A two-stage robust approach for minimizing the weighted number of tardy jobs with objective uncertainty

Minimizing the weighted number of tardy jobs on one machine is a classical and intensively studied scheduling problem. In this paper, we develop a two-stage robust approach, where exact weights are known after accepting the jobs to be performed, and before sequencing them on the machine. This assumption allows diverse recourse decisions to be taken in order to better adapt one’s mid-term plan. The contribution of this paper is twofold: First, we introduce a new scheduling problem and model it as a min-max-min optimization problem with mixed-integer recourse by extending existing models proposed for the deterministic case. Second, we take advantage of the special structure of the problem to propose two solution approaches based on results from the recent robust optimization literature: namely the finite adaptability (Bertsimas and Caramanis in IEEE Trans Autom Control 55(12):2751–2766, 2010) and a convexification-based approach (Arslan and Detienne in INFORMS J Comput 34(2):857–871, 2022). We also study the additional cost of the solutions if the sequence of jobs has to be determined before the uncertainty is revealed. Computational experiments are reported to analyze the effectiveness of our approaches.