Meta-Heuristic Solution Approaches for Traveling Salesperson Problem

International Journal of Applied Mathematics Electronics and Computers ISSN:2147-82282147-6799 http://dergipark.gov.tr/ijamec Original Research Paper Meta-Heuristic Solution Approaches for Traveling Salesperson Problem Omar Mohammed Ahmed AHMED1, Humar KAHRAMANLI2 Accepted : 27/08/2018 Published: 30/09/2018 DOI: 10.1039/b000000x Abstract: The traveling salesperson problem (TSP) is the NP-hard optimization problems which have been widely studied over the past years. TSP creates a Hamiltonian cycle where each node is visited once and only once to minimize the total traveled distance. TSPs are difficult to be solved using classical mathematical methods. Even with nowadays computers solving TSP problems with these methods takes very plenty of time. Therefore, many efficient optimization methods have been focused for academic proposes for the TSP all the times. Most of the TSP problems are now solved by meta-heuristic methods, that provides a satisfactory solutions in real-time. Metaheuristic algorithms were inspired from behaviors of animals and insects such as ants, bees, fish schools, bird flocks and mammals.This paper focuses on three meta-heuristic methods: Whale Optimization Algorithm (WOA), Particle Swarm Optimization (PSO) algorithm and Grey Wolf Optimizer (GWO). The problem for application was selected from TSPLIB. Probably the best implemented solutions were Whale Optimization Algorithm and Grey Wolf Optimizer which can be recommended as primary algorithm to solve the TSP or to start with the meta-heuristic solution. Keywords: Travelling salesperson problem, Meta-heuristic optimization, Whale Optimization Algorithm, Grey Wolf Optimizer, Particle Swarm Optimization. 1. Introduction The Travelling Salesperson Problem (TSP) is one of the complex and well-known NP-hard combinatorial optimization problems. It is easy to understand the TSP where it remains at the list of the one of the challenging problems of operational research. Its purpose is finding the shortest path for a salesperson who must visit N cities. Solving TSP has both of practical importance and academic interest, and it is an important topic of active research. A great number of methods have been invented to solve TSP problems. Some of them are Genetic Algorithms (GA) [1], Simulated Annealing (SA) [2], Tabu-Search Algorithm (TSA) [3], Ant Colony Optimization (ACO) Algorithm [4], Memetic Algorithm (MA) [5], Bee Colony Optimization (BCO) Algorithm [6], Firefly Algorithm (FA) [7], Cuckoo Search Algorithm (CSA) [8]. In spite of classical algorithms such as TS and SA are not that efficient to be used for solving optimization problems, Evolutionary Algorithms (EA) such as MA and GA gives appropriate solutions for complex optimization problems. TSP problem can be explained as follow: Give the shortest path that covers all cities along. Let’s assume that R = (N; S) be a graph where N is a collection of vertices and S is a collection of edges. Let C=(Cij ) be a cost (or distance) matrix related with S. In the TSP problem minimum distance loop (Hamiltonian loop or cycle) determination required, which is all the vertices are visited just one time. Assume that salesperson already knows Cij (i, j ∈ {1, 2, 3, … , 𝑁}) which indicates the distance between the ith and jth cities. The salesperson must select the route with the minimum travel distance. Besides it this tour must include all of the cities moreover each city must appear only one time. The salesperson could begin his route from any city, while he must _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 1 Graduate School of Natural Sciences, Selcuk University, Konya, TÜRKİYE 2 Department of Computer Engineering, Technology Faculty, Selcuk University, Konya, TÜRKİYE * Corresponding Author: return to the city where he began his tour. The need of quick find of satisfactory solutions to TSP has caused to the development of numerous methods such as metaheuristics. Meta-heuristic algorithms have showed effective performance in solving a large set of optimization problems. They have many advantages over classical methods such as flexibility and simplicity. Meta-heuristic methods are generally easy to implement and procced. In addition, these methods are very simple and flexible, and they are able to deal with many problems, both continuous and discrete moreover mixed. Nowadays, the techniques which is using for TSP divides into two main groups: approximation algorithm, and exact methods which guarantees obtaining the optimal solution. Approximation algorithms have the ability to obtain more accurate, therefore they are very appropriate to solve large-scale problems. These algorithms also divide into two groups: heuristic optimization techniques and local search algorithms. Heuristic optimization techniques search around the optimal solution. GA [1], SA [2], ACO [4], PSO [30], Artificial Neural Network (ANN) [9,10,11], Marriage in Honey Bees Optimization Algorithm [12] and Artificial Immune Algorithm (AIS) [13] are examples for heuristic optimization techniques. 2 - Opt [14], 3 Opt [15], LK [16], LKH [17] and Inver-over [18] are the examples for the local search algorithms. The second main category for solving TSP problems is exact methods which have the ability to obtain guarantee optimal solutions, but it leads to increasing in the problem’s scale, the required time for solving exponentially increases. The common exact techniques include dynamic branch and bound method [19], programming method [20]. In the past years, many researches could combine meta-heuristic algorithms with local search to develop a novel hybrid algorithm to solve TSP, such as LK and genetic operators [21], combined ant colony optimization algorithm with mutation strategy [22], combined technique of a 2-Opt and genetic algorithm [23]. These International Journal of Applied Mathematics Electronics and Computers (IJAMEC) 2018, 6(3), 21–26 | 21 combined algorithms can get satisfactory solution in less iteration. In addition, the above mentioned heuristic, metaheuristic algorithms and exact algorithms have been tested by number of developers on TSP successfully. This paper examines three nature-inspired (meta-heuristic) algorithms to solve TSP. Six benchmarks problem were selected to test the algorithms, and the obtained results show that the WOA and GWO achieve better results than PSO. The rest of the article is organized as follows: Section 2 gives detailed information of the meta-heuristic algorithms; section 3 gives brief explanation about applications, in section 4 simulation and comparisons are presented. In section 5 the work is concluded. 2. Method Three nature-inspired algorithms: GWO, WOA and PSO have been u (...truncated)