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