Вісник Національного технічного університету "ХПÌ": Системний аналіз, управління та інформаційні технології (Jul 2019)

TOTAL WEIGHTED TARDINESS MINIMIZATION FOR TASKS WITH A COMMON DUE DATE ON PARALLEL MACHINES IN CASE OF AGREEABLE WEIGHTS AND PROCESSING TIMES

  • Alexander Anatolievich Pavlov,
  • Elena Borisovna Misura,
  • Oleg Valentinovich Melnikov

DOI
https://doi.org/10.20998/2079-0023.2019.01.04
Journal volume & issue
no. 1
pp. 20 – 24

Abstract

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We consider tasks scheduling problem on identical parallel machines by the criterion of minimizing the total weighted tardiness of tasks. All tasks arrive for processing at the same time. Weights and processing times are agreeable, that is, a greater weight of a task corresponds to a shorter processing time. In addition, we have arbitrary start times of machines for tasks processing. The times may be less or greater than the due date or to coincide with it. The problem in this formulation is addressed for the first time. It can be used to provide planning and decision making in systems with a network representation of technological processes and limited resources. We give efficient PSC-algorithm with complexity that includes the poly­nomial component and the approximation algorithm based on permutations of tasks. The polynomial component contains sufficient signs of optimality of the obtained solutions and allows to obtain an exact solution by polynomial subalgorithm. In the case when the sufficient signs of opti­mality do not fulfill, we obtain approximate solution with an estimate of de­viation from the optimum for each individual problem instance of any prac­tical dimension. We show that a schedule obtained as a result of the problem solving can be split into two schedules: the schedule on machines which start time is less than or equal to the due date, and the schedule on machines which start after the due date. Optimization is only done in the first sched­ule. The second schedule is optimal by construction. Statistical studies of the PSC-algorithm showed its high efficiency. We solved problems with dimensions up to 40,000 tasks and up to 30 ma­chines. The average time to solve the problem by the algorithm using the most efficient types of permutations was 27.3 ms for this dimension. The average frequency of an optimal solution obtaining amounted to 90.3 %. The average deviation from an optimum was no more than 0.000251.

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