Interdiction Model Research Articles

An Optimization-Based Framework for the Identification of Vulnerabilities in Electric Power Grids Exposed to Natural Hazards.

This article proposes a novel mathematical optimization framework for the identification of the vulnerabilities of electric power infrastructure systems (which is a paramount example of critical infrastructure) due to natural hazards. In this framework, the potential impacts of a specific natural hazard on an infrastructure are first evaluated in terms of failure and recovery probabilities of system components. Then, these are fed into a bi-level attacker-defender interdiction model to determine the critical components whose failures lead to the largest system functionality loss. The proposed framework bridges the gap between the difficulties of accurately predicting the hazard information in classical probability-based analyses and the over conservatism of the pure attacker-defender interdiction models. Mathematically, the proposed model configures a bi-level max-min mixed integer linear programming (MILP) that is challenging to solve. For its solution, the problem is casted into an equivalent one-level MILP that can be solved by efficient global solvers. The approach is applied to a case study concerning the vulnerability identification of the georeferenced RTS24 test system under simulated wind storms. The numerical results demonstrate the effectiveness of the proposed framework for identifying critical locations under multiple hazard events and, thus, for providing a useful tool to help decisionmakers in making more-informed prehazard preparation decisions.

A Logic-based Decomposition Approach for Multi-Period Network Interdiction Models

This research details the creation of a large-scale optimization approach for solving an application of a multi-period bilevel network interdiction problem (NIP). In this class of multi-period NIP, interdiction activities must be scheduled to minimize the cumulative maximum flow over a finite time horizon. A logic-based decomposition (LBD) approach is proposed that utilizes constraint programming to exploit the scheduling nature of this multi-period NIP. Computational results–comparing solutions obtained using the proposed approach versus traditional mixed-integer programming approach–suggest that the LBD approach is more efficient in finding solutions for medium to large problem instances.

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.

A Cooperative Network Interdiction Model and Its Optimization Solution Algorithm

Service disruptions due to deliberate sabotage are serious threats to supply systems. To alleviate the loss of accessibility caused by such disruptions, identifying the system vulnerabilities that would be worth strengthening is a critical problem in the field of Critical Infrastructure Protection (CIP). Today’s supply systems tend to be attacked in a cooperative manner. A set of facilities of an infrastructure system would be struck by several attackers cooperatively to create the worst-case disruptions. In this case, the conventional critical infrastructure protection approach based on R-Interdiction Median models with Fortification (RIMF) does not work well because it does not consider the cooperative manner among attackers. To handle the problem, a Bi-objective Probabilistic Interdiction Median Model with Cooperative Attacks (BICA) is proposed in this work. It is formulated as a bi-objective nonlinear optimization model and solved by a modified multi-objective evolutionary algorithm. Experiments verify the high interdiction losses of disruptions caused by cooperative attacks without any protection. Compared with the conventional model, the experimental results show that the BICA model can model the cooperative attacking problem effectively and that its evolutionary-based algorithm can solve the model efficiently.

Analyzing the robustness of an array of wireless access points to mobile jammers

We present an approach for measuring the vulnerability of a wireless network. Our metric, n-Robustness, measures the change in a network’s total signal strength resulting from the optimal placement of n jammers by an attacker. Toward this end, we develop a multi-period mixed-integer programming interdiction model that determines the movement of n jammers over a time horizon so as to minimize the total signal strength of users during a sustained jamming attack. We compared several solution approaches for solving our model including a Lagrangian relaxation heuristic, a genetic algorithm, and a stage decomposition heuristic. We tested our approach on a wireless trace dataset developed as part of the Wireless Topology Discovery project at the University of California San Diego. We found that the Lagrangian approach, which performed best overall, finds a close-to-optimal solution while requiring much less time than solving the MIP directly. We then illustrate the behavior of our model on a small example taken from the dataset as well as a set of experiments. Through our experiments we conclude that the total signal power follows a sigmoid curve as we increase the number of jammers and access points. We also found that increasing access points only improves network robustness initially; after that the benefit levels off. In addition, we found that the problem instances we considered have an n-Robustness of between 39 and 69%, indicating that the value of the model parameters (e.g., number of jammers, number of time periods) has an effect on robustness.

Trilevel shortest path network interdiction with partial fortification

This paper presents a new formulation and solution method for partial fortification and interdiction of a trilevel shortest path problem that is a generalization of existing network interdiction models to a more practical setting. We consider a trilevel problem in which a defense plan is determined at the top level to reinforce some network components against a potential attack. The middle-level problem is solved to predict the most damaging potential attack plan to the hardened network, while the bottom-level problem is used to determine the shortest path in the surviving network. In the proposed model, partial interdiction and fortification are permitted to make the model more general and practical. In addition, the costs of interdiction and fortification are minimized in the attacker’s and defender’s objective functions, and the corresponding budget constraints are relaxed.We have devised a trilevel decomposition algorithm to solve the proposed model in a reasonable time. Furthermore, we transform the trilevel partial fortification problem into a bilevel problem using duality and propose a bilevel decomposition algorithm to solve it. We also study three different formulations for a trilevel shortest path interdiction problem with total fortification and solve these problems using a combination of duality, decomposition and relaxation methods. The numerical results show that the optimal objective values obtained by the proposed models are better than those of the total interdiction models. The solution times of the proposed algorithm applied on a set of test problems are significantly less than those of the existing algorithms. This implies that the budget relaxations in the proposed model improve the total traversing cost in the presence of partial attack and fortification.

Promoting Resiliency in Emergency Communication Networks

Police, fire, and emergency personnel rely on wireless networks to serve the public. Whether it is during a natural disaster, or just an ordinary calendar day, wireless nodes of varying types form the infrastructure that local, regional, and even national scale agencies use to communicate while keeping the population served safe and secure. In this article, Michael R. Bartolacci and Stanko Dimitrov present a network interdiction modeling approach that can be utilized for analyzing vulnerabilities in public service wireless networks; subject to hacking, terrorism, or destruction from natural disasters. They develop a case study for wireless networks utilized by the sheriff's department of Miami-Dade County in Florida in the United States. Finally, the authors' modeling approach—given theoretical budgets for the “hardening” of wireless network nodes and for would-be destroyers of such nodes—highlights parts of the network where further investment may prevent damage and loss of capacity.

Reliable Network Interdiction Models with Multiple Unit Costs

This paper proposes a reliable network interdiction model with multiple unit costs, which maximizes the minimum arrival cost of the invader to the sink by setting obstacles on some arcs with limited resources in the given network. In other words, given a graph with a source and a sink, several arcs will be selected with limited resources such that each path contains as many weights as possible. This model needs to be transferred into a bilevel program because its constraints can hardly be listed explicitly even for a graph with a moderate size, because the number of paths between any two given points increases exponentially according to the size of the graph. This bilevel model is equivalent to an integer model with a low degree number of constraints by converting the inner programming to a shortest path problem. We first prove that this problem is non-deterministic polynomial-time (NP)-hard. Secondly, we reduce the number of constraints to the first power from the exponential degree by using the dual technique. Lastly, the national railway network is used to show the feasibility of our method.

Analysis of budget for interdiction on multicommodity network flows

In this paper, we concentrate on computing several critical budgets for interdiction of the multicommodity network flows, and studying the interdiction effects of the changes on budget. More specifically, we first propose general interdiction models of the multicommodity flow problem, with consideration of both node and arc removals and decrease of their capacities. Then, to perform the vulnerability analysis of networks, we define the function F(R) as the minimum amount of unsatisfied demands in the resulted network after worst-case interdiction with budget R. Specifically, we study the properties of function F(R), and find the critical budget values, such as $$R_a$$Ra, the largest value under which all demands can still be satisfied in the resulted network even under the worst-case interdiction, and $$R_b$$Rb, the least value under which the worst-case interdiction can make none of the demands be satisfied. We prove that the critical budget $$R_b$$Rb for completely destroying the network is not related to arc or node capacities, and supply or demand amounts, but it is related to the network topology, the sets of source and destination nodes, and interdiction costs on each node and arc. We also observe that the critical budget $$R_a$$Ra is related to all of these parameters of the network. Additionally, we present formulations to estimate both $$R_a$$Ra and $$R_b$$Rb. For the effects of budget increasing, we present the conditions under which there would be extra capabilities to interdict more arcs or nodes with increased budget, and also under which the increased budget has no effects for the interdictor. To verify these results and conclusions, numerical experiments on 12 networks with different numbers of commodities are performed.

Methods for removing links in a network to minimize the spread of infections

Minimizing the spread of infections is a challenging problem, and it is the subject matter in many different fields such as epidemiology and cyber-security. In this paper, we investigate link removal as an intervention strategy and study the relative effectiveness of different link removal methods in minimizing the spread of infections in a network. With that in mind, we develop four connectivity-based network interdiction models and formulate these models as mixed integer linear programs. The first model minimizes the number of connections between infected and susceptible nodes; the second the number of susceptible nodes having one or more connections with infected nodes; the third the total number of paths between infected and susceptible nodes; and the fourth the total weight of the paths between infected and susceptible nodes. We also propose heuristic algorithms to solve the models. The network interdiction models act as link removal methods, i.e., each return a solution consisting of a set of links to remove in the network. We compare the effectiveness of these four methods with the effectiveness of an existing link removal method [25], a method based on link betweenness centrality [18], and random link removal method. Our results show that complete isolation of susceptible nodes from infected nodes is the most effective method in reducing the average number of new infections (reduce occurrence) under most scenarios, and the relative effectiveness of the complete isolation method increases with transmission probability. In contrast, removing the highest probability transmission paths is the most effective method in increasing the average time to infect half of the susceptible nodes (reduce speed) under most scenarios, and the relative effectiveness of this method decreases with transmission probability.

Conservation under uncertainty: optimal network protection strategies for worst‐case disturbance events

SummaryConservation goals are ideally set after a thorough understanding of potential threats; however, predicting future spatial patterns of threats, such as disturbance, remains challenging. Here, we develop a novel extension of network fortification–interdiction models (NFIM) that deals with uncertainty in future spatial patterns of disturbance by optimally selecting sites that will best mitigate a worst‐case scenario for a given magnitude of disturbance.This approach uses information on between‐patch movement probabilities and patch‐specific survival, which can be estimated from mark–recapture data, to optimize life expectancy. Optimization occurs in three interrelated stages: protection, followed by disturbance and then assessment.We applied the modelling approach to two mark–recapture data sets: roseate ternsSterna dougalliiin the north‐eastern United States and the Everglade snail kiteRostrhamus sociabilis plumbeusin Florida. We contrasted the results to a more conventional approach of protecting sites that maximize connectivity (by minimizing the distances among protected sites) and a bi‐objective model that maximizes connectivity and the number of individuals under protection.Protecting sites that best mitigate future worst‐case disturbance scenarios consistently resulted in higher predicted life expectancies than protecting patches that minimize dispersal distance. Predicted life expectancy was similar betweenNFIMand the bi‐objective model for the small roseate tern network, yet theNFIMpredicted higher life expectancy than any of the scenarios in the bi‐objective model in the snail kite network.Synthesis and applications. This application of interdiction models prescribed a combination of patches for protection that results in the least possible decrease in life expectancy. Our analyses of the snail kite and roseate tern networks suggest that managing to protect these prescribed patches by the network fortification ‐interdiction models (i.e. protecting against the worst‐case disturbance scenario) is more beneficial than managing patches that minimize dispersal distance or maximize the number of individuals under protection if the conservation goal is to ensure the long‐term persistence of a species.

A bi-level network interdiction model for solving the hazmat routing problem

In the current work, we considered the problem of hazardous material distribution where the distributer chooses the routes on the network, and a regulatory agency controls the behaviour of the distributer to traverse the specified routes. In these circumstances, the distributer sets to select some routes to minimise the total distributing costs. Mostly, this occurs due to selecting risky arcs in which more individuals are exposed to risk. To prevent this and increase the capability to deal with the risk of hazardous material transportation through roads, the regulatory agency obliges carriers to traverse through the most secure arcs, though imposing more distribution costs. The problem is modelled as a bi-level routing problem. The bi-level model is difficult to solve and may be ill-posed. Two meta-heuristic algorithms are proposed to solve the bi-level model, and some randomly generated problems are applied to show the applicability and efficiency of the proposed algorithms.

Vulnerability assessment and re-routing of freight trains under disruptions: A coal supply chain network application

In this paper, we present a two-stage mixed integer programming (MIP) interdiction model in which an interdictor chooses a limited amount of elements to attack first on a given network, and then an operator dispatches trains through the residual network. Our MIP model explicitly incorporates discrete unit flows of trains on the rail network with time-variant capacities. A real coal rail transportation network is used in order to generate scenarios to provide tactical and operational level vulnerability assessment analysis including rerouting decisions, travel and delay costs analysis, and the frequency of interdictions of facilities for the dynamic rail system.

Interdiction Analysis of Electric Grids Combining Cascading Outage and Medium-Term Impacts

This paper presents an improved interdiction model to identify maximal electric grid attacks. The contribution of the model is that it incorporates both short-term (seconds to minutes) and medium-term (minutes to days) impacts of the possible attack. The medium-term impacts are examined by an interdicted dc optimal power flow model. The short-term impacts are addressed by a cascading outage analysis model that uses a set of sequentially applied checkers to perform the simulation of the cascading outage events and assess the short-term impacts of a blackout subsequent to specified terrorist attacks. An integer programming heuristic is applied that can utilize standard optimization software (e.g. CPLEX) to solve master problems generated by the heuristic. The proposed model has been verified using the IEEE 300-Bus Test System and IEEE RTS 96 Test System. Discussions of the results and future research plans are also presented in this paper.

Securing a border under asymmetric information

AbstractWe study a stochastic interdiction model of Morton et al. IIE Transactions, 39 (2007):3–14 that locates radiation sensors at border crossings to detect and prevent the smuggling of nuclear material. In this model, an interdictor places sensors at customs checkpoints to minimize a potential smuggler's maximum probability of crossing a border undetected. We focus on a model variant in which the interdictor has different, and likely more accurate, perceptions of the system's parameters than the smuggler does. We introduce a model that is tighter and uses fewer constraints than that of Morton et al. We also develop a class of valid inequalities along with a corresponding separation procedure that can be used within a cutting‐plane approach to reduce computational effort. Computational results demonstrate the effectiveness of our approach.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 91–100, 2014

Using container inspection history to improve interdiction logistics for illicit nuclear materials

Strengthening the United States' ability to prevent adversaries from smuggling nuclear materials into the country is a vital and ongoing issue. The prospect of additional countries, such as Iran, obtaining the know-how and equipment to produce these special nuclear materials in the near future underscores the need for efficient and effective inspection policies at ports and border crossings. In addition, the reduction of defense and homeland security budgets in recent years has made it increasingly important to accomplish the interdiction mission with fewer funds. Addressing these complications, in this article, we present a novel two-port interdiction model. We propose using prior inspection data as a low-cost way of increasing overall interdiction performance. We provide insights into two primary questions: first, how should a decision maker at a domestic port use detection data from the foreign port to improve the overall detection capability? Second, what are potential limitations to the usefulness of prior inspection data—is it possible that using prior data actually harms decision making at the domestic port? We find that a boundary curve policy (BCP) that takes into account both foreign and domestic inspection data can provide a significant improvement in detection probability. This BCP also proves to be surprisingly robust, even if adversaries are able to infiltrate shipments during transit. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 433-448, 2013

Network interdiction via a Critical Disruption Path: Branch-and-Price algorithms

This paper presents an innovative approach to maximally disconnect a given network. More specifically, this work introduces the concept of a Critical Disruption Path, a path between a source and a destination vertex whose deletion minimizes the cardinality of the largest remaining connected component. Network interdiction models seek to optimally disrupt network operations. Existing interdiction models disrupt network operations by removing vertices or edges. We introduce the first problem and formulation that optimally fragments a network via interdicting a path. Areas of study in which this work can be applied include transportation and evacuation networks, surveillance and reconnaissance operations, anti-terrorism activities, drug interdiction, and counter human-trafficking operations. In this paper, we first address the complexity associated with the Critical Disruption Path problem, and then provide a Mixed-Integer Linear Programming formulation for finding its optimal solution. Further, we develop a tailored Branch-and-Price algorithm that efficiently solves the Critical Disruption Path problem. We demonstrate the superiority of the developed Branch-and-Price algorithm by comparing the results found via our algorithm with the results found via the monolith formulation. In more than half of the test instances that can be solved by both the monolith and our Branch-and-Price algorithm, we outperform the monolith by two orders of magnitude.

Exact interdiction models and algorithms for disconnecting networks via node deletions

This paper analyzes the problem of maximizing the disconnectivity of undirected graphs by deleting a subset of their nodes. We consider three metrics that measure the connectivity of a graph: the number of connected components (which we attempt to maximize), the largest component size (which we attempt to minimize), and the minimum cost required to reconnect the graph after the nodes are deleted (which we attempt to maximize). We formulate each problem as a mixed-integer program, and then study valid inequalities for the first two connectivity objectives by examining intermediate dynamic programming solutions to k-hole subgraphs. We randomly generate a set of test instances, on which we demonstrate the computational efficacy of our approaches.

Assessing the impact of vulnerability modeling in the protection of critical infrastructure

This paper examines the impact of arc metrics on the computational performance and spatial similarity in network interdiction modeling. Computational impact is measured in the number of iterations and total time required to reach an optimal solution. A combination of spatial analytical tools is offered as a methodology to assess the similarity in defense resource allocation when applying different arc interdiction metrics. An experimental design was devised and implemented using two real-world sub-networks of the Los Angeles County roadway system. This paper shows that arc metric selection has a limited effect on the spatial allocation of defense resources though metric choice does directly impact computation time. These results have direct implications to public policy and decision-making by enabling a modeler to increase his/her situational awareness and also their confidence in resource allocation decisions by selecting metrics that will improve their solution capabilities.

Economic analysis of the N−k power grid contingency selection and evaluation by graph algorithms and interdiction methods

Contingency analysis is important for providing information about the vulnerability of power grids. Many methods have been purposed to use topological structures of power grids for analyzing contingency states. Considering failures of buses and lines, we present and compare several graph methods for selecting contingencies in this paper. A new method, called critical node detection, is introduced for selecting contingencies consisting of failures on buses. Besides these methods, we include an interdiction model which provides the worst case contingency selection. Our measurement for contingency evaluation is to maximize the social benefit, or to minimize the generating and load shedding cost. Comparing with other measurements for contingency selection, our model is based on economic analysis and is reasonable for evaluating the selected contingency state. Additionally, a contingency consisting of both buses and lines is also studied.