preemptive 1-machine job scheduling, total weighted tardiness, heuristic, sorting approach, remaining processing periods, remaining available periods


Background. In preemptive job scheduling, total weighted tardiness minimization is commonly reduced to solving a combinatorial problem, which becomes practically intractable as the number of jobs and the numbers of their processing periods increase. To cope with this challenge, heuristics are used. A heuristic, in which the decisive ratio is the weighted reciprocal of the maximum of a pair of the remaining processing period and remaining available period, is closely the best one. However, the heuristic may produce schedules of a few jobs whose total weighted tardiness is enormously huge compared to the real minimum. Therefore, this heuristic needs further improvements, one of which already exists for jobs without priority weights with a sorting approach where remaining processing periods are minimized. Three other sorting approaches still can outperform it, but such exceptions are quite rare.

Objective. The goal is to determine the influence of the four sorting approaches and try to select the best one in the case where jobs have their priority weights. The heuristic will be applied to tight-tardy progressive idling-free 1-machine preemptive scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates.

Methods. To achieve the said goal, a computational study is carried out with applying each of the four heuristic approaches to minimize total weighted tardiness. For this, two series of 4151500 scheduling problems are generated. In the solution of a scheduling problem, a sorting approach can “win” solely or “win” in a group of approaches, producing the heuristically minimal total weighted tardiness. In each series, the distributions of sole-and-group “wins” are ascertained.

Results. The sole “wins” and non-whole-group “wins” are rare: the four sorting approaches produce schedules with the same total weighted tardiness in over 98.39 % of scheduling problems. Although the influence of these approaches is different, it is therefore not really significant. Each of the sorting approaches has heavy disadvantages leading sometimes to gigantic inaccuracies, although they occur rarely. When the inaccuracy occurs to be more than 30 %, this implies that 3 to 9 jobs are scheduled.

Conclusions. Unlike the case when jobs do not have their priority weights, it is impossible to select the best sorting approach for the case with job priority weights. Instead, a hyper-heuristic comprising the sorting approaches (i. e., the whole group, where each sorting is applied) may be constructed. If a parallelization can be used to process two or even four sorting routines simultaneously, the computation time will not be significantly affected.


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