A trilevel r-interdiction selective multi-depot vehicle routing problem with depot protection

The determination of critical facilities in supply chain networks has been attracting the interest of the Operations Research community. Critical facilities refer to structures including bridges, railways, train/metro stations, medical facilities, roads, warehouses, and power stations among others, which are vital to the functioning of the network. In this study we address a trilevel optimization problem for the protection of depots of utmost importance in a routing network against an intelligent adversary. We formulate the problem as a defender-attacker-defender game and refer to it as the trilevel r-interdiction selective multi-depot vehicle routing problem (3LRI-SMDVRP). The defender is the decision maker in the upper level problem (ULP) who picks u depots to protect among m existing ones. In the middle level problem (MLP), the attacker destroys r depots among the (m-u) unprotected ones to bring about the biggest disruption. Finally, in the lower level problem (LLP), the decision maker is again the defender who optimizes the vehicle routes and thereby selects which customers to visit and serve in the wake of the attack. All three levels have an identical objective function which is comprised of three components. (i) Operating or acquisition cost of the vehicles. (ii) Traveling cost incurred by the vehicles. (iii) Outsourcing cost due to unvisited customers. The defender aspires to minimize this objective function while the attacker tries to maximize it. As a solution approach to this trilevel discrete optimization problem, we resort to a smart exhaustive enumeration in the ULP and MLP. For the LLP we design a metaheuristic algorithm that hybridizes Variable Neighborhood Descent and Tabu Search techniques adapted to the Selective MDVRP (SMDVRP). The performance of this algorithm is demonstrated on 33 MDVRP benchmark instances existing in the literature and 41 SMDVRP instances generated from them. Numerical experiments on a large number of 3LRI-SMDVRP instances attest that our comprehensive method is effective in dealing with the defender-attacker-defender game on multi-depot routing networks.