Optimal Allocation of Protective Resources in Shortest-Path Networks

TLDR

Short‑path networks can suffer traffic delays or component loss when disrupted. The study introduces a game‑theoretic multilevel program that allocates protection resources to minimize the shortest‑path length after a worst‑case disruption. An implicit enumeration algorithm is developed to solve the multilevel problem to optimality, with heuristics for the lower‑level interdiction at each tree node and variable‑fixing rules to reduce problem size. Computational experiments show the method can find optimal protection strategies for sizable networks, and sensitivity analysis reveals how disruption level, resource availability, and arc characteristics affect the solution.

Abstract

This article introduces a game-theoretic approach for allocating protection resources among the components of a network so as to maximize its robustness to external disruptions. Specifically, we consider shortest-path networks where disruptions may result in traffic flow delays through the affected components or even in the complete loss of some elements. A multilevel program is proposed to identify the set of components to harden so as to minimize the length of the shortest path between a supply node and a demand node after a worst-case disruption of some unprotected components. An implicit enumeration algorithm is then developed to solve the multilevel problem to optimality. The approach is streamlined by solving the lower-level interdiction problem heuristically at each node of an enumeration tree and by using some variable fixing rules to reduce the dimension of the lower-level problems. A thorough computational investigation demonstrates that the proposed solution method is able to identify optimal protection strategies for networks of significant size. The paper is concluded with a study of the sensitivity of the solution approach to variations of the problem parameters such as the level of disruption and protective resources and the distribution of the arc lengths and delays.