Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Research on Stackelberg Game between the Supplier and the Retailer under VMI Cai Jianhu, Zhou Gengui College of Business and Administration, Zhejiang University of Technology, P.R.China, 310032 Abstract Now many small and mid-sized businesses are selling perishable products, and it is important to decide inventory quantity before the selling season. The two-echelon supply chain in our paper comprises one supplier and one retailer, and the products in our paper are perishable and sold over a single season. The paper discusses the vendor managed inventory (VMI) tactics. Under VMI, the supplier is in charge of the inventory and he should decide optimal inventory quantity before the selling season. A demand function is introduced, and wholesale price is assumed to be fixed. The retailer acts as a Stackelberg leader by setting retail price, and the supplier then decides his optimal inventory quantity. The equilibrium solutions of the Stackelberg game are gained. Key words Supply chain, Stackelberg game, VMI 1 Introduction Now many small and mid-sized businesses are selling perishable products, and the products’ selling seaon is always very short. Thus the businesses must decide inventory quantity before the selling season under uncertainty. The classical newsvendor model perfectly reflects the above decision environment and has offered some basic research conclusions. In classical model, there is only one decision maker, and there is only one decision variable, i.e. wholesale price. Thus, it is comparatively simply and easy to find the optimal inventory quantity. And newsvendor model has been discussed by many researchers [1-5] In reality, many enterprises are the members of a supply chain, and they must consider the other members’ responses. The two-echelon supply chain in our paper comprises one supplier and one retailer, and the supply chain’s competition structure must be analyzed. Traditionally, the retailer orders products from the supplier before the selling season, and the retailer should bear all inventory risk. And the main decision variable are wholesale price and inventory quantity. Lariviere and Porteus[6] constructs a Stackelberg game in which the supplier acts as the leader by setting wholesale price, and the retailer should choose his optimal order quantity. The equilibrium solutions are gained, and they find the supplier always can gain most profit of the whole supply chain. And the retailer can gain more profit by giving more demand information to the supplier. Many researches always discuss the cases in which the retailer bears all inventory risk. Now VMI is playing a more and more important role in supply chain management. According to some investigations, VMI tactics are more important than the JIT (Just in Time). Some retailers, such as Wal-Mart, Kmart, Dillard, JCPenney, succeed in applying VMI. Disney and Towill[7] find the bullwhip effect can be effectively reduced by applying VMI. It is worthwhile for us to study the decision models under VMI, and the supply chain’s competition structure should be especially discussed. In this paper, we discuss the inventory decisions models when the demand is relevant to the retail price. The supply chain operates under VMI, and the supplier takes charge of the inventory. The Stackelberg game is introduced, and the competition processes are analyzed. We find the equlibrium solutions are unique. 2 The Model 1410 Consider a supply chain with one supplier and one retailer, and the products are perishable and sold over a single selling season. The supply chain operates under VMI, and the supplier controls inventory and bears all inventory risk. Let w denote wholesale price per unit and v denote salvage value of every unsold unit and c denote marginal production cost for the supplier. Let d denote stochastic demand, which is relevant with the retail price p . The demand function can be expressed as: d = X − bp ,b > 0 (1) where X is the market potential for the products during the selling period. And we assume X is a normal distribution mean µ , variance σ , namely, X ~ N ( µ , σ ) . Because market potential is not negative, we always assume that µ is sufficiently large and σ is sufficiently small. Thus we 2 2 have d ~ N ( µ − bp, σ ) . Let f ( x ) and F ( x) denote the density and distribution functions 2 of d respectively. We further assume wholesale price w is exogenously fixed. Thus the decision variables are retail price p and inventory quantity q . The supplier incurs a shortage cost when the products are out of stock, and this shortage cost is s per unit. Assume v < c < w < p . + Define t = max(0, t ) . The game process is following: the retailer first decides unit retail price as the Stackelberg leader; ① ②the supplier then chooses inventory quantity. We will analyze the above game by backward induction. Given retail price, the supplier should decide his optimal inventory quantity. And the supplier’s expected profit function is: π s = E[(w − c ) min( q, d ) + (v − c )(q − d )+ − s(d − q )+ ] q = ( w + s − c )q − ( w + s − v ) ∫ F ( x)dx − s µ , (2) 0 We have ∂π s = ( w + s − c) − ( w + s − v ) F ( q ) ∂q ∂ 2π s = −( w + s − v ) f ( q ) < 0 ∂q 2 Thus π s is concave in q , let Φ( ∂π s w+ s −c = 0 , and we have F (q ) = , which can be expressed as: ∂q w+ s −v q − ( µ − bp) σ )= w+s−c w+s−v So that the optimal inventory quantity that the supplier will choose given retail price is: q1 = µ − bp + L1σ Where Φ ( L1 ) = (3) w+ s−c w+ s−v (4) the retailer determines his optimal retail price anticipating the response of the supplier, and the objective profit function of the retailer is: 1411 π r = ( p − w) E min(q1 , d ) − sE(d − q1 )+ (5) q1 = ( p + s − w)(q1 − ∫ F ( x)dx) − sµ 0 q1 ∂π r = q1 − ∫ F ( x)dx + ( p + s − w)(−b + bF (q1 )) 0 ∂p We have ∂ 2π r = −2b(1 − F (q1 )) − b2 ( p + s − w) f (q1 ) ≤ 0 ∂p2 Thus π r is concave in p , let ∂π r = 0 , we have: ∂p q1 µ − bp + L1σ − ∫ F ( x)dx + ( p + s − w)(−b + b 0 w+s−c )=0 w+s−v (6) where ∫ q1 0 q1 F ( x)dx = ( xF ( x ))0q1 − ∫ x f ( x)dx = L1σ 0 w+s−c + σφ ( L1 ) w+s−v (7) substitute (7) to (6) , and we can gain the retailer’s optimal retail price as following: p* = 1 [( w + s − v)( µ − σφ ( L1 )) + (c − v )( L1σ + bw − bs)] b(w + s + c − 2v) (8) * We further substitute p to (3), and we can gain the supplier’s optimal inventory quantity as following: q1* = µ − bp* + L1σ (9) * * Thus the equilibrium solution of this Stackelberg game is ( p , q1 ) , and we also can gain the supplier’s equilibrium expected profit π s and the retailer’s equilibrium expected profit π r . * * 3 Conclusion In this model, the retailer transfers inventory risk to the supplier. And the retailer also gains the Stackelberg leader status by first setting retail price and he always chooses a high retail price to gain most profit. Thus the supplier is under the high pressure. In reality, the supplier will try his best to improve his competition ability for more profit. For example, the supplier may set a profit level, under which the supplier will not take part in the trade with the retailer. Thus the retailer must consider the supplier’s competition ability while he chooses his optimal retail price. In this model, we only consider the situation in which there is only one retailer. And we should consider inventory transshipment among the retailers if there are multiple retailers, and the retailers also should compete with each other for more profit. And wholesale price is fixed in our paper,, the supplier has no chance to set wholesale price in this paper. In fact, the supplier always has the ability to set wholesale price, under which there is a three-stage game between the supplier and the retailer. In first stage, the supplier decides wholesale price. In second stage, the retailer chooses retail price. And in third stage, the supplier will choose his optimal inventory quantity. This game will be difficult to analyze by using backward induction. We should explore above new problems in the future. 1412 References [1] Gallego G, Moon I. The Distribution Free Newsboy Problem: Review and Extensions. Journal of the Operational Research Society, 1993, 44: 825~834 [2] Moutaz Khouja. The Single-Period (News-Vendor) Problem: Literature Review and Suggestions for Future Research. Omega, 1999, 27: 537~553 [3] Kabak I, Schiff A. Inventory Models and Management Objectives. Sloan Management Review, 1978, 19(2): 53~59 [4] Lau H. The Newsboy Problem under Alternative Optimization Objectives. Journal of Operational Research Society, 1980, 31: 525~535 [5] Li J, Lau H, Lau AH. A Two-Product Newsboy Problem with Satisfying Objective and Independent Exponential Demands. IIE Transactions, 1991, 23: 29~39 [6] Martin A. Larivier, Evan L. Porteus. Selling to the Newsvendor: An Analysis of Price-Only Contracts. Manufacturing & Service Operations Management, 2001, 3(4): 293~305 [7] S. M. Disney, D. R. Towill. Vendor-Managed Inventory and Bullwhip Reduction in a Two-Level Supply Chain. International Journal of Operations & Production Management, 2003, 23(6): 625~651 1413