Train stop scheduling problem: An exact approach using valid inequalities and polar duality

Abstract

Consider the problem of minimizing the number of train stops on a particular rail line. The objective is to assign passengers of each origin-destination pair to different trains in such a way that train capacity and passenger demand constraints are satisfied with minimal stoppages. The literature refers to this problem as the train stop scheduling problem. The problem has been extensively studied for decades, yet an exact solution approach has not been proposed. This paper proposes several valid inequalities to strengthen the mixed-integer programming formulation and solve the problem exactly in a reasonable amount of CPU time. The concept of polar duality has been utilized to find more complex valid inequalities which may be hard to find otherwise. Despite the problem’s high practical relevance, valid inequalities for the problem have not yet been studied in the literature. An aggregation procedure has been proposed to solve large size problem instances exactly. Computational study demonstrate the efficacy of the proposed valid inequalities.