Travelling salesman problem using reduced algorithmic Branch and bound approach P. Ranjana Hindustan Institute of Technology and Science

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Volume 118 No. 20 2018, 419-424 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Travelling salesman problem using reduced algorithmic Branch and bound approach P.

1 Volume 118 No , ISSN: (on-line version) url: ijpam.eu Travelling salesman problem using reduced algorithmic Branch and bound approach P. Ranjana Hindustan Institute of Technology and Science Abstract-Travelling salesman problem (TSP) is a classic algorithmic problem that focuses on optimization. TSP is an important problem because its solution can be used in other graph and network problems. The exact solutions to these problems cannot be obtained in polynomial time, so we need to go for the approximation algorithms which could give the solution in polynomial time. The aim of this paper is to find the shortest route using the branch and bound technique. There were several algorithms existing to solve the travelling salesman problem using branch and bound technique. Here a reduced branch and bound (RBB) algorithm is proposed which uses the graph to model the travelling salesman problem and reduced the branching and reduces the tour length. The proposed algorithm is proved to be efficient than the other branch and bound algorithms for even for lesser number of cities by giving the mere optimal solution and it gives the optimal solution for larger number of cities which could be applied for larger scale problems be defined as returning to the starting point after visiting all the point with the lease cost..the real world problem that is most widely used nowadays is to calculate the routes in google maps. Keyword: Algorithm, Branch and Bound, Optimization, Problem, Travelling Salesman. 1. INTRODUCTION Most of the optimisation problems are solved using computers, there were several algorithmic problem solving techniques applied to solve those problems. The most common technique applied to solve those problems are brute force technique, where it calculates every possible solution to the problem and tries to find the solution by selecting the best solution. But this will not be suitable for all types of problems, sometimes the problem will be of larger size and it is not possible to find the solution to these problems in a reasonable time. Problems of these types are known as NP hard problems. The travelling salesman problem is NP hard problem. The travelling salesman problem is a problem to find the shortest route travelled by the salesman. The salesman has to visit all the cities and end up in the starting city by minimising the distance travelled. If this problem is used for large number of cities it is difficult to get the optimal solution in a fairly reasonable time. II. TRAVELLING SALESMAN PROBLEM Travelling salesman problem in real world problem could Figure 1: TSP as Graph The travelling salesman problem could be solved mathematically, for this the problem is to be converted to graph theory, figure 1. represents the travelling salesman problem as a graph where the vertices v1,v2,v3,v4 are considered as cities and the edges connecting the cities e1,e2,e3,e4,e5,e6 were considered as the road connecting the cities. In order to use mathematical methods in solving the Travelling Salesman Problem one needs to translate it to the language of a graph theory. Travelling salesman problem referees to the salesman who travels all the cities and wants to find the shortest possible tour by visiting each city exactly once and returns back to the starting city. This problem can be represented as a graph with n nodes representing the cities and e edges representing the distance between the cities. The goal is to find the minimum Hamiltonian cycle. The Hamiltonian cycle is the shortest tour that visits each node exactly once and 419

2 return to the starting node. The Hamiltonian cycle is defined as a path in the graph that contains all the vertices of the graph. The objective function of TSP is to minimize the total distance travelled This is represented mathematically as in equation(1). ΣΣcijxij (1) Where cij is given by the cost taken by the distance travelled. For the symmetric travelling salesman problem the distance travelled can be given by for j=1 to n Σxij=1 for i =1 to n Σxji=1 xij= 0 or 1 for all i for all j In TSP problem with n nodes there are (n-1)! feasible solution, the problem is to find the optimal solution. There are many variants in TSP with vast applications, some of the main category are Symmetric travelling salesman, Asymmetric travelling salesman and Multiple Travelling sales man under these type the classification are dong based on single depot or multiple depot, number of salesman, number a cities the sales man has to visit and son on. Each variants of TSP many applications one such application is the global navigation system. The other applications and various of TSP can been seen in[6]. III.APPROACHES USED IN SOLVING TSP A. Brute force Approach Many problem solving approaches were used in solving TSP problems. One of the well-known and straight forward approach is the brute force approach. This approach gives the estimated time for solving tsp fir 16 nodes in more than 2 years. [1]. The time taken for solving the cities with more number of times is unpredictable, To find the exact solution for a problem of size n There are 2 n.n sub problems to solve in linear time so it take O(2 n n 2 ).so this cannot be applied for the Travelling salesman problem. Dynamic programming complexity is O(n22n-1), it but still better than exhaustive search B. Nearest Neighbourhood The Nearest neighbourhood approach looks at the distance of the nearest cities that have not been visited and returns to the starting city when all other cities are visited, the time taken to find the solution using this method is less than the brute force method, but this approach does not guarantee best solution as brute force approach. [7]. In this approach the salesman can start at any city. This algorithm is very easy to execute, but in worst case it produces the solution that is much longer than the optimal solution C. Greedy approach This is the first approach to solve TSP. This approach uses edges coming out of the node and chooses the edge with minimum cost. If these n cheap edges form a Hamiltonian cycle then we get an optimal solution. Greedy algorithm provides the upper bound the lower bound is not easy to complete.it will produce local optimal solution, but does not guarantee for global optimal solution. D. Branch and bound techniques Branch and bound technique is most widely used in travelling salesman problem by constructing a state space tree to find the optimal solution among all feasible solutions by takin the value of the objective function. Branch and bound was initially studies by [2] and a more description was provided by [3] in the applications of TSP. The branch and bound techniques gives all feasible solutions by solving the problem, by trying the practical solution ad starting the value in the upper bound for finding the optimal solutions [5]. The various approaches discussed above gives the exact solution, so these methods are exact methods. E..Research Gap Travelling salesman problem is considered as NP hard problem so it is so it is very difficult to get the solution in polynomial time. There were many heuristic and genetic algorithms to get the optimal solution, but they were not able to get the mere optimal solution for larger data [10].The approaches used to solve TSP are using exact and approximation methods. The branch and bound methods has the advantage of avoiding calculation of all partial 2trees by finding a practical solution a noting down its value as an upper bound for the optimum. During parallelisation the load balancing becomes difficult for the branch and bound. Also the branch and bound is limited to a small size network of cities. It will become prohibitive if the solution of the state space grows exponentially for larger size network. If the cities are connected by the edges with specified constraints in each node. Edges connect the two cities based on the constraints. A subtree of nodes generates a legal tour counting the leaf node. Many branch and bound techniques algorithms were used in solving the travelling salesman problem R [4] load balancing in branch and bound techniques is to be addressed. Since branching referees to the state space tree can be generated using BFS or DFS, here the algorithms developers the state space tree with DFS. IV. PROBLEM FORMULATION As the basic definitions of TS P is discussed. The problem is formulated using graph theory. If the problem is represented as graphs since it is easily understood and easily solved. So the problem is solved using the graph representation with the basic definition of the Travelling salesman problem. The second part gives the graph theories provided. In the third part algorithmic display of the proposed method is presented. Load balancing is also achieved by parallelization using highly efficient first branch and bound algorithm to solve the travelling salesman problem with 1024 processer is based on 1 tree relaxation problems.[8].the TSP problem is loved using the branch and bound techniques by improving the load balancing even when a new node is added dynamically. When jobs are assigned dynamically load balancing is done by massive parallel processing system but in real world this need a high performance energy efficient computing system[9].for problem construction we have an undirected graph G=(V,E), where V=(1,2,3,4) representing the set of node as shown in Figure 2. The depot node is node 1 and E represents the set of 420

3 edges (1,2,3,4,5,6) and each edge has the edge cost.the Figure 3.represents the graph with 8 nodes or 8 cities and figure 4 represents the graph with 16 cities. V. ALGORITHMIC IMPLEMENTATION To implement the problem the algorithms display of methods are used. The dataset is take from TSPLIB. It is a library for same instances for the TSP and related problems from various sources and various types. The TSP LIB, GR17 data set is used which has 12 nodes as cities and ATT 48 data set is used for 48 nodes and PR 226 for 226 nodes. It is applied on a symmetric travelling salesman (TSP) problem where the given set of node the cost of travelling from node i to node j is same as the cost to travel from node j to node. This is usually represented as undirected graph as given in the figure.1.the Asymmetric travelling salesman problem is the total cost of travelling from node i to node j is different from the total cost of travelling from node j to node i. This is usually represented as an undirected graph as given in the figure.1 A. TSP Solver Figure 2. With 4 cities Figure 3. With 8 cities TSP solver is implemented with JAVA code. The input to this is the data from the TSP Lib Library. The software parses and extracts the information like Weighted array- The graph is converted into an array Cities-The number of nodes and the name of the nodes in the graph Names: data file type TSP of ATSP Dimensions which denotes the number of cities Weight type- The type of weight between the nodes Distance type- The type of distance between the nodes. The reduced branch and bound algorithm is implemented in the TSP solver and compared with the other existing algorithms using branch and bound techniques for which the optimal solution is obtained. The problems are studies with problem of size4, 8 and 16 cities for the data for the above graph. The distance covered is calculated for thesis technique. The branch and bound algorithm is implemented by improvising it by reducing the edges when it is found that the feasible solutions could not be obtained. The reducing branch and bound algorithm works well even for less number of nodes. When the number of nodes was added then there will be a considerable increase. Here the Travelling salesman problem using the data table and reduced branch and bound method given the correct solution to the problem and this have be proved up to 50 cities. This method is used to solve the delivery of packets to certain address for which we know the exact distance between these addresses. This method deliverers the packet through the optimal route and calculates the time in units. B. Reduced Branch and Bound Algorithm The Reducing Branch and Bound algorithm works by selecting the smallest lower bound value using the DFS. The input to implement RBB algorithm is represented in figure 5 and also the output obtained from that is also represented below the figure. Figure 4. With 16 cities 421

Reduced Branch and Bound Algorithm (RBB) Solution tree is constructed by adding the edges in lexicographic order When a new node is added the decision tree will decide whether to include or exclude

4 Reduced Branch and Bound Algorithm (RBB) Solution tree is constructed by adding the edges in lexicographic order When a new node is added the decision tree will decide whether to include or exclude the tour If is represented as 0 or 1 If an edge is excluded then it is not possible to have adjacent edges The branch corresponding to those edges are reduced If an edge is includes there is a possibility of having adjacent edges Now the value is one so the algorithms branches and repeats the same stop. Figure 6a.With 4 cities Figure 6b.With 8 cities Figure 5. Input and output in TSP solver VI...RESULTS AND DISCUSSIONS From the above table it is found that for smaller solutions the time taken by the dataset is taken and we observe that for small data set the algorithms find the solution in less than one minute, for other larger steps reasonably good solutions are obtained. The Figure 6 shows the implementation travelling salesman for 4 cities, figure 6b, and shows the travelling salesman problem implementation for 8 cities and figure 6c shows the travelling salesman problem implementation using the proposed algorithm for 16 cities. Like this the TSP problem up to 50 cities is solved using the RBB algorithm and it is found that it performs better than the other BB approached and the performance increases if the number of cities are increased, thus if the number of cities are increased the proposed approach is able to produce the near optimal solution. Figure 6c. With 16 cities 422

5 VII. CONCLUSION Though many algorithmic problem solving techniques are available for solving the travelling salesman problem the branch and bound technique gives the optimal solution than the other approaches. The proposed method uses the branch and bound technique by reducing the branches and prepares the cost matrix at each step. We observe that it is able to obtain the minimum cost for the tour. It also gives the fact for expanding the node. The ides at initial stages with minimum number of nodes does not give s mere optimal solution for lesser number of nodes, but when the number of nodes are increased it will gives the optimal solution. REFERENCES [1] Calgor, H. (12 January 2017). The Brute force algorithms. [2] George B. Dantzig, D. R. (1954). Solution of a Large-Scale Traveling-Salesman. The road corporation. santa monica, california, [3] George B. Dantzig, D. R. (2010). Solution of a Large-Scale Traveling-Salesman. Springer-Verlag Berlin Heidelberg, [4] Mirta Mataija, M. R. ( Vol. 4 (2016.), No. 1, ). Solving the travelling salesman problem. Zbornik Veleučilišta u Rijeci,, pp [5] Pathak, h. C. (2012). Survey of Methods of Solving TSP along with its Implementation using Dynamic Programming Approach. International Journal of Computer Applications, Volume 52 - Number 4. [6] Rajesh Matai, S. P. (2101). Traveling Salesman Problem:An Overview of Applications, Formulations,. [7] StefanHougardy, M. (November 2015). On the nearest neighbor rule for the metric traveling salesman problem. Discrete Applied Mathematics, Pages [8] Tschoke, S., Lubling, R., & Monien, B. (n.d.). Solving the traveling salesman problem with a distributed branch-and-bound algorithm on a 1024 processor network. Parallel Processing Symposium, Proceedings., 9th International. [9] Ujaldón, A. L. (March 2016, Volume 19,). Dynamic load balancing on heterogeneous clusters for parallel ant colony optimization. Cluster Computing, Springer link, [10] Vikas Raman, N. S. ( 2017). Review of different heuristic algorithms for solving Travelling Salesman Problem. International Journal of Advanced Research in Computer Science, p

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Dynamic Capacity Routing in Networks with MTSP

Dynamic Capacity Routing in Networks with MTSP Ranjana Ponraj1*, George Amalanathan2 1 Hindustan 2 University, Chennai, Tamil Nadu, India. Periyar Maniammai University, Thanjavur, Tamil Nadu, India. *