© P. Pongcharoen ISA/1 Applying Designed Experiments to Optimise the Performance of Genetic Algorithms for Scheduling Capital Products P. Pongcharoen,

© P. Pongcharoen ISA/1

Applying Designed Experiments to Optimise the Performance of Genetic Algorithms

for Scheduling Capital Products

P. Pongcharoen, D.J. Stewardson, C. Hicks

and P.M. Braiden.

University of Newcastle upon Tyne


Scheduling

• “The allocation of resources over time to perform a collection of tasks” (Baker 1974)

• “Scheduling problems in their static and deterministic forms are extremely simple to describe and formulate, but are difficult to solve” (King and Spakis 1980)


Scheduling Problems

• Involve complex combinatorial optimisation

• For n jobs on m machines there are potentially (n!)m

sequences, e.g. n=5 m=3 => 1.7 million sequences.

• Most problems can only be solved by inefficient non-

deterministic polynomial (NP) algorithms.

• Even a computer can take large amounts of time to

solve only moderately large problems


Scheduling the Production of Capital Goods

• Deep and complex product structures

• Long routings with many types of operations

on multiple machines

• Multiple constraints such as assembly,

operation precedence and resource

constraints.


Product Structure 4

5

97

6

81

231234

230

235233 232226

229

238

240236

239

242

243

241

237

246

248244

247

245

228

Feature: 2 Products, 118 Machining, 17 Assembly and 17 machines


JSPFSPPSP All jobs are independent

Single-level scheduling

No assembly operations

Nagar(1995)

Assumption

Products have no structureIn reality

Most products require some type of assembly

"Strongly relationship between product structure & sequencing"

Fry et.al.(1989)


Conventional Optimisation Algorithms

• Integer Linear Programming• Dynamic Programming• Branch and Bound

These methods rely on enumerative search and are therefore only suitable for small problems


More Recent Approaches

• Simulated Annealing• Taboo Search• Genetic Algorithms

Characteristics :• Stochastic search.• Suitable for combinatorial optimisation problems.• Due to combinatorial explosion, they may not search

the whole problem space. Thus, an optimal solution is not guaranteed.


check and reordercomponents

SolutionSpace

Chromosome

Chromosome

Chromosome

::

Parent 1

Parent 2+ ==>

Offspring 1

Offspring 2

Parent 3 ==> Offspring 3

Mutation Operation

Crossover Operation

Genetic OperationPopulation

Fitness Measure

Offspring 1

Offspring 2

Offspring 3

FitnessTesting

random

encode

decoding

selection

next generation

Repair Process

Start

Terminate?

Stop

noyes

Chromosome

RouletteWheel

chromosomeselection

check and reorderoperations

identify and avoiddeadlock

check capacityand adjust timing

randomly

combine

genes

GA developed for production scheduling


Chromosome representation

P 110 1

P 11 0 3

P 120 1

P 120 2

P 110 2

Sub-chromosome 1

Sub-chromosome 2

Machine 1

Machine 2

P 120 2

P 110 2

P 110 1

P 11 0 3

P 120 1

Chromosome

P ...0 ...

P i0 j

Machine n

P i = Part or component number i

O j = Operation number j

Resource no. 1

Resource no. 2

P ...0 ...

Resource no. nSub-chromosome n

P i0 j


Crossover OperationsInitial Description Reference BCGA

CX Cycling crossover Oliver et al., 1987 ER Edge recombination Whitley et al., 1989EERX Enhanced edge recombination crossover Starkweather et al., 1991 AEX Alternating edges crossover Greffensette et al., 1985MPX Maximal preservation crossover Mühlenbein et al., 1992 1PX One point crossover Murata and Ishibuchi, 1994 OX Order crossover Davis, 1985 PBX Position based crossover Syswerda, 1991 IPX Independent position crossover Murata and Ishibuchi, 1994PMX Partial matching crossover Goldberg and Lingle, 1985 LOX Linear order crossover Falkenauer and Bouffoix, 1991 SCX Sub-tour chunk crossover Greffensette et al., 19852PEX Two points end crossover Murata and Ishibuchi, 1994 2PCX Two points centre crossover Murata and Ishibuchi, 1994 2PECX Two points end / centre crossover Murata and Ishibuchi, 1994DX Diagonal (three parents) crossover Eiben et al., 1989


Mutation Operations

Initial Description Reference BCGA2OAS Two operations adjacent swap Murata and Ishibuchi, 1994 3OAS Three operations adjacent swap Murata and Ishibuchi, 1994 2ORS Two operations random swap Murata and Ishibuchi, 1994 3ORS Three operations random swap Murata and Ishibuchi, 1994 IM Inverse mutation Goldberg, 1989 SOM Shift operation mutation Murata and Ishibuchi, 1994 CIM Centre Inverse mutation Tralle, 2000 E2ORS Enhanced two operations random swap Tralle, 2000


Fitness function

Minimise : Pe(Ec+Ep) + Pt(Tp)

Where Ec = max (0, Dc - Fc)

Ep = max (0, Dp - Fp)

Tp = max (0, Fp - Dp)


First Stage (Screening) ExperimentCrossover operatorsMutation

operators CX EERX MPX 1PX OX PBX PMX 2PCX

2OAS A D C B

3OAS B A D C

2ORS C B A D

3ORS D C B A

IM D C B A

SOM D C B A

E2ORS D C B A

CINV D C B A

Parameter settings Coded parameter settings

Combine P/G %C %M Combine P/G %C %M

A 60/20 0.9 0.18 A 1 1 1B 60/20 0.3 0.02 B 1 -1 -1C 20/60 0.3 0.18 C -1 -1 1D 20/60 0.9 0.02 D -1 1 -1


Analysis of Variance(Screening Experiment)

Source DF SS MS F PP/G 1 243.48 243.48 7.06 0.011%C 1 195.78 195.78 5.68 0.022%M 1 21.39 21.39 0.62 0.435COP 7 232.58 33.23 0.96 0.470MOP 7 164.23 23.46 0.68 0.688Seed 1 4.96 4.96 0.14 0.706Seed * %M 1 55.66 55.66 1.61 0.211Seed * %C 1 43.82 43.82 1.27 0.266Seed * P/G 1 76.15 76.15 2.21 0.145Error 42 1448.25 34.48Total 63 2486.30


Relative performance of COP and MOP(Screening Experiment)

CrossoverOperators

Mean StandardDeviation

MutationOperators

Mean StandardDeviation

EERX 103.2 2.28 2OAS 104.6 2.28CX 104.5 2.28 IM 105.4 2.28

PBX 105.3 2.28 CIM 105.4 2.281PX 107.0 2.28 SOM 105.9 2.28PMX 107.3 2.28 2ORS 106.1 2.28MPX 107.6 2.28 E2ORS 107.7 2.28OX 108.6 2.28 3ORS 108.0 2.28

2PCX 109.6 2.28 3OAS 110.0 2.28


Analysis of Variance(Second Stage Experiment)

Source DF SS MS F P%M 1 101.01 101.01 7.36 0.009%C 1 27.38 27.38 1.99 0.164MOP 1 116.42 116.42 8.84 0.005COP 1 138.58 138.58 10.09 0.003P/G 1 0.63 0.63 0.05 0.832Seed 3 54.14 18.05 1.31 0.280COP * P/G 1 56.91 56.91 4.14 0.047Seed * %C 3 107.51 35.84 2.61 0.061Error 51 700.30 13.73Total 63 1302.88


Relative Performance of COP and MOP(Second Stage Experiment)

Crossover and mutation operators Mean Standarddeviation

Enhanced Edge Recombination Crossover (EERX) 105.5 0.692Two Points Centre Crossover (2PCX) 108.4 0.692Two Operations Adjacent Swap (2OAS) 105.4 0.692Three Operations Adjacent Swap (3OAS) 108.5 0.692


Regression Analysis

Predictor Coefficient Standard deviation P valueConstant 106.975 0.485 0.000COP +1.4715 0.485 0.004MOP -1.5615 0.485 0.002%M -1.2563 0.485 0.012%C * P/G -0.9430 0.485 0.057

Penalty cost = £106,975+1,471.5(COP)-1,561.8(MOP)-1,256.3(%M)-943(%C*P/G)


Interaction Diagram for P/G and COP

Interaction diagram for P/G combination and

crossover operator

104

106

108

110

Crossover Operator

Pen

alty

Cos

t (£

1000

)

P/G 60/20 (+1)

P/G 20/60 (-1)

2PCXEERX


Conclusions• BCGA scheduling tool is influenced by a large

number of factors.• The investigation of the best genetic operators and

parameters requires an efficient experimental design to enable the work to be performed within a reasonable time.

• The sequential strategy has been very effective in minimising the amount of time and computational resources


Conclusions (continue)• The screening experiment reduced number of

crossover and mutation operators.• The second experiment showed that the choice of

operators was statistically significant.• It also found that the low level of P/G combination

produced the best results when used with the EERX crossover operator.

• The different findings emerging from previous work suggests that appropriate GA operators and parameters may be case dependent.


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© P. Pongcharoen ISA/1 Applying Designed Experiments to Optimise the Performance of Genetic Algorithms for Scheduling Capital Products P. Pongcharoen,