Abstract
This study addresses a two-machine job-shop scheduling problem with fixed lower and upper bounds on the job processing times. An exact value of the job duration remains unknown until completing the job. The objective is to minimize a schedule length (makespan). It is investigated how to best execute a schedule, if the job processing time may be equal to any real number from the given (closed) interval. Scheduling decisions consist of the off-line phase and the on-line phase of scheduling. Using the fixed lower and upper bounds on the job processing times available at the off-line phase, a scheduler may determine a minimal dominant set of schedules (minimal DS), which is based on the proven sufficient conditions for a schedule dominance. The DS optimally covers all possible realizations of the uncertain (interval) processing times, i.e., for each feasible scenario, there exists at least one optimal schedule in the minimal DS. The DS enables a scheduler to make the on-line scheduling decision, if a local information on completing some jobs becomes known. The stability approach enables a scheduler to choose optimal schedules for most feasible scenarios. The on-line scheduling algorithms have been developed with the asymptotic complexity O(n2) for n given jobs. The computational experiment shows the effectiveness of these algorithms.
Highlights
Many real-world production planning and scheduling problems have various uncertainties
Our approach to the solution of the uncertain job-shop scheduling problem J2|lij ≤ pij ≤ uij, ni ≤ 2|Cmax is based on the following remark
More than 80% of the instances from these three classes were optimally solved at the off-line phase of scheduling or at the on-line phases of scheduling provided that the maximal error δ of the given job processing times was no greater than 70%, i.e., for δ ∈ {5%, 10%, 15%, 20%, 30%, 40%, 50%, 60%, 70%}
Summary
Many real-world production planning and scheduling problems have various uncertainties. Different approaches are used for solving the uncertain planning and scheduling problems. A stability approach [1,2,3,4] for solving sequencing and scheduling problems with the interval uncertainty is based on the stability analysis of the optimal job permutations (schedules) to possible variations of the job processing times (durations). In this paper, this approach is applied to the uncertain two-machine job-shop scheduling problem, where a job processing time is only known once the job is completed.
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