Application of particle swarm optimization in 3 dimensional travelling salesman problem on the globe
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The document explores the application of Particle Swarm Optimization (PSO) to solve the 3-dimensional Traveling Salesman Problem (TSP) on the globe, emphasizing the unique challenges posed by spherical geometry. It outlines how the PSO algorithm can optimize routes for points on the Earth's surface, contributing to fields like transportation and aviation. The study includes experimental results demonstrating the efficiency of this approach for various data point scenarios.
Application of particle swarm optimization in 3 dimensional travelling salesman problem on the globe
- 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