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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
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Overview
1. Introduction
2. Problem formulation
3. Perturbation analysis (PA)
4. PA algorithm
5. Numerical examples
6. Conclusions
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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
. . .
. . .
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Introduction -- a simple structure
1
2 3
54
product
component
component
WIP
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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.
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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.
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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.
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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
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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 = ?
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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.
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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.
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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
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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
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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 ?
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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
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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
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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.
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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).
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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.
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Time comparison for gradient estimate• Time needed to obtain gradient estimator with K=1000.
time (second)
number of job
simulation method
PA method
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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)
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Example 1 -- two stage uniform distribution
• Convergence of planned parameters (S1 , S2)
(6.96, 8.44)
S1
S2
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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)
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Example 2 -- two stage exponential distribution
(7.22, 8.42)
• Convergence of planned parameters (S1 , S2)
S2
S1
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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
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Convergence of cost in PA+SA
J(S)
iteration number
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The maximum gradient in PA+SA
(+/-) max {|J(S)/Si |, i=1,…, n}
iteration number
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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)
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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.
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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.
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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.
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Convergence of cost in PA+SA
iteration number
J(S)784.9
120.7
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The maximum gradient in PA+SA
(+/-) max {|J(S)/Si |, i=1,…, n}
iteration number
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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
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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.
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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