Coalitions of firms in manufacturing networks: stability and optimality issues

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1 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, 2011 Coaltons of frms n manufacturng networks: stablty and optmalty ssues Jean-Claude Hennet*. Sona Mahjoub*,** * LSIS, CNRS-UMR 6168, Unversté Paul Cézanne, Faculté Sant Jérôme, Avenue Escadrlle Normande Némen, Marselle Cedex 20, France (Tel: +33(4) , 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: A supply network desgn problem s addressed under a prce elastc demand from the market. It s formulated as a bform game, combnng a strategc subgame wth a cooperatve subgame. The cooperatve subgame has a quadratc objectve functon. The results of ths game are the wnnng coalton of manufacturng frms and the payoff profle assocated wth t. The proposed payoff allocaton polcy s shown to be far, effcent and ndvdually ratonal. Coaltonal stablty s condtoned on the game convexty, whch s not always verfed. Global proft optmalty s obtaned f the manufacturng network s drectly facng the market. In the presence of an autonomous retaler between the manufacturng network and the market, the supply chan output s not globally optmal any more. 1. INTRODUCTION One of the man achevements of the European coordnated acton CODESNET (Vlla et al. 2009), has been to dentfy dfferent types of assocatons mplemented among the european frms, and specally SMEs (Small and Medum Enterprses), confrontng rough competton and the effects of globalzaton. Two man types of busness agreements have been observed: buyer-suppler contracts and resource sharng contracts. Networks of frms connected by contracts of the frst type, generally consttute market drven supply networks wth complementary sklls and products. On the contrary, contracts of the second type often connect frms wth smlar actvty and capablty, tryng to exchange experence and practce on the use of modern nformaton and management tools. Typcal examples of such assocatons can be found n the so-called dstrcts of northern Italy (Vlla et al. 2009). In fact, frms facng hazardous envronment are strongly motvated to combne ther strengths and compensate for ther weaknesses. The sharng of resources appears a key approach for ncreasng effcency, both globally and ndvdually. Actually, the types of network created manly dffer by the types of resources shared: manufacturng equpment n the frst case, nformaton systems n the second case. Based on ths unfyng asserton, ths study proposes to represent an assocaton of frms as a coalton able to generate the largest proft and to dstrbute t n a far manner that s, at least, provdng ts members wth a greater ncome than when he was alone. In terms of cooperatve game theory (see e.g. Osborne and Rubnsten, 1994), these propertes are called optmalty, effcency and ndvdual ratonalty. From a theoretcal vewpont, stablty of a coalton requres the stronger property of (global) ratonalty. Ths study presents a supply network that conssts of a prce reactve market, a retaler and a manufacturng network. It s shown that the proposed proft sharng polcy exhbts the propertes of effcency, farness and ndvdual ratonalty, but that, n general, t does not guarantee global ratonalty. Polces that are both effcent and globally ratonal are sad to belong to the core of the game. Implementaton of a core allocaton polcy s a necessary condton to guarantee coaltonal stablty. But farness s also a key property het determnes the acceptablty of an allocaton polcy. Then, the so-called game convexty property s presented as a necessary condton to guarantee both stablty and farness. It s also shown that global optmalty of the supply network can be obtaned only f the supply network s drectly facng the market. The presence of an autonomous retaler as an ntermedate actor between the market and the manufacturng network tends to decrease the global effcency of the supply network. 2. MANUFACTURING NETWORK OPTIMIZATION UNDER VARIABLE PRICES The paper analyzes the ssue of supply network formaton under prce elastc demands for fnal products. Ths problem can be nterpreted as a bform game that combnes a strategc game between a manufacturers network and the market, and Copyrght by the Internatonal Federaton of Automatc Control (IFAC) 6419

2 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, 2011 a cooperatve game wthn the manufacturng network. Bform games have been ntroduced by Brandenburger and Stuart (2007) to descrbe busness stuatons that combne strategc decsons and cooperatve agreements. In partcular, ths hybrd model has been adopted n the SCM (Supply Chan Management) lterature. In ths context, several authors (Anupnd et al. (2001), Plambeck and Taylor (2005) Chatan and Zemsky (2007)) have defned games composed of two stages, the frst one beng strategc and the second one cooperatve. In ths paper, the two stages are ntegrated through a Stackelberg mechansm n whch the manufacturng network decson antcpates the market reacton functon. In the strategc subgame, the manufacturng network s supposed to domnate the retaler, who acts as a Stackelberg follower. The consumers optmzaton problem then determnes the quanttes purchased to the retaler as functons of market prces. Integraton of the retaler s decsons and the market reacton functon n the manufacturers proft functon generates a quadratc producton game not yet studed n the lterature. Two versons of ths game are studed n the paper. The frst verson ncludes a retaler who acts as an ntermedate player between the manufacturng network and the market. In the second verson, the manufacturng network also plays the role of a retaler by drectly sellng ts products on the market. 2.1 The market game Consder a retaler sellng on a market a set of products numbered =1,,n. He buys the products to the manufacturng network who fxes the vector of unt wholesale prces,..,. In the market game between the retaler and the set of customers, the retaler plays frst, by proposng a prce vector,.., 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. For each fnal product sold on the market, the retaler faces a stochastc demand. 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 p T Let y ( y 1 y n ) be the expected equlbrum output vector of products durng a reference perod. Equaton (1) s called the (expected) demand curve. As n Larvere and Porteus (2001), the retaler faces the nverse demand curve drectly derved from (1) and correspondng to the equlbrum optmalty condtons of the market game. The products beng assumed ndependent, the nverse demand curve for each product =1,,n s: 1 p y Quanttes and prces beng nonnegatve, a necessary condton for equatons (2) to be vald s : p p MAX, wth MAX p (1) (2). (3) 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. (4) The prce vector, s obtaned from (2) n the form: 1 p Dag( ) y Dag( )1 (5) 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 Dag( ) y (6) The optmalty condton takes the followng form: 6420

3 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, (7) And snce 0, the crteron s strctly concave and admts a sngle optmal soluton. For each product, the optmal expected demand s: y w. (8) 2 2 The non negatvty of ths quantty derves from nequaltes (3) and (4). Accordngly, the proposed retal prce s derved from (2): w p. (9) The retaler s problem It s now assumed that the vector of wholesale prces, w, s mposed to the retaler by the manufacturer s network, who acts as a Stackelberg leader. It s related to the output vector y, by: 2. (10) w y Then, as a Stackelberg follower, the retaler reacts by choosng the retal prces (9) that maxmze hs expected proft. From (3), (9), (10), the retaler expected proft s:. (11) 2.3 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 to a market through a retaler. 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 j, 1.., owns a vector,,..,, 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. 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 ) S ( e ) S j j S such that: 1 0 f f j S. (12) 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:. (13) 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 manufacturers 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 (5), the coalton determnes to maxmze hs expected proft. The par of optmal vectors, can thus be determned by the manufacturers network. 2.4 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 es. The proft optmzaton problem can be formulated as follows: (14) Maxmze, Π M Subject to 0,1 For a gven vector w, problem (14) characterzes a cooperatve game, namely the Lnear Producton Game 6421

4 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, 2011 (LPG) studed n Owen (1975), Van Gellekom et al. (2000) and Hennet and Mahjoub (2010). In the bform game studed n ths paper, varables are decson varables wth optmal values related to the optmal output values through relatons (10). By substtutng equatons (10) nto the objectve functon of problem (14), we obtan a set of quadratc programmng problems (P) that defnes a quadratc producton game. Maxmze, Π M Subject to, 0,1 (P) 3. 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: (1) the proft maxmzaton problem for the manufacturng network consdered as a whole, (2) the coalton decson problem through the choce of vector e S, (3) the problem of proft allocaton to the members of the optmal coalton. 3.1 Manufacturng proft maxmzaton Consder a coalton S, S N. It s represented by ts characterstc vector. The value v( of coalton S, s defned as the maxmal proft that can be obtaned by ths coalton. It s obtaned as the soluton of the followng problem, denoted (P S ): Maxmze ΠM Subject to (P S ) wth gven, 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 3.1 The grand coalton generates the optmal manufacturng 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 3.1 does not mply optmalty of the grand coalton s the sense of the followng defnton: Defnton 3.2 Coalton optmalty The optmal cardnalty of the TU-game (N, v) * 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 *. For Quadratc Producton Games, as well as for Lnear Producton Games (Hennet and Mahjoub, 2010), t may happen that some coaltons wth a smaller cardnalty than N also yeld the optmal expected proft. 3.2 Proft allocaton n a coalton It 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 (15) A partcular allocaton polcy, ntroduced by Shapley (1953) has been shown to possess the best propertes n terms of farness. It s called the Shapley value, and defned by: s: 6422

5 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, z ( N, v) ( S ( r)) (16) 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. To study the stablty propertes of the Shapley value allocaton, the notons of core and game convexty are now ntroduced. Defnton 3.3 Core The core of a cooperatve game s the set of feasble payoff profles ( u ) N that are both effcent (Pareto optmal) (condton (17)) and ratonal (condton (18)): N u( N ) u v * (17) 1 u( v( S; S P(N ). (18) As n Glles (1959), core allocatons can be consdered stable snce they consttute: the set of feasble outcomes that cannot be mproved upon by any coalton of players. Defnton 3.4 Convexty A cooperatve game s convex f and only f: s : ( v*, v*, v * ). Ths QPG s not convex snce property S 1,2, T 1,3 (17) does not apply, for nstance, f. The vector of wholesale prces mposed by the manufacturng 7.5 network to the retaler s : w. Then, the optmal 9.17 vector of retal prces s: 8.75 p It s thus necessary 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 (10), (3) Compute the Shapley value allocaton (16) 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. v ( S T ) v( v( T ) v( S T ) S N, T N (19) The followng result apples (Shapley, 1971): Property 3.5 If a cooperatve game s convex, the Shapley value allocaton s n the core. Unfortunately, convexty s not guaranteed n general for the QPG, as llustrated n the followng counter-example. Property 3.6 The QPG s not convex n general Proof by a counter-example: Consder the followng numercal data: A,, B ,,, c 5 The unconstraned optmum of the proft functon s: v * for y* [ ]. Ths 1,2,3, 1,2, 1,3, and soluton s feasble for coaltons v( 0 for S 1, 2, 3, 2,3. For ths problem, the core allocaton s unque: (v*, 0, 0) and the Shapley allocaton 4. GLOBAL SUPPLY CHAIN OPTIMALITY The problem of globally optmzng the expected proft of the supply chan can be wrtten as follows: Maxmze Subject to 1 p Dag( ) y Dag( )1,, 0,1. (20) Ths problem s exactly formulated as the manufacturng producton game n whch the manufacturng network also plays the role of the retaler by drectly sellng the products on the market: Maxmze Subject to, 0,1. (P ) 6423

6 Preprnts of the 18th IFAC World Congress Mlano (Italy) August 28 - September 2, 2011 Note that problem (P ) dffers from problem (P) only by the coeffcent of the quadratc term n the crteron. Agan, the maxmal proft value s obtaned n partcular for the grand coalton, N. The global optmal proft of the supply network s the optmum of problem (P N ), obtaned by replacng n (P ) by e 1. N The manufacturng network generates the maxmal global proft and captures t entrely f t also plays the role of the retaler wth respect to the market. If the retaler s a dfferent and autonomous body, then, the global supply chan proft s smaller than the optmal one because the optmal output y for problem (P) dffers from the optmal one, y*, that solves problem (P ). Furthermore, even f the manufacturng network acts as a Stackelberg leader wth respect to the retaler, the manufacturng network s unable to capture the expected proft of the retaler, whch s always gven by (11):. Consder the partcular case when the retaler s dstnct from the manufacturng network and the unconstraned optmum of the manufacturng network proft functon, Π M satsfes constrants. Then, the optmal output values are c y for =1,,n. The global proft of the supply 4 chan s then dstrbuted among the manufacturng network and the retaler n the proportons 2/3, 1/3, snce 2 ( c ) ( c ) 16 and. In ths case, the global optmal s obtaned by producng * c twce the quanttes of the decentralzed case: y 2 f they are feasble. Assumng feasblty of these quanttes, the suboptmalty gap (loss of ncome) of the supply chan proft functon s equal to the proft value of the retaler: Π G 5. CONCLUSIONS ( ) 2 c. 16 The study reported n ths paper studes a network of manufacturers who acts as a Stackelberg leader relatvely to a retaler who controls a market by fxng the market prces and knowng the expected reacton curves of the customers. Ths stuaton has generated a Quadratc Producton Game that has been solved to optmalty wth a proft allocaton polcy that s far, effcent and ndvdually ratonal. However, t has also been shown that n general, ths game s not convex and therefore coaltonal stablty s not guaranteed by ths polcy. Global supply chan optmalty s also restrcted to the case of the network of enterprses actng both as the manufacturer and the retaler. REFERENCES Anupnd et al. (2001). A general framework for the study of decentralzed dstrbuton systems. Manufacturng & servce operatons management, 3(4), Brandenburger, A. and Stuart, H. (2007). Bform games. Management scence, 53(4), Chatan, O. and Zemsky, P. (2007). The horzontal scope of the frm: organzatonal tradeoffs vs. buyer-suppler relatonshps. Management scence, 53(4), 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.)), , Prnceton, NJ: Prnceton Unversty. Hennet, J.C. and Mahjoub, S. (2010). Toward the far sharng of proft n a supply network formaton, Internatonal Journal of Producton Economcs, 127(1), Larvere, M.A. and Porteus, E.L. (2001). Sellng to the newsvendor: an analyss of prce-only contracts. Manufacturng & servce operatons management, 3(4), 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, 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), Shapley, L. S. (1971). Cores of convex games. Internatonal Journal of Game Theory 1, Van Gellekom, J. R. G., Potters J. A. M., Rejnerse J. H., Engel M. C., and Tjs, S. H. (2000). Characterzaton of the Owen Set of Lnear Producton Processes, Games and Economc Behavor, vol. 32(1), pp Vlla A., Antonell D. Eds (2009), A European roadmap to SME networks development, Sprnger. 6424