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The ordinal optimisation of genetic control parameters for flow shop scheduling

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Abstract

Genetic algorithms (GAs) have been widely applied for many non-polynomial hard optimisation problems, such as flow shop and job shop scheduling. It is well known that the efficiency and effectiveness of a GA is highly depend on its control parameters, but setting suitable parameters often involves tedious trial and error. Currently, setting optimal parameters is still a substantial problem and is one of the most important and promising areas for GAs. In this paper, the determination of optimal GA control parameters with limited computational effort and simulation replication constraints, namely, population size, crossover and mutation probabilities, is firstly formulated as a stochastic optimisation problem. Then, the ordinal optimisation (OO) and the optimal computing budget allocation (OCBA) are applied to select the optimal GA control parameters, thereby providing a reasonable performance evaluation for hard flow shop scheduling problems. The effectiveness of the methodology is demonstrated by simulation results based on benchmarks.

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Abbreviations

n :

number of jobs

m :

number of machines

p i,j :

processing time of job i on machine j

C max, C̄ max :

makespan, average makespan

f :

fitness value

J, J̄ :

expectation and mean makespan for a certain parameter combination

L :

sample performance

P s :

population size

p c , p m :

crossover probability and mutation probability

P(k):

population at kth generation

X, X′, X″ :

an individual and temporary individual

r :

a random number between 0 and 1

θ=(P s , p c , p m ):

a parameter combination of P s, p c , p m

θ*:

the true best parameter combination

θ b :

the best parameter combination determined by the mean makespan

N e :

total number of evaluation for the GA

N g :

maximum generation number for the GA

N s :

the number of the alternative choice for the population size

N c :

the number of the alternative choice for the crossover probability

N m :

the number of alternative choice for the mutation probability

K=N s ×N c ×N m :

the total number of designs or parameter combinations

P(CS):

the probability of the correct selection

APCS :

the approximate value of P(CS)

P* :

the pre-defined confidence level for the correct selection

T :

the total simulation replications

T i :

the simulation replications for the ith parameter combination

\(\sigma ^{2}_{i} \) :

the sample variance of makespan for the ith parameter combination

δ b,i :

δ b,i =C̄ max(θ b )−C̄ max(θ i )

\(\sigma ^{2}_{{b,i}} \) :

\(\sigma ^{2}_{{b,i}} = {\sigma ^{2}_{b} } \mathord{\left/ {\vphantom {{\sigma ^{2}_{b} } {T_{b} }}} \right. \kern-\nulldelimiterspace} {T_{b} } + {\sigma ^{2}_{i} } \mathord{\left/ {\vphantom {{\sigma ^{2}_{i} } {T_{i} }}} \right. \kern-\nulldelimiterspace} {T_{i} }\)

s :

the iteration number for OCBA

l :

the iteration number for the genetic operators

k :

the integer for the counting generation

i,j :

the indication number

n 0 :

the initial simulation replications in OCBA

Δ:

the increment of simulation replications in OCBA

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Acknowledgements

The authors would like to thank Professor Y. C. Ho (Harvard University) for his helpful discussions. This research is partially supported by the National Science Foundation of China (60204008, 60374060) and 973 program (2002CB312200).

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Authors and Affiliations

  1. Department of Automation, CFINS, Tsinghua University, Beijing 100084, PR China

    L. Wang, L. Zhang & D.-Z. Zheng

Authors
  1. L. Wang
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  2. L. Zhang
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  3. D.-Z. Zheng
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Correspondence to L. Wang.

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Wang, L., Zhang, L. & Zheng, DZ. The ordinal optimisation of genetic control parameters for flow shop scheduling. Int J Adv Manuf Technol 23, 812–819 (2004). https://doi.org/10.1007/s00170-002-1509-6

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