Fuzzy models and modeling beyond Artificial Intelligence László T. Kóczy Faculty of Engineering, Széchenyi István University, Győr, Hungary E-mail:
[email protected] Department of Telecommunications and Media Informatics, Budapest University of Technology and Economics, Budapest, Hungary E-mail:
[email protected] In a great variety of application fields where decision making is based on a very large number of factors and/or vague or imprecise conditions it is necessary to introduce some new models and modeling techniques which go beyond the more traditional approaches offered by symbolic Artificial Intelligence. The key problem here is the combination of imprecision and uncertainty with high complexity which cannot be managed by any algorithm with tractable computational complexity not even within the limits of polynomiality [1]. A large portion of decision problems within management engineering falls into this class. In this paper a fuzzy approach is proposed which goes beyond traditional Artificial Intelligence using fuzzy rule based inference systems, while it utilizes other techniques, evolutionary algorithms, neural networks and traditional optimization along with their respective combinations. We hope the proposed tools will be useful for solving some of the real life problems in this area. Fuzzy rule based models were first proposed by Zadeh in 1973 [2] and later practically implemented with some technical innovations by Mamdani and Assilian in 1975 [3]. If…then… rules contain an arbitrary number of variables xi with antecedent descriptors, formulated either by linguistic terms or directly by convex and normal fuzzy sets (fuzzy numbers) in the “If” part, and one or several output variables yi and their corresponding consequent terms/membership functions in the “then” part. An alternative model was proposed by Takagi and Sugeno in 1985 [4] where the consequent part was represented by y=f(x) type functions. Later Sugeno and his team proposed the basic idea of hierarchical fuzzy models where the meta-level contained rules with symbolic outputs referring to further sub-rule bases on the lower levels [5]. All these types of fuzzy rule based models might be interpreted as sophisticated and human friendly ways of defining (approximate) mappings from the input space to the output space, the rule bases being technically fuzzy relations of X×Y but representing virtual “fuzzy graphs” of extended (vague) functions in the state space. A great advantage of all these models compared to more traditional symbolic rule models is the inherent interpolation property of the fuzzy sets providing a cover of the input space, with the kernel values of the antecedents corresponding to the areas (or points) where supposedly exact information is available – typical points of the graph, and vaguer “grey” areas bridging the gaps between these characteristic values by the partially overlapping membership functions or linguistic terms. It was however necessary in all initial types of fuzzy models that the antecedents formed a mathematically or semantically full cover, without any gaps between the neighboring terms. The “yellow tomato problem” proposed by ourselves pointed out that often it was not really necessary to have a fuzzy cover, since “yellow tomatoes” could be interpreted as
something between “red tomatoes” and “green tomatoes”, with properties lying also in between, the former category describing “ripe tomatoes”, the latter “unripe tomatoes”, yellow tomatoes being thus “half ripe”. Linear fuzzy rule interpolation proposed in 1990 provided a new algorithm extending the methods of reasoning and control to sparse fuzzy covers where gaps between adjoining terms could be bridged with the help of this new approach, delivering the correct answer both in the case of linguistic rules like the “Case of the yellow tomato” and in more formal situations where membership functions were defined only by mathematical functions [6, 7, 8]. This idea was later extended to many non-linear interpolation methods, some of them providing very practical means to deal with real life systems. In 1993 and in a more advanced and practically applicable way in 1997 we proposed the extension of fuzzy interpolation to a philosophically completely new area, the sub-models of the hierarchical rule base itself [9, 10]. Interpolation of sub-models in different sub-spaces raised new mathematical problems. The solution was given by a projection based approach with local calculations and subsequent substitution into the meta-level rule base, replacing sub-model symbols by actual fuzzy sub-conclusions. After an overview of these types of fuzzy models the lecture will deal with the problem of model identification. Often human experts provide the approximate rules – like they did in the hierarchical fuzzy helicopter control application by Sugeno referred to above, but more often, there are only input-output data observed on a black box type system as the starting point to construct the fuzzy rule structure and the rules themselves. Sugeno and Yasukawa proposed in 1993 [11] the use of fuzzy c-means (FCM) clustering originally introduced by Bezdek [12] for identifying output clusters in the data base, from where rules could be constructed by best fitting trapezoidal membership functions applied on the input projections of the output clusters. This method has a limited applicability, partly because of the conditions of FCM clustering, and partly because of the need to have a well structured behavior from the black box under observation. In the last few years our group has introduced a large variety of identification methods for various types of fuzzy models. The FCM based approach could be essentially extended to hierarchical models as well, and was applied to a real life problem (petroleum reservoir characterization [13, 14]) successfully. Another possibility was the bacterial evolutionary algorithm [15], a nature inspired optimization technique. In these kind of methods one candidate solution for the problem is considered as a so-called individual. The advantage of these algorithms is their ability to solve and quasi-optimize problems with non-linear, highdimensional, multi-modal, and discontinuous character. It has been shown that evolutionary algorithms are efficient tools for solving non-linear, multi-objective and constrained optimizations. These algorithms have the ability to explore large admissible spaces, without demanding the use of derivatives of the objective functions, such as the gradient-based training methods. Their principles are based on the search for a population of solutions, which tuning is done using mechanisms similar to biological recombination. The use of Levenberg-Marquardt (LM) [16, 17] optimization for finding the rule parameters was borrowed from the field of neural networks. These latter approaches all had their advantages and disadvantages, partly concerning convergence speed and accuracy, partly locality and globality of the optimum. The most recent results [18] showed that the combination of bacterial evolutionary algorithm and LM, called
Bacterial Memetic Algorithm delivered very good results compared to the previous ones, with surprisingly good approximations even when using very simple fuzzy models. Research on extending this new method to more complicated fuzzy models is going on currently. We are not only focusing on control engineering, but we also have some other topics related to logistics and marketing based on the (bacterial) evolutionary algorithms. For designing and developing products/services it is vital to know the relevancy of the performance generated by each technical attribute and how they can increase customer satisfaction. Improving the parameters of technical attributes requires financial resources, and the budgets are generally limited. Thus the optimum target is to achieve maximum customer satisfaction within given financial limits. Kano’s quality model [19] classifies the relationships between customer satisfaction and attribute-level performance and indicates that some of the attributes have a non-linear relationship to satisfaction rather power-function should be used. For the customers’ subjective evaluation these relationships are not deterministic and are uncertain. We proposed a method for fuzzy extension of Kano’s model and presented numerical examples in which for the approximation of the optimum the bacterial approach is used [20]. The other problem we are dealing with is the Traveling Salesman Problem (TSP) [21]. The aim here is to find the cheapest way of visiting all elements in a given set of cities and returning to the starting point. In solutions presented in the literature costs of travel between nodes (cities) are based on Euclidean distances, the problem is symmetric and the costs are constant. In our research a novel construction and formulation of the TSP is presented in which the requirements and features of practical application in road transportation and supply chains are taken into consideration. We are approaching the TSP with the bacterial memetic algorithm, where effective local search heuristics are used within the evolutionary cycle. Computational results present the efficiency of the method. This problem is discussed in more details in our technical paper published in this very same proceedings [22]. Acknowledgment This paper was supported by a Széchenyi University Main Research Direction Grant 2009 and the National Scientific Research Fund Grants OTKA T048832 and K75711. Special acknowledgment to Antonio Ruano and Cristiano Cabrita (University of Algarve, Faro, Portugal) to Tom D. Gedeon (Australian National University, Canberra) and Alex Chong (formerly with Murdoch University, Perth, Australia) and to János Botzheim, Péter Földesi and László Gál (Széchenyi István University, Győr, Hungary). References [1] A. V. Aho, J. E. Hopcroft, J. D. Ullman. Data Structures and Algorithms, Addison-Wesley Series in Computer Science and Information Processing, 1983. [2] L. A. Zadeh. Outline of a new approach to the analysis of complex systems, IEEE Trans. Syst., Man, Cybern., vol. SMC-1, pp. 28-44, 1973. [3] E. H. Mamdani, S. Assilian. An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud., vol. 7, pp. 1-13, 1975.
[4] T. Takagi, M. Sugeno. Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116-132, 1985. [5] M. Sugeno, T. Murofushi, J. Nishio, H. Miwa. Helicopter control based on fuzzy logic. Presented at the Second Fuzzy Symposium on Fuzzy Systems and Their Applications to Human and Natural Systems, Tokyo - Ochanomizu, 1991. [6] L. T. Kóczy, K. Hirota. Fuzzy inference by compact rules, in Proceedings of the International Conference on Fuzzy Logic and Neural Networks, Iizuka, Fukuoka, pp. 307-310, 1990. [7] L. T. Kóczy, K. Hirota. Approximate reasoning by linear rule interpolation and general approximation, International Journal of Approximate Reasoning, 9, pp. 197-225, 1993. [8] L. T. Kóczy, K. Hirota. Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases, Information Sciences, 71, pp. 169-201, 1993. [9] L. T. Kóczy, K. Hirota. Interpolation in structured fuzzy rule bases, in Proceedings of the IEEE International Conference on Fuzzy Systems, FUZZ-IEEE’93, San Francisco, pp. 803-808, 1993. [10] L. T. Kóczy, K. Hirota. Size Reduction by Interpolation in Fuzzy Rule Bases, IEEE Trans. Syst., Man, Cybern., Part B 27, pp. 14-25, 1997. [11] M. Sugeno, T. Yasukawa. A fuzzy-logic-based approach to qualitative modelling, IEEE Transactions on Fuzzy Systems, vol. 1, no. 1, pp. 7-31, 1993. [12] J. C. Bezdek. Recognition with Fuzzy Objective Function Algorithms, New York: Plenum Press, 1981. [13] T. D. Gedeon, P. M. Wong, Y. Huang, C. Chan. Two Dimensional Fuzzy-Neural Interpolation for Spatial Data, in Proceedings Geoinformatics'97: Mapping the Future of the AsiaPacific, Taiwan, vol. 1, pp. 159-166, 1997. [14] T. D. Gedeon, K. W. Wong, P. M. Wong, Y. Huang. Spatial Interpolation Using Fuzzy Reasoning, Transactions on Geographic Information Systems, vol. 7, no. 1, pp. 55-66, 2003. [15] N. E. Nawa, T. Furuhashi. Fuzzy System Parameters Discovery by Bacterial Evolutionary Algorithm, IEEE Tr. Fuzzy Systems vol. 7, pp. 608-616, 1999. [16] K. Levenberg. A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math., vol. 2., no. 2., pp. 164-168, 1944. [17] D. Marquardt. An Algorithm for Least-Squares Estimation of Nonlinear Parameters, J. Soc. Indust. Appl. Math., vol. 11, pp. 431-441, 1963. [18] J. Botzheim, C. Cabrita, L. T. Kóczy, A. E. Ruano. Fuzzy rule extraction by bacterial memetic algorithm, in Proceedings of the 11th World Congress of International Fuzzy Systems Association, IFSA 2005, Beijing, China, pp. 1563-1568, 2005. [19] N. Kano, N. Seraku, F. Takahashi, S. Tsuji. Attractive quality and must-be quality. Hinshitsu. The Journal of Japanese Society for Quality Control, pp. 39-48, 1984. [20] P. Földesi, L. T. Kóczy, J. Botzheim. Fuzzy solution for non-linear quality models, in Proceedings of the 12th International Conference on Intelligent Engineering Systems, INES 2008, Miami, Florida, pp. 269–275, 2008. [21] D. L. Applegate, R. E. Bixby, V. Chvátal, W. J. Cook. The Traveling Salesman Problem, A Computational Study, Princeton University Press, Princeton and Oxford, 2006. [22] P. Földesi, L. T. Kóczy, J. Botzheim, M. Farkas. Eugenic Bacterial Memetic Algorithm for Fuzzy Road Transport Traveling Salesman Problem, published in this very same proceedings.
