Single Machine Scheduling Problem with Interval Processing Times and Total Completion Time Objective

Yuri Sotskov,Natalja Egorova

doi:10.3390/a11050066

Abstract

We consider a single machine scheduling problem with uncertain durations of the given jobs. The objective function is minimizing the sum of the job completion times. We apply the stability approach to the considered uncertain scheduling problem using a relative perimeter of the optimality box as a stability measure of the optimal job permutation. We investigated properties of the optimality box and developed algorithms for constructing job permutations that have the largest relative perimeters of the optimality box. Computational results for constructing such permutations showed that they provided the average error less than 0 . 74 % for the solved uncertain problems.

Highlights

Since real-life scheduling problems involve different forms of uncertainties, several approaches have been developed in the literature for dealing with uncertain scheduling problems
Efficient algorithms are derived for finding a job permutation with the largest relative perimeter of the optimality box
We tested seven classes of harder problems, where the job permutation πk ∈ S constructed by Algorithm 3 outperforms the mid-point permutation πmid− p and the permutation πmax constructed by Algorithm MAX-OPTBOX

Summary

Introduction

Since real-life scheduling problems involve different forms of uncertainties, several approaches have been developed in the literature for dealing with uncertain scheduling problems. Job processing times are assumed to be random variables with the known probability distributions [1,2]. If one has no sufficient information to characterize the probability distribution of all random processing times, other approaches are needed [3,4,5]. A stability approach [10,11,12] is based on the stability analysis of the optimal schedules to possible variations of the numerical parameters. We apply the stability approach to a single machine scheduling problem with uncertain processing times of the given jobs. Efficient algorithms are derived for finding a job permutation with the largest relative perimeter of the optimality box.

Methods

Results

Conclusion