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New ideas in Lagrangian relaxation for a scheduling problem with the weighted tardiness criterion

Rudy Jarosław, Idzikowski Radosław, Smutnicki Czesław, Banaszak Zbigniew, Bocewicz Grzegorz

International Journal of Applied Mathematics and Computer Science

2024

Vol. 34, no. 2

235--251

We consider an extension of Lagrangian relaxation methods for solving the total weighted tardiness scheduling problem on a single machine. First, we investigate a straightforward relaxation method and decompose it into upper and lower subproblems. For the upper subproblem we propose an alternative solving method in the form of a local search metaheuristic. We also introduce a scaling technique by arbitrary numbers to reduce the complexity of the problem and confront it with greatest common divisor scaling. Next, we propose a novel alternative relaxation approach based on aggregating constraints. We discuss the properties and implementation of this new approach and a technique to further reduce its computational complexity. We perform a number of computer experiments on instances based on the OR-Library generation scheme to illustrate and ascertain the numerical properties of the proposed methods. The results indicate that for larger instances the proposed alternative relaxation and scaling approaches have a much better convergence rate with little to no decrease in solution quality. The results also show that the proposed local-search metaheuristic is a viable alternative to the existing solving methods.

Improving the TSAB algorithm through parallel computing

Rudy Jarosław, Pempera Jaroslaw, Smutnicki Czesław

Archives of Control Sciences

2020

Vol. 30, No. 3

411--435

In this paper, a parallel multi-path variant of the well-known TSAB algorithm for the job shop scheduling problem is proposed. Coarse-grained parallelization method is employed, which allows for great scalability of the algorithm with accordance to Gustafon’s law. The resulting P-TSAB algorithm is tested using 162 well-known literature benchmarks. Results indicate that P-TSAB algorithm with a running time of one minute on a modern PC provides solutions comparable to the ones provided by the newest literature approaches to the job shop scheduling problem. Moreover, on average P-TSAB achieves two times smaller percentage relative deviation from the best known solutions than the standard variant of TSAB. The use of parallelization also relieves the user from having to fine-tune the algorithm. The P-TSAB algorithm can thus beused as module in real-life production planning systems or as a local search procedure in other algorithms. It can also provide the upper bound of minimal cycle time for certain problems of cyclic scheduling.