Tri-level mixed-binary linear programming: Solution approaches and application in defending critical infrastructure

Decentralized decision-making is becoming more ubiquitous in different organizations that often follow a hierarchical structure. To model these problems, multi-level programming has been suggested as a suitable methodology for modeling the interaction between the different levels of decisions. However, multi-level programming, even for the case of bi-levels, is known to be strongly NP-hard. To address this computational challenge, we develop three different heuristic-based approaches for solving a specific class of tri-level programming problems, in which the leader has direct control over the follower’s decisions to a certain extent, with a common objective function shared at all levels. As expected, each heuristic type offers a trade-off between solution quality and computational time. To illustrate our solution approach, we present an application for defending critical infrastructure to improve its resilience against intentional attacks. In this context we use a defender-attacker-defender model and apply it to electrical power grids. We also propose a modified implementation of a widely adopted enumeration algorithm in this area, with a warm-starting solution technique that significantly enhanced the computational performance of the enumeration algorithm. We test our solution approaches on three electrical transmission networks and present the results of our numerical computations as well as some insights.