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     

New product introduction against a predator: A bilevel mixed‐integer programming approach

We consider a scenario with two firms determining which products to develop and introduce to the market. In this problem, there exists a finite set of potential products and market segments. Each market segment has a preference list of products and will buy its most preferred product among those available. The firms play a Stackelberg game in which the leader firm first introduces a set of products, and the follower responds with its own set of products. The leader's goal is to maximize its profit subject to a product introduction budget, assuming that the follower will attempt to minimize the leader's profit using a budget of its own. We formulate this problem as a multistage integer program amenable to decomposition techniques. Using this formulation, we develop three variations of an exact mathematical programming method for solving the multistage problem, along with a family of heuristic procedures for estimating the follower solution. The efficacy of our approaches is demonstrated on randomly generated test instances. This article contributes to the operations research literature a multistage algorithm that directly addresses difficulties posed by degeneracy, and contributes to the product variety literature an exact optimization algorithm for a novel competitive product introduction problem. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Algorithms for solving the conditional covering problem on paths

Consider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear‐time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial‐time, dynamic programming algorithm to find the optimal solution. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A three‐stage procurement optimization problem under uncertainty

This article examines a problem faced by a firm procuring a material input or good from a set of suppliers. The cost to procure the material from any given supplier is concave in the amount ordered from the supplier, up to a supplier‐specific capacity limit. This NP‐hard problem is further complicated by the observation that capacities are often uncertain in practice, due for instance to production shortages at the suppliers, or competition from other firms. We accommodate this uncertainty in a worst‐case (robust) fashion by modeling an adversarial entity (which we call the “follower”) with a limited procurement budget. The follower reduces supplier capacity to maximize the minimum cost required for our firm to procure its required goods. To guard against uncertainty, the firm can “protect” any supplier at a cost (e.g., by signing a contract with the supplier that guarantees supply availability, or investing in machine upgrades that guarantee the supplier's ability to produce goods at a desired level), ensuring that the anticipated capacity of that supplier will indeed be available. The problem we consider is thus a three‐stage game in which the firm first chooses which suppliers' capacities to protect, the follower acts next to reduce capacity from unprotected suppliers, and the firm then satisfies its demand using the remaining capacity. We formulate a three‐stage mixed‐integer program that is well‐suited to decomposition techniques and develop an effective cutting‐plane algorithm for its solution. The corresponding algorithmic approach solves a sequence of scaled and relaxed problem instances, which enables solving problems having much larger data values when compared to standard techniques. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

On efficiency of multistage channel with bargaining over wholesale prices

This article considers a multistage channel with deterministic price‐sensitive demand. Two systems for pricing decisions, that is, the bargaining system and the leader‐follower system, are compared. We characterize the necessary and sufficient conditions on the power structure, under which the solution of the bargaining system Pareto dominates that of the leader‐follower system. Also, under such conditions, we give a tight upper bound of channel efficiency of the bargaining system, which converges to 100% channel efficiency as the number of stages increases to infinity. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 449–459, 2016     

Optimal FCFS allocation rules for periodic‐review assemble‐to‐order systems

In Assemble‐To‐Order (ATO) systems, situations may arise in which customer demand must be backlogged due to a shortage of some components, leaving available stock of other components unused. Such unused component stock is called remnant stock. Remnant stock is a consequence of both component ordering decisions and decisions regarding allocation of components to end‐product demand. In this article, we examine periodic‐review ATO systems under linear holding and backlogging costs with a component installation stock policy and a First‐Come‐First‐Served (FCFS) allocation policy. We show that the FCFS allocation policy decouples the problem of optimal component allocation over time into deterministic period‐by‐period optimal component allocation problems. We denote the optimal allocation of components to end‐product demand as multimatching. We solve the multi‐matching problem by an iterative algorithm. In addition, an approximation scheme for the joint replenishment and allocation optimization problem with both upper and lower bounds is proposed. Numerical experiments for base‐stock component replenishment policies show that under optimal base‐stock policies and optimal allocation, remnant stock holding costs must be taken into account. Finally, joint optimization incorporating optimal FCFS component allocation is valuable because it provides a benchmark against which heuristic methods can be compared. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 158–169, 2015     

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017     

Robust shortest path planning and semicontractive dynamic programming

In this article, we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is reached with a minimum cost path under the worst possible instance of the uncertainty. Problems of this type arise, among others, in planning and pursuit‐evasion contexts, and in model predictive control. Our analysis makes use of the recently developed theory of abstract semicontractive dynamic programming models. We investigate questions of existence and uniqueness of solution of the optimality equation, existence of optimal paths, and the validity of various algorithms patterned after the classical methods of value and policy iteration, as well as a Dijkstra‐like algorithm for problems with nonnegative arc lengths.© 2016 Wiley Periodicals, Inc. Naval Research Logistics 66:15–37, 2019     

Franchising cooperation through chance cross‐constrained games

This paper explores the role of franchising arrangements in the context of game theory. We assume a single franchisor and a single franchisee channel and address the impact of fixed lump‐sum fees, royalties, wholesale price, and retail price on the franchise contracts. We start with the chance cross‐constrained noncooperative situation where the franchisor, as the leader, first specifies his/her strategies. The franchisee, as the follower, then decides on his/her decision. We then relax the assumption of franchisee's inability to influence the franchisor's decisions and discuss cooperative situation between the franchisor and the franchisee. Nash's bargaining model is utilized to select the best Pareto‐efficient payment scheme for the franchisor and the franchisee to achieve their cooperation. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 669–685, 2000     

Iterative methods for dynamic stochastic shortest path problems

We consider a routing policy that forms a dynamic shortest path in a network with independent, positive and discrete random arc costs. When visiting a node in the network, the costs for the arcs going out of this node are realized, and then the policy will determine which node to visit next with the objective of minimizing the expected cost from the current node to the destination node. This paper proposes an approach, which mimics the classical label-correcting approach, to compute the expected path cost. First, we develop a sequential implementation of this approach and establish some properties about the implementation. Next, we develop stochastic versions of some well-known label-correcting methods, including the first-in-first-out method, the two-queue method, the threshold algorithms, and the small-label-first principle. We perform numerical experiments to evaluate these methods and observe that fast methods for deterministic networks can become very slow for stochastic networks. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 769–789, 1998     

Interdiction models for delaying adversarial attacks against critical information technology infrastructure

Information technology (IT) infrastructure relies on a globalized supply chain that is vulnerable to numerous risks from adversarial attacks. It is important to protect IT infrastructure from these dynamic, persistent risks by delaying adversarial exploits. In this paper, we propose max‐min interdiction models for critical infrastructure protection that prioritizes cost‐effective security mitigations to maximally delay adversarial attacks. We consider attacks originating from multiple adversaries, each of which aims to find a “critical path” through the attack surface to complete the corresponding attack as soon as possible. Decision‐makers can deploy mitigations to delay attack exploits, however, mitigation effectiveness is sometimes uncertain. We propose a stochastic model variant to address this uncertainty by incorporating random delay times. The proposed models can be reformulated as a nested max‐max problem using dualization. We propose a Lagrangian heuristic approach that decomposes the max‐max problem into a number of smaller subproblems, and updates upper and lower bounds to the original problem via subgradient optimization. We evaluate the perfect information solution value as an alternative method for updating the upper bound. Computational results demonstrate that the Lagrangian heuristic identifies near‐optimal solutions efficiently, which outperforms a general purpose mixed‐integer programming solver on medium and large instances.     

A problem in network interdiction

Suppose we are given a network G=(V,E) with arc distances and a linear cost function for lengthening arcs. In this note, we consider a network-interdiction problem in which the shortest path from source node s to sink node t is to be increased to at least τ units via a least-cost investment strategy. This problem is shown to reduce to a simple minimum-cost-flow problem. Applications and generalizations are discussed, including the multiple-destination case.     

用非线性规划求解有限推力最优交会

利用非线性规划方法研究了航天器的有限推力最优交会问题。这种方法利用了近年来发展起来的直接优化技术,用分段多项式来表示整个轨道的状态和控制向量,将最优控制问题转化为非线性规划问题。在应用这种方法时,先将整个轨道分为若干推力段和无推力段,然后利用配置方法产生推力段的约束段,利用状态转移矩阵来产生无推力段的约束。最后,对共面轨道情况下的交会进行了数值仿真,验证了方法的有效性和鲁棒性。     

Pricing bridges to cross a river

We consider a pricing problem in directed, uncapacitated networks. Tariffs must be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs in the network are assumed to be given. There are n clients, the followers, and after the tariffs have been determined, the clients route their demands independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by applications in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX‐hard. Moreover, we analyze the effect of uniform pricing, proving that it yields both an m approximation and a (1 + lnD)‐approximation. Here, D is upper bounded by the total demand of all clients. In addition, we consider the problem under the additional restriction that the operator must not reject any of the clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless P = NP, and we prove the existence of an n‐approximation algorithm. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimal interdiction policy for a flow network

This paper analyzes the problem faced by a field commander who, confronted by an enemy on N battlefields, must determine an interdiction policy for the enemy's logistics system which minimizes the amount of war material flowing through this system per unit time. The resource utilized to achieve this interdiction is subject to constraint. It can be shown that this problem is equivalent to determining the set of arcs Z* to remove subject to constraint from a directed graph G such that the resulting maximal flow is minimized. A branch and bound algorithm for the solution to this problem is described, and a numerical example is provided.     

Optimal interdiction of a supply network

Under certain conditions, the re-supply capability of a combatant force may be limited by the characteristics of the transportation network over which supplies must flow. Interdiction by an opposing force may be used to reduce the capacity of that network. The effects of such efforts vary for differing missions and targets. With only a limited total budget available, the interdictor must decide which targets to hit, and with how much effort. An algorithm is presented for determining the optimum interdiction plan for minimizing network flow capacity when the minimum capacity on an arc is positive and the cost of interdiction is a linear function of arc capacity reduction.     

Outsourcing warranty repairs: Dynamic allocation

In this paper we consider the problem of minimizing the costs of outsourcing warranty repairs when failed items are dynamically routed to one of several service vendors. In our model, the manufacturer incurs a repair cost each time an item needs repair and also incurs a goodwill cost while an item is awaiting and undergoing repair. For a large manufacturer with annual warranty costs in the tens of millions of dollars, even a small relative cost reduction from the use of dynamic (rather than static) allocation may be practically significant. However, due to the size of the state space, the resulting dynamic programming problem is not exactly solvable in practice. Furthermore, standard routing heuristics, such as join‐the‐shortest‐queue, are simply not good enough to identify potential cost savings of any significance. We use two different approaches to develop effective, simply structured index policies for the dynamic allocation problem. The first uses dynamic programming policy improvement while the second deploys Whittle's proposal for restless bandits. The closed form indices concerned are new and the policies sufficiently close to optimal to provide cost savings over static allocation. All results of this paper are demonstrated using a simulation study. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Generalized orienteering problem with resource dependent rewards

We introduce a generalized orienteering problem (OP) where, as usual, a vehicle is routed from a prescribed start node, through a directed network, to a prescribed destination node, collecting rewards at each node visited, to maximize the total reward along the path. In our generalization, transit on arcs in the network and reward collection at nodes both consume a variable amount of the same limited resource. We exploit this resource trade‐off through a specialized branch‐and‐bound algorithm that relies on partial path relaxation problems that often yield tight bounds and lead to substantial pruning in the enumeration tree. We present the smuggler search problem (SSP) as an important real‐world application of our generalized OP. Numerical results show that our algorithm applied to the SSP outperforms standard mixed‐integer nonlinear programming solvers for moderate to large problem instances. We demonstrate model enhancements that allow practitioners to represent realistic search planning scenarios by accounting for multiple heterogeneous searchers and complex smuggler motion. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

On the uncapacitated K‐commodity network design problem with zero flow‐costs

Extending Sastry's result on the uncapacitated two‐commodity network design problem, we completely characterize the optimal solution of the uncapacitated K‐commodity network design problem with zero flow costs for the case when K = 3. By solving a set of shortest‐path problems on related graphs, we show that the optimal solutions can be found in O(n³) time when K = 3, where n is the number of nodes in the network. The algorithm depends on identifying a list of “basic patterns”; the number of basic patterns grows exponentially with K. We also show that the uncapacitated K‐commodity network design problem can be solved in O(n³) time for general K if K is fixed; otherwise, the time for solving the problem is exponential. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

A generic model of motor‐carrier fuel optimization

Fuel optimizers are decision models (software products) that are increasingly recognized as effective fuel management tools by U.S. truckload carriers. Using the latest price data of every truck stop, these models calculate the optimal fueling schedule for each route that indicates: (i) which truck stop(s) to use, and (ii) how much fuel to buy at the chosen truck stop(s) to minimize the refueling cost. In the current form, however, these models minimize only the fuel cost, and ignore or underestimate other costs that are affected by the models' decision variables. On the basis of the interviews with carrier managers, truck drivers, and fuel‐optimizer vendors, this article proposes a comprehensive model of motor‐carrier fuel optimization that considers all of the costs that are affected by the model's decision variables. Simulation results imply that the proposed model not only attains lower vehicle operating costs than the commercial fuel optimizers, but also gives solutions that are more desirable from the drivers' viewpoint. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008