On the Dual and Inverse Problems of Scheduling Jobs to Minimize the Maximum Penalty

Alexander A Lazarev,Nikolay Pravdivets,Frank Werner

doi:10.3390/math8071131

Alexander A Lazarev, Nikolay Pravdivets + Show 1 more

Open Access

https://doi.org/10.3390/math8071131

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Journal: Mathematics	Publication Date: Jul 10, 2020
Citations: 7	License type: CC BY 4.0

Affiliation: Otto-von-Guericke University Magdeburg

Abstract

In this paper, we consider the single-machine scheduling problem with given release dates and the objective to minimize the maximum penalty which is NP-hard in the strong sense. For this problem, we introduce a dual and an inverse problem and show that both these problems can be solved in polynomial time. Since the dual problem gives a lower bound on the optimal objective function value of the original problem, we use the optimal function value of a sub-problem of the dual problem in a branch and bound algorithm for the original single-machine scheduling problem. We present some initial computational results for instances with up to 20 jobs.

Highlights

We consider single-machine scheduling problems, where a set of n jobs N = {1, 2, . . . , n} has to be processed on a single machine starting at time τ
While the original problem 1|r j | Lmax is NP-hard in the strong sense [1], we prove that both the dual and inverse problems of this problem can be solved in polynomial time
Lazarev et al [20] considered the problem of minimizing maximum lateness and the makespan in the case of equal processing times and proposed a polynomial time approach for finding the Pareto-optimal set of feasible solutions

Summary

Introduction

A more recent branch and bound algorithm for this single-machine problem with release dates and precedence constraints has been given in [12] This algorithm uses four heuristics for finding initial upper bounds and a variable neighborhood search procedure. Lazarev et al [20] considered the problem of minimizing maximum lateness and the makespan in the case of equal processing times and proposed a polynomial time approach for finding the Pareto-optimal set of feasible solutions. They presented two approaches, the efficiency of which depends on the number of jobs and the accuracy of the input-output parameters.

The Dual Problem

The Inverse Problem of the Maximum Lateness Problem

Computational Results

Conclusions