The vulnerability of transportation infrastructure is critical to the optimal performance of public transit systems, particularly rail networks, which have consistently been targeted by terrorist attacks. This article aims to identify critical arcs vulnerable to attacks and to evaluate attacker and defender behavior under varying budget levels. We present a continuous nonlinear bi-objective optimization model in which the attacker seeks to maximize the average travel time and the weighted variance, whereas the defender aims to minimize them. We focus on defense measures that are rapidly reallocated and can be kept partially or fully unobservable (e.g., allocation of guards to protect the railway), motivating a simultaneous model. Durable, observable hardening investments are beyond our current scope and are more appropriately addressed with sequential models. To solve the model at scale, we employ a combination of the Lagrange relaxation method and the Frank-Wolfe algorithm to transform the Minimax model into a convex nonlinear minimization model. Results show that the proposed model successfully identifies critical arcs and demonstrates that increased defensive investment can significantly reduce attacker impact. A case study on Iran's railway reveals a linear, nonlinear, or peak-like attacker-defender behavior by considering budget thresholds beyond which attacks become ineffective.
Kouroshniya et al. (Thu,) studied this question.