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Supply network formation as a biform game Jean-Claude Hennet*. Sonia Mahjoub*,** * LSIS, CNRS-UMR 6168, Université Paul Cézanne, Faculté Saint Jérôme, Avenue Escadrille Normandie Niémen, 13397 Marseille Cedex 20, France (Tel: +33(4)91056016, e-mail: [email protected]). ** FIESTA , ISG Tunis, 41 rue de la liberté, 2000 Le Bardo, Tunisia (e-mail:[email protected]) Abstract: In the context of fixed market prices for the selected set of goods to be manufactured, supply network formation problems have been previously analyzed as cooperative linear production games. In particular, the profit sharing problem among partners of the winning coalition has been solved by a perfectly competitive solution, called the Owen set. Now, if an enterprise network decides to organize itself as a supply chain and imposes the wholesale price of its manufactured goods, then the supply chain design problem under a price elastic random demand from the market can be formulated as a biform game, combining a strategic subgame with a cooperative subgame. The decision variables of the strategic subgame are the wholesale prices and the retail prices of the goods, while the results of the cooperative subgame are the winning coalition and the payoff profile associated with it. The optimal global value function is then computed as the solution of a quadratic programming problem. In this scheme, the enterprise network plays the role of the Stackelberg leader and the retailer the role of the follower. The paper studies this type of biform games. In particular, it shows the existence of a payoff policy that is fair, efficient and individually rational. Copyright © 2010 IFAC 1 INTRODUCTION This paper proposes a new supply chain model based on game theory. In the context of international projects such as the European coordinated action CODESNET (2009), it has been observed that some networks of manufacturers have now organized themselves both internally, in a cooperative manner, by sharing their products and resources, and externally, as dominant strategic actors relatively to their suppliers and customers. The concept of co-opetition, coined by Brandenburger and Nalebuff (1996), can be useful to analyze such new structures of power and trade. However, in this paper, competition does not only emerge from the cooperative game between manufacturers. It is also the leading trend of the profit sharing mechanism between manufacturers and retailers. Biform games have been introduces by Brandenburger and Stuart (2007) to describe situations in which a supply chain agent needs to make strategic decisions in a competitive environment. This hybrid model has been adopted in several SCM literatures. In particular, Anupindi et al. (2001) analyzed a decentralized distribution system composed of independent retailers. In the first stage, before demand realization, each retailer makes its own decision on how much to order. In the second stage, after observing the demands, the retailers can cooperate by reallocating their inventories and allocating the corresponding additional profit. The authors have shown that this biform game has a nonempty core and have constructed an allocation mechanism based on dual solution and contained in the core of the game. Plambeck and Taylor (2005) studied a model with two original equipment manufacturers (OEMs) who sell their capacity to the contract manufacturer (CM). In the first stage, the OEM non-cooperatively choose their capacity and innovation levels. In the second cooperative stage, the manufacturers pool their capacity and negotiate the allocation of the additional profit obtained from capacity pooling. In Chatain and Zemsky (2007), a biform game approach is used to model a buyer-supplier relationship. First, suppliers make initial proposals and take organizational decisions. This stage is analyzed using a non-cooperative game theory approach. Then, suppliers negotiate with buyers who seek to outsource two tasks. In this stage, a cooperative game theory is applied to characterize the outcome of bargaining among the player over how to distribute the total surplus. Each supplier’s share of the total surplus is the product of its added value and its relative bargaining power. The quadratic production game of this paper is defined as a biform game that combines a strategic game between a manufacturers’ network and the market, and a cooperative game within the manufacturing network. In the strategic game, the manufacturing network is supposed to dominate the market, who acts as a Stackelberg follower. The consumers’ optimization problem determines the market prices on the basis of the wholesale prices imposed by manufacturing network. 2 SOME PRELIMINARIES ON GAME THEORY Biform games combine strategic games with cooperative games. Some preliminaries on both types of games are useful to understand this study. 2.1 Strategic games. Classically, a non-cooperative static game is a collection N , , where N 1,..., N is a finite set of players with N = card( N ), X is a set of decisions available for each represent the , ,…., player i, 1, … … , N and utility (or payoff) received by each player. The payoff of each player depends on the strategy chosen by all the players. Given an N-players game, player i aims to choose a strategy that maximizes his utility function π x , x , given that the other players’ strategy is summarized by decisions … … . Then, the best strategy of . , player i is defined by: A set of actions x , x , … . , x game if: is a Nash equilibrium of the x x i 1, … . , N. (1) use the locally optimal strategy x . From this definition, a Nash equilibrium is a set of actions from which no player can improve the value of his utility function by unilaterally deviating from it. Stackelberg games are strategic games with 2 players. They are also called leader-follower games. They are not in the normal form since they are dynamic with 2 steps. The leader plays first, anticipating the decision of the follower, and the follower has no other choice than to act optimally as anticipated by the leader. Such games generally reach a compromise situation, called the Stackelberg equilibrium. The leader’s optimal decision, denoted x , is computed recursively from the knowledge of the follower’s optimal response function x x : argmax π x , x x players, denoted N 1,..., N , with N = Card( N ). A coalition S is a subset of N : S N . The set P (N ) is the set of all the subsets of N . In a TU (Transferable Utility) cooperative (or coalitional) game in the sense of Von Neumann and Morgenstern (1944), each coalition S P (N ) is characterized by a value function v( S ) 0 . The value v(S ) is the maximal utility (or payoff) that can be obtained by coalition S. All the utilities are transferable (TU-game) in the sense that they are all shares of the global payoff. Each player i N seeks to maximize his utility function, which is the payoff that he can obtain from belonging to a coalition S N . Notation N \ S represents the set of players that belong to N but not to S . If S is the winning coalition, then any player j N \ S has a null payoff. Let v * be the maximal global payoff of the TU-game (N, v) : v* max v( S ) . S P (N ) (3) ui v * . With every coalition S we associate a payoff iN u(S ) defined by: In the normal form of the game, each player i selects his optimal strategy x assuming that all the other players also x Classically, a cooperative game involves a finite set of N A feasible payoff profile is a vector (ui ) iN such that Definition 2.1 Nash equilibrium argmax π x , x 2.2 TU-cooperative games. and x x x . (2) u(S ) ui . (4) iS Several properties will now be defined. Definition 2.2: Efficiency (Pareto optimality) The feasible payoff profile (u i ) iN is said to be efficient (or Pareto optimal) if and only if N u (N ) u i v * . (5) i 1 Definition 2.3: Rationality A feasible payoff profile (ui )iN is said to be rational if the payoff of every coalition S is larger than its value v(S ) : u ( S ) v( S ) S; S P (N ) . (6) Definition 2.4: Core Let y ( y1 y n )T be the output vector of products during a The core of a TU-game is the set of feasible payoff profiles reference period. Equation (8) is called the demand curve. As in Lariviere and Porteus (2001), the retailer faces the inverse demand curve obtained from the optimality conditions of the market game. The products being assumed independent, the inverse demand curve for each product i=1,…,n is: (u i ) iN that satisfy conditions (3) and (4). Namely, it is the set of feasible payoff profiles that are both efficient (Pareto optimal) and rational. As in Gillies (1959), the core is defined as:”the set of feasible outcomes that cannot be improved upon by any coalition of players”. pi 1 i yi i (9) i Quantities and prices being nonnegative, a necessary condition for equations (9) to be valid is : Definition 2.5: Optimal coalition The optimal cardinality of the TU-game (N, v) is: s* mincard ( S ) v( S ) v *. An optimal coalition of the p i p MAX , with p MAX i . (10) i i i TU-game (N, v) is a coalition S * N that satisfies By convention, condition (10) is always satisfied if p i is not v ( S * ) v * and card ( S ) s * . the actual retailer price for product i, but is obtained from the Definition 2.6. Convexity actual retailer price for product I, denoted p a through the A TU cooperative game is convex if and only if: following relation (11). i v( S T ) v( S ) v(T ) v( S T ) S N, T N (7) 3 3.1 THE SUPPLY CHAIN MODEL p i min( p a , p MAX ) i 3.2 The market game Consider a retailer selling on a market a set of products numbered i=1,…,n. In the market game between the retailer and the set of customers, the retailer plays first, by proposing a price vector p=(p1…pn)T and the market reacts by buying a quantity that depends on this price and on its habits and requirements. The supply-demand negotiation game can be represented as an iterative process. The current price p i (t ) is the decision variable fixed by the retailer and the currently purchased quantity, y i (t ) ,is the decision variable of the present and past quantities and prices purchased by customers at periods t,t-1, t-2,…t-h+1, with h the system memory, supposed finite. In a generic manner, we write: y i (t ) f i ( Y i (t 1), Pi (t) ) . For each product i=1,…,n, the market game is supposed to reach a stable equilibrium for which the expected quantity y i sold over a reference period, satisfies y i i p i i retailer over the reference period is : under conditions: wi p i . ∏ The price vector, p Diag ( 1 i is obtained from (9) in the form: ) y Diag ( i )1 i (13) diagonal terms mi , and 1 is the vector with all the components equal to 1, and the appropriate dimension (n in this case). The objective is to find the optimal vector ∏ , with: Diag ( 1 i )y 1 Diag ( that maximize i )y i The optimality condition takes the following form: (12) where Diag ( m i ) denotes a diagonal matrix with generic ∏ (8) The retailer’s problem For each final product sold on the market, the retailer faces a stochastic demand. Considering the price-dependent expected quantity sold, y i for i=1,…,n, the expected profit of the market. Different models of the market reaction function can be investigated. Let Y i (t) and P i (t) be the vectors of (11) i (14) ∏ 2 And since (15) 0, the criterion is strictly concave and admits a single optimal solution. For each product, the optimal expected demand is: yi i i wi 2 2 (16) The non negativity of this quantity derives from inequalities in (10) and (12). Accordingly, the proposed retail price is derived from (9): pi w i i 2 i 2 . (17) It is assumed that the vector of wholesale prices, w, is determined by the manufacturer’s network, who acts as a Stackelberg leader. It is related to the output vector y, by: (18) N vector eS 0,1 such that: (eS ) j 1 if j S (eS ) j 0 if j S . (20) be the amount of resource r available at enterprise j, B (( Brj )) RN , and Ari the amount of resource r Then, as a Stackelberg follower, the retailer reacts by choosing the retail prices (17) that maximize his expected profit. From (11), (17), (18), the retailer expected profit is: ∑ As in Van Gellekom et al. (2000), a coalition S is defined as a subset of the set N of N enterprises with characteristic For the R types of resources considered (r=1,…,R), let Brj 2 wi i y i i i ∏ the follower, the retailer can only accept or reject the manufacturer’s proposal. It is assumed that the retailer agrees to conclude any contract, provided that he obtains an expected profit greater than his opportunity cost which is set equal to zero by convention. After the manufacturers network has set the vector of wholesale prices, w, the retailer determines p (or equivalently ) to maximize his expected profit. Having anticipated the retailer’s reaction function (13), to maximize his expected profit. the coalition determines The pair of optimal vectors can thus be determined , by the manufacturers’ network. . (19) necessary to A (( Ari )) produce Rn one unit The manufacturers’ network produce commodities and sell them in a market. The N manufacturers compete to be partners in a coalition N . Each candidate enterprise is characterized by its production resources: manufacturing plants, machines, work teams, robots, pallets, storage areas, etc. Mathematically, each firm , 1 … … . . , of R owns a vector , ,….., , types of resources. These resources can be used, directly or indirectly to produce the vector ,….., of final products. The coalition incurs manufacturing costs ,….., per unit of each final product and sells the ,….., to products at the wholesale price vector the retailer who acts as an intermediate party between the manufacturers’ network and the final consumers. Under a wholesale price contracts, the coalition of manufacturers acts as the Stackelberg leader by fixing the wholesale price vector w as a take-it-or-leave-it proposal. As i, Resource capacity constraints for coalition S are thus written: 3.4 Consider a network of N firms represented by numbers in the set N 1,..., N . These firms are willing to cooperate to product . . 3.3 of (21) The manufacturers’ game At the manufacturing stage, two different problems must be solved: the strategic problem of selecting the wholesale price vector w, and the cooperative problem of optimizing the production vector y and the coalition characteristic vector eS . The profit optimization problem can be formulated as follows: Maximize , (22) Subject to 0,1 For a given vector w, problem (22) characterizes a cooperative game, namely the Linear Production Game (LPG) studied in Owen (1975) and Hennet and Mahjoub (2009). In the biform game studied in this paper, variables are decision variables with optimal values related to the through relations (18). Acting as the optimal output values Stackelberg leader in the strategic game with the retailer, the manufacturing network anticipates the optimal reaction of the retailer by substituting equations (18) into the objective function of problem (22). The obtained set of quadratic programming problems (P) defines a quadratic production game denoted (QPG). Maximize ∑ , Subject to (P) 0,1 , 4 THE QUADRATIC PRODUCTION GAME By assuming exogenous prices imposed by the market, the LPG describes a competitive economic situation. On the contrary, the quadratic production game (QPG) described in this paper is more appropriate to describe an oligopolistic situation in which the manufacturing network imposes its decisions to the retailer who himself has a dominant position over customers and imposes the retail prices. In this context, the QPG addresses the three following issues: 4.1 the profit maximization problem for the manufacturing network considered as a whole, the coalition decision problem through the choice of vector e S , the problem of profit allocation to the members of the optimal coalition. Global profit maximization Consider a coalition S, S N . The maximal profit that can be obtained by this coalition is obtained as the solution of the following problem, denoted (PS): Subject to (PS) , , 0,1 solution of problem (PS), denoted v(S ) , is obtained for the e N 1 . Note that matrices A and B in (PS) are componentwise nonnegative. For any set S N , e S e N and BeS BeN . Then, the optimal solution of (PS) is feasible for (PN) and the maximal expected profit can be obtained as the optimal solution of (PN). The global profit maximization problem can thus be solved through solving (PN) instead of (P), with the advantage of solving a problem in which all the variables are continuous. It can be noticed that property 4.1 does not imply optimality of the grand coalition is the sense of definition. It may happen that some coalitions with smaller cardinality than N also yield the optimal expected profit. 4.2 Profit allocation in a coalition From definition 2.4, any profit allocation policy in the core of the QPG is efficient and rational. Other properties can differentiate allocations. In particular, it is desirable to relate profit allocations of players to their marginal contribution to the value function. Classically (see e.g. Osborne and Rubinstein, 1994), the marginal contribution of player i to coalition S N with i S is defined by: i ( S ) v( S i ) v( S ) (23) A particular allocation policy, introduced by Shapley (1953) has been shown to possess the best properties in terms of balance and fairness. It is called the Shapley value, and defined by : 1 i (Si (r )) N ! r R (24) for each i in N where R is the set of all N! orderings of N , and Si (r ) is the set of players preceding i in the ordering r. Furthermore, the following result applies (Shapley, 1971): Formally, problem (PS) is similar to problem (P), except for the fact that in problem (P), is a vector of decision variables, while in problem (PS), vector is the fixed characteristic vector of the investigated coalition, S. The following result can be derived from the comparison of problems (PS) for different coalitions S N . The optimal output vector denoted Consider the grand coalition N . Its characteristic vector is i (N , v) ∑ Maximize Property 4.1 The grand coalition generates the optimal profit . Property 4.2 If the QPG is convex, the Shapley value allocation is in the core. Unfortunately, convexity is not guaranteed in general for the QPG, as it is illustrated in the example. It is then possible to differentiate coalitional rationality (not verified in general) from individual rationality. Finally, the manufacturers’ game can be solved in a fair, efficient and individually rational manner through the following steps: 1. Solve problem ( PN ) to obtain the maximal profit 2. 3. 4. and the optimal output vector y, Set the wholesale price vector w computed by (18), Set the market price vector p computed by (17), Compute the Shapley value allocation (24) to allocate the expected profit among the partners. Computation of the Shapley value allocation requires computing the solution of all the problems (PS) for S N , and this, of course, can be very time consuming for large sets of manufacturing partners. 4.3 A numerical example A very simple numerical example is constructed to illustrate 1 0.5 10 0 0 the approach: A , B , 0.7 0.8 0 10 20 2 20 5 , , , c . The unconstrained optimum of 3 40 5 the QPG is: v* 32.29 for y* [2.5 6.25] . This solution is feasible for coalitions 1,2,3, 1,2 , 1,3 , and v( S ) 0 for S 1, 2, 3, 2,3. For this problem, the core allocation is unique: (v*, 0, 0) and the Shapley allocation is: Brandenburger A.M. and Nalebuff B. (1996). Co-Opetition: A revolution mindset that combines competition and cooperation, Currency Doubleday, New York. Brandenburger, A. and Stuart, H. (2007). Biform games. Management science, 53(4), 537-549 Chatain, O. and Zemsky, P. (2007). The horizontal scope of the firm: organizational tradeoffs vs. buyer-supplier relationships. Management science, 53(4), 550-565. CODESNET (2009), A European roadmap to SME networks development, A. Villa, D. Antonelli Eds, Springer. Gillies, D.B. (1959). Solutions to general non-zero-sum games. In:. Contributions to the theory of Games vol. IV. Annals of math studies, vol. 40. (Tucker, A.W., Luce, R.D., (Eds.)), 47–85., Princeton, NJ: Princeton University. Hennet, J.C. and Mahjoub, S. (2009). A Cooperative Approach to Supply Chain Network Design, Preprints of the 13th IFAC Symposium on Information Control Problems in Manufacturing (INCOM'09), 1545-1550. does not apply, for instance, if S 1,2, T 1,3. The Lariviere, M.A. and Porteus, E.L. (2001). Selling to the newsvendor: an analysis of price-only contracts. Manufacturing & service operations management, 3(4), 293-305. vector of wholesale prices imposed by the manufacturing network to the retailer is: w 7.5 . Then, the optimal 9.17 Osborne, M.J. and Rubinstein, A. (1994). A course in game theory. The MIT Press, Cambridge, Massachussetts, U.S.A, London, England. ( 2 v*, 1 v*, 1 v * ). This QPG is not convex since property (7) 3 6 6 vector of retail prices is: p 8.75 . 11.25 5 Owen, G. (1975). On the core of linear production games. Mathematical Programming, 9, 358–370. CONCLUSIONS In the study reported in this paper, the network of manufacturers acts as a Stackelberg leader relatively to the retailer. This situation has generated a QPG not yet studied in the literature, but still relatively easy to solve. It has been shown that in general, this game is not convex and therefore that coalitional rationality and fairness of the allocation policy are not always compatible. The reverse case, when the retailer is the Stackleberg leader, gives rise to a different biform game that seems to be more difficult, since the reaction function of the manufacturers’ network cannot be expressed analytically. This problem still seems to be open. 6 REFERENCES Anupindi et al. (2001). 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