Abstract
We study a single-machine scheduling problem to minimize the total completion time of the given set of jobs, which have to be processed without job preemptions. The lower and upper bounds on the job duration is the only information that is available before scheduling. Exact values of the job durations remain unknown until the completion of the jobs. We use the optimality region for the job permutation as an optimality measure of the optimal schedule. We investigate properties of the optimality region and derive O ( n ) -algorithm for calculating a quasi-perimeter of the optimality set (i.e., the sum of lengths of the optimality segments for n given jobs). We develop a fast algorithm for finding a job permutation having the largest quasi-perimeter of the optimality set. The computational results in constructing such permutations show that they are close to the optimal ones, which can be constructed for the factual durations of all given jobs.
Highlights
A lot of real-life scheduling problems involve different forms of uncertainties
Job durations are assumed to be random variables with the specific probability distributions known before scheduling [1,2]
A stability approach [10] is based on the stability analysis of the optimal schedules to possible variations of the job durations
Summary
A lot of real-life scheduling problems involve different forms of uncertainties. For dealing with uncertain scheduling problems, several approaches have been developed in the literature. We apply the stability approach to the single-machine scheduling problem with interval durations of the given jobs. Given a scenario p ∈ T and a permutation πk ∈ S, let Ci = Ci (πk , p) denote the completion time of the job Ji in the schedule determined by the permutation πk. The above uncertain scheduling problem is denoted as 1| piL ≤ pi ≤ pU i | ∑ Ci using the three-field notation α| β|γ [12], where α denotes the processing system, β characterizes conditions for processing the jobs and γ determines the criterion
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