A new approximation algorithm for unrelated parallel machine scheduling with release dates

Zhi Pei (), Mingzhong Wan and Ziteng Wang
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Zhi Pei: Zhejiang University of Technology
Mingzhong Wan: Zhejiang University of Technology
Ziteng Wang: Northern Illinois University

Annals of Operations Research, 2020, vol. 285, issue 1, No 18, 397-425

Abstract: Abstract In the current study, an unrelated parallel machine scheduling problem with release dates is considered, which is to obtain a job assignment with minimal sum of weighted completion times. Although this problem is NP-hard in the strong sense, which renders the optimality seeking a formidable task within polynomial time, a 4-approximation algorithm based on the constant worst-case bound is devised and proved in comparison with the existing 16/3-approximation (Hall et al. in Math Oper Res 22(3):513–544, 1997). In the newly proposed algorithm, the original scheduling problem is divided into several sub-problems based on release dates. For each sub-problem, a convex quadratic integer programming model is constructed in accordance with the specific problem structure. Then a semi-definite programming approach is implemented to produce a lower bound via the semi-definite relaxation of each sub-problem. Furthermore, by considering the binary constraint, a branch and bound based method and a local search strategy are applied separately to locate the optimal solution of each sub-problem. Then the solution of the original scheduling problem is derived by integrating the outcomes of the sub-problems via the proposed approximation algorithm. In the numerical analysis, it is discovered that the proposed methods could always obtain a scheduling result within $$1\%$$1% of the optimal solution. And an asymptotic trend could be observed where the quality of solutions improves even further as the scale of the scheduling problem increases.

Keywords: Unrelated parallel machine scheduling; Release dates; Semi-definite programming; Approximation algorithm; Branch and bound (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10479-019-03346-4

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