Total Weighted Tardiness Minimization for Tasks With a Common Due Date on Parallel Machines in Case of Agreeable Weights and Processing Times

Abstract

We consider n tasks scheduling problem on m 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 i n systems with a network representation of technological processes and limited resources. We give efficient PSC-algorithm with ( log ) 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.

Highly efficient scheduling algorithms for identical parallel machines with sufficient conditions for optimality of the solutions

“…It follows clearly from Theorem 3 that the following properties, which were proved in [7,9,10] for the TTP problem, are also true for the TTPR problem.…”

“…This section will use content from [9,10]. Proofs of theorems and statements that are not given here can be found in [7,9,10].…”

“…Algorithm 𝐴 init for initial TTPR schedule building is given in [9,10] as follows: 1. Renumber the jobs of the set 𝐽 in non-decreasing order of processing times.…”

“…It was shown in [9] that the TTPR problem can be solved with a PSC-algorithm [7,10] designed for the problem variant with equal machine release times (the TTP problem, involving the minimisation of the total tardiness of jobs on identical parallel machines). The work in [10] describes PSC-algorithms for the TTP and TTPR problems and explores their efficiency using a statistical approach.…”

“…The work in [10] describes PSC-algorithms for the TTP and TTPR problems and explores their efficiency using a statistical approach. Other papers [7,13] propose a generalised form of the TTPR problem for the case where the machine release times may exceed the common due date (the TTPRE problem).…”