Stewardson-Strasburg CUSD #5A A Rookie’s View to the First 60 Days of Becoming a New PBIS District.
P. Pongcharoen, D.J. Stewardson, C. Hicks and P.M. Braiden. University of Newcastle upon Tyne
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Transcript of P. Pongcharoen, D.J. Stewardson, C. Hicks and P.M. Braiden. University of Newcastle upon Tyne
© 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
© P. Pongcharoen ISA/2
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)
© P. Pongcharoen ISA/3
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
© P. Pongcharoen ISA/4
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.
© P. Pongcharoen ISA/5
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
© P. Pongcharoen ISA/6
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)
© P. Pongcharoen ISA/7
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
© P. Pongcharoen ISA/8
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.
© P. Pongcharoen ISA/9
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
© P. Pongcharoen ISA/10
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
© P. Pongcharoen ISA/11
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
© P. Pongcharoen ISA/12
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
© P. Pongcharoen ISA/13
Fitness function
Minimise : Pe(Ec+Ep) + Pt(Tp)
Where Ec = max (0, Dc - Fc)
Ep = max (0, Dp - Fp)
Tp = max (0, Fp - Dp)
© P. Pongcharoen ISA/14
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
© P. Pongcharoen ISA/15
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
© P. Pongcharoen ISA/16
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
© P. Pongcharoen ISA/17
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
© P. Pongcharoen ISA/18
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
© P. Pongcharoen ISA/19
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)
© P. Pongcharoen ISA/20
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
© P. Pongcharoen ISA/21
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
© P. Pongcharoen ISA/22
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.
© P. Pongcharoen ISA/23
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