Exact algorithms for solving the constrained parallel-machine scheduling problems with divisible processing times and penalties

In this paper, we address the constrained parallel-machine scheduling problem with divisible processing times and penalties (the CPS-DTP problem), which is a further generalization of the parallel-machine scheduling problem with divisible processing times (the PS-DT problem). Concretely, given a set M of m identical machines and a set J of n independent jobs, each job has a processing time and a penalty, the processing times of these n jobs are divisible, and we implement these n jobs under the requirement that each job in J must be either continuously executed on one machine with its processing time, or rejected with its penalty that we must pay for. We may consider three versions of the CPS-DTP problem, respectively. (1) The constrained parallel-machine scheduling problem with divisible processing times and total penalties (the CPS-DTTP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize the of such a schedule T for jobs in A plus the summation of penalties paid for jobs not in A; (2) The constrained parallel-machine scheduling problem with divisible processing times and maximum penalty (the CPS-DTMP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize themakespan of such a schedule T for jobs in A plus maximum penalty paid for jobs not in A; (3) The constrained parallel-machine scheduling problem with divisible processing times and bounded penalty (the CPS-DTBP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement and the summation of penalties paid for jobs not in A is no more than a fixed bound, the objective is to minimize the makespan of such a schedule T for jobs in A. As our main contributions, we design three exact algorithms to solve the CPS-DTTP problem, the CPS-DTMP problem and the CPS-DTBP problem, and these three algorithms run in time O((n log n+ nm)C), O(n(2) log n) and O((n log n+nm) logC), respectively, where C is the optimal value of same instance for the PS-DT problem.