Inventory Management (Deterministic Model) Coordination Mechanisms for a Distribution System with One Supplier and Multiple Retailers Prof. Dr. Jinxing Xie Department of Mathematical Sciences Tsinghua University, Beijing 100084, China http://faculty.math.tsinghua.edu.cn/~jxie Email: jxie@ math.tsinghua.edu.cn Voice: (86-10)62787812 Fax: (86-10)62785847 Office: Rm. 1202, New Science Building 1 Background: Supply Chain Structure  Fangruo Chen, Awi Federgruen, Y.S. Zheng. Coordination Mechanisms for a Distribution System with One Supplier and Multiple Retailers, MS 2001, 47/5, 693-708 T0 Ti , pi (or di) w ci , h i hi  h i  h0  0 K 0 , c0 , h0  (d ) i K  Ki  K s i d i ( pi )  pi (d i ) r i 2 Notation 3 Notation 4 Problems: Coordination Mechanism Centralized Decision  Decentralized Decision Can order-quantity discount scheme coordinate the supply chain (the distribution system)? Definition. A contract (scheme, mechanism) is said to coordinate the supply chain if the set of supply chain optimal actions is a Nash equilibrium, i.e., no firm has a profitable unilateral deviation from the set of supply chain optimal actions. 5 Decentralized Decision Model 6 Decentralized Decision: Stackelberg Game Retailer i’s problem di (w) Ti (w) (12) The supplier’s problem 7 Decentralized Decision: Algorithm (1) (…,-2,-1,0,1,2,…) Fangruo Chen; Awi Federgruen; Yu-Sheng Zheng. NEAR-OPTIMAL PRICING AND REPLENISHMENT STRATEGIES FOR A RETAIL / DISTRIBUTION SYSTEM. OR, 2001, 49/6, 839-853. 8 Centralized Decision Model 9 Centralized Solution (…,-2,-1,0,1,2,…) (1) (1) 10 Fangruo Chen; Awi Federgruen; Yu-Sheng Zheng. OR, 2001, 49/6, 839-853 Order-Quantity Discount 11 Proof 12 Proof 13 Proof 14 Order-Quantity Discount: Demand rates are exogenously given 15 Proof 16 Proof: Incremental Order-Quantity Discount 17 Proof 18 Proof: All-Unit Order-Quantity Discount 19 Proof 20 Proof 21 Proof 22 Note 23 Coordination Mechanisms (1) 24 Coordination Mechanisms (2) Pay to the supplier: retailer i 25 Coordination Mechanisms (3) 26 Coordination Mechanisms (4) 27 Property of the Contract Notes • The scheme is by no means unique. • If {Kis } are retailer-specific, the pricing scheme fails to be uniform across all retailers. 28 Proof: Retailers’ Problem 29 Proof: Retailers’ Problem 30 Proof: Supplier’s Problem Average payment from retailers to supplier Average cost of the supplier Profit function of the supplier 31 Proof: Supplier’s Problem 32 When a new retailer comes … 33 When a new retailer comes … Proof. (Omitted) 34 Review of this lecture  Distribution system  centralized vs. decentralized control  order-quantity discount scheme  coordination mechanism  Any other simpler coordination mechanisms?  How about other system?  general system  information asymmetry 35