[Reddy, 2(10): October, 2013] ISSN: 2277-9655 Impact Factor: 1.852 http: // www.ijesrt.com (C) International Journal of Engineering Sciences & Research Technology [2634-2645] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY An Alternate Travelling Salesman Problem P. Madhu Mohan Reddy *1 , E. Sudhakara 1 , S. Sreenadh 1 , S. V. K. Varma 1 *1 Department of Mathematics, Sri Venkateswara University, Tirupati-517 502, Andhra Pradesh, India mmrphdsv@gmail.com Abstract We consider Lexi-Search Approach using Pattern Recognition Technique for a Travelling Sales Man Problem (TSP) in which he wants to visit m cities, where m is even. Let N be the set of n stations defined as N= {1, 2, 3, 4…n} and N1UN2=N. The city ‘1’ taken as the home city and it is in N1. He has to starts from head quarter city {1} which is in N1 from there he visits a city in N2. In this way the salesman visits m cities alternatively and m ≤ n. D (i, j) be the distance or cost matrix. A salesman starts for his tour from a home city (say 1) and come back to it after completing the all the m cities. There is a restriction that he must visit the N1, N2 groups alternatively. An exact algorithm is proposed for this TSP. The algorithm solves the problem on identify the key patterns which optimize the objective of the cost/distance. Hence the objective of the problem is to find a tour with minimum total distance while completing all the m cities alternatively by above considerations. Keywords: Pattern Recognition, Lexi-Search Approach, Travelling Sales Man Problem. Introduction The travelling salesman problem is one of the oldest combinatorial programming problems (Flood[2]- 1956 and Croes[3]-1958) and can be stated as follows. There are n cities and the distance between any ordered pair of cities is known. Starting from one of the cities a salesman is to visit the other cities only once and return to the starting city. The objective is to find a tour in such a way that the total length of the tour is minimum. The Traveling Salesman Problem is one of the most intensively studied problems in computational mathematics and is a kind of mathematical puzzle with a long enough history Dantzig, Fulkerson and Johnson- 1954 and Tutte-1940. Many solution procedures have been developed such as Flood [2]-1956, Hardgrave & Nembanser[12]-1962 and Little[13] et al-1963 for the travelling salesman problem. One more generalization which is called the “Truncated Travelling Salesman Problem” (Sundara Murthy [9] -1979). There are many algorithms for usual one man travelling salesman problem developed by researchers from time to time. But the problem has not received much attention in its restricted context. However, literature which is available with regard to the TSP with variations is discussed (Das [7] - 1976, 78, Jaillet [8] - 1985, Pandit[5] - 1961, 63, 64, 65, Murthy & Srivastava et al [10] 1969. The Generalized Travelling Salesman Problem was first addressed by Srivastava[10] et al (1970).Time Dependent Travelling Salesman Problem Was also attempted by Bhavani[11] – 1997, Sobhan Babu[6] – 2000, Naganna[4] – 2001 and Balakrishna[1] – 2006 and the simple combinatorial structure of the TDTSP were taken into account. With sufficient ingenuity, one can also formulate problems of this type as a non trivial Integer programming problem, as a zero– one programming problem more generally (Balas et.al. – 1991 and Glover[14] – 1965). Combinatorial structure may not be clear in some problems because of the NP– Hard nature, but one can always identify the relevant Cartesian product and the feasibility criterion Problem Description In this paper we consider Pattern Recognition Based Lexi-Search Approach for A Travelling Sales Man Problem (TSP) in which he wants to visit m cities. The objective is to find a Tour such that total length of the tour minimum. An exact algorithm is proposed for this TSP. The algorithm solves the problem on identify the key patterns which optimize the objective of the cost/distance. The algorithm calculates the solution incrementally for deferent cities have visited by the salesman and the best combination is taken as the solution. Let N be the set of n stations defined as N= {1, 2, 3, 4…n} and N 1 UN 2 =N. The city ‘1’ taken as the home city and in N 1 . He has to starts from head quarter city {1} which is in N 1 from there he visits a city in N 2. In this way the salesman visits m cities alternatively and m ≤ n. D be the distance or cost matrix associated with the node pair of cities i and j. A salesman starts for his tour from a home city (say 1) and come back to it after