Efficiency and equity in the multi organization scheduling problem

Abstract

We consider the multi organization scheduling problem (MOSP) [19]: given N organizations owning, each of them, one set of tasks and machines, the aim is to compute a schedule which gathers all the tasks on all the machines, and such that the makespan is minimized. A rationality constraint must be fulfilled: no organization should increase its makespan (the completion time of its last task) compared to the case where it schedules its own tasks (and only its own tasks) on its own machines. We show that cooperation (sharing machines and tasks) can benefit to all the organizations simultaneously, since they may decrease their makespans by a factor of N. We present an algorithm which is (1+ϵ)-approximate, while the makespan of each organization is increased by a factor at most (1+ϵ). We also study to which extent the rationality constraint (or a relaxed constraint) increases the makespan, compared to problem (P||Cmax) where there is no such a constraint. Finally, we introduce a new problem, which focus on equity: the aim is to return a schedule which fulfills the rationality constraint and which maximizes the factor by which each organization has decreased its makespan. We give an optimal algorithm for this problem in a particular case, and show that it is NP-hard and hard to approximate in the general case. We complete this paper by an efficient heuristic for this problem.