# The Best Approximation of Generalized Fuzzy Numbers Based on Scaled Metric

Authors: Tofigh Allahviranloo; Rasoul Saneifard; Rahim Saneifard; Farzad Kiani; Samad Noeiaghdam; Vediyappan Govindan
Journal: Journal of Mathematics
 (2022)

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## Abstract

The ongoing study has been vehemently allocated to propound an amelioratedα-weighted generalized approximation of an arbitrary fuzzy number. This method sets out to lessen the distance between the original fuzzy set and its approximation. In an effort to elaborate the study, formulas are designed for computing the ameliorated approximation by using a multitude of examples. The numerical samples will be exemplified to illuminate the improvement of the nearest triangular approximation (Abbasbandy et al., Triangular approximation of fuzzy numbers using α-weighted valuations, Soft Computing, 2019). A variety of features of the ameliorated approximation are then proved.


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## Body

## 1. Introduction

Fuzzy numbers have matured in many disciplines dealing with vagueness and imperfect information in decision-making, expert systems, and determining other related areas.The importance of a fuzzy number has been recognized considering the uppermost representation of information is numeric. Since its inception, a large circle of researchers has focused on refinements of computing to the nearest interval of triangular, trapezoidal, or parametric approximation of a fuzzy number. New developments, to the extent that they are relevant to this field, have been served very fast. Defuzzification methods have been extensively surveyed and applied in areas of applications such as fuzzy expert systems. The noteworthy idea of defuzzification is expected to encapsulate a fuzzy number to get a typical value from a given fuzzy set. The end goal is to deliver a correspondence from the set of all fuzzy quantities into a set of crisp values. Recently, a wide range of defuzzification procedures has been proposed in the literature that should facilitate the study [1]. Detyniecki and Yager have studied the ranking fuzzy numbers using α-weighted valuations in [2]. In [3], the nearest interval approximation of a fuzzy number has been discussed by Grzegorzewski. A novel approach of defuzzification is presented by Ma et al. [4]. Yeger and Filev [5, 6] have studied the ranking fuzzy numbers using valuations and instantiation of possibility distributions.Carlsson et al. announced an interval-valued possibilistic mean [7], and Chanas explicated an interval approximation of a fuzzy number [8]. Abbasbandy and Asady featured a procedure of the nearest trapezoidal fuzzy number to a general fuzzy number [9]. In addition, Delgado et al. [1] and Abbasbandy et al. [10] circulated symmetrical and nonsymmetrical trapezoidal representation of fuzzy numbers and triangular approximations of fuzzy numbers. The simple and more assessment method applied in these works needs to be established as a moderation of a distance between an arbitrary fuzzy number and its nearest approximation.The previously mentioned authors examined the advanced generalized approximation of fuzzy quantities assumed by introducing a metric on the set of fuzzy numbers in a minimization process. The expected result was the computation of the core of the triangular fuzzy quantity. To conduct the study, the researcher goes through the following sections. In Section2, we are going to expound a number of notions of fuzzy numbers and announce a weighted distance between fuzzy numbers and the refined proposed approximation method in [10]. In Section 3, the weighted triangular approximation of a fuzzy number will be investigated in terms of weighted distance to reach its general calculating formula. It is advantageous and absolutely necessary to go through a number of reasonable properties in Section 4. Examples are served in Section 5. Section 6 comprises a conclusion. Numerous references are included in the final section.


## 2. Preliminaries

The vital definitions of a fuzzy number [10] have been created as follows:Definition 1.
Letℜ be the set of whole real numbers. A fuzzy number is a fuzzy set such as A:ℜ⟶I=0,1 that satisfies(1)
A is upper semicontinuous(2)
Ax=0 outside various interval a,d(3)
There are real numbersb,c such that a≤b≤c≤d then(a)
Ax escalates on a,b(b)
Ax descend onc,d(c)
Ax=1, b≤x≤cDefinition 2.
A crucial sample of a fuzzy quantity is the generalized fuzzy quantityu centered at uC with widths σ>0 and β>0.(1)1σx−uc+σ,uc≤x≤uc+β,1βuc+β−x,uc−σ≤x≤uc,0,otherwise.
The parametric form of the (1) is u¯α=uc−σ+σα and u¯α=uc+β−βα. If the triangular fuzzy number in E is contributed by an ordered triple u=ul,uc,ur∈ℜ3 in particular ul≤uc≤ur, then uα=u¯α,u¯α=uc−1−αuc−ul,uc+1−αur−uc.


### 2.1. Assessment Procedure

In this section, we provide the particular valuation methods. It is often too appropriate to deal with the problem of comparison of fuzzy numbers associating with a fuzzy numberu some representative value, Valu. There is a conspicuous case in point to compare the fuzzy subsets in terms of these single representative values.(2)Valu=∫01Averageuαdα.Equation (2) is elicited. A popularized classification for a set of assessment relations,(3)Valu=∫01Averageuαfαdα∫01fαdα,where f is a mapping from 0,1 to 0,1. In the increasing family, we take it to the consideration f:α⟶fα=αq, by q≥0. Two satisfying specific cases are about to be observed (the extremes):(i)
Forq=0, we observe equal to 1. Thereupon, the valuation is given by the original Equation (2).(ii)
Ifq⟶∞, the Dirac relation is attained. As a result, the valuation will be(4)Valu=Averageu1,which is categorized in place of the average of the core. Conforming to our investigation, it is highly desirable to pay extra attention to the higherα-level sets as we have the larger q. In the limiting case, when q approaches infinity, we just use u1 (the core of u). The decreasing family which is the complementary case of the increasing family is shown as follows:(5)f:α⟶fα=1−αq,byq≥0.Here, anew, the authors have two favorite instances (the extremity):(i)
Forq=0, the authors obtain f equal to 1. The estimate expressed in Equation (2) will be promised.(ii)
Forq⟶∞, we obtain the Dirac function. This implies that the valuation is expressed as follows:(6)Valu=Averageu0,which is the average of the support. In the decreasing family, for q≥0, we are placing more emphasis on the lower α-level sets. The larger q has more emphasis given to the lower α− cut sets.The best generalized fuzzy quantitiesTu to u in [10] can be shown as follows:(7)tl=16+4w6B−2wtc−9∫01u¯α1−αdα+9+12w∫01u¯α1−αdα,tc=Valu=Averageu1,tr=16+4w6B−2wtc−9∫01u¯α1−αdα+9+12w∫01u¯α1−αdα.Such thatB=Valu.


## 2.1. Assessment Procedure

In this section, we provide the particular valuation methods. It is often too appropriate to deal with the problem of comparison of fuzzy numbers associating with a fuzzy numberu some representative value, Valu. There is a conspicuous case in point to compare the fuzzy subsets in terms of these single representative values.(2)Valu=∫01Averageuαdα.Equation (2) is elicited. A popularized classification for a set of assessment relations,(3)Valu=∫01Averageuαfαdα∫01fαdα,where f is a mapping from 0,1 to 0,1. In the increasing family, we take it to the consideration f:α⟶fα=αq, by q≥0. Two satisfying specific cases are about to be observed (the extremes):(i)
Forq=0, we observe equal to 1. Thereupon, the valuation is given by the original Equation (2).(ii)
Ifq⟶∞, the Dirac relation is attained. As a result, the valuation will be(4)Valu=Averageu1,which is categorized in place of the average of the core. Conforming to our investigation, it is highly desirable to pay extra attention to the higherα-level sets as we have the larger q. In the limiting case, when q approaches infinity, we just use u1 (the core of u). The decreasing family which is the complementary case of the increasing family is shown as follows:(5)f:α⟶fα=1−αq,byq≥0.Here, anew, the authors have two favorite instances (the extremity):(i)
Forq=0, the authors obtain f equal to 1. The estimate expressed in Equation (2) will be promised.(ii)
Forq⟶∞, we obtain the Dirac function. This implies that the valuation is expressed as follows:(6)Valu=Averageu0,which is the average of the support. In the decreasing family, for q≥0, we are placing more emphasis on the lower α-level sets. The larger q has more emphasis given to the lower α− cut sets.The best generalized fuzzy quantitiesTu to u in [10] can be shown as follows:(7)tl=16+4w6B−2wtc−9∫01u¯α1−αdα+9+12w∫01u¯α1−αdα,tc=Valu=Averageu1,tr=16+4w6B−2wtc−9∫01u¯α1−αdα+9+12w∫01u¯α1−αdα.Such thatB=Valu.


## 3. The Best Generalized Fuzzy Number

We shall now propose the triangular approximation of a fuzzy number by the minimization technique, and we will use a metric to calculate the core of the generalized fuzzy quantityTu. Taking u as a general fuzzy number, this paper is trying to probe the nearest generalized fuzzy quantity:(8)Tu=tlu,tcu,tru=tl,tc,tr.Initially, we compute the core ofTu. This process can be extended with respect to the following minimization technique:(9)MinimizeS=Valu−ValTu2.With (3) and parametric form of Tu as T¯uα,T¯uα, subsequent form is provided as follows:(10)ValTu=∫01T¯uα+T¯uα/2αqdα∫01αqdα=12∫012tc+1−αtr−2tc+tlαqdα∫01αqdα=q+122tcq+1+tr−2tc+tl1q+1−1q+2=tc+tr−2tc+tl2q+2.To accomplish the core ofTu, the partial derivative will be shown as follows:(11)∂s∂tc=2Valu−tc−tr−2tc+tl2q+2−1+1q+2.And we solve∂s/∂tc=0 which results in(12)tc=Valuq+2q+1−tr−tl2q+1.Looking at the1-cut set, and extra q, the higher level sets would be produced when fα=αq, the core of Tu to be clarified. This is the case when q⟶∞ one obtains(13)tc=Valu=Averageu1.To calculate the core of the generalized fuzzy quantityTu, the subsequent mathematical programming problem is constructed:(14)MinP=D12u,Tu+wD22u,Tu,D1u,Tu=Valu−ValTu.Giving consideration to(15)D2u,Tu=∫01fαu¯α−T¯uα2dα+∫01fαu¯α−T¯uα2dα1/2.Andw≥0 is an extra number that is modified by the determinant. One may define fα=k+1/2αq, in such a way ∫01fαdα=1/2. Working with (3) in view of fα=1−αq, as to(16)ValTu=∫01T¯uα+T¯uα/21−αqdα∫011−αqdα=∫012tc+1−αtr−2tc+tl1−αqdα2∫011−αqdα.One can write(17)ValTu=tcq+2+q+12q+2tl+tr.Including(18)P=Valu−ValTu2+w∫01fαu¯α−T¯uα2dα+w∫01fαu¯α−T¯uα2dα,P=B−tcq+2+q+12q+2tl+tr2+w∫01fαu¯α−tc−1−αtc−tl2dα+w∫01fαu¯α−tc+1−αtr−tc2dα.Such thatB=Valu. To gain the core of Tu, the authors must compute partial derivatives with respect to tl and tr. In the case of fα=α, we can make the following observation:(19)∂P∂tl=2B−tcq+2−q+1tl+tr2q+2−q+12q+2−2w∫01αu¯α−tc−1−αtc−tl1−αdα,∂P∂tr=2B−tcq+2−q+1tl+tr2q+2−q+12q+2−2w∫01αu¯α−tc+1−αtr−tc1−αdα.So as to lightenP, we obtain ∂P/∂tl=0, ∂P/∂tr=0 and then, q⟶∞ we compute:(20)−B+tl+tr2−2w∫01αu¯α1−α−tc1−α+tc−tl1−α2dα=0,−B+tl+tr2−2w∫01αu¯α1−α−tc1−α−tr−tc1−α2dα=0.If the above system is solved, we achieve the generalized fuzzy numberTu which is the best to u as follows:(21)tlu=16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα,tc=Valu=Averageu1,tru=16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα.Now, we need to go through the subsequent theorem.Theorem 1.
Assume thatT and T′ are the nearest triangular approximations of fuzzy number u, such that Tu is the nearest triangular fuzzy number tou, defined in Equation (21), and T′u is the triangular fuzzy number which is the nearest to u that is defined in Equation (7). Suppose that Tu=tl,tc,tr and T′=tl′,tc′,tr′, then du,Tu≤Du,T′u.Proof.
The researchers are familiar with the functionfα=k+1/2αk is nonnegative and increasing on 0,1 with f0=0 and ∫01fαdα=1/2. Then, we have fα≤1/2≤1, ∀α∈0,1. Hence,(22)∫01fαD22u,Tudα1/2≤∫01fαD22u,T′udα1/2≤∫01D22u,T′udα1/2,du,Tu≤Du,Tu.Theorem 2.
The best generalized fuzzy quantities to a given fuzzy quantityu exist and are unique. Now, let’s go to the following proof.Proof.
Let(23)2q+122q+42+23w2q+122q+422q+122q+422q+122q+42+23w.
Be the Hessian matrix for (3.11). Asw,q are positive, the mentioned matrix is a positive definite matrix. Therefore, the result is a tool to minimize and isolate the result.Theorem 3.
The best triangular approximation is a fuzzy number.Proof.
LetTu be the nearest triangular approximation to a given fuzzy number u. Should Tu in (21), be a fuzzy number, the satisfied conditions for Definition 2 can properly be met. Let the parametric form of Tu be T¯u,T¯u; thus, the first and second conditions will be satisfied; T¯u=tc−1−αtc−tl is a bounded monotonic increasing left continuous function, and T¯u=tc+1−αtr−tc is a bounded monotonic decreasing left continuous function. We will attempt to prove that it will satisfy the third condition, i.e., T¯u<T¯u; therefore, tc−1−αtc−tl≤tc+1−αtr−tc must hold. So, we should show that tl≤tr. We have knowledge that(24)u¯α≤u¯α⇒αu¯α1−α≤αu¯α1−α,0≤α≤1⇒∫01αu¯α1−αdα≤∫01αu¯α1−αdα⇒36+12w∫01αu¯α1−αdα≤36+12w∫01αu¯α1−αdα,w≥0.
So far, we have(25)−36∫01αu¯α1−αdα≤−36∫01αu¯α1−αdα,(26)6B−wtc−36∫01αu¯α1−αdα≤6B−wtc−36∫01αu¯α1−αdα.
By summing up (24) and (26), the following can be worked out:(27)6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα≤6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα.
Multiplying the above relation by1/6+w shows the following results:(28)16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα≤16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα.
tlu≤tru are introduced. This operation completes the proof.


## 4. Attribute

In this part, for creditworthiness, we regard quite a number of desirable features of the approximation suggested in the foregoing section.Theorem 4.
Ifu=ul,uc,ur is a generalized fuzzy number, then Tu=u.Proof.
In view of the fact thatu is a generalized fuzzy number, with (21), we get(29)tcu=Valu=Averageu1=uc⇒tcu=uc.
AndB=Averageu0=ul+ur/2. Therefore, by (21), the following will be obtained:(30)tru=16+w6ul+ur2−wuc−36∫01αuc−1−αuc−ul1−αdα+36+12w∫01αuc+1−αur−uc1−αdα,tru=16+w3ul+3ur−wtc−36uc12+ul12+36+12wuc12+ur12=16+w3ul+3ur−wuc−3uc−3ul+3uc+3ur+wuc+wur=16+w6+wur=ur,and(31)tlu=16+w6ul+ur2−wuc−36∫01αuc+1−αur−uc1−αdα+36+12w∫01αuc−1−αuc−ul1−αdα.
In a similar way, by simplification, we gettlu=ul. We then obtain tru=ur and tlu=ul which finalize the proof.Theorem 5.
The best generalized triangular approximation is invariant to translation, such asTu+z=Tu+z for all z∈R and u∈Ε.Proof.
Assumez represents an arbitrary crisp number and u specifies a fuzzy quantity. By (21), we realize(32)tcu+z=Averageu1+z=Averageu1+z=tc+z,and(33)B′=Averageu0+z=Averageu0+z=B+z,tlu+z=16+w6B+z−wtcu+z−36∫01αu¯α+z1−αdα+36+12w∫01αu¯α+z1−αdα=16+w6B+6z−wtc−wz−36∫01αu¯α1−αdα−6z+36+12w∫01αu¯α1−αdα+6z+2wz=16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα+6+wz=6+wz6+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα=tlu+z.
It is obvious from (21) that similar computations ending in tru+z=tru+z will be accomplished. This should implyTu+z=Tu+z, which proves the translation invariance.Theorem 6.
The best generalized approximation is a scale invariant. There is an excellent case in point:Tμu=μTu,∀μ∈ℜ\0,u∈Ε.Proof.
Letμ typify a nonzero crisp value. By (21), the authors gain tcμu=Averageμu1=μAverageu1=μtcu and B′=Averageμu0=μB. Then,(34)tlμu=16+w6μB−wtcμu−36∫01αμu¯α1−αdα+36+12w∫01αμu¯α1−αdα=μtlu.
One can pinpoint thattrμu=μtru. As a result, we earn Tμu=μTu, which consummate the proof.Theorem 7.
The best-generalized approximation is monotonic ifu,v are symmetric and Averageu1=Averagev1.Proof.
Letu=ul,uc,ur and v=vl,vc,vr denote two arbitrary fuzzy numbers. Furthermore, let Tu=tl,tc,tr and Tv=tl′,tc′,tr′ be two triangular fuzzy numbers which should be computed using u and v, sequentially. We assume that u⊆v; accordingly, u,v believed to be symmetric; hence, v¯α≤u¯α and u¯α≤v¯α for each and every α∈0,1. Since Averageu1=Averagev1, proved by (21), we observe tcu=tcv, i.e., tc=tc′. Also, Averageu0=Averagev0, i.e., B=B′. Consequently, by (9), the following will be worked out:(35)tl=16+w6B−wtc−36∫01αu¯α1−αdα+36+12w∫01αu¯α1−αdα≥16+w6B′−wtc′c−36∫01αv¯α1−αdα+36+12w∫01αv¯α1−αdα=tl′.
In this regard, we are allowed to presenttr≤tr′, which leads to Tu⊆Tv, and the criterion of monotony holds.


## 5. Numerical Example

Example 1.
Offeru as a fuzzy quantity of the following parametric applying uα=2α−2,1−α, α∈0,1. Now, designate w=10, then having (21) and (7), we demonstrate Table 1.
The accomplished results are exhibited in Figure1.Table 1
Example1.
tltctrEquation−1.98043478300.7195652174(7)−1.93035714300.698214285(21)Figure 1
The best generalized fuzzy quantity forw=10, solid curve: u; dashed curve: equation (7); dotted curve: equation (21).Example 2.
Assume thatu enumerate a fuzzy quantity of the following parametric subject to uα=α1/2−1,1−α1/2, α∈0,1. By (21) and (7), considering setting w=10, we observe Table 2.
The attained results are pictured in Figure2.
Another example will be helpful.Table 2
Results of Example2.
tltctrEquation−0.700.7(7)−0.62857142800.628571428(21)Figure 2
The best generalized fuzzy quantity forw=10, solid curve: u; dashed curve: equation (7); dotted curve: equation (21).Example 3.
Assume thatu is a fuzzy quantity of the next parametric; reconcile to uα=1+α1/2,30−27α1/2, α∈0,1. By (21) and (7), and setting w=10, we accomplish Table 3. (See Figure 3).Table 3
Results of Example3.
tltctrEquation1.5586956522.522.65869565(7)2.6821428572.522.28214286(21)Figure 3
The best generalized fuzzy quantity forw=10, solid curve: u; dashed curve: equation (7); dotted curve: equation (21).


## 6. Conclusion

This paper attempted to suggest an advanced method for theα-scaled generalized approximation of general fuzzy quantities. The given procedure yields generalized fuzzy quantities that leave no doubts about their being straightforward and unequivocal. Then, feasible properties of the α-scaled generalized approximation of a fuzzy quantity were ferreted out.



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