一个简单的分布式配送模型 谢金星 清华大学数学科学系 Tel: 010-62787812 Email:jxie@math.tsinghua.edu.cn http://faculty.math.tsinghua.edu.cn/~jxie

简要提纲 分布式配送的基本概念 简单的数学模型 求解与分析 扩展：带时间窗口的情形 小结

Centralized Warehousing (CW) vs. Distributed Distribution (DD) CW: The components for a product is stocked together in a central warehouse, and orders are integrated at that warehouse. DD: The components for a product is stocked separately, near their manufacturing sites or landing points, and orders integration occurs at one of many non-stocking merge centers distributed over the region, using a merge-in-transit process. (e.g. Use 40 merge centers to cover USA) “DD” first introduced by L.R. Kopczak (Stanford) in 1995.

References Li J. and Xie J., "Performance Analysis of Distributed-Distribution System Under Uniform Distribution of Transportation Time", OR Transactions, Vol. 7, No.1, 2003, 83-90. (in Chinese) Li J. and Xie J., "Performance Analysis of Distributed-Distribution System With Time Window Under Uniform Distribution of Transportation Time", Systems Engineering: Theory & Practice, Vol.22, No.3, 2002, 80-87. (in Chinese) Li J. and Xie J., "Performance Analysis of Distributed-Distribution System Under Normal Distribution of Transportation Time", Control Theory & Applications, Vol.20, No.3, 2003, 464-466. (in Chinese)

Distributed distribution: Example

A simple model for two components Transportation time: fi – fixed part (assumed to be 0) Xi – i.i.d., Uniform on [0, a] Holding cost h1 <= h2

A simple model for two components TM: A customer requests a delivery in TM from now on : Expected service level (order filled before needed) Problem: When to transport part 1 and 2, respectively? Decision variable: (t, T ) (Noticing h1 <= h2) We can assume T + t <= TM, T <= a (why?)

A simple model for two components Objective: holding cost in the merge center f (T )

A simple model for two components Constraints: service level g (t, T )

The model

Solving the model f(T) is convex in T, the unconstrained minimum point is For the service level, t = TM - T

Equivalent model

Solving the model Step 1: Calculate Step 2: Calculate Step 3: Optimal T* = min (T0*, T1*).

Discussion Min Max Application: For given TM and , can you accept the order? If no, how to negotiate with the customer?

Generalization: time window (JIT) But the analytical expression for g(t,T) is more complicated

Summary Thanks you! CW vs. DD A simple model and solving procedure Discussion and applications Generalized to the case with time window Other generalizations? More components? More constraints? More cost considerations? …… Thanks you!