Working Paper Article Version 2 This version is not peer-reviewed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 Vakhania, N.; Werner, F. A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines. Preprints 2021, 2021010142

## Abstract

We consider the problem of scheduling $n$ jobs with identical processing times and given release as well as delivery times on $m$ uniform machines. The goal is to minimize the makespan, i.e., the maximum full completion time of any job. This problem is well-known to have an open complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for the problem which is based on the earlier introduced algorithmic framework blesscmore (branch less and cut more''). We extend the analysis of the so-called behavior alternatives developed earlier for the version of the problem with identical parallel machines and show how the earlier used technique for identical machines can be extended to the uniform machine environment if a special condition on the job parameters is imposed. The time complexity of the proposed algorithm is $O(\gamma m^2 n\log n)$, where $\gamma$ can be either $n$ or the maximum job delivery time $q_{\max}$. This complexity can even be reduced further by using a smaller number $\kappa<n$ in the estimation describing the number of jobs of particular types. However, this number $\kappa$ becomes only known when the algorithm has terminated.

## Subject Areas

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

Comment 1
Commenter: Frank Werner
Commenter's Conflict of Interests: Author
Comment: Based on a problem with Lemma 1 and its earlier proof, we updated this and made the corresponding adaptations in the abstract, introduction and conclusions.  Normally we do not update a preprint, but here we think it is necessary to correct this mistake.
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