A search game with a protector

A classic problem in Search Theory is one in which a searcher allocates resources to the points of the integer interval [1, n] in an attempt to find an object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We model the situation as a two‐person non‐zero‐sum game so that we can take into account the fact that using resources can be costly. It is shown that this game has a unique Nash equilibrium when the searcher's probability of finding an object located at point i is of the form (1 − exp (−λ_ix_i)) exp (−μ_iy_i) when the searcher and protector allocate resources x_i and y_i respectively to point i. An algorithm to find this Nash equilibrium is given. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47:85–96, 2000     

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

For many combinatorial optimization problems that are NP-hard, a number of special cases exist that can be solved in polynomial time. This paper addresses the issue of solving one such problem, the well-known m-median problem with mutual communication (MMMC), by exploiting polynomially solvable special cases of the problem. For MMMC, a dependency graph is defined that characterizes the structure of the interactions between decision variables. A Lagrangian decomposition scheme is proposed that partitions the problem into two or more subproblems, each having the same structure as the original problem, but with simpler dependency graphs. The dual problems are solved using subgradient or multiplier adjustment methods. An efficient method of adjusting the multiplier values is given. Computational results are reported that show the method to be quite effective. In addition, applications of the approach to other difficult location problems is discussed. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 791–815, 1998     

Pareto-optimality in classical inventory problems

In this paper the inventory problem with backorders both deterministic and stochastic is studied using trade-off analysis in the context of vector optimization theory. The set of Pareto-optimal solutions is geometrically characterized in both the constrained and unconstrained cases. Moreover, a new way of utilizing Pareto-optimality concepts to handle classical inventory problems with backorders is derived. A new analysis of these models is done by means of a trade-off analysis. New solutions are shown, and an error bound for total inventory cost is provided. Other models such as multi-item or stochastic lead-time demand inventory problems are addressed and their Pareto-optimal solution sets are obtained. An example is included showing the additional applicability of this kind of analysis to handle parametric problems. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 83–98, 1998     

Damage accumulation and probability of kill for gun and target engagements

Determination of the gunfire probability of kill against a target requires two parameters to be taken into consideration: the likelihood of hitting the target (susceptibility) and the conditional probability of kill given a hit (vulnerability). Two commonly used methods for calculating the latter probability are (1) treating each hit upon the target independently, and (2) setting an exact number of hits to obtain a target kill. Each of these methods contains an implicit assumption about the probability distribution of the number of hits‐to‐kill. Method (1) assumes that the most likely kill scenario occurs with exactly one hit, whereas (2) implies that achieving a precise number of hits always results in a kill. These methods can produce significant differences in the predicted gun effectiveness, even if the mean number of hits‐to‐kill for each distribution is the same. We therefore introduce a new modeling approach with a more general distribution for the number of hits‐to‐kill. The approach is configurable to various classes of damage mechanism and is able to match both methods (1) and (2) with a suitable choice of parameter. We use this new approach to explore the influence of various damage accumulation models on the predicted effectiveness of weapon‐target engagements.     

Optimal search for a moving target with the option to wait

We investigate the problem in which an agent has to find an object that moves between two locations according to a discrete Markov process (Pollock, Operat Res 18 (1970) 883–903). At every period, the agent has three options: searching left, searching right, and waiting. We assume that waiting is costless whereas searching is costly. Moreover, when the agent searches the location that contains the object, he finds it with probability 1 (i.e. there is no overlooking). Waiting can be useful because it could induce a more favorable probability distribution over the two locations next period. We find an essentially unique (nearly) optimal strategy, and prove that it is characterized by two thresholds (as conjectured by Weber, J Appl Probab 23 (1986) 708–717). We show, moreover, that it can never be optimal to search the location with the lower probability of containing the object. The latter result is far from obvious and is in clear contrast with the example in Ross (1983) for the model without waiting. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Solving partially observable agent‐intruder games with an application to border security problems

In this paper, we introduce partially observable agent‐intruder games (POAIGs). These games model dynamic search games on graphs between security forces (an agent) and an intruder given possible (border) entry points and high value assets that require protection. The agent faces situations with dynamically changing, partially observable information about the state of the intruder and vice versa. The agent may place sensors at selected locations, while the intruder may recruit partners to observe the agent's movement. We formulate the problem as a two‐person zero‐sum game, and develop efficient algorithms to compute each player's optimal strategy. The solution to the game will help the agent choose sensor locations and design patrol routes that can handle imperfect information. First, we prove the existence of ?‐optimal strategies for POAIGs with an infinite time horizon. Second, we introduce a Bayesian approximation algorithm to identify these ?‐optimal strategies using belief functions that incorporate the imperfect information that becomes available during the game. For the solutions of large POAIGs with a finite time horizon, we use a solution method common to extensive form games, namely, the sequence form representation. To illustrate the POAIGs, we present several examples and numerical results.     

A continuous game of ambush

The article considers a two-person competitive problem in which a traveller wishes to choose a path across a rectangle from left to right in such a way as to avoid being ambushed by his adversary who has placed obstacles within the rectangle. Our results supplement those that have already been obtained by Ruckle and they indicate that, under certain conditions, the players need to adopt rather sophisticated strategies if they are to act optimally. This suggests that a complete solution to the problem could be difficult.