Yuan-Ze University A Genetic Algorithm with Injecting Artificial Chromosomes for Single Machine Scheduling Problems Pei-Chann Chang, Shih-Shin Chen, Qiong-Hui Ko, Chin-Yuan Fan.

Evolutionary Algorithm with Probability Models Single Machine Scheduling (Ei+Ti) Injecting Artificial Chromosome Parameter Selection and Method Comparisons The EAPM might be effective  Introduction  Problem Statement  Methodology  Empirical Results  Conclusions Contents

Introduction Research Framework Scheduling Single Machine Earliness/ Tardiness Problem Exact Algorithm Heuristics Meta-Heurisitc Genetic Algorithm Algorithm

Introduction Mutation exploits local information of current chromosomes. Crossover mates two individuals into two new offspring so that it explores the solution space. Selection is to preserve better chromosomes to be survived. Selection Crossover Mutation

Introduction ProblemAlgorithms  Continuous  Combinatorial  Single/Multi Objective  Nature Behavior Probability Model Improvements EAPM Memetic GAs Sexual VEGA NSGA II SPEA 2

EAPM Primary Steps Main Procedures Three general steps Selection is required Characteristics Crossover is not used. Mutation is not used. An explicit probability model Step 3 generates a population of chromosomes by probability model. Step 2 extracts gene information from population. Step 1 is to evaluate chromosomes’ fitness and to select better chromosomes.

Evolutionary Algorithm with Probability Models Ackley Bajula & Davies Muhlenbein and Paaß 1987 Feedback from population Voting Population-Base Incremental Learning (PHIL) Combining Optimizers with Mutual Information Tree (COMIT) 1999 Compact Genetic Algorithm (cGA) Replace crossover and mutation operator Zhang et al. Chang et al. Zhang classified these algorithms into EDA. For extensive review of evolutionary algorithm base on probability models, please refer to Larrañaga and Lozano and 2007 Artificial Chromosome Single/Multi Objective problem 2005 Guided Mutation or Mutation Matrix

Problem Statement n i=1 Min Z = Σ(α i E i +β i T i ) s.t. Σx ij =1 j =1 to n Σx ij =1 i =1 to n C i － d i － E i + T i = 0 x ij {0,1} n i=1 n j=1 A A 20 Jobs 30 Jobs 40 Jobs 50 Jobs 60 Jobs 90 Jobs Testing instances: Sourd (2005)

Main Procedure Population: The population used in the Genetic Algorithm Generations: The number of generations startingGen: It determines when does the AC works interval: The frequency to generate artificial chromosomes 1.Initiate Population 2.ConstructInitialPopulation(Population) 3.RemovedIdenticalSolution() 4.counter  0 5.while counter < generations do 6. Evaluate Objectives and Fitness() 7. FindEliteSolutions(i) 8. if counter < startingGen or counter % interval != 0 do 9. Selection with Elitism Strategy() 10. Crossover() 11. Mutation() 12. TotalReplacement() 13. else 14. CalculateAverageFitness() 15. CollectGeneInformation() 16. GenerateArtificialChromsomomes() 17. Replacement(μ+λ) 18. End if 19. counter  counter end while

Genetic Operators  Selection:  tournament selection has better convergence and computational time-complexity properties than others. (Goldberg Deb, 1991)  Crossover:  Murata and Ishibuchi (1994) reported that two- point crossover is effective in scheduling problems.  Mutation:  Swap mutation operator is used because of its simplicity.

Artificial Chromosome  Extract Chromosome Information  Proportional Selection  Replacement

Step 1  To extract the population information.  A data structure called dominance matrix store it.

Step 2  Job assignment by probability selection

Empirical Results  Sourd (2005) provided single machine Ei/Ti instances.  Parameter settings: By Design of Experiment (DOE)  Replications: 30 times Hybrid Algorithm Artificial Chromosomes Genetic Algorithm with Dominance Properties (ACGADP) Simple Genetic Algorithm (SGA) Artificial Chromosome Genetic Algorithm (ACGA) Genetic Algorithm with Dominance Properties (GADP)

Parameter settings Population Size: 100 Crossover Rate: 0.9 Mutation Rate: 0.5 SGA Starting Generation: 250 Interval: 50 ACGA

Convergence Diagram

Results: The average objective of 20 jobs InstanceSGAGADPACGAACGADP sks222a sks225a sks228a sks252a sks255a sks258a sks282a sks285a sks288a

Results: The average objective of 30 jobs InstanceSGAGADPACGAACGADP sks322a sks325a sks328a sks352a sks355a sks358a sks382a sks385a sks388a

Results: The average objective of 40 jobs InstanceSGAGADPACGAACGADP sks422a sks425a sks428a sks452a sks455a sks458a sks482a sks485a sks488a

Summary Relative Average Error Ratio SGA ACGA GADP ACGADP 9.971% 0.251% 0.173% 0.109%

Results: The average objective of 50 jobs InstanceSGAGADPACGAACGADP sks522a sks525a sks528a sks552a sks555a sks558a sks582a sks585a sks588a

Results: The average objective of 60 jobs InstanceSGAGADPACGAACGADP sks622a sks625a sks628a sks652a sks655a sks658a sks682a sks685a sks688a

Results: The average objective of 90 jobs InstanceSGAGADPACGAACGADP sks922a sks925a sks928a sks952a sks955a sks958a sks982a sks985a sks988a

ANOVA SourceDFSeq SSAdj SSAdj MSFP instance E E method36.52E E instance*me thod E Error E Total E+12

Pair-wise Comparison Duncan Grouping Mean N Method A SGA B GADP B C B ACGADP C C ACGA ACGA The worst solution quality GADPSGA Better than SGA and works efficiently Outperform other algorithms.

Conclusions Artificial Chromosomes Genetic Operators Probability Model Genetic Algorithm with injecting artificial chromosomes

Conclusions Effective Simple Hybrid Methods The benefits of ACGA ACGA outperform others It is easy to implement It can be applied with other meta-heuristic

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