The Traveling Salesman Problem (often called TSP) is a classic algorithmic problem in the field of computer science and operations research. It is focused on blogger.com this context, better solution often means a solution that is cheaper, shorter, or blogger.com is a mathematical problem. It is most easily expressed as a graph describing the locations of a set of nodes Feb 14, · The traveling salesman problem (TSP) involves finding the shortest path that visits n specified locations, starting and ending at the same place and visiting the other n Jun 07, · In both high school and university, you will be required to write research papers. Yes — papers in the plural. And that’s the first reason you may want to turn to Custom Writing and ask us, “write my research paper.” The very first undertaking, when assigned a paper, is to choose from the colossal list of topics for research papers that are available
Travelling salesman problem - Wikipedia
The traveling salesman problem TSP is the most popular and most studied non-deterministic polynomial NP hard problem that has been used in various fields of research paper traveling salesman problem and technology. Due to the NP-hard nature, research paper traveling salesman problem is very difficult to solve this problem effectively and efficiently.
For this reason, diverse appropriate optimization algorithms have been designed and developed in the last few decades. Among these algorithms, heuristic algorithms are much more suitable to tackle with this complex problem.
In this paper, we propose a hybrid heuristic algorithm to solve the symmetric TSP problem by combining the search mechanism of repetitive nearest neighbor RNN heuristic and simulated annealing SA heuristic algorithms, research paper traveling salesman problem. In fact, a set of better routes are generated step by step by the RNN algorithm and these routes are improved through the iterative improvement process of the SA algorithm.
The proposed algorithm is tested on a set of benchmark symmetric TSP datasets and compared with the basic RNN and SA algorithms as well as some other hybrid algorithms existing in the literature. It is demonstrated by the experimental results that the proposed algorithm is more effective than both the basic RNN and SA algorithms, and the obtained optimum results are in good agreement with the corresponding best-known optimum results. In addition, the proposed algorithm outperforms some other hybrid algorithms in terms of solution quality.
Combinatorial OptimizationTraveling Salesman ProblemRepetitive Nearest Neighbor AlgorithmSimulated Annealing AlgorithmNeighborhood StructureHybrid Algorithm. The traveling salesman problem TSP is a well-known combinatorial optimization problem that has been extensively studied in various fields of science and technology such as mathematics, artificial intelligence, physics, operations research, and biology.
It is the problem of finding the possible shortest route among a list of cities, where each city is included once and only once and finally returns to the starting city. It is believed that the history of the TSP problem was discovered in in Vienna [1].
However, a formal description of the TSP problem was formulated by Dantzig et al. in [2]. Since then it has been used in modeling diverse real-world problems, such as designing hardware devices, microchips, and radio electronic devices, data association, data transmission in computer networks, DNA sequencing, vehicle routing, job scheduling, clustering of data arrays, image processing and pattern recognition, analysis of the structure of crystals, transportation, logistics, supply chain management, etc.
The TSP problem is easy to understand but often very difficult to solve as it contains all features of the combinatorial optimization problem. In fact, it has been proven to be a non-deterministic polynomial NP research paper traveling salesman problem problem [5]. By NP-hard, we mean those problems for which no polynomial time algorithm exists to effectively solve them. Indeed, the executive time of any existing algorithm for solving the TSP problem is increased super-polynomially or, perhaps exponentially with the number of cities [4].
Thus, the study on improving the solution algorithm of the TSP problem has important theoretical, engineering, and practical significance. Depending on the distance cost matrix D, the TSP problem can be categorized as symmetric or asymmetric. The graph G is called a symmetric TSP if all research paper traveling salesman problem edges of G are symmetrical edges, i. In this study, we consider the asymmetric TSP problem. Thus, the objective function Z of the TSP problem can be formulated as follows [3] :.
Equation 2 ensures that each location j is occupied by only one city, research paper traveling salesman problem, whereas Equation 3 guarantees that each city i is assigned one exact position. In contrast, Equation 5 assures that each city in the final route will be visited once and that no sub-routes will be formed. Due to the applicability and complexity, various researchers have been designed and developed different optimization algorithms in the last few decades to deal with the TSP problem.
Among these algorithms, heuristic algorithms are the most successful and widely used search mechanism for solving the TSP problem. Heuristic algorithms, research paper traveling salesman problem, however, offer a satisfactory solution but are often meet with the problem of premature converge.
As a result, the search process easily falls into the trapped of local optimum condition and is unable to jump the solution into another promising search space. For this reason, many researchers turned into developing hybrid optimization algorithms through the integration of the superiority of two or more heuristic optimization algorithms.
The aim of this paper is to design and implement a hybrid optimization algorithm called RNN-SA for solving symmetric TSP problem, which uses the search process of RNN algorithm and SA algorithm, research paper traveling salesman problem. The proposed RNN-SA algorithm performs the search procedure by two stages. It first generates a set of feasible routes step by step based on the procedure of RNN. Then, these routes are used as the initial solutions of the SA algorithm and are improved iteratively in an effective iterative improvement process.
The proposed algorithm is implemented on a collection of benchmark TSP datasets. The proposed RNN-SA gets better results than both the RNN and SA algorithms and performs research paper traveling salesman problem than some other hybrid optimization algorithms.
The rest of this paper is organized as follows. In Section 2, we review some research paper traveling salesman problem hybrid algorithms for solving the TSP problem. In Section 3, we briefly introduce the optimization approaches under consideration in this study that make up the proposed hybrid algorithm.
In Section 4, we discuss the proposed hybrid RNN-SA algorithm in detail, research paper traveling salesman problem. In Section 5, we present experimental results, result analysis, and performance comparisons. The conclusion of the paper is summarized in Section 6.
In this section, we review some recently published hybrid optimization procedures that use different strategies to develop sophisticated solution methods for solving the TSP problem.
Geng et al. proposed an adaptive hybrid algorithm called ASA-GS by combining the problem solving efforts of the SA algorithm and greedy search GS mechanism to solve the symmetric TSP problem [6]. They use a greedy search strategy in the optimization procedure of general SA algorithm to speed up the solution convergence rate.
Their testing experiments demonstrated that the ASA-GS performs well on both small and large-scale TSP datasets and even outperforms some recently developed optimization algorithms. In fact, this composite algorithm found a better trade-off between solution quality and computation time for solving symmetric TSP problem. Utilize the benefit of the genetic algorithm GASA, ant colony optimization ACO and particle swarm optimization PSO a hybrid algorithm named GSA-ACO-PSO is reported by Chen and Chien to tackle symmetric TSP problem [7].
In this optimization framework, a set of feasible solutions are generated by the ant colony system and these feasible solutions are adopted as the initial solution of the GA procedure. Then, research paper traveling salesman problem, the GA is executed with SA mutation techniques to achieve better solutions. On the other hand, the role of the particle swarm optimization process is to facilitate the exchange of pheromone information among the populations in the ant colony system after a predefined number of cycles, research paper traveling salesman problem.
The experimental evaluations indicated that this hybrid algorithm exhibits better performance than some other related optimization algorithms. Deng et al. developed a hybrid algorithm with the help of evolutionary concepts of GA, ACO and PSO algorithms to solve the symmetric TSP problem [3]. In the implementation process of this algorithm, a series of sub-optimal solutions are first generated through the combination of wholeness, randomicity, and rapidity of the PSO and GA techniques.
After that, the resulting solutions are exploited based on the ACO algorithm by utilizing the benefit of the parallel, positive feedback and higher accuracy. Their algorithm evaluation illustrated that it performs better than some other evolutionary Research paper traveling salesman problem solving algorithms.
In [8], Zhan et al. proposed a new version of the SA algorithm named list-based SA algorithm LBSA in order to solve the symmetric TSP problem. In this approach, a list-based cooling schedule is adopted to control temperature reduction in the basic SA algorithm. Their experimental results indicated that the LBSA finds good approximate solutions and outperforms some other state-of-the-art algorithms.
A hybrid optimization algorithm by combining the superiority of the symbiotic organism search SOS and SA algorithms named SOS-SA is reported by Ezugwu et al.
In this optimization framework, research paper traveling salesman problem, the initial feasible solutions for the SOS algorithm are generated by applying the conventional SA algorithm, research paper traveling salesman problem. After that, these research paper traveling salesman problem are modified and improved through the three intelligent optimization phases of SOS algorithm.
Comparative results demonstrated that the SOS-SA framework can research paper traveling salesman problem TSP optimal solutions and show competitive behavior with other state-of-the-art optimization algorithms. Hore et al. reported a hybrid variable neighborhood search HVNS algorithm to solve both symmetric and asymmetric TSPs [4]. They accomplish the search procedure by two stages such as it first generates an initial feasible solution through a route construction based greedy approach, and then improves this solution iteratively by using various neighborhood structure with stochastic stopping criteria.
The algorithm evaluation found that it performs better than the conventional optimization algorithms, and VNS-1 and VNS-2 algorithms as well. In [10], research paper traveling salesman problem, Tsai et al. proposed a hybrid algorithm called ACOMAC by introducing multiple ant clans MAC idea in the process of ACO algorithm.
They also discussed ant colony system ACS for solving the TSP problem. In this work, multiple nearest neighbor NN and dual nearest neighbor DNN are combined separately with both ACOMAC and ACS to enhance their performance. In this section, we give a brief introduction of the optimization techniques under consideration in this study that make up the proposed hybrid algorithm for solving TSP problem. The RNN optimization algorithm and the SA optimization algorithm are briefly discussed in the following subsections consecutively.
The RNN is the route construction algorithm that is an extension of the well-known Nearest Neighbor NN research paper traveling salesman problem [11]. The NN algorithm attempts to construct the route based on the connections of nearest neighbors.
It starts with a city chosen at random as the starting city of the route and then includes the next city which is located closest to the last city.
The performance of this algorithm is highly sensitive to the choice of starting city. To remedy this, the RNN algorithm was developed. It performs better than the NN algorithm for solving TSP problem. However, there is time complexity of order O n 3 while the algorithm running time of NN is reported as O n 2 [12].
In fact, the RNN algorithm constructs a set of routes step by step through some strategies. Its route construction procedure can be described as a sequence of the following steps. In research paper traveling salesman problem first step, the search engine generates n sub-routes and each sub-route consists of one single city.
The set of constructed sub-routes can be expressed as follows:. Step Each sub-route obtained from the first step is extended the network in this step by adding a different nearest city from the remaining cities. Thus, the set of possible n sub-routes is constructed on the basis of Equation 7. In the Equation 7research paper traveling salesman problem, d c ic j is the Euclidean distance between the cities c i and c jwhich is calculated based on the formula presented in Equation 8.
Let x iy i and x jy j be the Cartesian coordinate of the location of the cities c i and c j. Then, the formula for calculating the Euclidean distance between c i and c j is as follows:. Step In this step, each 2-city sub-route of R 2 is enlarged through a nearest city that is not yet in the route. Thus, the set of possible 3-city sub-routes is constructed as bellows:. In this way, the route construction mechanism is continued until all the cities are included in each route.
Finally, we get a set of feasible n routes and from there a best route is searched out. In this paper, the RNN method is considered to generate a set of n feasible TSP routes.
The Asymmetric Traveling Salesman Problem
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This paper gives an introduction to the Traveling Salesman Problem that includes current research. Additionally, the algorithms are used to nd a route The traveling salesman problem can be divided into two types: the problems where there is a path between Feb 14, · The traveling salesman problem (TSP) involves finding the shortest path that visits n specified locations, starting and ending at the same place and visiting the other n Jun 07, · In both high school and university, you will be required to write research papers. Yes — papers in the plural. And that’s the first reason you may want to turn to Custom Writing and ask us, “write my research paper.” The very first undertaking, when assigned a paper, is to choose from the colossal list of topics for research papers that are available
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