On an optimization problem in sensor selection

We address the following sensor selection problem. We assume that a dynamic system possesses a certain property, call it Property D, when a set Gamma of sensors is used. There is a cost c(A) associated with each set A of sensors that is a subset of Gamma. Given any set of sensors that is a subset of Gamma, it is possible to determine, via a test, whether the resulting system-sensor combination possesses Property D. Each test required to check whether or not Property D holds incurs a fixed cost. For each set of sensors A that is a subset of Gamma there is an a priori probability p(A) that the test will be positive, i.e., the system-sensor combination possesses Property D. The objective is to determine a test strategy, i.e., a sequence of tests, to minimize the expected cost, associated with the tests, that is incurred until a least expensive combination of sensors that results in a system-sensor combination possessing Property D is identified. We determine conditions on the sensor costs c(A) and the a priori probabilities p(A) under which the strategy that tests combinations of sensors in increasing order of cost is optimal with respect to the aforementioned objective.