Production, Manufacturing and LogisticsNash game model for optimizing market strategies, configuration of platform products in a Vendor Managed Inventory (VMI) supply chain for a product family
Introduction
VMI (Vendor Managed Inventory) is an inventory cooperation scheme in supply chains. Under a VMI system, the vendor decides on the appropriate inventory levels for each product of itself and its retailers, and the appropriate inventory policies to maintain these levels (Simchi-Livi et al., 2008). An early classical successful case for VMI is the partnership between Wal-Mart and Procter & Gamble (P & G). This partnership had dramatically improved P & G’s on-time deliveries and Wal-Mart’s sales by 20–25% and in the inventory turnover by 30% (Buzzell and Ortmeyer, 1995). Barilla, a pasta manufacturer, adopted VMI in 1988, which made an inventory reduction by near 50% at its retailers (Hausman, 2003). In practice, In VMI systems, many vendors are manufacturers, like Barilla, Dell and HP (Tyan and Wee, 2003). They buy materials or modules from their suppliers (e.g. ink boxes, cables, plastics, circuit board for HP) to produce modules (e.g. subassemblies) and then products (e.g. notebooks or printers). With VMI, the products are distributed to retailers whose inventories are managed by the manufacturers to meet the requirements from end customers. In VMI systems, except inventory management, some degree of autonomies is retained for individual enterprises to respond to changing environments. For example, enterprises enjoy the right of determining product prices and advertising investments in promoting products. Therefore, a question under concern is how each enterprise takes advantage of such autonomies in advertising and pricing to maximize its profit in a VMI system. However, it only receives little attention in literature except few papers like Yu et al. (2009b).
This paper focuses on the above problem in a three-level supply chain (multiple suppliers, one manufacturer, and multiple retailers) with VMI partnership between the manufacturer and retailers (depicted in Fig. 1). From upstream suppliers, the manufacturer selects suppliers and purchases raw materials or/and product components to produce product modules. Each module has multiple alternatives (options) which can in turn be selected to assemble multiple substitutable products. The products are transported to different downstream retailers, and are finally sold in markets where the retailers and products compete with each other. The product demand, then the profit, of one retailer is not only dependent on the product price and advertising investment of the retailer itself, but also dependent on all product prices and advertising investments from the other retailers.
In the supply chain, retailers are able to determine their individual retail prices and advertising investments according to their own market environments so as to maximize their individual profits. The manufacturer is able to determine its advertising investment for its brand reputation, replenishment cycles for raw materials and finished products, platform product configuration and quantity discount to maximize its profit.
The approach we take in this research is to model the supply chain as a dual Nash game which is composed of two sub games. One is between the competing retailers in the retail sector and the other is between the manufacturer and individual retailers. The manufacturer and its retailers are of equal and interdependent status. Every enterprise’s decision varies with the decisions of the other enterprises. The equilibrium is obtained when all enterprises are not willing to change their decisions. This equilibrium is called as a Nash equilibrium. We propose an overall algorithm to solve this model by integrating analytic method, GA (genetic algorithm) and iterative method.
The dual Nash game model and the proposed solution algorithm constitute a powerful decision support for solving the competitive supply chain configuration problem. Its use is demonstrated and tested through numerical examples. Several sensitivity analyses are conducted to assess what effects the competition between the products and retailers has on their business performance and what instruments the manufacturer and retailers have in order to adjust their business objectives. Mutual impacts of decision variables and parameters of different enterprises in the supply chain with fair competition are examined.
The remainder of this paper is organized as follows: Section 2 briefly summarizes the relevant researches in the literature. Section 3 formulates the profit maximization problems of the retailers and the manufacturer respectively, and mathematically defines the supply chain problems as a dual Nash game. Section 4 proposes a solution algorithm for the dual Nash game. Section 5 gives a numerical example and conducts some sensitive analyses. Section 6 concludes the paper.
Section snippets
Literature review
The research literature related to this paper can be divided into those on supply chain configuration (SCC), and those on Nash games in supply chain management. Although these areas are studied by many researchers, models simultaneously dealing with the above aspects are complex and sparse, especially those on games happening both among retailers, and between the manufacturer and retailers who produce or sell multiple substitutable products.
Dual Nash game model
This section mainly proposes a dual command Nash game model after we first define research context and notations.
Solution algorithm for Nash equilibrium
To obtain the equilibrium, we have to obtain: (i) the optimal response of individual enterprises to the decision results of the other enterprises and (ii) the equilibrium conditions of the Nash equilibrium. This section defines the response functions and the equilibrium conditions in Section 4.1, and discusses the calculation of the RR-Nash equilibrium in Section 4.2. The manufacturer’s optimal response function is developed in Section 4.3, and the algorithm to find the dual Nash game
Numerical example
The example supply chain is sketched in Fig. 6. The details of the BOM (bills of materials) of the four product variants are shown in Fig. 7, from which the parameter values of and are assigned. The values of the other parameters are given in Appendix A (see the Supplementary data of this paper).
The computational parameters in the GA are population size = 120, number of generations , crossover probability = 0.7, and . The algorithm programming is coded in
Conclusion and future work
This paper discusses a VMI supply chain where a manufacturer and multiple retailers interact with each other in order to maximize their own profits. We have proposed a dual Nash game model, consisting of RR-Nash game between retailers and MR-Nash game between the manufacturer and all retailers as a whole. We have developed an algorithm to find the dual Nash equilibrium efficiently. Different from other research in the literature, we have conducted a comprehensive investigation into how the
Acknowledgements
We are grateful to the Grants for national outstanding young researchers (NWO VENI#451-07-017) in the Netherlands, from NSFC (#70501027, #70725001, and #70629002.) and from Hong Kong University Research Committee for the financial supports.
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