<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M1" height="11.8958pt" version="1.1" viewBox="-0.0657574 -11.83 12.0508 11.8958" width="12.0508pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-91" d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"/></g></svg>-Distance Based IF-THEN Rules

Aliev, R. A.; Huseynov, O. H.; Zulfugarova, R. X.

doi:https://doi.org/10.1155/2016/1673537

The Scientific World Journal

On this page

Abstract Introduction Preliminaries Conclusion References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 1673537 | https://doi.org/10.1155/2016/1673537

-Distance Based IF-THEN Rules

R. A. Aliev,^1,2O. H. Huseynov,³and R. X. Zulfugarova³

Academic Editor: Vilém Novák

Received18 Dec 2015

Revised04 Feb 2016

Accepted08 Feb 2016

Published27 Apr 2016

Abstract

Decision making, reasoning, and analysis in real-world problems are complicated by imperfect information. Real-world imperfect information is mainly characterized by two features. In view of this, Professor Zadeh suggested the concept of a -number as an ordered pair of fuzzy numbers and , the first of which is a linguistic value of a variable of interest, and the second one is a linguistic value of probability measure of the first one, playing a role of its reliability. The concept of distance is one of the important concepts for handling imperfect information in decision making and reasoning. In this paper, we, for the first time, apply the concept of distance of -numbers to the approximate reasoning with -number based IF-THEN rules. We provide an example on solving problem related to psychological issues naturally characterized by imperfect information, which shows applicability and validity of the suggested approach.

1. Introduction

Decision making, reasoning, and analysis in real-world problems are complicated by imperfect information. Real-world imperfect information is mainly characterized by two features. On the one hand, real-world information is often described on a basis of perception, experience, and knowledge of a human being. In turn, these operate with linguistic description carrying imprecision and vagueness, for which fuzzy sets based formalization can be used. On the other side, perception, experience, and knowledge of a human being are not sources of the truth. Therefore, the reliability is a degree of a partial confidence of a human being, which is naturally partial. This partial reliability is also naturally imprecise and can be formalized as a fuzzy value of probability measure. In order to ground the formal basis for dealing with real-world information, Zadeh suggested the concept of a -number [1] as an ordered pair of continuous fuzzy numbers used to describe a value of a random variable , where is a fuzzy constraint on values of and is a fuzzy reliability of and is considered as a value of probability measure of . Nowadays a series of works devoted to -numbers and their application in decision making, control, and other fields [2–13] exists. A general and computationally effective approach to computation with discrete -numbers is suggested in [14–16]. The authors provide motivation of the use of discrete -numbers mainly based on the fact that NL-based information is of a discrete framework. The suggested arithmetic of discrete -numbers includes basic arithmetic operations and important algebraic operations.

The concept of distance is one of the important concepts for decision making and reasoning [17, 18]. In this paper, we for the first time apply the concept of distance of -numbers to the approximate reasoning with -number based IF-THEN rules. An approximate reasoning refers to a process of inferring imprecise conclusions from imprecise premises [17–38]. As one can see, this process often takes place in various fields of human activity including economics, decision analysis, system analysis, control, and everyday activity. The reason for this is that information relevant to real-world problems is, as a rule, imperfect. According to Zadeh, imperfect information is information which in one or more respects is imprecise, uncertain, incomplete, unreliable, vague, or partially true [39]. We can say that in a wide sense approximate reasoning is reasoning with imperfect information.

The paper is structured as follows. In Section 2, we present some prerequisite material including definitions of a discrete fuzzy number, a discrete -number, and probability measure of a discrete fuzzy number. In Section 3, we propose several distance measures for -numbers. In Section 4, we describe the statement of the problem and the suggested approach to reasoning with -rules on the basis of distance of -numbers. In Section 5, we illustrate an application of the suggested approach to a real-world problem which involves modeling of psychological aspects. Section 6 concludes.

2. Preliminaries

2.1. Main Definitions

Definition 1 (a discrete fuzzy number [40–43]). A fuzzy subset of the real line with membership function is a discrete fuzzy number if its support is finite; that is, there exist with , such that and there exist natural numbers , with satisfying the following conditions:(1) for any natural number with ;(2) for natural numbers with ;(3) for natural numbers with .

Definition 2 (a discrete random variable and a discrete probability distribution [44]). A random variable, , is a variable whose possible values are outcomes of a random phenomenon. A discrete random variable is a random variable which takes only a countable set of its values .
Consider a discrete random variable with outcomes space . A probability of an outcome , denoted , is defined in terms of a probability distribution. A function is called a discrete probability distribution or a probability mass function ifwhere and .

Definition 3 (arithmetic operations over discrete random variables [44, 45]). Let and be two independent discrete random variables with the corresponding outcome spaces and and the corresponding discrete probability distributions and . The probability distribution of , , is the convolution of and which is defined for any , , , as follows:

Definition 4 (probability measure of a discrete fuzzy number [46]). Let be discrete random variable with probability distribution . Let be a discrete fuzzy number describing a possibilistic restriction on values of . A probability measure of denoting is defined as

Definition 5 (a scalar multiplication of a discrete fuzzy number [16]). A scalar multiplication of a discrete fuzzy number by a real number is the discrete fuzzy number , whose -cut is defined aswhereand the membership function is defined as

Definition 6 (addition of discrete fuzzy numbers [40–43]). For discrete fuzzy numbers , , their addition is the discrete fuzzy number whose -cut is defined aswhere , , , , and the membership function is defined as

Definition 7 (a discrete -number [15, 16]). A discrete -number is an ordered pair of discrete fuzzy numbers and . plays a role of a fuzzy constraint on values that a random variable may take. is a discrete fuzzy number with a membership function , , playing a role of a fuzzy constraint on the probability measure of , , .

3. Distance between Two -Numbers

Denote by the space of discrete fuzzy sets of . Denote by the space of discrete fuzzy sets of .

Definition 8 (the supremum metric on [47]). The supremum metric on is defined aswhere is the Hausdorff distance.
is a complete metric space [47, 48].

Definition 9 (fuzzy Hausdorff distance [16]). The fuzzy Hausdorff distance between is defined aswherewhere is the value which is within -cut and 1-cut. is a complete metric space.

Denote by the space of discrete -numbers:

Definition 10 (supremum metrics on [16]). The supremum metrics on are defined as is a complete metric space. This follows from the fact that is a complete metric space.
has the following properties:

Definition 11 (fuzzy Hausdorff distance between -numbers [16]). The fuzzy Hausdorff distance between -numbers is defined as

Definition 12 (-valued Euclidean distance between discrete -numbers [16]). Given two discrete -numbers , -valued Euclidean distance between and is defined as

4. -Valued IF-THEN Rules Based Reasoning

A problem of interpolation of -rules termed as -interpolation was addressed by Zadeh as a challenging problem [33]. This problem is the generalization of interpolation of fuzzy rules [49]. The problem of -interpolation is given below.

Given the following -rules, if is and so on and is , then is , if is and so on and is , then is , if is and so on and is then is ,and a current observation is and so on and is ,find the -value of . Here is the number of -valued input variables and is the number of rules.

The idea underlying the suggested interpolation approach is that the ratio of distances between the resulting output and the consequent parts is equal to one between the current input and the antecedent parts [49]. This implies for -rules that the resulting output is computed aswhere is the -number valued consequent of the th rule, , are coefficients of linear interpolation, and is the number of -rules. , where is the distance between current th -number valued input and the th -number valued antecedent of the th rule. Thus, computes the distance between a current input vector and the vector of the antecedents of th rule.

In this paper, we will consider discrete -numbers. The operations of addition and scalar multiplication of discrete -numbers are described below.

Addition of Discrete -Numbers. Let and be discrete -numbers describing imperfect information about values of variables and . Consider the problem of computation of addition . The first stage is the computation addition of discrete fuzzy numbers on the basis of Definition 6.

The second stage involves stage-by-stage construction of which is related to propagation of probabilistic restrictions. We realize that, in -numbers and , the “true” probability distributions and are not exactly known. In contrast, the information available is represented by the fuzzy restrictions:which are represented in terms of the membership functions as

Thus, one has the fuzzy sets of probability distributions of and with the membership functions defined as

Therefore, we should construct these fuzzy sets. , , is a discrete fuzzy number which plays the role of a soft constraint on a value of a probability measure of . Therefore, basic values , , , of a discrete fuzzy number , , are values of a probability measure of , . Thus, given , we have to find such probability distribution which satisfies

At the same time, we know that has to satisfy

Thus, the following goal programming problem should be solved to find :subject to

For each and each denote and , . As and are known and are unknown, we see that problem (23)-(24) is nothing but the following goal linear programming problem:subject to

Having obtained the solution , , of problems - for each , recall that , . As a result, , , is found, and, therefore, distribution is obtained. Thus, to construct , we need to solve simple problems -. Let us mention that in general, problems - do not have a unique solution. In order to guarantee existence of a unique solution, the compatibility conditions can be included:

This condition implies that the centroid of is to coincide with that of .

Probability distributions , , naturally induce probabilistic uncertainty over the result . This implies, given any possible pair of the extracted distributions, the convolution , , is to be computed as follows:Given , the value of probability measure of can be computed:However, the “true” is not exactly known as the “true” and are described by fuzzy restrictions. In other words, the fuzzy sets of probability distributions and induce the fuzzy set of convolutions , , with the membership function defined assubject towhere is operation.

As a result, fuzziness of information on described by induces fuzziness of the value of probability measure as a discrete fuzzy number . The membership function is defined as subject toAs a result, is obtained as .

Scalar Multiplication of Discrete -Numbers. Let us consider a scalar multiplication of a discrete -number : , . The resulting is found as follows. is determined based on Definition 5.

In order to construct , at first probability distributions , , should be extracted by solving a linear programming problem analogous to -. Next, we realize that , , induce probability distributions , , related to as follows:such thatThe fuzzy set of probability distributions with membership function induces the fuzzy set of probability distributions with the membership function defined astaking into account (32)-(33).

Next, we compute probability measure of , given . Given a fuzzy restriction on described by , we construct a fuzzy number with the membership function :subject to

As a result, is obtained as .

Let us now consider the special case of the considered problem of -rules interpolation, suggested in [50, 51].

Given the -rulesand a current observationfind the -value of .

For this case, as the reliabilities of the -number based consequents of the considered rules are equal, , according to formula (17) the -number valued output of the -rules, , is computed aswhere and as both inputs and the antecedents of the considered -rules are of a special -number; that is, they are represented by discrete fuzzy numbers with the reliability equal to 1.

5. An Application

Let us consider modeling of a fragment of a relationship between the student motivation, attention, anxiety, and educational achievement [52]. The information on the considered characteristics is naturally imprecise and partially reliable. Indeed, one deals mainly with intangible, nonmeasurable mental indicators. For this reason, the use of -rules, as rules with -number valued inputs and outputs based on linguistic terms from a predefined codebook, is adequate way for modeling of this relationship. This rules will help to evaluate a student with given -number based evaluations of the characteristics. Consider the following -rules: The 1st rule: If motivation is , attention is , and anxiety is , then achievement is . The 2nd rule: If motivation is , attention is , and anxiety is , then achievement is .

Here, the pairs (·,·) are -numbers where uppercase letters denote the following linguistic terms: , High; , Low; , Medium; , Good; , Excellence; , Usually. The codebooks containing linguistic terms of values of antecedents and consequents are given in Figures 1, 2, 3, and 4. The codebook for the degrees of reliability of values of antecedents and consequents is shown in Figure 5.

The considered -numbers are given below.

The 1st rule inputs:The 1st rule output:The 2nd rule inputs:

The 2nd rule output:

Consider a problem of reasoning within the given -rules by using the suggested -interpolation approach. Let the current input information for motivation, attention, and anxiety be described by the following -numbers , , and , respectively:

-interpolation approach based reasoning consists of two main stages.

() For each rule compute dist as distance between the current input -information , , and and -antecedents of -rules base , , and , . For simplicity, we will use the supremum metric (13):

Consider computation of for the 1st and 2nd rules. Thus, we need to determine , where values , , and are computed on the basis of (13). We have obtained the results:

Thus, the distance for the 1st rule is

Analogously, we computed the distance for the 2nd rule as

() Computation of the aggregated output for -rules base by using linear -interpolation:

The obtained interpolation coefficients are and . The aggregated output is defined as

We have obtained the following result:

In accordance with the codebooks shown in Figures 4 and 5, we have achievement is “High” with the reliability being “Usually.” This linguistic approximation is made by using similarity measure between the obtained output and fuzzy sets in the codebooks.

6. Conclusion

A concept of a -number suggested by Zadeh is a key to computation with imprecise and partial reliable information. In this paper, we propose applying distance of -numbers to approximate reasoning within IF-THEN rules with -numbers-based antecedents and consequents.

A real-world application of the suggested research has been provided to illustrate its validity and potential applicability.

Competing Interests

The authors declare that they have no competing interests.

References

L. A. Zadeh, “A note on $Z$ -numbers,” Information Sciences, vol. 181, no. 14, pp. 2923–2932, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
R. A. Aliev and L. M. Zeinalova, “Decision making under Z-information,” in Human-Centric Decision-Making Models for Social Sciences, P. Guo and W. Pedrycz, Eds., Studies in Computational Intelligence, pp. 233–252, Springer, 2014.
View at: Google Scholar
R. Banerjee and S. K. Pal, “The Z-number enigma: a study through an experiment,” in Soft Computing State of the Art Theory and Novel Applications, R. R. Yager, A. M. Abbasov, M. Z. Reformat, and S. N. Shahbazova, Eds., vol. 291 of Studies in Fuzziness and Soft Computing, pp. 71–88, Springer, Berlin, Germany, 2013.
View at: Publisher Site | Google Scholar
B. Kang, D. Wei, Y. Li, and Y. Deng, “A method of converting Z-number to classical fuzzy number,” Journal of Information & Computational Science, vol. 9, no. 3, pp. 703–709, 2012.
View at: Google Scholar
B. Kang, D. Wei, Y. Li, and Y. Deng, “Decision making using Z-numbers under uncertain environment,” Journal of Computational Information Systems, vol. 8, no. 7, pp. 2807–2814, 2012.
View at: Google Scholar
E. S. Khorasani, P. Patel, S. Rahimi, and D. Houle, “An inference engine toolkit for computing with words,” Journal of Ambient Intelligence and Humanized Computing, vol. 4, no. 4, pp. 451–470, 2013.
View at: Publisher Site | Google Scholar
J. Lorkowski, R. A. Aliev, and V. Kreinovich, “Towards decision making under interval, set-valued, fuzzy, and Z-number uncertainty: a fair price approach,” in Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 2244–2253, Beijing, China, July 2014.
View at: Publisher Site | Google Scholar
S. K. Pal, S. Dutta, R. Banerjee, and S. S. Sarma, “An insight into the Z-number approach to CWW,” Fundamenta Informaticae, vol. 124, no. 1-2, pp. 197–229, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. D. Springer, The Algebra of Random Variables, John Wiley & Sons, New York, NY, USA, 1979.
View at: MathSciNet
S. Tadayon and B. Tadayon, “Approximate Z-number evaluation based on categorical sets of probability distributions,” in Recent Developments and New Directions in Soft Computing, vol. 317 of Studies in Fuzziness and Soft Computing, pp. 117–134, Springer, Berlin, Germany, 2014.
View at: Publisher Site | Google Scholar
R. R. Yager, “On a view of Zadeh's Z-numbers,” in Advances in Computational Intelligence: 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Catania, Italy, July 9–13, 2012, Proceedings, Part III, vol. 299 of Communications in Computer and Information Science, pp. 90–101, Springer, Berlin, Germany, 2012.
View at: Publisher Site | Google Scholar
L. T. Kóczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal of Approximate Reasoning, vol. 9, no. 3, pp. 197–225, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
L. A. Zadeh, “Methods and systems for applications with Z-numbers,” United States Patent no. US 8,311,973 B1, November 2012.
View at: Google Scholar
R. A. Aliev, A. V. Alizadeh, O. H. Huseynov, and K. I. Jabbarova, “Z-number-based linear programming,” International Journal of Intelligent Systems, vol. 30, no. 5, pp. 563–589, 2015.
View at: Publisher Site | Google Scholar
R. A. Aliev, A. V. Alizadeh, and O. H. Huseynov, “The arithmetic of discrete Z-numbers,” Information Sciences, vol. 290, pp. 134–155, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
R. A. Aliev, O. H. Huseynov, R. R. Aliyev, and A. A. Alizadeh, The Arithmetic of Z-Numbers. Theory and Applications, World Scientific, Singapore, 2015.
View at: Publisher Site | MathSciNet
L. T. Kóczy, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal of Approximate Reasoning, vol. 9, no. 3, pp. 197–225, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
L. A. Zadeh, “Interpolative reasoning in fuzzy logic and neural network theory,” in Proceedings of the 1st IEEE International Conference on Fuzzy Systems, San Diego, Calif, USA, 1992.
View at: Google Scholar
S. K. Pal and D. P. Mandal, “Fuzzy logic and approximate reasoning an overview,” The Journal of the Institution of Electronics and Telecommunication Engineers, India, vol. 37, no. 5-6, pp. 548–560, 1991.
View at: Google Scholar
D. Dubois and H. Prade, “Fuzzy sets in approximate reasoning, Part 1: inference with possibility distributions,” Fuzzy Sets and Systems, vol. 100, no. 1, pp. 73–132, 1999.
View at: Publisher Site | Google Scholar
R. R. Yager, “On the theory of approximate reasoning,” Controle Automacao, vol. 4, pp. 116–125, 1994.
View at: Google Scholar
L. A. Zadeh, “Fuzzy logic and approximate reasoning,” Synthese, vol. 30, no. 3-4, pp. 407–428, 1975.
View at: Publisher Site | Google Scholar
R. R. Yager, “Deductive approximate reasoning systems,” IEEE Transactions on Knowledge and Data Engineering, vol. 3, no. 4, pp. 399–414, 1991.
View at: Publisher Site | Google Scholar
R. R. Yager, “Fuzzy sets and approximate reasoning in decision and control,” in Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 415–428, IEEE, March 1992.
View at: Google Scholar
K. K. Dompere, Fuzziness and Approximate Reasoning. Epistemics on Uncertainty, Expectation and Risk in Rational Behavior, vol. 237 of Studies in Fuzziness and Soft Computing, Springer, Berlin, Germany, 2009.
View at: Publisher Site
R. Aliev and A. Tserkovny, “Systemic approach to fuzzy logic formalization for approximate reasoning,” Information Sciences, vol. 181, no. 6, pp. 1045–1059, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
E. H. Mamdani, “Application of fuzzy logic to approximate reasoning using linguistic synthesis,” IEEE Transactions on Computers, vol. 26, no. 12, pp. 1182–1191, 1977.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. A. Zadeh, “Outline of a new approach to the analysis of complex system and decision processes,” IEEE Transactions on Systems, Man and Cybernetics, vol. 3, pp. 28–44, 1973.
View at: Google Scholar
L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975, Part II, vol. 8, no. 4, pp. 301–357, Part III, vol. 9, no. 1, pp. 43–80.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Toward a generalized theory of uncertainty (GTU)—an outline,” Information Sciences, vol. 172, no. 1-2, pp. 1–40, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
L. A. Zadeh, “Is there a need for fuzzy logic?” Information Sciences, vol. 178, no. 13, pp. 2751–2779, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications tomodeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985.
View at: Google Scholar
M. Sugeno, Fuzzy Control, North-Holland Publishing, 1988.
Y. Tang and X. Yang, “Symmetric implicational method of fuzzy reasoning,” International Journal of Approximate Reasoning, vol. 54, no. 8, pp. 1034–1048, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Luo and N. Yao, “Triple I algorithms based on Schweizer-sklar operators in fuzzy reasoning,” International Journal of Approximate Reasoning, vol. 54, no. 5, pp. 640–652, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
L. Zhang and K.-Y. Cai, “Optimal fuzzy reasoning and its robustness analysis,” International Journal of Intelligent Systems, vol. 19, no. 11, pp. 1033–1049, 2004.
View at: Publisher Site | Google Scholar
Z. Q. Wu, M. Masaharu, and Y. Shi, “An improvement to Kóczy and Hirota's interpolative reasoning in sparse fuzzy rule bases,” International Journal of Approximate Reasoning, vol. 15, no. 3, pp. 185–201, 1996.
View at: Publisher Site | Google Scholar | MathSciNet
L. A. Zadeh, “Computing with words and perceptions—a paradigm shift,” in Proceedings of the IEEE International Conference on Information Reuse and Integration (IRI '09), pp. 450–452, Las Vegas, Nev, USA, August 2009.
View at: Publisher Site | Google Scholar
J. Casasnovas and J. V. Riera, “On the addition of discrete fuzzy numbers,” WSEAS Transactions on Mathematics, vol. 5, no. 5, pp. 549–554, 2006.
View at: Google Scholar | MathSciNet
J. Casasnovas and J. V. Riera, “Weighted means of subjective evaluations,” in Soft Computing and Humanities in Social Sciences, R. Seizing and V. Sanz, Eds., vol. 273 of Studies in Fuzziness and Soft Computing, pp. 323–345, Springer, Berlin, Germany, 2012.
View at: Publisher Site | Google Scholar
W. Voxman, “Canonical representations of discrete fuzzy numbers,” Fuzzy Sets and Systems, vol. 118, no. 3, pp. 457–466, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
G. Wang, C. Wu, and C. Zhao, “Representation and operations of discrete fuzzy numbers,” Southeast Asian Bulletin of Mathematics, vol. 28, pp. 1003–1010, 2005.
View at: Google Scholar
M. Charles, J. Grinstead, and L. Snell, Introduction to Probability, American Mathematical Society, 1997.
R. C. Williamson and T. Downs, “Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds,” International Journal of Approximate Reasoning, vol. 4, no. 2, pp. 89–158, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. A. Zadeh, “Probability measures of fuzzy events,” Journal of Mathematical Analysis and Applications, vol. 23, no. 2, pp. 421–427, 1968.
View at: Publisher Site | Google Scholar | MathSciNet
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, Theory and Applications, World Scientific, Singapore, 1994.
View at: Publisher Site | MathSciNet
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, Theory and Applications, World Scientific, River Edge, NJ, USA, 1994.
View at: Publisher Site | MathSciNet
L. T. Kóczy and K. Hirota, “Rule interpolation by α-level sets in fuzzy approximate reasoning,” Bulletin for Studies and Exchanges on Fuzziness and Its Applications, vol. 46, pp. 115–123, 1991.
View at: Google Scholar
R. Aliev and K. Memmedova, “Application of Z-number based modeling in psychological research,” Computational Intelligence and Neuroscience, vol. 2015, Article ID 760403, 7 pages, 2015.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Z-numbers—a new direction in the analysis of uncertain and complex systems,” in Proceedings of the 7th IEEE International Conference on Digital Ecosystems and Technologies (IEEE-DEST '13), July 2013.
View at: Google Scholar
K. Mammadova, “Modeling of impact of pilates on students performance under Z-information,” in Proceedings of the 8th World Conference on Intelligent Systems for Industrial Automation (WCIS '14), pp. 171–178, Tashkent, Uzbekistan, 2014.
View at: Google Scholar

Copyright

Copyright © 2016 R. A. Aliev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

1367

Downloads

716

Citations