Supply network formaton as a bform game Jean-Claude Hennet*. Sona Mahjoub*,** * LSIS, CNRS-UMR 6168, Unversté Paul Cézanne, Faculté Sant Jérôme, Avenue Escadrlle Normande Némen, 13397 Marselle Cedex 20, France (Tel: +33(491056016, e-mal: jean-claude.hennet@lss.org. ** FIESTA, ISG Tuns, 41 rue de la lberté, 2000 Le Bardo, Tunsa (e-mal:mahjoubsona@yahoo.fr Abstract: In the context of fxed market prces for the selected set of goods to be manufactured, supply network formaton problems have been prevously analyzed as cooperatve lnear producton games. In partcular, the proft sharng problem among partners of the wnnng coalton has been solved by a perfectly compettve soluton, called the Owen set. Now, f an enterprse network decdes to organze tself as a supply chan and mposes the wholesale prce of ts manufactured goods, then the supply chan desgn problem under a prce elastc random demand from the market can be formulated as a bform game, combnng a strategc subgame wth a cooperatve subgame. The decson varables of the strategc subgame are the wholesale prces and the retal prces of the goods, whle the results of the cooperatve subgame are the wnnng coalton and the payoff profle assocated wth t. The optmal global value functon s then computed as the soluton of a quadratc programmng problem. In ths scheme, the enterprse network plays the role of the Stackelberg leader and the retaler the role of the follower. The paper studes ths type of bform games. In partcular, t shows the exstence of a payoff polcy that s far, effcent and ndvdually ratonal. Copyrght 2010 IFAC 1 INTRODUCTION Ths paper proposes a new supply chan model based on game theory. In the context of nternatonal projects such as the European coordnated acton CODESNET (2009, t has been observed that some networks of manufacturers have now organzed themselves both nternally, n a cooperatve manner, by sharng ther products and resources, and externally, as domnant strategc actors relatvely to ther supplers and customers. The concept of co-opetton, coned by Brandenburger and Nalebuff (1996, can be useful to analyze such new structures of power and trade. However, n ths paper, competton does not only emerge from the cooperatve game between manufacturers. It s also the leadng trend of the proft sharng mechansm between manufacturers and retalers. Bform games have been ntroduces by Brandenburger and Stuart (2007 to descrbe stuatons n whch a supply chan agent needs to make strategc decsons n a compettve envronment. Ths hybrd model has been adopted n several SCM lteratures. In partcular, Anupnd et al. (2001 analyzed a decentralzed dstrbuton system composed of ndependent retalers. In the frst stage, before demand realzaton, each retaler makes ts own decson on how much to order. In the second stage, after observng the demands, the retalers can cooperate by reallocatng ther nventores and allocatng the correspondng addtonal proft. The authors have shown that ths bform game has a nonempty core and have constructed an allocaton mechansm based on dual soluton and contaned n the core of the game. Plambeck and Taylor (2005 studed a model wth two orgnal equpment manufacturers (OEMs who sell ther capacty to the contract manufacturer (CM. In the frst stage, the OEM non-cooperatvely choose ther capacty and nnovaton levels. In the second cooperatve stage, the manufacturers pool ther capacty and negotate the allocaton of the addtonal proft obtaned from capacty poolng. In Chatan and Zemsky (2007, a bform game approach s used to model a buyer-suppler relatonshp. Frst, supplers make ntal proposals and take organzatonal decsons. Ths stage s analyzed usng a non-cooperatve game theory approach. Then, supplers negotate wth buyers who seek to outsource two tasks. In ths stage, a cooperatve game theory s appled to characterze the outcome of barganng among the player over how to dstrbute the total surplus. Each suppler s share of the total surplus s the product of ts added value and ts relatve barganng power. The quadratc producton game of ths paper s defned as a bform game that combnes a strategc game between a
manufacturers network and the market, and a cooperatve game wthn the manufacturng network. In the strategc game, the manufacturng network s supposed to domnate the market, who acts as a Stackelberg follower. The consumers optmzaton problem determnes the market prces on the bass of the wholesale prces mposed by manufacturng network. 2 SOME PRELIMINARIES ON GAME THEORY Bform games combne strategc games wth cooperatve games. Some prelmnares on both types of games are useful to understand ths study. 2.1 Strategc games. Classcally, a non-cooperatve statc game s a collecton N,, where N 1,..., N s a fnte set of players wth N = card( N, X s a set of decsons avalable for each player, 1,,N and,,., represent the utlty (or payoff receved by each player. The payoff of each player depends on the strategy chosen by all the players. Gven an N-players game, player ams to choose a strategy that maxmzes hs utlty functon π x,x, gven that the other players strategy s summarzed by decsons. Then, the best strategy of player s defned by:,. Defnton 2.1 Nash equlbrum A set of actons x,x,.,x s a Nash equlbrum of the game f: x x argmax π x,x 1,., N. (1 In the normal form of the game, each player selects hs optmal strategy x assumng that all the other players also use the locally optmal strategy x. From ths defnton, a Nash equlbrum s a set of actons from whch no player can mprove the value of hs utlty functon by unlaterally devatng from t. Stackelberg games are strategc games wth 2 players. They are also called leader-follower games. They are not n the normal form snce they are dynamc wth 2 steps. The leader plays frst, antcpatng the decson of the follower, and the follower has no other choce than to act optmally as antcpated by the leader. Such games generally reach a compromse stuaton, called the Stackelberg equlbrum. The leader s optmal decson, denoted x, s computed recursvely from the knowledge of the follower s optmal response functon x x : x argmax π x,x x and x x x. (2 2.2 TU-cooperatve games. Classcally, a cooperatve game nvolves a fnte set of N players, denoted N 1,..., N, wth N = Card( N. A coalton S s a subset of N : S N. The set P (N s the set of all the subsets of N. In a TU (Transferable Utlty cooperatve (or coaltonal game n the sense of Von Neumann and Morgenstern (1944, each coalton S P (N s characterzed by a value functon v ( 0. The value v ( s the maxmal utlty (or payoff that can be obtaned by coalton S. All the utltes are transferable (TU-game n the sense that they are all shares of the global payoff. Each player N seeks to maxmze hs utlty functon, whch s the payoff that he can obtan from belongng to a coalton S N. Notaton N \ S represents the set of players that belong to N but not to S. If S s the wnnng coalton, then any player j N \ S has a null payoff. Let v * be the maxmal global payoff of the TU-game (N, v : v* max v(. (3 SP ( N A feasble payoff profle s a vector ( u N such that u v *. Wth every coalton S we assocate a payoff N u ( defned by: u u S (. (4 Several propertes wll now be defned. Defnton 2.2: Effcency (Pareto optmalty The feasble payoff profle ( u N s sad to be effcent (or Pareto optmal f and only f N u( N u v *. (5 1 Defnton 2.3: Ratonalty A feasble payoff profle ( u N s sad to be ratonal f the payoff of every coalton S s larger than ts value v ( : u( v( S; S P (N. (6
Defnton 2.4: Core The core of a TU-game s the set of feasble payoff profles ( u N that satsfy condtons (3 and (4. Namely, t s the set of feasble payoff profles that are both effcent (Pareto optmal and ratonal. As n Glles (1959, the core s defned as: the set of feasble outcomes that cannot be mproved upon by any coalton of players. Defnton 2.5: Optmal coalton The optmal cardnalty of the TU-game (N, v s: * s mn card( v( v *. An optmal coalton of the TU-game (N, v s a coalton S* N that satsfes v( S * v * and card( s *. Defnton 2.6. Convexty A TU cooperatve game s convex f and only f: v ( S T v( v( T v( S T S N, T N (7 3 THE SUPPLY CHAIN MODEL 3.1 The market game Consder a retaler sellng on a market a set of products numbered =1,,n. In the market game between the retaler and the set of customers, the retaler plays frst, by proposng a prce vector p=(p 1 p n T and the market reacts by buyng a quantty that depends on ths prce and on ts habts and requrements. The supply-demand negotaton game can be represented as an teratve process. The current prce p (t s the decson varable fxed by the retaler and the currently purchased quantty, y (t,s the decson varable of the market. Dfferent models of the market reacton functon can be nvestgated. Let Y (t and P (t be the vectors of present and past quanttes and prces purchased by customers at perods t,t-1, t-2, t-h+1, wth h the system memory, supposed fnte. In a generc manner, we wrte: y ( t f (Y (t 1, P (t. For each product =1,,n, the market game s supposed to reach a stable equlbrum for whch the expected quantty y sold over a reference perod, satsfes y (8 p T Let y ( y 1 y n be the output vector of products durng a reference perod. Equaton (8 s called the demand curve. As n Larvere and Porteus (2001, the retaler faces the nverse demand curve obtaned from the optmalty condtons of the market game. The products beng assumed ndependent, the nverse demand curve for each product =1,,n s: p 1 y Quanttes and prces beng nonnegatve, a necessary condton for equatons (9 to be vald s : MAX p p, wth MAX p (9. (10 By conventon, condton (10 s always satsfed f p s not the actual retaler prce for product, but s obtaned from the actual retaler prce for product I, denoted through the followng relaton (11. a MAX p mn( p, p (11 3.2 The retaler s problem For each fnal product sold on the market, the retaler faces a stochastc demand. Consderng the prce-dependent expected quantty sold, y for =1,,n, the expected proft of the retaler over the reference perod s : under condtons: w p. (12 The prce vector, s obtaned from (9 n the form: 1 p Dag( y Dag( 1 (13 where Dag ( m denotes a dagonal matrx wth generc dagonal terms m, and 1 s the vector wth all the components equal to 1, and the approprate dmenson (n n ths case. The objectve s to fnd the optmal vector that maxmze, wth: 1 Dag( y 1 Dag( y (14 The optmalty condton takes the followng form: p a
2 (15 And snce 0, the crteron s strctly concave and admts a sngle optmal soluton. For each product, the optmal expected demand s: (16 y w 2 2 The non negatvty of ths quantty derves from nequaltes n (10 and (12. Accordngly, the proposed retal prce s derved from (9: w p 2 2. (17 It s assumed that the vector of wholesale prces, w, s determned by the manufacturer s network, who acts as a Stackelberg leader. It s related to the output vector y, by: 2 (18 w y Then, as a Stackelberg follower, the retaler reacts by choosng the retal prces (17 that maxmze hs expected proft. From (11, (17, (18, the retaler expected proft s:. (19 3.3 The manufacturers network Consder a network of N frms represented by numbers n the set N 1,...,N. These frms are wllng to cooperate to produce commodtes and sell them n a market. The N manufacturers compete to be partners n a coalton N. Each canddate enterprse s characterzed by ts producton resources: manufacturng plants, machnes, work teams, robots, pallets, storage areas, etc. Mathematcally, each frm owns a vector,,..,,, 1.., of R types of resources. These resources can be used, drectly or ndrectly to produce the vector,.., of fnal products. The coalton ncurs manufacturng costs,.., per unt of each fnal product and sells the products at the wholesale prce vector,.., to the retaler who acts as an ntermedate party between the manufacturers network and the fnal consumers. Under a wholesale prce contracts, the coalton of manufacturers acts as the Stackelberg leader by fxng the wholesale prce vector w as a take-t-or-leave-t proposal. As the follower, the retaler can only accept or reject the manufacturer s proposal. It s assumed that the retaler agrees to conclude any contract, provded that he obtans an expected proft greater than hs opportunty cost whch s set equal to zero by conventon. After the manufacturers network has set the vector of wholesale prces, w, the retaler determnes p (or equvalently to maxmze hs expected proft. Havng antcpated the retaler s reacton functon (13, the coalton determnes to maxmze hs expected proft. The par of optmal vectors, can thus be determned by the manufacturers network. As n Van Gellekom et al. (2000, a coalton S s defned as a subset of the set N of N enterprses wth characterstc vector e 0, 1 N ( e ( e S S j j S such that: 1 0 f f j S. (20 j S For the R types of resources consdered (r=1,,r, let B rj be the amount of resource r avalable at enterprse j, RN B (( B rj, and A r the amount of resource r necessary to produce one unt of product, Rn A (( A r. Resource capacty constrants for coalton S are thus wrtten:. (21 3.4 The manufacturers game At the manufacturng stage, two dfferent problems must be solved: the strategc problem of selectng the wholesale prce vector w, and the cooperatve problem of optmzng the producton vector y and the coalton characterstc vector e S. The proft optmzaton problem can be formulated as follows: Maxmze, Subject to (22 0,1 For a gven vector w, problem (22 characterzes a cooperatve game, namely the Lnear Producton Game (LPG studed n Owen (1975 and Hennet and Mahjoub (2009. In the bform game studed n ths paper, varables are decson varables wth optmal values related to the optmal output values through relatons (18. Actng as the Stackelberg leader n the strategc game wth the retaler, the
manufacturng network antcpates the optmal reacton of the retaler by substtutng equatons (18 nto the objectve functon of problem (22. The obtaned set of quadratc programmng problems (P defnes a quadratc producton game denoted (QPG. Maxmze, Subject to (P, 0,1 4 THE QUADRATIC PRODUCTION GAME By assumng exogenous prces mposed by the market, the LPG descrbes a compettve economc stuaton. On the contrary, the quadratc producton game (QPG descrbed n ths paper s more approprate to descrbe an olgopolstc stuaton n whch the manufacturng network mposes ts decsons to the retaler who hmself has a domnant poston over customers and mposes the retal prces. In ths context, the QPG addresses the three followng ssues: the proft maxmzaton problem for the manufacturng network consdered as a whole, the coalton decson problem through the choce of vector e S, the problem of proft allocaton to the members of the optmal coalton. 4.1 Global proft maxmzaton Consder a coalton S, S N. The maxmal proft that can be obtaned by ths coalton s obtaned as the soluton of the followng problem, denoted (P S : Maxmze Subject to (P S,, 0,1 Formally, problem (P S s smlar to problem (P, except for the fact that n problem (P, s a vector of decson varables, whle n problem (P S, vector s the fxed characterstc vector of the nvestgated coalton, S. The followng result can be derved from the comparson of problems (P S for dfferent coaltons S N. The optmal soluton of problem (P S, denoted v (, s obtaned for the output vector denoted. Property 4.1 The grand coalton generates the optmal proft Consder the grand coalton N. Its characterstc vector s e N 1. Note that matrces A and B n (P S are componentwse nonnegatve. For any set S N, es en and Be S Be N. Then, the optmal soluton of (P S s feasble for (P N and the maxmal expected proft can be obtaned as the optmal soluton of (P N. The global proft maxmzaton problem can thus be solved through solvng (P N nstead of (P, wth the advantage of solvng a problem n whch all the varables are contnuous. It can be notced that property 4.1 does not mply optmalty of the grand coalton s the sense of defnton. It may happen that some coaltons wth smaller cardnalty than N also yeld the optmal expected proft. 4.2 Proft allocaton n a coalton From defnton 2.4, any proft allocaton polcy n the core of the QPG s effcent and ratonal. Other propertes can dfferentate allocatons. In partcular, t s desrable to relate proft allocatons of players to ther margnal contrbuton to the value functon. Classcally (see e.g. Osborne and Rubnsten, 1994, the margnal contrbuton of player to coalton S N wth S s defned by: v( ( v( S S (23 A partcular allocaton polcy, ntroduced by Shapley (1953 has been shown to possess the best propertes n terms of balance and farness. It s called the Shapley value, and defned by : 1 ( N, v ( S ( r (24 N! rr for each n N where R s the set of all N! orderngs of N, and S (r s the set of players precedng n the orderng r. Furthermore, the followng result apples (Shapley, 1971: Property 4.2 If the QPG s convex, the Shapley value allocaton s n the core. Unfortunately, convexty s not guaranteed n general for the QPG, as t s llustrated n the example. It s then possble to dfferentate coaltonal ratonalty (not verfed n general from ndvdual ratonalty. Fnally, the manufacturers game can be solved n a far, effcent and ndvdually ratonal manner through the followng steps:
1. Solve problem ( P N to obtan the maxmal proft and the optmal output vector y, 2. Set the wholesale prce vector w computed by (18, 3. Set the market prce vector p computed by (17, 4. Compute the Shapley value allocaton (24 to allocate the expected proft among the partners. Computaton of the Shapley value allocaton requres computng the soluton of all the problems (P S for S N, and ths, of course, can be very tme consumng for large sets of manufacturng partners. 4.3 A numercal example A very smple numercal example s constructed to llustrate 1 0.5 10 0 0 the approach: A,, 0.7 0.8 B 0 10 20 2 20 5,,,. 3 40 c 5 The unconstraned optmum of the QPG s: v * 32.29 for y* [2.5 6.25]. Ths soluton s feasble for coaltons 1,2,3, 1,2, 1,3, and v( 0 for S 1, 2, 3, 2,3. For ths problem, the core allocaton s unque: (v*, 0, 0 and the Shapley allocaton s: ( 2 1 1 v*, v*, v*. Ths QPG s not convex snce property (7 3 6 6 does not apply, for nstance, f S 1,2, T 1,3. The vector of wholesale prces mposed by the manufacturng network to the retaler s: 7.5 w. Then, the optmal 9.17 vector of retal prces s: 8.75 p. 11.25 5 CONCLUSIONS In the study reported n ths paper, the network of manufacturers acts as a Stackelberg leader relatvely to the retaler. Ths stuaton has generated a QPG not yet studed n the lterature, but stll relatvely easy to solve. It has been shown that n general, ths game s not convex and therefore that coaltonal ratonalty and farness of the allocaton polcy are not always compatble. The reverse case, when the retaler s the Stackleberg leader, gves rse to a dfferent bform game that seems to be more dffcult, snce the reacton functon of the manufacturers network cannot be expressed analytcally. Ths problem stll seems to be open. 6 REFERENCES Anupnd et al. (2001. A general framework for the study of decentralzed dstrbuton systems. Manufacturng & servce operatons management, 3(4, 349-368. Brandenburger A.M. and Nalebuff B. (1996. Co-Opetton: A revoluton mndset that combnes competton and cooperaton, Currency Doubleday, New York. Brandenburger, A. and Stuart, H. (2007. Bform games. Management scence, 53(4, 537-549 Chatan, O. and Zemsky, P. (2007. The horzontal scope of the frm: organzatonal tradeoffs vs. buyer-suppler relatonshps. Management scence, 53(4, 550-565. CODESNET (2009, A European roadmap to SME networks development, A. Vlla, D. Antonell Eds, Sprnger. Glles, D.B. (1959. Solutons to general non-zero-sum games. In:. Contrbutons to the theory of Games vol. IV. Annals of math studes, vol. 40. (Tucker, A.W., Luce, R.D., (Eds., 47 85., Prnceton, NJ: Prnceton Unversty. Hennet, J.C. and Mahjoub, S. (2009. A Cooperatve Approach to Supply Chan Network Desgn, Preprnts of the 13th IFAC Symposum on Informaton Control Problems n Manufacturng (INCOM'09, 1545-1550. Larvere, M.A. and Porteus, E.L. (2001. Sellng to the newsvendor: an analyss of prce-only contracts. Manufacturng & servce operatons management, 3(4, 293-305. Osborne, M.J. and Rubnsten, A. (1994. A course n game theory. The MIT Press, Cambrdge, Massachussetts, U.S.A, London, England. Owen, G. (1975. On the core of lnear producton games. Mathematcal Programmng, 9, 358 370. Plambeck, E.L. and Taylor, T.A. (2005. Sell the Plant? The mpact of contract manufacturng on nnovaton, capacty and proftablty. Management scence, 51(1, 133-150. Shapley, L. S. (1953. A value for n-person games. In Contrbutons to the Theory of Games II. (A.W.Tucker and R.D. Luce (eds., Prnceton Unversty Press, 307-317. Shapley, L. S. (1971. Cores of convex games. Internatonal Journal of Game Theory 1, 11-26. Von Neumann, J. and Morgenstern, O. (1944. Theory of games and economc behavor, Prnceton, NJ: Prnceton Unversty Press.