单产品、无能力限制的批量问题

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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: [email protected]
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
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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
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