Minimizing the weighted number of late jobs on a single machine with equal processing times and agreeable release and due dates

Abstract

Consider the problem of a set of n jobs which needs to be processed by a single ma chine. The processing time for each job is identical to all the others, and predefined. Once on the machine, preemptions are not allowed. Every job has a release date be fore which it can not be scheduled. They also have due dates, before which they are supposed to have completed processing by the machine. The jobs are weighted, and the goal is to find a schedule which maximize the sum of weights of jobs complete in time. Baptiste [1] approached a generic instance of this problem with a dynamic programming solution which runs in O(n 7 ) time. We use an additional hypothesis related to release and due dates: for any job j, its release date is denoted by rj and its due date by dj . We say that a set of jobs and release and due dates are agreeable if, and only if, for two jobs j1 and j2, rj1 < rj2 ⇔ dj1 < dj2 . We model this problem as an integer linear programming and run in general solvers like glpsol and CPLEX. Finally, we present an alternative solution inspired on Baptiste’s original dynamic programming to solve only instances whose release and due dates are "agreeable" like we defined earlier. Our solution outperforms the original and the solvers when the set of dates is agreeable, running in O(n 3 ) time.