De-Cycling Cyclic Scheduling Problems

An elegant way to tackle a problem that you cannot solve is to cast it to a problem that you can solve very well. Cyclic Scheduling problems are very similar to Resource Constrained Project Scheduling Problems (RCPSP), except that the project activities are repeated over time. Due to the similarity, reducing Cyclic Scheduling problems to RCPSPs seems an appealing approach. In this paper we discuss four methods to perform the reduction. The first two are existing techniques. The remaining ones are novel and include the first (to the best of our knowledge) equivalent RCPSP formulation of a cyclic problem. We compare the presented approaches in an experimental evaluation. Introduction The classical Resource Constrained Project Scheduling Problem (RCPSP) consists in ordering a set A of activities ai, connected by a set E of precedence constraints (ai, aj). The two sets form the so-called Project Graph. Each activity has a fixed duration di and requires some amount of one or more resources with limited capacity. For sake of simplicity we consider here a single resource with capacity c and refer to the requirements as rqi. The objective is to minimize the makespan. Cyclic scheduling problems (see e.g. [Draper et al., 1999; Hanen, 1994]) are very similar, with the main difference that activities are repeated indefinitely over time. We refer as 〈i, ω〉 to the ω-th execution of activity ai, where ω ∈ Z is called execution number. A schedule is an assignment of a start time s(i, ω) to each 〈i, ω〉 and has in principle infinite size. From a practical perspective, one is typically interested in finding a periodic schedule, where: s(i, ω) = s(i, 0) + ω · λ (1) where λ ≥ 0 is the period and measures the solution quality: the lower the period, the better the schedule. As a second difference w.r.t. the RCPSP, precedence constraints may link activity executions with different ω. In detail, each arc (ai, aj) ∈ E is labeled with a delay δi,j and means that 〈j, ω〉 must wait for the end of 〈i, ω − δi,j〉. In Copyright c © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. a0