Solving Defender-Attacker-Defender Models for Infrastructure Defense

In Operations Research, Computing, and Homeland Defense, R.K. Wood and R.F. Dell, editors, INFORMS, Hanover, MD, pp. 28-49.The article of record as published may be located at http://dx.doi.org10.1287/ics.2011.0047This paper (a) describes a defender-attacker-defender sequential game model (DAD) to plan defenses for an infrastructure system that will enhance that system's resilience against attacks for an intelligent adversary, (b) describes a realistic formulation of DAD for defending a transportation network, (c) develops a decomposition algorithm for solving this instance of DAD and others, and (d) demonstrates the solution of a small transportation-network example. A DAD model generally evaluates system operation through the solution of an optimization model, and the decomposition algorithm developed here requires only that this system-operation model be continuous and convex. For example, our transportation-network example incorporates a congestion model with a (convex) nonlinear objective function and linear constraints

Optimization Approaches To Protect Transportation Infrastructure Against Strategic and Random Disruptions

Past and recent events have proved that critical infrastructure are vulnerable to natural catastrophes, unintentional accidents and terrorist attacks. Protecting these systems is critical to avoid loss of life and to guard against economical upheaval. A systematic approach to plan security investments is paramount to guarantee that limited protection resources are utilized in the most effcient manner. This thesis provides a detailed review of the optimization models that have been introduced in the past to identify vulnerabilities and protection plans for critical infrastructure. The main objective of this thesis is to study new and more realistic models to protect transportation infrastructure such as railway and road systems against man made and natural disruptions. Solution algorithms are devised to effciently solve the complex formulations proposed. Finally, several illustrative case studies are analysed to demonstrate how solving these models can be used to support effcient protection decisions

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

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%

Efficient Algorithms for Solving Facility Problems with Disruptions

This study investigates facility location problems in the presence of facility disruptions. Two types of problems are investigated. Firstly, we study a facility location problem considering random disruptions. Secondly, we study a facility fortification problem considering disruptions caused by random failures and intelligent attacks.We first study a reliable facility location problem in which facilities are faced with the risk of random disruptions. In the literature, reliable facility location models and solution methods have been proposed under different assumptions of the disruption distribution. In most of these models, the disruption distribution is assumed to be completely known, that is, the disruptions are known to be uncorrelated or to follow a certain distribution. In practice, we may have only limited information about the distribution. In this work, we propose a robust reliable facility location model that considers the worst-case distribution with incomplete information. Because the model imposes fewer distributional assumptions, it includes several important reliable facility location problems as special cases. We propose an effective cutting plane algorithm based on the supermodularity of the problem. For the case in which the distribution is completely known, we develop a heuristic algorithm called multi-start tabu search to solve very large instances.In the second part of the work, we study an r-interdiction median problem with fortification that simultaneously considers two types of disruption risks: random disruptions that happen probabilistically and disruptions caused by intentional attacks. The problem is to determine the allocation of limited facility fortification resources to an existing network. The problem is modeled as a bi-level programming model that generalizes the r-interdiction median problem with probabilistic fortification. The lower level problem, that is, the interdiction problem, is a challenging high-degree non-linear model. In the literature, only the enumeration method is applied to solve a special case of the problem. By exploring the special structure property of the problem, we propose an exact cutting plane method for the problem. For the fortification problem, an effective logic based Benders decomposition algorithm is proposed

Optimizing dynamic investment decisions for railway systems protection

Past and recent events have shown that railway infrastructure systems are particularly vulnerable to natural catastrophes, unintentional accidents and terrorist attacks. Protection investments are instrumental in reducing economic losses and preserving public safety. A systematic approach to plan security investments is paramount to guarantee that limited protection resources are utilized in the most efficient manner. In this article, we present an optimization model to identify the railway assets which should be protected to minimize the impact of worst case disruptions on passenger flows. We consider a dynamic investment problem where protection resources become available over a planning horizon. The problem is formulated as a bilevel mixed-integer model and solved using two different decomposition approaches. Random instances of different sizes are generated to compare the solution algorithms. The model is then tested on the Kent railway network to demonstrate how the results can be used to support efficient protection decisions

Models, Theoretical Properties, and Solution Approaches for Stochastic Programming with Endogenous Uncertainty

In a typical optimization problem, uncertainty does not depend on the decisions being made in the optimization routine. But, in many application areas, decisions affect underlying uncertainty (endogenous uncertainty), either altering the probability distributions or the timing at which the uncertainty is resolved. Stochastic programming is a widely used method in optimization under uncertainty. Though plenty of research exists on stochastic programming where decisions affect the timing at which uncertainty is resolved, much less work has been done on stochastic programming where decisions alter probability distributions of uncertain parameters. Therefore, we propose methodologies for the latter category of optimization under endogenous uncertainty and demonstrate their benefits in some application areas. First, we develop a data-driven stochastic program (integrates a supervised machine learning algorithm to estimate probability distributions of uncertain parameters) for a wildfire risk reduction problem, where resource allocation decisions probabilistically affect uncertain human behavior. The nonconvex model is linearized using a reformulation approach. To solve a realistic-sized problem, we introduce a simulation program to efficiently compute the recourse objective value for a large number of scenarios. We present managerial insights derived from the results obtained based on Santa Fe National Forest data. Second, we develop a data-driven stochastic program with both endogenous and exogenous uncertainties with an application to combined infrastructure protection and network design problem. In the proposed model, some first-stage decision variables affect probability distributions, whereas others do not. We propose an exact reformulation for linearizing the nonconvex model and provide a theoretical justification of it. We designed an accelerated L-shaped decomposition algorithm to solve the linearized model. Results obtained using transportation networks created based on the southeastern U.S. provide several key insights for practitioners in using this proposed methodology. Finally, we study submodular optimization under endogenous uncertainty with an application to complex system reliability. Specifically, we prove that our stochastic program\u27s reliability maximization objective function is submodular under some probability distributions commonly used in reliability literature. Utilizing the submodularity, we implement a continuous approximation algorithm capable of solving large-scale problems. We conduct a case study demonstrating the computational efficiency of the algorithm and providing insights

Passenger railway network protection: A model with variable post-disruption demand service

Protecting transportation infrastructures is critical to avoid loss of life and to guard against economic upheaval. This paper addresses the problem of identifying optimal protection plans for passenger rail transportation networks, given a limited budget. We propose a bi-level protection model which extends and refines the model previously introduced by Scaparra et al, (Railway infrastructure security, Springer, New York, 2015). In our extension, we still measure the impact of rail disruptions in terms of the amount of unserved passenger demand. However, our model captures the post-disruption user behaviour in a more accurate way by assuming that passenger demand for rail services after disruptions varies with the extent of the travel delays. To solve this complex bi-level model, we develop a simulated annealing algorithm. The efficiency of the heuristic is tested on a set of randomly generated instances and compared with the one of a more standard exact decomposition algorithm. To illustrate how the modelling approach might be used in practice to inform protection planning decisions, we present a case study based on the London Underground. The case study also highlights the importance of capturing flow demand adjustments in response to increased travel time in a mathematical model

Responsive Contingency Planning for Supply Chain Disruption Risk Mitigation

Contingent sourcing from a backup resource is an effective risk mitigation strategy under major disruptions. The production volumes and speeds of the backup resource are important protection design considerations, as they affect recovery. The objective of this dissertation is to show that cost-effective protection of existing supply networks from major disruptions result from planning appropriate volume and response speeds of a backup production facility prior to the disruptive event by considering operational aspects such as congestion that may occur at facilities. Contingency strategy are more responsive and disruption recovery periods can be shortened through such prior planning. The dissertation focuses on disruption risk arising from intelligent or pre-meditated attacks on supply facilities. An intelligent attacker has the capability to create worst case loss depending on the protection strategy of a given network. Since the attacker seeks the maximum loss and the designer tries to identify the protection scheme which minimizes this maximum loss, there exists an interdependence between attack and protection decisions. Ignoring this characteristic leads to suboptimal mitigation solutions under such disruptions. We therefore develop a mathematical model which utilizes a game theoretic framework of attack and defense involving nested optimization problems. The model is used to decide optimal selection of backup production volume and the response speeds, the facilities to build such capability within the available budget. The reallocation of demands from a disrupted facility to an undisrupted facility in a contingency strategy leads to congestion of the undisrupted facility, which may result in longer lead times and reduced throughput during disruption periods, thereby limiting the effectiveness of a contingency strategy. In the second part of the dissertation, we therefore analyze congestion effects in responsive contingency planning. The congestion cost function is modeled and integrated into the mathematical model of responsive contingency planning developed in the first part of the dissertation. The main contribution of this dissertation is that a decision tool has been developed to plan protection of an existing supply networks considering backup sourcing through gradual capacity acquisition. The solution methodology involving recursive search tree has been implemented which allows exploring protection solutions under a given budget of protection and multiple combinations of response speeds and production capacities of a backup facility. The results and analysis demonstrate the value of planning for responsive contingency in supply chains subject to risks of major disruptions and provide insights to aid managerial decision making