## Application of AHP for Traveling Salesman Problem

Application of AHP for Traveling Salesman Problem Sunarin Chanta and Ornurai Sangsawang Department of Industrial Management, Faculty of Industrial Tec...
Author: Esmond Payne
Application of AHP for Traveling Salesman Problem Sunarin Chanta and Ornurai Sangsawang Department of Industrial Management, Faculty of Industrial Technology and Management, King Mongkut’s Institute of Technology North Bangkok, Thailand. Abstract. The objective of this research is to determine the best route for recording volume usage of water supply. Mathematical Approach applied in traveling salesman problem is developed to solving this problem. In order to investigate the best route, we must consider both quantitative and qualitative factors influencing route selection. Branch and Bound method, the Nearest Neighbor Heuristic, Cheapest Insertion Heuristic, and Two-Way Exchange Improvement Heuristic are selected to minimize total distance. The Analytic Hierarchy Process is applied for quantitative and qualitative factors evaluation in a systematic criterion. The presented methodology can provide managers with a more effective and efficient method for making route recording decisions in a real-world. Keywords: Traveling salesman problem, Analytic hierarchy process, Branch and bound, Shortest path.

1. Introduction The traveling salesman problem (TSP) is a problem in discrete or combinatorial optimization. The problem statement is given a number of cities and the distances of traveling from any city to any other city to illustrate the shortest round-trip route that each city once then returns to starting city. Usually, the total distance is related to the total cost causing fuel consumption, but in real world problems it’s not true. The shortest route is not the route that is lowest cost in all cases. There are not only distance that implicated to cost but also quantitative and qualitative factors influencing these problems such as slope, congestion, condition of traffic, etc. Moreover, qualitative factors such as convenience of operations or satisfaction of operators is the important factor to be considerated. Matematical models and heuristics were developed to solving TSP broadly. Nearest Neighbor Heuristic is the original heuristic that everyone known, and then the other intuitive heuristics had emeraged in orderly such as Insertion Heuristics, Improving Solutions, etc. Most of these heuristics give the satisfaction answer, but they can not give the optimization as mathematical model. The solution of mathematical model is optimized but it bases on distance or cost. It’s not practical in operation. There are many important factors involving operation that should be considered for determining the best rout. The decision maker have to make a trade off among these factors in route selection problem. An analytical approach frequently selected for solving a

multicriteria decision problems is the Analytical Hierarchy Process(AHP), first introduced by Saaty[1]. In this study, the AHP is used for selection the best route from methodologies for TSP. Three heuristics, the Nearest Neighbor heuristic, the Cheapest Insertion heuristic, and Twoway Exchange Improvement heuristic are selected to illustrate the best route and one well-known mathematical model, Branch and Bound method is compared with them. 2. Methodology 2.1 The Nearest Neighbor Heuristic The salesman starts at any city in tour and then visits the city nearest to the starting city. From there he visits the nearest city that was not visited until all cities in tour are visited, and the salesman returns to the start. It’s can be represented that[3] Step1: Start at node n1 Step2: If all nodes have been visited (j=n) then exit Else select a node which is nearest to the current node with respect to the distance(Dij); Dij = Min{Dij | i ≤ j ≤ n} and repeat this step Dij = a distance from node i to j ; 1 ≤ i,j ≤ n ; 0 ≤ Dij ≤ ∞ ; i = 1,…, n ni = a node i in network n = total nodes in network

Proceedings of the International Conference on Computer and Industrial Management, ICIM, October 29-30, 2005, Bangkok, Thailand

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2.2 The Cheapest Insertion Heuristic An intuitive approach to the TSP is to start with a subtour, a tour on small subsets of nodes, and then extend this tour by inserting one of the remaining nodes between two neighboring nodes in the subtour until all nodes have been inserted. It’s can be represented that[3] Step1: Construct the initial subtour Sk Step2: If number of nodes in subtour = total nodes in network(k=n) then exit Else select the cheapest node which is the lowest increase length of the tour to insert to subtour; Sk = Min{Sk+1 | Sk+1=∑ijDij}; 1≤i