1. Introduction
The searching for a lost target either located or moved is often a time-critical issue, that is, when the target is very important. The primary objective is to find and search for the lost target as soon as possible. The searching for lost targets has recently applications such as the search for a goldmine underground, the search for Landmines and navy mines, the search for the cancer cells in the human body, the search for missing black box of a plane crash in the depth of the sea of ocean, the search for a damaged unit in a large linear system such as telephone lines, and mining system, and so on [1] [2] [3] . Search problem when the lost target is located or moved on the real line has been considered in [4] - [9] . The coordinated search technique discussed on the real line when the located target has symmetric or unsymmetric distribution as in [10] [11] [12] . Also, the coordinated search for a located target in the plane has been examined in [13] [14] [15] [16] . Recently, [17] and [18] proposed and studied a modern search model in the three-dimensional space to find a 3-D randomly located target by one searcher, two searchers and four searchers.
2. Problem Formulations
One of the most complicate problems when a mother loses her son in a way of multiple ways, here the primary objective is finding the lost son, as soon as possible in a minimum time. The survival rate of the son in this region gradually decreases, so the search team must organize itself quickly to begin the mission of the searching for the lost son immediately. Also, when the target is serious as a car, which filled by explosives, and it moves on one road from disjoint roads, and then the search effort must be unrestricted and we can use more than searcher to detect the target at right time.
The search team which consists of 4 searchers will organize itself on 2 straight lines to find the lost target as soon as possible. We clarify a modern technique by collaboration between each two searchers to find the lost person in minimum time. This problem can be characterized as follows.
2.1. The Searching Framework
The space of search: 2 disjoint lines.
The target: The target moves with a random walk motion on one of 2 disjoint straight lines.
The means of search: Looking for the lost target performed by two searches on each line. The searchers start searching for the target from the origins of the two lines with continuous paths and with equal speeds. In addition, the search spaces (2 straight lines) are separated into many distances.
2.2. The Searching Technique
Assume that we have two searchers S1 and S2 that start together looking for the lost target from O1 on L1. The two searchers coordinate their search about the lost target, where the searcher S1 searches to the right and goes from the O1 to H1, and the searcher S2 searches to the left and goes from O1 to −H1, the two searchers S1 and S2 reach to H1 and −H1 in the same time of G1. Then they come back to O1 again in the same time of G2. If one of the two searchers do not find the lost target, then the two searchers S1 and S2 begin the new cycle search for the lost target, where they go from O1 to H2 and −H2, respectively and they will reach to H2 and −H2 in the same time of G3. Then they come back to O1 again in the same time of G4 and so on. Also, we have two other searchers S3 and S4 start together looking for the lost target from O2 on the second line L2, the searcher S3 searchers to the right and goes from O2 to
, and the searchers S4 searches to the left and goes to the left and goes from O2 to
, the two searchers S3 and S4 reach to
and
in the same time of
. Then they come back to O2 again in the same time of
. If one of the two searchers not find the lost target, then the two searchers S3 and S4 begin the new cycle search for the lost target, where they go from O2 to
and
, respectively and they will reach to
and
in the same time of
, then they come back to O2 again in the same time of
, and so on. The four searchers return to the O1 and O2 after searching successively common distances until the target is found.
2.3. The Movement of the Target and the Searchers
A target is assumed to move randomly on one of two disjoint lines according to a stochastic process
. Assume that
is a sequence of independent identically distributed random variables such as for any
:
and
, where
. For
,
,
.
We assume the searchers S1 and S2 begin their search path from O1 on L1 with speeds V1, and the searchers S3 and S4 begin their search path from O2 on L2 with speeds V2, following the search paths which are functions
and
on L1 and
and
on L2, respectively, such that:
, (1)
and
, (2)
where V1 and V2 are constants in
and
. Let the set of all search paths of the two searchers S1 and S2, which satisfy condition (1), be respectively by
and
respectively and the set of all search paths of the searchers S3 and S4 which satisfy condition (2), be represented by
and
, respectively. we represented to the path of S1 and S2 by
where
, where
.
The search plan of the four searchers be represented by
, where
is the set of all search plan.
We assume that
if the target moves on L1 and
if the target moves on L2 such that
. There is a known probability measure
on
which describes the location of the target, where v1 is probability measure induced by the position of the target on L1, while v2 on L2. The first meeting time valued in
defined as
where Z0 is a random variable representing the initial position of the target and valued in
(or
) and independent of
.
At the beginning of the search suppose that the lost target is existing on any integer point on L1 but more than H1 or less than
or the lost target is existing on an integer point on L2 but more than
or less than
. Let
be the first meeting time between S1 and the target and
be the first meeting time between S2 and the target and
be the first meeting time between S3 and the target and
be the first meeting time between S4 and the target. The main objective is to find the search plan
such that
. In this case
is said to be a finite search plan, and if
, where E terms to expectation value, then we call
is an optimal search plan. Given
, if z is:
, where
is integer, then
2.4. Finite Search Plan
Let
be positive integers such that
,
,
, where
and
are the least positive integers and
.
We shall define the sequences
for the searcher S1 on the first line L1 and
for the searcher S3 on the second line L2 and the search plans with speeds 1 as follows:
on L1,
on L2.
We shall define the search path as follows:
for any
, if
, then
,
and
.
Also, if
, then
,
and
.
We define the notion
on L1,
on L2,
the searchers S1 and S2 return to the origin of L1 after searching successively common distances
, and
, respectively and the searchers S3 and S4 return to the origin of L2 after searching successively common distances
, and
, respectively until the target is found.
Theorem 1: If
is a search plan defined above, then the expectation
if finite if
,
,
,
,
,
,
,
and
. (3)
are finite.
Proof: Assume that X and Y are independent of
, if
, then
until the first meeting between S1 and the target on L1, also if
, then
until the first meeting between S2 and the target on L2. We can apply this assumption on the second line by replacing X by Y and
by
respectively. Hence, for any
,
hence
(4)
to solve Equation (4) we shall find the value of
,
,
and the value of
as the following
We get
(5)
also,
(6)
(7)
We get
(8)
(9)
substituting by (5), (6), (7) and (8) in (4) we can get
hence
where,
,
,
,
,
,
,
,
and
.
Lemma 1: For any
, let
for
, and
. Let
be a strictly increasing sequence of integers with
,
,
For more details see [1] .
Theorem 2: The chosen search plan satisfies
where,
,
,
,
,
,
,
, and
are linear function.
Proof: This theorem will prove for
and
, and by similar way we can prove the other cases
and
1) if
, then
and if
, then
,
2) if
, then
,
and if
, then
,
from Theorem (2), see (Mohamed [1] ) we obtain
and
Let us define the following
1)
, where
is a sequence of (i. i. d. r. v.)
, where
is a sequence of (i. i. d. r. v.).
2)
,
.
3)
,
,
4)
is an integer such that
, and
is an integer such that
,
5)
, and
,
and
6)
,
,
then
and
satisfies the condition of the renewal equation, for more details see [19] .
If
and
then by Theorem (2) see (Mohamed [1] )
and
are non increasing and we can apply Lemma (2) to obtain
and
Since
and
satisfied the condition of the renewal equation, hence
and
is bounded for all j by a constant, so
,
and
.
Theorem 3: If
is a finite search plan, then
is finite.
Proof: If
, then
and so
,
then, we conclude that
and
,
or
and
.
On the first line L1 if
, then
with probability one and hence
.
If
, but
, then
and
.
On the second line L2 if
, then
with probability one, by the same way we can get
is finite on the second line L2.
3. Existence of an Optimal Search Plan
Theorem 4: Let for any
, let
be a process. The mapping
is lower semi-continuous on
.
Proof: Let
be the indicator function of the set
by the Fatou Lebesque theorem see (Stone [16] ) we get
,
for any sequence
in
is sequentially compact [20] , thus the mapping
is lower semi continuous on
, then this mapping attains its minimum.
4. Conclusions
We have described a new kind of search technique to find a lost moving target on one of two disjoint lines. The motion of the four searchers on the two lines in the quasi-coordinated search technique is independent, and this helps us to find the lost target without waste of time and cost, especially if this target is valuable as the search for lost children. Actually we calculated the finite search plan. Also; we proved the existence of an optimal search plan which minimizes the expected value of the first meeting time between one of the searchers and the target.
In the future work, we will introduce an important search problem, looking for a randomly moving target as a general case and the searchers will begin their mission from any point on the line.