Distributionally robust single machine scheduling with release and due dates over Wasserstein balls

Abstract

Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the ${\ell }_1$-norm and ${\ell }_2$-norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.