استراتژی های غیر مشارکتی برای تولید و تعیین اندازه دسته تولید کالای در حال حمل و نقل در یک سیستم فروشنده چند خریداری

The supply chain structure examined in this paper consists of a single vendor (or manufacturer) with multiple heterogeneous buyers (or retailers). A continuous deterministic model is presented. To satisfy buyers demands, the vendor will deliver the product in JIT shipments to each buyer. The production rate is constant and sufficient to meet the buyers’ demands. The product is shipped in discrete batches from the vendor's stock to buyers’ stocks and all shipments are realized instantaneously. Special production–replenishment policies of the vendor and the buyers are analyzed. That is, the production batch is transferred to each buyer in several sub-batches in each production distribution cycle (PDC). This paper offers game model without prices, where agents minimize individual costs. It is a non-cooperative (1+N)-person game model with agents (a single vendor and N-buyers) choosing numbers and sizes of transferred batches. The model describes inventory patterns and cost structure of PDC. It is proved that there exist Nash equilibria in several types of sub-games of the considered game.

One of the major tasks in supply chain management is to coordinate the processes in the supply chain to obtain lower system-wide cost. In general, a supply chain is composed of independent partners with individual costs. For this reason, each firm (partner) is interested in minimizing its own cost independently. Both, in practice and in the literature considerable attention is paid to the coordination of flows between distinct entities (as supplier, manufacturer, transporter, buyer, etc.) in supply chain. The idea of joint optimization for vendor and buyer was initiated by Goyal (1976) and Banerjee (1986). A basic policy is any feasible policy where deliveries are made only when the buyer has zero inventory. Several authors incorporated policies in which sizes of successive shipments from the vendor to the buyer within a production cycle either increases by a factor (equal to the ratio of production rate to the demand rate) or are equal in size. For one buyer case, Hill (1999) shows that in the optimal PDC, the production batch first is transferred in increasing size and then in equals size of sub-batches. In most papers dealing with integrated inventory models, the transportation cost is considered only as a part of fixed setup or replenishment cost. Ertogral et al. (2007) have studied how the results of incorporation of transportation cost into the model influence on better decision making under equal size shipment policies. A fundamental advance in the two-side cost structure is in recognizing how delivery-transportation costs apply to both sides. David and Eben-Chaime (2003) and Kelle et al. (2003) suggested such a separation for policies with respect to equal in size deliveries. Some ideas for partition of delivery-transportation costs in more general case was given in Bylka (2003). However, there is an additional set of problems involved in implementing policies (strategies) with respect to whether and how the agents participate in the delivery-transportation costs in multiple buyers case. For the case with deliveries of equal sizes, some solutions can be find in a number of papers, including Banerjee et al. (2007), Chan and Kingsman (2007) and Tang et al. (2008). This paper presents a solution in the case with non-equal size deliveries. Other related papers have been developed by Siajadi et al. (2006) and Abdul-Jalbar et al. (2007). A comprehensive literature review of related works in this field is presented in Sarmah et al. (2006). The general result of this type of papers is that cooperations reduce the total system cost. It is not a typical case that suppliers and buyers coordinate their production and ordering–shipment policies. Some researchers, for example Kelle et al. (2003) have presented quantitative results which can serve as a motivation and negotiating tool for suppliers and buyers to coordinate their decisions. It is natural that potential savings in cooperation (in centralized case) cannot be ignored. Competitive pressures drive profits down. Perhaps, it forces firms to reduce costs while maintaining excellent customer service. Most studies on game theory models of supply chain consider agents which maximize individual profit functions (with respect to purchase and sale prices). Bylka, 2003 and Bylka, 2009 investigated equilibrium strategies in non-cooperative game under the assumption, that only the division of the transportation cost with respect to the numbers of shipments is centrally coordinated or negotiated before the game. The research presented in this paper offers a game model without prices, where agents minimize individual costs. It is a non-cooperative game model with agents (a single vendor and N-buyers) choosing numbers and sizes of transferred batches. It is a generalization of the paper Bylka (2009). The remainder of the paper is organized as follows. In Section 2, the model describing inventory patterns and cost structure under a production distribution cycle (PDC) is developed. Then, it is assumed that the players (the vendor and the buyers) choose decisions through a given sub-game of considered game. The existence of Nash equilibrium strategies is proved in 3 and 4.

The problem of how a vendor and multiple buyers interacts with each other in order to minimize their individual costs in a production distribution system was presented. While the vendor and each buyer incur their inventory holding cost, the transportation (or, more general, delivery associated) costs all partners are charged. Even if the system contains mechanism which completely recognize individual costs, the join optimal strategy may be questioned. A mathematical analysis of policies for integrated vendor–buyers inventory system (with respect to their possible competition) was presented. Such policies were used as strategies in a non-cooperative constrained game View the MathML sourceΓ=(Π˜,φ˜,V˜). In this game each agent has two decision variables in its strategy—the size and the number of controlled shipments. The vendor, choosing the production bath, decides about the length of the PDC. With respect to the question about existence of equilibrium multi-strategies, a satisfactory answer has been obtained for two extremal cases—(1) only the vendor controls all shipments and (2) only the buyers control all shipments. Namely, exact forms of Nash equilibrium strategies in the sub-games were presented. As the third case, the following two progressive coordination model in production distribution management was considered. The first step—an agreement with respect to the number of controlled shipments (the class of strategies View the MathML sourceΠ˜k˜). The second step—a constrained non-cooperation game, where sizes of transferred batches are decision variables. The games considered have different classes of admissible policies indexed by View the MathML sourcek˜. In each game, the agents independently choose strategies to minimize agent’ costs. The existence of Nash and Stackelberg equilibrium strategies in sub-games was proved. The equilibria in such games are different as well in the total costs as in agents’ participation in such costs. One question whether there exists an equilibrium multi-strategy in the main game ΓΓ is still open. In numerical simulations of the one buyer case we notice that the total costs of optimal cooperative policies are close to total costs of equilibrium policies, however, under the conditions that (1) classes of admissible strategies are well fitted for the optimal policies in cooperative case and (2) the holding costs satisfy the assumption A3. Theoretical justification of them is beyond the scope of the paper.