The real life problems deal with imperfectly specified knowledge and some degree of imprecision, uncertainty or inconsistency is embedded in the problem specification. The well-founded theory of fuzzy sets is a special way to model the uncertainty. The rules in a fuzzy model contain a set of propositions, each of which restricts a fuzzy variable to a single fuzzy value by means of the predicate equivalency. That way, each rule covers a single fuzzy region of the fuzzy grid. The proposed system of this thesis extends this structure to provide more general fuzzy rules, covering the input space as much as possible. In order to do this, new predicates are considered and a Max-Min Ant System is proposed to learn such fuzzy rules. Ant system is a general purpose algorithm inspired by the study of behavior of ant colonies. It is based on cooperative search paradigm that is applicable to the solution of combinatorial optimization problem. In this thesis we consider the combinatorial optimization issue of travelling salesman problem (TSP) which evaluates more generic Fuzzy rules provided by Max-Min Ant System (MMAS). The existing ant colony system (ACS) was a distributed algorithm applied to the travelling salesman problem (TSP). In ACS, a set of cooperating agents called ants cooperate to find good solutions for TSPs (but here, Ants search their path randomly). Ants cooperate using an indirect form of communication mediated by pheromone they deposit on the edges of the TSP problem in symmetric instances. However most of the TSP issues carry both symmetric and asymmetric instances.

1. S. Abe and M. S. Lan, “Fuzzy rules extraction directly from numerical data for function approximation,” IEEE Trans. Syst. Man Cybern., vol. 25. no. 1. pp. 119-129, 2005.
2. J.Casillas, O. Cordon, and F. Herrera, “COR: A methodology to improve ad hoc data-driven linguistic rule learning methods by inducing cooperation among rules”, IEEE Trans. Syst. Man Cybern. Part B-Cyber., vol. 32, no. 4, pp. 526-537, 2002.
3. J. L. Castro, J. J. Castro-Schez, and J. M. Zurita, “Learning maximal structure rules in fuzzy logic for knowledge acquisition in expert systems”, Fuzzy Sets Syst., vol. 101, pp. 331-342, 1999.
4. M. Dorigo, A. Colorni, and V. Maniezzo, “The ant system: Optimization by a colony of cooperating agents”, IEEE Trans. Syst. Man Cyber. B, vol. 26. no, pp. 29-41, 1996.
5. M.Dorigo and L. M. Gambardella, “Ant colony systems: A cooperative learning approach to the travelling salesman problem”, IEEE Trans. Evol. Comput., vol. 1. pp. 53-66, 1997.
6. M. Dorigo and T. Stuzle, Ant Colony Optimization. MIT Press, 2004.
7. Freisleben, B. Merz, “A genetic local search algorithm for solving symmetric and asymmetric travelling salesman problems”, May 1996. pp. 616—621
8. H. Ishibuchi, K. Nozaki, N. Yamamoto, and H. Tanaka, “Selecting fuzzy if-then rules for classification problems using genetic algorithms”, IEEE Trans. Fuzzy Syst., vol. 3, no. 3, pp. 260-270, 2005.
9. Lin, Shen, Kernighan, B. W. "An Effective Heuristic Algorithm for the Travelling-Salesman Problem", Vol. 21, No. 2. (1973), pp. 498-516.
10. Pablo Carmona, Juan Luis Castro, “Using Ant Colony Optimization for Learning Maximal Structure Fuzzy Rules”, 2005 IEEE International Conference on Euzzy Systems.
11. A. Piegat, “Fuzzy Modeling and Control”, Heidelberg, Germany: Physical-Verlag, 2001.
12. T. Stutzle and H. H. Hoos, “MAX-MIN ant system”, Futur Gener. Compu. Syst., Vol. 16. no. 8. pp. 889-914, 2000.
13. L.X. Wang and J. M. Mendel, “Generating fuzzy rules by learning from examples”, IEEE Trans. Syst. Man Cyber., vol. 22, no. 6, pp. 1415-1427, 1992.

MMAS Algorithm travelling salesman problem (TSP) ant colony system (ACS)