# Z-Distance Based IF-THEN Rules

Authors: R. A. Aliev; O. H. Huseynov; R. X. Zulfugarova
Journal: The Scientific World Journal
 (2016)

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## 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 aZ-number as an ordered pair Z=(A,B) of fuzzy numbers A and B, 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 Z-numbers to the approximate reasoning with Z-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.


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## Body

## 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 aZ-number [1] as an ordered pair Z=(A,B) of continuous fuzzy numbers used to describe a value of a random variable X, where A is a fuzzy constraint on values of X and B is a fuzzy reliability of A and is considered as a value of probability measure of A. Nowadays a series of works devoted to Z-numbers and their application in decision making, control, and other fields [2–13] exists. A general and computationally effective approach to computation with discrete Z-numbers is suggested in [14–16]. The authors provide motivation of the use of discrete Z-numbers mainly based on the fact that NL-based information is of a discrete framework. The suggested arithmetic of discrete Z-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 Z-numbers to the approximate reasoning with Z-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 Section2, we present some prerequisite material including definitions of a discrete fuzzy number, a discrete Z-number, and probability measure of a discrete fuzzy number. In Section 3, we propose several distance measures for Z-numbers. In Section 4, we describe the statement of the problem and the suggested approach to reasoning with Z-rules on the basis of distance of Z-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 subsetA of the real line R with membership function μA:R→[0,1] is a discrete fuzzy number if its support is finite; that is, there exist x1,…,xn∈R with x1<x2<⋯<xn, such that supp⁡(A)={x1,…,xn} and there exist natural numbers s, t with 1≤s≤t≤n satisfying the following conditions:(1)
μA(xi)=1 for any natural number i with s≤i≤t;(2)
μA(xi)≤μA(xj) for natural numbers i,j with 1≤i≤j≤s;(3)
μA(xi)≥μA(xj) for natural numbers i,j with t≤i≤j≤n.Definition 2 (a discrete random variable and a discrete probability distribution [44]).
A random variable,X, is a variable whose possible values x are outcomes of a random phenomenon. A discrete random variable is a random variable which takes only a countable set of its values x.
Consider a discrete random variableX with outcomes space {x1,…,xn}. A probability of an outcome X=xi, denoted P(X=xi), is defined in terms of a probability distribution. A function p is called a discrete probability distribution or a probability mass function if(1)PX=xi=pxi,where p(xi)∈[0,1] and ∑i=1np(xi)=1.Definition 3 (arithmetic operations over discrete random variables [44, 45]).
LetX1 and X2 be two independent discrete random variables with the corresponding outcome spaces X1={x11,…,x1i,…,x1n1} and X2={x21,…,x2i,…,x2n2} and the corresponding discrete probability distributions p1 and p2. The probability distribution of X12=X1∗X2, ∗∈{+,-,·,/}, is the convolution p12=p1∘p2 of p1 and p2 which is defined for any x∈{x1∗x2∣x1∈X1,x2∈X2}, x1∈X1, x2∈X2, as follows:(2)p12x=∑x=x1∗x2p1x1p2x2.Definition 4 (probability measure of a discrete fuzzy number [46]).
LetX be discrete random variable with probability distribution p. Let A be a discrete fuzzy number describing a possibilistic restriction on values of X. A probability measure of A denoting P(A) is defined as(3)PA=∑i=1nμAxipxi=μAx1px1+μAx2px2+⋯+μAxnpxn.Definition 5 (a scalar multiplication of a discrete fuzzy number [16]).
A scalar multiplication of a discrete fuzzy numberA by a real number λ∈R is the discrete fuzzy number A1=λA, whose α-cut is defined as(4)A1α=x∈λ·supp⁡A∣min⁡λAα≤x≤max⁡λAα,where(5)λ·supp⁡A=λx∣x∈supp⁡A,min⁡λAα=min⁡λx∣x∈Aα,max⁡λAα=max⁡λx∣x∈Aα,and the membership function is defined as(6)μλAx=sup⁡α∈0,1∣x∈λAα.Definition 6 (addition of discrete fuzzy numbers [40–43]).
For discrete fuzzy numbersA1, A2, their addition A12=A1+A2 is the discrete fuzzy number whose α-cut is defined as(7)A12α=x∈supp⁡A1+supp⁡A2∣min⁡A1α+A2α≤x≤max⁡A1α+A2α,where supp⁡(A1)+supp⁡(A2)={x1+x2∣xj∈supp⁡(Aj),j=1,2}, min⁡{A1α+A2α}=min{x1+x2∣xj∈Ajα,  j=1,2},   max⁡{A1α+A2α}=max⁡{x1+x2∣xj∈Ajα,j=1,2}, and the membership function is defined as(8)μA1+A2x=sup⁡α∈0,1∣x∈A1α+A2α.Definition 7 (a discreteZ-number [15, 16]).
A discreteZ-number is an ordered pair Z=(A,B) of discrete fuzzy numbers A and B. A plays a role of a fuzzy constraint on values that a random variable X may take. B is a discrete fuzzy number with a membership function μB:{b1,…,bn}→[0,1], {b1,…,bn}⊂[0,1], playing a role of a fuzzy constraint on the probability measure of A, P(A)=∑i=1nμA(xi)p(xi), P(A)∈supp⁡(B).


## 2.1. Main Definitions

Definition 1 (a discrete fuzzy number [40–43]).
A fuzzy subsetA of the real line R with membership function μA:R→[0,1] is a discrete fuzzy number if its support is finite; that is, there exist x1,…,xn∈R with x1<x2<⋯<xn, such that supp⁡(A)={x1,…,xn} and there exist natural numbers s, t with 1≤s≤t≤n satisfying the following conditions:(1)
μA(xi)=1 for any natural number i with s≤i≤t;(2)
μA(xi)≤μA(xj) for natural numbers i,j with 1≤i≤j≤s;(3)
μA(xi)≥μA(xj) for natural numbers i,j with t≤i≤j≤n.Definition 2 (a discrete random variable and a discrete probability distribution [44]).
A random variable,X, is a variable whose possible values x are outcomes of a random phenomenon. A discrete random variable is a random variable which takes only a countable set of its values x.
Consider a discrete random variableX with outcomes space {x1,…,xn}. A probability of an outcome X=xi, denoted P(X=xi), is defined in terms of a probability distribution. A function p is called a discrete probability distribution or a probability mass function if(1)PX=xi=pxi,where p(xi)∈[0,1] and ∑i=1np(xi)=1.Definition 3 (arithmetic operations over discrete random variables [44, 45]).
LetX1 and X2 be two independent discrete random variables with the corresponding outcome spaces X1={x11,…,x1i,…,x1n1} and X2={x21,…,x2i,…,x2n2} and the corresponding discrete probability distributions p1 and p2. The probability distribution of X12=X1∗X2, ∗∈{+,-,·,/}, is the convolution p12=p1∘p2 of p1 and p2 which is defined for any x∈{x1∗x2∣x1∈X1,x2∈X2}, x1∈X1, x2∈X2, as follows:(2)p12x=∑x=x1∗x2p1x1p2x2.Definition 4 (probability measure of a discrete fuzzy number [46]).
LetX be discrete random variable with probability distribution p. Let A be a discrete fuzzy number describing a possibilistic restriction on values of X. A probability measure of A denoting P(A) is defined as(3)PA=∑i=1nμAxipxi=μAx1px1+μAx2px2+⋯+μAxnpxn.Definition 5 (a scalar multiplication of a discrete fuzzy number [16]).
A scalar multiplication of a discrete fuzzy numberA by a real number λ∈R is the discrete fuzzy number A1=λA, whose α-cut is defined as(4)A1α=x∈λ·supp⁡A∣min⁡λAα≤x≤max⁡λAα,where(5)λ·supp⁡A=λx∣x∈supp⁡A,min⁡λAα=min⁡λx∣x∈Aα,max⁡λAα=max⁡λx∣x∈Aα,and the membership function is defined as(6)μλAx=sup⁡α∈0,1∣x∈λAα.Definition 6 (addition of discrete fuzzy numbers [40–43]).
For discrete fuzzy numbersA1, A2, their addition A12=A1+A2 is the discrete fuzzy number whose α-cut is defined as(7)A12α=x∈supp⁡A1+supp⁡A2∣min⁡A1α+A2α≤x≤max⁡A1α+A2α,where supp⁡(A1)+supp⁡(A2)={x1+x2∣xj∈supp⁡(Aj),j=1,2}, min⁡{A1α+A2α}=min{x1+x2∣xj∈Ajα,  j=1,2},   max⁡{A1α+A2α}=max⁡{x1+x2∣xj∈Ajα,j=1,2}, and the membership function is defined as(8)μA1+A2x=sup⁡α∈0,1∣x∈A1α+A2α.Definition 7 (a discreteZ-number [15, 16]).
A discreteZ-number is an ordered pair Z=(A,B) of discrete fuzzy numbers A and B. A plays a role of a fuzzy constraint on values that a random variable X may take. B is a discrete fuzzy number with a membership function μB:{b1,…,bn}→[0,1], {b1,…,bn}⊂[0,1], playing a role of a fuzzy constraint on the probability measure of A, P(A)=∑i=1nμA(xi)p(xi), P(A)∈supp⁡(B).


## 3. Distance between TwoZ-Numbers

Denote byF the space of discrete fuzzy sets of R. Denote by F[a,b] the space of discrete fuzzy sets of [a,b]⊂R.Definition 8 (the supremum metric onD [47]).
The supremum metricd on F is defined as(9)dA1,A2=sup⁡dHA1α,A2α∣0<α≤1,A1,A2∈F,where dH is the Hausdorff distance.
(
F
,
d
) is a complete metric space [47, 48].Definition 9 (fuzzy Hausdorff distance [16]).
The fuzzy Hausdorff distancedfH between A1,A2∈F is defined as(10)dfHA1,A2=⋃α∈0,1αdfHαA1,A2,where(11)dfHαA1,A2=supα≤α-≤1⁡dHA1α-,A2α-,where α- is the value which is within α-cut and 1-cut. (F,dfH) is a complete metric space.Denote byZ the space of discrete Z-numbers:(12)Z=Z=A,B∣A∈F,B∈F0,1.Definition 10 (supremum metrics onZ [16]).
The supremum metrics onZ are defined as(13)DZ1,Z2=dA1,A2+dB1,B2;(Z,D) is a complete metric space. This follows from the fact that (F,d) is a complete metric space.
D
Z
1
,
Z
2 has the following properties:(14)DZ1+Z,Z2+Z=DZ1,Z2,DZ2,Z1=DZ1,Z2,DλZ1,λZ2=λDZ1,Z2,λ∈R,DZ1,Z2≤DZ1,Z+DZ,Z2.Definition 11 (fuzzy Hausdorff distance betweenZ-numbers [16]).
The fuzzy Hausdorff distancedfHZ between Z-numbers Z1=(A1,B1),Z2=(A2,B2)∈Z is defined as(15)dfHZZ1,Z2=dfHA1,A2+dfHB1,B2.Definition 12 (Z-valued Euclidean distance between discrete Z-numbers [16]).
Given two discreteZ-numbers Z1=(A1,B1),Z2=(A2,B2)∈Z, Z-valued Euclidean distance dE(Z1,Z2) between Z1 and Z2 is defined as(16)dEZ1,Z2=Z1-Z22.


## 4.Z-Valued IF-THEN Rules Based Reasoning

A problem of interpolation ofZ-rules termed as Z-interpolation was addressed by Zadeh as a challenging problem [33]. This problem is the generalization of interpolation of fuzzy rules [49]. The problem of Z-interpolation is given below.Given the followingZ-rules,ifX1 is ZX1,1=(AX1,1,BX1,1) and so on and Xm is ZXm,1=(AXm,1,BXm,1), then Y is ZY=(AY,1,BY,1),ifX1 is ZX1,2=(AX1,2,BX1,2) and so on and Xm is ZXm,2=(AXm,2,BXm,2), then Y is ZY=(AY,2,BY,2),ifX1 is ZX1,n=(AX1,n,BX1,n) and so on and Xm is ZXm,n=(AXm,n,BXm,n) then Y is ZY=(AY,n,BY,n),and a current observationX1 is ZX1′=(AX1′,BX1′) and so on and Xm is ZXm′=(AXm′,BXm′),find the Z-value of Y. Here m is the number of Z-valued input variables and n 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 Z-rules that the resulting output ZY′ is computed as(17)ZY′=∑j=1nwjZY,j=∑j=1nwjAY,j,BY,j,where ZY,j is the Z-number valued consequent of the jth rule, wj=1/ρj/∑j=1n1/ρj, j=1,…,n are coefficients of linear interpolation, and n is the number of Z-rules. ρj=∑i=1mD(ZXi′,ZXi,j), where D is the distance between current ith Z-number valued input and the ith Z-number valued antecedent of the jth rule. Thus, ρj computes the distance between a current input vector and the vector of the antecedents of jth rule.In this paper, we will consider discreteZ-numbers. The operations of addition and scalar multiplication of discrete Z-numbers are described below.Addition of Discrete
Z
-Numbers. Let Z1=(A1,B1) and Z2=(A2,B2) be discrete Z-numbers describing imperfect information about values of variables X1 and X2. Consider the problem of computation of addition Z12=Z1+Z2. The first stage is the computation addition of discrete fuzzy numbers A12=A1+A2 on the basis of Definition 6.The second stage involves stage-by-stage construction ofB12 which is related to propagation of probabilistic restrictions. We realize that, in Z-numbers Z1=(A1,B1) and Z2=(A2,B2), the “true” probability distributions p1 and p2 are not exactly known. In contrast, the information available is represented by the fuzzy restrictions:(18)∑k=1n1μA1x1kp1x1kisB1,∑k=1n2μA2x2kp2x2kisB2,which are represented in terms of the membership functions as(19)μB1∑k=1n1μA1x1kp1x1k,μB2∑k=1n2μA2x2kp2x2k.Thus, one has the fuzzy sets of probability distributions ofp1 and p2 with the membership functions defined as(20)μp1p1=μB1∑k=1n1μA1x1kp1x1k,μp2p2=μB2∑k=1n2μA2x2kp2x2k.Therefore, we should construct these fuzzy sets.Bj, j=1,2, is a discrete fuzzy number which plays the role of a soft constraint on a value of a probability measure of Aj. Therefore, basic values bjl∈supp⁡Bj, j=1,2, l=1,…,m, of a discrete fuzzy number Bj, j=1,2, are values of a probability measure of Aj, bjl=P(Aj). Thus, given bjl, we have to find such probability distribution pjl which satisfies(21)bjl=μAjxj1pjlxj1+μAjxj2pjlxj2+⋯+μAjxjnjpjlxjnj.At the same time, we know thatpjl has to satisfy(22)∑k=1njpjlxjk=1,pjlxjk≥0.Thus, the following goal programming problem should be solved to findpj:(23)μAjxj1pjlxj1+μAjxj2pjlxj2+⋯+μAjxjnjpjlxjnj⟶bjl,subject to(24)pjlxj1+pjlxj2+⋯+pjlxjnj=1,pjlxj1,pjlxj2,…,pjlxjnj≥0.For eachl=1,…,m and each k=1,…,nj denote ck=μAj(xjk) and vkl=pjl(xjk), k=1,…,nj. As ck and bjl are known and vkl are unknown, we see that problem (23)-(24) is nothing but the following goal linear programming problem:23′c1v1l+c2v2l+⋯+cnvnl⟶bjl,subject to24′v1l+v2l+⋯+vnl=1,v1l,v2l,…,vnl≥0.Having obtained the solutionvkl, k=1,…,nj, of problems 23′-24′ for each l=1,…,m, recall that vkl=pjl(xjk), k=1,…,nj. As a result, pjl(xjk), k=1,…,nj, is found, and, therefore, distribution pjl is obtained. Thus, to construct μpjl, we need to solve m simple problems 23′-24′. Let us mention that in general, problems 23′-24′ do not have a unique solution. In order to guarantee existence of a unique solution, the compatibility conditions can be included:(25)∑k=1njxjkpjlxjk=∑k=1njxjkμAjxjk∑k=1njμAjxjk.This condition implies that the centroid ofAj is to coincide with that of pjl.Probability distributionspjl(xjk), k=1,…,nj, naturally induce probabilistic uncertainty over the result X=X1+X2. This implies, given any possible pair p1l,p2l of the extracted distributions, the convolution p12s=p1l∘p2l, s=1,…,m2, is to be computed as follows:(26)p12x=∑x1+x2=xp1lx1p2lx2,∀x∈X12;x1∈X1,x2∈X2.Given p12s, the value of probability measure of A12 can be computed:(27)PA12=∑k=1nμA12x12kp12x12k.However, the “true” p12s is not exactly known as the “true” p1l and p2l are described by fuzzy restrictions. In other words, the fuzzy sets of probability distributions p1l and p2l induce the fuzzy set of convolutions p12s, s=1,…,m2, with the membership function defined as(28)μp12p12=maxp1,p2⁡μp1p1∧μp2p2,subject to(29)p12=p1∘p2,μpjpj=μBj∑k=1njμAjxjkpjlxjk,where ∧ is min operation.As a result, fuzziness of information onp12s described by μp12 induces fuzziness of the value of probability measure P(A12) as a discrete fuzzy number B12. The membership function μB12 is defined as (30)μB12b12s=sup⁡μp12sp12s,subject to(31)b12s=∑kp12sxkμA12xk.As a result, Z12=Z1+Z2 is obtained as Z12=(A12,B12).Scalar Multiplication of Discrete
Z
-Numbers. Let us consider a scalar multiplication of a discrete Z-number ZX=(AX,BX): ZY=λ·ZX, λ∈R. The resulting ZY=(AY,BY) is found as follows. AY=λAX is determined based on Definition 5.In order to constructBY, at first probability distributions pX,l, l=1,…,m, should be extracted by solving a linear programming problem analogous to 23′-24′. Next, we realize that pX,l, l=1,…,m, induce probability distributions pY,l, l=1,…,m, related to ZY as follows:(32)pY=pYy1∖y1+pYy2∖y2+⋯+pYyn∖yn,such that(33)yk=λxk,pYyk=pXxk.The fuzzy set of probability distributions pX with membership function μpX(pX,l)=μB~X∑k=1nμA~X(xk)pX,l(xk) induces the fuzzy set of probability distributions pY,l with the membership function defined as(34)μpYpY,l=μpXpX,l,taking into account (32)-(33).Next, we compute probability measure ofAY, given pY. Given a fuzzy restriction on pY described by μpY, we construct a fuzzy number BY with the membership function μBY:(35)μBYbY,l=sup⁡μpYpY,l,subject to(36)bY,l=∑kpY,lxkμAYxk.As a result,ZY=λ·ZX is obtained as ZY=(AY,BY).Let us now consider the special case of the considered problem ofZ-rules interpolation, suggested in [50, 51].Given theZ-rules(37)IfXisAX,1thenYisAY,1,BIfXisAX,2thenYisAY,2,B⋮IfXisAX,nthenYisAY,n,Band a current observation(38)XisAX,BX,find the Z-value of Y.For this case, as the reliabilities of theZ-number based consequents of the considered rules are equal, BY,k=B, according to formula (17) the Z-number valued output of the Z-rules, ZY′=(AY′,BY′), is computed as(39)ZY′=∑j=1nwjZY,j=∑j=1nwjAY,j,BY,j=∑j=1nwjAY,j,B,where wj=1/ρj/∑k=1n1/ρk and ρj=∑i=1mD(ZXi′,ZXi,j)=∑i=1mD((AXi′,1),(AXi,j,1))=∑i=1md(AXi′,AXi,j) as both inputs and the antecedents of the considered Z-rules are of a special Z-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 Z-rules, as rules with Z-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 Z-number based evaluations of the characteristics. Consider the following Z-rules:The 1st rule: If motivation is (M,U), attention is (H,U), and anxiety is (L,U), then achievement is (E,U).The 2nd rule: If motivation is (M,U), attention is (M,U), and anxiety is (M,U), then achievement is (G,U).Here, the pairs (·,·) areZ-numbers where uppercase letters denote the following linguistic terms: H, High; L, Low; M, Medium; G, Good; E, Excellence; U, 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.Figure 1
Linguistic terms for a value of motivation.Figure 2
Linguistic terms for a value of attention.Figure 3
Linguistic terms for a value of anxiety.Figure 4
Linguistic terms for a value of achievement.Figure 5
Linguistic terms for reliability of antecedents and consequents.The consideredZ-numbers are given below.The 1st rule inputs:(40)ZAM=02.6+0.53.3+14+0.54.7+05.4,ZBU=00.7+0.50.75+10.8+0.50.85+00.9;ZAH=057.5+0.568.75+180+190,ZBU=00.7+0.50.75+10.8+0.50.85+00.9;ZAL=01.19+0.51.6+12+0.52.4+02.8,ZBU=00.7+0.50.75+10.8+0.50.85+00.9.The 1st rule output:(41)ZAVH=080+0.585+190+0.595+0100,ZBU=00.7+0.50.75+10.8+0.50.85+00.9.The 2nd rule inputs:(42)ZAM=02.6+0.53.3+14+0.54.7+05.4,ZBU=00.7+0.50.75+10.8+0.50.85+00.9;ZAM=035+0.546.25+157.5+0.568.75+080,ZBU=00.7+0.50.75+10.8+0.50.85+00.9;ZAM=02+0.52.4+12.8+0.53.2+03.6,ZBU=00.7+0.50.75+10.8+0.50.85+00.9.The 2nd rule output:(43)ZAH=070+0.575+180+0.585+090,ZBU=00.7+0.50.75+10.8+0.50.85+00.9.Consider a problem of reasoning within the givenZ-rules by using the suggested Z-interpolation approach. Let the current input information for motivation, attention, and anxiety be described by the following Z-numbers Z1=(ZA1,ZB1), Z2=(ZA2,ZB2), and Z3=(ZA3,ZB3), respectively:(44)ZA1=02.5+0.53+13.5+0.54+04.5,ZB1=00.6+0.50.65+10.7+0.50.75+00.8;ZA2=025+0.535+145+0.555+065,ZB2=00.6+0.50.65+10.7+0.50.75+00.8;ZA3=01.3+0.52.3+13.3+0.53.65+04,ZB3=00.6+0.50.65+10.7+0.50.75+00.8.Z-interpolation approach based reasoning consists of two main stages.(1) For each rule computedist as distance ρj between the current input Z-information Z1=(ZA1,ZB1), Z2=(ZA2,ZB2), and Z3=(ZA3,ZB3) and Z-antecedents of Z-rules base Zj1=(Aj1,Bj1), Zj2=(Aj2,Bj2), and Zj3=(Aj3,Bj3), j=1,2. For simplicity, we will use the supremum metric DZi,Zji (13): (45)ρj=∑i=13DZi,Zji.Consider computation ofρj for the 1st and 2nd rules. Thus, we need to determine ρj=∑j=13DZj,Z1j, where values DZ1,Z11, DZ2,Z12, and DZ3,Z13 are computed on the basis of (13). We have obtained the results:(46)DZ1,Z11=dHA1,A11+dHB1,B11=0.9+0.1=1,DZ2,Z12=40.1,DZ3,Z13=1.4.Thus, the distance for the 1st rule is(47)ρ1=42.5.Analogously, we computed the distance for the 2nd rule as(48)DZ1,Z2,1=1,DZ2,Z2,2=15.1,DZ3,Z2,3=0.8,ρ2=16.9.(2) Computation of the aggregated output ZY for Z-rules base by using linear Z-interpolation:(49)ZY=w1ZY,1+w2ZY,2,w1=1/ρ11/ρ1+1/ρ2,w2=1/ρ21/ρ1+1/ρ2.The obtained interpolation coefficients arew1=0.28 and w2=0.72. The aggregated output ZY is defined as(50)ZY=0.28ZY1+0.72ZY2=AY,BY.We have obtained the following result:(51)ZAY=072.8+0.578.2+182.6+0.584+089,ZBY=00.68+0.50.73+10.78+0.50.81+00.84.In accordance with the codebooks shown in Figures4 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 aZ-number suggested by Zadeh is a key to computation with imprecise and partial reliable information. In this paper, we propose applying distance of Z-numbers to approximate reasoning within IF-THEN rules with Z-numbers-based antecedents and consequents.A real-world application of the suggested research has been provided to illustrate its validity and potential applicability.



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*Source: 1673537-2016-04-27.xml*