A Sorting Improvement in the Heuristic Based on Remaining Available and Processing Periods to Minimize Total Tardiness in Progressive Idling-Free 1-Machine Preemptive Scheduling

Abstract

Background. In preemptive job scheduling, which is a part of the flow-shop sequencing tasks, one of the most crucial goals is to obtain a schedule whose total tardiness would be minimal. Total 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 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 tardiness is 25 % greater than the minimum or even worse. Therefore, this heuristic needs a corrective branch which would further try to minimize total tardiness under certain conditions. Objective. The goal is to ascertain what is to be corrected in the heuristic so that the total tardiness value could be obtained lesser. 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. In this scheduling problem, the inaccuracy of finding the minimal total tardiness has the strongest negative impact, so this is almost the worst case, which defines the accuracy limit of the heuristic and positively serves just as the principle of minimax guaranteeing decreasing losses in the worst conditions. Methods. The heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job. To achieve the said goal, three other sorting approaches are presented and a computational study is carried out with applying each of the four heuristic approaches to minimize total tardiness. For this, two series of 266000 and 1064000 scheduling problems are generated. Results. The earliest-job sorting ensures a heuristically minimal total tardiness value in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases. Nevertheless, a sorting approach with minimizing remaining processing periods produces a heuristically minimal total tardiness for almost any scheduling problem. If an exception occurs, this sorting approach “loses” to the other sorting approaches very little. Moreover, the exceptions are quite rare as it has been registered just a one scheduling problem (out of 31914 cases followed by a sole “win” of a heuristic version) whose minimal total tardiness is achieved by the earliest-job sorting. Conclusions. The best heuristic version is that one which uses the sorting approach with minimizing remaining processing periods. This, however, is confirmed only for the case where jobs do not have any priorities. The case when jobs have their priority weights is to be yet analyzed.

Sorting Approaches in the Heuristic Based on Remaining Available and Processing Periods to Minimize Total Weighted Tardiness in Progressive Idling-Free 1-Machine Preemptive Scheduling

“…Despite its rather probabilistic nature, the random sorting has had almost a quarter of all sole "wins". In the case of TPIF1MP scheduling problems without priority weights, sole "wins" constitute about 2.39 % to 2.44 % of the volume of generated problems [13]. Here, in the first series, the percentage is just 0.5814 % (for 24136 sole "wins" out of 4151500 generated scheduling problems).…”

“…So, in the case when ( 11) is true, the RPP-RAP heuristic sorts maximal decisive ratios in subset (8) by release dates, where the scheduling preference is given to the earliest job (whose release date is the least). However, along with this earliest-job sorting, the other three approaches to sorting might be used: the RPP-or-due-date sorting, the min-RPP sorting, and random sorting [13].…”

“…Therefore, this heuristic needs further improvements. One of such improvements is suggested in article [13] which considers minimization of total tardiness (i. e., without job priority weights) in TPIF1MP 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. Whereas the RPP-RAP heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job (this is the earliest-job sorting), three other sorting approaches are presented in [13].…”

“…One of such improvements is suggested in article [13] which considers minimization of total tardiness (i. e., without job priority weights) in TPIF1MP 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. Whereas the RPP-RAP heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job (this is the earliest-job sorting), three other sorting approaches are presented in [13]. In accordance with article [13], the earliest-job sorting ensures a heuristically minimal total tardiness in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases.…”

“…Whereas the RPP-RAP heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job (this is the earliest-job sorting), three other sorting approaches are presented in [13]. In accordance with article [13], the earliest-job sorting ensures a heuristically minimal total tardiness in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases. Nevertheless, a sorting approach with minimizing remaining processing periods produces a heuristically minimal total tardiness for almost any scheduling problem.…”