A GENETIC A LGORITHM FOR PARALLEL MACHINE TOTAL TARDINESS PROBLEM

1. A GENETIC ALGORITHM FOR PARALLEL MACHINE TOTAL TARDINESS PROBLEM M. Furkan Kıraç Ümit Bilge Müjde Kurtulan Department of Industrial Engineering Boğaziçi University EURO / INFORMS İstanbul 2003 July 06-10

2. Objective • Genetic Algorithms are rooted from a strong idea with a simplebasic mechanics that involves only the process of copying strings and swapping partial strings. • Implicit parallelism which traverse the search space climbing many hills in parallel. • However GAs are prone to premature convergence and impose numerous parameters to fine-tune. • In this study, a genericadaptive control mechanismto slow down or prevent this premature convergence and reduce the parameter dependence of a BasicGenetic Algorithm (GA) is developed and implemented over a hard to solve problem: TheParallel Machine Total Tardiness Problem (PMTT). • The fundamental elements of GA are investigatedand the solutionstrategy developed is benchmarked with the literature for performance evaluation. EURO / INFORMS İstanbul 2003 July 06-10

3. Outline • Problem definition and characteristics for PMTT • Basic Genetic Algorithm (GA) approach to PMTT and experimentation • Adaptive Control Mechanism over Basic GA and experimentation • Results compared to literature • Conclusions EURO / INFORMS İstanbul 2003 July 06-10

4. Parallel Machine Total Tardiness Problem • ‘n’ independent jobs to be scheduled on ‘m’ uniform parallel machines • Each job has • a distinct ready timeri • a distinct due date di • an integer processing time pi • Sequence dependent setup time sij • Objective is to minimize the total tardiness of all the jobs, ∑Ti, • Ti is the respective tardiness of job i calculated as Ti = max{0, Ci - di} • Ci is the completion time of job i EURO / INFORMS İstanbul 2003 July 06-10

5. Problem Characteristics • In most studies from the literature the general assumption is that • the machines are identical • all jobs are available at time zero • and setup times do not exist • These assumptions are far too simplistic when confronted with the real world situations • In this study, these features are also incorporated into the model to approach the problem with real world situations • Each machine in our problem set has a speed factor associated with it. Machines are not identical. EURO / INFORMS İstanbul 2003 July 06-10

6. Chromosome Encoding for PMTT • The chromosome representation used encodes each job in the schedule as a gene on the chromosome • Machine sequences are separated by an asterisk (*) on the chromosome EURO / INFORMS İstanbul 2003 July 06-10

7. Details of Basic GA Algorithm • Initial population:Random population + solutions generated by list scheduling heuristics such as EDD, SPT, SST, ERT • Parent selection:Ranking Roulette WheelLess bias is introduced since the fitness values are based on a ranking of the total tardiness values • Crossover operator: Uniform order-based crossoverThe crossover operator generate a binary string where the number of “1”s and “0”s can be controlled. This binary string is used as a template to combine the genetic information and properties of the two parents. • Mutation operator:Swap operationConsists of swapping two randomly selected genes. EURO / INFORMS İstanbul 2003 July 06-10

8. Crossover operator EURO / INFORMS İstanbul 2003 July 06-10

9. Transient Population Generation • The population generation method is Transient • Creates a transient phase in the progress from one generation to the next • Transient population consists of the old population and the new offspring, where • N is population size • Nc is number of children produced • To keep the population size constant, Nc individuals need to be eliminated • Gives a greater chance of survival to the old populationmembers as long as they are fit enough EURO / INFORMS İstanbul 2003 July 06-10

10. SORTED TRANSIENT POPULATION consisting of 150 individuals N = 100 Nc = 50 Transient Population Elimination Basic Elimination Best 53 individuals preserved 48 individuals eliminated Worst 2 individuals eliminated EURO / INFORMS İstanbul 2003 July 06-10

11. Analysis of Basic GA • GA has a high number of parameters that can be regulated for higher performance, but this introduces the difficulty of fine-tuning the parameters • Population Size • New Generation Creation Method • Fitness Evaluation Method • Parent Selection method • Crossover Probability & Operator • Mutation Probability & Operator • Mutation Strength • … • GA is prone to the risk of premature convergence • i.e. the population converges to a set of good performing and highly similar members or • to an individual without having much chance of generating representatives of diverse hyperplanes of the solution space EURO / INFORMS İstanbul 2003 July 06-10

12. Unstable ? Why not control it ? • The weakness of GAs can be attributed to the high sensitivity of the GA parameters • strong parameter dependence affects the robustness • Therefore, the GA can be termed as unstable from the control theory point of view • When a system is defined as unstable, the natural attitude is to try to control it • Classical control theory proposes closed-loop systems for robust control of a system EURO / INFORMS İstanbul 2003 July 06-10

13. CONTROLLER  output reference error + - sensor K Closed-loop Control Systems • A closed-loop system is one that considers the output of the previous state as a feedback input for the successive state • In this study, a control mechanism consisting of two complementary subcomponents is devised EURO / INFORMS İstanbul 2003 July 06-10

14. Adaptive Control over Basic GA • Preliminary experiments performed with Basic GA indicate that the problem under study favors rather high mutation rates • high diversity within the GA search • Therefore, the population diversity is the first performance indicator to be controlled for higher performance • aims to overcome the risk of premature convergence due to the dominance of some fit individuals • Additionally, a training mechanism is developed • designed to operate on the weak offspring in the population to bring them to a level of maturity EURO / INFORMS İstanbul 2003 July 06-10

15. Diversity Control • An adaptive mechanism to control the population diversity whenever it deviates from a threshold value is developed • The operating principle is simple • in that whenever the population diversity falls below a given percentage, the control mechanism is triggered • A set of diversifying operations are performed on the population • At the end of these moves the population diversity increases and the Basic GA is resumed until diversity falls below the threshold level EURO / INFORMS İstanbul 2003 July 06-10

16. Control for Population Diversity EURO / INFORMS İstanbul 2003 July 06-10

17. Effect of Adaptive Diversity Control Bar charts showing the population distribution BEFORE The instant when the diversity threshold is reached and the control mechanism is triggered AFTERBy the operation of diversity control, the peak consisting of converged individuals is suppressed and the population distribution is smoothed # of individuals 66 # of individuals 22 tardiness 800000 tardiness 160000 EURO / INFORMS İstanbul 2003 July 06-10

18. Training • In order to further exploit the recombining strength of the crossover operator, an adaptation from real life occurrences is introduced at this stage • This is called “training” based on the argument that a newborn child is not capable of surviving in the environment without first going through training • This concept is extended to encompass the entire set of unfit individuals in the population instead of just the offspring EURO / INFORMS İstanbul 2003 July 06-10

19. Training Parameters • The trigger of training is a performance measure of the system that stimulates steepest descent when the search stagnates for a proportion of the entire search duration • This proportion is set to be 1.0%, i.e. 100 non-improving generations • the duration of the training session applied over each of the individuals(number of iterations for which steepest descent will take over ) • the number of individuals to be educated EURO / INFORMS İstanbul 2003 July 06-10

20. Effect of Training Control AFTER In other words, the function of training can be defined as decreasing the skewness in the population distribution. BEFORE The function of the training phase is to improve the fitness of the worst population members so that the population distribution curve is smoothed out # of individuals 25 # of individuals 26 tardiness 500000 tardiness 180000 EURO / INFORMS İstanbul 2003 July 06-10

21. Effect of Diversity andTraining Control EURO / INFORMS İstanbul 2003 July 06-10

22. Experimentation • The problem set used for experimentation consists of parallel machine scheduling problems of 40, and 60 jobs, developed and tested by Sivrikaya-Şerifoğlu, F. and G.Ulusoy to study a GA • The same problem set is addressed by Bilge,Ü., F.Kıraç, M. Kurtulan and P. Pekgün in a deterministic TS approach • These problem sets are as follows: • Instances with n = 40, and n = 60 were randomly generated (n: number of jobs) • Number of machines, m, is 2 or 4 • 20 distinct instances generated for each group. EURO / INFORMS İstanbul 2003 July 06-10

23. Performance Measure • K is the number of problem instances over which the values are evaluated (20 in this case) • Performance measure used in this study is a comparative relative measure which takes the best-known TS values for the problem instances reported in the literature [Bilge et al.] as a basis where, i = 1, 2, 3, 4, 5 denotes different replications • j = 1, 2, …, 20 denotes the instance number in a given problem set EURO / INFORMS İstanbul 2003 July 06-10

24. Performance Ratio (PR) Best Known Result TS GA • This ratio is used for a comparison of the relative achievements obtained via each metaheuristic • The aim in this study is to obtain a ratio as low as possible • A ratio greater than 1.0 means that the GA’s performance is worse than the TS presented in [Bilge et al.] on the average. • A ratio of 1.0 means that the average behavior of the GA is comparable to the average behavior of the TS presented in [Bilge et al.] • A ratio less than 1.0 means that the average results obtained by the GA is better than the TS presented in [Bilge et al.] • A ratio less than 0.0 means that the best known values in the literature are improved by the GA . EURO / INFORMS İstanbul 2003 July 06-10

25. Performance of Adaptive GA Diversity Non-Mutants = 10 out of 100 (Best fit individuals) Number of Trainees = 20 out of 100 (Worst fit individuals) Training Duration = 15 (Steepest Descent Steps) EURO / INFORMS İstanbul 2003 July 06-10

26. Improved Best Known Results Those values marked with a (*) are contributed by the adaptive GA algorithm devised in this study EURO / INFORMS İstanbul 2003 July 06-10

27. Conclusion • The major enhancement brought to the GA concept in this study is the genericadaptive control mechanism which aims to better exploit its strengths by diminishing its high parameter dependence • Population diversity is selected as the system output upon which the adaptive GA approach is based • In order to achieve a closed-loop form for the controller over the Basic GA, two complementary control strategies that operate upon different triggers are implemented • They complement each other such that wheneverone of them is triggered, the result causes the other strategy to be triggered. EURO / INFORMS İstanbul 2003 July 06-10

28. Conclusion • Our usage of steepest descent algorithm as the base of the training control mechanism is somewhat different from its proposed applications in the literature. Most studies propose climbing heuristics after the GA has converged to various local optima. This strategy can still be implemented over our approach. • Different control mechanisms and triggers can be developed for faster and more effective traversal of the search space. We only provided a certain way of forming a valid closed loop control system. EURO / INFORMS İstanbul 2003 July 06-10