Bicriterion scheduling with a negotiable common due window and resource-dependent processing times

Abstract

We investigate scheduling problems with a negotiable common due window where the job processing times are controllable by allocating extra resources to process the jobs and the resource amounts can be either discrete or continuous. We adopt a bicriterion analysis, where one criterion is a cost function consisting of the weighted numbers of early and late jobs, and due window assignment cost (DWAC), whereas the other criterion is the total resource consumption cost (TRSC). We investigate four problems resulting from different treatments of the two criteria as follows: P1, which minimizes the sum of the two criteria; P2 and P3, which minimize one of the two criteria subject to a constraint on the value of the other criterion, respectively; and P4, which identifies the set of Pareto-optimal points of the two criteria. We show that P1 is polynomially solvable, while P2–P4 with both resource types are all NP-hard. With the discrete resource type, we propose pseudo-polynomial-time algorithms for P4, establishing that P2–P4 are all binary NP-hard. With the continuous resource type, we provide an optimal algorithm for P2–P4 by solving a series of mixed integer linear programming (MILP) models. We also provide a two-dimensional fully polynomial-time approximation scheme (FPTAS) to approximate the Pareto set. Finally, we perform computational experiments to verify the effectiveness of the developed solution procedures.