# Scheduling Jobs and Preventive Maintenance Activities on Parallel Machines

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3 Subject to Min h i E i + w i T i generated and represented by a table of size of N genes. A gene corresponds to a machine on which the job having as number the index of the gene in the table is executed. The following figure shows the used chromosome s representation: P p i+1 t=1 x it = 1 i = 1... M (1) t s=max(t p i+1,1) P p i+1 t=1 P p i+1 t=1 P = x is 1 t = 1... P (2) tx it + E i d j1 + 1 i = 1... M (3) tx it T i d j2 + 1 i = 1... M (4) N M p p i + p m j ; x it {0, 1} i = 1... M (5) j=1 In this model, the equations (1) ensure that each job is processed once. The second inequalities are resources constraints indicating that at most just one job can be handled at a time slot. The third and the fourth inequalities decide for each maintenance task to start early, tardy or on time. Finally, the fifth constraints are integrity constraints. For this formulation, the MIP solver ILOG Cplex 10.1 is able to solve all our small instances with a size less or equal to 5 maintenance tasks in a time period less than 3 seconds. 3.2 An evolutionary algorithm for P m, h j1 w i C i Evolutionary algorithms are probabilistic algorithms used to provide excellent solutions in reasonable time for hard and time complex problems. In a genetic algorithm, many new solutions are created at each iteration (called also generation). A new obtained solution (called also a child solution) emanates from the crossing of two solutions from the previous generation (called parents solutions). Sometimes, the child solution may undergo a mutation operation before it passes for evaluation (fitness). Generally, a genetic algorithm is defined by the correspondingly elements: Initial population, cross-over operation, mutation operation and the replacement strategy. The initial population The initial population consists of a set of N p initial solutions called individuals or chromosomes. The chromosomes of a population are usually randomly generated. Sometimes they are determined by specific rules. Generally, the population s size should not be large to not increase the computational time of obtaining the final solution. On the other hand, it should not be small to not affect the quality of the final solution. For our proposed genetic algorithm, we generate an initial population of size of 100 chromosomes (Np=100). Each chromosome is randomly This chromosome s representation is translated as follows in a solution for the problem : We first establish the WSPT sequence. After that, we test if the first job of the WSPT sequence (job 1) may be assigned before the maintenance task of the machine determined by the first gene in the table. If it is possible, we assign it. If it is not possible we assign it on the same machine after the maintenance task. We repeat the procedure until assigning the task N of the WSPT sequence. The crossover operation: In the cross-over operation, two parents should be selected from the population of the current generation to generate at least one child. In our genetic algorithm, only 90% of the best chromosomes participate in the crossover operation. At each cross-over operation, two children are produced. The principle of parent s selection is as follows: First the 1 st parent and the 45 th parent are selected. After that, the second and the 46 th parents, etc... until the 44 th parent and the 90 th parent. After the selection of each two parents, we randomly generate an integer k from the uniform [1, N].The first child C 1 inherits the subsequence [1,..., k] of its genes from the first parent and its remaining empty genes [k+1,..., N] from the second parent. The second child C 2 inherits the subsequence [1,..., k] of its genes from the second parent. After that, we fill its remaining empty genes [k+1,..., N] from the first parent. The cross-over operator used to generate the children in our genetic algorithm is called one opts cross-over operator. The mutation operation: In the mutation operation, an individual is randomly chosen from the population to undergo a small modification. In our algorithm, we apply the mutation on ten individuals randomly chosen from the population. The slight modification consists of permutation of two genes randomly selected. The replacement strategy: In a genetic algorithm, the number of new solutions increases from a generation to another. Keeping all generated solutions of the iterations should amplify the population size. This fact may increase the computational time for obtaining the final solution and also the algorithm may be memory consuming. A replacement strategy that consists of replacing the individuals (solutions) in the population ISSN: ISBN:

5 Finally the earliness and the tardiness penalties are drawn from the uniform distribution [1,...,10]. 5 instances have been generated for each combination of N, M, T and R. The algorithms were coded in C language and implemented on a Pentium IV-500 personal computer using concert technology technique with Cplex 10.1 to solve the Dual Lagrangian model of the proposed lower bound. Table 1. Mean gaps value between the Genetic Algorithm and the lower bound M N Mean Gap (%) Max Gap(%) Min Gap(%) ,97 0, ,1257 0,47 0, ,0401 0,18 0, ,9977 2,22 0, ,3508 1,05 0, ,1266 0,29 0, ,7388 3,33 0, ,6775 1,41 0, ,2647 0,50 0, ,7841 5,24 1, ,0929 1,78 0, ,4460 0,88 0,23 The first column of table 1 represents the machine number M. The second column represents the jobs number N. The third, the fourth and the fifth columns are respectively the average gap between the solution of the genetic algorithm and the based Lagrangian lower bound, the maximum found gap and the minimum found gap. From column 3, we can observe that for all produced upper bounds, the average gap decreases when the job number increases. From column 4 and 5, the maximum observable gap is equal to 5,24% while the minimum gap is equal to 0,01%. Hence, we can conclude that both proposed genetic algorithm and lower bound produce good quality upper and lower bounds values for the second sub-problem for all instances. Table 2. Mean computational time for obtaining a Genetic Algorithm solution M N Time (s) , , , , , , , , , , , ,826 According to this table, we observe that for all generated instances, the maximum mean computational time does not exceed 1 second. Hence, we can confirm that the proposed genetic algorithm is efficient in terms of quality solution and computational time. 5 Conclusion In this paper we have proposed an upper bound for the problem of scheduling N jobs on M parallel machines where each machine should be maintained once during the planning horizon. We have simultaneously minimized the total sum of the job s weighted completion times and the preventive maintenance cost. The proposed upper bound solution is obtained by a hierarchical method that consists of decomposing the problem into two sub-problems: The first sub-problem is called earliness-tardiness minimization problem with a particular framework costs. The second sub-problem is the problem of scheduling jobs on parallel machines. For the first sub-problem, we have proposed a linear model that allows the availabilities of the machines during the planning horizon with a minimal maintenance cost. For the second sub-problem, a Genetic algorithm and lower bound were proposed. Computational results show that the genetic algorithm and the lower bound for the second subproblem of the hierarchical method produce excellent initial solutions that may be used to perform the B&B algorithm. Acknowledgements: This work has been funded by Conseil Régional Champagne Ardenne. References: [1] EH. Aghezzaf, MA. Jamali and D. Ait-Kadi, An integrated production and preventive maintenance planning model, European Journal of Operational Research. 181, 2007, pp [2] EH. Aghezzaf and NM. Najid, Integrated production planning and preventive maintenance in deteriorating production systems,information Sciences. 178, 2008, pp [3] WL. Eastman, S. Even and IM. Isaacs, Bounds for optimal scheduling of n jobs on m processors,management science 11, 1964,pp [4] R. Mellouli, S. Cherif, C. Chou and I. Kacem, Identical parallel-machine scheduling under availability constraints to minimize the sum of completion times,european Journal of Operational Research 197, 2009, pp [5] HG. Graves, C-Y. Lee, Scheduling Maintenance and Semiresumable Jobs on a Single Machine,Naval Research Logistics 46, 1999, pp [6] I. Kacem, C. Chengbin, S. Ahmed, Single-machine scheduling with an availability constraint to minimize the weighted sum of the completion times,computers & Operations Research 35, 2008,pp ISSN: ISBN:

6 [7] W. Kubiak, J. Blazewicz, P. Formanowicz, J. Breit and G. Schmidt, Two-machine flow shops with limited machine availability,european Journal of Operational Research 136, 2002, pp [8] C-Y. Lee, Minimising the makespan in the two machine scheduling scheduling problem with an availability constraint,operational Research Letters 20, 2000, pp [9] C-Y. Lee, Z-L. Chen, Scheduling Jobs and Maintenance Activities on Parallel Machines,Naval Research Logistics 47, 2000, pp [10] G. Li, Single machine earliness and tardiness scheduling,european Journal of Operational Research 26, 1997, pp [11] C-F. Liaw, A branch and bound algorithm for the single machine earliness and tardiness scheduling problem, Computers & Operations Research 26, 1999, pp [12] F. Sourd, S. Keded-Sidhoum and Y. Rio Solis, Lower bounds for the earliness-tardiness scheduling problem on parallel machines with distinct due dates, European Journal of Operational Research 189, 2008, pp [13] W.E. Smith, Various optimizers for single-stage production, Naval Research Logistics Quarterly 3,1956, pp ISSN: ISBN:

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