An optimal investigation in two stage search with recognition errors

We consider an optimal search plan for a movable target. A searcher conducts a two-stage search consisting of a broad search and an investigating search. In the broad search phase, the searcher seeks for many uncertain signals featuring target. They are called contacts. There are three possible entities of the origin of a contact. They are a true target, false object or noise. When a contact is gained, the searcher executes the investigating search in the following step. He gives a judgment on the contact after it. At the moment, the searcher could make the errors of the first and the second kind. We model the search procedure taking account of these three possible entities and the two kinds of human errors and derive conditions of the optimal investigating time so as to maximize the detection probability of a target. By numerical examples, we show some optimal investigating plans and explain the properties of these optimal plans.