Stochastic Scheduling on Unrelated Machines

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. Here, the processing time of job j on machine i is governed by random variable Pij , and its realization becomes known only upon job completion. With wj being the given weight of job j, we study the objective to minimize the expected total weighted completion time E [∑ j wjCj ] , where Cj is the completion time of job j. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee (3+∆)/2+ε. 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 rij , our bound is (2 + ∆) + ε. We also show that the dependence of the performance guarantees on ∆ is tight. Via ∆ = 0, currently best known bounds for deterministic scheduling on unrelated machines are contained as special case. 1998 ACM Subject Classification F.2.2 Sequencing and Scheduling, G.2.1 Combinatorial Algorithms, G.1.6 Optimization