On Maximum Discounted Effort Reward Search Problem

Abstract

In this paper, we formulate a new search model for detecting two related targets that randomly located in a finite set of different cells or randomly moved through those cells. We assume that the search effort at each fixed number of time intervals is a random variable with a normal distribution. Rather than minimizing the expected effort of detecting two related targets, the proposed mathematical model allows us to include the search effort as a function with fuzzy parameter (discounted parameter). Another feature of this paper is considering a fuzzy extension of a stochastic optimization problem, which is interesting. We present an algorithm that gives the optimal distribution of an effort which makes the discounted effort reward of finding the targets is maximum. Two numerical examples are illustrated to show the effectiveness of this model by setting some parameters to represent some situations, such as detecting the enemy ships, fighters and the landmines in the war.the objective is to find a search strategy that minimizes the expected cost until the target will be found. In other search problems, Ahlswede and Wegener (1987) and Stone (1975) found a search strategy that maximizes the total probability of successful search. Assaf and Zamir (1987), Assaf and Sharlin (1994) and Sharlin (1987) investigated the discrete search problems when there are more than one stationary hidden targets. Song and Teneketizs (2004) determined the optimal search strategies with multiple sensors that maximize the total probability of successful search where the target is hidden in one of the finite set of different cells. On the other hand, Assaf and Sharlin (1994) investigated a discrete approach when the target moves randomly in a finite set of different cells.The optimal search problem is a decision making process that involves two dynamic partakers: the searcher who owns a sensor and the targets that are searched. In our case, the target movement does not depend on the searcher (i.e., we do not have an evading target). Many authors aimed to find the optimal search path of the searcher, represented by the cells, it visits over the known time periods that minimizes the no detection probability and the search effort. This search problem can be modeled in a continuous way as in Mathews (2008), Gan and Sukkarieh (2010) and Sarmiento et al. (2009) or in a discrete form as in Eagle and Yee (1990) and Yang et al. (2002), depending on the searching region representation, the sensor type and the searcher possible actions. Recently, Hong et al. (2009a, 2009b proposed an approximation algorithm for the optimal search path. This algorithm optimizes an approximate instead of exact path detection probability computed using the conditional probabilities that take an account of the search history of a fixed length window of known periods. Then, finding the maximum detection probability search path can be formulated as a shortest-path problem on an acyclic layered network whose number of layers is the order of search duration. More recently, ...

Cooperative search model for finding a brownian target on the real line

“…In (11) we find that the expected cost of the first meeting time between one of the searchers and the Brownian target is less than 22.9148 unit of time.…”

“…Also, they found the optimal search plan which minimizes this expected value in the case of the target position has a symmetric and asymmetric distributions. In an earlier work, the searching problem for a lost target has been studied extensively in many variations, mostly by El-Hadidy et al [5][6][7][8][9][10][11][12][13], Kagan and Ben-Gal [14], Guerrier and Holcman [15], Palyulin et al [16], Radmard and Croft [17], Stone et al [18].…”

Existence of cooperative search technique to find a Brownian target

“…Besides that, they found the optimal search plan that minimizes this expected value. For different kinds of search plans on the line, plan, and space, the reader can see, El-Hadidy et al [11][12][13][14][15][16][17][18][19][20], Kagan and Ben-Gal [21], Guerrier and Holcman [22], Palyulin et al [23], Radmard and Croft [24], Stone et al [25], and Jia et al [26].…”

Existence of a finite multiplicative search plan with random distances and velocities to find a d-dimensional Brownian target

“…The main objective of these earlier works is to obtain the conditions that make the first meeting time between one of the searchers and the moving target finite. On the other hand, when the target is located, some earlier works discussed many different search strategies with deterministic distances and velocities to find this target in minimum time on the line, plane and space , such as [17][18][19][20][21][22][23][24][25].…”