Unrelated Machine Scheduling with Stochastic Processing Times

Two important characteristics encountered in many real-world scheduling problems are heterogeneous processors and a certain degree of uncertainty about the sizes of jobs. In this paper we address both, and study for the first time a scheduling problem that combines the classical unrelated machine scheduling model with stochastic processing times of jobs. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee (3+∆)/2+ε for the stochastic version of the unrelated parallel machine scheduling problem with the weighted sum of completion times objective. Here, ε > 0 is arbitrarily small, and ∆ is an upper bound on the squared coefficient of variation of the processing times. When jobs also have individual release dates, our bound is (2 + ∆) + ε. We also show that the dependence of the performance guarantees on ∆ is tight. Via ∆ = 0, the currently best known bounds for deterministic scheduling on unrelated machines are contained as special case.