Concepts in search theory have developed since World War II. The study of search plans has found considerable interest among searchers due to its wide applications in our life. Searching for lost targets either located or moved is often a time-critical issue, especially when the target is very important . In many commercial and scientific missions at sea, it is of crucial importance to find lost targets underwater. We illustrate a technique known as coordinated search, that completely characterizes the search for a randomly located target on a plane. The idea is to avoid wasting time looking for a missing target. Two searchers or robots start from the center of a circle to search out a lost target, the first searcher looks for the target on the right side of the circular area, and the second one looks for it on the left side. The time taken to detect the target is obtained by assuming the target’s position has a symmetric distribution. The procedures to facilitate the detection of the target are presented as an algorithm and as a flowchart. An application demonstrates the applicability of this search technique and the associated decrease in search cost. Its effectiveness is illustrated by numerical results, which indicates considerable promise.
Search theory rests on the oldest area of operations research. The initial steps made by Koopman in the Anti-submarine Warfare Operations Research Group of the U.S. Navy during World War II for submarine detection are still widely used for this purpose. The theory has also been used by the Navy to search for objects such as the H-bomb lost in the ocean near Palomares, Spain, in 1966, the submarine Scorpion lost in 1968, and numerous lesser-known objects. The U.S. Coast Guard uses the search theory to plan some of its more complicated search and rescue efforts.
To mention some important models of search plans, we should begin with the linear search strategy, which has many life and mission applications, such as searching for a damaged unit in a large linear system (electrical power lines, telephone lines, and gas support lines), whether a linear system is independent or intersecting (see El-Rayes et al. [
We present a coordinated algorithm, with the important advantage that it avoids wasting time searching for the target. The mission is carried out by two searchers starting from the origin of a known circular area on the sea surface. The first searcher looks for the target on the right side of the circular area, and the second one looks for it on the left side. It is important to note that neither searcher returns to the origin. We use modern means of communication and sign language to save much effort and time.
On May 19, 2020, maritime authorities announced that they had lost contact with a small ship on its way to the Socotra Archipelago port in the Indian Ocean, carrying two families in addition to the ship’s crew. They suddenly received a call from the missing ship, which enabled them to determine its coordinates. Navy ships were then sent out to search for the missing ship. We present a coordinated algorithm to solve this kind of problem, with the advantage of a detection time saving element, as the probability distribution function of the ship’s location is known to the searchers (sensors).
In this model, the search for the missing ship is carried out according to coordinated movement between the two searchers. After each move, both send signals to a marine ship’s signal reception center by radio telex. There are two types of detection:
1. Perfect detection: One of the searchers detects the lost ship in the specified search section and sends a positive sign.
2. False detection: Neither searcher detects the lost ship in the specified search section, and one or both sends a positive sign.
Let (X, Y) be independent random variables representing the position of the target, with cumulative distribution function (CDF)
Searchers s_{1} and s_{2} start searching together for the lost ship (target) from the origin (0, 0) with equal speeds
Any track has width
Searchers s_{1} and s_{2} follow a coordinated search path to find the target. Let E_{i} and F_{i} be the search paths of s_{1} and s_{2}, respectively, i = 1, 2,…, n , and let L_{i} and M_{i} be their respective search sectors. Let t_{1} and t_{2} be the respective times taken by s_{1} and t_{2} to search each sector in paths E_{i} and F_{i}, respectively.
The search proceeds as follows.
The searchers s_{1} and s_{2} follow the search path. Let E_{i} be the path for the first searcher, where i ≥ 0, t_{1} is the time for the first searcher, L_{i} is the sector which s_{1} searches, i = 1, 2, …. , n, and F_{i} the path of the second searcher, where i ≥ 0, M_{i} is the sector which s_{2} searches, i = 1,2,…..n, and the two searchers move from (0,0) to detect the target. The first search path e_{1} of s_{1} is as follows: searcher s_{1} goes to (0,
Searcher s_{2} goes to (0,
where i = 1, 2, 3, …, n,
Proof:
1) For s_{1}:
If the target is in sector L1, then
If the target is in sector L2, then
If the target is in sector L3, then
2) For s_{2}:
If the target is in sector M1, then
If the target is in sector M2, then
If the target is in sector M3, then
Then
The calculation of the expected value of the time to detect the lost target is described in the following algorithm and flowchart.
Main algorithm |
initialize searchers |
Velocities |
i = 1; |
for (i to k) |
distance |
if (i = odd number) then |
move S1 toward point (0, r_{i}) with distance |
S1 completes searching in sector |
else |
move S1 toward point (0,-r_{i}) with distance |
S1 completes searching in sector |
end if |
calculate time for step |
if (target not found in S1 || S2 ) then |
print S1 + “not found target in” + L_{i}) |
print S2 + “not found target in” + M_{i}) |
else |
if (target found in S1 ) then |
print green signal+ |
else |
print green signal+ |
exit a for loop |
end if |
End for |
End Main |
Searchers s_{1} and s_{2} have a mission to search for a lost ship on the sea surface. The ship’s location follows a standard bivariate normal distribution for two independent random variables
So,
The first search if r: 0
The Second search if r:
Special cases:
If
Hence,
By considering the values of i, r, and
i | r | ||
---|---|---|---|
0 | 0 | 10 | 0 |
0 | 1 | 10 | 0.115721 |
0 | 2 | 10 | 0.978161 |
0 | 3 | 10 | 1.46451 |
0 | 4 | 10 | 1.91147 |
1 | 0 | 10 | 0 |
1 | 1 | 10 | 0.115721 |
1 | 2 | 10 | 0.757269 |
1 | 3 | 10 | 1.42546 |
1 | 4 | 10 | 1.90942 |
2 | 0 | 10 | 0 |
2 | 1 | 10 | 0.314292 |
2 | 2 | 10 | 1.68174 |
2 | 3 | 10 | 0.0430129 |
2 | 4 | 10 | 0.000121085 |
3 | 0 | 10 | 0 |
3 | 1 | 10 | 0.135183 |
3 | 2 | 10 | 0.442948 |
3 | 3 | 10 | 7.03006 |
3 | 4 | 10 | 0.00409326 |
4 | 0 | 10 | 0 |
4 | 1 | 10 | 0.0163944 |
4 | 2 | 10 | 0.0016467 |
4 | 3 | 10 | 0.117145 |
4 | 4 | 10 | 18.3028 |
0 | 0 | 20 | 0 |
0 | 1 | 20 | 0.101895 |
0 | 2 | 20 | 0.911795 |
0 | 3 | 20 | 1.35793 |
0 | 4 | 20 | 1.83993 |
We thank the reviewers for careful checking and LetPub (