### Proof

```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?