Particle Swarm Optimization (PSO), one of the meta-heuristic methods used to solve optimization problems, Eberhart and Dr. It is an intuitive optimization technique that is categorized by population-based herd intelligence developed by Kennedy. Sharing information in birds and fish, like people speaking or otherwise sharing information, points to social intelligence. The PSO was developed by inspiring birds to use each other in their orientation and inspired by the social behavior of fish swarms. In PSO, the individuals forming the population are called particles, each of which is assumed to move in the state space, and each piece carries its potential solution. Each piece can remember the best situation and the particles can exchange information among themselves

1. APPLICATION OF PARTICLE SWARM
OPTIMIZATION IN 3-DIMENSIONAL
TRAVELLING SALESMAN PROBLEM
ONTHE GLOBE
Maad M. Mijwel
2016

2. Travelling Salesman Problem
• TSP is a problem that must be solved by a salesperson who must travel back to
the starting point by traveling all cities in the list with minimal cost.
• Criteria to be optimized in the problem can be cost, time, money or distance
values
• The Traveling Salesman Problem (TSP) can be expressed as a Hamiltonian cycle
that is used in modeling data in computer science and handled within the
framework of graph theory.

3. Particle Swarm Optimization
• Particle Swarm Optimization (PSO), one of the meta-heuristic methods used to solve
optimization problems, Eberhart and Dr. It is an intuitive optimization technique that is
categorized by population-based herd intelligence developed by Kennedy.
• Sharing information in birds and fish, like people speaking or otherwise sharing information,
points to social intelligence.
• The PSO was developed by inspiring birds to use each other in their orientation and inspired by
the social behavior of fish swarms.
• In PSO, the individuals forming the population are called particles, each of which is assumed to
move in the state space, and each piece carries its potential solution.
• Each piece can remember the best situation and the particles can exchange information
among themselves.

4. IMPLEMENTING THEWORKER AND IMPORTANT
• In this work,TSP is solved by PSO algorithm for randomly placed points on the sphere from 3D shapes.
• With this study, it is possible to make use of this method for each criterion to optimize the flights that
the aircraft, such as jet or plane, moving on the surface of the earth that first comes to mind with
spherical similarity.

5. Sphere Mathematics
• The sphere is a three-dimensional object, an object created by the
equidistant points from a fixed point in space. Each point spreads at
an equal distance (radius r) from the center of the globe in three
dimensions (x, y, z) is located on the surface of the globe.
• 𝑟 = 𝑥2 + 𝑦2 + 𝑧2

6. • The circle passing through the center of the globe and bounded by the sphere
is large.The great circle on earth is the equator of the world.
• The larger circle becomes more important when it is made aware of the
shortest distance between two points on the sphere along its lower section.
• The shortest path between two points is called geodesic.
Sphere Mathematics

7. Mathematical Presentation of Points on the Sphere
• Inclined echolastic surfaces are two-dimensional objects, whose positions on the surface are defined
by the u and v parameters. A coordinate position on the surface is represented by the parametric
vector function in the function of the u and v parameters for the Cartesian coordinate values x, y, z.
• 𝑃 (𝑢) = (𝑥 (𝑢, 𝑣) 𝑦 (𝑢, 𝑣) 𝑧 (𝑢, 𝑣))
• Each coordinate value is a function of two surface parameters u and v that can vary from 0 to 1.
• Coordinates are expressed by the following equations for a spherical surface originating at the center,
where r is the radius of curvature.
• 𝑥 (𝑢, 𝑣) = 𝑟.cos (2π𝑢) .sin (π𝑣)
• 𝑦 (𝑢, 𝑣) = 𝑟.sin (2π𝑢) .sin (π𝑣)
• 𝑧 (𝑢, 𝑣) = 𝑟.cos (π𝑣)
• The u parameter describes the constant latitude lines on the surface and the v parameter describes the
fixed longitude lines.

8. • x, y, z coordinates for different values of u and v parameters:
u v x y z
0 0 0 0 1
0 0.5 1 0 6.123233e-17
0 1 1,224646e-16 0 -1
0.5 0 0 0 1
0.5 0.5 -1 1.224646e-16 6.123233e-17
0.5 1 -1.224646e-16 1.499759e-32 -1
1 0 0 0 1
1 0.5 1 -2.449293e-16 6.123233e-17
1 1 1.224646e-16 -2.999519e-32 -1
Mathematical Presentation of Points on the Sphere

9. Finding the shortest distance between two points on the globe
The shortest distance between two points (p1, p2) on a spherical
surface is the arc length of the large space.
That is, V1 and V2 can be used as the angle of theta (θ) in radians
between two vectors.
V1 · V2 = P1XP2X + P1YP2Y + P1ZP2Z
the shortest path formula is as follows:
θ = arccos( V1 · V2 )
The problem is that EuclideanTSP is different.
Because the shortest distance between ((pi and pj) ) 2 points is calculated as the 3D Euclidean distance in the
3D EuclideanTSP, our problem is calculated using the arc length.
The distance matrix of the points on the sphere is the same as the symmetric TSP. Distance (pi : pj) =
Distance (pj : pi)

10. Unit Sphere Surface-On-TSP Solution Using PSO
• TheTSP to be applied on the sphere differs from normalTSP problems.
• The salesperson can only navigate points located on the surface of the sphere.
• The only different restriction on this problem is that the points are on the surface, not inside the circle.
• In this study, solutions were developed for a certain number of point clusters using PSO.

11. PSO's probing-adapted general structure
For each particle
Bring fragment to start position
End
Do
For each particle
Calculate eligibility value (turn length)
If the eligibility value (lap length) is better than pbest,
Set the current tour length as new pbest
End
Set the best of the pbest values found by all the particles to
the gbest of all the particles (the shortest lap length)
For each particle
Calculate the velocities of particles
Update particle locations
End
While the maximum number of iterations or until a minimum
error condition is reached.

12. • According to this general structure, initially the starting individuals of the PSO algorithm are created
randomly.
• A distance matrix is created in which all the points of each of the points are kept at distances.
• With the distance matrix, the fitness values of each individual are calculated.
• Individuals with minimum turn length from individuals are identified as global best.
• After finding these two best values; particle, velocity and position are updated in order.
   1
1 1 2 2. . . . .k k k k k k k k
i i i i iv wv c rand pbest x c rand gbest x
    
1 1k k k
i i ix x v 
 
Unit Sphere Surface-On-TSP Solution Using PSO

13. Experimental Results
Simulation results are presented in unit sphere (r = 1)
N = 100, 150, 200, 250, 300, 350, 400 points.The simulations were repeated 100 times for each value of
N.At each trial, a random point cloud was created.
Point Numbers
100 150 200 250 300 350 400
133,1503 205,3276 280,1300 354,6136 428,9046 503,5720 578,7485
132,2149 203,9083 278,0525 352,9967 426,2686 501,8394 575,7905
131,6228 202,8926 277,5403 351,1674 425,9008 500,4234 574,9534
130,8192 201,8222 276,3966 350,2219 424,7062 499,4872 573,5802
129,7220 200,9317 275,0707 349,7115 423,4063 498,6680 572,6015
Average sphericalTSP lap lengths calculated with PSO for N = 100, 150, 200, 250, 300, 350, 400 points on a curtain
surface.

14. 0
100
200
300
400
500
600
700
10 20 30 40 50
TourLengths
Number of Evolution
PSO
100
150
200
250
300
350
400
Experimental Results

15. If two points are to be visited and they are opposite points on the unit sphere (to return to the starting
point), the GlobalTSP lap length is approximately 2𝜋𝑟 = 6,283,185.
Experimental Results

16. Transparent and solid views of Minimum Rounds for Randomly Placed 100, 250 and 400 Points on the Globe
Experimental Results

17. Conclusions and Recommendations
• The adaptation of theTSP to the sphere and the proposed method are especially important for the movement
planning on the surface of the earth.
• Planes etc. moving through the earth's surface. vehicles can benefit from this method in optimizing problems
such as cost-time in the travel of coordinates for certain reasons such as transportation, travel, defense.
• It will be useful in this work to understand particle behaviors on every global object in the real world.
• In future studies, other methods used in theTSP solution (eg Ant Colony Optimization-ACO) can be tested in
the globalTSP solution.Also PSO and ACO etc. GlobalTSP problems can be addressed by using hybrid and
hierarchical methods.
• This method can also be tested on the geometric shapes that can actually be adapted as well as on the curtain.

18. Teşekkür Ederim.
Maad M. Mijwel
2016