An approximation scheme for the bi-scenario sum of completion times trade-off problem

An influential aspect of any scheduling problem is the processing time of the tasks (jobs), which can be deterministic, stochastic, or even uncertain. Scheduling according to unique and known processing times (a.k.a. nominal values) may be naïve, since many production systems are subject to inherent uncertainty. We offer a novel approach for scheduling under uncertainty, inspired by scenario-based optimization. This new approach copes with uncertainty by simultaneously optimizing a given scheduling criterion under two different scenarios of processing times. We term the new problem the bi-scenario trade-off problem. We study the sum of completion times criterion for which we develop a 2-approximation algorithm and show that it is asymptotically tight. We use this approximation to develop the polynomial time approximation scheme that approximates the Pareto-optimal set of solutions and introduce data-dependent analysis to examine the sensitivity of the model parameters.