Satisfiability tests and time‐bound adjustmentsfor cumulative scheduling problems

This paper presents a set of satisfiability tests and time-bound adjustment algorithms that can be applied to cumulative scheduling problems. An instance of the Cumulative Scheduling Problem (CuSP) consists of (1) one resource with a given capacity and (2) a set of activities, each having a release date, a deadline, a processing time and a resource capacity requirement for each activity. The problem is to decide whether there exists a start time assignment to all activities such that at no point in time the capacity of the resource is exceeded and all timing constraints are satisfied. The Cumulative Scheduling Problem can be seen as a relaxation of the decision variant of the Resource-Constrained Project Scheduling Problem. We present three necessary conditions for the existence of a feasible schedule. Two of them are obtained by polynomial relaxations of the CuSP. The third one is based on energetic reasoning. We show that the second condition is closely related to the subset bound, a well-known lower bound of the m-Machine Problem. We also present three algorithms, based on the previously mentioned necessary conditions, to adjust release dates and deadlines of activities. These algorithms extend the time-bound adjustment techniques developed for the One-Machine Problem. They have been incorporated in a branch and bound procedure to solve the Resource-Constrained Project Scheduling Problem. Computational results are reported.