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Prof. Dr. Jinxing Xie Department of Mathematical Sciences

Inventory Management (Deterministic Model): Dynamic Lot-Sizing Problem & Capacitated Lot-Sizing Problem. 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

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Prof. Dr. Jinxing Xie Department of Mathematical Sciences

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  1. Inventory Management (Deterministic Model):Dynamic Lot-Sizing Problem & Capacitated Lot-Sizing Problem 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

  2. Review: EOQ and ELSP • EOQ (EPQ / EMQ) • Deterministic, statistic demand (not time-varying) • Single stage (uncapacitated), infinite planning horizon • ELSP (Economic Lot-Sizing Problem): • Multiple products • Single stage (Single Capacitated Machine) • Multiple stage: • Echelon Inventory; Powers-of-Two Policies • How about finite horizon case? • Constant demand: Equal cycles, or EOQ approximation • Dynamic demand (time-varying): Lot-sizing Problem

  3. 单产品、无能力限制的批量问题 （Single-level Uncapacitated Lotsizing） 某工厂生产某种产品用以满足市场需求,且已知在时段t中的市场需求为dt . 在某时段t, 如果开工生产, 则生产开工所需的生产准备费为st , 单件产品的生产费为ct . 在某时段t期末, 如果有产品库存, 单件产品的库存费为ht . (假设这些参数非负) 假设初始库存为0, 不考虑能力限制, 工厂应如何安排生产, 可以保证按时满足生产, 且使总费用最小?

  4. 单产品、无能力限制的批量问题 d(t) 0 T t

  5. 整数（０－１）规划模型: 非线性／线性？ 假设在时段t, 产品的生产量为xt , 期末产品的库存为It (I0 =0); 用二进制变量yt表示在时段t工厂是否进行生产准备. （假设不允许缺货） xt <=M*yt, yt＝0 or 1, M充分大

  6. 单产品、无能力限制的批量问题 假设费用均非负，则在最优解中 ，即 注：当ct为常数，目标函数可变为 定理(Zero-switch Property; Zero-Inventory Property) 一定存在满足条件 的最优解. 可以只考虑

  7. w12 w23 w34 w11 w22 w33 w44 1 3 2 4 5 w24 w13 w14 单产品、无能力限制的批量问题 记wij为第i时段生产 　时所导致的费用(包括生产准备费、生产费和库存费), 即 其中 网络：从所有节点i到j (> i)连一条弧, 弧上的权为wi,j-1 , 如T=4时： 即从节点1到5找一条最短路

  8. 动态规划求解 用ft表示当t时段初始库存为0时，从t时段到T 时段的子问题的最优费用值 (即从节点t到T+1的最短路长) 最优值（费用）为 f1 . 计算复杂性为 1990(OPERIONS RESEARCH), 1991(Management Science): 对s, c, h 与t无关的情形，找到O(T)的算法；否则找到O(T logT )的算法

  9. for i=1,2,…,T { A=0; B=0; C=0; for j=i,i+1,…,T { A=A+dj; if (j>i) B=B+hj-1; C=C+B*dj; if (A=0) wij=0; else wij=si+ci*A+C; } } 注：如何计算wij？ in O(T2)? 记 算法（计算wi,j）in O(T2)

  10. 0 x1 x4 x2 x3 1 3 2 4 I1 I2 I3 d1 d2 d3 d4 单产品、无能力限制的批量问题：另一种建模方法 • 模型扩展： • 提前期非０ • 允许缺货 • 价格折扣 • 非线性成本 • Inflation • 有限能力 • 多级系统 • …… 凹费用（concave cost）最小费用流问题

  11. Lot-sizing in Serial System • Serial system （Love，1972, MS):

  12. Serial System 1 2 N

  13. Serial System

  14. N=3 n=4 Serial System

  15. Serial System

  16. Serial System: Algorithm Design

  17. Serial System: Dynamic Programming

  18. Serial System: Computational complexity

  19. Multi-stage system • Serial system （Love，1972, MS): • Assembly system： IN-TREE (1984, MS):

  20. Multi-stage system • Distribution system • General

  21. Earlier researches in the field

  22. General multi-stage system • When production capacity is INFINITE, • Dynamic lot-sizing problem (DLSP) (also called uncapacitated CLSP, since DLSP sometimes refers to Discrete Lot-Sizing Problem) • When production capacity is incorporated, then problem is much more difficult (strongly NP-hard) • Capacitated lot-sizing problem (CLSP)

  23. HGA for General CLSP

  24. General CLSP Model

  25. Review of this lecture: DLSP & CLSP • Finite horizon, Dynamic demand • Single stage (WW algorithm) • Serial system (Love) • Assembly system • Distribution system • General system • What can be generalized to DLSP and CLSP • Zero-Switch Policy • Nested Policy • Echelon Inventory

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