Job scheduling for maximum revenue on uniform, parallel machines with major and minor setups and job splitting

Abstract

We consider a revenue maximization scheduling problem where jobs can be split on parallel, uniform machines. By assuming that only major and minor setups exist between jobs and that these can be grouped into families, we reformulate the scheduling problem as a continuous knapsack problem with setup time. We then create the Revenue Rate Heuristic to solve the maximum revenue scheduling problem for both the partial and binary revenue cases. This makes use of a revenue rate – the ratio between total revenue that can be obtained from a job and the time required to achieve the revenue. The Revenue Rate Heuristic schedules the job with the highest revenue rate first and updates all the revenue rates accordingly based on existing scheduled families and remaining capacity. In cases where there are no setup times or if there is only one machine and an ordering of job revenue and production time, we are able to show that the Revenue Rate Heuristic is optimal. Numerical studies show that the Revenue Rate Heuristic performs well with an average optimality gap of 3.47% across a large set of parameters and when the setup times across families are different from each other. In a case study based on firm-level data, we also show that the average optimality gap remains small.