Upload
hisa
View
24
Download
1
Tags:
Embed Size (px)
DESCRIPTION
Job Release-Time Design in Stochastic Manufacturing Systems Using Perturbation Analysis. By : Dongping Song Supervisors : Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering University of Newcastle upon Tyne March, 2000. Overview. 1. Introduction 2. Problem formulation - PowerPoint PPT Presentation
Citation preview
Job Release-Time Design in Stochastic Manufacturing Systems
Using Perturbation Analysis
By: Dongping Song
Supervisors: Dr. C.Hicks & Dr. C.F.Earl
Department of MMM Engineering
University of Newcastle upon Tyne
March, 2000
Overview
1. Introduction
2. Problem formulation
3. Perturbation analysis (PA)
4. PA algorithm
5. Numerical examples
6. Conclusions
Introduction -- a real example
Number of jobs = 113; Number of resources=13.
8 opers
. . .9 opers
. . .7 opers
. . .11 opers
. . .16 opers
. . .12 opers 10 opers
. . .15 opers
. . . 12 opers
. . .
. . .
Introduction -- a simple structure
1
2 3
54
product
component
component
WIP
Introduction -- job release times
job 4
job 5
job 3
job 2
S 2S 4
S 5
S 3
due dateS 1
job 1
waiting
earliness
waiting
• Si -- job release times
• Result in waiting time if {Si } is not well designed.
Introduction -- backwards scheduling
job 4
job 5
job 3
job 2
S 2S 4
S 5
S 3
due dateS 1
job 1
Not good if uncertain processing times or finite resource capacity.
distribution of completion time
tardy probability
Introduction -- uncertainty problem
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
Processing times follow probability distributions.
job 4
job 5
job 3
job 2
S 2S 4
S 5
S 3
S 1
job 1
Introduction -- resource problem
Job 2 and job 3 use the same resource job 2 is delayed, job 1 is delayed resulting in waiting times and tardiness.
job 2
job 1
waiting
waiting
tardiness
waiting
due date
Problem formulation
• Find optimal S=(S1, S2, …, Sn) to minimise expected total cost:
J(S) = EWIP holding costs + product earliness costs + product tardiness costs)}
• Key step of stochastic approximation is:
J(S)/Si = ?
Perturbation analysis -- references
• Ho,Y.C. and Cao, X.R., 1991, Perturbation
Analysis of Discrete Event Dynamic Systems,
Kluwer.
• Glasserman,P., 1991, Gradient Estimation Via
Perturbation Analysis, Kluwer.
• Cassandras,C.G. 1993, Discrete Event Systems:
Modeling and Performance Analysis, Aksen.
Perturbation analysis -- general problem
• Consider to minimise: J() = EL(,)
J(.) -- system performance index.
L(.) -- sample performance function.
-- a vector of n real parameters.
-- a realization of the set of random sequences.
• PA aims to find an unbiased estimator of gradient -- J()/i , with as little computation as possible.
Perturbation analysis -- main idea
• Based on a single sample realization
• Using theoretical analysis
sample function gradient
• CalculateL(,)/i , i = 1, 2, …, n
• Exchange E and :
? EL(,)/i L(,)/i
= J()/i
PA algorithm -- concepts
• Sample realization for {Si}-- nominal path (NP)
• Sample realization for {Si+Sj ji} --
perturbed path (PP), where is sufficiently small.
• All perturbed paths are theoretically constructed
from NP rather than from new experiments
PA algorithm -- Perturbation rules
• Perturbation generation rule -- When PP starts to deviate from NP ?
• Perturbation propagation rule -- How the perturbation of one job affects the processing of other jobs?
-- along the critical paths
-- along the critical resources
• Perturbation disappearance rule -- When PP and NP overlaps again ?
PA algorithm -- Perturbation rules
• If S2 is perturbed to be S2+ .
• Cost changes due to the perturbation.
job 4
job 5
job 3
job 2
S 2S 4
S 5
S 3
due dateS 1
job 1
+S 2
perturbation generation
perturbation disappearance
PA algorithm -- Perturbation rules• If S3 is perturbed to be S3+ .
• Cost changes due to the perturbation.
job 4
job 5
job 3
job 2
S 2S 4
S 5
S 3
due dateS 1
job 1
+S 3
perturbation generation
perturbation propagation
PA algorithm -- gradient estimate
• From PP and NP to calculate sample function gradient : L(S,)/Si
-- usually can be expressed by indicator functions.
• Unbiasedness of gradient estimator:
EL(S,)/Si = J(S)/Si
Condition: processing times are independent
continuous random variables.
Stochastic approximation
• Iteration equation: k+1 = k+1 + kJk
step size gradient estimator of J
• Robbins-Monro (RM) algorithm: if EJk = J.
• Kiefer-Wolfowitz (KW) algorithm: if Jk is finite
difference estimate.
• RM is faster than KW (Fu and Hu, 1997).
Time comparison for gradient estimate
• Finite difference estimator of gradient:
• PA estimator of gradient
1
1K
L S L Si l l
l
K ( , ) ( , )
1
1KL S Si l i
l
K
( , ) /
-- where 1, 2, …, K is a sequence of sample processes.
Time comparison for gradient estimate• Time needed to obtain gradient estimator with K=1000.
time (second)
number of job
simulation method
PA method
Example 1 -- two stage uniform distribution
S2 S1 J(S) J/S2J/S1
Yano, 87 6.95 8.40 3.73 0.000 0.000
PA + SA 6.96 8.44 3.78 0.006 0.004
• Two stage serial system with uniform distributions
12
• Compare with theoretical results (Yano, 1987)
Example 1 -- two stage uniform distribution
• Convergence of planned parameters (S1 , S2)
(6.96, 8.44)
S1
S2
Example 2 -- two stage exponential distribution
S2 S1 J(S) J/S2J/S1
Yano, 87 7.26 8.39 6.71 0.000 0.000
PA + SA 7.22 8.42 6.70 0.002 0.008
• Two stage serial system with exponential distributions
• Compare with theoretical results (Yano, 1987)
Example 2 -- two stage exponential distribution
(7.22, 8.42)
• Convergence of planned parameters (S1 , S2)
S2
S1
Example 3 -- multi-stage system• Assume: Normal distribution for processing times;
Infinity capacity model.
• Product structure:
11 124 5
3 10
1
2 9
7 8
6
Convergence of cost in PA+SA
J(S)
iteration number
The maximum gradient in PA+SA
(+/-) max {|J(S)/Si |, i=1,…, n}
iteration number
Compare with simulated annealing
time(second)
J(S)
Compare the convergence of cost over time (second).
simulated annealing
PA+SA method
Where simulated annealing uses four different settings (initial step sizes and number for check equilibrium)
Example 4 -- complex system
8 opers
. . .9 opers
. . .7 opers
. . .11 opers
. . .16 opers
. . .12 opers 10 opers. . .
238
15 opers
. . . 12 opers
. . .
. . .
228
229
230
231 234
226:15 232:12
243 247
242 246
245237
239
226:1 232:1
233:12
233:1
235:10 236:16 240:11
235:1 236:1 240:1
241:7
241:!
244:9
244:1
244:8
244:1
• Assume: Normal distribution and finite capacity model.
Resource constraintsResources Job sequences
1000: 247, 243, 239, 234, 231, 246, 242, 238, 230, 245, 237, 229, 228.
1211: 236:1, 236:2, 236:3, 236:4, 236:5, 236:6, 236:7, 226:1, 236:8, 226:2, 226:3, 226:4, 226:5, 226:6, 236:11, 226:7, 232:1, 226:8, 235:1, 232:2, 236:12, 235:2, 226:9, 232:3, 235:3, 240:1, 235:4, 240:2, 226:10, 232:5, 236:13, 233:2, 235:5, 240:3, 233:3, 235:6, 240:4, 232:7, 226:11, 233:4, 235:7, 240:5, 232:8, 233:5, 235:8, 240:6, 232:9, 233:6, 240:7, 226:12, 232:10, 235:9, 240:8, 233:8, 240:9, 233:9, 226:13, 235:10, 240:10, 236:15, 226:14, 240:11, 236:16, 226:15.
1212: 236:9, 236:10, 232:4, 232:6, 236:14, 232:11, 232:12.
1511: 233:1, 233:7, 233:11.
Resource constraintsResources Job sequences
1129: 233:10. 1224: 233:12. 1222: 244:1, 244:3, 244:5, 241:1, 241:2, 241:3, 248:2, 248:3, 248:5, 248:6.1113: 244:2, 241:4, 241:5, 248:4.1115: 241:6, 241:7.1315: 244:4.1226: 244:6, 244:7.1125: 244:8, 248:7, 248:8.1411: 244:9, 248:1.
Total number of jobs: 113; Number of resources: 13.
Convergence of cost in PA+SA
iteration number
J(S)784.9
120.7
The maximum gradient in PA+SA
(+/-) max {|J(S)/Si |, i=1,…, n}
iteration number
Compare with simulated annealing
time(minute)
J(S)
Compare the convergence of cost over time (minute).
simulated annealing
PA+SA method
with four different settings
Conclusions• Effective algorithm to design job release times.
• Can deal with complex systems beyond the ability of analytical methods.
• Faster to obtain gradient estimator than simulation method
• Faster than simulated annealing to optimise parameters
• Not depend on particular distributions and can include other stochastic factors.
Further Work
• Convexity of the cost function and global
optimization problem
• The effect of different job sequences on job
release time design
• Further compare with other optimisation
methods