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Poster Paper Proc. of Int. Joint Colloquium on Emerging Technologies in Computer Electrical and Mechanical 2011

Comparison of Various Crossover Techniques of Genetic Algorithm for Optimum Job Sequencing Based on Minimum Elapsed Time Seema1 and Sunil Kumar2 1

Assistant Professor (Computer Engineering), Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo, Distt. Bathinda (Punjab) India Email: garg_seema238@yahoo.co.in 2 Assistant Professor (Mechanical Engineering), Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo, Distt. Bathinda (Punjab) India, Email: sunilbaghla@yahoo.co.in

job shop production both sequencing and layout. Omar et al [2] proposed GA based algorithm designed on well-known priority rules such as shortest processing time and longest processing time using critical block neighborhood and distance measure. Raghavendra et al. [3] applied GA to workload problems on the basis of shortest processing time and longest processing time for loading the work-in-process jobs with the objective of minimize the imbalance of workload among the parallel machines. Hasan and Sarkar [4] proposed a GA for JSSP’s with an objective of minimizing make-span while satisfying a number of hard constraints after introduction of number of priority rules such as partial reordering, gap reduction and restricted swapping to improve the performance of the GA. Yamada and Nakano [5] developed GA based methodology to calculate a set of completion times for each operation that satisfies some constraints for travelling salesman and job sequencing problems and succeeded in optimizing the problem to determine the schedule which minimizes total travel time.

Abstract—Classical methods like johnson method can only be applied to find optimum job sequence for job sequencing problem if certain inequalities are satisfied, otherwise they don’t give any valid solution to the problem especially when large number of jobs are processed on large number of machine (100 x 100 problem) . In present paper, Genetic Algorithm is proposed to solve the job sequencing problems for which the classical methods are not able to provide any solution. The classical johnson algorithm and proposed Genetic Algorithm are implemented using C programming for various job sequencing problems. The comparison of results of both algorithms will demonstrate the validity and strength of Genetic Algorithms as compared to classical Johnson method. The designed Genetic Algorithm is also implemented for job sequencing problem using different crossover techniques to compare the effectiveness of these techniques for calculation of the optimum job sequence while machining. Index terms: Job Sequencing, Crossover, Mutation, Fitness, Chromosome, offspring

I. INTRODUCTION III. TECHNIQUES FOR JOB SEQUENCING

Genetic Algorithms can be applied to optimization and constraint satisfaction problems, which are NP-hard in nature e.g. Job sequencing, traveling salesman, space allocation and map coloring problems etc. Solution to job sequencing problems is very important as it helps to reduce inventory, improve productivity and efficiency of any enterprise. The Genetic Algorithms are based on phenomenon of natural selection and natural genetics. In Genetic Algorithms, a population of some feasible solutions for a given problem is maintained. The population undergoes evolution in the form of natural selection. In each generation relatively “good” solutions reproduce and relatively “bad” solutions die, to be replaced by offspring of the good.

Two types of Algorithms exist for NP-Hard problems discussed below. 1. Exact Algorithms that find exact solutions and converge to optimum value at faster rate e.g. Johnson Method, graphical method etc. They will work reasonably fast only for relatively small problem sizes. 2. Heuristic Algorithms that provide, not exact but probably good solutions, satisfying some constraints. They provide solution close to exact/optimum solution satisfying a set of boundary conditions and constraints e.g. Genetic Algorithm, Particle Swarm Optimization etc. A. Johnson Method for Job Sequencing The Johnson method is used to find the optimum job sequence for job sequencing problems. Certain inequalities are to be satisfied for conversion of higher machine problems to 2-machine problems. The objective is to determine the sequence of jobs which will allow all jobs to be performed so that the time, from the beginning of the first job till the

II. LITERATURE REVIEW Taha et al [1] explained that due to the combinatorial nature of the solution spaces, genetic algorithm is applied for both sequencing and layout. A case study is also reported to demonstrate the feasibility of the proposed approach in © 2011 ACEEE DOI: 02.CEM.2011.01.554

Poster Paper Proc. of Int. Joint Colloquium on Emerging Technologies in Computer Electrical and Mechanical 2011 completion of the last job, is minimum. The Johnson method to solve the ‘nx2’ problem include the inequalities satisfaction. If Minimum time (Aj, Bj+1) equals Minimum time (Aj+1+, Bj), where A and B the two machines on which the ‘n’ jobs are to be processed, then ‘job j’ would be indifferent to job ‘j+1’ any of them could precede the other. This rule can be extended to find the optimal sequence of a set of given jobs. Starting with any sequence S0, the optimal sequence S* can be obtained by successive interchange of consecutive jobs by applying this rule. In operational terms, the algorithm can be stated in a stepwise manner as below: (a) Select the least processing time, for all jobs on all machine (b) if the minimum is for Ar, for the rth job on machine A, do the rth job on the first machine. (c) If the minimum is for Bs, for the sth job on machine B, do the sth job in the end. (d) In case of tie between Ar and Bs, perform the rth job first & sth in last. (e) If there is tie between 2 or more timing in either of the series, select either the jobs involved and perform the first or last. Repeat above steps until all jobs are assigned. The higher order problem involving three or more machines is first converted into 2-machine problem and then solved using above mentioned algorithm to get the solution.

the jobs is described with an array. Each element of the array represents the number of the job. B. Fitness Function The objective function, the function to be optimized, provides the mechanism for evaluating the fitness of each chromosome. In JSP, the objective is to calculate time for each sequence of the job processing and total time elapsed for all sequences in the population. The fitness function is defined as n fit (fx) =

M (ji) T (ji)

...(1)

i=1 where M is the machine and T is the time elapsed on the machine. C. Selection Method In present algorithm, steady-state selection is used. The best fit chromosomes are selected for crossover and mutation to generate new offspring’s according to phenomenon of natural selection. D. Crossover Operator In present algorithm, three crossover techniques namely Order Crossover, Position Crossover and Partially Mapped Crossover are investigated. The GA is implemented on the problem-1 & problem-2 using above three crossover techniques to find optimum job sequence using C language on Intel P-IV processor with 1GB RAM. Table 1 shows the results of optimal job sequence, processing time and computer time for run of the computer program.

B. Genetic Algorithm for Job Sequencing Genetic Algorithm is modeled after the processes of evolution and genetic recombination. The building blocks of the algorithm are named after genetic elements. Genetic Algorithm are inspired by Darwin’s theory of the survival of the fittest, which states that in nature, competition among individuals for scant resources such as food and space results in the fittest individuals dominating over weaker ones. Only the fittest individual will survive and reproduce, while the weaker individuals die out. Genes are the binary encoding of each problem variable, and of the genes as a string are referred to as a chromosome. The GA procedure is detailed below. 1. Generate population of chromosomes 2. Evaluate the fitness of each chromosome in the population. Find the sequence incurring minimum processing time. 3. Update new population by repeating the following a. Select two parent chromosomes from the population for crossover with crossover probability to form new offspring sequences. b. Select one parent chromosomes from the population for mutation with mutation probability to form new offspring sequence. c. Replace old offspring with new off springs. 4. This process repeats until either a desired fitness has been achieved or a set number of generations have occurred.

TABLE 1:CROSSOVER TECHNIQUES FOR JOB SEQUENCING PROBLEMS

The result show that all the crossover techniques are equally accurate as far as finding optimal sequence and processing time on machine is concerned but Genetic Algorithm using partially mapped crossover takes less computer time resulting in faster execution of the program. Based on the above investigation, partially mapped crossover (PMX) is used for generation of the child offspring’s in present methodology. Under PMX, two strings are aligned and two crossing sites are picked uniformly at random along the strings. These two points define a matching section that is used to affect a cross through ‘position-by-position’ exchange operations.

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Poster Paper Proc. of Int. Joint Colloquium on Emerging Technologies in Computer Electrical and Mechanical 2011 Consider the parents with cross-points indicated by |: P1 = (9 8 4 | 5 6 7 | 1 3 2 10) and P2 = (8 7 1 | 2 3 10 | 9 5 4 6)

inequality or condition is required to be satisfied to apply GA. The output of the GA program demonstrates the above fact. The GA results in valid optimum sequence of problem-2 for which Johnson method failed to generate any optimum sequence.

Two resulting child sequences after crossover after performing PMX are S1 = (9 8 4 | 2 3 10 | 1 6 5 7) and S2 = (8 10 1| 5 6 7 | 9 2 4 3)

VI. CONCLUSION Genetic Algorithm is efficient method for optimization of NP-hard problems e.g. job sequencing and traveling salesman problems, where the conventional methods cease to converge and give sub-optimal or no solution. An attempt is made to solve Job Sequencing problems using Genetic Algorithms. They eliminate the inaccuracies that occur in classical methods. Genetic Algorithm solves the problems of job sequencing iteratively for which Johnson method failed to generate any optimum sequence.

E. Mutation Operator In present implementation, swap mutation operator is used. In swap mutation, two alleles at random are picked and their positions are swapped preserving most of adjacency information. String before Mutation String after mutation 1 0 0 %1 0 1 %1 1 0 1 1 %1 0 0 %0 1 site 1 site 2 site 1 site 2 Fig 2: Swap Mutation

VI. APPENDIX-I Problem-1

VI. RESULTS & DISCUSSIONS A. Results The Johnson Method and Genetic Algorithm were implemented on problem-1 and problem-2 (appendix-I). The GA was implemented using partially mapped crossover and swap mutation using the program designed on turbo C++ compiler. Both the problems (6 jobs and 3 machines) are attempted for population size of 1000 and 5000 generations. The results of both implementations are detailed below. Output of GA based method (Problem-1) OPTIMAL SEQUENCE FOR 6-JOB, 3-MACHINE IS: 4=>3=>1=>6=>2=>5, Time: 77 hrs, Computer Time: 5 sec Output of Johnson Method (Problem-1) OPTIMAL SEQUENCE FOR 6-JOB, 3-MACHINE IS: 4=>3=>1=>6=>2=>5, Time: 77 hrs,Computer Time: 19 sec Output of GA based method (Problem-1) OPTIMAL SEQUENCE FOR 6-JOB, 3-MACHINE IS 4=>1=>6=>2=>5=>3,Time: 77hrs, Computer Time: 6 sec Output of Johnson Method (Problem-1) INEQUALITIES NOT SATISFIED, METHOD FAILS

Time matrix:

8 6 4 6 3 1

13 14 9 12 8 13

3 12 5 2

9 6 4 6

13 14 9 12

Problem 2:

Time matrix:

VI. REFERENCES [1]. Z. Taha, F. Tahriri, A. Zuhdi, “Sequencing and Layout Optimization in Virtual Production Line,” Journal of Quality, Vol. 18, No. 4, pp: 351-374, 201 [2]. M. Omar, A. Baharum, Y.A. Hasan, “A Job-shop scheduling problem using Genetic Algorithm,” IMT-GT Regional Conference on Mathematics, Statistics and applications, 2006 [3]. B.V. Raghavendra, A.N.N. Murthi, M. Jayaram, “Job Sequence to minimize the workload Imbalance on parallel machines through Genetic Algorithm”, International Journal of Engineering Science and Technology, , pp: 51-59, 2011 [4]. S.M.K. Hasan, R. Sarkar., “A with Priority Rules for Solving Job-Shop Scheduling Problems”, IEEE Proceedings, 2 010 [5]. T. Yamada, R. Nakano, “Genetic Algorithms for Job-Shop Scheduling Problems”, Modern Heuristic for Decision Support, pp. 67-81, 1997 [6]. T. Yamada, R. Nakano, “Scheduling by Genetic local search with Multi-step Crossover”. In 4th PPSN, pp. 960-969, 1997 [7]. M. Mitchell, “An Introduction to Genetic Algorithms”, MIT Press, Cambridge, MA, 1996 [8]. G. Syswerda, “Schedule optimization using Genetic Algorithms”, Handbook of Genetic Algorithms, pp. 332-349, 1991 [9]. J.C. Werner, M.E. Aydin, T.C. Fogarty, “Evolving genetic algorithm for Job Shop Scheduling problems”, Proceedings of ACDM, University of Plymouth, UK, 2010 [10]. T. Yamada, R. Nakano, “Job-shop scheduling”, IEEE control engineering: The Institution of Electrical Engineers, India, 1997

B. Discussions The results of implementation on problem-1 reveal that the output of both implementation is same i.e. both algorithm results in same optimum job sequence that validate the GA for job sequencing problems as it generate same output sequence as Johnson method did. The only difference lies in the computer time taken for execution of the computer program. The GA based program taken almost 75% lesser time for execution of the program. The results of implementation on problem-2 demonstrate the power of GA as compared to Johnson method. The Johnson method is not able to solve the Job Sequencing Problem because the inequality to convert the higher order problem to two-machine problem is not satisfied. So Johnson method fails for this class of problem. In contrary to it, the GA can be implemented, due to its iterative nature, for all types of problems as no © 2011 ACEEE DOI: 02.CEM.2011.01. 554

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