Optimal Multiplicative Generalized Linear Search Plan for a Discrete Random

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 R: 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 first meeting time between one of the searchers and the target finite. Furthermore, we show the existence of the optimal search plan that minimizes the expected value of the first meeting time and find it. The effectiveness of this model is illustrated using numerical example.


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 (1) the smallest cost and (2) 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  0 which has known probability distribution function.A searcher starts looking for the target at a point  0 (| 0 | < ∞).The searcher moves continuously along the line in both directions of the starting point  0 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 () ∈ Φ(), where Φ() 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 onedimensional 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 point on the real line, and the initial position of the target has any distribution.El-Rayes and Mohamed [6] have shown the existence of a search plan which minimizes the expected value of the first meeting time between the searcher and the randomly moving target with imposed conditions.Fristedt and Heath [7] derived the conditions for optimal search path which minimizes the cost of effort of finding a randomly moving target on the real line.Ohsumi [8] presented an optimal search plan for a target moving with Markov process along one of -nonintersecting arcs to a safe destination within a time limit, where the target starts at a safe base and tries to pass.El-Rayes et al. [9] illustrated this problem when the target moves on the real line with a Brownian motion, and the searcher starts the search from the origin.Recently, Mohamed et al. [10] discussed this problem for a Brownian target motion on one of -intersected real lines in which any information of the target position is not available to the searchers all the time.Mohamed et al. formulate a search model and find the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite.Furthermore, they showed the existence of the optimal search plan that minimizes the expected value of the first meeting time and found it.
On the other hand, when the target is located somewhere on the real line according to a known probability distribution, the searcher searches for it with known velocity and tries to find it in minimal expected time.It is assumed that the searcher can change the direction of its motion without any loss of time.The target can be detected only if the searcher reaches the target.In an earlier work, this problem has been studied extensively in many variations, mostly by Beck et al. [11][12][13][14][15][16][17], Franck [18], Rousseeuw [19], Reyniers [20,21], and Balkhi [22,23].
Furthermore, Mohamed et al. [24,25] have got more interesting results when they studied this problem to find a randomly located target in the plane.The target has symmetric or asymmetric distribution and with less information about this target available to the searchers.More recently, Mohamed and El-Hadidy [26] studied this problem when the target moves with parabolic spiral in the plane and starts its motion from a random point.Also, Mohamed and El-Hadidy [27] disscussed this problem in the plane when the target moves randomly with conditionally deterministic motion.Some problems of search may impose using more than one searcher such as when we search for a valuable target (e.g., person lost on one of -disjoint roads) or search for a serious target (e.g., a car filled with explosives which moves randomly in one of -disjoint roads).Thus, the main contributions of this paper center around studying the problem of searching for a one-dimensional random walker target that is moving on one of a system of -disjoint continuous real lines in R  (i.e., not intersected continuous real lines in the -space).The problem that is studied here is very interesting where we argue to give the conditions on a strategy (or trajectories) of -searchers, one on each line, that make the expected value of the first meeting time between one of the searchers and the target be finite and minimum.In this problem there exists a complication of such analysis.This complication is due to the fact that the searchers do not know the initial position of the target but only know its probability distribution.Otherwise, the problem would be reduced to determine the strategy of just one searcher on the same target's line.This problem is already tackled in [6,7].This work focuses on the necessary conditions for the existence of finite and optimal search plan that finds a random walker target.
The optimal search plan that is proposed here shows that the special structure of the search problem can be exploited  to obtain the efficient solution.For example, the search plan for a criminal drunk leaves its cache and walks up and down through one of -disjoint streets, totally disoriented.This paper is organized as follows.In Section 2 we formulate the problem.We display some properties that the search model should satisfy in Section 3. The search plan and the conditions that make the expected value of the first meeting time between one of the searchers and the target be finite is discussed in Section 4. The existence of optimal search plan that minimizes the expected value of the first meeting time is presented in Section 5.The optimal search plan is studied in Section 6.In Section 7 we illustrate the effectiveness of this model using numerical example.Finally, the paper concludes with a discussion of the results and directions for future research.

Problem Description
The problem under study can be formally described as follows.We have -searchers   ,  = 1, 2, . . ., , that start the searching process from any point rather than the origin of the line   ,  = 1, 2, . . ., , respectively, as in Figure 1.Each of the searchers moves continuously along its line in both directions of the starting point.The searcher   would conduct its search in the following manner.Start at  0 and go to the left (right) as far as  1 .Then, turn back to explore the right (left) part of  0 as far as  2 .Retrace the steps again to explore the left (right) part of  1 as far as  3 and so forth.In this paper, we need to determine   ,  = 1, 2, . . ., ,  = 1, 2, . .., that minimize the first meeting time between one of the searchers and the moving target.
Let  be a set of integer numbers and  + a nonnegative part of .We also assume that {  } ≥1 are sequences of independent identically distributed random variables in   ,  = 1, 2, . . ., , respectively.In addition, we let a value of "−1" indicate a step to the left and a value "1" a step to the right, so that, for any  ≥ 1, we have the dimensional probability vectors, [( Supposing that, for  > 0,  ∈  + , () = ∑  =1   , (0) = 0 and  0 is a random variable that represent the target initial position, valued in 2 or (2 + 1) and independent of (),  ≥ 0; if   > 0 and   such that 0 ≤  1 = (  +   )/2 ≤   , where  1 is an integer, then The target is assumed to move randomly on one of disjoint real lines according to the process {(),  ∈  + }, where  + is the set of positive integer numbers and () is a one-dimensional random walk motion.The initial position of the target is unknown but the searchers know its probability distribution (i.e., the probability distribution of the target is given at time 0) and the process {(),  ∈  + } which controls the target's motion.

Properties of the Search Model
In this section we start to discuss some properties that the search model for a one-dimensional random walk target should satisfy.
The searcher   would conduct its search as in the above manner that is detailed in Section 2. Consequently, for any line   ,  = 1, 2, . . ., ,  ∈  + , we have the following.
Case 1.If we consider that Q is a set of positive even numbers such that Q = {2, 4, 6, . . ., n}, for F ∈ Q,  ∈  + , then we have if Case 2. Also, if  is a set of positive odd numbers such that  = {1, 3, 5, . . ., m}, for F ∈ ,  ∈  + , then we have for any if It is clear that the search path of   depends on   ,   and  ∈  + .Since  0 ∈  then the first meeting time is done at   ∈  where   is an integer number.For this reason we will prove the conditions (properties) that make   meet the target at   as in the following theorems.
(iii) By using (ii), if   ≥ 0 then we have , where [  ] means the greatest integer less than or equal to   ,  = 1, 2, . . ., .It is sufficient to show that We have the following cases.

Existence of a Finite Search Plan
In this section, we find the conditions that make the search plan be finite.In addition, we will discuss that under what these conditions are indeed finite.This is a crucial issue related to the existence of a finite search plan.So, we will provide useful theorems that help us to do it.

Optimal Search Plan
In this section, we will find the optimal search plan that minimize the expected value of the first meeting time.

An Illustrative Example
In this section, we clarify the effectiveness of this model by considering the following example.
Example 1. Suppose that a criminal drunk (target) leaves its cache and walks up and down through one of -disjoint streets, totally disoriented.We model the streets as real lines with cache at the point  0 .In addition, assume that the criminal drunk takes unit steps, so we may record his position with an integer.Thus, for example, if he takes 5 steps to the left, he will be at a position  0 − 5.If we consider the previous conditions in Theorems 2 and 3 hold, then the optimal expected value of the first meeting time depends on the optimal values of the distances   that the searcher   should do it and the optimal values of   ,  = 2, 4, 6.Using the constraint  (2−1) ≤  ≤  (2) ,  = 1, 2, . . ., , 1 ≤  ≤ F/2, from Case 2, such that  = 2, 4, 6 and for  = 1, 2, in MONLP(III), and also using (38), we obtain the optimal values of   that give the optimal values of   ,   in Table 1.
It is clear that the optimal distances   that   should do it depend on  where   =   and   =   (   − 1).From the previous numerical calculations and by considering  0 = 10, we get Figure 2 that shows the projection of the target's motion (one-dimensional random walk) on the   path with  = 1, 2 where the current position on the line represented in the vertical axis versus the time steps at horizontal axis.

Discussion and Conclusions
A multiplicative generalized linear search plan for a onedimensional random walk target on one of -disjoint real lines has been presented, where the target initial position is given by a random variable  0 .Therefore, the target will be meet if    < ∞, where () = ( 1 (),  2 (), . . .,   ()) and    is the expected value of the first meeting time between one of the searchers and the target.We discuss some properties that the search model should satisfy in Theorems 2 and 3. We introduce the proof of conditions that make a search plan finite in Theorem 4, based on the continuity of the search plan and the conditions in Theorems 2 and 3 to show that    < ∞.We provide more analysis by using Lemma 5, Theorems 2 and 3 in Theorem 6 to show that the search plan () is finite if the conditions 2 ĥ() ≤ (||), 2 q() ≤ L(||), if  0 > 0 and 1 ŵ() ≤ (||), 1 ĥ() ≤ M(||) if  0 < 0, where (||), L(||), (||), and M(||) are vectors of linear functions.We use Theorem 7 to show that if there exist a finite search plan then the expected value of the target initial position (| 0 |) is finite.It will also be interesting to see a direct consequence of Theorems 4, 6, and 7 satisfying the existence of a finite search plan if and only if (| 0 |) is finite.The existence of an optimal search plan has been proved in Theorem 10.
In future research, it seems that the proposed model will be extendible to the multiple searchers case by considering the combinations of movement of multiple targets.

Figure 2 :
Figure 2: (a) and (b) give the optimal expected value of the first meeting time between   and the target that moves with discrete random walk in one dimension.