Modelling Rural Police Patrol

Naval Research Logistics

Atkinson

Glazebrook

2014

Consider a patrol problem, where a patroller traverses a graph through edges to detect potential attacks at nodes. An attack takes a random amount of time to complete. The patroller takes one time unit to move to and inspect an adjacent node, and will detect an ongoing attack with some probability. If an attack completes before it is detected, a cost is incurred. The attack time distribution, the cost due to a successful attack, and the detection probability all depend on the attack node. The patroller seeks a patrol policy that minimizes the expected cost incurred when, and if, an attack eventually happens. We consider two cases. A random attacker chooses where to attack according to predetermined probabilities, while a strategic attacker chooses where to attack to incur the maximal expected cost. In each case, computing the optimal solution, although possible, quickly becomes intractable for problems of practical sizes. Our main contribution is to develop efficient index policies-based on Lagrangian relaxation methodology, and also on approximate dynamic programming-which typically achieve within 1% of optimality with computation time orders of magnitude less than what is required to compute the optimal policy for problems of practical sizes.

“…It is possible to compute the optimal value in (25). To do so, it is helpful to first write the dual of the linear program defined by (6)- (7) in Section 2.2.…”

Section: Appendix B: the Strategic-attacker Casementioning

confidence: 99%

Optimal patrol to uncover threats in time when detection is imperfect

Naval Research Logistics

Atkinson

Glazebrook

2014

“…Earlier works focused on allocating police patrol resources among different areas to maximize the overall performance (Chaiken and Dormont 1978, Chelst 1978, Larson 1972, Olson and Wright 1975. Besides police patrol in urban areas, there are specialized patrol models for rural areas (Birge and Pollock 1989) and on highways (Lee et al 1979, Taylor et al 1985. These earlier works assumed that the frequencies of crimes at different locations remain constant and are known to the patrol force.…”

Section: Introductionmentioning

confidence: 99%

A Graph Patrol Problem with Random Attack Times

Atkinson

Chung

et al. 2013

Operations Research

This paper presents a patrol problem, where a patroller traverses a graph through edges to detect potential attacks at nodes. To design a patrol policy, the patroller needs to take into account not only the graph structure, but also the different attack time distributions, as well as different costs incurred due to successful attacks, at different nodes. We consider both random attackers and strategic attackers. A random attacker chooses which node to attack according to a probability distribution known to the patroller. A strategic attacker plays a two-person zero-sum game with the patroller. For each case, we give an exact linear program to compute the optimal solution. Because the linear programs quickly become computationally intractable as the problem size grows, we develop index-based heuristics. In the random-attacker case, our heuristic is optimal when there are two nodes, and in a suitably chosen asymptotic regime. In the strategic-attacker case, our heuristic is optimal when there are two nodes if the attack times are deterministic taking integer values. In our numerical experiments, our heuristic typically achieves within 1% of optimality with computation time orders of magnitude less than what is required to compute the optimal policy.

“…Earlier works focus on allocating police patrol resources among different areas to maximize the overall performance (Chaiken and Dormont, 1978;Chelst, 1978;Larson, 1972;Olson and Wright, 1975;Szechtman et al, 2008;Portugal and Rocha, 2013). Besides police patrol in urban areas, there are specialized patrol models for rural areas (Birge and Pollock, 1989) and on highways (Lee et al, 1979;Taylor et al, 1985). These earlier works assume that the frequencies of crimes at different locations remain constant and are known to the patrol force.…”

Section: Introductionmentioning

confidence: 99%

Technical Note—Optimal Patrol of a Perimeter

2022

Operations Research

A defender dispatches patrollers to circumambulate a perimeter to guard against potential attacks. The defender decides on the time points to dispatch patrollers and each patroller’s direction and speed, as long as the long-run rate at which patrollers are dispatched is capped at some constant. An attack at any point on the perimeter requires the same amount of time, during which it will be detected by each passing patroller independently with the same probability. The defender wants to maximize the probability of detecting an attack before it completes, while the attacker wants to minimize it. We study two scenarios, depending on whether the patrollers are undercover or wear a uniform. Conventional wisdom would suggest that the attacker gains advantage if he can see the patrollers going by so as to time his attack, but we show that the defender can achieve the same optimal detection probability by carefully spreading out the patrollers probabilistically against a learning attacker.