An Implicit Optimization Approach for Survivable Network Design

We consider the problem of designing a network of minimum cost while satisfying a prescribed survivability criterion. The survivability criterion requires that a feasible flow must still exists (i.e. all demands can be satisfied without violating arc capacities) even after the disruption of a subset of the network's arcs. Specifically, we consider the case in which a disruption (random or malicious) can destroy a subset of the arcs, with the cost of the disruption not to exceed a disruption budget. This problem takes the form of a tri-level, two-player game, in which the network operator designs (or augments) the network, then the attacker launches a disruption that destroys a subset of arcs, and then the network operator attempts to find a feasible flow over the residual network. We first show how this can be modeled as a two-stage stochastic program from the network operator's perspective, with each of the exponential number of potential attacks considered as a disruption scenario. We then reformulate this problem, via a Benders decomposition, to consider the recourse decisions implicitly, greatly reducing the number of variables but at the expense of an exponential increase in the number of constraints. We next develop a cut-generation based algorithm. Rather than \emph{explicitly} considering each disruption scenario to identify these Benders cuts, however, we develop a bi-level program and corresponding separation algorithm that enables us to \emph{implicitly} evaluate the exponential set of disruption scenarios. Our computational results demonstrate the efficacy of this approach

A Synthesis of Optimization Approaches for Tackling Critical Information Infrastructure Survivability

Over the years, Critical Infrastructures (CI) have revealed themselves to be extremely disaster-prone, be the disasters nature-based or man-made. This paper focuses on a specific category of CI: Critical Information Infrastructures (CII), which are commonly deemed to include communication and information net-works. The majority of all the other CI (e.g. electricity, fuel and water supply, transport systems, etc.) are crucially dependent on CII. Therefore, problems associated with CII that disrupt the services they are able to provide (whether to a single end-user or to another CI) are of increasing interest. This paper discusses some recent developments in optimization models regarding CII’s ability to with-stand disruptive events within three main spheres: network survivability assessment, network resource allocation strategy and survivable design

A bi-level model and solution methods for partial interdiction problem on capacitated hierarchical facilities

Due to the importance of gaining high levels of customer satisfaction in today's competitive world, making appropriate decisions in the face of malicious attacks is valued highly by many organizations. In this paper, to predict and handle the destructive effects of an intentional attack on capacitated nested hierarchical facilities, a bi-level partial interdiction problem is proposed. In this problem, there is an interdictor who can attack facilities partially in different levels. Subsequently, the system defender could respond to the customers’ demand in two different ways, namely through the remaining system facilities and the outsourcing option. The goal of the defender is to minimize the satisfaction cost of all customers’ demand under the interdictor's attacking scenario. This problem can be modeled as a bi-level programming model in which an interdictor and the system defender play the role of the leader and the follower, respectively. Due to the inherent complexity of the bi-level programming models, we develop a heuristic approach, namely “FDS”, to obtain near optimal solutions within a reasonable running time. In each iteration of the FDS, an interdiction scenario is produced heuristically and, thereupon CPLEX solver is called to solve the lower level of the model. To evaluate the effectiveness of the proposed model, a comparison between the cost of customers’ demand satisfaction in both absence and presence of the bi-level model is drawn. Computational results show that for those instances in which the optimal solutions are available, the proposed model can, on average, achieve a saving of 7.94%

Multiple Allocation Hub Interdiction and Protection Problems: Model Formulations and Solution Approaches

In this paper, we present computationally efficient formulations for the multiple allocation hub interdiction and hub protection problems, which are bilevel and trilevel mixed integer linear programs, respectively. In the hub interdiction problem, the aim is to identify a subset of r critical hubs from an existing set of p hubs that when interdicted results in the maximum post-interdiction cost of routing flows. We present two alternate ways of reducing the bilevel hub interdiction model to a single level optimization problem. The first approach uses the dual formulation of the lower level problem. The second approach exploits the structure of the lower level problem to replace it by a set of closest assignment constraints (CACs). We present alternate sets of CACs, study their dominance relationships, and report their computational performances. Further, we propose refinements to CACs that offer computational advantages of an order-of-magnitude compared to the one existing in the literature. Further, our proposed modifications offer structural advantages for Benders decomposition, which lead to substantial computational savings, particularly for large problem instances. Finally, we study and solve large scale instances of the trilevel hub protection problem exactly by utilizing the ideas developed for the hub interdiction problem

Robust Design of Distribution Networks Considering Worst Case Interdictions

Multi-echelon facility location models are commonly employed to design transportation systems. While they provide cost-efficient designs, they are prone to severe financial loss in the event of the disruption of any of its facilities. Additionally, the recent crisis in the world motivates OR practitioners to develop models that better integrate disruptive event in the design phase of a distribution network. In this research, we propose a two-echelon capacitated facility location model under the risk of a targeted attack, which identifies the optimal location of intermediate facilities by minimizing the weighted sum of pre and post interdiction flow cost and the fixed cost of opening intermediate facilities. The developed model results in a tri-level Mixed Integer Programming (MIP) formulation, reformulated in a two-level MIP. Hence, we prescribe solution methods based on Bender Decomposition as well as two variants that enhance the speed performance of the algorithm. The results reveal the importance of selecting backup facilities and highlight that premium paid to design a robust distribution network is negligible given the benefit of reducing the post-interdiction cost when a disruptive event occurs

Survivable Network Design Under Optimal and Heuristic Interdiction Scenarios

Abstract: We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer's network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows

Multi-Level Multi-Objective Programming and Optimization for Integrated Air Defense System Disruption

The U.S. military\u27s ability to project military force is being challenged. This research develops and demonstrates the application of three respective sensor location, relocation, and network intrusion models to provide the mathematical basis for the strategic engagement of emerging technologically advanced, highly-mobile, Integrated Air Defense Systems. First, we propose a bilevel mathematical programming model for locating a heterogeneous set of sensors to maximize the minimum exposure of an intruder\u27s penetration path through a defended region. Next, we formulate a multi-objective, bilevel optimization model to relocate surviving sensors to maximize an intruder\u27s minimal expected exposure to traverse a defended border region, minimize the maximum sensor relocation time, and minimize the total number of sensors requiring relocation. Lastly, we present a trilevel, attacker-defender-attacker formulation for the heterogeneous sensor network intrusion problem to optimally incapacitate a subset of the defender\u27s sensors and degrade a subset of the defender\u27s network to ultimately determine the attacker\u27s optimal penetration path through a defended network

Exact Algorithms for Mixed-Integer Multilevel Programming Problems

We examine multistage optimization problems, in which one or more decision makers solve a sequence of interdependent optimization problems. In each stage the corresponding decision maker determines values for a set of variables, which in turn parameterizes the subsequent problem by modifying its constraints and objective function. The optimization literature has covered multistage optimization problems in the form of bilevel programs, interdiction problems, robust optimization, and two-stage stochastic programming. One of the main differences among these research areas lies in the relationship between the decision makers. We analyze the case in which the decision makers are self-interested agents seeking to optimize their own objective function (bilevel programming), the case in which the decision makers are opponents working against each other, playing a zero-sum game (interdiction), and the case in which the decision makers are cooperative agents working towards a common goal (two-stage stochastic programming). Traditional exact approaches for solving multistage optimization problems often rely on strong duality either for the purpose of achieving single-level reformulations of the original multistage problems, or for the development of cutting-plane approaches similar to Benders\u27 decomposition. As a result, existing solution approaches usually assume that the last-stage problems are linear or convex, and fail to solve problems for which the last-stage is nonconvex (e.g., because of the presence of discrete variables). We contribute exact finite algorithms for bilevel mixed-integer programs, three-stage defender-attacker-defender problems, and two-stage stochastic programs. Moreover, we do not assume linearity or convexity for the last-stage problem and allow the existence of discrete variables. We demonstrate how our proposed algorithms significantly outperform existing state-of-the-art algorithms. Additionally, we solve for the first time a class of interdiction and fortification problems in which the third-stage problem is NP-hard, opening a venue for new research and applications in the field of (network) interdiction

Synthesis, Interdiction, and Protection of Layered Networks

This research developed the foundation, theory, and framework for a set of analysis techniques to assist decision makers in analyzing questions regarding the synthesis, interdiction, and protection of infrastructure networks. This includes extension of traditional network interdiction to directly model nodal interdiction; new techniques to identify potential targets in social networks based on extensions of shortest path network interdiction; extension of traditional network interdiction to include layered network formulations; and develops models/techniques to design robust layered networks while considering trade-offs with cost. These approaches identify the maximum protection/disruption possible across layered networks with limited resources, find the most robust layered network design possible given the budget limitations while ensuring that the demands are met, include traditional social network analysis, and incorporate new techniques to model the interdiction of nodes and edges throughout the formulations. In addition, the importance and effects of multiple optimal solutions for these (and similar) models is investigated. All the models developed are demonstrated on notional examples and were tested on a range of sample problem sets

Locating and Protecting Facilities Subject to Random Disruptions and Attacks

Recent events such as the 2011 Tohoku earthquake and tsunami in Japan have revealed the vulnerability of networks such as supply chains to disruptive events. In particular, it has become apparent that the failure of a few elements of an infrastructure system can cause a system-wide disruption. Thus, it is important to learn more about which elements of infrastructure systems are most critical and how to protect an infrastructure system from the effects of a disruption. This dissertation seeks to enhance the understanding of how to design and protect networked infrastructure systems from disruptions by developing new mathematical models and solution techniques and using them to help decision-makers by discovering new decision-making insights. Several gaps exist in the body of knowledge concerning how to design and protect networks that are subject to disruptions. First, there is a lack of insights on how to make equitable decisions related to designing networks subject to disruptions. This is important in public-sector decision-making where it is important to generate solutions that are equitable across multiple stakeholders. Second, there is a lack of models that integrate system design and system protection decisions. These models are needed so that we can understand the benefit of integrating design and protection decisions. Finally, most of the literature makes several key assumptions: 1) protection of infrastructure elements is perfect, 2) an element is either fully protected or fully unprotected, and 3) after a disruption facilities are either completely operational or completely failed. While these may be reasonable assumptions in some contexts, there may exist contexts in which these assumptions are limiting. There are several difficulties with filling these gaps in the literature. This dissertation describes the discovery of mathematical formulations needed to fill these gaps as well as the identification of appropriate solution strategies