# Optimal Multiplicative Generalized Linear Search Plan for a Discrete Random Walker

Authors: Abd-Elmoneim Anwar Mohamed; Mohamed Abd Allah El-Hadidy
Journal: Journal of Optimization
 (2013)

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## Abstract

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 ofn-disjoint real lines in ℝn: 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.


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## Body

## 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(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 variableX0 which has known probability distribution function. A searcher starts looking for the target at a point H0(|H0|<∞). The searcher moves continuously along the line in both directions of the starting point H0 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 Eτϕ,that is, the expected value of the first meeting time τϕ between the searcher and the target. The problem is to find a search plan ϕ(t), such that Eτϕ<∞; in this case, we call that ϕ(t) is a finite search plan and if Eτϕ*<Eτϕfor all ϕ(t)∈Φ(t), where Φ(t) terms to the class of all search plans then we call that ϕ*(t) 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 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 K-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 n-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 [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 ofn-disjoint roads) or search for a serious target (e.g., a car filled with explosives which moves randomly in one of n-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 n-disjoint continuous real lines in ℝn (i.e., not intersected continuous real lines in the n-space). The problem that is studied here is very interesting where we argue to give the conditions on a strategy (or trajectories) of n-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 ofn-disjoint streets, totally disoriented.This paper is organized as follows. In Section2 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 haven-searchers Si,i=1,2,…,n, that start the searching process from any point rather than the origin of the line Li,i=1,2,…,n, respectively, as in Figure 1. Each of the searchers moves continuously along its line in both directions of the starting point. The searcher Si would conduct its search in the following manner. Start at δi0 and go to the left (right) as far as Hi1. Then, turn back to explore the right (left) part of δi0 as far as Hi2. Retrace the steps again to explore the left (right) part of Hi1 as far as Hi3 and so forth. In this paper, we need to determine Hij,i=1,2,…,n,j=1,2,…, that minimize the first meeting time between one of the searchers and the moving target.Figure 1
The search planϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t)) of the searchers Si,i=1,2,…,n.LetI be a set of integer numbers and I+ a nonnegative part of I. We also assume that {Xij}j≥1 are sequences of independent identically distributed random variables in Li,i=1,2,…,n, 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 j≥1, we have the n-dimensional probability vectors,[P(X1j=1)=p1,P(X2j=1)=p2,…,P(Xnj=1)=pn]and  [P(X1j=-1)=q1,P(X2j=-1)=q2,…,P(Xnj=-1)=qn], where [p1,p2,…,pn]  +  [q1,q2,…,qn]  =  [1,1,…,1].Supposing that, fort>0, t∈I+, W(t)=∑j=1t‍Xij,W(0)=0 and X0 is a random variable that represent the target initial position, valued in 2I or (2I+1) and independent of W(t),t≥0; if ξi>0 and Xij such that 0≤Ki1=(ξi+xi)/2≤ξi, where Ki1 is an integer, then
(1)P(W(ξi)=Ki1)={(ξiKi1)piKi1qiξi-Ki1,0,ifKi1doesnotexist.The target is assumed to move randomly on one ofn-disjoint real lines according to the process {W(t),t∈I+}, where I+ is the set of positive integer numbers and W(t) 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 {W(t),t∈I+} which controls the target’s motion.Figure1 gives an illustration of the search plan that then-searchersSi,i=1,2,…,n, follow it. Moreover, it is to be noted that this search plan ϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t)) is a combination of continuous functions ϕi(t) with speed vi and given by ϕi(t):I+→I,  i=1,2,…,n, such that
(2)|ϕi(t1)-ϕi(t2)|<vi|t1-t2|∀t1,t2∈I+,|ϕi(t1)-ϕi(t2)|lϕi(0)=0,i=1,2,…,n.The first meeting timeτϕ is a random variable valued in I+, and it is defined by
(3)τϕ={inf{t:eitheroneofϕi(t)=X0+W(t),i=1,2,…,n},∞,ifthesetisempty,
where X0∈I, that is, a random variable independent of W(t), and it represents the initial position of the target. We suppose that X0=Xi if the target moves on Li,i=1,2,…,n. Also assume that the set of all search plans of the searchers Si with speeds vi,i=1,2,…,n, respectively, satisfying condition (2) be Φvi(t),i=1,2,…,n. The problem is to find a search plan ϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t))∈Φ(t) for the n-searchers such that Eτϕ<∞, where Eτϕ is the expected value of the first meeting time between one of the searchers and the target, and Φ(t)={(ϕ1(t),ϕ2(t),…,ϕn(t));ϕi(t)∈Φvi(t),i=1,2,…,n}, is called a class of all sets of the search plans.Assuming thatϑi, λi,i=1,2,…,n, are positive numbers such that ϑi>1,Ci=(ϑi-1)/(ϑi+1)>|pi-qi|,λi=bϖi,b=1,2,…, and ϖi(1-Ci)/2 is positive number also. In this problem, we assume that vi=1,i=1,2,…,n. In addition, we define the sequences {Gij},{dij}, and {Hij},i=1,2,…,n,j=1,2,… by Gij=λi(ϑij-1),dij=(-1)j+1Ci[Gij+1+(-1)j+1], and Hij=dij+δi0. We also use the notations ψi(Gi(2j+1))=W(Gi(2j+1))-Ki1(Gi(2j+1)), ψ^i(Gi(2j))=W(Gi(2j))+Ki2(Gi(2j)), where Ki1(Gi(2j+1))=CiGi(2j+1), Ki2(Gi(2j))=CiGi(2j) which are positive functions and the following remark.Remark 1.
If0<ϰ§<1,§=1,2,…,κ, then ∏§=1κϰ§<∑§=1κ‍ϰ§.There are known probability measuresγi,i=1,2,…,n, such that γ1+γ2+⋯+γn=1 on L1∪L2∪⋯∪Ln. And, they describe the location of the target, where γi is induced by the position of the target on Li.The objective is to obtain the conditions that make the search planϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t)) be finite (i.e., Eτϕ<∞). We also need to show the existence of the optimal search plan ϕ*(t)=(ϕ1*(t),ϕ2*(t),…,ϕn*(t)) 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 searcherSiwould conduct its search as in the above manner that is detailed in Section 2. Consequently, for any line Li,i=1,2,…,n, t∈I+, we have the following.Case 1.
If we consider thatℚ is a set of positive even numbers such that ℚ={2,4,6,…,n^}, for ℱ∈ℚ,j∈I+, then we have
(4)⋯<Hi(j+2)<0<Hij<Hi(j-2)<⋯<Hi2<δi0<Hi1<Hi3<⋯;
for anyt∈I+, if Gi(2j-1)≤t≤Gi(2j),1≤j≤ℱ/2, then
(5)ϕi(t)=Hi(2j+1)-δi0-[t-Gi(2j-1)];
if Gij≤t≤Gi(j+1), j≥ℱ+1, then
(6)ϕi(t)=Hij+(-1)j[t-Gij];
if Gi(2j)≤t≤Gi(2j+1), 1≤j≤ℱ/2, then
(7)ϕi(t)=δi0-Hi(2j)+[t-Gi(2j)].Case 2.
Also, ifO is a set of positive odd numbers such that O={1,3,5,…,m^}, for ℱ∈O,j∈I+, then we have
(8)⋯<Hi2<δi0<Hi1<Hi3<⋯<Hi(j-2)<Hij<0<Hi(j+2)<⋯;
for any t∈I+, if Gi(2j-1)≤t≤Gi(2j), 1≤j≤(ℱ+1)/2, then
(9)ϕi(t)=δi0-Hi(2j-1)-[t-Gi(2j-1)];
if Gij≤t≤Gi(j+1), j≥ℱ+1, then
(10)ϕi(t)=Hij+(-1)j[t-Gij];
if Gi(2j)≤t≤Gi(2j+1), 1≤j≤(ℱ-1)/2, then
(11)ϕi(t)=Hi(2j)-δi0+[t-Gi(2j)].
It is clear that the search path ofSi depends on λi, ϑi and j∈I+. Since X0∈I then the first meeting time is done at xi∈I where xi is an integer number. For this reason we will prove the conditions (properties) that make Si meet the target at xi as in the following theorems.Theorem 2.
IfCi is a rational number different from 1 to −1 and, for all ξi≥1,V(ξi)=(W(ξiϑi)-Ciξiϑi)/2 then(i)
there exists a sequence{Yij}j≥1 of independent identically distributed random variables such that V(ξi)=∑j=1Γ‍Yij, and the distribution of Yij is concentrated on the integers E(Yij)  =  (ϑi/2)[E(Xij)-Ci] and the probability vector [p(Y1j=ℱ)>0,p(Y2j=ℱ)>0,…,p(Ynj=ℱ)>0], if and only if -ϑi(1+Ci)/2≤ℱ≤ϑi(1-Ci)/2.(ii)
The probability vector,[p(V(ξ1)=x1)>0,p(V(ξ2)=x2)>0,…,p(V(ξn)=xn)>0], holds if and only if xi is an integer such that -ξiϑi(1+Ci)/2≤xi≤ξiϑi(1-Ci)/2.(iii)
IfCi≠pi-qithen there exist constants ri1 and ri2 depending on Ci,pi such that, for any xi∈ℝ, where ℝ is the set of real numbers, if ξi>ri1xi+ri2 and if xi≥0 we have [p(0≤V(ξ1+1)≤x1),p(0≤V(ξ2+1)≤x2),…,p(0≤V(ξn+1)≤xn)]  ≤  [p(0≤V(ξ1)≤x1),p(0≤V(ξ2)≤x2),…,p(0≤V(ξn)≤xn)], and if xi<0 we have, [p(x1≤V(ξ1+1)<0),p(x2≤V(ξ2+1)<0),…,p(xn≤V(ξn+1)<0)]  ≤  [p(x1≤V(ξ1)<0),p(x2≤V(ξ2)<0),…,p(xn≤V(ξn)<0)].Proof.
(i) DefineYij=∑ℱ=1ϑi‍((Xi(ℱ+(j-1)ϑi)-Ci)/2),j≥1, and [p(Y1j=x1),p(Y2j=x2),…,p(Ynj=xn)]=[p(W(ϑ1))=2x1+ϑ1C1,p(W(ϑ2))=2x2+ϑ2C2,…,p(W(ϑn))=2xn+ϑnCn].
Consequently, if[p(Y1j=x1)>0,p(Y2j=x2)>0,…,p(Ynj=xn)>0], then from (1) we have xi+ϑi(1+Ci)/2 that is an integer, and since xi is an integer also then we have 0≤xi+ϑi(1+Ci)/2≤ϑi. Therefore, -ϑi(1+Ci)/2≤xi≤ϑi(1-Ci)/2. If xi=ℱ then we have [p(Y1j=ℱ)>0,p(Y2j=ℱ)>0,…,p(Ynj=ℱ)>0], if and only if -ϑi(1+Ci)/2≤ℱ≤ϑi(1-Ci)/2. In addition, by using (1), E(∑ℱ=1ϑi‍((Xi(ℱ+(j-1)ϑi)-Ci)/2))=(ϑi/2)[E(Xij)-Ci] is proved.
(ii) SinceV(ξi)=(W(ξiϑi)-Ciξiϑi)/2 then we have[p(V(ξ1)=x1),p(V(ξ2)=x2),…,p(V(ξn)=xn)]=[p((W(ξ1ϑ1)-C1ξ1ϑ1)/2=x1),p((W(ξ2ϑ2)-C2ξ2ϑ2)/2=x2),…,p((W(ξnϑn)-Cnξnϑn)/2=xn)]=  [p(W(ξ1ϑ1))=2x1+C1ξ1ϑ1),p(W(ξ2ϑ2))=2x2+C2ξ2ϑ2),…,p(W(ξnϑn))=2xn+Cnξnϑn)], and using (1) the prove is completed.
(iii) By using (ii), ifxi≥0 then we have [p(0≤V(ξ1)≤x1),p(0≤V(ξ2)≤x2),…,p(0≤V(ξn)≤xn)]=[∑j=0[x1]‍p(V(ξ1)=j),∑j=0[x2]‍p(V(ξ2)=j),…,∑j=0[xn]‍p(V(ξn)=j)], and if xi<0 we get [p(x1≤V(ξ1+1)<0),p(x2≤V(ξ2+1)<0),…,p(xn≤V(ξn+1)<0)]=[∑j=[x1]0‍p(V(ξ1)=j),∑j=[x2]0‍p(V(ξ2)=j),…,∑j=[xn]0‍p(V(ξn)=j)], where [xi] means the greatest integer less than or equal toxi, i=1,2,…,n. It is sufficient to show that
(12)[p(V(ξ1+1)=ℱ),p(V(ξ2+1)=ℱ),…,p(V(ξn+1)=ℱ)]≤[p(V(ξ1)=ℱ),p(V(ξ2)=ℱ),…,p(V(ξn)=ℱ)]ifξi>ri1xi+ri2.
We have the following cases.
Case (a). If Ci<-1, then, from (ii), [p(V(ξ1)=ℱ)>0,p(V(ξ2)=ℱ)>0,…,p(V(ξn)=ℱ)>0], if and only if 2ℱ/ϑi(1-Ci)≤ξi≤-2ℱ/ϑi(1+Ci). We take ri1=-2/ϑi(1+Ci),ri2=0  and then ξi>ri1|ℱ|⇒ξi>-2ℱ/ϑi(1+Ci) that leads to [p(V(ξ1)=ℱ)=0,p(V(ξ2)=ℱ)=0,…,p(V(ξn)=ℱ)=0]. Consequently, (12) holds.
Case (b). If Ci>1, hence [p(V(ξ1)=ℱ)>0,p(V(ξ2)=ℱ)>0,…,p(V(ξn)=ℱ)>0], if and only if -2ℱ/ϑi(1+Ci)≤ξi≤2ℱ/ϑi(1-Ci). From (ii) putting ri1=-2/ϑi(1-Ci),ri2=0 then ξi>ri1|ℱ|⇒ξi>2ℱ/ϑi(1-Ci) that leads to[p(V(ξ1)=ℱ)=0,p(V(ξ2)=ℱ)=0,…,p(V(ξn)=ℱ)=0].
Then, (12) holds.
Case (c). If  -1<Ci<1, then let αi=(1-Ci)/2,βi=1-αi and Δi=(αi/qi)αi(βi/pi)βi, where [p1,p2,…,pn]+[q1,q2,…,qn]=[1,1,…,1]. In addition, put ri2=1/(Δi-1),  ri1=Δimax[(1/αi,1/βi)ϑi(Δi-1)]. Since E(Xij)≠Ci and 0<αi<1 then αi≠qi and then Δi>1. Consequently, ri1 and ri2 are positive and well defined. Assuming that ξi>ri1|ℱ| then  -βiξiϑi<ℱ<αiξiϑi; therefore, from (ii), we have [p(V(ξ1)=ℱ)>0,p(V(ξ2)=ℱ)>0,…,p(V(ξn)=ℱ)>0].
It remains to prove that if[p(V(ξ1)=ℱ)>0,p(V(ξ2)=ℱ)>0,…,p(V(ξn)=ℱ)>0] and ξi>ri1|ℱ|+ri2 then (2) holds.
Considering thatZij=(Xij-Ci)/2 then [p(Z1j=α1)=p1,p(Z2j=α2)=p2,…,p(Znj=αn)=pn],  [p(Z1j=-β1)=q1,p(Z2j=-β2)=q2,…,p(Znj=-βn)=qn], and [p(V(ξ1)=ℱ),p(V(ξ2)=ℱ),…,p(V(ξn)=ℱ)]=[p(∑j=1ξ1ϑ1‍Z1j=ℱ),p(∑j=1ξ2ϑ2‍Z2j=ℱ),…,p(∑j=1ξnϑn‍Znj=ℱ)]. By using (1) we have
(13)[p(V(ξ1+1)=ℱ)p(V(ξ1)=ℱ),p(V(ξ2+1)=ℱ)p(V(ξ2)=ℱ),…,p(V(ξn+1)=ℱ)p(V(ξn)=ℱ)]=[∏j=1ϑ1β1ξ1ϑ1+jΔ1(ξ1ϑ1+j+(jα1+ℱ)/β1)×∏j=1ϑ1α1ξ1ϑ1+j+ϑ1β1Δ1(ξ1ϑ1+j+(iβ1-ℱ)/α1),∏j=1ϑ2β2ξ2ϑ2+jΔ2(ξ2ϑ2+j+(jα2+ℱ)/β2)×∏j=1ϑ2α2ξ2ϑ2+j+ϑ2β2Δ2(ξ2ϑ2+j+(iβ2-ℱ)/α2),…,∏j=1ϑnβnξnϑn+jΔn(ξnϑn+j+(jαn+ℱ)/βn)×∏j=1ϑnαnξnϑn+j+ϑnβnΔn(ξnϑn+j+(iβn-ℱ)/αn)].
Since ξi>ri1|ℱ|+ri2 then every term is strictly less than 1. Consequently, [p(V(ξ1+1)=ℱ),p(V(ξ2+1)=ℱ),…,p(V(ξn+1)=ℱ)]<[p(V(ξ1)=ℱ),p(V(ξ2)=ℱ),…,p(V(ξn)=ℱ)].Theorem 3.
IfE(Xij)<Ci,Ci∈ℝ (the set of real numbers), then there exists εi,0<εi<1, such that [p(W(ξ1)),p(W(ξ2)),…,p(W(ξn))]≤[ε1ξ1,ε2ξ2,…,εnξn], for all ξi,i=1,2,…,n.Proof.
Forζi>0,[p(W(ξ1)≥C1ξ1),p(W(ξ2)≥C2ξ2),…,p(W(ξn)≥Cnξn)]=[p(exp{ζ1(W(ξ1)-C1ξ1)}≥1),p(exp{ζ2(W(ξ2)-C2ξ2)}≥1),…,p(exp{ζn(W(ξn)-Cnξn)}≥1)]≤[E(exp{ζ1(W(ξ1)-C1ξ1)}),E(exp{ζ2(W(ξ2)-C2ξ2)}),…,E(exp{ζn(W(ξn)-Cnξn)})]=[{f1(ζ1)}ξ1,{f2(ζ2)}ξ2,…,{fn(ζn)}ξn], where[{f1(ζ1)},{f2(ζ2)},…,{fn(ζn)}]=[E(exp{ζ1(X1j-C1)}),E(exp{ζ2(X2j-C2)}),…,E(exp{ζn(Xnj-Cn)})]=[p1exp{ζ1(1-C1)}+q1exp{ζ1(-1-C1)},p2exp{ζ2(1-C2)}+q2exp{ζ2(-1-C2)},…,pnexp{ζn(1-Cn)}+qnexp{ζn(-1-Cn)}]; if E(Xij)<Ci,i=1,2,…,n, then [{f1′(0)},{f2′(0)},…,{fn′(0)}]<[0,0,…,0], and since [{f1(0)},{f2(0)},…,{fn(0)}]<[1,1,…,1], then [minζ1>0{f1(ζ1)},minζ2>0{f2(ζ2)},…,minζn>0{fn(ζn)}]=[ε1,ε2,…,εn]<[1,1,…,1].
By similar arguments ifE(Xij)>Ci,i=1,2,…,n, then there exist εi<1 such that [p(W(ξ1)),p(W(ξ2)),…,p(W(ξn))]≤[ε1ξ1,ε2ξ2,…,εnξn] for all ξi,i=1,2,…,n.


## 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.
Letγi be the measure defined on ℝ by Xij,i=1,2,…,n, and if ϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t)) is a search plan defined previously, the expectation Eτϕ is finite if
(14)∫-∞δi0[∑j=1ℱ/2‍ϑi2jp(ψi(Gi(2j))<-xi)+∑j=ℱ/2+1∞‍ϑi2jp(ψ~i(Gi(2j))≤-xi)]γi(dxi),∫δi0∞∑j=1∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi)γi(dxi), are finite when δi0>0. And if δi0<0, thenEτϕis finite if
(15)∫δi0∞[∑j=(ℱ+1)/2∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi)+∑j=1(ℱ-1)/2‍ϑi2j+1p(ψ~i(Gi(2j+1))≤-xi)]γi(dxi),∫-∞δi0∑j=1∞‍ϑi2jp(ψ~i(Gi(2j))<-xi)γi(dxi), are finite.Proof.
The hypothesesXi and δi0 are valued in 2I or (2I+1) then Xi+W(t) is greater than ϕi(t) until the first meeting; also if Xi is smaller than δi0 then Xi+W(t) is smaller than ϕi(t) until the first meeting. Since τϕi>t,i=1,2,…,n, are mutually exclusive events then p(τϕi>t)=p(τϕ1>t or τϕ2>t or ⋯ or τϕn>t)=∑i=1n‍p(τϕi>t), and, for any j≥0 we have
(16)p(τϕi>Gi(2j+1))≤∫-∞δi0p(Xi+W(Gi(2j))<Hi(2j)∣Xi=xi)γi(dxi)+∫δi0∞p(Xi+W(Gi(2j+1))>Hi(2j+1)∣Xi=xi)γi(dxi).
Using the notationψ~i(Gi(2j))=W(Gi(2j))+CiGi(2j), we obtain W(Gi(2j))-Hi(2j)<-Xi=xi; then W(Gi(2j))-(-1)2j+1Ci[Gi(2j)+1+(-1)2j+1]<-xi leads to W(Gi(2j))-(-1)Ci[Gi(2j)+1+(-1)]=W(Gi(2j))+Ci[Gi(2j)]=ψ~i(Gi(2j))<-xi. Similarly, by using the notation ψi(Gi(2j+1))=W(Gi(2j+1))-CiGi(2j+1), we get ψi(Gi(2j+1))>-xi. Consequently,
(17)p(τϕi>Gi(2j+1))≤∫-∞δi0p(ψ~i(Gi(2j))<-xi)γi(dxi)+∫δi0∞p(ψi(Gi(2j+1))>-xi)γi(dxi),
also,
(18)p(τϕi>Gi(2j))≤∫-∞δi0p(ψ~i(Gi(2j))<-xi)γi(dxi)+∫δi0∞p(ψi(Gi(2j-1))>-xi)γi(dxi).
From Remark1, we obtain
(19)Eτϕ=∫0∞p(τϕ>t)dtEτϕ≤∑j=0∞‍∫G1jG1(j+1)p(τϕ1>t)dt+⋯EτϕEτϕ+∑j=0∞‍∫GnjGn(j+1)p(τϕn>t)dt,
where Gi0=0; then
(20)Eτϕ≤∑j=0∞‍∫G1jG1(j+1)p(τϕ1>G1j)dt+⋯+∑j=0∞‍∫GnjGn(j+1)p(τϕn>t)dt=∑i=1n‍∑j=0∞‍(Gi(j+1)-Gij)p(τϕi>Gij)=∑i=1n‍∑j=0∞‍[λi(ϑij+1-1)-λi(ϑij-1)]p(τϕi>Gij)=∑i=1n‍∑j=0∞‍λiϑij(ϑi-1)p(τϕi>Gij)=∑i=1n‍[λi(ϑi-1)∑j=0∞‍ϑijp(τϕi>Gij)]=∑i=1n‍[ϑi2p(τϕi>Gi2)+ϑi3p(τϕi>Gi3)+⋯)λi(ϑi-1)×(p(τϕi>D)+ϑip(τϕi>Gi1)mmmn+ϑi2p(τϕi>Gi2)+ϑi3p(τϕi>Gi3)+⋯)].
Ifδi0<0, then we get(21)Eτϕ≤∑i=1n‍[∫-∞δi0p(ψ~i(Gi(ℱ+2))<-xi)γi(dxi)}λi(ϑi-1)×(∫-∞δi0p(ψ~i(Gi(ℱ+2))<-xi)γi(dxi)}p(τϕi>0)+ϑip(τϕi>Gi1)mmmmm+ϑi2{∫δi0∞p(ψi(Gi1)>-xi)γi(dxi)mmmmmmmmn+∫-∞δi0p(ψ~i(Gi2)<-xi)γi(dxi)}mmmmm+ϑi3{∫δi0∞p(ψi(Gi3)>-xi)γi(dxi)mmmmmmmmn+∫-∞δi0p(ψ~i(Gi2)<-xi)γi(dxi)}mmmmm+ϑi4{∫δi0∞p(ψi(Gi3)>-xi)γi(dxi)mmmmmmmmn+∫-∞δi0p(ψ~i(Gi4)<-xi)γi(dxi)}mmmmm+⋯mmmmm+ϑiℱ{∫δi0∞p(ψi(Gi(ℱ-1))>-xi)γi(dxi)mmmmmmmmn+∫-∞δi0p(ψ~i(Giℱ)<-xi)γi(dxi)}mmmmm+ϑiℱ+1{∫δi0∞p(ψi(Gi(ℱ+1))>-xi)γi(dxi)mmmmmmmmm+∫-∞δi0p(ψ~i(Giℱ)<-xi)γi(dxi)}mmmmm+ϑiℱ+2{∫δi0∞p(ψi(Gi(ℱ+1))>-xi)γi(dxi)mmmmmmmmm+∫-∞δi0p(ψ~i(Gi(ℱ+2))<-xi)γi(dxi)}mmmmm+ϑiℱ+3{∫δi0∞p(ψi(Gi(ℱ+3))>-xi)γi(dxi)mmmmmmmmm+∫-∞δi0p(ψ~i(Gi(ℱ+2))<-xi)γi(dxi)}mmmmm+⋯∫-∞δi0)];thus,(22)Eτϕ≤∑i=1n‍[∫δi0∞λi(ϑi-1)×(∫δi0∞p(τϕi>0)+ϑip(τϕi>Gi1)+ϑi2(ϑi+1)×∫-∞δi0p(ψ~i(Gi2)<-xi)γi(dxi)+⋯+ϑiℱ(ϑi+1)×∫-∞δi0p(ψ~i(Giℱ)<-xi)γi(dxi)+⋯+ϑi3(ϑi+1)∫δi0∞p(ψi(Gi3)>-xi)γi(dxi)+ϑi5(ϑi+1)×∫δi0∞p(ψi(Gi5)>-xi)γi(dxi)+⋯+ϑiℱ+3(ϑi+1)×∫δi0∞p(ψi(Gi(ℱ+3))>-xi)γi(dxi)+⋯)].
Leads to(23)Eτϕ≤∑i=1n‍[∫δi0∞λi(ϑi-1)×[∫δi0∞1g^i+(ϑi+1)×(∫-∞δi01w^i(xi)γi(dxi)+∫δi0∞1q^i(xi)γi(dxi)+∫δi0∞1h^i(xi)γi(dxi))]]=∑i=1n‍λi(ϑi-1)(1g^i)+∑i=1n‍[∫δi0∞λi(ϑi-1)(ϑi+1)×(∫δi0∞∫-∞δi01w^i(xi)γi(dxi)mmmmmn+∫δi0∞1q^i(xi)γi(dxi)mmmmmn+∫δi0∞1h^i(xi)γi(dxi))],
where
(24)1g^i=p(τϕi>0)+ϑip(τϕi>Gi1)+ϑi2∫δi0∞p(ψi(Gi1)>-xi)γi(dxi),1w^i(xi)=∑j=1∞‍ϑi2jp(ψ~i(Gi(2j))<-xi),1q^i(xi)=∑j=1(ℱ-1)/2‍ϑi2j+1p(ψ~i(Gi(2j+1))≤-xi),1h^i(xi)=∑j=(ℱ+1)/2∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi).
ThenEτϕ is finite if
(25)∫δi0∞[∑j=(ℱ+1)/2∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi)+∑j=1(ℱ-1)/2‍ϑi2j+1p(ψ~i(Gi(2j+1))≤-xi)]γi(dxi),∫-∞δi0∑j=1∞‍ϑi2jp(ψ~i(Gi(2j))<-xi)γi(dxi), are finite.
By similar way ifδi0>0, then Eτϕ is finite if
(26)Eτϕ≤∑i=1n‍[∫δi0∞λi(ϑi-1)(2g^i)+λi(ϑi-1)(ϑi+1)×(∫-∞δi02w^i(xi)γi(dxi)+∫-∞δi02q^i(xi)γi(dxi)+∫δi0∞2h^i(xi)γi(dxi))],
where
(27)2g^i=p(τϕ1>0)+ϑip(τϕi>Gi1)+ϑi2∫δi0∞p(ψi(Gi1)>-xi)γi(dxi),2w^i(xi)=∑j=1ℱ/2‍ϑi2jp(ψi(Gi(2j))<-xi),2q^i(xi)=∑j=ℱ/2+1∞‍ϑi2jp(ψ~i(Gi(2j))≤-xi),2h^i(xi)=∑j=1∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi).
AndEτϕ is finite if
(28)∫-∞δi0[∑j=1ℱ/2‍ϑi2jp(ψ~i(Gi(2j))<-xi)+∑j=ℱ/2+1∞‍ϑi2jp(ψ~i(Gi(2j))≤-xi)]γi(dxi),∫δi0∞∑j=1∞‍ϑi2j+1p(ψi(Gi(2j+1))>-xi), are finite.Lemma 5 (see El-Rayes et al. [9]).
Ifam~≥0,am~+1≤am~,m~=1,2,…, and {d-m~},m~=1,2,…, are a strictly increasing sequence of integer numbers withd-0=0, then, for any k~=1,2,…,
(29)∑m~=k~∞‍[d-m~+1-d-m~]ad-m~+1≤∑m~=d-k~∞‍am~≤∑m~=k~∞‍[d-m~+1-d-m~]ad-m~,
where ∑m~=k~∞‍[d-m~+1-d-m~]ad-m~+1,∑m~=d-k~∞‍am~, and ∑m~=k~∞‍[d-m~+1-d-m~]ad-m~ are vectors in formulas and they are taken to be row vectors. These vectors are defined as follows:
(30)∑m~=k~∞‍[d-m~+1-d-m~]ad-m~+1=[∑m~=k~∞‍[d-1(m~+1)-d-1m~]a1d-1(m~+1),∑m~=k~∞‍[d-2(m~+1)-d-2m~]a2d-2(m~+1),…,∑m~=k~∞‍[d-n(m~+1)-d-nm~]and-n(m~+1)],∑m~=d-k~∞‍am~=[∑m~=d-1k~∞‍a1m~,∑m~=d-2k~∞‍a2m~,…,∑m~=d-nk~∞‍anm~],∑m~=k~∞‍[d-m~+1-d-m~]ad-m~=[∑m~=k~∞‍[d-1(m~+1)-d-1m~]a1d-1m~,∑m~=k~∞‍[d-2(m~+1)-d-2m~]a2d-2m~,…,∑m~=k~∞‍[d-n(m~+1)-d-nm~]and-nm~].Now, we will discuss under what conditions of the chosen search plan should be satisfied to make the previous integrals in Theorem4 are indeed finite. This is a crucial issue related to the existence of a finite search plan.Theorem 6.
The chosen search plan should satisfy2h^(x)≤L(|x|), 2q^(x)≤L~(|x|), if δi0>0 and 1w^(x)≤M(|x|), 1h^(x)≤M~(|x|) if δi0<0, where L(|x|),L~(|x|),M(|x|), and M~(|x|) are vectors of linear functions given by 1w^(x)=[1w^1(x1),1w^2(x2),…,1w^n(xn)],1h^(x)=[h^11(x1),1h^2(x2),…,1h^n(xn)],2h^(x)=[h^12(x1),2h^2(x2),…,2h^n(xn)],2q^(x)=[q^12(x1),2q^2(x2),…,2q^n(xn)],L(|x|)=[L1(|x1|),L2(|x2|),…,Ln(|xn|)],L~(|x|)=[L~1(|x1|),L~2(|x2|),…,L~n(|xn|)],M(|x|)=[M1(|x1|),M2(|x2|),…,Mn(|xn|)] and M~(|x|)=[M~1(|x1|),M~2(|x2|),…,M~n(|xn|)].Proof.
We will prove this theorem for2h^(x) when δi0>0, where 2h^(x)=[h^12(x1),2h^2(x2),…,2h^n(xn)]=[∑j=1∞‍ϑ12j+1p(ψ1(G1(2j+1))>-x1),∑j=1∞‍ϑ22j+1p(ψ2(G2(2j+1))>-x2),…,∑j=1∞‍ϑn2j+1p(ψn(Gn(2j+1))>-xn)], and we obtain the following cases.
(I) Ifxi>δi0, then we have [h^12(x1),2h^2(x2),…,2h^n(xn)]=[h^12(δ10),2h^2(δ20),…,2h^n(δn0)]+[∑j=1∞‍ϑ12j+1p(-x1<ψ1(G1(2j+1))≤-δ10),∑j=1∞‍ϑ22j+1p(-x2<ψ2(G2(2j+1))≤-δ20),…,∑j=1∞‍ϑn2j+1p(-xn<ψn(Gn(2j+1))≤-δn0)].
(II) If0≤xi≤δi0, then we have [h^12(x1),2h^2(x2),…,2h^n(xn)]=[h^12(0),2h^2(0),…,2h^n(0)]+[∑j=1∞‍ϑ12j+1p(-x1<ψ1(G1(2j+1))≤0),∑j=1∞‍ϑ22j+1p(-x2<ψ2(G2(2j+1))≤0),…,∑j=1∞‍ϑn2j+1p(-xn<ψn(Gn(2j+1))≤0)].
(III) Ifxi≤δi0, then we get [h^12(x1),2h^2(x2),…,2h^n(xn)]=[h^12(0),2h^2(0),…,2h^n(0)]-[∑j=1∞‍ϑ12j+1p(0<ψ1(G1(2j+1))≤-x1),∑j=1∞‍ϑ22j+1p(0<ψ2(G2(2j+1))≤-x2),…,∑j=1∞‍ϑn2j+1p(0<ψn(Gn(2j+1))≤-xn)].
From (II) we have[h^12(x1),2h^2(x2),…,2h^n(xn)]≤[h^12(0),2h^2(0),…,2h^n(0)], and for xi≥δi0[h^12(x1),2h^2(x2),…,2h^n(xn)]=[h^12(δ10),2h^2(δ20),…,2h^n(δn0)]+[∑j=1∞‍ϑ12j+1p(-x1<ψ1(G1(2j+1))≤-δ10),∑j=1∞‍ϑ22j+1p(-x2<ψ2(G2(2j+1))≤-δ20),…,∑j=1∞‍ϑn2j+1p(-xn<ψn(Gn(2j+1))≤-δn0)], but [h^12(x1),2h^2(x2),…,2h^n(xn)]=[h^12(0),2h^2(0),…,2h^n(0)]+[∑j=1∞‍ϑ12j+1p(-x1<ψ1(G1(2j+1))≤0),∑j=1∞‍ϑ22j+1p(-x2<ψ2(G2(2j+1))≤0),…,∑j=1∞‍ϑn2j+1p(-xn<ψn(Gn(2j+1))≤0)]. Consequently, from Theorem 3, we get [h^12(0),2h^2(0),…,2h^n(0)]=[∑j=1∞‍ϑ12j+1p(ψ1(G1(2j+1))>0),∑j=1∞‍ϑ22j+1p(ψ2(G2(2j+1))>0),…,∑j=1∞‍ϑn2j+1p(ψn(Gn(2j+1))>0)]≤[∑j=1∞‍ϑ12j+1ε1G1(2j+1),∑j=1∞‍ϑ22j+1ε2G2(2j+1),…,∑j=1∞‍ϑn2j+1εnGn(2j+1)]≤[ϑ13/(ϑ12-1),ϑ23/(ϑ22-1),…,ϑn3/(ϑn2-1)],0<εi<1.
Suppose the following holds:(i)
d-iξi=Gi(2ξi+1)/σi=μi(ϑi2ξi+1-1),(ii)
μi(ξi)=ψi(ξiσi)/2=∑j=1ξi‍Xij, where {Xij},j=1,2,…, is a sequence of independent identically distributed random variables,(iii)
ai(ξi)=p(-xi/2<μi(ξi)≤0)=∑ℱ=0|xi|/2‍p(-(ℱ+1)<μi(ξi)≤-ℱ),(iv)
m~ is an integer such that d-im~=ri1|xi|+ri2,(v)
α~i=ϑi2/μi(ϑi-1),(vi)
Ui(ℱ,ℱ+1)=∑ξi=0∞‍p(-(ℱ+1)<μi(ξi)≤-ℱ).
Thus, from Theorem2, we have ai(ξi) is nonincreasing if ξi>d-iξi. By applying Lemma 5, we have [h^12(x1),2h^2(x2),…,2h^n(xn)]-[h^12(0),2h^2(0),…,2h^n(0)]=[∑j=1∞‍ϑ12j+1p(-x1<ψ1(G1(2j+1))≤0),∑j=1∞‍ϑ22j+1p(-x2<ψ2(G2(2j+1))≤0),…,∑j=1∞‍ϑn2j+1p(-xn<ψn(Gn(2j+1))≤0)]=[∑ξ1=1m~‍ϑ12ξ1+1a1(d-1ξ1),∑ξ2=1m~‍ϑ22ξ2+1a2(d-2ξ2),…,∑ξn=1m~‍ϑn2ξn+1an(d-nξn)]  +  [∑ξ1=m~+1∞‍ϑ12ξ1+1a1(d-1ξ1),∑ξ2=m~+1∞‍ϑ22ξ2+1a2(d-2ξ2),…,∑ξn=m~+1∞‍ϑn2ξn+1an(d-nξn)]  ≤  [∑ξ1=1m~‍ϑ12ξ1+1,∑ξ2=1m~‍ϑ22ξ2+1,…,∑ξn=1m~‍ϑn2ξn+1]  +  [α~1∑ξ1=m~+1∞‍(d-1ξ1-d-1(ξ1-1))a1(d-1ξ1),α~2∑ξ2=m~+1∞‍(d-2ξ2-d-2(ξ2-1))a2(d-2ξ2),…,α~n∑ξn=m~+1∞‍(d-nξn-d-n(ξn-1))an(d-nξn)]   ≤  [∑ξ1=1m~‍ϑ12ξ1+1,∑ξ2=1m~‍ϑ22ξ2+1,…,∑ξn=1m~‍ϑn2ξn+1]  +  [α~1∑ξ1=d-1m~∞‍a1(ξ1),α~2∑ξ2=d-2m~∞‍a2(ξ2),…,α~n∑ξn=d-nm~∞‍an(ξn)]   ≤  [∑ξ1=1m~‍ϑ12ξ1+1,∑ξ2=1m~‍ϑ22ξ2+1,…,∑ξn=1m~‍ϑn2ξn+1]  +  [α~1∑ℱ=0|x1|/2‍U1(ℱ,ℱ+1),α~2∑ℱ=0|x2|/2‍U2(ℱ,ℱ+1),…,α~n∑ℱ=0|xn|/2‍Un(ℱ,ℱ+1)].
SinceUi(ℱ,ℱ+1) satisfies the conditions of renewal theorem as in Feller [28], then Ui(ℱ,ℱ+1) is bounded for all ℱ,i=1,2,…,n, by a constant so 2h^(x)=[q1(x1),q2(x2),…,qn(xn)]≤[h^12(δ10),2h^2(δ20),…,2h^n(δn0)]+[Λ112,2Λ21,…,2Λn1]+[Λ122|x1|,2Λ22|x2|,…,2Λn2|xn|]=[L1(|x1|),L2(|x2|),…,Ln(|xn|)]=L(|x|), then, 2h^(x)≤L(|x|). Similar to the previous analysis way we can prove that q^2(x)≤L~(|x|) and also 1w^(x)≤M(|x|), 1h^(x)≤M~(|x|)  ifδi0<0. Therefore, the proof is completed.Theorem 7.
If there exists a finite search planϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t))∈Φ(t), then E(|X0|) is finite.Proof.
ForEτϕi<∞,i=1,2,…,n, we have p  (τϕ is finite) = 1 and so p (one of τϕi is finite, i=1,2,…,n) = 1. Therefore,
(31)p(⋃i=1n(τϕi<∞))=∑i=1n‍p(τϕi<∞)-∑i,β-p((τϕi<∞)∩(τϕβ-<∞))+⋯+(-1)np(⋂i=1n(τϕi<∞)).
Sinceτϕi<∞, i=1,2,…,n, are mutually exclusive events, then
(32)∑i=1n‍p(τϕi<∞)=1.
Consequently,p  (τϕi is finite) = 1 or p  (τϕθ is finite) = 1  for all i≠θ,i,θ=1,2,…,n. If p  (τϕi is finite) = 1 then X0=ϕi(τϕi)-W(τϕi) with probability one and |X0|≤|ϕi(τϕi)|+|W(τϕi)|≤τϕi+|W(τϕi)| that leads to E(|X0|)≤Eτϕi+E(|W(τϕi)|). ButW(τϕi)<τϕi, then E(|W(τϕi)|)<Eτϕi, and E(|X0|) is finite.
On the other hand, ifp  (τϕθ is finite) = 1 then X0=ϕθ(τϕθ)-W(τϕθ) with probability one, and by the same manner we can get E(|X0|) is finite.Remark 8.
A direct consequence of Theorems4, 6, and 7 satisfies the existence of a finite search plan if and only if E(|X0|) is finite.


## 5. Existence of an Optimal Search Plan

The goal of the searching strategy could minimize the expected value of the first meeting time between one of the searchers and the target. Therefore, the main problem here is to find a search pathsϕi(t),i=1,2,…,m. If such a search paths exists, we call it optimal search paths.Definition 9.
Let{ϕih(t)}h≥1∈Φ1(t),i=1,2,…,m, be sequences of search plans; we say that ϕih(t) converges to ϕi(t) as h tends to ∞ if and only if, for any t∈I+,ϕih(t) converges to ϕi(t) uniformly on every compact space (El-Rayes et al. [9]).Theorem 10.
Let for anyt∈I+ and W(t) be a one-dimensional random walk. Then, the mapping,
(33)(ϕ1(t),ϕ2(t),…,ϕn(t))⟶Eτϕ∈I+,
is lower semicontinuous on Φ1(t).Proof.
LetB(ϕi,t) be the indicator function of the set {τϕi>t,i=1,2,…,n}; by the Fatou-Lesbesque theorem, we get
(34)Eτϕ=E[∑t=0∞‍B(ϕi,t)]Eτϕ=E[∑t=0∞‍limh→∞infB(ϕih(t),t)]Eτϕ≤∑t=0∞‍limh→∞infE(1{τϕh>t}),
for any sequences ϕih→ϕi in Φ1(t), where Φ1(t) is sequentially compact; see El-Rayes et al. [9]. Thus the mapping (ϕ1(t),ϕ2(t),…,ϕn(t))→Eτϕ is lower semicontinuous mapping on Φ1(t) and this completed the proof.


## 6. Optimal Search Plan

In this section, we will find the optimal search plan that minimize the expected value of the first meeting time.Sinceτϕ depends on infimum either one of the functions ϕi(t),i=1,2,…,n, then we need to minimize Eτϕ. That happens when we obtain the optimal search plan ϕ*(t)={ϕ1*(t),ϕ2*(t),…,ϕn*(t)}, where ϕi(t) is a function on a distance Hij, i=1,2,…,n, 1≤j≤ℱ/2. Therefore, it can be seen that the vector (ϕ1(t),ϕ2(t),…,ϕn(t)) aren-distinct objective functions. Since the speed of the n-searchers (search team) is vi=1,i=1,2,…,n, then the objective functions ϕi(t),i=1,2,…,n, are distinct set of constrains given from Cases 1 and 2. In this problem, we consider the constraint Gi(2j-1)≤t≤Gi(2j),i=1,2,…,n,1≤j≤ℱ/2. Then, the problem takes the form
(35)min(ϕ1(t),ϕ2(t),…,ϕn(t))subjecttot∈T={ℱ2t∈I+∣Gi(2j-1)≤t≤Gi(2j),mmmmmmmmmmmni=1,2,…,n,1≤j≤ℱ2},
where t is an n-dimensional vector of the decision variables, ϕ1(t),ϕ2(t),…,ϕn(t) aren-distinct objective functions of the decision vector of inequality constraints, and T is the feasible set of constrained decisions.Butvi=1,i=1,2,…,n, that leads to t=Hij, i=1,2,…,n,1≤j≤ℱ/2; then the above problem is represented as the following multiobjective nonlinear programming problem (MONLP):
(36)MONLP:min(Ξ1(H1j),Ξ2(H2j),…,Ξn(Hnj)),subjecttoGi(2j-1)≤Hij≤Gi(2j),Hij∈I+,i=1,2,…,n,1≤j≤ℱ2,whereΞi(Hij)=Hi(2j+1)-δi0-[t-Gi(2j-1)].However, on formulating the MONLP which closely describes and represents the real decision situation, various factors of the real system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and the constraints involve many variables and parameters whose possible values may be assigned by the experts. In the conventional approach, such parameters are changed according to the variables in an experimental and/or subjective manner through the experts understanding the nature of the variables.SubstitutingGij=bϖi(ϑij-1) in the above MONLP we have
(37)MONLP(I):min(Ξ1(ϑ1;ϖ1),Ξ2(ϑ2;ϖ2),…,Ξn(ϑn;ϖn)),subjecttobϖi(ϑi2j-1-1)≤Hij,Hij≤bϖi(ϑi2j-1),Hij∈I+,i=1,2,…,n,1≤j≤ℱ2,whereΞi(Hij)=Hi(2j+1)-δi0whereΞi(Hij)=-[t-bϖi(ϑi2j-1-1)],
where the variable ϑi takes +ve numbers greater than 1 and the parameter ϖi is the least positive integer such that ϖi(1-(ϑi-1)/(ϑi+1))/2 is positive number; Hij≥0for all i=1,2,…,n,1≤j≤ℱ/2. Furthermore, Hij is a function of the variable ϑi and the parameter ϖi, where
(38)Hij=(-1)j+1Ci[Gij+1+(-1)j+1]+δi0=(-1)j+1(ϑi-1ϑi+1)[bϖi(θij-1)+1+(-1)j+1]+δi0.Then, the previous MONLP(I) takes the form(39)MONLP(II):min(Ξ1(ϑ1;ϖ1),Ξ2(ϑ2;ϖ2),…,Ξn(ϑn;ϖn)),subjecttobϖi(ϑi2j-1-1)≤(h1-1)(ϑi-1ϑi+1)×[bϖi(ϑij-1)+h1]+δi0,bϖi(ϑi2j-1)≥(h1-1)(ϑi-1ϑi+1)bϖi(ϑi2j-1)≥×[bϖi(ϑij-1)+h1]bϖi(ϑi2j-1)≥+δi0,whereΞi(ϑi;ϖi)=(h3)(ϑi-1ϑi+1)whereΞi(ϑi;ϖi)=×[bϖi(ϑi2j-1-1)+h4]whereΞi(ϑi;ϖi)=-2δi0+(h2)(ϑi-1ϑi+1)whereΞi(ϑi;ϖi)=×[bϖi(ϑij-1)+h5]whereΞi(ϑi;ϖi)=+bϖi(ϑi2j-1-1),h1=1+(-1)j+1,h2=(-1)j+2,h3=(-1)2j+1,h4=1+(-1)2j,h5=1+(-1)j,ϑi>1,ϖi>0∀i=1,2,…,n,b=1,2,…,1≤j≤ℱ2.Choosing a certain weights℧i,i=1,2,…,n, such that ∑i=1n‍℧i=1 and the certain parameters ϖi=1,i=1,2,…,n, then the previous MONLP(II) takes the form(40)MONLP(III):min℧i{(h3)(ϑi-1ϑi+1)MONLP(III):min℧im×[b(ϑi2j-1-1)+h4]MONLP(III):min℧im-2δi0+(h2)(ϑi-1ϑi+1)MONLP(III):min℧im×[b(ϑij-1)+h5]MONLP(III):min℧im+b(ϑi2j-1-1)(ϑi-1ϑi+1)},MONLP(III):subjecttob(ϑi2j-1-1)≤(h1-1)(ϑi-1ϑi+1)×[b(ϑij-1)+h1]+δi0,MONLP(III):subjecttob(ϑi2j-1)≥(h1-1)(ϑi-1ϑi+1)×[b(ϑij-1)+h1]+δi0,1-ϑi<0,∀b=1,2,…,1≤j≤ℱ2.From the Kuhn-Tucker conditions we have(41)12{2(ϑi+1)2[(h3-1)[b(ϑi2j-1-1)+h4]mmmmmmm+(h2-1)[b(ϑij-1)+h5]]+(ϑi-1ϑi+1)b[(h3-1)(2j-1)ϑi2j-2+(h2-1)jϑij-1]+b(2j-1)ϑi2j-22(ϑi+1)2}+u1{b(2j-1)ϑi2j-2-(h1-1)ϑi+1×[2ϑi+1[b(ϑij-1)+h1]+(ϑi-1)jbϑij-1]}+u2{-b(2j-1)ϑi2j-2+(h1-1)ϑi+1×[2ϑi+1[b(ϑij-1)+h1]+(ϑi-1)jbϑij-1]}-u3=0,(42)u1{b(ϑi2j-1-1)(ϑi+1)-(h1-1)(ϑi-1)×[b(ϑij-1-1)+h1]}=0,(43)u2{(h1-1)(ϑi-1)[b(ϑij-1)+h1]-b(ϑi2j-1-1)(ϑi+1)}=0,(44)u3{1-ϑi}=0.Solving (41)–(44), we found the optimal values of ϑi that give the optimal distances Hijfor all b=1,2,…,1≤j≤ℱ/2, and they are given by
(45)ϑi=Υ^+Λ^b(1-2j)ϑi2j-2-1,ϑi>1,
where
(46)Υ^=2[(h3-1)[b(ϑi2j-1-1)+h4]+(h2-1)[b(ϑij-1)+h5]],Λ^=b(h3-1)(2j-1)ϑi2j-b(h3-1)(2j-1)ϑi2j-2+b(h2-1)jϑij+1-b(h2-1)jϑij-1.


## 7. 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 ofn-disjoint streets, totally disoriented. We model the streets as real lines with cache at the point X0. 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 X0-5. 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 Hij that the searcher Si should do it and the optimal values of Gij,j=2,4,6. Using the constraint Gi(2j-1)≤t≤Gi(2j),i=1,2,…,n,1≤j≤ℱ/2,from Case 2, such that j=2,4,6 and for b=1,2, in MONLP(III), and also using (38), we obtain the optimal values of ϑi that give the optimal values of Hij,Gij in Table 1.Table 1
The optimal values ofϑi,Hij and Gij,j=2,4,6,b=1,2.
b
j
ϑ
i
G
i
j
H
i
j
1
2
1.8836
0.8836
0.27075
+
δ
i
0
4
2.5265
1.5265
0.6607
-
δ
i
0
6
2.6982
1.6982
0.7798
+
δ
i
0
2
2
2.5226
3.0452
1.3162
+
δ
i
0
4
2.7897
3.5794
2.2963
-
δ
i
0
6
2.8652
3.7304
1.8
+
δ
i
0It is clear that the optimal distancesHij that Si should do it depend on b where λi=bϖi and Gij=λi(ϑij-1).From the previous numerical calculations and by consideringδi0=10, we get Figure 2 that shows the projection of the target’s motion (one-dimensional random walk) on the Si path with b=1,2 where the current position on the line represented in the vertical axis versus the time steps at horizontal axis.(a) and (b) give the optimal expected value of the first meeting time betweenSi and the target that moves with discrete random walk in one dimension.
(a)
b
=
1
(b)
b
=
2


## 8. Discussion and Conclusions

A multiplicative generalized linear search plan for a one-dimensional random walk target on one ofn-disjoint real lines has been presented, where the target initial position is given by a random variable X0. Therefore, the target will be meet if Eτϕ<∞, where ϕ(t)=(ϕ1(t),ϕ2(t),…,ϕn(t)) and Eτϕ 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 Eτϕ<∞. We provide more analysis by using Lemma 5, Theorems 2 and 3 in Theorem 6 to show that the search plan ϕ(t) is finite if the conditions 2h^(x)≤L(|x|),2q^(x)≤L~(|x|), if δi0>0 and 1w^(x)≤M(|x|), 1h^(x)≤M~(|x|) if δi0<0, where L(|x|), L~(|x|), M(|x|), and M~(|x|) 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 E(|X0|) 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 E(|X0|) is finite. The existence of an optimal search plan has been proved in Theorem 10.To find the optimal distancesHij,i=1,…,n,  j=1,2,…, that give the optimal search plan ϕ*(t)=(ϕ1*(t),ϕ2*(t),…,ϕn*(t)) we solve the multiobjective nonlinear programming problem (MONLP) which contains the variables ϑi and the parameters λi,i=1,…,n. We use the Kuhn-Tucker conditions to obtain the optimal values ofϑithat can give the optimal distances, after substituting (38), and then minimize the expected value of the first meeting time. The effectiveness of this model is illustrated using numerical example.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.



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*Source: 706176-2013-07-30.xml*