On the asymptotic optimality of a simple on-line algorithm for the stochastic single-machine weighted completion time problem and its extensions

We consider the stochastic single-machine problem, when the objective is to minimize the expected total weighted completion time of a set of jobs that are released over time. We assume that the existence and the parameters of each job including its release date, weight, and expected processing times are not known until its release date. The actual processing times are not known until processing is completed. We analyze the performance of the on-line nonpreemptive weighted shortest expected processing time among available jobs (WSEPTA) heuristic. When a scheduling decision needs to be made, this heuristic assigns, among the jobs that have arrived but not yet processed, one with the largest ratio of its weight to its expected processing time. We prove that when the job weights and processing times are bounded and job processing times are mutually independent random variables, WSEPTA is asymptotically optimal for the single-machine problem. This implies that WSEPTA generates a solution whose relative error approaches zero as the number of jobs increases. This result can be extended to the stochastic flow shop and open shop problems, as well as models with stochastic job weights.