A genetic algorithm approach for the single machine scheduling problem with linear earliness and quadratic tardiness penalties

• Authors: Jorge M. S. Valente;José Fernando Gonçalves
• Venue: Computers and Operations Research
• Year: 2009

In this paper, we consider the single machine scheduling problem with linear earliness and quadratic tardiness costs, and no machine idle time. We propose a genetic approach based on a random key alphabet. Several genetic algorithms based on this approach are presented. These versions differ on the generation of the initial population, as well as on the use of local search. The proposed procedures are compared with existing heuristics, as well as with optimal solutions for the smaller instance sizes. The computational results show that the performance of the proposed genetic approach is improved by the addition of a local search procedure, as well as by the insertion of simple heuristic solutions in the initial population. Indeed, the genetic versions that include either or both of these features not only provide significantly better results, but are also much faster. The genetic versions that use local search are clearly superior to the existing heuristics, and the improvement in performance over the best existing procedure increases with both the size and difficulty of the instances. These genetic procedures are also quite close to the optimum, and provided an optimal solution for most of the test instances. Scope and purpose: This paper considers a single machine scheduling problem with linear earliness and quadratic tardiness costs, and no machine idle time. Scheduling with early and tardy penalties has received considerable attention from the scheduling community, due to its practical importance. Indeed, early/tardy scheduling problems are compatible with the concepts of Just-in-Time production and supply chain management, which have been adopted by many organizations. Single machine scheduling environments actually occur in several practical applications. Also, the performance of many production systems is often determined by the schedules for a single bottleneck machine. Furthermore, the study of single machine problems frequently provides results that prove useful for more complex scheduling environments. The assumption that no machine idle time is allowed is also appropriate for many production settings. In fact, idle time should be avoided when the machine has limited capacity or high operating costs, and when starting a new production run involves high set-up costs or times. In this paper, we present several algorithms based on a genetic approach that uses a random key alphabet. The various versions of the genetic algorithm differ on the generation of the initial population, as well as on the use of local search. These procedures are compared with existing heuristics, as well as with optimal solutions for some instance sizes. The computational results show that inserting solutions generated by simple heuristics in the initial population, and using a local search procedure, enhances the performance of the proposed genetic approach. In fact, the addition of one or both of these features improves both the solution quality and the speed of the genetic algorithm. The genetic versions that apply local search clearly outperform the existing heuristics, and are quite close to the optimum solutions. Also, the improvement over the best existing procedure increases with both the size and the difficulty of the test instances.