Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines

Version 1 : Received: 6 January 2021 / Approved: 8 January 2021 / Online: 8 January 2021 (10:59:29 CET)
Version 2 : Received: 16 March 2021 / Approved: 16 March 2021 / Online: 16 March 2021 (15:08:24 CET)

How to cite: Vakhania, N.; Werner, F. A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines. Preprints 2021, 2021010142 (doi: 10.20944/preprints202101.0142.v1). Vakhania, N.; Werner, F. A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines. Preprints 2021, 2021010142 (doi: 10.20944/preprints202101.0142.v1).

Abstract

The problem of sequencing $n$ equal-length non-simultaneously released jobs with delivery times on $m$ uniform machines to minimize the maximum job completion time is considered. To the best of our knowledge, the complexity status of this classical scheduling problem remained open up to the date. We establish its complexity status positively by showing that it can be solved in polynomial time. We adopt for the uniform machine environment the general algorithmic framework of the analysis of behavior alternatives developed earlier for the identical machine environment. The proposed algorithm has the time complexity $O(\gamma m^2 n\log n)$, where $\gamma$ can be any of the magnitudes $n$ or $q_{\max}$, the maximum job delivery time. In fact, $n$ can be replaced by a smaller magnitude $\kappa<n$, which is the number of special types of jobs (it becomes known only upon the termination of the algorithm).

Subject Areas

scheduling; uniform machines; release time; delivery time; time complexity; algorithm

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