Multitasking scheduling with shared processing

Recently, the problem of multitasking scheduling has raised a lot of interest in the service industries. Hall et al. (Discrete Applied Mathematics, 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. With a team being modeled as a single machine, the processing sharing of the machine is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio, to the primary jobs. In this paper, we generalize this model to parallel machines and allow the fraction of the processing capacity assigned to routine jobs to vary from one to another. The objectives are minimizing makespan and minimizing the total completion time of primary jobs. We show that for both objectives, there is no polynomial time approximation algorithm unless P=NP if the sharing ratios are arbitrary for all machines. Then we consider the problems where the sharing ratios on some machines have a constant lower bound. For each objective, we analyze the performance of the classical scheduling algorithms and their variations and then develop a polynomial time approximation scheme when the number of machines is a constant.