A robust cardinality-constrained model to address the machine loading problem

Several deterministic models have been proposed in the literature to solve the machine loading problem (MLP), which considers a set of product types to be produced on a set of machines using a set of tool types, and determines the quantity of each product type to be produced at each time period and the corresponding machine tool loading configuration. However, processing times are subject to random increases, which could impair the quality of a deterministic solution. Thus, we propose a robust MLP counterpart, searching for an approach that properly describes the uncertainty set of model parameters and, at the same time, ensures practical application. We exploit the cardinality-constrained approach, which considers a simple uncertainty set where all uncertain parameters belong to an interval, and allows tuning the robustness level by bounding the number of parameters that assume the worst value. The resulting plans provide accurate estimations on the minimum production level that a system achieves even in the worst conditions. The applicability of the robust MLP and the impact of robustness level have been tested on several problem variants, considering single- vs multi-machine and single- vs multi-period MLPs. We also consider the execution of the plans in a set of scenarios to evaluate the practical implications of MLP robustness. Results show the advantages of the robust formulation, in terms of improved feasibility of the plans, identification of the most critical tools and products, and evaluation of the maximum achievable performance in relation to the level of protection. Moreover, low computational times guarantee the applicability of the proposed robust MLP counterpart.