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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

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Background: Supply Chain Structure Fangruo Chen, Awi Federgruen, Y.S. Zheng. Coord

ination Mechanisms for a Distribution System with One Supplier and Multiple Retailers, MS 2001, 47/5, 693-708

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T0 wTi , pi (or di)

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Notation

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Notation

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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)?

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Decentralized Decision Model

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Decentralized Decision: Stackelberg Game

Retailer i’s problem

The supplier’s problem

di (w)

Ti (w)

(12)

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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, 839-853.

(1)(…,-2,-1,0,1,2,…)

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Centralized Decision Model

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Centralized Solution

Fangruo Chen; Awi Federgruen; Yu-Sheng Zheng. OR, 2001, 49/6, 839-853

(1)

(1)(…,-2,-1,0,1,2,…)

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Order-Quantity Discount

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Proof

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Proof

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Proof

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Order-Quantity Discount:Demand rates are exogenously given

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Proof

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Proof: Incremental Order-Quantity Discount

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Proof

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Proof: All-Unit Order-Quantity Discount

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Proof

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Proof

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Proof

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Note

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Coordination Mechanisms (1)

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

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Coordination Mechanisms (3)

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Coordination Mechanisms (4)

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Property of the Contract

Notes

• The scheme is by no means unique.

• If {Kis } are retailer-specific, the pricing scheme f

ails to be uniform across all retailers.

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Proof: Retailers’ Problem

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Proof: Retailers’ Problem

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Proof: Supplier’s ProblemAverage payment from retailers to supplier

Average cost of the supplier

Profit function of the supplier

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Proof: Supplier’s Problem

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When a new retailer comes …

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When a new retailer comes …

Proof. (Omitted)

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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