VIEWS: 150 PAGES: 8 CATEGORY: Emerging Technologies POSTED ON: 5/16/2012 Public Domain
(IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 Optimization of Membership Functions Based on Ant Colony Algorithm Parvinder Kaur Shakti Kumar Amarpartap Singh Department of Electronics & Computational Intelligence Department of Electronics & Communications Laboratory, Communications SLIET, Longowal, Punjab, INDIA IST Kalawad, Haryana, INDIA SLIET, Longowal, Punjab, INDIA parvinderbhalla@gmail.com shaktik@gmail.com amarpartapsingh@yahoo.com Abstract—In fuzzy model identification membership function both antecedent and consequent parts [3]. Very recently, in tuning plays an important role towards error minimization. This fact in parallel with this work, fuzzy neural networks with paper proposes a ACO based strategy for membership function evolving structure have been developed [6]. Various tuning. The algorithm was implemented on a standard rapid orthogonal transformation methods [7]-[10] have been battery charger data set. The simulation results were compared proposed for selecting important fuzzy rules from a given rule with other three algorithms available in the literature. It was base. Another rule base optimization method through the observed that the proposed algorithm outperforms the other exhaustive search techniques was suggested by Arun et al. in three algorithms on mean squared error (MSE) performance [11, 12]. K.Nozaki et.al [13] proposed a method for basis. automatically generating fuzzy if-then rules from numerical Keywords—Ant Colony Algorithm; Fuzzy Membership data. Wang and Mendel [14] proposed a new approach to function. combine the fuzzy rule bases generated from the numerical data and the linguistic fuzzy rules. I. INTRODUCTION Genetic algorithms (GAs) have also been used [15, 16] for A mathematical model is constructed by analyzing input- optimizing fuzzy membership functions and fuzzy rule base. output measurements from the system. Very often, there exists H.S. Hwang [17] and S.J. Kang et al. [18] proposed an another important information source in the form of approach for design of the optimal rule base using knowledge from human experts, known as linguistic evolutionary programming. Evolutionary programming information. The linguistic information provides qualitative simultaneously evolves the structure and the parameter of the instructions and descriptions about the system and is fuzzy rule base. The particle swarm optimization (PSO) especially useful when the input-output measurements are algorithm, like other evolutionary algorithms, is a stochastic difficult to obtain. The ability to deal simultaneously both with algorithm that uses a population of potential solution (called linguistic information and numerical information in a particles) to probe the search space. Arun Khosla et al. [19], systematic and efficient manner is one of the most important applied the PSO algorithm for identification of optimized advantages of fuzzy models [1, 2]. The principles of fuzzy fuzzy models from the available data. modeling were outlined by Zadeh in 1965 when he gave the Ant colony optimization (ACO) [20] is a metaheuristic that concept of grade of membership and published his seminal belongs to the group of swarm intelligence based techniques. paper on fuzzy sets that lead to the birth of fuzzy logic In a number of experiments presented in [20]-[22] Dorigo et technology [1]. In the beginning the concepts of fuzzy sets and al. illustrated the complex behaviour of ant colonies. The fuzzy logic encountered criticism from technical and scientific application of ant-inspired algorithms to rule induction is a community. However, a large number of successful industrial relatively recent area of research, but is gaining increasing fuzzy logic applications generated an increased interest in interest. A first attempt to apply ACO to fuzzy modeling was fuzzy logic. There is hardly any field that has not been made by Casillas et al. in [23]. However, the ACO algorithm influenced with the emergence of fuzzy logic. is not used for generating fuzzy rules, but for assigning rule A typical tendency until early 1990s was to rely on existing conclusions. In their problem graph the fixed number of nodes expert knowledge and to just tune fuzzy sets’ parameters using are fuzzy rule antecedents found by a deterministic method gradient-based methods or genetic algorithms (GAs) [3]. In from the training set. An ant goes round the problem graph, the late 1990s, so-called data-driven or rule/knowledge visiting each and every node in turn and probabilistically extraction methods were introduced. The attempt was to assigns a rule conclusion to each. The recent applications of identify the model structure and parameters based primarily on ACO to fuzzy modeling are [24]-[30]. data [4, 5]. The techniques used are mainly clustering, linear Although various techniques [31]-[44] have been suggested least squares and/or non-linear optimization for fine-tuning of for fuzzy model identification, yet there is no uniformly 38 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 accepted formulation, which carries out the modeling functions and parameters of consequent part of rules. The effectively and efficiently. There are no sound guidelines for parameter identification is basically an optimization problem the choice of membership functions. More extensive empirical with an objective function. investigation is needed in this area before a general conclusion Model validation involves testing the model based on some can be made about membership functions. performance criterion. In this paper a new technique based on ACO for dealing with the problem of membership function optimization is III. ANT COLONY OPTIMIZATION presented. With this aim the paper is set up as follows. In ALGORITHM Section 2 a brief introduction to fuzzy systems modeling is Ants as individuals are unsophisticated living beings. presented. Section 3 provides a brief account of ACO However, their collective behavior exhibits intelligent algorithm. Optimization of membership functions through behavior. It is this foraging behaviour that has so far inspired ACO is presented in Section 4. Section 5 represents the application of optimization algorithm called Ant experimental results considering battery charger problem. Colony Optimization to rule induction [20, 21]. Many Finally, conclusions are drawn in section 6. experiments [22] with ant colonies have been conducted in order to determine how ants are able to find the shortest II. FUZZY SYSTEMS MODELING path between their nest and a food source. It is believed that Fuzzy modeling is the task of identifying the parameters of this ability arises from their stigmergic interaction with each fuzzy inference system so as to achieve a desired behaviour. other. They communicate by leaving behind them a chemical The fuzzy model identification process involves the question substance called a pheromone, effectively changing the of providing a methodology for development i.e. a set of common environment. In making decisions about which path techniques for obtaining the fuzzy model from information to take, ants are guided by the amount of pheromone laid on and knowledge about the system. a path – the greater the amount of pheromone on a path the The problem of fuzzy model identification includes the higher is the probability that an individual ant will choose following issues [2-4]: that path. Ant Colony Optimization (ACO) is a paradigm for Selecting the type of fuzzy model. designing metaheuristic algorithms for combinatorial Selecting input and output variables for the model. optimization problems. Choosing the structure of membership functions. Determining the number of fuzzy rules. A Simple-ACO (S-ACO) algorithm for the shortest path Identifying the parameters of antecedent and consequent problem membership functions. S-ACO is a didactic tool to explain the basic mechanisms Identifying the consequent parameters of rules. underlying ACO algorithms. This algorithm adapts the real ant’s behavior to the solution of shortest path problems on Defining some performance criteria for evaluating fuzzy graphs. Following is the details on how to implement S-ACO models. on shortest path problem [21]. These issues can be grouped into three subproblems: structure identification, parameter estimation and model validation as Nomenclature: shown in figure 1. If the performance of the model obtained is Lk = Length of ant k’s path not satisfactory, the model structure is modified and the = evaporation constant, 0,1 parameters are re-estimated till the performance is satisfactory [2, 3]. = increment in pheromone quantity = 1 k Lk Linguistic N ik = neighborhood of ant k when at node i. Information Satisfied Structure Parameter Model Identification Estimation Validation = a constant = 2 Numerical Information Step1: Ants’ Path-Searching Behavior Each ant builds, starting from the source node, a solution to Not Satisfied the problem by applying a step-by-step decision policy. At each node, local information stored on the node itself or on its Figure 1. Fuzzy Model Identification Process outgoing arcs is read (sensed) by the ant and used in a Structure identification involves finding the important input stochastic way to decide which node to move to next. At the variables from all possible input variables, specifying beginning of the search process, a constant amount of membership functions, partitioning the input space and pheromone (e.g., ij 1 ) is assigned to all the arcs. When knowledge representation in the form of fuzzy if-then rules. located at a node i an ant k uses the pheromone trails ij to Parameter estimation involves identifying the best values for a set of model parameters. There are two types of parameters in compute the probability of choosing j as next node: a fuzzy model: parameters of antecedent membership 39 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 ij membership functions, rule-base and hence the corresponding system behaviour. ACO algorithms like other evolutionary pij lN k il , if j N ik ; k (3) algorithms have the capability to find optimal or near optimal solution in a given complex search space and can be used to i if j N i k 0, modify /learn the parameters of fuzzy model. Evolutionary In S-ACO the neighborhood of a node i contains all the algorithms offer a number of advantages over other search nodes directly connected to node i in the graph, except for the methods as they integrate elements of directed and stochastic predecessor of node i. In this way the ants avoid returning to search. These algorithms do not require any knowledge about the same node they visited immediately before node i. An ant the characteristics of the search space. Moreover, due to repeatedly hops from node to node using this decision policy parallel nature of the evolutionary algorithms, the possibility until it eventually reaches the destination node. Due to to reach a global minimum (or maximum) is high. differences among the ants’ paths, the time step at which ants The application of ACO for membership functions reach the destination node may differ from ant to ant. optimization involves a number of important considerations. The first step in applying such an algorithm is to completely Step2: Path Retracing and Pheromone Update encode a fuzzy system into a weighted graph. The next When ant k reaches the destination node, the ant switches important step is to define an appropriate objective function. from the forward mode to the backward mode and then The objective function is supposed to represent the quality of retraces step by step the same path backward to the source solution and act as interface between optimization algorithm node. An additional feature is that, before starting the return and the problem under consideration. Mean Square Error trip, an ant eliminates the loops it has built while searching for (MSE), as defined in (6), has been used for rating the quality its destination node. During its return travel to the source the of fuzzy model. The ideal value of MSE would be zero. ant k deposits an amount of pheromone on arcs it has k N 2 yk ~k visited. In particular, if ant k is in the backward mode and it 1 MSE = y (6) traverses the arc (i, j), it changes the pheromone value ij as N k 1 follows: where, ij ij k (4) yk = Actual output as available in data set ~k = Computed output of the model y By this rule an ant using the arc connecting node i to node j increases the probability that forthcoming ants will use the N = number of data points taken for model validation same arc in the future. The value of can be constant or k function of the path length-the shorter the path the more For the purpose of encoding, consider a multi-input single- pheromone is deposited by an ant. output system with n number of inputs with labels x1, x2,……………, xn and the number of fuzzy sets for these inputs are Step3: Pheromone Trail Evaporation m1, m2,……………., mn respectively and the output variable is In the last step, for each edge in the graph, evaporate represented through t number of fuzzy sets. Our encoding is pheromone trails with exponential speed. Pheromone trail based on the following assumptions: evaporation can be seen as an exploration mechanism that i) Fixed number of triangular membership functions are avoids quick convergence of all the ants towards a sub optimal used for both input and output variables and placed path. In S-ACO, pheromone trails are evaporated by applying symmetrically over corresponding universes of discourse. the following equation to all the arcs: The universe of discourse or simply universe is the ij 1 ij (5) working range of variable. ii) First and last membership functions of each input and output variable are represented with z-type and sigma- Step4: Termination Condition type membership functions respectively. The program stops if at least one of the following ii) Complete rule-base is considered, where all possible termination conditions applies: combinations of input membership functions of all the 1.) if end of edge is the terminal node; input variables are considered for rule formulation. 2.) a maximum number of algorithm iteration has been iii) Overlapping between the adjacent membership functions reached. for all the variables is ensured through some predefined constraints. IV. OPTIMIZATION OF MEMBERSHIP a) Encoding Mechanism for Tuning of the Fuzzy Membership FUNCTIONS THROUGH ACO Functions The fuzzy model identification can be formulated as a In fuzzy model identification the foremost task is parameter search and optimization problem in high-dimensional space, estimation of antecedent part of the model, which consists of where each point corresponds to a fuzzy system i.e. represents determination of the input variables, centers and spreads of the 40 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 membership functions. In many cases, the parameters Ei= Ei - (Ei – Ei-1) * wk associated with fuzzy membership functions are defined in an If (i = 1) ,then arbitrary manner. Given a performance measure, the selection Ei= Ei - (Ei – xmin) * wk of membership function parameters alters the behavior of the controller. Naturally, it is appropriate to use those parameters The above equation makes each membership function move to that lead to optimum performance. the left. ACO will be used to find the optimum values of fuzzy A random number is generated to move membership functions membership function parameters. This is achieved by left or right. evaluating a performance measure while tuning or altering these parameters. In general for input variable # n Let’s assume that a variable is represented by three fuzzy Ei= Ei + (Ei+1 – Ei) * wk sets as in fig.2. The vertices are indicated by Ei’s, where E1 If (i = mn) ,then (i=1) represent vertex of first fuzzy set and so on. Ei= Ei + (xmax – Ei) * wk E1 E2 E3 where i=1,2…… mn and Ei= Ei - (Ei – Ei-1) * wk If (i = 1) ,then Ei= Ei - (Ei – xmin) * wk ACO Representation: In order to find the optimal values for fuzzy membership functions using ACO, first encoded the above problem into a xmin xmax weighted graph as shown in fig.3. Parameters to be modified Input Variable # n Figure 2. Representation of a variable with 3 membership functions with overlapping between the adjacent membership functions Ei (i=1) Ei (i=2) Ei (i= mn -1) Ei (i= mn) Then the constraints to ensure the overlap between the w1 w2 adjacent membership functions for all the input variables for w3 the Sugeno fuzzy model can be represented as below: ...... xmin ≤ E1< E2< E3<….< Em1 ≤ xmax where m1, m2,……………., mn represents number of fuzzy sets for w5 n input variables and xmin and xmax are the minimum and maximum values of the variable respectively. Figure 3. Representation of membership functions in Ant’s Graph For the adjustment of membership functions the following Each fuzzy set represents one graph. For each fuzzy set we equations are defined: have different parallel paths which will move each membership function to the left or right depending on wk. The Input Variable #1 value of the parameters of membership function has to be chosen in such a way so as to minimize error according to Ei= Ei + (Ei+1 – Ei) * wk expression (9). If (i = m1) ,then Ei= Ei + (xmax – Ei) * wk Problem Formulation: Figure 4 represent a Sugeno type fuzzy system. It is clear where i=1,2…… m1, k=1,2………etc. from fig. that such systems consist of 4 major modules i.e. fuzzifier, rule composition module (fuzzy ―MIN‖ operators), The above equation makes each membership function move to implication module (multipliers in this case), and the right. Here wk decides the percentage of movement. defuzzification module. 41 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 W1 C1 Any minimization technique may not be applicable if the problem is very complex. We apply Simple Ant Colony Z MIN 4 MUL optimization S-ACO algorithm to evaluate rule base. MIN 4 wi ci Tr MUL Crisp i MIN 3 output V. APPLICATION EXAMPLE: BATTERY MUL CHARGER S 2 MIN MUL The suggested approach has been applied for identification wi Z MIN 1 MUL of fuzzy model for the rapid Nickel-Cadmium (Ni-Cd) battery S MIN MUL charger [45]. The main objective of development of this charger was to charge the batteries as quickly as possible but Fuzzifier Composi- Implica- tion tion without doing any damage to them. Input-output data consisting of 561 points, obtained through experimentation is 0.1 W6 available at http://www.research.4t.com. For this charger, the two input variables used to control the charging rate (Ct) are Figure 4: Sugeno type Fuzzy System absolute temperature of the batteries (T) and its temperature gradient (dT/dt). Charging rates are expressed as multiple of The overall computed output, in the case of a Sugeno type rated capacity of the battery, e.g. C/10 charging rate for a system, can be written as follows: battery of C=500 mAh is 50 mA [46]. The input and output variables identified for rapid Ni-Cd battery charger along with Computed output = i(Wi * Ci) / Wi (7) their universes of discourse are listed in Table 1. The number of fuzzy rules can be defined as below: Table 1 n m Input and Output variables for rapid Ni-Cd battery charger alongwith their R= i universes of discourse i 1 But these R rules are due to combinations of membership INPUT VARIABLES MINIMUM MAXIMUM functions of various inputs and these are incomplete as we VALUE VALUE could have knowledge only about antecedent part and Temperature (T)[0C] 0 50 consequents are yet unknown. Because for any set of inputs, Temperature Gradient 0 1 Wi are easily computed by fuzzifier and rule composing (dT/dt)[0C/sec] modules, the right hand side of output expression (7) can be evaluated if we could choose the proper values for Cis. OUTPUT VARIABLE For a given data set of a system, W is are known. Find the Charging Rate (Ct)[A] 0 8C appropriate values of Ci such that the difference between the computed output and the actual output as given in data is minimum. The block diagram for the system to be identified is given in figure 5. Ocomputed = W1* C1 + W2* C2 + ………+ WR* Cj W1 + W2 + ………+ WR (8) We compare this computed output with actual output as given in data set and find the error. Let the error be defined as follows: Error E = Actual output (as given in data set) – Computed output (as given in equation 8). Figure 5: Battery Charger Fuzzy Model The Sugeno type model for battery charger with two inputs Now the whole problem of rule base generation boils down and single output variable is shown in figure 6. Let us assume to a minimization problem as stated below: that the temperature with the universe of discourse ranging Minimize objective function E from 0-50 degree centigrade has been partitioned into 3 fuzzy E = OActual – OComputed sets namely temperature low, med (medium), and temperature Subject to the constraint that Ci {specified set of high. The temperature gradient is partitioned into two fuzzy consequents}. (9) sets (membership functions) namely low and high as shown in 42 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 figure 7. Initially set the parameters of membership functions Simulation Results: of input variables using modified FCM clustering technique The methodology presented has been implemented as a [47] as shown in figure 7. Once fuzzification of the inputs is Matlab m-file. Set of operating parameters as listed in Table 2, carried out, we get the 6 combinations of input membership were used for the identification of above model. Fig. 8 shows functions (3*2 = 6) representing 6 antecedents of rules as the optimized membership functions of the inputs given in figure 6. These 6 rules form the rulebase for the ―temperature‖ and ―temperature gradient‖ using S-ACO. The system under identification. The rulebase is yet incomplete as simulation results are presented in Table 3. It is clear from the for each rule the consequent need to be found out. From the results (500 iterations) that the fuzzy model without tuning of given dataset of table 1 we find that the there are only 5 membership functions (initial parameters setting using consequents that form the set of consequents from where we modified FCM [47]) leads to a mean square error of 0.14. have to choose one particular element as the consequent for a With tuning (using proposed technique) this error reduced to particular rule. The specified set of consequents in this case 0.0023. Further as the number of iterations increases system are C1= trickle = 0.1 Amp, C2=Low = 1 Amp, C3= Med = 2 performance gets better. Weighted average defuzzification Amp, C4= High= 3 Amp and, C5= Ultrafast = 4 Amp. We have technique was selected for Singleton fuzzy model [2]. to choose parameters of antecedent and consequents in such a way so as to fulfill condition given by expression (9). Table 2 ACO algorithm parameters for fuzzy model identification of Battery Charger Parameter Value Number of Ants 40 Iterations 500 α (a constant) 2 (evaporation constant) 0.4 k (Pheromone deposit factor) 0.1 Crisp output Figure 6: Sugeno type Fuzzy Model for Battery Charger Figure 8: Membership functions Optimized by S-ACO Algorithm Figure 7: Membership functions before Optimization 43 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 Table 3 [5] T. Takagi and M. Sugeno, ―Fuzzy identification of systems and its Simulation Results applications to modeling and control,‖ IEEE Transactions on Systems, Number of MSE of Fuzzy MSE of Fuzzy Man and Cybernetics, Vol. 15, pp.116-132, 1985. Iterations system system [6] H.Ishibuchi et al., ―Neural Networks that learn from Fuzzy if then rules,‖ IEEE Trans. on Fuzzy Systems, Vol.1, pp.85-97, 1993. (without tuning of (with tuning using [7] J.Yen and L.Wang, ―An SVD-based fuzzy model reduction strategy,‖ membership S-ACO) Proceedings of the Fifth IEEE International conference on Fuzzy functions) Systems, New Orleans, LA, pp. 835-841, 1996. 100 0.19 0.0183 [8] J.Yen and L.Wang, ―Application of statistical information criteria for optimal fuzzy model construction,‖ IEEE Transactions on Fuzzy 500 0.14 0.0023 Systems, Vol. 6, No.3, pp. 362-372, 1998. [9] J.Yen and L.Wang, ―Simplifying fuzzy rule-based models using orthogonal transformation methods,‖ IEEE Transactions on Systems, Man and Cybernetics, Vol.29, 1999. Table 4 [10] Y.Yam, P.Baranyi and C.T. Yang, ―Reduction of Fuzzy Rule Base via Comparison of the Proposed Approach with Other Algorithms Singular Value Decomposition,‖ IEEE Transactions on Fuzzy Systems, (Battery Charger) Vol.7, No.2, pp.120-132, 1999. Mean Square [11] Arun Khosla, Shakti Kumar, K.K. Aggarwal, ―Hardware Reduction for Algorithm Error Fuzzy based systems via Rule Reduction Through Exhaustive Search Technique‖, National Seminar on emerging convergent technologies and Hybrid Learning [47] 0.1321 systems (SECTAS-2002), Dayalbag Educational Institute, Agra, India, March 1-2, 2002, pp 381-385. [12] Arun Khosla, Shakti Kumar, K.K. Aggarwal, ―Optimizing Fuzzy Rule Genetic Algorithm [48] 0.130 Base Through State Reduction‖, National Seminar on emerging convergent technologies and systems (SECTAS-2002), Dayalbag Educational Institute, Agra, India, March 1-2, 2002, pp. 415-419. Particle Swarm Optimization [49] 0.1123 [13] Ken Nozaki, Hisao Ishibuchi and H.Tanaka, ―A simple but powerful heuristic method for generating fuzzy rules from numerical data,‖ Fuzzy Proposed Approach (S-ACO) 0.0023 Sets and Systems, Vol.86, pp. 251-270, 1997. [14] Li-Xin Wang and Jerry M. Mendel, ―Generating fuzzy rules by Learning from Examples,‖ IEEE Transactions on Systems, Man and Cybernetics, Vol.22, No.6, pp. 1414-1427, 1992. VI. CONCLUSIONS [15] A.Homaifar and E.Mc.Cormick, ―Simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms,‖ This paper has presented an ACO based membership IEEE Transactions on Fuzzy Systems, Vol.3, No.2, pp. 129-139, 1995. function tuning approach. We assumed that an identified [16] Y.Shi, R. Eberhart and Y.Chen, ―Implementation of Evolutionary Fuzzy model was available to us. For this given model we tuned the Systems,‖ IEEE Transactions on Fuzzy Systems, Vol.7, No.2, pp. 109- membership functions of antecedents to minimize the MSE. 119, 1999. In order to evaluate MSE we first encoded the problem [17] H.S. Hwang, ―Automatic design of fuzzy rule base for modeling and control using evolutionary programming,‖ IEE Proceedings- Control appropriately into a weighted graph whose edge lengths Theory Applications, Vol. 146, No. 1, pp. 9-16, 1999. represented percentage of movement for fuzzification. The [18] S.J. Kang, C.H. Woo, H.S. Hwang and K.B. Woo, ―Evolutionary Design difference between computed output (i(Wi * Ci) / Wi ) and of Fuzzy Rule Base for Nonlinear System Modeling and Control,‖ IEEE Transactions on Fuzzy Systems, Vol. 8, No.1, pp. 37-45, 2000. the actual output as given in the training example gives the [19] Arun Khosla, Shakti Kumar, K.K.Aggarwal, Jagatpreet Singh, ―Particle error. This error was used to update the pheromone trail. Swarm Optimizer for building fuzzy models,‖ Proceeding of one week Smaller the error more the amount of pheromone that being workshop on applied soft computing SOCO-2005, Haryana Engg.College, Jagadhri, India, July 25-30, pp 43-71, 2005. deposited on the path. This allows artificial ants to choose a [20] Marco Dorigo and Thomas Stutzle, Ant Colony Optimization, Eastern path with higher pheromone deposit with higher probability. Economy Edition, PHI, 2005. Finally all the ants followed a path that has the high [21] Marco Dorigo, Vittorio Maniezzo and Alberto Colorni, ―The Ant pheromone deposit leading to shortest path i.e. path with least System: Optimization by a colony of cooperating agents‖ IEEE Transactions on Systems, Man, and Cybernetics–Part B, Vol.26, No.1, error. This lead to optimized membership functions. pp.1-13, 1996. Simulation results shows that the proposed approach [22] M. Dorigo and L.M. Gambardella, Ant colony system: a cooperative outperforms the other three algorithms in terms of mean learning approach to the traveling salesman problem, IEEE Transaction square error. on Evolutionary Computation, 1(1) (1997), pp. 53-66, 1997. [23] J. Casillas, O. Cordon and F. Herrera, ―Learning fuzzy rules using ant colony optimization algorithms,‖ Proc. 2nd Int. Workshop Ant REFERENCES Algorithms, 2000, pp. 13-21. [1] L.A.Zadeh, ―Fuzzy Sets,‖ Information and Control, Vol.8, pp. 338-353, [24] R.S. Parpinelli, H.S. Lopes and A.A. Freitas, ―An ant colony algorithm 1965. for classification rule discovery,‖ in Data Mining: A Heuristic [2] John Yen and Reza Langari, ―Fuzzy Logic Intelligence, Control and Approach, pp. 190-208, H.A. Abbass, R.A. Sarkar. Idea Group Information,‖ Prentice Hall, New Jersey, 1999. Publishing, 2002. [3] Plamen A. et al., ―Identification of Evolving Fuzzy Rule-Based [25] Bo Liu, H.A. Abbass and B.McKay, ―Classification rule discovery with Models,‖ IEEE Transactions on Fuzzy Systems, Vol. 10, No.5, pp.667- Ant Colony Optimization,‖ Proc. of the IEEE/WIC Int’l conf. on 677, 2002. Intelligent Agent Technology (IAT’03), 2003. [4] M. Sugeno and T. Yasukawa, ―A fuzzy logic based approach to [26] M. Galea and Q. Shen, ―Fuzzy rules from ant-inspired computation,‖ qualitative modeling,‖ IEEE Transactions on Fuzzy Systems, Vol. 1, Proc. IEEE Int’l Conf. Fuzzy Systems, pp. 1691-1696, 2004. No.1, pp.7-31, 1993. 44 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 [27] P. Carmona and J. L. Castro, ―Using ant colony optimization for [39] Eghbal G. Mansoori, M.J. Zolghadri and S.D. Katebi, ―SGERD: A learning maximal structure fuzzy rules,‖ Proc. IEEE Int. Conf. Fuzzy steady-state genetic algorithm for extracting fuzzy classification rules Systems, pp. 999-999, 2005. from data,‖ IEEE Transactions on Fuzzy Systems, Vol.16, No.4, pp. [28] H.Nobahari and Seid H. Pourtakdoust, ―Optimization of fuzzy rule bases 1061-1071, Aug. 2008. using continuous Ant Colony System,‖ Proceeding of the first [40] Z. Ning, Y S. Ong, K.W. Wong and K.T. Seow, ―Parameter International Conference on Modeling, Simulation and Applied identification using Memetic algorithms for fuzzy systems,‖ Proc. of the Optimization, Sharjah, U.A.E., Feb. 2005. fourth Int’l conf. on intelligent technologies (Intech’03), pp 833-839, [29] R.Martinez, O. Castillo and J.Soria, ―Parameter tuning of membership 2003. functions of a Type-1 and Type-2 fuzzy logic controller for an [41] Shakti K., P. Bhalla, ―Fuzzy Rulebase Generation from Numerical Data autonomous wheeled mobile robot using Ant Colony Optimization,‖ using Ant Colony Optimization,‖ MAIMT- Journal of IT & Proceedings of the 2009 IEEE International Conference on Systems, Management. Vol.1, No.1 May - Oct. 2007, pp. 33-47. Man and Cybernetics, San Antonio, TX, USA, Oct. 2009. [42] Shakti Kumar and Parvinder Kaur, ―Fuzzy Rulebase Generation: A [30] C. Juang and Po-Han Chang, ―Designing fuzzy-rule-based systems using Biogeography Based Optimization Approach,‖ 3rd International continuous Ant-Colony Optimization,‖ IEEE Transactions on Fuzzy Conference on Intelligent Systems and Networks (IISN-2009), Feb 14- Systems, Vol. 18, No.1, Feb. 2010. 16, 2009, ISTK, Jagadhri, Haryana, India, pp. 425-428. [31] A.A.A. Esmin, A.R. Aoki, G. Lambert-Torres, ―Particle swarm [43] Shakti Kumar, Parvinder Kaur and Amarpartap Singh ―Soft Computing optimization for fuzzy membership functions optimization,‖ IEEE Int’l Approaches to Fuzzy System Identification: A Survey,‖ 3rd International Conf. on Syst., Man and Cybern., vol. 3, Oct. 2002. Conference on Intelligent Systems and Networks (IISN-2009), Feb 14- [32] Seema Chopra, Ranjit Mitra and Vijay Kumar, ―Reduction of Fuzzy 16, 2009, ISTK, Jagadhri, Haryana, India, pp.402-411. Rules and Membership Functions and its application to Fuzzy PI and PD [44] Shakti Kumar, Parvinder Kaur, Amarpartap Singh, ―Fuzzy Rulebase type controllers,‖ Int’l journal of Control, Automation, and Systems, Generation from numerical data using Biogeography Based vol.4, no.4, pp. 438-447, Aug. 2006. Optimization Approach,‖ Journal of Institution of Engineers IE (I), Vol. [33] Hyong-Euk Lee, Kwang-Hyun Park and Z.Z.Bien, ―Iterative Fuzzy 90, pp.8-13, July 2009. Clustering Algorithm with Supervision to construct probabilistic Fuzzy [45] Arun Khosla, Shakti Kumar, K.K. Aggarwal, ―Design and Development Rule Base from numerical data,‖ IEEE Transactions on Fuzzy Systems, of RFC-10: A Fuzzy Logic Based Rapid Battery Charger for Nickel- Vol. 16, No.1, pp.263-277, Feb. 2008. Cadmium Batteries. HiPC (High Performance Computing)‖, Workshop [34] P. Carmona, J.L. Castro and J. M. Zurita, ―FRIwE: Fuzzy rule on Soft Computing, Bangalore, 2002, pp. 9-14. identification with exceptions,‖ IEEE Transactions on Fuzzy Systems, [46] Linden D., ―Handbook of Batteries, Mc.Graw Hill Inc., 1995. Vol. 12, No.1, pp.140-151, Feb. 2004. [47] Arun Khosla, Shakti Kumar and K. K. Aggarwal, ―Fuzzy Controller for [35] B. Apolloni, A. Brega, D.Malchiodi, G. Palmas and A. M. Zanaboni, Rapid Nickel-Cadmium Batteries Charger through Adaptive Neuro- ―Learning rule representations from data,‖ IEEE Transactions on Fuzzy inference system (ANFIS) Architecture,‖ Proceedings of 22nd Systems, Man and Cybernetics- Part A, Vol. 36, No. 5, pp. 1010-1028, International Conference of the North American Fuzzy Information Sep. 2006. Processing Society, Chicago, Illinois, USA, July 24–26, 2003, pp. 540– [36] Xiao-Jun Zeng and M.G. Singh, ―Knowledge bounded least squares 544. method for the identification of fuzzy systems,‖ IEEE Transactions on [48] Shakti Kumar, ―Introduction to Fuzzy Logic Based Systems,‖ Systems, Man and Cybernetics- Part C, Vol. 33, No. 1, pp. 24-32, Feb. Proceedings of Workshop on Intelligent System Engineering (WISE- 2003. 2010), 2010. [37] S. B. Morphet, L.B. Morphet, ―Combining single input/single output [49] Arun Khosla, Shakti Kumar and K. K. Aggarwal, ―A Framework for fuzzy decision trees,‖ IEEE Int’l Conf. on Fuzzy Syatems, Vancouver, identification of Fuzzy models through Particle Swarm Optimization Canada, pp. 1792-1798, July 2006. Algorithm,‖ IEEE Indicon 2005, Dec. 11-13, 2005, pp. 388-391. [38] T. Pal and Nikhil R. Pal, ―SOGARG: A self organized genetic algorithm based rule generation scheme for fuzzy controllers,‖ IEEE Transactions on Evolutionary Computation, vol. 7, no. 4, Aug. 2003. 45 http://sites.google.com/site/ijcsis/ ISSN 1947-5500