Scheduling with release dates and preemption to minimize multiple max-form objective functions

Abstract

In this paper, we study multi-agent scheduling with release dates and preemption on a single machine, where the scheduling objective function of each agent to be minimized is regular and of the maximum form (max-form). The multi-agent aspect has three versions, namely ND-agent (multiple agents with non-disjoint job sets), ID-agent (multiple agents with an identical job set), and CO-agent (multiple competing agents with mutually disjoint job sets). We consider three types of problems: The first type (type-1) is the constrained scheduling problem, in which one objective function is to be minimized, subject to the restriction that the values of the other objective functions are upper bounded. The second type (type-2) is the weighted-sum scheduling problem, in which a positive combination of the objective functions is to be minimized. The third type (type-3) is the Pareto scheduling problem, for which we aim to find all the Pareto-optimal points and their corresponding Pareto-optimal schedules. We show that the type-1 problems are polynomially solvable, and the type-2 and type-3 problems are strongly NP-hard even when all jobs’ release dates are zero and processing times are one. When the number of the scheduling criteria is fixed and they are all lateness-like, such as minimizing Cmax, Fmax, Lmax, Tmax, and WCmax, where WCmax is the maximum weighted completion time of the jobs, the type-2 and type-3 problems are polynomially solvable. To address the type-3 problems, we develop a new solution technique that guesses the Pareto-optimal points through some elaborately constructed schedule-configurations.