D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering
-
Upload
vanna-petty -
Category
Documents
-
view
27 -
download
0
description
Transcript of D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering
![Page 1: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/1.jpg)
Planning operation start timesfor the manufacture of capital products
with uncertain processing times and resource constraints
D.P. Song, Dr. C.Hicks & Dr. C.F.Earl
Department of MMM Engineering
University of Newcastle upon Tyne
ISAC, Newcastle upon Tyne, on 8-10 Sept., 2000.
![Page 2: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/2.jpg)
Overview
1. Introduction
2. Problem formulation
3. Perturbation Analysis (PA) method
4. Simulated Annealing (SA) method
5. Case studies
6. Conclusions
![Page 3: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/3.jpg)
Introduction -- a real example
Number of operations = 113; Number of resources=13.
8 opers
. . .9 opers
. . .7 opers
. . .11 opers
. . .16 opers
. . .12 opers 10 opers
. . .15 opers
. . . 12 opers
. . .
. . .
![Page 4: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/4.jpg)
Introduction -- a simple example
1
2 3
54
Final assembly
Component
Component
Subassembly
![Page 5: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/5.jpg)
Introduction -- operation start times
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
waiting
earliness
waiting
• Si -- part or operation start times
• Result in waiting times if {Si } is not well designed.
![Page 6: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/6.jpg)
Introduction -- backward scheduling
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
This seems perfect, but we may have uncertain processing time and finite resource capacity.
![Page 7: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/7.jpg)
Introduction -- uncertainty problem
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
Distribution of completion time
tardy probability
Uncertainty results in a high probability of tardiness.
Distribution of processing time
![Page 8: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/8.jpg)
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
S 1
part 1
Introduction -- resource problem
Part 2 and part 3 use the same resource part 2 is delayed, part 1 is delayed results in waiting times and tardiness.
part 2
part 1
waiting
waiting
tardiness
waiting
due date
![Page 9: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/9.jpg)
Problem formulation
• Find optimal S=(S1, S2, …, Sn) to minimise expected total cost:
J(S) = EWIP holding costs + product earliness costs + product tardiness costs)}
• Assumption: operation sequences are fixed.
• Key step of Stochastic Approximation is: J(S)/Si = ?
![Page 10: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/10.jpg)
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.
![Page 11: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/11.jpg)
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
![Page 12: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/12.jpg)
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
• Two concepts: nominal path and perturbed path
![Page 13: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/13.jpg)
PA algorithm -- Perturbation rules
• Perturbation generation rule -- When PP starts to deviate from NP ?
• Perturbation propagation rule -- How the perturbation of one part affects the processing of other parts?
-- along the critical paths
-- along the critical resources
• Perturbation disappearance rule -- When PP and NP overlaps again ?
![Page 14: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/14.jpg)
PA algorithm -- Perturbation rules
• Cost changes due to the perturbation.
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
+S 2
perturbation generation
perturbation disappearance
• If S2 is perturbed to be S2+ .
![Page 15: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/15.jpg)
PA algorithm -- Perturbation rules• If S3 is perturbed to be S3+ .
• Cost changes due to the perturbation.
part 4
part 5
part 3
part 2
S 2S 4
S 5
S 3
due dateS 1
part 1
+S 3
perturbation generation
perturbation propagation
![Page 16: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/16.jpg)
PA algorithm -- gradient estimate
• From PP and NP 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.
![Page 17: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/17.jpg)
Stochastic Approximation
• Iteration equation: k+1 = k+1 + kJk
step size gradient estimator of J
• Combine PA and Stochastic Approximation => PASA algorithm to optimise operation start times
![Page 18: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/18.jpg)
Simulated Annealing algorithm
• Random local search method
• Ability to approximate the global optimum
• Outer loop -- cooling temperature (T) until T=0.
• Inner loop -- perform Metropolis simulation with fixed T to find equilibrium state
![Page 19: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/19.jpg)
Simulated Annealing algorithm
• In our problem, a solution = (S1, S2, …, Sn).
• A neighborhood of a solution can be obtained
by making changes in Si.
• New solution is adjusted to meet precedence and resource constraints; non-negative.
• Cost is evaluated by averaging a set of sample processes.
![Page 20: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/20.jpg)
Simulated Annealing algorithmInitialisation
Metropolis simulation with fixedtemperature T
Adjust the solution
Evaluate cost function
Improvement
Accept newsolution
Accept new solutionwith a probability
Check for equilibrium
Stop criteria at outer loop
Return optimal solution
Coolingtemperature T
Yes No
Yes
Yes
No
No
Generate new solution
![Page 21: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/21.jpg)
Example 1 -- multi-stage system
• Product structure and resource constraints• Assume: Normal distributions for processing times.
4 5 11 12
3 10
1
2 9
7 8
6
Resouce code: Operation sequence
1001: 6, 11002: 10, 21003: 3, 91004: 4, 111005: 5, 121006: 8, 7
• There is no analytical methods to solve this problem.
![Page 22: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/22.jpg)
Convergence of cost in PASA
J(S)
Using Perturbation Analysis Stochastic Approximation to optimise operation start times.
![Page 23: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/23.jpg)
Euclidean norm of gradient in PASA
• Euclidean norm = ||Jn||
![Page 24: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/24.jpg)
Compare PASA with Simulated Annealing
Compare the convergence of costs over CPU time (second).
Where Simulated Annealing uses four different settings (initial temperature and number for check equilibrium)
Method J(S)
SA1 23.94
SA2 23.93
SA3 23.92
SA4 24.11
----------------
PASA 23.90
![Page 25: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/25.jpg)
Example 2 -- 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
248:8
248:1
• Complex product structure with Normal distributions.
![Page 26: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/26.jpg)
Resource constraintsResources Operation 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.
![Page 27: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/27.jpg)
Resource constraintsResources Operation 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.
Number of resources: 13. Total number of operations: 113.
![Page 28: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/28.jpg)
Convergence of cost in PASA
Using Perturbation Analysis Stochastic Approximation to optimise operation start times.
J(S)
![Page 29: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/29.jpg)
Euclidean norm of gradient in PASA
||Jn||
![Page 30: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/30.jpg)
Compare PASA with Simulated Annealing
Compare the convergence of costs over CPU time (minute).
with four different settingsMethod J(S)
SA1 121.74
SA2 124.60
SA3 121.78
SA4 124.90
----------------
PASA 120.79
![Page 31: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/31.jpg)
Conclusions
• Both PASA and SA can deal with complex systems beyond the ability of analytical methods.
• PASA is much faster and yields better solutions than Simulated Annealing in case studies
• SA is more robust and flexible, does not require any assumption on uncertainty
![Page 32: D.P. Song, Dr. C.Hicks & Dr. C.F.Earl Department of MMM Engineering](https://reader036.fdocuments.in/reader036/viewer/2022062422/56812b97550346895d8fb7d6/html5/thumbnails/32.jpg)
Further Work
• Optimise both operation sequences and start times
• Integrate Perturbation Analysis with SA or Evolution algorithms
• Extend to dynamic planning problems such as incremental planning and re-planning