Approximating multi-scenario efficient solutions for an uncertain bi-criteria parallel machine scheduling problem

Abstract

This paper studies a bi-criteria parallel machine scheduling problem with uncertain processing times to minimize makespan and total completion time. We adopt a scenario-based approach for uncertainty and assume that a scenario index can take any real value in an interval. This index value can represent, for example, the amount of congestion in a network. We assume that the job processing times are non-decreasing polynomial functions of fixed degrees with respect to the scenario index. To tackle multi-criteria optimization in the presence of uncertainty, researchers have proposed the concept of multi-scenario efficient set (Botte and Schöbel 2019, Engau and Sigler 2020). We adopt this framework and want to compute, in polynomial time, an approximation of this set with theoretical performance guarantees. Based on a dynamic programming framework, with carefully designed states, we obtain a Fully Polynomial-Time Approximation Scheme (FPTAS) for the problem.