Optimal Multiplicative Generalized Linear Search Plan for a Discrete Random Walker

This paper formulates a search model that gives the optimal search plan for the problem of finding a discrete random walk target in minimum time. The target moves through one of n-disjoint real lines in : we have n-searchers starting the searching process for the target from any point rather than the origin. We ？find the conditions that make the expected value of the fi？rst meeting time between one of the searchers and the target fi？nite. Furthermore, we show the existence of the optimal search plan that minimizes the expected value of the fi？rst meeting time and fi？nd it. The effectiveness of this model is illustrated using numerical example. 1. Introduction The search problem for a randomly moving target has a remarkable importance in our life due to its great applicability. This problem is very interesting because it may arise in many real world situations such as searching for randomly moving persons or targets on roads. That mathematical analysis collected all the researches derived from searching applications of the II World War. They solved with beauty two complementary objectives of the search: find the target with the smallest cost and in minimum time. Readers are referred to Koopman [1] for the early works and to Benkoski et al. [2] and Frost and Stone [3] for a more recent survey. In the linear search problem, the target moves on the real line according to a known random process, and its initial position is given by the value of a random variable which has known probability distribution function. A searcher starts looking for the target at a point . The searcher moves continuously along the line in both directions of the starting point until the target is met. The searcher would change its direction at suitable points many times before meeting its goal. Thus, we consider that the path length which we represented is the cost of the search. The aim of the searcher is to minimize , that is, the expected value of the first meeting time between the searcher and the target. The problem is to find a search plan , such that ; in this case, we call that is a finite search plan and if for all , where terms to the class of all search plans then we call that is an optimal search plan. MC Cabe [4] found a finite search plan for a one-dimensional random walk target when the searcher starts the search from the origin, and the initial position of the target has a standard normal distribution. Mohamed [5] discussed the existence of a finite search plan for a one-dimensional random walk target in general case which means that the search may start from any