Searching and Patrolling Dispersed Locations

Mellor, Edward and Glazebrook, Kevin and Shone, Robert (2025) Searching and Patrolling Dispersed Locations. PhD thesis, Lancaster University.

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Abstract

There are many real-world scenarios in which hidden objects or targets must be found. In searching for these targets, it may be necessary to traverse great distances, and practical search strategies should take into account the costs of moving between different locations. In this thesis, we present a discrete-time search model and a continuous-time patrol model that both explicitly take costs of movement into account. The search problem involves looking for a target which has been hidden in one of finitely many geographically distinct locations according to a known probability distribution. A searcher moves between these locations in order to find the target. For each location, a search takes some known amount of time to complete and independently finds the target with a known probability if the target is there. The searcher aims to minimise the expected total amount of time needed to find the target. The version of our problem without travel times can be solved to optimality using Gittins indices, which direct the searcher to always search the location that yields the maximal rate of target discovery. When travel times are included, the problem becomes much more challenging because the searcher becomes less willing to move to another location due to the potential future travel time back to the current location. In addition, when choosing the next destination, the searcher needs to take into account not only the distance to each destination, but also each destination’s distance to all other locations. We draw upon restless bandit theory to derive an index heuristic that takes travel times into account, and show that this heuristic has a tendency to leave a location prematurely. Subsequently, we use a range of methods to improve the index heuristic and demonstrate its strong performance via computational experiments. The patrol problem also involves searching for a target (now perhaps best thought of as an attacker) among finitely many geographically distinct locations. Rather than being present at the start of the search, these attackers arrive over time. It is therefore necessary for the patroller to patrol continuously, visiting all locations infinitely often, to catch these attackers. Once the patroller arrives at a location, they can spend any amount of time searching for targets with some detection rate at that location, before moving to a different location. The objective of the patroller is to minimize the expected time an attacker stays undetected at a location regardless of where the attack occurs. In the special case where all travel times are set to zero, we elucidate an optimal cyclic policy in which the patroller allocates a fixed fraction of their effort to each location continuously. As in the case of the search problem, this patrol problem becomes significantly more challenging when travel times are introduced. We define two cycle types based on common patrol practice for perimeter patrol and border patrol, respectively, and derive formulae for the expected time to detecting an attack in each case. We also provide an algorithm for finding the best parameters for each cycle type subject to some unimodality conditions. We give several examples where these cycle types perform well and numerically demonstrate that the optimal patrol policy depends highly on the structure and parameters of each patrol problem.