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 1 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 2 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? 3 单产品、无能力限制的批量问题 d(t) 0 T t 4 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) T min z ( st y t ct xt ht I t ) t 1 s.t. I t 1 xt I t d t , t 1,2, , T , 1, yt 0, xt 0, I 0 0, xt <=M*yt, yt =0 or 1, M充分大 xt , I t 0, xt 0, t 1,2, , T , t 1,2, , T . 5 单产品、无能力限制的批量问题 I 0 IT 0 假设费用均非负,则在最优解中 T T x d t 1 ,即 t t 1 t 注:当ct为常数,目标函数可变为 T z ( st yt ht I t ) t 1 定理 (Zero-switch Property; Zero-Inventory Property) 一定存在满足条件 I t 1 xt 0(1 t T ) 的最优解. 可以只考虑 xt 0, dt , dt dt 1 ,, dt dt 1 dT 6 单产品、无能力限制的批量问题 记wij为第i时段生产 xi di di 1 d j 0 时所导致的费用(包 j 1 括生产准备费、生产费和库存费), 即 wij si ci xi ht I t t i 其中 I t d t 1 d i 2 d j (i t j 1) 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: w12 1 w11 2 w23 w22 w13 w14 3 w34 w33 4 w24 w44 5 即从节 点1到5 找一条 最短路 7 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) fT 1 0, f min [ w f ] t t 1 T 1 t 最优值(费用)为 f1 . 计算复杂性为 O(T 2 ) 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 8 到O(T logT )的算法 注:如何计算wij? in O(T2)? 记 d i j d i d i 1 d j ; for i=1,2,…,T { A=0; B=0; C=0; for j=i,i+1,…,T hi j hi hi 1 h j { A=A+dj; si ci d i j j d t hi ,t 1 , if d i j 0 t i wij 0, if d i j 0 if (j>i) B=B+hj-1; C=C+B*dj; if (A=0) wij=0; else 算法(计算wi,j) in O(T2) wij=si+ci*A+C; } } 9 单产品、无能力限制的批量问题:另一种建模方法 T d t 1 模型扩展: t 0 • 提前期非0 • 允许缺货 x1 1 d1 x2 x4 x3 I1 2 d2 I2 3 d3 • 价格折扣 • 非线性成本 I3 4 d4 凹费用(concave cost)最小费用流问题 • Inflation • 有限能力 • 多级系统 • …… 10 Lot-sizing in Serial System Serial system (Love,1972, MS): 11 Serial System 1 2 N 12 Serial System 13 Serial System N=3 n=4 14 Serial System 15 Serial System: Algorithm Design 16 Serial System: Dynamic Programming 17 Serial System: Computational complexity 18 Multi-stage system Serial system (Love,1972, MS): Assembly system: IN-TREE (1984, MS): 19 Multi-stage system Distribution system General 20 Earlier researches in the field 21 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) 22 HGA for General CLSP 23 General CLSP Model 24 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 25