1 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 , China math.tsinghua.edu.cn Voice: (86-10) Fax: (86-10) Office: Rm. 1202, New Science Building

2 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, T 0 w T i, p i (or d i )

3 Notation

4

5 Problems: Coordination Mechanism 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. Centralized Decision  Decentralized Decision Can order-quantity discount scheme coordinate the supply chain (the distribution system)?

6 Decentralized Decision Model

7 Decentralized Decision: Stackelberg Game Retailer i’s problem The supplier’s problem d i (w) T i (w) (12)

8 Decentralized Decision: Algorithm Fangruo Chen; Awi Federgruen; Yu-Sheng Zheng. NEAR-OPTIMAL PRICING AND REPLENISHMENT STRATEGIES FOR A RETAIL / DISTRIBUTION SYSTEM. OR, 2001, 49/6, (1) (…,-2,-1,0,1,2,…)

9 Centralized Decision Model

10 Centralized Solution Fangruo Chen; Awi Federgruen; Yu-Sheng Zheng. OR, 2001, 49/6, (1) (…,-2,-1,0,1,2,…)

11 Order-Quantity Discount

12 Proof

13 Proof

14 Proof

15 Order-Quantity Discount: Demand rates are exogenously given

16 Proof

17 Proof: Incremental Order-Quantity Discount

18 Proof

19 Proof: All-Unit Order-Quantity Discount

20 Proof

21 Proof

22 Proof

23 Note

24 Coordination Mechanisms (1)

25 Coordination Mechanisms (2) Pay to the supplier: retailer i

26 Coordination Mechanisms (3)

27 Coordination Mechanisms (4)

28 Property of the Contract Notes The scheme is by no means unique. If {K i s } are retailer-specific, the pricing scheme fails to be uniform across all retailers.

29 Proof: Retailers’ Problem

30 Proof: Retailers’ Problem

31 Proof: Supplier’s Problem Average payment from retailers to supplier Average cost of the supplier Profit function of the supplier

32 Proof: Supplier’s Problem

33 When a new retailer comes …

34 When a new retailer comes … Proof. (Omitted)

35 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