A Branch-and-bound Algorithm for Parallel Machine Scheduling Problems

In this paper, a Branch-and-Bound (B&B) algorithm is presented to solve the problem of minimizing the maximum completion time Cmax on unrelated parallel machines with machine eligibility restrictions when job preemption is not allowed. A customized lower bound, a search and a branching strategy are developed for the B&B. A machine eligibility factor is also introduced to represent the percentage of jobs eligible across all machines. Multiple instances of the problem with different machine, job, and eligibility factor configurations are generated and solved. The B&B algorithm is able to solve up to 8 machines and 40 jobs within a reasonable time. To evaluate the performance of the B&B algorithm, the number of nodes examined is used as a measure of performance. It was established through experimentation that the B&B algorithm’s performance increased with increasing eligibility factor.