  Recently, the problem of multitasking scheduling has attracted a lot of
attention in the service industries where workers frequently perform multiple
tasks by switching from one task to another. Hall, Leung and Li (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. The processing
sharing 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. 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.
