration of the game.

8 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 Therefore existene of Nash equilibrium points redues the negotiation proess to a one-shot exhange of information. In many real situations the equilibrium is not unique and the first player imposes the outome of the game. The partiular equilibrium reahed in suh an asymmetri game is alled takelberg equilibrium. Outside the full information ontext the outome of a game generally depends on who plays first and how the players negotiate. o far the general ase of sequential games with inomplete information has not reeived muh attention in the literature. 3.2 Equilibrium states The analysis naturally starts from the risk neutral situation in whih the individual goal of eah player is uniquely to maximize his/her gain or to minimize his/her loss. Consider the ase of m deision variables q... q m and n players with utility funtions π q... q... π q... q m n m. It an be assumed that eah player i has some deision variables imposed unontrollable to him and others that he an hoose set of ontrollable variables Γ i. An equilibrium state orresponds for eah player to partial optimality with respet to his ontrollable variables : π i q *... q * j.. q* m π i q *... q j. q * m q j Γi In the ase of a ontinuous deision set inequality leads to a neessary ondition of the following type: i q j q *... q * j.. q * m 0. 2 Additionally eah player has a minimal expetation on his/her utility funtion: 0 i q *... q * j.. q * m π i π 3 In pratial ases it soon appears that without a oordination mehanism no agreement an be

9 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 obtained by negotiation sine the set of onstraints and 3 is generally inompatible. As a onsequene of this observation the main purpose of a ontrat in our study will be to modify the utility funtions of the players: -- to generate a not empty feasible set of values Σ for q... q that satisfies the set of onstraints m 3 -- to identify a point q *... q * j.. q * m Σ that satisfies the optimality onditions. Furthermore a suessful negotiation requires not only the non-emptiness of Σ but also its reahability with onvergene to q*... q* j.. q* m 3.3 Coordination through ontrats A ontrat an be haraterized by several parameters that strengthen the ommitments of partners through risk profit or ost sharing and/or move the equilibrium state toward a better global performane. The values of ontrat parameters are generally determined through a negotiation proess between the partners. Aording to [6] a oordinating ontrat is one that results in a Pareto-optimal solution aeptable to eah agent. A ontrat negotiation results in a game generally involving a leader and a follower. Examples are apaity games ruled by the produer [3] utoff poliies deided by the supplier [5]. The purpose of this study is to determine the funtions and parameters to be integrated in the utility funtions of the ators so as to move their loal optimal deision toward a mutually preferred equilibrium state. 4 Deision integration through ontrats 4. Coordination of deisions in an enterprise network The supply hains onsidered in this study have a distributed nature that results in heterogeneous

10 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 deision enters. An analysis is thus needed to identify possible soures of onflits and to orret them through ontrats. The viewpoint of eah type of ator: ustomer produer supplier will be haraterized by a utility funtion and one speifi deision variable. The ustomer will deide on the most aeptable value of the delivery time the produer will hose the end-produt inventory level and the supplier will set the prodution apaity. The optimal values of deision variables will be onditioned by some parameters that define the terms of the ontrats. By indiretly ontrolling the deision the ator who determines the ontrat parameters ats a takelberg leader while the one who optimally selets the value of the deision variable on the basis of the ontrat atually ats as a takelberg follower. Along this sheme several types of ontrats will be proposed to drive the system lose to its global equilibrium while maintaining the deentralized deisional struture. uh ontrats are applied to the simple model presented above. However they imply variables of deision and mehanisms of ompensation whih ould easily apply to the real supply hains subjet to parametri adjustments obtained by simulation. 4.2 The ustomer viewpoint For the ustomer the utility funtion depends on the quality of the produt that is both its added-value and its lead-time. Using the relations 3 and 4 the expeted lead time of ustomer orders is defined by: W ProbI 0 τ / L/ 4 Let a denote the unit value of the produt and ϕ L the ost funtion of bakorders for the ustomer. L π a p ϕ L 5 In this setion the values of are supposed exogenously given. Under the rational assumption that funtion ϕ L is monotonously inreasing in L maximization of ustomer satisfation redues to minimizing his amount of bakorders L. Thus for the ustomer the optimal value of L is 0. It is

11 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 obtained for and/or. It is lear that suh a ustomer utility funtion is not aeptable and even impossible to satisfy for the produer and the supplier. A ontrat then has to be negotiated between the ustomer and the produer. uppose now from a realisti point of view that L following result an be shown. Proposition Under a linear prie adjustment mehanism p utility funtion in the interval [ L ] ϕ is stritly onvex in the interval [ L] 0. 0 ϕ 0 with 0 0. Then the η there is a unique maximal value of the ustomer Proof Consider the linear prie adjustment mehanism 0 0 L p 0 η : p L p η W p η / with p > 0 η 0 6 The new ustomer utility funtion is given by: π trit onvexity of L 0 > ' L a p0 + ηl ϕ L 7 ϕ in the interval [ L ] 0 implies strit onavity of L π in [ L ] ' 0. ' Furthermore ϕ L η L ' and 0 > 0. Then by onavity the value of L [ 0 L ] * for whih ' π L is maximal is unique. It is ' L * L if L 0 ' or uniquely defined by L* 0 if ' L < 0. Notes:

12 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 ' It is natural to assume L < 0 sine for large values of L the marginal inonveniene on leadtime dominates the marginal utility of prie disount. Two possible tehniques an be used to implement this ontrat: either the prie is onstant but there is a penalty that depends on the average amount of bakorders or the prie varies as a funtion of the observed lateness of delivery. Both ases orrespond on the average to the ontrat desribed by 6. In this setion the value of is supposed onstant. However a dependene on p is introdued in the sequel to ounterat a prie inrease if and therefore L derease. uh a prie inrease would tend to further derease and therefore to be eonomially unprodutive. From the global supply hain viewpoint the value of L* maximizing the onsumer utility funtion under the priing mehanism 6 should also be globally optimal for the whole supply hain. Assuming 0 ' L < 0 the hoie of the disount oeffiient η > 0 should then satisfy : ' L* 0 η* ϕ L*. 8 The diffiulty in this oordination mehanism is determination of the marginal inrease of ϕ inonveniene L*. In a supply-hain system ustomers and produers determine the ontrat parameters through negotiation. In the ase that the internal models of the partners are their private information the value of the lead-time ompensation η is determined through a bargaining game of alternating offers in whih the players alternate making offers until one is aepted. To desribe this kind of a game we need to introdue the variable time. The first move of the game ours in period 0 when the produer makes a proposal for the value η and asks for an answer onerning the amount of bakorders L*η that the ustomer is willing to selet under ontrat p 0 η. In period the ustomer makes a proposal by

13 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 announing his optimal amount of bakorders L * η and the orresponding prie pl * η whih he is ready to pay. The produer may aept or rejet this proposal in period 2. Aeptane ends the game while rejetion leads to a new proposal in whih the produer adjusts the value of η and proposes an updated ontrat p 0 η. The game ontinues in this fashion until an agreement or if no offer is ever aepted until the disagreement event. We note that the partners prefer an agreement rather than the disagreement event and they seek to reah an agreement as soon as possible sine time is valuable. Under these assumptions the bargaining game of the partners an be modeled as an extensive game with perfet information. It an reah a Nash equilibrium that is defined using the dependene 5 of L on. In the equilibrium the optimal value of the lead-time ompensation η* should satisfy: * η* + L* η* 9 0 L 0 To illustrate this setion an example of inonveniene funtion ϕ L 5 + Ar tan has been π 2 seleted. Fig.2 represents the evolution with L in the range [ L ] 0 with L 0 of the ustomer utility funtion under fixed prie p 8.2 2a and under the priing mehanism 2b with p a 5 η 0.5. In this example the optimal value of L for the ustomer is: L* 7.

14 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 Utility utility b 6 5 2a delay Delay Fig.2 Customer utility funtion with fixed prie 2a and delay dependent prie 2b 4.3 The produer viewpoint Assume that has been determined by the market as a result of the ontrat [ p 0 η] agreed by the ustomer and the produer. Then the supplier deision variable is determined by negotiation with the produer and the internal optimization problem of the produer redues to seleting her optimal inventory level *. Introdution of 6 in the expression of π p yields: π p + [ p0 p ] h + b + η h T. Under a prie-only ontrat the payment transfer T does not depend on. Then the produer profit funtion is onave in and the optimal inventory apaity is given by the lassial newsvendor formula: h Log h + b + η Log h + b + η Log h * η 20 Log Log Log Aording to formula 20 * is a dereasing funtion of as shown on Fig.3. Thus as the

15 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 prodution apaity of the supplier inreases the produer an redue her optimal inventory level *. * * mu Fig.3 Evolution of * with h0.5 b5η The supplier viewpoint If we now assume that is given that L takes the value L* determined by the ustomer and the value * determined by the produer then the supplier should reserve the apaity to assure supply with rate * determined to optimize π s. From expression 8 it learly appears that the nature of the transfer payment is essential in maximization of the supplier utility funtion. In the ase of a T delay independent prie the apaity to be installed by the supplier would tend to be whih is learly unaeptable by the ustomer Fig. and by the produer Fig.2. A delay-dependent transfer payment will then be onsidered. As in [4] it seems appropriate to selet a transfer payment linearly dereasing with the delivery delay τ. This assumption defines a ontrat rk with delivery delay dependent prie. Parameter k is to be deided by the produer for a given referene prie r. From 3 and with k as a saling oeffiient the average transfer takes the value: k T r 2

16 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 Under this ontrat the derivative of the supplier profit funtion with respet to is: k 2 Therefore for > the supplier profit funtion is onave and reahes its maximal value for : k * + 22 The transfer payment then takes the value: k T * r 23 Another onstraint to be satisfied is positivity of π s whih requires: k r v Under ontrat rk the transfer payment is independent of. The supplier-produer system an then be oordinated by the produer who ats as the takelberg leader by seleting parameter k for a given prie r for whih the supplier if he ats rationally will install the apaity * given by 22. Note that any stabilizing value of * * > an be obtained by 22 through an appropriate hoie of k. Also the optimal value of * given by 20 an be hosen by the produer without any negotiation. But then the profit optimizing produer imposes the largest value of * aeptable by the supplier to minimize her referene inventory level * and the assoiated inventory osts. The limit border to the pair of values rk aepted by the supplier is obtained for the minimal aeptable profit π s. In partiular ondition 24 orresponds to π s 0. Evolution of profit values with are shown in Fig.4. For the numerial values of Fig.2 the produer referene inventory level is: *6. Condition 24 then

17 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 gives the maximal value of k aeptable by the supplier for r3 and v : k0.25. For the hoie k0.25 imposed by the produer the optimal value of *.5 is obtained from 22. π π p π Fig.4 upplier and produer profit urves 5 Conlusions In enterprise networks deentralized deisions are generally less effiient than a entralized mehanism maximizing a global utility funtion. In partiular when deisions are deentralized with different utility funtions a system with a dominant ator usually leads to a takelberg game in whih the leader gets the maximal value of his/her utility funtion while the followers are maintained at their minimal aeptable satisfation level. However in spite of this unbalane suh an equilibrium may orrespond to global optimal onditions provided that the ontrats between the partners allow for a shift of loal equilibria toward globally optimal values. In this respet different ontrats have been shown to possess this property: delay adjustment mehanisms for retail pries delay dependent pries for delivery from the supplier.

18 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 Referenes [] Arda Y. and J.C. Hennet Inventory ontrol in a multi-supplier system International Journal of Prodution Eonomis Vol 04/2 pp [2] Buzaott J. and J.G. hantikumar tohasti Models of Manufaturing ystems Prentie Hall New Jersey 993. [3] G.P. Cahon upply hain oordination with ontrats in upply Chain Management: Design Coordination and Operation A.G. de Kok and.c. Graves Ed. Amsterdam: Elsevier 2003 pp [4] Cahon G. and M. Lariviere Contrating to Assure upply : How to hare Demand Foreasts in a upply Chain Management iene vol.47 No5 pp [5] G.P. Cahon and. Netessine Game theory in supply hain analysis in Handbook of Quantitative upply Chain Analysis: Modeling in the ebusiness Era D. mihi-levi.d. Wu and Z. hen Ed. Boston: Kluwer 2004 pp [6] Cahon G. F. Zhang.. Prouring fast delivery: sole-souring with information asymmetry Management iene 526 pp [7] Corbett C.J. D. Zhou and C.. Tang Designing upply Contrats : Contrat Type and Information Asymmetry Management iene. 504 pp [8] Gan X..P.ethi H.Yan Coordination of supply hains with risk averse agents Prodution and Operations Management 3 2 pp [9] Gupta D. et W. Weerawat upplier-manufaturer oordination in apaitated two-stage supply hains. European Journal of Operational Researh vol. 75 pp [0] Gupta D. Weerawat W. et N. elvaraju upply ontrats for diret-to-market manufaturers. Tehnial report University of Minnesota.

19 aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 [] Jemai Z. and F. Karaesmen L influene de la deentralization sur la performane d une haîne logistique à deux étages in Frenh Pro. MOIM 03 pp [2] M.J. Osborne and A. Rubinstein A Course in Game Theory The MIT Press 994.

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