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Journal of Optimization
Volume 2013 (2013), Article ID 706176, 13 pages
Optimal Multiplicative Generalized Linear Search Plan for a Discrete Random Walker
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
Received 18 February 2013; Accepted 14 June 2013
Academic Editor: Bijaya Panigrahi
Copyright © 2013 Abd-Elmoneim Anwar Mohamed and Mohamed Abd Allah El-Hadidy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  for the early works and to Benkoski et al.  and Frost and Stone  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  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  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  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  derived the conditions for optimal search path which minimizes the cost of effort of finding a randomly moving target on the real line. Ohsumi  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.  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.  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. [11–17], Franck , Rousseeuw , 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  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  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 (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.
2. Problem Description
The problem under study can be formally described as follows. We have -searchers , that start the searching process from any point rather than the origin of the line , 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 and go to the left (right) as far as . Then, turn back to explore the right (left) part of as far as . Retrace the steps again to explore the left (right) part of as far as and so forth. In this paper, we need to determine , 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 are sequences of independent identically distributed random variables in , respectively. In addition, we let a value of “” indicate a step to the left and a value “” a step to the right, so that, for any , we have the -dimensional probability vectors,and , where + = .
Supposing that, for, , and is a random variable that represent the target initial position, valued in or and independent of ; if and such that , where 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 ) and the process which controls the target’s motion.
Figure 1 gives an illustration of the search plan that the-searchers, follow it. Moreover, it is to be noted that this search plan is a combination of continuous functions with speed and given by , , such that
The first meeting time is a random variable valued in , and it is defined by where , that is, a random variable independent of , and it represents the initial position of the target. We suppose that if the target moves on . Also assume that the set of all search plans of the searchers with speeds , respectively, satisfying condition (2) be . The problem is to find a search plan for the -searchers such that , where is the expected value of the first meeting time between one of the searchers and the target, and , is called a class of all sets of the search plans.
Assuming that , , are positive numbers such that , and is positive number also. In this problem, we assume that . In addition, we define the sequences , and by , and . We also use the notations , , where , which are positive functions and the following remark.
Remark 1. If , then .
There are known probability measures , such that on . And, they describe the location of the target, where is induced by the position of the target on .
The objective is to obtain the conditions that make the search plan be finite (i.e., ). We also need to show the existence of the optimal search plan and find it, that is, to give the minimum expected value of the first meeting time.
3. 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 searcherwould conduct its search as in the above manner that is detailed in Section 2. Consequently, for any line , , we have the following.
Case 1. If we consider that is a set of positive even numbers such that , for , then we have for any, if , then if , , then if , , then
Case 2. Also, if is a set of positive odd numbers such that , for , then we have
for any , if , , then
if , , then
if , , then
It is clear that the search path of depends on , and . Since 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.
Theorem 2. If is a rational number different from to −1 and, for all then(i)there exists a sequence of independent identically distributed random variables such that , and the distribution of is concentrated on the integers = and the probability vector , if and only if .(ii)The probability vector, , holds if and only if is an integer such that .(iii)Ifthen there exist constants and depending on such that, for any , where is the set of real numbers, if and if we have ≤ , and if we have, ≤ .
Proof. (i) Define , and .
Consequently, if , then from (1) we have that is an integer, and since is an integer also then we have . Therefore, . If then we have , if and only if . In addition, by using (1), is proved.
(ii) Since then we have= , and using (1) the prove is completed.
(iii) By using (ii), if then we have , and if we get , where means the greatest integer less than or equal to, . It is sufficient to show that
We have the following cases.
Case (a). If , then, from (ii), , if and only if . We take and then that leads to . Consequently, (12) holds.
Case (b). If , hence , if and only if . From (ii) putting then that leads to.
Then, (12) holds.
Case (c). If , then let and , where . In addition, put , . Since and then and then . Consequently, and are positive and well defined. Assuming that then ; therefore, from (ii), we have .
It remains to prove that if and then (2) holds.
Considering that then , , and . By using (1) we have Since then every term is strictly less than . Consequently, .
Theorem 3. If (the set of real numbers), then there exists , such that , for all .
Proof. For, where; if , then , and since , then .
By similar arguments if , then there exist such that for all .
4. 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.
Theorem 4. Let be the measure defined on by , and if is a search plan defined previously, the expectation is finite if , are finite when . And if , thenis finite if , are finite.
Proof. The hypotheses and are valued in or then is greater than until the first meeting; also if is smaller than then is smaller than until the first meeting. Since , are mutually exclusive events then or or or , and, for any we have
Using the notation , we obtain ; then leads to . Similarly, by using the notation , we get . Consequently, also,
From Remark 1, we obtain where ; then
If , then we getthus,
Leads to where
Then is finite if , are finite.
By similar way if , then is finite if where
And is finite if , are finite.
Lemma 5 (see El-Rayes et al. ). If , and , are a strictly increasing sequence of integer numbers with, then, for any , where , and are vectors in formulas and they are taken to be row vectors. These vectors are defined as follows:
Now, we will discuss under what conditions of the chosen search plan should be satisfied to make the previous integrals in Theorem 4 are indeed finite. This is a crucial issue related to the existence of a finite search plan.
Theorem 6. The chosen search plan should satisfy , , if and , if , where , and are vectors of linear functions given by and .
Proof. We will prove this theorem for when , where , and we obtain the following cases.
(I) If , then we have .
(II) If , then we have .
(III) If , then we get .
From (II) we have , and for , but . Consequently, from Theorem 3, we get