Fixed interval scheduling with third-party machines

We study a problem of schedulingnjobs on machines of two types: in-house machines and third-party machines. Scheduling on in-house machines incurs no additional costs, while using third-party machines implies costs depending on their number and the time of usage. Each job has a fixed time interval for being processed which can be divided and allocated among several machines, as long as there is only one machine processing the job at any time. No machine can process more than one job at a time. Jobs can be rejected, and they are of different importance that is reflected in the weight of each job. The objective is to find a subset of the jobs and the number of third-party machines for any period of time so that the accepted jobs can be feasibly scheduled, the total weight of the accepted jobs is maximized, and the total machine usage costs does not exceed a given upper bound. We also study a similar problem in which the objective is to maximize the total time at which at least one job is processed. Both problems are encountered in situations in which certain activities with given start and completion times have to be serviced by human operators. Examples are air traffic control and the monitoring safe vehicle unloading. Other examples are the employment of subcontractors in agriculture, construction or transportation. We will present NP-hardness proofs, polynomial and pseudo-polynomial optimal algorithms and an approximation algorithm for these problems and their special cases. These problems admit graph-theoretical interpretations associated with finding independent sets and a proper vertex coloring in interval graphs.