Minimizing the number of machines in the Fixed Interval Scheduling Problem with energy constraints

Abstract

This paper deals with a new variant of the Fixed Interval Scheduling Problem that includes additional energy constraints. Specifically, given a set of independent jobs, having fixed start and end times and requiring a predefined amount of energy for their processing, all jobs should be processed on a set of independent machines operating with plug-in batteries. Each battery is defined by its maximum storage capacity. The stored energy is consumed by the machine when the jobs are processed. If a machine does not have enough energy to process a job, its battery can be charged using specific devices called chargers. Thus, charging tasks should be executed on each machine in order to retrieve enough energy to carry out jobs. When charging should be undertaken, each charging task has to be simultaneously processed on the machine and on a processor that can deliver at most a given amount of energy. This processor can execute more than one charging task at once, provided that the total energy delivered does not exceed the available energy. Our objective is to study the complexity and the approximability of this new fixed interval scheduling problem with energy constraints where the objective is to minimize the number of machines required to carry out all jobs. In this paper, we study the complexity of this problem and propose an approximation algorithm having a performance guarantee of 3/2.