Fare evasion in transit networks

Abstract

Public transit systems in major urban areas usually operate under deficits and therefore require significant subsidies. An important cause of this deficit, particularly in the developing world, is the high fare evasion rate mainly due to an ineffective control policy or the lack of it. In this paper we study new models for optimizing fare inspection strategies in transit networks based on bilevel programming. In the first level, the leader (the network operator) determines probabilities for inspecting passengers at different locations, while in the second level, the followers (the fare-evading passengers) respond by optimizing their routes given the inspection probabilities and travel times. To model the followers’ behavior we study both a nonadaptive variant, in which passengers select a path a priori and continue along it throughout their journey, and an adaptive variant, in which they gain information along the way and use it to update their route. For these problems, which are interesting in their own right, we design exact and approximation algorithms, and we prove a tight bound of 3/4 on the ratio of the optimal cost between adaptive and nonadaptive strategies. For the leader’s optimization problem, we study a fixed-fare and a flexible-fare variant, where ticket prices may or may not be set at the operator’s will. For the latter variant, we design an LP-based approximation algorithm. Finally, employing a local search procedure that shifts inspection probabilities within an initially determined support set, we perform an extensive computational study for all variants of the problem on instances of the Dutch railway and the Amsterdam subway network. This study reveals that our solutions are within 5% of theoretical upper bounds drawn from the LP relaxation. We also derive exact nonlinear programming formulations for all variants of the leader’s problem and use them to obtain exact solutions for small instance sizes.