Chapter 6 TRAVELLING SALESMAN PROBLEM

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1 98 hapter 6 TRVLLING SLSMN PROLM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pair wise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem by Menger (1930) and is one of the most intensively studied problems in optimization. However it was unnoticed till Menger (1994) published a book where he narrated the foundation of mathematical problem for the travelling salesman problem. It is used as a benchmark for many optimization methods. ven though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. If a slight modification is made in the problem, it appears as a sub-problem in many areas, such as genome sequencing. In these applications, the concept city represents, for example, customers, soldering points, or N fragments, and the concept distance represents traveling times or cost, or a similarity measure between N fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. In the theory of computational complexity, the decision version of TSP belongs to the class of NP-complete problems. Thus, it is assumed that there is no efficient algorithm for solving TSP problems. In other words, it is likely that the worst case running time for any algorithm for TSP increases exponentially with the number of cities, so

2 99 even some instances with only hundreds of cities will take many PU years to solve exactly. The origins of the traveling salesman problem are unclear. handbook for traveling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment. Hamilton (1800) and Kirkman (1800) expressed the concept of Mathematical problems related to the travelling salesman problem. The general form of the TSP appears to have been first studied by mathematicians notably by Menger (1930). Further Menger (1930) also defines the problem related with salesman ship based on brute-force algorithm, and observes the non-optimality of the nearest neighbor heuristic. However Whitney (1930) introduced the name travelling salesman problem. uring the period 1950 to 1960, the travelling salesman problem started getting popularity in scientific circle is especially in urope and the US. Many researchers like antzig, Fulkerson and Johnson (1954) at the RN orporation in Santa Monica expressed the problem as an integer linear program and developed the cutting plane method for its solution. With these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. In the following decades, the problem was studied by many researchers from mathematics, science, chemistry, physics, and other sciences. Karp (1972) showed that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a scientific explanation for the apparent computational difficulty of finding optimal tours. 6.2 pplication of New lternate Method of ssignment Problem in TSP We have so far already discussed the algorithm and examples for solving an assignment problem using a new alternate method in hapter 5. Now in this chapter we discuss how the new alternate method for solving an assignment

3 100 problem can be applied for Travelling Salesman Problem. For this we have considered an example related with travelling salesman problem and explain in detail how to find optimal solution using new alternate method of assignment problem. xample salesman has to visit five cities.,, and. The distances (in hundred kilometers) between the five cities are shown in Table 6.1. From ity Table 6.1 To city If the salesman starts from city and has to come back to city, which route should he select so that total distance traveled become minimum? Solution onsider the effective matrix. This is shown in Table 6.2. From ity Table 6.2 To city In this matrix first, we will take first row which is referred a city. We select that column (assignment) for which it contains minimum distance. For this example, incase of first row, column (assignment) has the minimum value. In the similar way, we select all the rows and find the minimum value for the respective columns. These are given in Table 6.3.

4 101 olumn 1(ity) Table 6.3 olum 2(ssignment) In this table, we observed that assignment occur only once with city. That is city is unique for city and hence we assign city to. This is shown in Table 6.4. Table However, for other job assignment occur more than once. Hence they are not unique. So how other job will be assigned further we discuss below. Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.5. Table 6.5 olumn 1(ity) olum 2(ssignment) Since assignment occur with city and. Hence first we take the difference between the value of and next minimum value (here tie is happens). Here

5 102 maximum difference is 2 for and hence we assign to city. This is shown in Table 6.6. Table Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.7. olumn 1(ity) Table 6.7 olum 2(ssignment), Since assignment occur with city, and. Hence we take the difference between the value of and next minimum value, here the maximum difference is 3 for Job. nd hence we assign to city. This is shown in Table 6.8. Table Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.9. Table 6.9 olumn 1(ity) olumn 2(ssignment)

6 103 Here, we cannot assign to city. Therefore we only assign to city. Then obviously, we have no other choice rather to assign for ity. Finally, we can assign all the cities along with distance which is shown in Table Table 6.10 olumn 1(ity) olum 2(ssignment) istance Total 26 This solution is happened to be same as that of Hungarian method. Hence we can say that the minimum value is still 26 in both the methods. So this solution is optimal. However our method seems to be very simple, easy and takes very few steps in solving the method. 6.3 onclusion In this chapter, we have applied new alternate method of an assignment problem for solving Travelling salesman problem where it is shown that this method also gives optimal solution. Moreover the optimal solution obtained using this method is same as that of optimal solution obtained by Hungarian method. So we conclude that the Hungarian method and our method give same optimal solution. However the technique for solving Travelling Salesman problem using our method is more simple and easy as it takes few steps for the optimal solution.

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