A Genetic Algorithm with Injecting Artificial Chromosomes for Single Machine Scheduling Problems

1. A Genetic Algorithm with Injecting Artificial Chromosomes for Single Machine Scheduling Problems

2. Contents Evolutionary Algorithm with Probability Models • Introduction Single Machine Scheduling (Ei+Ti) • Problem Statement Injecting Artificial Chromosome • Methodology Parameter Selection and Method Comparisons • Empirical Results The EAPM might be effective • Conclusions http://ppc.iem.yzu.edu.tw

3. Scheduling • Single Machine • Earliness/Tardiness • Exact Algorithm • Heuristics • Meta-Heurisitc • Genetic Algorithm Algorithm Problem Research Framework Introduction http://ppc.iem.yzu.edu.tw

4. Introduction Selection is to preserve better chromosomes to be survived. Mutation exploits local information of current chromosomes. Mutation Selection Crossover Crossover mates two individuals into two new offspring so that it explores the solution space. http://ppc.iem.yzu.edu.tw

5. EAPM Memetic Sexual GAs NSGA II VEGA SPEA 2 Introduction Problem Algorithms • Nature Behavior • Probability Model • Improvements • Continuous • Combinatorial • Single/Multi Objective http://ppc.iem.yzu.edu.tw

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

7. Zhang classified these algorithms into EDA. For extensive review of evolutionary algorithm base on probability models, please refer to Larrañaga and Lozano. Zhang et al. Chang et al. • 2005 • Guided Mutation or • Mutation Matrix • 2005 and 2007 • Artificial Chromosome • Single/Multi Objective problem Evolutionary Algorithm with Probability Models Muhlenbein and Paaß Bajula & Davies Ackley • 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 http://ppc.iem.yzu.edu.tw

8. n n n i=1 i=1 j=1 Problem Statement Min Z = Σ(αiEi+βiTi) s.t. Σxij=1 j =1 to n Σxij=1 i =1 to n Ci－di－Ei + Ti = 0 xij {0,1} 20Jobs 30Jobs 40Jobs A 50Jobs A 60Jobs 90Jobs Testing instances: Sourd (2005)http://www-poleia.lip6.fr/~sourd/ http://ppc.iem.yzu.edu.tw

9. Main Procedure • Initiate Population • ConstructInitialPopulation(Population) • RemovedIdenticalSolution() • counter  0 • while counter < generationsdo • Evaluate Objectives and Fitness() • FindEliteSolutions(i) • if counter < startingGen or counter % interval != 0 do • Selection with Elitism Strategy() • Crossover() • Mutation() • TotalReplacement() • else • CalculateAverageFitness() • CollectGeneInformation() • GenerateArtificialChromsomomes() • Replacement(μ+λ) • Endif • counter  counter + 1 • endwhile 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 http://ppc.iem.yzu.edu.tw

10. 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. http://ppc.iem.yzu.edu.tw

11. Artificial Chromosome • Extract Chromosome Information • Proportional Selection • Replacement http://ppc.iem.yzu.edu.tw

12. Step 1 • To extract the population information. • A data structure called dominance matrix store it. http://ppc.iem.yzu.edu.tw

13. Step 2 • Job assignment by probability selection http://ppc.iem.yzu.edu.tw

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

15. ACGA • Starting Generation: 250 • Interval: 50 Parameter settings SGA • Population Size: 100 • Crossover Rate: 0.9 • Mutation Rate: 0.5 http://ppc.iem.yzu.edu.tw

16. Convergence Diagram http://ppc.iem.yzu.edu.tw

17. Results: The average objective of 20 jobs http://ppc.iem.yzu.edu.tw

18. Results: The average objective of 30 jobs http://ppc.iem.yzu.edu.tw

19. Results: The average objective of 40 jobs http://ppc.iem.yzu.edu.tw

20. Summary Relative Average Error Ratio 0.173% ACGA 9.971% SGA ACGADP GADP 0.109% 0.251% http://ppc.iem.yzu.edu.tw

21. Results: The average objective of 50 jobs http://ppc.iem.yzu.edu.tw

22. Results: The average objective of 60 jobs http://ppc.iem.yzu.edu.tw

23. Results: The average objective of 90 jobs http://ppc.iem.yzu.edu.tw

24. ANOVA http://ppc.iem.yzu.edu.tw

25. The worst solution quality Outperform other algorithms. Better than SGA and works efficiently SGA GADP ACGA Pair-wise Comparison http://ppc.iem.yzu.edu.tw

26. Conclusions Artificial Chromosomes Genetic Algorithm with injecting artificial chromosomes Genetic Operators Probability Model http://ppc.iem.yzu.edu.tw

27. Conclusions Effective HybridMethods Simple • ACGA outperform others • It is easy to implement • It can be applied with other meta-heuristic The benefits of ACGA http://ppc.iem.yzu.edu.tw

28. Thank You !