# Optimization of Membership Functions Based on Ant Colony Algorithm by ijcsiseditor

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(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

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(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
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

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   lN 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

 yk   ~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)              yk  = 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

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(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.

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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

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

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(IJCSIS) International Journal of Computer Science and Information Security,
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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
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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
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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.
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