Apply Ordinal Optimization to Optimize the Job-Shop Scheduling Under Uncertain Processing Times

The job shop scheduling problem is generally divided into two types according to production environments, the job shop scheduling problem with deterministic processing times and the job shop scheduling problem with uncertain processing times. Regarding the job shop scheduling problem with deterministic processing times, the shop parameters such as processing times are constant throughout the realization of a schedule. In the job shop scheduling problem with uncertain processing times, the actual processing time is uncertain until the operation is completed. In practice, the process may take less or more time than originally scheduled, which makes it a challenging task. In this paper, an approach integrating the artificial immune system into ordinal optimization is designed to look for a near-optimal schedule in a relatively short time. The job shop scheduling problem with uncertain processing times is first formulated as a stochastic constraint optimization problem to minimize the summation of tardiness penalty costs and storage costs. Next, the artificial immune system supported by a rough estimate is adopted to determine a selected subset within a limited computing time. Finally, the optimal computing budget allocation is utilized to look for a near-optimal schedule. The proposed approach is applied to two test examples: the medium-size with six jobs and six machines, and the large size with ten jobs and ten machines. The uncertain processing times are modeled by three probability distributions: truncated normal, exponential, and uniform. Test results demonstrate that the near-optimal schedule obtained by the proposed approach has significant improvements in terms of both efficiency and solution quality.