Abstract
Most studies on scheduling problem have assumed that the machine is continuously available. However, the machine is likely to be unavailable for many practical reasons (e.g. breakdown and planned preventive maintenance). Once a machine is not available, the optimal scheme in the deterministic scenario may not be the optimal choice. To address the actual scenario, a scheduling problem with two identical parallel machines, one of which requires periodic maintenance, is considered in this article. For uncertainties in reality, the processing and maintenance time are considered as uncertain variables. A novel method to study the worst-case ratio in an uncertain scenario is proposed. It is shown that the worst-case ratio of the longest processing time algorithm is 3/2 at a high confidence level. Given theoretical analysis results, an modified algorithm is proposed. Finally, numerical experiments are used to verify the feasibility of the modified algorithm.
Highlights
In recent years, scheduling problems with machine availability constraints have aroused increasing attentions
For the existence of non-deterministic phenomena in practice, processing and maintenance time were considered uncertain variables, which is not consistent with problems studied in existing literature
To analyze the uncertain two-parallelmachine scheduling problem, jobs are sorted by uncertain measure at a relatively high confidence level a which is over 0:8
Summary
In recent years, scheduling problems with machine availability constraints have aroused increasing attentions. According to Koulamas and Kyparisis,[5] a delayed-start longest processing time (LPT) algorithm was proposed for an identical parallel-machine scheduling problem. According to Li et al.,[9] two mathematical programming models were developed for parallel-machine scheduling with periodic maintenance, and two improved heuristic algorithms based on worst-case analysis of LPT and LS were proposed. It is assumed that all jobs can be sorted in a non-increasing order of their processing times at a confidence level a, and the LPT algorithm is used to solve the uncertain two-parallel-machine problem. If two jobs are ranked by their processing times at the confidence level a which is more than 0.8, the uncertain worst-case ratio of the LPT algorithm is 3/2 at the confidence level a: MfCLPT ł (3=2)COPT g ø a.
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