We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the ROADEF/EURO challenge 2020 1 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [CRC TRR 154], Fonds De La Recherche Scientifique [PDR T0098.18], and Bundesministerium für Bildung und Forschung. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0105 .

Abstract The commodity constrained split delivery vehicle routing problem (C‐SDVRP) is a routing problem where customer demands are composed of multiple commodities. A fleet of capacitated vehicles must serve customer demands in a way that minimizes the total routing costs. Vehicles can transport any set of commodities and customers are allowed to be visited multiple times. However, the demand for a single commodity must be delivered by one vehicle only. In this work, we developed a heuristic with a performance guarantee to solve the C‐SDVRP. The proposed heuristic is based on a set covering formulation, where the exponentially‐many variables correspond to routes. First, a subset of the variables is obtained by solving the linear relaxation of the formulation by means of a column generation approach which embeds a new pricing heuristic aimed to reduce the computational time. Solving the linear relaxation gives a valid lower bound used as a performance guarantee for the heuristic. Then, we devise a restricted master heuristic to provide good upper bounds: the formulation is restricted to the subset of variables found so far and solved as an integer program with a commercial solver. A local search based on a mathematical programming operator is applied to improve the solution. We test the heuristic algorithm on benchmark instances from the literature. The comparison with the state‐of‐the‐art heuristics for solving the C‐SDVRP shows that our approach significantly improves the solution time, while keeping a comparable solution quality and improving some best‐known solutions. In addition, our approach is able to solve large instances with 100 customers and six commodities, and also provides very good quality lower bounds. Furthermore, an instance of the C‐SDVRP can be transformed into a CVRP instance by simply duplicating each customer as many times as the requested commodities and by assigning as demand the demand of the single commodity. Hence, we compare heuristics for the C‐SDVRP against the state‐of‐the‐art heuristic for the Capacitated Vehicle Routing Problem (CVRP). The latter approach revealed to have the best performance. However, our approach provides solutions of comparable quality and has the interest of providing a performance guarantee.