1. Meta Heuristics Team Project
Students:
Dukai
Liwenfei

2. Introduction
The traveling salesman problem (TSP) is a typical example of a vary hard combinational optimization problem. The problem is to find the shortest tour that passed through each vertex in a give graph exactly once.
We try to develop two solutions to Traveling Salesman Problem (TSP) using Simulated annealing (SA) and Genetic algorithm (GA), respectively. In the Traveling Salesman Problem, the goal is to find the shortest distance between N different cities.
We will generate our best solutions by using the C++ Language.

3. Genetic Algorithm (GA)
In the traveling salesman problem we wish to find a tour of all nodes in a weighted graph so that the total weight is minimized. A genetic algorithms can be used to find a solution is much less time. There are a couple of steps to solving the traveling problem using a GA and we will describing them step by step.

4. Genetic Algorithm (GA) process
As a problem in Genetic algorithm, the traveling salesman problem is handled as follow steps:
First, create a group many random tours in what is called a population. This algorithm uses a greedy initial population that gives preference to linking cities that are close to each other.
Second, pick two of the shorter tours parents in the population and combine them to make two new child tours. Hopefully, these children tour will be better than either parents.

5. Genetic Algorithm (GA) process
Third, a small percentage of the time, the child tours are mutated . This is done to prevent all tours in the population from looking identical.
Fourth, the new child tours are inserted into the population replacing two of the longer tours. The size of the population remains the same.
Fifth, new children tours are repeatedly created until the desired goal is reached.

6. GA Analyzing and Programming
Firstly, by analyzing the general structure of TSP problem with Genetic algorithm (GA), programming with C++ language will be considered and produced.
Secondly, according to the C++ programming , the optimal solution of shortest path could be solved.
Additionally, using graph is also a helpful method in making algorithm and programming.

7. Simulated Annealing (SA)
This program uses the Simulated annealing algorithm to solve a form of the traveling salesman problem, which is to find the shortest (or a nearly shortest) path connecting a set of sites, such as cities or delivery points.

8. Simulated Annealing(SA) Process
As a problem in Simulated annealing, the traveling salesman problem is handled as follow steps:
Configuration: The cities are numbered i = N and each has coordinates (xi, yi). A configuration is a permutation of the number N, interpreted as the order in which the cities are visited.
Rearrangements: The moves consist of two types:
(a) A section of path is removed and then replaced with the same cities running in the opposite order; or

9. Simulated Annealing(SA) Process
(b) A section of path is removed. And then replaced in between two cities on another, randomly chosen, part of the path.
Objective Function. In the simplest form of the problem, E is taken just as the total length of journey.
Annealing schedule. This requires experimentation. We generate some random rearrangements, and use them to determine the range of values of total length that will be encountered from move to move.

10. SA Analyzing and Programming
Firstly, by analyzing the general structure of TSP problem with Simulated annealing (SA), programming with C++ language will be considered and produced.
Secondly, according to the C++ programming , the optimal solution of shortest path could be solved.
Additionally, using graph is also a helpful method in making algorithm and programming.

11. Reference
[1]. S. Kirkpatrick, C.D. Gelatt and M.P. Vechi, Optimization by simulated annealing, Science 220 (May 13, 1983).
[2]. F. Glover, Tabu search, Center for Applied Artificial Intelligence, Graduate School of Business, University of Colorado, Boulder, 1988.
[3]. E.L. Lawler, J.K. Lenstra and A.H.G. Rimnooy Kan, eds.,The Traveling Salesman Problem (North- Holland, 1985).
[4]. R.L. Karg and G.L. Thompson, A heuristic approach to solving travelling-salesman problems, Management Science 10 (1964).

12. Reference
[5]. G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, Solution of a large-scale travelling-salesman problem, Operations Research 2 (1954).
[6]. Byrant A. Julstrom, Very greedy crossover in a genetic algorithm for the traveling salesman problem, Proceedings of the 1995 ACM symposium on Applied computing, p , February 26-28, 1995, Nashville, Tennessee, United States.
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13. Reference
[8]. S. Lin and B.W. Kernighan, An effective heuristic algorithm for the traveling salesman problem, Oper. Res. 21 (1973).
[9]. R.G. Parker and R.L. Rardin, The traveling salesman problem: an update of research, Naval Research Logistics Quarterly 30 (1983).
[10]. M. Padberg and G. Rinaldi, Optimization of a 532-city symmetric traveling salesman problem by branch and cut, Operations Research Letters 6, no. 1 (1987).

14. Reference
[11]. Freisleben.B. and Merz.P , A genetic local search algorithm for solving symmetric and asymmetric traveling salesman problems, IEEE,1996.
[12]. M. Held and R.M. Karp, The travelling salesman problem and minimum spanning trees, part I, Operations Research 18 (1970) 1138–1162; Part II, Mathematical Programming 1 (1971).
[13]. Dorigo, M., A cooperative learning approach to the traveling salesman problem; Gambardella, L.M. Evolutionary Computation, IEEE, 1997

15. Reference
[14]. M. Malek, M. Guruswamy, H. Owens and M. Pandya, A hybrid algorithm technique, Technical Report, Dept. of Computer Sciences, The University of Texas at Austin, TR-89-06, 1989.
[15]. See, e.g., Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by V. Privman, (World Scientific, Singapore, 1990).
[16]. Kengo Katayama , Hiroyuki Narihisa, A new iterated local search algorithm using genetic crossover for the traveling salesman problem, Proceedings of the 1999 ACM symposium on Applied computing, 1999, San Antonio, Texas, United States.