Optimization of strategic defenses to provide specified post-attack production capacities

This paper presents a model for choosing a minimum-cost mix of strategic defenses to assure that specified production capacities for several economic sectors survive after a nuclear attack. The defender selects a mix of strategic defenses for each of several geographic regions. The attacker chooses an allocation of attacking weapons to geographic regions, within specified weapon inventories. The attack is optimized against any economic sector. This formulation allows the defense planner the capability to assess the results of the optimal defense structure for a “worst case” attack. The model is a mathematical program with nonlinear programming problems in the constraints; an example of its application is given and is solved using recently developed optimization techniques.     

Allocating capacity in parallel queues to improve their resilience to deliberate attack

We develop models that lend insight into how to design systems that enjoy economies of scale in their operating costs, when those systems will subsequently face disruptions from accidents, acts of nature, or an intentional attack from a well‐informed attacker. The systems are modeled as parallel M/M/1 queues, and the key question is how to allocate service capacity among the queues to make the system resilient to worst‐case disruptions. We formulate this problem as a three‐level sequential game of perfect information between a defender and a hypothetical attacker. The optimal allocation of service capacity to queues depends on the type of attack one is facing. We distinguish between deterministic incremental attacks, where some, but not all, of the capacity of each attacked queue is knocked out, and zero‐one random‐outcome (ZORO) attacks, where the outcome is random and either all capacity at an attacked queue is knocked out or none is. There are differences in the way one should design systems in the face of incremental or ZORO attacks. For incremental attacks it is best to concentrate capacity. For ZORO attacks the optimal allocation is more complex, typically, but not always, involving spreading the service capacity out somewhat among the servers. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Timely exposure of a secret project: Which activities to monitor?

A defender wants to detect as quickly as possible whether some attacker is secretly conducting a project that could harm the defender. Security services, for example, need to expose a terrorist plot in time to prevent it. The attacker, in turn, schedules his activities so as to remain undiscovered as long as possible. One pressing question for the defender is: which of the project's activities to focus intelligence efforts on? We model the situation as a zero‐sum game, establish that a late‐start schedule defines a dominant attacker strategy, and describe a dynamic program that yields a Nash equilibrium for the zero‐sum game. Through an innovative use of cooperative game theory, we measure the harm reduction thanks to each activity's intelligence effort, obtain insight into what makes intelligence effort more effective, and show how to identify opportunities for further harm reduction. We use a detailed example of a nuclear weapons development project to demonstrate how a careful trade‐off between time and ease of detection can reduce the harm significantly.     

Dynamic server assignment policies for assembly‐type queues with flexible servers

We seek dynamic server assignment policies in finite‐capacity queueing systems with flexible and collaborative servers, which involve an assembly and/or a disassembly operation. The objective is to maximize the steady‐state throughput. We completely characterize the optimal policy for a Markovian system with two servers, two feeder stations, and instantaneous assembly and disassembly operations. This optimal policy allocates one server per station unless one of the stations is blocked, in which case both servers work at the unblocked station. For Markovian systems with three stations and instantaneous assembly and/or disassembly operations, we consider similar policies that move a server away from his/her “primary” station only when that station is blocked or starving. We determine the optimal assignment of each server whose primary station is blocked or starving in systems with three stations and zero buffers, by formulating the problem as a Markov decision process. Using this optimal assignment, we develop heuristic policies for systems with three or more stations and positive buffers, and show by means of a numerical study that these policies provide near‐optimal throughput. Furthermore, our numerical study shows that these policies developed for assembly‐type systems also work well in tandem systems. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal motion planning in rapid‐fire combat situations with attacker uncertainty

This article provides a modeling framework for quantifying cost and optimizing motion plans in combat situations with rapid weapon fire, multiple agents, and attacker uncertainty characterized by uncertain parameters. Recent developments in numerical optimal control enable the efficient computation of numerical solutions for optimization problems with multiple agents, nonlinear dynamics, and a broad class of objectives. This facilitates the application of more realistic, equipment‐based combat models, which track both more realistic models, which track both agent motion and dynamic equipment capabilities. We present such a framework, along with a described algorithm for finding numerical solutions, and a numerical example.     

Attack and defense of ICBMs deceptively based in a number of identical areas

On-site verification of ICBMs in the context of an arms control agreement might involve a situation where an inspector would choose one or more of a number of identical areas to inspect and would have confidence that the other areas had the same characteristics. This article considers optimal attack and defense of missiles deceptively based in a number of identical areas. The attacker may allocate warheads across areas as he desires and uniformly within areas. The defender may allocate interceptors across areas as he desires and either uniformly or preferentially within areas. The effect of restricting the defender to uniform allocation across areas is explored for various assumptions. Robustness of surviving missiles with respect to the number of attacking warheads is studied. Results are presented for a wide range of cases.     

Dual‐ascent and primal heuristics for production‐assembly‐distribution system design

This article proposes two dual‐ascent algorithms and uses each in combination with a primal drop heuristic embedded within a branch and bound framework to solve the uncapacitated production assembly distribution system (i.e., supply chain) design problem, which is formulated as a mixed integer program. Computational results indicate that one approach, which combines primal drop and dual‐ascent heuristics, can solve instances within reasonable time and prescribes solutions with gaps between the primal and dual solution values that are less than 0.15%, an efficacy suiting it for actual large‐scale applications. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Stochastic framework for partially degradation systems with continuous component degradation‐rate‐interactions

Many conventional models that characterize the reliability of multicomponent systems are developed on the premise that for a given system, the failures of its components are independent. Although this facilitates mathematical tractability, it may constitute a significant departure from what really takes place. In many real‐world applications, system components exhibit various degrees of interdependencies, which present significant challenges in predicting degradation performance and the remaining lifetimes of the individual components as well as the system at large. We focus on modeling the performance of interdependent components of networked systems that exhibit interactive degradation processes. Specifically, we focus on how the performance level of one component affects the degradation rates of other dependent components. This is achieved by using stochastic models to characterize how degradation‐based sensor signals associated with the components evolve over time. We consider “Continuous‐Type” component interactions that occur continuously over time. This type of degradation interaction exists in many applications, in which interdependencies occur on a continuum. We use a system of stochastic differential equations to capture such “Continuous‐Type” interaction. In addition, we utilize a Bayesian approach to update the proposed model using real‐time sensor signals observed in the field and provide more accurate estimation of component residual lifetimes. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 286–303, 2014     

A duel complicated by false targets and uncertainty as to opponent type

An attacker, being one of two types, initiates an attack at some time in the interval [-T, 0]. The a priori probabilities of each type are known. As time elapses the defender encounters false targets which occur according to a known Poisson process and which can be properly classified with known probability. The detection and classification probabilities for each type attacker are given. If the defender responds with a weapon at the time of attack, he survives with a probability which depends on the number of weapons in his possession and on attacker type. If he does not respond, his survival probability is smaller. These probabilities are known, as well as the current number of weapons in the defender's possession. They decrease as the number of weapons decreases. The payoff is the defender's survival probability. An iterative system of first-order differential equations is derived whose unique solution V₁(t),V₂(t),…,V_k(t) is shown to be the value of the game at time t, when the defender has 1, 2,…, k,… weapons, respectively. The optimal strategies are determined. Limiting results are obtained as t→-∞, while the ratio of the number of weapons to the expected number of false targets remaining is held constant.     

Exact algorithms and bounds for the dynamic assignment interdiction problem

We study a multi‐stage dynamic assignment interdiction (DAI) game in which two agents, a user and an attacker, compete in the underlying bipartite assignment graph. The user wishes to assign a set of tasks at the minimum cost, and the attacker seeks to interdict a subset of arcs to maximize the user's objective. The user assigns exactly one task per stage, and the assignment costs and interdiction impacts vary across stages. Before any stage commences in the game, the attacker can interdict arcs subject to a cardinality constraint. An interdicted arc can still be used by the user, but at an increased assignment cost. The goal is to find an optimal sequence of assignments, coupled with the attacker's optimal interdiction strategy. We prove that this problem is strongly NP‐hard, even when the attacker can interdict only one arc. We propose an exact exponential‐state dynamic‐programming algorithm for this problem as well as lower and upper bounds on the optimal objective function value. Our bounds are based on classical interdiction and robust optimization models, and on variations of the DAI game. We examine the efficiency of our algorithms and the quality of our bounds on a set of randomly generated instances. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 373–387, 2017     

Higher‐order Markov chain models for categorical data sequences*

In this paper we study higher‐order Markov chain models for analyzing categorical data sequences. We propose an efficient estimation method for the model parameters. Data sequences such as DNA and sales demand are used to illustrate the predicting power of our proposed models. In particular, we apply the developed higher‐order Markov chain model to the server logs data. The objective here is to model the users' behavior in accessing information and to predict their behavior in the future. Our tests are based on a realistic web log and our model shows an improvement in prediction. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Generation and storage dispatch in electricity networks with generator disruptions

We present methods for optimizing generation and storage decisions in an electricity network with multiple unreliable generators, each colocated with one energy storage unit (e.g., battery), and multiple loads under power flow constraints. Our model chooses the amount of energy produced by each generator and the amount of energy stored in each battery in every time period in order to minimize power generation and storage costs when each generator faces stochastic Markovian supply disruptions. This problem cannot be optimized easily using stochastic programming and/or dynamic programming approaches. Therefore, in this study, we present several heuristic methods to find an approximate optimal solution for this system. Each heuristic involves decomposing the network into several single‐generator, single‐battery, multiload systems and solving them optimally using dynamic programming, then obtaining a solution for the original problem by recombining. We discuss the computational performance of the proposed heuristics as well as insights gained from the models. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 493–511, 2015     

Two‐stage component test plans for testing the reliability of a series system

A system reliability is often evaluated by individual tests of components that constitute the system. These component test plans have advantages over complete system based tests in terms of time and cost. In this paper, we consider the series system with n components, where the lifetime of the i‐th component follows exponential distribution with parameter λ_i. Assuming test costs for the components are different, we develop an efficient algorithm to design a two‐stage component test plan that satisfies the usual probability requirements on the system reliability and in addition minimizes the maximum expected cost. For the case of prior information in the form of upper bounds on λ_i's, we use the genetic algorithm to solve the associated optimization problems which are otherwise difficult to solve using mathematical programming techniques. The two‐stage component test plans are cost effective compared to single‐stage plans developed by Rajgopal and Mazumdar. We demonstrate through several numerical examples that our approach has the potential to reduce the overall testing costs significantly. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 95–116, 2002; DOI 10.1002/nav.1051     

A componentwise index of service measurement in multi‐component systems

In this paper we present a componentwise delay measure for estimating and improving the expected delays experienced by customers in a multi‐component inventory/assembly system. We show that this measure is easily computed. Further, in an environment where the performance of each of the item delays could be improved with investment, we present a solution that aims to minimize this measure and, in effect, minimizes the average waiting time experienced by customers. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 50: 2003     

军事攻防中的多属性资源分配对策模型

针对资源受限情形下的两阶段攻防资源分配问题,提出一种基于多属性决策的资源分配对策模型。防守者首先将有限的防护资源分配到不同的目标上,继而进攻者选择一种威胁组合方式对目标实施打击。基于博弈论相关知识,模型的求解结果可以使防守者最小化自身损失,使进攻者最大化进攻收益。同时,针对模型的特点,给出了一些推论和证明。通过一个示例验证了模型的合理性以及相关推论的准确性,能够为攻、防双方规划决策提供辅助支持。     

DEA models for two‐stage processes: Game approach and efficiency decomposition

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). This tool has been utilized by a number of authors to examine two‐stage processes, where all the outputs from the first stage are the only inputs to the second stage. The current article examines and extends these models using game theory concepts. The resulting models are linear, and imply an efficiency decomposition where the overall efficiency of the two‐stage process is a product of the efficiencies of the two individual stages. When there is only one intermediate measure connecting the two stages, both the noncooperative and centralized models yield the same results as applying the standard DEA model to the two stages separately. As a result, the efficiency decomposition is unique. While the noncooperative approach yields a unique efficiency decomposition under multiple intermediate measures, the centralized approach is likely to yield multiple decompositions. Models are developed to test whether the efficiency decomposition arising from the centralized approach is unique. The relations among the noncooperative, centralized, and standard DEA approaches are investigated. Two real world data sets and a randomly generated data set are used to demonstrate the models and verify our findings. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

An attacker‐defender model for analyzing the vulnerability of initial attack in wildfire suppression

Wildfire managers use initial attack (IA) to control wildfires before they grow large and become difficult to suppress. Although the majority of wildfire incidents are contained by IA, the small percentage of fires that escape IA causes most of the damage. Therefore, planning a successful IA is very important. In this article, we study the vulnerability of IA in wildfire suppression using an attacker‐defender Stackelberg model. The attacker's objective is to coordinate the simultaneous ignition of fires at various points in a landscape to maximize the number of fires that cannot be contained by IA. The defender's objective is to optimally dispatch suppression resources from multiple fire stations located across the landscape to minimize the number of wildfires not contained by IA. We use a decomposition algorithm to solve the model and apply the model on a test case landscape. We also investigate the impact of delay in the response, the fire growth rate, the amount of suppression resources, and the locations of fire stations on the success of IA.     

A branch and bound algorithm for computing optimal replacement policies in consecutive k‐out‐of‐n‐systems

This paper presents a branch and bound algorithm for computing optimal replacement policies in a discrete‐time, infinite‐horizon, dynamic programming model of a binary coherent system with n statistically independent components, and then specializes the algorithm to consecutive k‐out‐of‐n systems. The objective is to minimize the long‐run expected average undiscounted cost per period. (Costs arise when the system fails and when failed components are replaced.) An earlier paper established the optimality of following a critical component policy (CCP), i.e., a policy specified by a critical component set and the rule: Replace a component if and only if it is failed and in the critical component set. Computing an optimal CCP is a optimization problem with n binary variables and a nonlinear objective function. Our branch and bound algorithm for solving this problem has memory storage requirement O(n) for consecutive k‐out‐of‐n systems. Extensive computational experiments on such systems involving over 350,000 test problems with n ranging from 10 to 150 find this algorithm to be effective when n ≤ 40 or k is near n. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 288–302, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10017     

Repairable consecutive‐k‐out‐of‐n: F system with Markov dependence

In this article, a model for a repairable consecutive‐k‐out‐of‐n: F system with Markov dependence is studied. A binary vector is used to represent the system state. The failure rate of a component in the system depends on the state of the preceding component. The failure risk of a system state is then introduced. On the basis of the failure risk, a priority repair rule is adopted. Then the transition density matrix can be determined, and the analysis of the system reliability can be conducted accordingly. One example each of a linear and a circular system is then studied in detail to explain the model and methodology developed in this paper. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 18–39, 2000