Parallel-machine rescheduling with job unavailability and rejection

We study the scheduling problem where a set of jobs has already been scheduled for processing on identical parallel machines to minimize the total completion time under the assumption that all the jobs are available at time zero. However, before processing begins, some jobs are delayed and become unavailable at time zero, so all the jobs need to be rescheduled with a view to not causing excessive schedule disruption with respect to the original schedule. To reduce the negative impact of job unavailability and achieve an acceptable service level, one option in rescheduling the jobs is to reject a subset of the jobs at a cost (the rejection cost). Three criteria are involved: the total completion time of the accepted jobs in the adjusted schedule, the degree of disruption measured by the maximum completion time disruption to any accepted job between the original and adjusted schedules, and the total rejection cost. The overall objective is to minimize the former criterion, while keeping the objective values of the latter two criteria to no greater than the given limits. We present two exact methods to solve the problem: (i) A dynamic programming based approach, establishing that the problem is NP-hard in the ordinary sense when the number of machines is fixed. (ii) An enhanced branch-and-price method that includes several features such as execution of the differential evolution algorithm for finding good initial feasible solutions and solving the pricing sub-problem, inclusion of reduced cost fixing during the inner iterations of the algorithm, and use of a heuristic procedure for constructing a good integer feasible solution. We perform extensive computational experiments to assess the efficiency of the proposed algorithms. The computational results demonstrate that the incorporated enhancements greatly improve the performance of the algorithm.