Parallel machine scheduling with earliness and tardiness penalties

Abstract

In the parallel machine scheduling problem with earliness and tardiness penalties (PMSP_E/T) considered here, a set of independent jobs with sequence-dependent setups is given to be scheduled on a set of parallel machines (processors) in a non-preemptive fashion such that the sum of the weighted earliness and tardiness values of all jobs is minimized. The due dates of the jobs are distinct which complicates the problem. In addition, each job has its own arrival time which brings the model closer to reality but complicates it further. The weights for earliness and tardiness are common to all jobs and are unequal in general. Two genetic algorithm approaches are employed to attack this problem; one with a crossover operator which is developed to solve multi-component combinatorial optimization problems of which PMSP_E/T is an instance, and the other with no crossover operator. Results of tests on 960 randomly generated problems indicate that genetic algorithms provide an efficient algorithm for PMSP_E/T; that neighborhood exchange type of search can yield relatively better results in small and easy instances of the problem but the genetic algorithm with the crossover operator outperforms such search in larger-sized, more difficult problems; and that the recombinative power of the genetic algorithm with the crossover operator improves with increasing problem size and difficulty making it ever more attractive for applications of larger sizes. Scope and purpose The parallel machine scheduling problem is an important and difficult problem. Traditionally, the problem has consisted of scheduling of a set of independent jobs on identical parallel machines (processors) with the aim of minimizing maximum job completion. In line with current trends towards just-in-time manufacturing strategies, where both early and tardy finishing of job processing are undesired, objectives related to earliness and tardiness penalties have become increasingly popular. Yet, already the scheduling of independent jobs with a common due date on a single machine is an NP-hard problem. Research efforts have therefore concentrated on heuristic approaches. This paper presents one such approach. Genetic algorithms (GAs) have become very popular as search algorithms due to their effectiveness and efficiency in large and complex search spaces. In this study, a genetic algorithm approach is developed to attack the problem of scheduling of a set of independent jobs on parallel machines. To model the actual practice more closely, assumptions such as distinct due dates and arrival times for jobs, different processing rates for machines and sequence-dependent set-up times are incorporated into the problem formulation. To the best knowledge of the authors, parallel machine scheduling problems of this scope have not been treated in the literature before. Computational results on a large test bed illustrate the effectiveness of GAs on this problem in general and the potentials of GAs with a new crossover operator (MCUOX – multi-component uniform order-based crossover) for problems of increasing sizes and difficulty.