Selection of fuzzy if‐then rules by a genetic method

Electronics and Communications in Japan, Part 3, Vol. 77, No. 2, 1994 Translated from Denahi Joho Tsushin Gakkai Ronbunshi, Vol. 76-A, No. 10, October 1993, pp. 1465-1473

Selection of Fuzzy I f -7kn Rules by a Genetic Method

Hisao Ishibuchi, Member, Ken No&, Naohisa Yamamoto, and Hideo Tanaka, Nonmembers

College of Engineering, University of Osaka Prefecture, Sakai, Japan 593

SUMMARY

This paper proposes a genetic algorithm-based method for constructing a fuzzy classification system with fuzzy ifthen rules. In the proposed method, a rule selection problem for constructing a compact fuzzy system with high classification power is formulated as a combinatorial optimization problem with two objectives: to maximize the classification rate and to minimize the number of rules. Then a method of implementing genetic algorithms is proposed for the application to this problem and its effectiveness is demonstrated by computer simulations. The implementation of genetic algorithms in this paper employs an approach where a set of fuzzy if-then rules is coded as a single individual.

Key words: Fuzzy classification systems; pattern classification; fuzzy if-then rules; genetic algorithms.

1. Introduction

Fuzzy information processing systems based on fuzzy if-then ples were studied mainly in the field of fuzzy control [ l , 27. Fuzzy &hen rules in fuzzy control problems were obtained from linguistic knowledge of domain experts in many cases. Recently, several ap- proaches to the rule generation and the learning from numerical information (i.e., input-output data) have been proposed. For example, genetic algorithm-based methods

II ...... (. ............ :. . . . . . . . , . . . .... : . . .

. . : . ,! . . . . . j . . : . . .

. : . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 1. Fuzzy partition by a fuzzy grid.

were proposed by Nomura et al. [4, 51, Nishiyama et al. [6], Takahama et al. [7], Karr [&lo], and Thrift [ll]. In these studies, genetic algorithms were employed for generating fuzzy if-then rules and tuning membership functions.

To construct a fuzzy classification system based on fuzzy ij-then rules, Ishibuchi et al. [12, 131 proposed a method for automatically generating fuzzy if-then rules from training patterns. In the fuzzy classification system in [12, 131, a pattern space is divided by a fuzzy grid such as Fig. 1 and a fuzzy if-then rule is generated for each fuzzy subspace. In this approach, the performance of a fuzzy classification system depends strongly on the choice of a fuzzy partition. For example, if the fuzzy

94 ISSN 1042-0967/94/00024094 @ 1994 Scripta Technica, Inc.

1.0

2

0

0.0 1 .o x1

Fig. 2. An example of classification problem.

partition is too fine, the classification rate of training patterns is high but that of test patterns, which are not employed for the rule generation, is low. By contrast, if the fuzzy partition is too coarse, complicated pattern classification is impossible. Therefore, the choice of an appropriate fuzzy partition is necessary for constructing a fuzzy classification system with high performance.

Here, let us consider a twoclass classification problem as given in Fig. 2. In Fig. 2, patterns in Class 1 and Class 2 are denoted by closed circles and open circles, respectively. As we can see from Fig. 2 intu- itively, the left-half of the pattern space requires a fine fuzzy partition but the right-half requires a coarse one. Therefore, it is difficult to choose an appropriate fuzzy partition when a simple fuzzy grid such as that in Fig. 1 is employed.

To handle this difficult, Ishibuchi et al. [12, 131 proposed the concept of distributed fuzzy rules where different fuzzy partitions were employed simultaneously in a single fuzzy classification system (see Fig. 3). Fig- ure 3 shows that all the fuzzy if-then rules corresponding to six fuzzy partitions are employed simultaneously in fuzzy inference. This approach solves the forementioned difficulty of the choice of an appropriate fuzzy partition by simultaneously employing coarse fuzzy partitions as well as fine fuzzy partitions. However, this approach creates a new difficulty in that the number of fuzzy if- then rules increases because multiple fuzzy partitions are employed.

This paper proposes a method for constructing a compact fuzzy system with high classification power by selecting minimum necessary rules from many fuzzy if- then rules corresponding to multiple fuzzy partitions.

Fig. 3. Illustration of distributed fuzzy if-rhen rules.

More precisely, first the rule selection problem for constructing a fuzzy classification system is formulated as a combinatorial optimization problem. The objective functions of this problem are the maximization of the classification rate and the minimization of the number of selected rules. Next, an implementation method of genetic algorithms is proposed for the application to this problem. In the proposed method, a set of fuzzy if-then rules is a single individual and the fitness of each individual is defined as a weighted sum of the two objective functions. Finally, the effectiveness of the proposed method is demonstrated by computer simulations.

2. Pattern Classification by Fuzzy r f - l k n Rules

Here, as a preliminary for selecting fuzzy ifthen rules by a genetic method, a generation method of fuzzy if--then rules by Ishibuch et al. [12, 131 is described briefly and its difficulty is clarified by a numerical example.

For simplicity of explanation, let a pattern space be the two-dimensional unit square [0, 13 x [0, 11. It is assumed that m patterns xp = (xp,, x,,), p = 1, 2, .... m, are given as training patterns from M classes: Class 1 (a), Class 2 (a), ..' Class M (CM).

When each axis of the two-dimensional pattern space is partitioned into K fuzzy subsets {A,", A,", .-, AKK}, the following rules are employed in a fuzzy classification system based on fuzzy if-then rules:

95

(a) K=2

i l ......... ....... *.. .... ... .... .. .... . ..L ....... . .

R?z :So RA :SO $& :s12

..,.....,_,.. .. / . ............. . . . . . . . a .... .....

R?, :ss' R;, :sa :El :sII

(b) K = 3

Fig. 4. Labels of fuzzy if-then rules.

Rule R$ : If X I is AP and x2 is A: then (a, 5 2 ) belongs to C6 with CF= CF; (1)

where RiiK is the label of the rule, AiK and Aj" are antecedent fuzzy sets, CuK is the consequent class, and CFOK

is the grade of certainty of the rule. Figure 4 shows the relation between the fuzzy partition of each axis of the pattern space and the labels of fuzzy rules for the cases of K = 2 and K = 3. The labels si's in Fig. 4 are the ordering of fuzzy rules, which will be employed in genetic algorithms in section 4.

While any shape of fuzzy sets can be employed as antecedent fuzzy sets, triangular fuzzy sets such as shown in Fig. 4 are used in this paper. The membership

function piK@) of the triangular fuzzy set AiK is as follows:

pl( s) = max( 1 - It - arl / b K , 01, i=l, 2, , K (2)

where u r is the center and b " is the spread of the triangular fuzzy set A:, which are defined as follows:

The consequent class CiiK and the grade of certainty CFUK of the fuzzy ifthen rule in Eq. (1) are determined automatically from the training patterns xp (p = 1, 2, .-, rn) by the following procedure.

Generation of procedure of fuzzy if-then rules

Step 1. Calculate pa for the t-th class (Ct) as follows (t = 1, 2, .-, M):

Step 2. Find Class X (CX) that has the maximum value of Po. That is, find CX such that

If two or more classes take the maximum value, let Cii" = + for the sake of convenience because the consequent class Cii" cannot be determined uniquely. If Class X (CX) is determined uniquely, let the consequent class ciiK IX Class x (cx).

Step 3. If the consequent class Cii" of the fuzzy if- rhetr rule is +, let CFUK = 0. If CiiK is not $, the grade of certainty CF~J" is calculated as follows:

M

t = l Cl *cx

a= z Pct/(M-l)

(7)

Fuzzy if-then rules with 4 in the consequent part are dummy rules, which are generated only for the sake of convenience of the programming of genetic algorithms in section 4. Therefore, those fuzzy rules have no effect on fuzzy inference.

96

( a )K=2

(b)K=3

(d)K=5

(e)K=6

Fig. 5. Generated fuzzy ifthen rules.

Figure 5 shows the generated fuzzy if-then rules for the classification problem in Fig. 2 by setting K = 2 - 7. In Fig. 5, each area in the pattern space denotes that the following fuzzy if-then rule is generated at the corresponding fuzzy subspace.

Hatched area: The consequent of the generated fuzzy i f fhen rule in this area is Class 1 (closed circles).

Dotted area: The consequent of the generated fuzzy if--then rule in this area is Class 2 (open circles).

White area: The consequent of the generated fuzzy if-then rule in this area is 4, i.e., a dummy rule is generated.

From Fig. 5, we can see that many dummy rules are generated in the case of a fine fuzzy partition (i.e.,

a large value of K ) . This is because there is no training pattern in the fuzzy subspace A: X A: and then Pcr becomes 0 for all classes.

Let S,, be the set of all the generated fuzzy if- then rules by the rule generation procedure; S, is defined as

where L is an integer that denotes the maximum number of fuzzy sets used for the fuzzy partition of each axis. Let us assume that a rule set S, which is a subset of S' (i.e., S G Su), is employed in a fuzzy classification system. In this case, the classification of an unknown pattern xp (xpl , xp2) is performed as follows.

Classification procedure of an unknown pattern xp = ( X p p XPJ

Step 1. Calculate aa for the t-th class (Ct) as follows ( t = 1, 2, 1.1, M):

( ~ c ~ = r n a x ( ~ , K ( ~ p l ) . ~ ~ ( ~ P * ) . CF:ICixr=Ct : R,KE S) (10)

Step 2. Find Class X (CX) that has the maximum value of cia. That is, find CXsuch that

If two or more classes take the maximum value, then the classification of xp is rejected as an unclassifiable pattern. If Class X (CX) is determined uniquely, assign xp to that class.

In the most basic fuzzy classification system, the following set of fuzzy if-then rules corresponding to a single value of K is employed:

Classification rules by the rule set S in Eq. (12) are shown in Fig. 6 for K = 2 - 7. This means that each figure in Fig. 6 is the result by fuzzy if-then rules corresponding to each fuzzy partition in Fig. 5. In Fig. 6, painted areas are unclassifiable areas where the classification of unknown patterns is rejected because there are no fuzzy if-then rules but dummy rules (see Fig. 5).

From Fig. 6, we can see that there are misclassi- fied patterns when the fuzzy partition is coarse (i.e., the

97

(a)K=2 p$q 0 0 0 .

0 .

(b)K=3

0 0

0 0 0 5' (c)K=4 0

0 0

(d)K=5

(e)K=6 Fl 0

0 0

0 .

( f )K=7

Fig. 6. Classification results.

value of K is small). We can see also that there are unclassifiable areas when the fuzzy partition is fine (ie., the value of K is large).

As the foregoing results show, it is very difficult to choose an appropriate value of K when a fuzzy classification system is based on fuzzy if-then rules generated by a single fuzzy partition. To solve this difficulty, distributed fuzzy if-then rules by Ishibuchi et al. [12, 131 employed an approach where a classification system was based on fuzzy if-then rules generated by various fuzzy partitions. For example, Fig. 7 shows the classification result based on all the fuzzy ifthen rules corresponding to K = 2 - 7. In Fig. 7, a good classification result with no unclassifiable area was obtained because fuzzy &hen rules obtained by multiple fuzzy partitions of various sizes were employed simultaneously. However, there is the difficulty that the number of fuzzy if-then rules is enormous.

0

Fig. 7. Classification result by distributed fuzzy if-then rules.

3. Formulation of a Combinatorial OptMization Problem

A generation method of fuzzy if-then rules from training patterns was described briefly in the last section. Here, the rule section problem is formulated as a combinatorial optimization problem for finding an appropriate subset S (S s S A d from the set S,, of all the generated fuzzy if-then rules.

Let NCP(S) be the number of training patterns that are classified correctly by the rule set S. The number of' fuzzy if-then rules in the rule set S is denoted by I Sl . Since the aim of this paper is to construct a compact fuzzy classification system with high performance, the maximization of NCP(S) and the minimization of I SI are considered as objective functions. That is, the following two-objective combinatorid optimization problem is formulated:

Combinatorial optimization problem

Objective functions: Maximize NCP(S) and minimization IS( .

Constraint condition: S s SAW

This problem is to select fuzzy if-then rules which should be included in the rule set S to correctly classify maximum patterns by minimum rules. The total number of feasible solutions of this problem is enormous, i.e., it is 2 where N is the number of rules in S,,. There fore, it is impossible from a practical viewpoint to exam ine all the feasible solutions and an efficient method is required for finding a good solution by searching only a part of the feasible solutions. As an efficient search method, this paper employs genetic algorithms [3].

98

4. Optimization by Genetic Algorithms (2) Selection operation

4.1. Definition of fitness

Since the rule selection problem in the last section is a two-objective combinatorial optimization problem, it is not possible to directly apply genetic algorithms to that problem. Therefore, the two objective functions are unified as follows by introducing weights WNcp and Ws (WhIcp. w, > 0).

Objective function: Maximize WNcp - NCP(S) - WS. 1st.

Using this objective function, the fitness function f(s) of the rule set S is defined as

4.2. Definition of individual

When applying genetic algorithms to a particular combinatorial optimization problem, each feasible solution should be represented as an individual that is a string of symbols. Therefore, this paper represents the rule set S, which is treated as an individual, by a string of the length N (N = 1 S,, I ) as follows:

s =sIs2"'sN, (14)

where s, = 1 denotes that the r-th rule belongs to S; sr = 0 denotes that the r-th rule is a dummy rule; and sr = -1 denotes that the r-th rule does not belong to S. Here it is assumed that each rule is ordered as s,, s,, -, sN, as shown in Fig. 4.

4.3. Genetic operations

Genetic algorithms in this paper, which are simple GAS [3] with an elitist strategy, consist of the following genetic operations.

(1) Generation of an initial population

Assign 0 to all the dummy rules and randomly assign 1 or -1 to the other fuzzy if-zhen rules. By this operation, a population of the initial generation is generated.

Let Y'$ be the population of the t-th generation. Select each individual S with the following selection probability for generating a new population of the next generation by a cross-over operation:

(3) Cross-over operation

Select a pair of individuals according to the selection probability. Then generate anew two individuals by exchanging elements before a randomly selected cutting point. This cross-over operation is iterated to completely generate a new population of the next generation.

(4) Mutation operation

Apply the following mutation operation with the mutation probability to each s,. of the individuals generated by the cross-over operation:

By this mutation operation, mutations of 1 -+ -1 and -1 -+ 1 may happen. The value of sr of dummy rules (i.e., s, = 0), however, never changes by this mutation.

5 . Elitist strategy

An elitist strategy is to preserve the best individual of the previous generation (i.e., an individual with the maximum fitness) to the next generation with no change. By this operation, the best individual generated before the t-th generation always exists at that generation.

5. Computer Simulation

5.1. Simulation results for a numerical example

The genetic algorithm described in the last section was applied with the following conditions:

The number of individuals in each population: 10

Total generations (stopping condition): lo00

Mutation probability: P,,, = 0.01

Values of weights: Whcp = 10, Ws = 1

99

0 .

(a)K=2

lo I 1 0.

3 Ol0 . I I ...... L P A J .::::61 . . . . . . .

(c)K=4

(d)K=5

(e)K=6

b

(f)Boundary

Fig. 8. Simulation result with the proposed method.

S, = {RuKli = 1, 2, -., K; j = 1, 2, ..., K; K = 2, 3, 4, 5 , 6 ) .

That is, the genetic algorithm was applied to the rule selection problem for selecting a rule set S from the fuzzy @hen rules corresponding to K = 2 - 6 shown in Figs. 5(a) to (e). For the maximum value of K, we used the minimum number of fuzzy sets of each axis by which all the training patterns could be classified correctly (from Fig. 6, we can see that all the training patterns were classified correctly when K = 6). The total number of rules, i.e., I SAul , in this problem is 90, and the total number of rule sets S is 290 = 1.2 X l p . Among them, 10,OOO rule sets (= 10 individuals X loo0 generations) were examined in the genetic algorithm.

With the forementioned conditions, ten trials were performed with different initial populations. Rule sets S that could correctly classify all the training patterns were obtained in all ten trials. The maximum, minimum, and

average numbers of fuzzy if-then rules in the ten trials were 8, 5, and 6.5, respectively.

An example of fuzzy if-then rules included in a selected rule set S by the forementioned computer simulations are shown in Fig. 8 together with the classification result by those fuzzy if-then rules. Five fuzzy &hen rules, which are denoted by hatched and dotted areas in Fig. 8(a) to (e), were selected and the classification result in Fig. 8(f) was obtained. From Fig. 8(e), which was depicted by classifying the pattern space using the selected five fuzzy if-then rules, we can see that all the training patterns were classified correctly. Furthermore, we can see that there is no unclassifiable area in Fig. 8(f) because no painted area exists in that figure (see Figs. 6(e) and ( f ) where unclassifiable areas were shown by painted areas). This is because the pattern space is cov- ered entirely with the five fuzzy ij-then rules in Fig. 8. This is clear if we check the subspace with positive membership values of the antecedent fuzzy sets of each fuzzy $-then rule. For example, we can see from the shape of the antecedent fuzzy sets A; and A,’ that the fuzzy ifthen rule in Fig. 8(a) corresponding to R212 in Fig. 4(a) covers all the pattern space. We can see also that the fuzzy if-then rule in Fig. 801) corresponding to f?,: in Fig. 4(b) covers the right-half of the pattern space.

In general, it is not always true that fuzzy if-then rules selected by the proposed method cover all the pattern space. However, it is expected from the following reasons that uncovered areas (i.e., unclassifiable areas) may not be large.

(1) To classify many patterns by a small number of rules, fuzzy if-then rules corresponding to coarse fuzzy partitions such as Figs. 8(a) and (b) should be selected for classifying many patterns by a single rule. Fuzzy ij-then rules corresponding to coarse fuzzy partitions cover a large area of the pattern space as mentioned in the foregoing.

(2) If a small number of fuzzy $-then rules corresponding to fine fuzzy partitions were selected, unclassifiable areas would exist. Such a rule set, however, may not have a high fitness value from the viewpoint of the number of rules and that of correctly classified patterns.

(3) If we use bell-shaped fuzzy sets, which spread infinitely to both sides, as antecedent fuzzy sets, selected fuzzy if-then rules always cover all the pattern space.

Table 1. Simulation results with the elitist strategy (the number of fuzzy if-then rules in the

rule set s)

Mutation Population size probability 5 10 50

0.1 15.0 16.1 15.6 0.01 6.1 6.5 7.4

0.001 8.2 6.6 6.0

Table 2. Simulation results with the elitist strategy (the number of correctly classified patterns)

Mutation Population size probability 5 10 50

0.1 20.0 20.0 20.0

Table 3. Simulation results without the elitist strategy (the number of fuzzy if-then rules in

the rule set s)

Mutation PoDulation size probability 5 10 50

25.9 24.6 23.0 0.1 0.0 1 28.5 26.0 20.5

0.001 34.2 29.3 23.1

Table 4. Simulation results without the elitist strategy (the number of correctly classified

patterns)

Mutation Population size probabiIity 5 10 50

0.1 20.0 20.0 20.0

0.01 19.9 20.0 20.0 0.001 19.7 20.0 20.0

0.01 20.0 20.0 20.0 0.00 1 19.5 19.7 19.6

From the comparison of Fig. 8 with Figs. 5 and 6, we can see the effectiveness of the proposed method. That is, we can see that a good classification result was obtained by a small number of fuzzy if-then rules in the proposed method. In addition, it is possible to interpret the five fuzzy if-then rules in Fig. 8 linguistically. For example, the fuzzy if-then rule in Fig. 8(a) corresponding to the coarsest fuzzy partition can be interpreted as "If x, is large and x, is small, then Class 1 (closed circle)." It is an advantage of classification systems based on fuzzy if-then rules that linguistic knowledge can be obtained from numerical information.

Next, ten trials were performed from different initial populations with each of the following conditions to examine the relation between the performance of genetic algorithms and parameter specifications in genetic operations.

The number of individuals in each population: 5 , 10, 50

Mutation probability: P,,, = 0.1, 0.01, 0.001

The other parameter specifications such as the number of generations, the values of weights and the rule set SAu

were not changed. Simulation results (averages of the ten trials) for each condition are shown in Tables 1 and 2. From Table 2, we can see that the rule set S with high classification power was obtained in almost all conditions regardless of parameter specifications. On the other hand, we can see from Table 1 that the number of fuzzy if-then rules in the rule set S was affected strongly by the mutation probability.

In addition, a simulation was run without the elitist strategy. The results are shown in Tables 3 and 4. It is clear from a comparison of Tables 1 and 3 that the elitist strategy enhances the genetic algorithm.

5.2. Simulation results

The proposed method also was applied to the classification problem of iris by Fisher [14] to examine its classification power for multiclass classification problems on multidimensional pattern spaces. The classification problem of iris is a three-class problem on a

101

fourdimensional pattern space where 50 patterns in each class are given.

In the same manner as in the last section, the genetic algorithm with the elitist strategy was applied to the classification problem of ins with the following parameter specifications: the number of individuals in each population was 10, the total number of generations was 1O00, the values of weights were WNcp = 10 and Ws = 1, and the mutation probability was P,,, = 0.1, 0.01, 0.001. The set of fuzzy $then rules corresponding to K = 2 - 6 was employed as the rule set S,. The total number of fuzzy $then rules in Su was 2' + 3' + 44 + S4 + 64 = 2274 (it was 692 if dummy rules were excluded).

Average results of five trials for each value of the probability are as follows.

Average results when P, = 0.1

The average number of rules: 277.2

The average number of correctly classified patterns: 149.8

Average results when P,,, = 0.01



Average results when P,,, = 0.001



From these results, we can see that almost all patterns were classified correctly and that the number of rules was reduced to 20 to 40 percent of S,.

Biasing the mutation probability was effective for reducing the number of rules. For example, the following average results of five trials were obtained by a biased mutation probability: 0.001 for the mutation of s,= -1+s ,=1and0 .01for thato f s ,=1+s ,= -1 .


The average number of correctly classified patterns: 149.2

As these results indicate, good results were obtained by using a large probability for the mutation to reduce the number of rules when the total number ot' fuzzy if-then rules was enormous.

6. Conclusions

In this paper, a genetic algorithm-based method was proposed for constructing a compact and high-performance fuzzy classification system with fuzzy if-then rules. First, the rule selection problem to choose fuzzy if-then rules in a fuzzy classification system was formulated as a combinatonal optimization problem. Next, an implementation method of genetic algorithms for this problem was proposed. Then it was shown by computer simulations that high-performance fuzzy classification systems based on a small number of fuzzy if-then rules were obtained by the proposed method.

In computer simulations, the effectiveness of the elitist strategy and the biased mutation probabilities was demonstrated. Discussions on how to appropriately speci- fy some parameters such as the shape of antecedent fuzzy sets and the number of fuzzy sets on each axis (i.e., the value of K ) are left for future work. In this paper, the selection problem of fuzzy if-then rules was handled by the approach where a set of rules was coded as an individual. An implementation of genetic algorithms with the coding of each rule as an individual also is left for future work together with the examination of the performance of such an approach.

REFERENCES

1. M. Sugeno. An introductory survey of fuzzy control. Information Sciences, 36, pp. 59-83 (1985).

2. C. C. Lee. Fuzzy logic in control systems: Fuzzy logic controller-Parts I and 11. IEEE Trans. on Systems, Man, and Cybernetics, SMC-20, 2, pp. 404-435 (1990).

3. D. E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison- Wesley, Reading, PA, USA (1989).

4. H. Nomura, S. Araki, I. Hayashi, and N. Wakami. A Learning Algorithm of Fuzzy Reason- ing Rule: A Structure Identification of Antecedent Part by Genetic Algorithm. Proc. of FAN Sympo- sium (Osaka, Papan), pp. 25-30 (Oct. 1991).

5 . H. Nomura, I. Hayashi, and N. Wakami. A Self- Tuning Method of Fuzzy Reasoning by Genetic

102

Algorithm. Proc. of the 1992 International Fuzzy Systems and Intelligent Control Conference, Louis- ville, KY, USA, pp. 236-245 (March 1992). T. Nishiyama, T. Takagi, R. R. Yager, and S. Nakanishi. Automatic Generation of Fuzzy Infer- ence Rules by Genetic Algorithm. Proc. of 8th Fuzzy System Symposium (Hiroshima, Japan), pp. 237-240 (May 1992). T. Takahama, S. Miyamoto, H. Ogura, and M. Nakamura. Acquisition of Fuzzy Control Rules by Genetic Algorithms. Proc. of 8th Fuzzy System Symposium (Hiroshima, Japan), pp. 241-244 (May 1992).

8. C. Karr. Genetic Algorithms for Fuzzy Control- lers. A1 Expert, February, pp. 26-33 (1991).

9. C. Karr. Applying Genetics to Fuzzy Logic. A1 Expert, March, pp. 38-43 (1991).

10. C. Karr. Design of an Adaptive Fuzzy Logic

6.

7.

Controller Using a Genetic Algorithm. Proc. of the Fourth International Conference on Genetic Algo- rithms, San Diego, USA, pp. 450-457 (July 1991).

11. P. Thrift. Fuzzy Logic Synthesis with Genetic Algorithms. Proc. of the Fourth International Conference on Genetic Algorithms, San Diego,

12. H. Ishibuchi, K. Nozaki, and H. Tanaka. Pattern recognition using distributed fuzzy rules. Trans. Institute of Systems, Control, and Information Engineers, 4, 12, pp. 517-526 (1991).

13. H. Ishibuchi, K. Nozaki, and H. Tanaka. Distrib- uted representation of fuzzy rules and its application to pattern classification. Fuzzy Sets and Sys- tems, 51, pp. 21-32 (1992).

14. R. A. Fisher. The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7, pp.

CA, USA, pp. 509-513 (July 1991).

179-188 (1936).

AUTHORS (from left to right)

Hisao Ishibuchi graduated in 1985 from the Department of Precise Mechanics Engineering, Kyoto University, where he received a Master’s degree in 1987. He then became a Research Associate and served as a Lecturer in 1993 and an Associate Professor in 1994 in the Department of Industrial Engineering, University of Osaka Prefecture, where he obtained a Dr. of Eng. degree. His current research includes applications of fuzzy set theory and neural networks. He is a member of IEEE; IFSA; and INNS.

Ken Nozaki graduated in 1992 from the Department of Industrial Engineering, University of Osaka Prefecture, where he is studying for a Master’s degree. His current research includes fuzzy set theory, neural networks, genetic algorithm-based pattern classification, and approximate classification. He is a member of the Institute of Systems, Control, and Information Engineers and the Japan Industrial Management Association.

Naohisa Yamamoto graduated in 1992 from the Department of Industrial Engineering, University of Osaka Prefecture, where he is currently studying for a Master’s degree. His current research includes fuzzy scheduling and genetic algorithms.

103

AUTHORS (continued)

Hide0 Tanaka received a B.S. from Kobe University in 1962. He spent two years working for the Central Research Laboratory of Daikin Corporation. He then received an M.S. and a Ph.D. from Osaka City University in 1966 and 1969, respectively. Since then, he has been with the Department of Industrial Engineering, University of Osaka Prefecture. He has been a Professor since 1987. His current research includes regression analysis and discriminant analysis based on possibility theory. He is a member of IEEE and IFSA.

104

Selection of fuzzy if‐then rules by a genetic method

Documents

Transcript of Selection of fuzzy if‐then rules by a genetic method