Mixed satisfiability tests for multiprocessor scheduling with release dates and deadlines

Abstract

We investigate classical satisfiability tests for $\text{P|r}{}_{\mathrm{i}}\text{,}\text{d}{}_{\mathrm{i}}\text{̃}\text{|−}$. Our motivation is to use the complementarity of classical preemptive relaxation and energetic reasoning. Thus minimum capacity constraints, based on the notion of mandatory parts of activities, are added to the classical max-flow formulation.

Introduction

In the identical parallel machine problem, a set I of activities has to be processed on m identical parallel machines. Each activity, A_i∈I, has a release date r_i, a processing time p_i and a deadline $\text{d}{}_{\mathrm{i}}\text{̃}$. All the data are assumed to be deterministic and integer. Each machine processes at most one job at a time and each job cannot be processed by more than one machine. We also assume that preemption is not allowed, and that machines are available from time zero onwards. Our aim is either to find a schedule in which activities are processed within their time-window $\text{[r}{}_{\mathrm{i}}\text{,}\text{d}{}_{\mathrm{i}}\text{̃}\text{]}$, without violating the machine constraint, or to prove that there exists no such solution. Classically, this problem is denoted by $\text{P|r}{}_{\mathrm{i}}\text{,}\text{d}{}_{\mathrm{i}}\text{̃}\text{,|−}$, which is known to be $\text{NP}$-hard in the strong sense. In this paper, we address some techniques to detect that there exists no feasible solution.

Our main motivation for studying this problem is the fact that it can be seen as a relaxation of some fundamental complex scheduling problems as Hybrid Flowshop and RCPSP. It corresponds to the decisional variant of P|r_i|L_max [6] and then it is strongly connected to the P|r_i,q_i|C_max, for which several efficient lower bounds have been presented, e.g., [4], [5], [6].

In this paper, we focus on two satisfiability tests for $\text{P|r}{}_{\mathrm{i}}\text{,}\text{d}{}_{\mathrm{i}}\text{̃}\text{|−}$: the first one is based on the preemptive relaxation, that can be solved in polynomial time, the second one is the energetic reasoning. Our basic motivation is to use the complementarity of those two approaches, using the notion of mandatory parts of activities into the classical max-flow formulation.

In Section 2, both preemptive approach (PR) and energetic reasoning (ER) applied to the $\text{P|r}{}_{\mathrm{i}}\text{,}\text{d}{}_{\mathrm{i}}\text{̃}\text{|−}$ are recalled and examples illustrating the drawbacks of those approaches are presented. In Section 3, we propose a new model that is based on a max-flow with minimum capacity. In Section 4, we show that our approach dominates recent satisfiability tests the presentation of which does not mention energetic reasoning, even if they implicitly use it.