Properties of Some Classes of Structured Minmaxmin Problems

Minmaxmin problems are well suited for representing some significant decision making problems, where both strategic and tactical decisions are to be made, at different points of time, in the presence of uncertain scenarios. We survey some basic properties and introduce some classes of structured minmaxmin problems. The main focus is on linear and bilinear minmaxmin problems, which reduce to classic nonsmooth optimization problems. Moreover, two classes of examples are introduced to highlight the practical role of such formulations. The first one is related to the optimal capacity planning of a production–distribution system, and the second one deals with product pricing and distribution in a profit-maximization framework. Finally, focusing on the capacity-planning and product-distribution problem, a computational study has been carried out to illustrate the practical performance of a cutting-plane and proximity-control algorithm for solving the resulting convex nonsmooth minmaxmin model. The numerical results confirm the robustness of the approach and its scalability with respect to both the network size and the number of scenarios.