On the best location of signal detectors

An event may occur anywhere in a planar area or on a linear region such as a route. One or more detectors are to be located within this region with the objective of maximizing the smallest probability of the detection of an event anywhere in the region. In other words, the minimum protection in the region is to be maximized. The probability that an event is detected by a detector is a decreasing function of the distance. For example, the probability may decrease with some power (say, 2) of distance, or this decrease could be approximately exponential with distance. Two solution procedures are proposed for the problem on a line segment: a mathematical programming model and a specially designed algorithm. The problem in an area is solved by a univariate search, a Demjanov-type algorithm, a mathematical programming model, and simulated annealing. Computational experience is reported.