Cardinal, Median Value, Variance and Covariance of Exponential Fuzzy Numbers with Shape Function and its Applications in Ranking Fuzzy Numbers

S. Rezvani

doi:10.1080/18756891.2016.1144150

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Volume 9, Issue 1, January 2016, Pages 10 - 24

Cardinal, Median Value, Variance and Covariance of Exponential Fuzzy Numbers with Shape Function and its Applications in Ranking Fuzzy Numbers

Authors

S. Rezvani¹^{, *}^,salim_rezvani@yahoo.com

^*Salim Rezvani. Department of Mathematics, Imam Khomaini Mritime University of Nowshahr, Nowshahr, Iran.

Corresponding Author

S. Rezvanisalim_rezvani@yahoo.com

Received 11 October 2014, Accepted 29 October 2015, Available Online 1 January 2016.

DOI: 10.1080/18756891.2016.1144150 How to use a DOI?
Keywords: Covariance; Exponential Fuzzy Numbers; Mathematical expectation; Possibilistic Mean Value; Ranking Fuzzy Number; Variance
Abstract: In this paper, the researcher proposed a method to cardinal, median value, variance and covariance of exponential fuzzy numbers with shape function . The covariance used in this method is obtained from the exponential trapezoidal fuzzy number, first by finding mathematical expectation and then calculating the variance of each exponential fuzzy numbers by E(A), E²(A) and then finding the possibilistic covariance between fuzzy numbers A and B. We are going to utilize Dubois and another researcher dominance possibility and necessity indices, within a exponential fuzzy numbers with shape function and its applications, in the case of measure of possibilistic correlation between fuzzy numbers A and B by their ranking possibility of their interaction compared to their possibilistic fuzzy number. Finally, we proposed a new method for ranking exponential fuzzy numbers with possibilistic variance between fuzzy numbers A and B. This approach helps avoiding any approximation that may exist due to incorrect comparing between fuzzy numbers and can effectively rank various fuzzy numbers, their images and overcome the shortcomings of the previous techniques and also the proposed approach is very simple and easy to apply in the real life problems. For the validation, the results of the proposed approach are compared with different existing approaches.
Copyright: © 2016. the authors. Co-published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Comparison of fuzzy numbers is considered one of the most important topics in fuzzy logic theory. In 1965, Zadeh [25] introduced the concept of fuzzy set theory to meet those problems. The early and most important work in the field of comparing fuzzy numbers has been presented by Dubois and Prade [10]. In most of cases in our life, the data obtained for decision making are only approximately known. Fuzzy numbers allow us to make the mathematical model of linguistic variable or fuzzy environment. In addition to a fuzzy environment, ranking is a very important decision making procedure. On the other hand, the dominance possibility indices, which have been introduced by Dubois and Prade, were utilized in the field of fuzzy mathematical programming and the field of stochastic fuzzy mathematical programming in another researcher. In 1978, Dubois and Prade defined any of the fuzzy numbers as a fuzzy subset of the real line [11]. The graded mean integration representation of generalized fuzzy number was introduced, by Chen and Hsieh [5], it also had been compared with some other different methods for representation with several different representation methods. ranking fuzzy values with satisfaction function investigated by Lee et al. [15]. Probability distributions can be interpreted as carriers of incomplete information [13], and possibility distributrions can be interpreted as carriers of imprecise information. In 1987 Dubois and Prade [11] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. In 2005 Carlsson, Fuller and Majlender [3] introduced a measure of possibilistic correlation between fuzzy numbers A and B by their joint possibility distribution C as an average measure of their interaction (introduced earlier by Fuller and Majlender in [12]) compared to their respective marginal variances. C. Carlsson [8] introduced possibilistic mean value, variance, covariance and correlation of fuzzy numbers. S. Rezvani in [16,23] introduced correct ordering of generalized and normal trapezoidal fuzzy numbers in divergence and Median Value. In 2015 Rezvani defined a ranking fuzzy numbers for variance as values are calculated by finding expected values using the probability density function corresponding to the membership functions of the given fuzzy number and provides the correct ordering of exponential trapezoidal fuzzy numbers [24]. Moreover, S. Rezvani ([16]–[24]) evaluated the system of ranking fuzzy numbers.

This ranking used in these field was based on formulating a possibility function, whether in the case of trapezoidal fuzzy numbers.The researcher proposed a method to cardinal, median value, variance and covariance of exponential fuzzy numbers with shape function . The covariance used in this method is obtained from the exponential trapezoidal fuzzy number, first by finding mathematical expectation and then calculating the variance of each exponential fuzzy numbers by E(A), E²(A) and then finding the possibilistic covariance between fuzzy numbers A and B. Finally, we proposed a new method for ranking exponential fuzzy numbers with possibilistic variance between fuzzy numbers A and B. This method can effectively rank various fuzzy numbers, their images and overcome the shortcomings of the previous techniques and also the proposed approach is very simple and easy to apply in the real life problems. For the validation, the results of the proposed approach are compared with different existing approaches.

2. Preliminaries

Generally, a generalized fuzzy number A is described as any fuzzy subset of the real line R, whose membership function μ_A satisfies the following conditions,

(i)
μ_A is a continuous mapping from R to the closed interval [0,1],
(ii)
μ_A(x) = 0,−∞ < u ≤ a,
(iii)
μ_A(x) = L(x) is strictly increasing on [a,b],
(iv)
μ_A(x) = w,b ≤ x ≤ c,
(v)
μ_A(x) = R(x) is strictly decreasing on [c,d],
(vi)
μ_A(x) = 0,d ≤ x < ∞

Where 0 < w ≤ 1 and a,b,c, and d are real numbers. We call this type of generalized fuzzy number a trapezoidal fuzzy number, and it is denoted by A = (a,b,c,d;w)_LR .

When w = 1, this type of generalized fuzzy number is called normal fuzzy number and is represented by A = (a,b,c,d)_LR.

However, these fuzzy numbers always have a fix range as [c,d] . A exponential fuzzy number A with shape function:

(1)fA(x)={w[e−[(b−x)/(b−a)]]na≤x≤b,wb≤x≤c,w[e−[(x−c)/(d−c)]]nc≤x≤d,

where 0 < w ≤ 1, a,b are real numbers, and c,d are positive real numbers. We denote this type of generalized exponential fuzzy number as A = (a,b,c,d; w)_E. Especially, when w = 1, we denote it as A = (a,b,c,d)_E .

we define the representation of generalized exponential fuzzy number based on the integral value of graded mean h-level as follow. Let the generalized exponential fuzzy number A = (a,b,c,d)_E, where 0 < w ≤ 1, and c,d are positive real numbers, a,b are real numbers as in formula (1). Now, let two monotonic shape functions be

(2)L(x)=w[e−[(b−x)/(b−a)]]n, R(x)=w[e−[(x−c)/(d−c)]]n

3. Possibilistic Mean Value of Exponential Fuzzy Numbers

In this section some important results, that are useful for the proposed methods, are proved.

Definition 1. [25].

Cardinality of a fuzzy number A is the value of the integral

(3)card A=∫abA dx=∫01(bα−aα) dα

Now, we use of above definition in exponential trapezoidal fuzzy numbers.

Theorem 1.

Cardinality of a exponential trapezoidal fuzzy number A characterized by (1) is the value of the integral

(4)card A=wn(b−a+d−c)−wn[(b−a)e−n+(d−c)e−n]+w(c−b)

Proof:

card A=∫abA dx=∫abw[e−[(b−x)/(b−a)]]ndx+∫bcw dx+∫cdw[e−[(x−c)/(d−c)]]ndx=w(b−a)n[1−e−n]+w(c−b)+w(d−c)n[1−e−n]=wn(b−a+d−c)−wn[(b−a)e−n+(d−c)e−n]+w(c−b).

The median value of a fuzzy set A was introduced as a possible scalar representative value of A.

Definition 2. [4].

The median value of a fuzzy number A characterized by (1) is the real number m_A from the support of A such that

(5)∫amAA dx=∫mAdA dx,

For practical purposes expression (5) can be rewritten as

(6)∫amAA dx=0.5cardA

The article can classify fuzzy numbers with respect to the ”distribution” of their cardinality as follows: a fuzzy number A is called

(I)
a fuzzy number with equally heavy tails if ∫abA dx=∫cdA dx ,
(II)
a fuzzy number with light tails if max {∫abA dx,∫cdA dx}≤0.5∫adA dx ,
(III)
a fuzzy number with heavy left tail if ∫abA dx>0.5∫adA dx ,
(IV)
a fuzzy number with heavy right tail if ∫cdA dx>0.5∫adA dx .

Now the article will study location of the median value m_A in the support of A. The article will also identify the fuzziness of m_A determined by its membership grade A(m_A).

Theorem 2.

If A is a exponential fuzzy number with light tails then

(7)mA=w2(b+c)+w2n(d−c−b+a)+w2n[(b−a)e−n−(d−c)e−n]

and A(m_A) = 1 .

Proof:

mA=w2(b+c)+0.5(∫cdA(x) dx−∫abA(x) dx)=w2(b+c)+0.5(∫cdw[e−[(x−c)/(d−c)]]ndx−∫abw[e−[(b−x)/(b−a)]]ndx)=w2(b+c)+0.5(w(d−c)n[1−e−n]−w(b−a)n[1−e−n])=w2(b+c)+w2n(d−c−b+a)+w2n[(b−a)e−n−(d−c)e−n].

Corollary 1.

If A has equally heavy tails then mA=w2(b+c) and A(m_A) = 1.

Theorem 3.

The median value of a exponential fuzzy number A characterized by (1) is

(8)∫amAA dx=w2n(b−a+d−c)−w2n[(b−a)e−n+(d−c)e−n]+w2(c−b)

Proof:

With use of theorem 1.,

∫amAA dx=0.5cardA=12[wn(b−a+d−c)−wn[(b−a)e−n+(d−c)e−n]+w(c−b)]=w2n(b−a+d−c)−w2n[(b−a)e−n+(d−c)e−n]+w2(c−b).

4. Covariance of Exponential Fuzzy Numbers

Definition 3. [26]

The random variable ξ has absolutely continuous distribution F_ξ (x), if there a function of density for this distribution, i.e.

(9)∃fξ(u)≥0:Fξ(x)=∫−∞xfξ(u) du

Properties of density:

(i).
f_ξ (x) = F′_ξ (x),
(ii).
∫−∞∞fξ(u) du=1 .

The most common examples of absolutely continuous distribution are the normal, uniform and exponential distributions.

The important numerical property of the random variable is the mathematical expectation.

Definition 4. [26]

In discrete case mathematical expectation of random variable is equal to sum of the products of probabilities of random variable and its values in each element of the space, i.e.:

(10)E=∑w∈Ωp(w)*ξ(w).

And for the absolutely continuous distribution the mathematical expectation is equal to integral over the space of density functions multiplied by the value of a random variable, i.e.

(11)E=∫x∈Ωxf(x) dx.

Theorem 4.

The mathematical expectation of a exponential fuzzy number A characterized by (1) is

(12)E(A)=w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2

Proof:

E(A)=∫−∞∞xA dx=∫abwx[e−[(b−x)/(b−a)]]ndx+∫bcwx dx+∫cdwx[e−[(x−c)/(d−c)]]ndx=[w(b−a)n(b−ae−n)+w(b−a)2n2(e−n−1)]+w(c2−b2)2+[w(d−c)2n2(1−e−n)+w(d−c)n(c−de−n)]=w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2.

Definition 5.

Let A be fuzzy number. We define the variance of A by

(13)Var(A)=E(x−E(x))2=E(x2)−E2(x)

Theorem 5.

let A be fuzzy number. Then

(14)Var(A)=[wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))]−[w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2]2

Proof:

We know of equ.(12) that

(15)E(x)=w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2

So we should obtain E(x²),

E(x2)=∫−∞∞x2A dx=∫abwx2[e−[(b−x)/(b−a)]]ndx+∫bcwx2 dx+∫cdwx2[e−[(x−c)/(d−c)]]ndx=[w(b−a)n(b2−a2e−n)−2(b−a)n[w(b−a)n(b−ae−n)+w(b−a)2n2(e−n−1)]]+w(c3−b3)3+[w(d−c)n(c2−d2e−n)+2(d−c)n[w(d−c)2n2(1−e−n)+w(d−c)n(c−de−n)]]=wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))

So

(16)E(x2)=wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))

Therefore with use equs((15),(16)) we have

Var(A)=E(x−E(x))2=E(x2)−E2(x)=wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))−{w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2}2.

Definition 6.

The covariance between fuzzy numbers A and B is defined as

(17)Cov(x,y)=E(x.y)−E(x)E(y)

Theorem 6.

let A be exponential fuzzy number. Then

(18)Cov(A,A)=Var(A)

Proof:

Let A = (a,b,c,d)_E, A exponential fuzzy number with shape function is

fA(x)={w[e−[(b−x)/(b−a)]]na≤x≤b,wb≤x≤c,w[e−[(x−c)/(d−c)]]nc≤x≤d,

From (17) we deduce

(19)Cov(A,A)=E(A.A)−E(A)E(A)

first E(A.A),

(20)E(A.A)=∫−∞∞x.xA dx=∫−∞∞x2A dx=E(x2)

Of (16) we know

E(x2)=wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b)),

Now E(A)E(A),

(21)E(A)E(A)=E2(A)

Of (15) we know

E(x)=w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2

So we deduce

E2(x)={w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2}2

Putting value of (20) and (21) in (19) yields

Cov(A,A)=wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)+2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))−{w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2}2=E(x2)−E2(x)=Var(A).

So Cov(A,A) = Var(A) .

Corollary 2.

let A,B and C be three exponential trapezoidal fuzzy number and let λ, μ ∈ [0, 1], Then the following properties hold,

(i)
Cov(A,B) = Cov(B,A),
(ii)
Cov(A,λ + μA) = μVar(A),
(iii)
Cov(λA + μB,C) = λCov(A,C) + μCov(B,C).

Theorem 7.

let A and B be two exponential trapezoidal fuzzy number and let λ, μ ∈ [0, 1], then

(22)Var(λA+μB)=λ2Var(A)+μ2Var(B)+2λμCov(A,B)

Proof:

From (18) we deduce

(23)Var(λA+μB)=Cov(λA+μB,λA+μB)

Therefore

(24)Cov(λA+μB,λA+μB)=Cov(λA,λA)+Cov(λA,μB)+Cov(μB,λA)+Cov(μB,μB)

To prove Theorem 7, we need the following.

(i)
Cov(λA,λA),
(ii)
Cov(λA, μB),
(iii)
Cov(μB,λA),
(iv)
Cov(μB, μB).

(i) Cov(λA,λA)=Var(λA)=E(λ2A2)−(E(λA))2E(λA)=∫−∞∞xλA dx==∫abwxλ[e−[(b−x)/(b−a)]]ndx+∫bcwxλ dx+∫cdwxλ[e−[(x−c)/(d−c)]]ndx=λw(e−n−1)n2[(b−a)2−(d−c)2]+λwn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+λw(c2−b2)2.E(λ2x2)=∫−∞∞λ2x2A dx=∫abwλ2x2[e−[(b−x)/(b−a)]]ndx+∫bcwλ2x2 dx+∫cdwλ2x2[e−[(x−c)/(d−c)]]ndxE(λ2x2)=λ2wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2λ2w(1−e−n)n3((d−c)3+(b−a)3)+2λ2wn2((d−c)2(c−de−n)+(b−a)2(ae−n−b))⇒E(λ2A2)−(E(λA))2=[λ2wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2λ2w(1−e−n)n3((d−c)3+(b−a)3)]−[λw(e−n−1)n2[(b−a)2−(d−c)2]+λwn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+λw(c2−b2)2]2=λ2[wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)]−λ2[w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2]2=λ2([wn(b−a+d−c)(b2+c2−e−n(a2+d2))+2w(1−e−n)n3((d−c)3+(b−a)3)]−[w(e−n−1)n2[(b−a)2−(d−c)2]+wn[(b−a)(b−ae−n)+(d−c)(c−de−n)]+w(c2−b2)2]2)=λ2(E(A2)−E2(A))=λ2Var(A)

So

(25)⇒(i) Cov(λA,λA)=λ2(E(A2)−E2(A))=λ2Var(A)

The same way, we deduce

(26)(iv) Cov(μB,μB)=Var(μB)=E(μ2B2)−(E(μB))2=μ2(E(B2)−E2(B))=μ2Var(B)

Now, prove for (ii) Cov(λA, μB),

From (17) we deduce

Cov(λA,λB)=E(λA.μB)−E(λA)E(μB)=λμE(A.B)−λE(A)μE(B)=λμE(A.B)−λμE(A)E(B)=λμ(E(A.B)−E(A)E(B))=λμCov(A,B)

So

(27)(ii) Cov(λA,μB)=λμCov(A,B)

The same way, we deduce

Cov(μB,λA)=μλE(B.A)−μλE(B)E(A)=μλ(E(B.A)−E(B)E(A))=μλCov(B,A)

By using of Corollary 2.(first properties), we know Cov(B,A) = Cov(A,B), Then

(28)(iii) Cov(μB,λA)=λμCov(A,B)

Putting (25), (26), (27) and (28) in (24) yields

(29)Cov(λA+μB,λA+μB)=λ2Var(A)+μ2Var(B)+λμCov(A,B)+λμCov(A,B)

Putting (29) in (23) yields

Var(λA+μB)=λ2Var(A)+μ2Var(B)+2λμCov(A,B).

5. Ranking Variance in Exponential Fuzzy Numbers

Let A = (a₁,b₁,α₁,β₁;w₁) and B = (a₂,b₂,α₂,β₂;w₂) be two generalized exponential trapezoidal fuzzy number, then use the following steps to compare A,B

*
step 1: Find E(x)_A and E(x)_B,
*
step 2: Find E(x²)_A and E(x²)_B,
*
step 3: Find Var(A) and Var(B),
*
step 4: Use of following definition for comparing fuzzy numbers.

Definition 7.

Let A, B be trapezoidal fuzzy numbers, then

(i)
A > B ⇔ Var(A) > Var(B),
(ii)
A < B ⇔ Var(A) < Var(B),
(iii)
A ∼ B ⇔ Var(A) ∼ Var(B).

Remark 1.

In this section suppose n = 1. A fuzzy number A with shape function where n > 0, will be denoted by A = (a,b,α,β;w)_n, if n = 1, we simply write A = (a,b,α,β;w), which is known as a trapezoidal fuzzy number.

5.1. Results

In this section, seven sets of fuzzy numbers are compared using the proposed approach and existing approaches. The results are shown in Table 1.

Approaches	Ex.1	Ex.2	Ex.3	Ex.4	Ex.5	Ex.6	Ex.7
Cheng [7]	A < B	A ∼ B	Error	A ∼ B	A > B	A < B < C	Error
Chu [9]	A < B	A ∼ B	Error	A < B	A > B	A < B < C	Error
Chen [4]	A < B	A < B	A < B	A < B	A > B	A < C < B	A > B
Abbasbandy [1]	Error	A ∼ B	A < B	A ∼ B	A < B	A < B < C	A > B
Chen [6]	A < B	A < B	A < B	A < B	A > B	A < B < C	A > B
Kumar [14]	A > B	A ∼ B	A < B	A < B	A > B	A < B < C	A > B
S. Rezvani [21]	A > B	A > B	A < B	A < B	A < B	A < B < C	A > B
Proposed approach	A > B	A < B	A > B	A ∼ B	A < B	A < B < C	A > B

Table (1):

A comparison of the ranking results for different approaches

Example 1.

Let A = (0.2,0.4,0.6,0.8;0.35) and B = (0.1,0.2,0.3,0.4;0.7) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.0798 and E(x)_B = 0.0392,
*
step 2: E(x²)_A = 0.0443 and E(x²)_B = 0.011,
*
step 3: Var(A) = 0.0443 − (0.0798)² = 0.0379 and Var(B) = 0.011 − (0.0392)² = 0.0095,
*
step 4: Var(A) > Var(B) then A > B.

Example 2.

Let A = (0.1,0.2,0.4,0.5;1) and B = (0.1,0.3,0.3,0.5;1) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.098 and E(x)_B = 0.076,
*
step 2: E(x²)_A = 0.0247 and E(x²)_B = 0.043,
*
step 3: Var(A) = 0.0247 − (0.098)² = 0.0151 and Var(B) = 0.043 − (0.076)² = 0.0372,
*
step 4: Var(A) < Var(B) then A < B.

Example 3.

Let A = (0.1,0.2,0.4,0.5;1) and B = (1,1,1,1;1) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.098 and E(x)_B = 0,
*
step 2: E(x²)_A = 0.0247 and E(x²)_B = 0,
*
step 3: Var(A) = 0.0247 − (0.098)² = 0.0151 and Var(B) = 0,
*
step 4: Var(A) > Var(B) then A > B.

Example 4.

Let A = (−0.5,−0.3,−0.3,−0.1;1) and B = (0.1,0.3,0.3,0.5;1) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = −0.076 and E(x)_B = 0.076,
*
step 2: E(x²)_A = 0.043 and E(x²)_B = 0.043,
*
step 3: Var(A) = 0.043 − (−0.076)² = 0.0372 and Var(B) = 0.043 − (0.076)² = 0.0372,
*
step 4: Var(A) ∼ Var(B) then A ∼ B.

Example 5.

Let A = (0.3,0.5,0.5,1;1) and B = (0.1,0.6,0.6,0.8;1) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.2753 and E(x)_B = 0.2097,
*
step 2: E(x²)_A = 0.2714 and E(x²)_B = 0.2484,
*
step 3: Var(A) = 0.2714 − (0.2753)² = 0.1954 and Var(B) = 0.2484 − (0.2097)² = 0.2044,
*
step 4: Var(A) < Var(B) then A < B.

Example 6.

Let A = (0,0.4,0.6,0.8;1) and B = (0.2,0.5,0.5,0.9;1) and C = (0.1,0.6,0.7,0.8;1) be three generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.2464 and E(x)_B = 0.2411 and E(x)_C = 0.2348,
*
step 2: E(x²)_A = 0.1615 and E(x²)_B = 0.2247 and E(x²)_C = 0.253,
*
step 3: Var(A) = 0.1615 − (0.2464)² = 0.1005 and Var(B) = 0.2247 − (0.2411)² = 0.1667 and Var(C) = 0.253 − (0.2348)² = 0.198,
*
step 4: Var(A) < Var(B) < Var(C) then A < B < C.

Example 7.

Let A = (0.1,0.2,0.4,0.5;1) and B = (−2,0,0,2;1) be two generalized trapezoidal fuzzy number, then

*
step 1: E(x)_A = 0.098 and E(x)_B = 0,
*
step 2: E(x²)_A = 0.0247 and E(x²)_B = −3.36,
*
step 3: Var(A) = 0.0247 − (0.098)² = 0.0151 and Var(B) = −3.36 − (0)² = −3.36,
*
step 4: Var(A) > Var(B) then A > B.

The results of example (8) are shown in Table 2.

Authors	Fuzzy number	Set 1	Set 2	Set 3	Set 4	Set 5
Wang et al.2009	A₁	0	0	0	0	0.792
	A₂	0	0	0.444	1.100	0.784
	A₃		1.857	0.444
Ranking results		A₁ ∼ A₂	A₁ ∼ A₂ < A₃	A₁ < A₂ ∼ A₃	A₁ < A₂	A₂ < A₁

Abbasbandy, p = 1,2006	A₁	3	8	8	8	8
	A₂	5	8.500	10	7.500	8.500
	A₃		9.500	9
Ranking results		A₁ < A₂	A₁ < A₂ < A₃	A₁ < A₃ < A₂	A₂ < A₁	A₁ < A₂

Abbasbandy, p = 2,2006	A₁	2.309	5.889	6.218	5.944	5.889
	A₂	4.472	6.377	7.916	5.598	6.377
	A₃		6.831	7.257
Ranking results		A₁ < A₂	A₁ < A₂ < A₃	A₁ < A₃ < A₂	A₂ < A₁	A₁ < A₂

Cheng.1998	A₁	1.725	4.031	4.358	3.707	4.031
	A₂	3.027	4.035	5.025	3.768	4.035
	A₃		4.694	5.020
Ranking results		A₁ < A₂	A₁ < A₂ < A₃	A₁ < A₃ < A₂	A₁ < A₂	A₁ < A₂

Chu.2002	A₁	0.741	2	1.986	1.986	2
	A₂	1.200	2.118	2.500	1.908	2.118
	A₃		2.374	2.222
Ranking results		A₁ < A₂	A₁ < A₂ < A₃	A₁ < A₃ < A₂	A₂ < A₁	A₁ < A₂

Y. Deng.2006	A₁	0.707	1.667	1.850	1.546	1.667
	A₂	1.354	1.690	2.850	2.086	1.690
	A₃		1.922	2.167
Ranking results		A₁ < A₂	A₁ < A₂ < A₃	A₁ < A₃ < A₂	A₁ < A₂	A₁ < A₂

Cheng.1998	A₁	0.133	0.167	0.397	0.242	0.167
	A₂	0.667	0.292	0.433	0.151	0.292
	A₃		0.083	0.630
Ranking results		A₂ < A₁	A₂ < A₁ < A₃	A₃ < A₂ < A₁	A₁ < A₂	A₂ < A₁

Proposed method	A₁,A₂,A₃	A₁ < A₂	A₁ < A₂ < A₃	A₂ < A₃ < A₁	A₂ < A₁	A₁ < A₂

Table (2):

The results of ranking fuzzy numbers in Example 8.

Example 8.

Consider the following sets of fuzzy sets:

Set 1 :
A₁ = (1,1,1,3), A₂ = (1,1,1,7).
Set 2 :
A₁ = (2,4,4,6), A₂ = (1,5,5,6), A₃ = (3,5,5,6).
Set 3 :
A₁ = (2,3,3,8), A₂ = (2,3,7,8), A₃ = (2,3,3,10).
Set 4 :
A₁ = (1,5,5,5), A₂ = (2,3,5,5).
Set 5 :
A₁ = (2,4,4,6), A₂ = (1,5,5,6).

According to the deviation degree method, we have:

In Set 1 :
- *
  step 1: E(x)_A = 0.31 and E(x)_B = 7.14,
- *
  step 2: E(x²)_A = 5.8 and E(x²)_B = 58.68,
- *
  step 3: Var(A) = 5.8−(0.31)² = 5.7 and Var(B) = 58.68 − (7.14)² = 7.7,
- *
  step 4: Var(A) < Var(B) then A < B.
In Set 2 :
- *
  step 1: E(x)_A = 10.08 and E(x)_B = 11.85 and E(x)_C = 8.67,
- *
  step 2: E(x²)_A = 69.12 and E(x²)_B = 60.37 and E(x²)_C = 80.79,
- *
  step 3: Var(A) = 69.12 − (10.08)² = −95.61 and Var(B) = 60.37 − (11.85)² = −80.05 and Var(C) = 80.79 − (8.67)² = 5.62,
- *
  step 4: Var(A) < Var(B) < Var(C) then A < B < C.
In Set 3 :
- *
  step 1: E(x)_A = 17.58 and E(x)_B = 26.3 and E(x)_C = 27.6,
- *
  step 2: E(x²)_A = 113.28 and E(x²)_B = 71.76 and E(x²)_C = 196.48,
- *
  step 3: Var(A) = 113.28 − (17.58)² = −195.78 and Var(B) = 71.76 − (26.3)² = −619.93 and Var(C) = 196.48 − (27.6)² = −565.28,
- *
  step 4: Var(B) < Var(C) < Var(A) then B < C < A.
In Set 4 :
- *
  step 1: E(x)_A = 8.44 and E(x)_B = 9.63,
- *
  step 2: E(x²)_A = 94 and E(x²)_B = 20.01,
- *
  step 3: Var(A) = 94 − (8.44)² = 22.77 and Var(B) = 20.01 − (9.63)² = −72.72,
- *
  step 4: Var(B) < Var(A) then B < A.
In Set 5 :
- *
  step 1: E(x)_A = 10.08 and E(x)_B = 11.85,
- *
  step 2: E(x²)_A = 69.12 and E(x²)_B = 60.37,
- *
  step 3: Var(A) = 69.12 − (10.08)² = −95.61 and Var(B) = 60.37 − (11.85)² = −80.05,
- *
  step 4: Var(A) < Var(B) then A < B.

6. Conclusion

In this paper, we proposed a new method to ranking exponential fuzzy numbers by variance of fuzzy numbers. The variance used in this method is obtained from the mathematical expectation and then calculating by E(A), E²(A) and then finding the possibilistic variance between in ranking fuzzy numbers A and B. The main advantage of the proposed approach is that the proposed approach provides the correct ordering of generalized and normal trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. This method can effectively rank various fuzzy numbers.

Figures

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 9 - 1
Pages: 10 - 24
Publication Date: 2016/01/01
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.1080/18756891.2016.1144150 How to use a DOI?
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TY  - JOUR
AU  - S. Rezvani
PY  - 2016
DA  - 2016/01/01
TI  - Cardinal, Median Value, Variance and Covariance of Exponential Fuzzy Numbers with Shape Function and its Applications in Ranking Fuzzy Numbers
JO  - International Journal of Computational Intelligence Systems
SP  - 10
EP  - 24
VL  - 9
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.1080/18756891.2016.1144150
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