# A New Distance Measure and Ranking Method for Generalized Trapezoidal Fuzzy Numbers

Authors: T. Allahviranloo; M. A. Jahantigh; S. Hajighasemi
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

This study presents an approximate approach for ranking fuzzy numbers based on the centroid point of a fuzzy number and its area. The total approximate is determined by convex combining of fuzzy number’s relative and its area that is based on decision maker’s optimistic perspectives. The proposed approach is simple in terms of computational efforts and is efficient in ranking a large quantity of fuzzy numbers. A group of examples by Bortolan and Degani (1985) demonstrate the accuracy and applicability of the proposed approach. Finally by this approach, a new measure is introduced between two fuzzy numbers.


---

## Body

## 1. Introduction

The fuzzy set theory pioneered by Zadeh [1] has been extensively used. Fuzzy numbers or fuzzy subsets of the real line R are applied to represent the imprecise numerical measurements of different alternatives. Therefore, comparing the different alternatives is actually comparing the resulting fuzzy numbers.Also in many applications of fuzzy set theory to decision making, we need to know the best select from a collection of possible solutions. This selection process may require that we rank or order fuzzy numbers.Several researchers presented ranking methods [2–7]. Various techniques are applied to compare the fuzzy numbers. Some of the exiting approaches are difficult to understand and have suffered from different plights, for example, the lack of discrimination, producing counterintuitive orderings, and ultimately resulting in inconsistent ordering if a new fuzzy number is added. Also nearly all approaches should acquire membership functions of fuzzy numbers before the ranking is performed; however, this may be infeasible in real applications. Furthermore, accuracy and efficiency should be of priority concern in the ranking process if ranking a large amount of fuzzy numbers. In light of the above discussion, specially in [6], Chen and Lu have proposed an approximate approach for ranking fuzzy numbers, in that they worked with dominance.Also we studied here the approach is determined by convex combining the centroid point(x0,y0) of x0 and area s of a fuzzy number that performs simple arithmetic operations for the ranking purpose, and it can be applied to rank the combination case of some fuzzy numbers and crisp numbers and the case of discrete fuzzy numbers and it is useful in ranking a large quantity of fuzzy numbers. Comparing the proposed approximate approach with the existing methods using both Bortolan and Degani’s examples [3].The methods of measuring the distance between fuzzy numbers have became important due to the significant applications in diverse fields like remote sensing, data mining, pattern recognition, multivariate data analysis and so on. Several distance measures for precise numbers are well established in the literature. Several researchers focused on computing the distance between fuzzy numbers [2, 3, 5, 7–10]. Here we introduce a new distance between two trapezoidal fuzzy numbers by the new approach proposed for them.This paper is organized as follows.In Section2, the basic concept of fuzzy number operation is brought. Section 3 introduces the ranking approach and presents some comparative examples which demonstrate the accuracy of the proposed approach over the exiting methods. A new measure between fuzzy numbers is defined in Section 4. Concluding remarks are finally made in Section 5.


## 2. Preliminaries

A fuzzy number is a convex fuzzy subset of real lineR and is completely defined by its membership function. Let A be a fuzzy number, whose membership function A(x) can generally be defined as
(1)A(x)={AL(x),fora≤x<bw,forb≤x<cAR(x),forc<x≤d0,Otherwise,
where 0<w≤1 is a constant, AL:[a,b)→[0,w] is monotonic increasing continuous from the right function, and AR:(c,d]→[0,w] is monotonic decreasing continuous from the left function. If w=1, then A is a normal fuzzy number; otherwise, it is said to be a nonnormal fuzzy number. If the membership function A(x) is piecewise linear and continuous, then A is referred to as a trapezoidal fuzzy number and is usually denoted by A=(a,b,c,d;w) or A=(a,b,c,d) if w=1. In this case, AL(x)=(w/(b-a))(x-a), a≤x<b, and AR(x)=(w/(c-d))(x-d), c≤x<d. In particular, when b=c, the trapezoidal fuzzy number is reduced to a triangular fuzzy number denoted by A=(a,b,d;w) or A=(a,b,d) if w=1 [11].Definition 1 (see [11]).
Consider a general trapezoidal fuzzy numberA=[a,b,c,d;w], whose membership function is defined as
(2)A(x)={wx-ab-a,fora≤x<bw,forb≤x<cwd-xd-c,forc<x≤d0,Otherwise,
where 0<w≤1. The centroid point (x0,y0) of A has been determined by
(3)x0=13[a+b+c+d-dc-ab(d+c)-(a+b)],y0=w3[1+c-d(d+c)-(a+b)].Definition 2.
Consider a general trapezoidal fuzzy numberA=[a,b,c,d;w]; the area of A determined by
(4)s=w2((d+c)-(a+b)).


## 3. The Ranking Method

LetA be a trapezoidal generalized fuzzy number. We define the particular convex combination of the fuzzy number’s relative locations on the x-axis, x0, and the fuzzy number’s area, s, in Definitions 1 and 2, respectively, as follows
(5)Dλ(A)=λx0+(1-λ)s,
where the index λ∈[0,1].The following choices are related to find approximate approach for rankings fuzzy numbers based on decision maker’s opinions.A decision maker is able to denote his preference for the fuzzy number’s relative locations on thex-axis, x, or the fuzzy number’s area, s, with ordering suitable index λ in obtaining approximate approach for ranking fuzzy numbers. If the decision maker has a preference for x0 then λ must be chosen in (1/2,1]. Also if the area s is a choice then λ must be chosen in [0,1/2).It is clear that forλ=0, the choice is just s and for λ=1 is x0. If λ=1/2 then both x0 and s have the same effects.Now a decision maker can rank a pair of fuzzy numbers,A and B, using Dλ(A) and Dλ(B) based on the following rules: (i)
Dλ(A)<Dλ(B), then A≺B,(ii)
Dλ(A)=Dλ(B), then A≈B,(iii)
Dλ(A)>Dλ(B), then A≻B.We can see calculation results of the proposed method in Table1 for the twelve-pair sets in Figure 1 that were used in [4].Table 1
Case studies using Boltolan and Degani's examples in Figure1.
Chang
Kerre
Lee-Lee
Liou-Wang
Jain
Proposed
O
G
γ
=
1
γ
=
.
5
γ
=
0
K
=
1
K
=
2
K
=
.
5
λ
=
1
λ
=
.
5
λ
=
0
A
1
.02
.80
0
0
.15
.1
.05
.18
.03
.40
.1
.1
.1
A
2
.18
1
1
1
.95
.9
.85
.90
.84
.95
.9
.5
.1
B
1
.14
.80
0
0
.75
.7
.65
.72
.55
.84
.7
.4
.1
B
2
.18
1
1
1
.95
.9
.85
.90
.84
.95
.9
.5
.1
C
1
.16
.85
0
0
.85
.8
.75
.81
.70
.90
.8
.45
.1
C
2
.09
1
1
1
.98
.95
.93
.95
.92
.97
.95
.5
.5
D
1
.04
.80
0
0
.25
.2
.15
.32
.12
.55
.2
.15
.1
D
2
.08
.80
0
.5
.45
.4
.35
.55
.33
.72
.1
.25
.4
D
3
.14
1
1
1
.75
.7
.65
.89
.80
.94
.1
.4
.7
E
1
0
.89
0
0
.5
.25
0
.09
0
.26
.03
.04
.05
E
2
.12
.85
0
.5
.65
.6
.55
.63
.42
.78
.6
.35
.1
E
3
.10
1
1
1
1
.98
.95
1
1
1
.96
.5
.05
F
1
.40
.96
.67
1
.75
.59
.43
.66
.53
.78
.55
.43
.32
F
2
.34
.89
.32
.32
.75
.59
.43
.69
.51
.81
.61
.47
.32
G
1
.58
.51
.40
.23
.75
.4
.05
.66
.53
.78
.41
.55
.7
G
2
.12
.89
.60
1
.65
.6
.55
.63
.42
.78
.6
.35
.1
H
1
.58
.42
.40
.13
.75
.4
.05
.66
.53
.78
.41
.55
.7
H
2
.14
.95
.70
1
.75
.7
.65
.72
.54
.84
.7
.4
.1
I
1
.46
1
.65
1
.95
.8
.65
.90
.84
.95
.76
.53
.3
I
2
.41
.86
.35
.61
.85
.7
.55
.76
.65
.86
.7
.5
.3
I
3
.38
.76
.20
.18
.75
.6
.45
.66
.54
.78
.63
.46
.3
J
1
.28
1
.65
1
.8
.7
.6
.82
.71
.89
.7
.45
.2
J
2
.37
.91
.34
.60
.8
.65
.5
.82
.71
.89
.63
.46
.3
J
3
.52
.75
.24
.23
.8
.58
.35
.82
.71
.89
.57
.51
.45
K
1
.56
1
.62
1
.85
.63
.4
.90
.82
.94
.62
.53
.45
K
2
.33
.85
.38
.62
.7
.55
.4
.69
.56
.80
.56
.43
.3
K
3
.20
.75
.24
.21
.6
.5
.4
.64
.45
.77
.5
.35
.2
L
1
.29
.91
.50
1
.65
.5
.35
.73
.60
.83
.5
.4
.3
L
2
.10
.91
.50
1
.55
.5
.45
.67
.48
.80
.5
.3
.1Figure 1Comparisons of the proposed method to the exiting methods for the twelve-pair sets in Figure1 are given in Tables 2 and 3.Table 2
Comparison of other methods for the examples in Figure1.
Chang
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
3
≺
E
2
F
1
≻
F
2
Kerre
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≈
D
2
≺
D
3
E
2
≺
E
1
≺
E
3
F
1
≻
F
2
Lee-Lee
O
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≈
D
2
≺
D
3
E
1
≈
E
2
≺
E
3
F
1
≻
F
2
G
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≻
F
2
Liou-Wang
γ=1
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≈
F
2
γ=.5
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≈
F
2
γ=0
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≈
F
2
Jain
k
=
1
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≻
F
2
k
=
2
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≻
F
2
k
=
.
5
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≈
F
2
Proposed
λ=1
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≻
F
2
λ=.5
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≺
D
2
≺
D
3
E
1
≺
E
2
≺
E
3
F
1
≻
F
2
λ=0
A
1
≺
A
2
B
1
≺
B
2
C
1
≺
C
2
D
1
≈
D
2
≈
D
3
E
1
≈
E
3
≺
E
2
F
1
≈
F
2Table 3
Comparison of other methods for the examples in Figure1.
Chang
G
1
≻
G
2
H
1
≻
H
2
I
3
≺
I
2
≺
I
1
J
1
≺
J
2
≺
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2
Kerre
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≺
K
2
≺
K
1
L
1
≈
L
2
Lee-Lee
O
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≺
K
2
≺
K
1
L
1
≈
L
2
G
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≺
K
2
≺
K
1
L
1
≈
L
2
Liou-Wang
γ=1
G
1
≺
G
2
H
1
≈
H
2
I
3
≺
I
2
≺
I
1
J
1
≈
J
2
≈
J
3
K
3
≺
K
2
≺
K
1
L
1
≺
L
2
γ=.5
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≺
K
2
≺
K
1
L
1
≈
L
2
γ=0
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≈
K
2
≈
K
1
L
1
≺
L
2
Jain
k
=
1
G
1
≻
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
1
≈
J
2
≈
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2
k
=
2
G
1
≻
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
1
≈
J
2
≈
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2
k
=
.
5
G
1
≈
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
1
≈
J
2
≈
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2
Proposed
λ=1
G
1
≺
G
2
H
1
≺
H
2
I
3
≺
I
2
≺
I
1
J
3
≺
J
2
≺
J
1
K
3
≺
K
2
≺
K
1
L
1
≈
L
2
λ=.5
G
1
≻
G
2
H
1
≻
H
2
I
3
≺
I
2
≺
I
1
J
1
≺
J
2
≺
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2
λ=0
G
1
≻
G
2
H
1
≻
H
2
I
1
≈
I
2
≈
I
3
J
1
≺
J
2
≺
J
3
K
3
≺
K
2
≺
K
1
L
1
≻
L
2


## 4. A New Measure between Two Trapezoidal Generalized Fuzzy Numbers

LetA and B be two arbitrary trapezoidal fuzzy numbers with x0 and s defined by (1) and (2), respectively. The distance between A and B is defined as
(6)d(A,B)=Dλ(A)-Dλ(B).A decision maker can order a pair of trapezoidal fuzzy numbers,A and B, too using d(A,B) based on the following rules: (i)
d(A,B)>0, then A≻B,(ii)
d(A,B)=0, then A≈B,(iii)
d(A,B)<0, then A≺B.


## 5. Some Properties

Some valuable properties are described in the following, which are useful in ranking a large quantity of generalized trapezoidal fuzzy numbers simultaneously. Assume that there are three different generalized trapezoidal fuzzy numbers,A, B, and C, to be ranked: (i)
d(A,A)=0(ii)
d(A,B)=-d(B,A),(iii)
ifd(A,B)>0, and d(B,C)>0 then d(A,C)>0,(iv)
more than two fuzzy numbers can be ranked by comparing with the benchmark fuzzy number. letA be the benchmark, d(A,B)=a, and d(A,C)=b. By using the previous two properties, obviously d(A,B)=d(A,C)-d(B,C)=b-a. Therefore, if b>a, then d(B,C)>0; that is, B≻C,(v)
the ranking if more than two fuzzy numbers has the robustness property, that is,(7)ifd(A,B)<ε,then|d(A,C)-d(B,C)|<ε. This equation suggests that the dominance difference between one fuzzy number and the other two fuzzy numbers is insignificant, if the two fuzzy numbers are close to each other. These properties hold since d(A,C)-d(B,C)=d(A,B).


## 6. Conclusion

Ranking fuzzy numbers is a critical task in a fuzzy decision making process. Particularly, when ranking a large quantity of fuzzy numbers and only limited information about them can be obtained, an effective, efficient, and accurate ranking method becomes necessary. The proposed ranking approach only considers convex combining of the centroid point(x0,y0) of x0 and area s of a fuzzy number with a decision maker’s optimistic perspectives, that is, actually the measurement of the degree of difference between two fuzzy numbers. This ranking is compared with some exiting ranking methods.Here too, a new distance measure has been introduced for computing crisp distances for fuzzy numbers.



---

*Source: 623757-2013-05-14.xml*