On Scheduling Inclined Jobs on Multiple Two-Stage Flowshops

We study scheduling on multiple two-stage flowshops in which each job has to pass through an R-operation and a T-operation. Motivated by the current research in data centers, we consider two restricted versions of the problem in which the jobs are inclined: one restricts that for each job, the R-operation consumes no less time than the T-operation, while the other assumes that the T-operation consumes no less time than the R-operation for each job. For the first case, we present an online 2-competitive algorithm and an offline 11/6-approximation algorithm. For the second case, we give an online 5/2-competitive algorithm, and prove, for the offline setting, that the problem can be reduced to the problem in the first case.