A Lagrangian Heuristic for Robustness, with an Application to Train Timetabling

Finding robust yet efficient solutions to optimization problems is a major practical issue that received large attention in recent years. Starting with stochastic programming, many of the approaches to robustness lead to a significant change in the problem formulation with respect to the non-robust (nominal) case. Besides requiring a much larger computational effort, this often results into major changes of the associated software. Lagrangian heuristics form a wide family of methods that work well in finding efficient (i.e., low cost) solutions for many problems. These methods approximately solve a relaxation of the problem at hand through an iterative Lagrangian optimization scheme, and apply several times a basic heuristic driven by the Lagrangian dual information (typically, the current Lagrangian costs) so as to hopefully update the current best feasible solution. In this context, the underlying Lagrangian optimization iterative method has the main purpose of producing “increasingly reliable” Lagrangian costs, while diversifying the search in the last iterations, when the Lagrangian bound is very close to convergence. The purpose of this paper is to propose, for the first time to the best of our knowledge, a very simple modification of the Lagrangian optimization scheme capable of dealing with robustness. This modification is based on two simple features: (a) we modify the problem formulation by introducing artificial parameters intended to “control” the solution robustness; and (b) during Lagrangian optimization, we dynamically change the weight of the control parameters so as to produce subproblems where robustness becomes more and more important. In this way, during the process we can easily collect a set of, roughly speaking, “Pareto optimal” heuristic solutions that have a different tradeoff between robustness and efficiency, and leave the final user the choice of the ones to analyze in more details—e.g., through a time-consuming validation tool. As a proof-of-concept, our approach is applied to the well-known aperiodic Train Timetabling Problem (TTP) on a corridor, and is computationally analyzed on real-world test cases from the Italian Railways, showing that it produces within much shorter computing times solutions whose quality is comparable with those produced by existing approaches to robustness. This proves the effectiveness for the specific application, suggesting that a simple modification of existing Lagrangian heuristics is a very promising way to deal with robustness in many other cases.