Proportionate Flow Shop Games

In a proportionate flow shop problem several jobs have to be processed through a fixed sequence of machines and the processing time of each job is equal on all machines. By identifying jobs with agents, whose costs linearly depend on the completion time of their jobs, and assuming an initial processing order on the jobs, we face an additional problem: how to allocate the cost savings obtained by ordering the jobs optimally? In this paper, PFS games are defined as cooperative games associated to proportionate flow shop problems. It is seen that PFS games have a nonempty core. Moreover, it is shown that PFS games are convex if the jobs are initially ordered in decreasing urgency. For this case an explicit expression for the Shapley value and a specific type of equal gain splitting rule which leads to core elements of the PFS game are proposed