Scheduling jobs and maintenance activities on parallel machines

Abstract

In this article, we deal with the problem of scheduling N production jobs on M parallel machines. Each machine should be maintained once during the planning horizon. We consider the case where the maintenance of the machines should start at time zero and the resources that ensure the maintenance are not sufficient. For such a reason, the maintenance tasks must be continuously run during the planning horizon. We aim to find a schedule composed of the production jobs and the maintenance tasks for which the total sum of the jobs’ weighted completion times and the preventive maintenance cost are minimized. We optimally solve the problem by an integer linear programming method. We also propose a heuristic method computed in two phases. Computational experiments are performed on randomly generated instances and the results show that the proposed methods produce satisfactory solutions for the problem.

Appendix: Description of the genetic algorithm

Initial population. The initial population consists of a set of Np initial solutions called individuals or chromosomes. The chromosomes of a population are generally generated in a random way. Generally, the population size should not be too large in order to avoid to increase the computational time spent for obtaining the final solution. However, it should not be too small in order to ensure a satisfactory quality of the final solution. For our proposed genetic algorithm, we generate an initial population of 100 chromosomes (Np = 100). Every chromosome is randomly generated.

Chromosome representation. Two types of chromosome representation are tested. Each representation consists in a table of N genes. In the first representation, a gene is a production job and the gene index is the order of the corresponding job. In the second representation it corresponds to a machine on which is performed the production job having as number the index of the gene. The two chromosome representations are shown in Figs. 2 and 3.

Fig. 2

First chromosome representation

Fig. 3

Second chromosome representation

The first chromosome representation is translated into a feasible solution for the problem as follows. First, we test if the first production job in the chromosome can be assigned to a machine before its maintenance task. If the assignment is not possible, we find the machine with the earliest completion time after its maintenance task and we assign the production job to this machine. If the assignment is possible, then we assign the production job and we consider the assignement of the next job. We repeat the same procedure until all the production jobs in the chromosome are assigned. Before the evaluation of the chromosome, we check if the WSPT order is respected in all the partial sequences before and after the maintenance tasks.

For the second chromosome representation we use the following rule. First, we establish the WSPT sequence. Then, we test if the first production job of the WSPT sequence (job 1) can be assigned before the maintenance task of the machine determined by the first gene in the chromosome. If the assignement is possible, then we assign the production job. If it is not possible, then we assign it on the same machine, determined by the gene, after the maintenance task. We repeat the same procedure until all the N production jobs of the WSPT sequence are assigned.

Crossover operator. In the crossover operator, two parents should be selected from the population of the current generation to generate at least one child. In our genetic algorithm, 90 % of the best chromosomes participate in the crossover operation. At each crossover operation, two children are produced. The principle of the selection is as follows. First, the 1st parent and the 45th parent are selected. Then, the 2nd and the 46th parents, etc $\ldots$. until the 44th parent and the 90th parent. For every two selected parents, we randomly generate an integer v from {1, 2, $\ldots$, N}. The next steps differ according to the type of the chromosome representation.

For the first representation type, the first child C1 inherits the subsequence of the first v genes from the first parent and the second child C2 inherits the subsequence of the first v genes from the second parent. Then, we fill the remaining empty genes of child C1 according to their order of appearance in the second parent. Finally we fill the remaining empty genes of child C2 according to their order of appearance in the first parent.

For the second representation, the first child C1 inherits the subsequence of the first v genes from the first parent and the last (N − v) genes from the second parent. The second child C2 inherits the subsequence of the first v genes from the second parent and the last (N − v) genes from the first parent. The crossover operator used to generate the children in our genetic algorithm is called the standard ONE-OPT crossover operator.

Mutation operator. In the mutation operation, an individual is randomly chosen from the population to undergo a small modification. Generally, the main purposes of using the mutation are the possibility to leave a local optimum and to diversify the next population. In our algorithm, we apply the mutation to 10 individuals randomly chosen from the population. The slight modification consists in exchanging two genes randomly selected. The procedure is the same for the two types of representation.

Replacement strategy. In a genetic algorithm the number of new solutions increases from a generation to another. Keeping all generated solutions of the iterations will 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 in replacing the individuals (solutions) in the population of the worst fitness by the new children with better evaluations may be a good solution for obtaining a better performance. In our implementation, we have taken at each generation all the best 100 chromosomes.