# Methods in Ranking Fuzzy Numbers: A Unified Index and Comparative Reviews

Authors: Thanh-Lam Nguyen
Journal: Complexity
 (2017)

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## Abstract

Fuzzy set theory, extensively applied in abundant disciplines, has been recognized as a plausible tool in dealing with uncertain and vague information due to its prowess in mathematically manipulating the knowledge of imprecision. In fuzzy-data comparisons, exploring the general ranking measure that is capable of consistently differentiating the magnitude of fuzzy numbers has widely captivated academics’ attention. To date, numerous indices have been established; however, counterintuition, less discrimination, and/or inconsistency on their fuzzy-number rating outcomes have prohibited their comprehensive implementation. To ameliorate their manifested ranking weaknesses, this paper proposes a unified index that multiplies weighted-mean and weighted-area discriminatory components of a fuzzy number, respectively, called centroid value and attitude-incorporated left-and-right area. From theoretical proof of consistency property and comparative studies for triangular, triangular-and-trapezoidal mixed, and nonlinear fuzzy numbers, the unified index demonstrates conspicuous ranking gains in terms of intuition support, consistency, reliability, and computational simplicity capability. More importantly, the unified index possesses the consistency property for ranking fuzzy numbers and their images as well as for symmetric fuzzy numbers with an identical altitude which is a rather critical property for accurate matching and/or retrieval of information in the field of computer vision and image pattern recognition.


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## Body

## 1. Introduction

It has been well recognized that uncertainty inevitably exists in several real-world phenomena due to the inherent errors or impreciseness of measurement tools, methods, and uncontrollable conditions [1, 2]. In managing the uncertainty and vagueness, the fuzzy set theory has been widely considered as a powerful tool [3, 4]. And many scholars have made special efforts in proposing more and more effective approaches to deal with practical problems in the fuzzy environment. Since the inception of the fuzzy set theory, Soliman and Mantawy [5] showed that five major strongly connected branches have been developed, including fuzzy mathematics, fuzzy logic and artificial intelligence, fuzzy systems, uncertainty and information, and fuzzy decision-making. Their subbranches have also been established; for example, fuzzy differential equations [6–14] and fuzzy integrodifferential equations [15–22] are of fuzzy mathematics while fuzzy-number ranking, the focus of this paper, is of fuzzy decision-making. Specifically, based on its feasible mathematical capacity for representing the imprecise information in practice, we have observed many successful cases spreading in disparate disciplines, such as robot selection [23], supplier selection [24], logistics center allocation [25], facility location determination [26], choosing mining methods [27], manufacturing process monitoring [1, 2, 28–31], cutting force prediction [32], firm-environmental knowledge management [33, 34], green supply-chain operation [35], and weapon procurement decision [36]. Apparently, to find their best alternative, those decisive problems are evaluated under resource constraints and with to some extent linguistic preference of multiattribute, which is realized from users’ perspectives, as well as subjective quantification of multiple characteristics, which is assessed from decision-makers [2, 3, 37–39]. In these cases, fuzzy-data comparisons and rankings are inevitable.As the fuzzy data (fuzzy numbers) can overlap with each other and are represented by possibility distributions, their comparison and ordering, not akin to that of real numbers which can be linearly ordered, become challenging and cumbersome. Generally, to rank fuzzy quantities, a set of fuzzy numbers, through a specific defuzzification measure, is converted into real numbers, where a natural order between them is definitive [40]. However, even when ordering for a set of single fuzzy numbers, this defuzzification procedure does lose a certain amount of fuzziness/imprecision information existing in the original data [1, 40–47], not to mention the ordering for problems of multicriteria decision-making, where sets of fuzzy numbers have experienced some mathematical operations [48]; therefore, much endeavor has been attempted to minimize loss of information, a fundamental problem for fuzzy-data analysis.Jain [49] in 1977 first launched a fuzzy set rating procedure for multiple-aspect decision-making. Since then, exploring a general ranking measure, capable of consistently differentiating the magnitude of fuzzy numbers, has widely captivated academics’ attention [50]. Nowadays, a majority of diverse improved approaches/indices established from wide-range perspectives focus on either compensating their predecessors’ failures in certain reasonable properties for ordering of fuzzy quantities [43, 44] or resolving the counterintuitive, indiscriminate, and/or inconsistent rating outcomes among certain types of fuzzy numbers [42, 51–54].In general, the existing ranking measures can be classified into two main categories:(i)
Indices that value the fuzzy number itself such as center-, area-, and deviation-driven ordering measures(ii)
Indices that not only evaluate the fuzzy number itself, but also gauge decision-maker’s attitude in regard to specific purposes such as confidence and riskIn category one, Yager [55] and Lee and Li [56] first borrowed statisticalcenter-oriented measures for assessing fuzzy numbers, where the former constructed a centroid (weighted mean) index and the latter developed mean and standard deviation indices; however, Cheng [57] pointed out their inefficient manipulation of the fuzzy numbers that possesses unusually large or small data (outliers) and mean-and-spread values. To cope with the inefficiencies, R. Saneifard and R. Saneifard [58], Zhang et al. [59], Bodjanova [60, 61], and Yamashiro [62] suggested a median index, a resistant measure of the center, to take into account data located on the tails; Cheng [57] proposed coefficient-of-variation and distance indices; but both indices were later criticized for some inconsistent ordering among specific types of fuzzy numbers [63]. Based on the area between the centroid point and the original point, Chu and Tsao [63] succeeded in establishing anarea-driven ranking index; unfortunately, because of its inherent computation flaw, the area index was questioned by Wang and Lee [64] who illustrated some numerical examples to show its counterintuitive results and further provided a compelling revised index to resolve the problem. Nonetheless, Wang and Lee’s area index does have its own deficiency of ordering correctness when encountering fuzzy numbers with identical centroid points [65]. By defining fuzzy-number maximal and minimal reference sets, Wang et al. [66] first introduced adeviation-driven ordering index by combining right-and-left deviation degree with the coefficient of relative variation; not surprisingly, this index was argued (1) bearing mathematical incapability with zero value in the denominator [53] and pointed out (2) leaving substantial room for improvement under some special occasions such as fuzzy numbers with the same left, right, and total utilities [39] as well as ranking fuzzy numbers’ images [46].Emphatically, the aforementioned drawbacks plagued on thisdeviation-driven ordering index have somewhat reignited the development ofcategory two, initially proposed by Liou and Wang [67] in 1992, and contrived ranking measures that not only evaluate the fuzzy number itself, but also consider decision-maker’s attitude in relation to specific purposes. The evidence can be seen in the most recent works; for example, to remove shortages of Wang et al.’s deviation-degree index [66], Wang and Luo [39] incorporated decision-maker’s attitude towards risk into left-and-right area between fuzzy-number points and the positive-and-negative ideal points; to improve Liou and Wang’s index [67], Yu and Dat [48] incorporated decision-maker’s attitude regarding confidence into left-right-total integral value subjected to fuzzy-number median value. More recently, Das and Guha [68] proposed a new ranking approach by computing the centroid point of trapezoidal intuitionistic fuzzy numbers (TrIFN) and applied it to solve multicriteria decision-making problems in combination with expert’s degree of satisfaction. However, their formulas fail to effectively work when their TrIFN (a,b,c,d) becomes either (a,a,c,d) or (a,b,c,c) or the satisfaction/dissatisfaction degree takes a value of zero. In addition, as shown in Table 1, certain shortcomings such as counterintuition, less reliability, inconsistency, complex/laborious computation, and indecisive ranking results have been found to be existing in several current ranking approaches.Table 1
The ranking performance assessments for some representative indices as opposed to the unified index.
Section
Example
Evaluated fuzzy numbers
Compared references
Shortcomings (cf. the index)
Section4.1
Example15
A~1=(1,4,5)
Yu & Dat [48]
More laborious in computation
A
~
2
=
(
2,3
,
6
)
Section4.1
Example16
A
~
1
=
(
5,6
,
7
)
Chu & Tsao [63]
Counterintuition
A
~
2
=
(
5.9,6
,
7
)
Cheng [57]
Counterintuition
A
~
3
=
(
6,6
,
7
)
Yu & Dat [48]
Less reliability
Section4.1
Example17
A
~
1
=
(
1,3
,
5
)
Liou & Wang [67], Yu & Dat [48]
InconsistencyCounterintuition at λ=0
A~2=(2,3,4)
A
~
3
=
(
1,4
,
6
)
Section4.2
Example18
A
~
1
=
(
1,5
,
5
)
Zhang et al. [73]
Computation complexity
A
~
2
=
(
2,3
,
5,5
)
Inconsistency
Section4.2
Example19
A
~
1
=
(
0,3
,
6
)
Ky Phuc et al. [38], Asady [46]
Computation complexityIndecisive ranking for (A~1,A~3)
A
~
2
=
(
-
1,0
,
2
)
A
~
3
=
(
0,2
,
4,6
)
Section4.2
Example20
A
~
1
=
(
-
12,1
,
2
)
Abbasbandy & Hajjari [74], Nasseri & Sohrabi [75]
Counterintuition
A
~
2
=
(
-
23
/
12,1
/
12,13
/
12
)
A
~
3
=
(
-
6,0
,
1,1
)
Section4.3
Example22
A
~
1
=
(
1,2
,
5
)
Ky Phuc et al. [38], Asady [46], Zhang et al. [73]
More elaborate in computation
A
~
2
=
(
1,2
,
2,4
)Ostensibly, as opposed to the prolific ranking indices to date that have been presented incategory one, the established ranking indices related tocategory two are still few, leaving a wide range of topics for further investigation. Based on the integration of the two categories, this paper proposes a unified index that multiplies weighted mean and weighted area, two discriminatory components of a fuzzy number, respectively, called centroid value(the category one measurement) and attitude-incorporated left-and-right area(the category two measurement). According to comprehensively comparative studies from triangular, triangular-and-trapezoidal mixed, and nonlinear fuzzy numbers, the unified index demonstrates obtrusive ranking benefits with respect to intuition support, computational easiness, consistency, and reliability capability.Aside from the Introduction, the remainder of this paper is organized into four sections as follows. Section2 provides preliminary definitions and remarks for the research. The proposed unified index is described in Section 3, whose comparative studies with some existing ranking indices are done with several literature-exemplary fuzzy numbers in Section 4. Summary and conclusions make up the last section.


## 2. Preliminaries

The following definitions and remarks are mainly adopted from Zimmermann [69] and Lee [70].Definition 1 (fuzzy subset).
LetR be a nonempty set. The fuzzy subset A~ of R is defined by a function ξA~:R→[0,1]. ξA~ is called amembership function.Definition 2 (α-cut set).
Theα-cut set of A~, denoted by A~αc, is defined by A~αc=x∈R:ξA~x≥α for all α∈(0,1]. The 0-cut set A~0c is defined as the closure of the set x∈R:ξA~x>0.Definition 3 (α-level set).
Theα-level set of A~, denoted by A~α, is defined by A~α=x∈R:ξA~x=α for all α∈[0,1].Definition 4 (fuzzy number).
A fuzzy numberA~=(a,b,c,d;w) is described as any fuzzy subset of the real line R with the membership function ξA~(x) which is given by(1)ξA~x=ξA~Lx,a≤x<bw,b≤x≤cξA~Rx,c<x≤d0,otherwise,where 0≤w≤1 is a constant and ξA~L(x), ξA~R(x) are continuous functions on [0,1].
A fuzzy number has the following properties:(i)
A~ is normal if there exists an x∈R such that ξA~(x)=1; that is, w=1.(ii)
ξA~(x) is fuzzy convex; that is, ξA~(tx+(1-t)y)≥minξA~x,ξA~y for t∈[0,1].(iii)
ξA~(x) is upper semicontinuous; that is, x∈R:ξA~x≥α is a closed subset of R for each α∈(0,1].(iv)
The0-level set A~0 is a closed and bounded subset of R.SinceA~α⊂A~0 for each α∈(0,1], condition (iv) shows that the α-level sets A~α are bounded subsets of R for all α∈(0,1]. It is well known that condition (ii) is satisfied if and only if the α-level set A~α is a convex subset of R. Therefore, from conditions (i)–(iv), it is implied that if A~ is a fuzzy number, then the α-level set of A~ is a closed, bounded, and convex subset of R, that is, a closed interval in R, denoted by A~α=[A~αL,A~αU].Remark 5.
LetA~ be a fuzzy number. Then, the following statements hold true: (i)
A~αL≤A~αU for all α∈[0,1].(ii)
A~αL is increasing with respect to α∈[0,1]; that is, A~αL≤A~βL for 0≤α<β≤1.(iii)
A~αU is decreasing with respect to α∈[0,1]; that is, A~αU≥A~βU for 0≤α<β≤1.Remark 6.
LetA~ be a fuzzy number such that its membership function is strictly increasing on interval [a,b] and strictly decreasing on interval [c,d]. From the fact of strict monotonicity, ξA~L(x) and ξA~R(x) are continuous functions on [0,1]. This implies that A~ is also a real fuzzy number.Definition 7 (the image of a fuzzy number [4]).
Letn fuzzy numbers be A~i=(ai,bi,ci,di;wi)(i=1,n¯). Then, the image of A~i is A~i′=(-di,-ci,-bi,-ai;wi), as shown in Figure 1.Figure 1
A
~
i
′ is the image of A~i.


## 3. A Unified Index

Based on integration of the two aforementioned categories for ranking fuzzy numbers, a unified index, which combines centroid value (weighted mean) and attitude-incorporated left-and-right area (weighted area), is proposed in this section.Definition 8 (centroid value (a center-driven measure that belongs to category one)).
Centroid value of a fuzzy numberA~i=(ai,bi,ci,di;wi) for i=1,n¯, symbolized by CVi, is defined as [3, 4, 38, 63, 65, 71](2)CVi=∫aidixξA~ixdx∫aidiξA~ixdx.From the statistical point of view, it is the weighted mean of A~i, meaning that when A~i=(a,a,a,a;wi), we can accordingly have CVi=a.Definition 9 (left-and-right areas (an area-driven measure that belongs to category one)).
Left-and-right areas of a fuzzy numberA~i for i=1,n¯, denoted by SiL and SiR, are given by(3)SiL=∫0wigA~iLydy,SiR=∫0wigA~iRydy,where gA~iL(y) and gA~iR(y) stand for inverse functions of the left-and-right membership functions, ξA~iL(x) and ξA~iR(x), respectively, and visual views of SiL and SiR are shown in Figure 2 [72].Figure 2
Left areaSiL and right area SiR.Now, a fuzzy-number measure belonging to category two is presented. It also contemplates decision-maker’s attitude as regards data revelation, called attitude-incorporated left-and-right area, signified byAAiλ.(4)AAiλ=λSiR+1-λSiL,where λ∈[0,1] is level of optimism reflecting a data-revelation optimism degree of a decision-maker, where the larger the λ set by the decision-maker is, the more optimistic attitude the decision-maker has on the data revelation. Two extreme cases are λ=0, meaning the decision-maker is completely pessimistic, and λ=1, meaning the decision-maker is completely optimistic. Case λ=1/2 reflects a neutral decision attitude. From the mathematical viewpoint, (4) can be seen as a weighted-area value of A~i.For boosting the fuzzy-number discrimination power, let us consider an index namedUIiλ by multiplying two size-discriminatory values of a fuzzy number; that is,(5)UIiλ=CVi+εiλSiR+1-λSiL.U
I
i
λ is calledunified index. And, εi initially takes a very small real number which is quantifiable and rational for comparing the targeted fuzzy numbers whose centroid values take a value of zero, CVi=0. It is used to provide consistent ranking power when CVi=0. Particularly, this paper suggests using εi=wi×10-9 so that we can efficiently rank fuzzy numbers that have similar centroids but different height.Remark 10.
Consider the ranking of two fuzzy numbers,A~i and A~j. Given the data-optimistic level λ, from (5), we obtain their realized unified indices, UIiλ and UIjλ. Then, the following decisions can be made: (i)
At the data-optimistic levelλ, if UIiλ>UIjλ, then A~i≻A~j.(ii)
At the data-optimistic levelλ, if UIiλ<UIjλ, then A~i≺A~j.(iii)
At the data-optimistic levelλ, if UIiλ=UIjλ, then A~i≃A~j.Now, we will prove the unified index’s consistency property when ranking fuzzy numbers and their images. Without loss of generality,CVi≠0 is considered in the following.Proposition 11.
LetA~i′=(-di,-ci,-bi,-ai;wi) be the image of a fuzzy number A~i=(ai,bi,ci,di;wi) for i=1,n¯. Its centroid value is CVi′=-CVi, left-and-right areas are Si′R=SiL and Si′L=SiR, attitude-incorporated left-and-right area is AAi′λ=AAi1-λ and AAi′1-λ=AAiλ, and unified index is UIi′λ=-UIi1-λ and UIi′1-λ=-UIiλ.Proof.
From (2),(6)CVi′=∫-di-aixξA~i′xdx∫-di-aiξA~i′xdx=-∫aidixξA~ixdx∫aidiξA~ixdx=-CVi.Based on (3),(7)Si′L=∫0wigA~i′Lydy=∫0wigA~iRydy=SiRSi′R=∫0wigA~i′Rydy=∫0wigA~iLydy=SiL.According to (4) and with the above results, Si′R=SiL and Si′L=SiR, we further have(8)AAi′λ=λSi′R+1-λSi′L=λSiL+1-λSiR=AAi1-λ.Similarly,(9)AAi′1-λ=1-λSi′R+λSi′L=1-λSiL+λSiR=AAiλ.Finally, regarding (5) and the aforementioned outcomes, we can simply obtain(10)UIi′λ=CVi′λSi′R+1-λSi′L=-UIi1-λ,UIi′1-λ=CVi′1-λSi′R+λSi′L=-UIiλ.We complete the proof.Proposition 12.
Let a set of fuzzy numbers beA~k=(ak,bk,ck,dk;wk) and their images A~k′=(-dk,-ck,-bk,-ak;wk), k=1,n¯. For a pairwise comparison of A~i and A~j for i,j∈k, two statements hold true: (1) UIiλ>UIjλ if and only if UIi′1-λ<UIj′1-λ and (2) UIiλ<UIjλ if and only if UIi′1-λ>UIj′1-λ.Proof.
ConsiderUIiλ>UIjλ. From Proposition 11, we have the results UIiλ=-UIi′1-λ and UIjλ=-UIj′1-λ. Thus, UIi′1-λ<UIj′1-λ. On the other hand, consider UIi′1-λ<UIj′1-λ. According to Proposition 11, UIi′1-λ=-UIiλ and UIj′1-λ=-UIjλ. Hence, UIiλ>UIjλ. Overall, the proof is completed.Remark 13.
Let a set of fuzzy numbers beA~k=(ak,bk,ck,dk;wk) and their images A~k′=(-dk,-ck,-bk,-ak;wk), k=1,n¯. As regards Remark 10 and Propositions 11 and 12, the following decisions can be made for a pairwise comparison of A~i and A~j, for i,j∈k.(i)
At the data-optimistic levelλ, if UIiλ>UIjλ, which is equivalent to UIi′1-λ<UIj′1-λ, then A~i≻A~j, which is equivalent to A~i′≺A~j′.(ii)
At the data-optimistic levelλ, if UIiλ<UIjλ, which is equivalent to UIi′1-λ>UIj′1-λ, then A~i≺A~j, which is equivalent to A~i′≻A~j′.(iii)
At the data-optimistic levelλ, if UIiλ=UIjλ, which is equivalent to UIi′1-λ=UIj′1-λ, then A~i≃A~j, which is equivalent to A~i′≃A~j′.Finally, the following theory is very useful for ranking “symmetric” fuzzy numbers with an identical altitude.Theorem 14.
Consider a set of “symmetric” fuzzy numbers,A~k=(ak,bk,ck,dk;wk), and their images A~k′=(-dk,-ck,-bk,-ak;wk), k=1,n¯. By using the unified index, the pairwise comparison of A~i and A~j for i,j∈k is λ=0.5, A~i≃A~j(A~i′≃A~j′), λ∈[0,0.5), A~i≺A~j(A~i′≻A~j′), and λ∈(0.5,1], A~i≻A~j(A~i′≺A~j′).Proof.
(i) SinceA~i=(ai,bi,ci,di;w) and A~j=(aj,bj,cj,dj;w) for i,j=1,n¯ are symmetric, we have ai+di=aj+dj. Moreover, from (2),(11)CVi=∫aidixξA~ixdx∫aidiξA~ixdx=ai+di2,CVj=∫ajdjxξA~jxdx∫ajdjξA~jxdx=aj+dj2.
Therefore,CVi=CVj.
(ii) According to (3) and (4), we have(12)AAiλ=λSiR+1-λSiLAAjλ=λSjR+1-λSjL,where(13)SiL=∫0wgA~iLydy,SiR=∫0wgA~iRydySjL=∫0wgA~jLydy,SjR=∫0wgA~jRydy.
Due to the symmetry, we haveAAiλ<AAjλ for λ∈[0,0.5), AAiλ=AAjλ when λ=0.5, and vice versa.
(iii) From (i), (ii), and (5), we have (i)
λ∈[0,0.5), UIiλ<UIjλ,(ii)
λ=0.5, UIiλ=UIjλ,(iii)
λ∈(0.5,1], UIiλ>UIjλ.
Finally, according to Remark13, we complete the proof.


## 4. Comparative Studies

In this section, several fuzzy-number examples, which are popular in the literature for a wide range of fuzzy-number comparative studies, are used to compare ranking performance between the unified index and some up-to-date representative indices from the publications. To make it easier to follow the whole discussion of comparison, Table1 briefly shows the evaluated types of fuzzy numbers, reference sources, and critical shortcomings of the references. Detailed explanations about performance shortages for existing indices in contrast with the proposed index are subsequently described in Examples 15~22.It can be noted that, based on Propositions11 and 12 and Remark 13, the unified index fulfills the consistency property for ranking the fuzzy numbers and their partnered images; for conciseness, in several examples, the consistency of image-ranking results is not mentioned or shown on the result tables.


### 4.1. Ranking of Normal Triangular Fuzzy Numbers

This subsection focuses on the ranking of normal triangular fuzzy numbers with some special shape which are recognizably difficult to discriminate in the literature. First, a case with two congruent fuzzy numbers is employed for checking index’s computation easiness; then, the work is extended on three similar fuzzy numbers for contrasting indices’ ranking consistency and intuition satisfaction; finally, an example, which includes a slight move-away fuzzy number and two fuzzy numbers with an identical center value and geometric enlargement relationship, is examined with respect to ranking indices’ reliability and consistency.Example 15.
Rank two fuzzy numbersA~1=(1,4,5) and A~2=(2,3,6) as shown in Figure 3 [48], which are congruent, but overlapping after flipping and sliding movement. Here, the proposed unified index is contrasted with the most recent work published by Yu and Dat [48] in 2014 as regards computation simpleness.
According to the unified index in (5), we simply have the results shown in Table 2, A~1≺A~2 (A~1′≻A~2′) at any arbitrary level-of-optimism attitude of data revelation from the decision-maker, λ∈[0,1]. Yu and Dat [48] advocated the identical ranking result in this case; however, their computation of median values before ranking these two fuzzy numbers is procedure-laborious in practice as reported by some predecessors [58–62].
By the same token, when comparing two normal triangular fuzzy numbersB~1=(0.1,0.6,0.8) and B~2=(0.2,0.5,0.9), taken from [76] and based on the proposed approach, we always have B~1≺B~2, which is coherent with that in [57, 63, 77–80]. However, the approaches by R. Chutia and B. Chutia [81] and Deng [82] lead to a counterintuitive result B~2≺B~1.Table 2
Ranking results for Example15.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
8.333
9.167
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.1
9.000
9.900
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.2
9.667
10.633
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.3
10.333
11.367
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.4
11.000
12.100
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.5
11.667
12.833
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.6
12.333
13.567
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.7
13.000
14.300
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.8
13.667
15.033
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.9
14.333
15.767
A
~
1
≺
A
~
2 and A~1′≻A~2′
1.0
15.000
16.500
A
~
1
≺
A
~
2 and A~1′≻A~2′Figure 3
Fuzzy numbersA~1 and A~2 in Example 15.Example 16.
Consider three triangle fuzzy numbers,A~1=(5,6,7), A~2=(5.9,6,7), and A~3=(6,6,7) [39], which are similar and covered with the same right-hand side as displayed in Figure 4. By human instinct, they are easily being discriminated; that is, for the fuzzy numbers and their images, the intuitive and consistent rankings are A~1≺A~2≺A~3 and A~1′≻A~2′≻A~3′. Therefore, this example is capable of judging the indices’ performance if intuition- and consistency-satisfied.
We first check the unified index. Based on (5), Propositions 11 and 12, and Remark 13, the ranking results, listed in Table 3 for the fuzzy numbers and their images, affirm the intuitive and consistent outcomes, A~1≺A~2≺A~3 and A~1′≻A~2′≻A~3′.
In the literature, while many support the intuitive results for ranking the fuzzy numbers [39, 46, 57, 66, 83, 84], Chen [85] and Chu and Tsao [63] provide a different consequence as A~1≺A~3≺A~2 and Cheng [57] gives A~3≺A~2≺A~1, so their counterintuitions are apparent.
Moreover, due to scarcity of methods in the literature for consistently ranking their images, a recent work from Yu and Dat [48] claimed to bridge the gap. Unfortunately, when λ=1, their approach leads to a disparate ranking, A~1≃A~2≃A~3 (A~1′≃A~2′≃A~3′), indicating that their index as a whole somewhat lacks reliability.Table 3
Ranking results for Example16.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
33.000
37.485
38.000
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.1
33.600
37.831
38.317
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.2
34.200
38.178
38.633
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.3
34.800
38.524
38.950
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.4
35.400
38.871
39.267
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.5
36.000
39.217
39.583
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.6
36.600
39.564
39.900
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.7
37.200
39.910
40.217
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.8
37.800
40.257
40.533
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.9
38.400
40.603
40.850
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
1.0
39.000
40.950
41.167
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′Figure 4
Fuzzy numbersA~1, A~2, and A~3 in Example 16.Example 17.
Again, examine three fuzzy numbers,A~1=(1,3,5), A~2=(2,3,4), and A~3=(1,4,6), as shown in Figure 5. Visibly, A~3=(1,4,6) is right way out A~1 and A~2, so there is no dispute that a capable index should rate A~3 (A~3′) as the largest (smallest). The challenging one is to distinguish A~1 and A~2 (A~1′ and A~2′) due to their symmetry with respect to x=3, identical centroid value, and their geometric enlargement relationship. Actually, majority of the existing ranking measures in category one (evaluating the fuzzy number itself) rank A~1≃A~2, and their image ranking is not available. Therefore, this example is to compare the proposed unified index with the category two ranking measures (not only evaluating the fuzzy number itself, but also gauging decision-maker’s attitude in regard to specific purposes such as confidence and risk), initiated by Wang and Luo [39], Yu and Dat [48], Yu et al. [65], and Liou and Wang [67], in terms of ranking indices’ reliability and consistency.
First, we check the unified index’s results in Table4. Regardless of λ∈[0,1], A~3 (A~3′) is always the largest (smallest), which confirms human intuition. For the ranking of A~1 and A~2, dividing from λ=0.5, A~1≃A~2 (A~1′≃A~2′); the upper part λ∈[0,0.5), A~1≺A~2 (A~1′≻A~2′); the lower part λ∈(0.5,1], A~1≻A~2 (A~1′≺A~2′). Although this result has been proved in Theorem 14, there are still some insightful conclusions to be addressed.
First, this finding is consistent with that of Wang and Luo [39] and Yu et al. [65]. In fact, with respect to the unified index, these results are reasonable because the chosen λ value manifests the decision-maker’s optimism towards revelation of left- and right-area data. λ∈(0.5,1] implies that the right-area data is more preferred by the decision-maker; λ∈[0,0.5) represents the notion that the decision-maker is more optimistic regarding the left-area data; λ=0.5 indicates that the decision-maker is neutral towards preference of data location.
Then, we evaluate the indices proposed by Yu and Dat [48] and Liou and Wang [67]. While Yu and Dat’s work confirms most of the results in Table 4, it does exhibit an apparent counterintuition issue at λ=0, where it suggests that A~3 does not dominate A~2; that is, A~2≃A~3 (A~2′≃A~3′). Moreover, Liou and Wang’s index [67] not only afflicts the same shortage of Yu and Dat’s index, but also has shown inconsistent results for ranking the fuzzy numbers and their images due to the index’s limited definition and generalization.Table 4
Ranking results at different optimism levels in Example17.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
6.000
7.500
9.167
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.1
6.600
7.800
10.083
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.2
7.200
8.100
11.000
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.3
7.800
8.400
11.917
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.4
8.400
8.700
12.833
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.5
9.000
9.000
13.750
A
~
1
≃
A
~
2
≺
A
~
3 and A~1′≃A~2′≻A~3′
0.6
9.600
9.300
14.667
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.7
10.200
9.600
15.583
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.8
10.800
9.900
16.500
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.9
11.400
10.200
17.417
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
1.0
12.000
10.500
18.333
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′Figure 5
Fuzzy numbersA~1, A~2, and A~3 in Example 17.


### 4.2. Ranking for Normal Triangular-and-Trapezoid Mixed Fuzzy Numbers

Here, the proposed unified index is used to broaden the ranking comparisons to normal triangular-and-trapezoid mixed fuzzy numbers. The cases from the literature that have one trapezoid mixed with one triangular fuzzy number, followed by two examples with two triangular fuzzy numbers, are investigated.Example 18.
Compare a triangular fuzzy numberA~1=(1,5,5) overlapping with a trapezoidal fuzzy number A~2=(2,3,5,5), as shown in Figure 6. Of ten existing measures that have been studied in this case, three (30%) support A~1≺A~2 [30, 66, 86] and seven (70%) stand for A~1≻A~2 [47, 53, 63, 73, 74, 83, 87]. Clearly, this stark contrast outcome is intriguing for further investigation. Therefore, in this example, we first attempt to explain the predecessors’ conflicting consequence by using the unified index. Then, the index itself will be compared with the recent work proposed by Zhang et al. in 2014 [73] to lay out their result similarity as well as their performance with regard to computation easiness and image consistency.
Table5 is the ranking results of using the unified index, where λ∈[0,0.8], A~1≻A~2 and λ∈0.9,1, A~1≺A~2. Once more, the chosen λ value manifests the decision-maker’s optimism towards revelation of the left-and-right area of fuzzy data. From the λ-probability point of view, around 80% support A~1≻A~2 and 20% favor A~1≺A~2. In fact, this result, providing a level-of-optimism attitude-based explanation for conflicts among the comparison, is interesting to be approximate with aforementioned percentages obtained from the literature conclusions. Moreover, it is also similar to Zhang et al.’s [73] result who uses a preference-probability relation to explain the uncertainty level of the comparison; with seven intricate and somewhat complicated steps, they concluded A~1≻A~2 with a confidence degree of 73% and A~1≺A~2 with 27%.
Finally, it is worth mentioning that as opposed to the unified index, Zhang et al.’s [73] seven-step algorithm for ranking fuzzy numbers not only suffers a computation-complexity problem, but also lacks capacity for ranking the fuzzy-number image.Table 5
Ranking results at different optimism levels in Example18.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
11.000
9.333
A
~
1
≻
A
~
2
0.1
11.733
10.267
A
~
1
≻
A
~
2
0.2
12.467
11.200
A
~
1
≻
A
~
2
0.3
13.200
12.133
A
~
1
≻
A
~
2
0.4
13.933
13.067
A
~
1
≻
A
~
2
0.5
14.667
14.000
A
~
1
≻
A
~
2
0.6
15.400
14.933
A
~
1
≻
A
~
2
0.7
16.133
15.867
A
~
1
≻
A
~
2
0.8
16.867
16.800
A
~
1
≻
A
~
2
0.9
17.600
17.733
A
~
1
≺
A
~
2
1.0
18.333
18.667
A
~
1
≺
A
~
2Figure 6
Fuzzy numbersA~1 and A~2 in Example 18.Example 19.
Taken from [38] and shown in Figure 7, one trapezoid fuzzy number, A~3=(0,2,4,6), mingled with two triangular fuzzy numbers, A~1=(0,3,6) and A~2=(-1,0,2), is considered in this example. Noticeably, A~2 left distances away from A~1 and A~3, so there is no argument that a reliable index should discriminate A~2 (A~2′) as the smallest (largest). The question is the rating result of the triangular fuzzy number A~1 and the trapezoid fuzzy number A~3 and their images. Therefore, this example is to compare the unified index with the recent works of Asady in 2010 and Ky Phuc et al. [38] in 2012 who proposed a deviation-degree ranking measure.
First, we check the unified index’s results in Table6. Regardless of λ∈[0,1], A~2 (A~2′) is always the smallest (largest), which confirms human intuition. For the ranking of A~1 and A~3, dividing from λ=0.5, A~1≃A~3 (A~1′≃A~3′); the upper part λ∈[0,0.5), A~1≻A~3 (A~1′≺A~3′); the lower part λ∈(0.5,1], A~1≺A~3 (A~1′≻A~3′). Literally, this finding (refer to Theorem 14) is consistent with two fuzzy numbers with the same attitude and symmetry, shown in Example 17.
Then, we evaluate Ky Phuc et al.’s [38] and Asady’s [46] deviation-degree index. Despite the exhausted computation, its capability can only provide the partial result, “A~1≻A~2” and “A~3≻A~2,” leaving undecided ranking for A~1 and A~3. Actually, as mentioned in Section 1, the deviation-degree index, belonging to the category one ranking measure, has the limitation for ranking the fuzzy numbers akin to A~1 and A~3 that are overlapping and each has axis-of-symmetry property.Table 6
Ranking results of the three fuzzy numbers in Example19.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
4.500
0.167
3.000
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.1
5.400
0.183
4.200
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.2
6.300
0.200
5.400
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.3
7.200
0.217
6.600
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.4
8.100
0.233
7.800
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.5
9.000
0.250
9.000
A
~
1
≃
A
~
3
≻
A
~
2 and A~1′≃A~3′≺A~2′
0.6
9.900
0.267
10.200
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.7
10.800
0.283
11.400
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.8
11.700
0.300
12.600
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.9
12.600
0.317
13.800
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
1.0
13.500
0.333
15.000
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′Figure 7
Fuzzy numbersA~1, A~2, and A~3 in Example 19.Example 20.
Additionally, let us consider one trapezoidal fuzzy number,A~3=(-6,0,1,1), blended with two triangular fuzzy numbers, A~1=(-12,1,2) and A~2=(-23/12,1/12,13/12), which are adapted from [66] and shown in Figure 8. Unlike the previous challenging one that is with a symmetric and triangle-embedded trapezoid shape, they are all left-skewed fuzzy numbers and easy to be distinguished by human perception; that is, A~1≺A~3≺A~2. Hence, for this subsection, this example is capable of judging the indices’ performance if intuition-satisfied.
The result in Table7, obtained with the unified index, clearly shows that A~1≺A~3≺A~2, which is identical to previous works in [46, 47, 63, 64, 66, 87]. However, counter results are claimed by Abbasbandy and Hajjari [74] who ranked them as A~1≃A~2≃A~3 and Nasseri and Sohrabi [75] who suggested A~2≺A~3≺A~1. Both works’ counterintuition is obvious.Table 7
Ranking results of the three fuzzy numbers in Example20.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
-
18.333
-
0.229
-
4.125
A
~
1
≺
A
~
3
≺
A
~
2
0.1
-
16.833
-
0.221
-
3.850
A
~
1
≺
A
~
3
≺
A
~
2
0.2
-
15.333
-
0.213
-
3.575
A
~
1
≺
A
~
3
≺
A
~
2
0.3
-
13.833
-
0.204
-
3.300
A
~
1
≺
A
~
3
≺
A
~
2
0.4
-
12.333
-
0.196
-
3.025
A
~
1
≺
A
~
3
≺
A
~
2
0.5
-
10.833
-
0.188
-
2.750
A
~
1
≺
A
~
3
≺
A
~
2
0.6
-
9.333
-
0.179
-
2.475
A
~
1
≺
A
~
3
≺
A
~
2
0.7
-
7.833
-
0.171
-
2.200
A
~
1
≺
A
~
3
≺
A
~
2
0.8
-
6.333
-
0.163
-
1.925
A
~
1
≺
A
~
3
≺
A
~
2
0.9
-
4.833
-
0.154
-
1.650
A
~
1
≺
A
~
3
≺
A
~
2
1.0
-
3.333
-
0.146
-
1.375
A
~
1
≺
A
~
3
≺
A
~
2Figure 8
Fuzzy numbersA~1, A~2, and A~3 in Example 20.Example 21.
Now, two special cases taken from R. Chutia and B. Chutia [81] are considered. The first set includes A~1=(0.1,0.1,0.1,0.1;0.8) and A~2=(-0.1,-0.1,-0.1,-0.1;1.0) which were ranked as A~1>A~2; and the second one includes B~1=(1,1,1,1;0.5) and B~2=(1,1,1,1;1.0) which were ranked as B~1<B~2. The proposed unified index also leads to similar conclusions as in [37, 78, 81, 88–90], indicating that the index can effectively work with crisp numbers as well.


### 4.3. Ranking for Nonlinear Fuzzy Numbers

Finally, although empirical phenomenon and human perception are rather unlikely to gather the nonlinear fuzzy numbers, this more general type can be justifiable for investigating the index’s computation easiness as well as adaptability.Example 22.
Let us consider two fuzzy numbers shown in Figure9, adapted from Liou and Wang [67]: A~1=(1,2,5) and A~2=(1,2,2,4) with a nonlinear membership function(14)fA~2x=1-x-221/2,1≤x≤2,1-14x-221/2,2≤x≤4,0,otherwise.
In this nonlinear case, by using the unified index, the conclusions in Table8, A~1≻A~2 (A~1′≺A~2′) for λ∈[0,1], do not add much complexity for the computation. Obviously, previous proposed measures in [53, 63, 66, 67, 76] possess the same conclusion and computation easiness. However, in recent works, Ky Phuc et al. [38], Asady [46], and Zhang et al. [73], their indices become more complicated and elaborate for ranking the nonlinear fuzzy numbers as well as their images.Table 8
Ranking results at different optimism levels in Example22.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
6.750
2.945
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.1
7.650
3.516
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.2
8.550
4.087
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.3
9.450
4.658
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.4
10.350
5.230
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.5
11.250
5.801
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.6
12.150
6.372
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.7
13.050
6.943
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.8
13.950
7.515
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.9
14.850
8.086
A
~
1
≻
A
~
2 and A~1′≺A~2′
1.0
15.750
8.657
A
~
1
≻
A
~
2 and A~1′≺A~2′Figure 9
Fuzzy numbersA~1 and A~2 in Example 22.


## 4.1. Ranking of Normal Triangular Fuzzy Numbers

This subsection focuses on the ranking of normal triangular fuzzy numbers with some special shape which are recognizably difficult to discriminate in the literature. First, a case with two congruent fuzzy numbers is employed for checking index’s computation easiness; then, the work is extended on three similar fuzzy numbers for contrasting indices’ ranking consistency and intuition satisfaction; finally, an example, which includes a slight move-away fuzzy number and two fuzzy numbers with an identical center value and geometric enlargement relationship, is examined with respect to ranking indices’ reliability and consistency.Example 15.
Rank two fuzzy numbersA~1=(1,4,5) and A~2=(2,3,6) as shown in Figure 3 [48], which are congruent, but overlapping after flipping and sliding movement. Here, the proposed unified index is contrasted with the most recent work published by Yu and Dat [48] in 2014 as regards computation simpleness.
According to the unified index in (5), we simply have the results shown in Table 2, A~1≺A~2 (A~1′≻A~2′) at any arbitrary level-of-optimism attitude of data revelation from the decision-maker, λ∈[0,1]. Yu and Dat [48] advocated the identical ranking result in this case; however, their computation of median values before ranking these two fuzzy numbers is procedure-laborious in practice as reported by some predecessors [58–62].
By the same token, when comparing two normal triangular fuzzy numbersB~1=(0.1,0.6,0.8) and B~2=(0.2,0.5,0.9), taken from [76] and based on the proposed approach, we always have B~1≺B~2, which is coherent with that in [57, 63, 77–80]. However, the approaches by R. Chutia and B. Chutia [81] and Deng [82] lead to a counterintuitive result B~2≺B~1.Table 2
Ranking results for Example15.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
8.333
9.167
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.1
9.000
9.900
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.2
9.667
10.633
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.3
10.333
11.367
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.4
11.000
12.100
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.5
11.667
12.833
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.6
12.333
13.567
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.7
13.000
14.300
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.8
13.667
15.033
A
~
1
≺
A
~
2 and A~1′≻A~2′
0.9
14.333
15.767
A
~
1
≺
A
~
2 and A~1′≻A~2′
1.0
15.000
16.500
A
~
1
≺
A
~
2 and A~1′≻A~2′Figure 3
Fuzzy numbersA~1 and A~2 in Example 15.Example 16.
Consider three triangle fuzzy numbers,A~1=(5,6,7), A~2=(5.9,6,7), and A~3=(6,6,7) [39], which are similar and covered with the same right-hand side as displayed in Figure 4. By human instinct, they are easily being discriminated; that is, for the fuzzy numbers and their images, the intuitive and consistent rankings are A~1≺A~2≺A~3 and A~1′≻A~2′≻A~3′. Therefore, this example is capable of judging the indices’ performance if intuition- and consistency-satisfied.
We first check the unified index. Based on (5), Propositions 11 and 12, and Remark 13, the ranking results, listed in Table 3 for the fuzzy numbers and their images, affirm the intuitive and consistent outcomes, A~1≺A~2≺A~3 and A~1′≻A~2′≻A~3′.
In the literature, while many support the intuitive results for ranking the fuzzy numbers [39, 46, 57, 66, 83, 84], Chen [85] and Chu and Tsao [63] provide a different consequence as A~1≺A~3≺A~2 and Cheng [57] gives A~3≺A~2≺A~1, so their counterintuitions are apparent.
Moreover, due to scarcity of methods in the literature for consistently ranking their images, a recent work from Yu and Dat [48] claimed to bridge the gap. Unfortunately, when λ=1, their approach leads to a disparate ranking, A~1≃A~2≃A~3 (A~1′≃A~2′≃A~3′), indicating that their index as a whole somewhat lacks reliability.Table 3
Ranking results for Example16.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
33.000
37.485
38.000
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.1
33.600
37.831
38.317
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.2
34.200
38.178
38.633
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.3
34.800
38.524
38.950
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.4
35.400
38.871
39.267
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.5
36.000
39.217
39.583
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.6
36.600
39.564
39.900
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.7
37.200
39.910
40.217
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.8
37.800
40.257
40.533
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.9
38.400
40.603
40.850
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
1.0
39.000
40.950
41.167
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′Figure 4
Fuzzy numbersA~1, A~2, and A~3 in Example 16.Example 17.
Again, examine three fuzzy numbers,A~1=(1,3,5), A~2=(2,3,4), and A~3=(1,4,6), as shown in Figure 5. Visibly, A~3=(1,4,6) is right way out A~1 and A~2, so there is no dispute that a capable index should rate A~3 (A~3′) as the largest (smallest). The challenging one is to distinguish A~1 and A~2 (A~1′ and A~2′) due to their symmetry with respect to x=3, identical centroid value, and their geometric enlargement relationship. Actually, majority of the existing ranking measures in category one (evaluating the fuzzy number itself) rank A~1≃A~2, and their image ranking is not available. Therefore, this example is to compare the proposed unified index with the category two ranking measures (not only evaluating the fuzzy number itself, but also gauging decision-maker’s attitude in regard to specific purposes such as confidence and risk), initiated by Wang and Luo [39], Yu and Dat [48], Yu et al. [65], and Liou and Wang [67], in terms of ranking indices’ reliability and consistency.
First, we check the unified index’s results in Table4. Regardless of λ∈[0,1], A~3 (A~3′) is always the largest (smallest), which confirms human intuition. For the ranking of A~1 and A~2, dividing from λ=0.5, A~1≃A~2 (A~1′≃A~2′); the upper part λ∈[0,0.5), A~1≺A~2 (A~1′≻A~2′); the lower part λ∈(0.5,1], A~1≻A~2 (A~1′≺A~2′). Although this result has been proved in Theorem 14, there are still some insightful conclusions to be addressed.
First, this finding is consistent with that of Wang and Luo [39] and Yu et al. [65]. In fact, with respect to the unified index, these results are reasonable because the chosen λ value manifests the decision-maker’s optimism towards revelation of left- and right-area data. λ∈(0.5,1] implies that the right-area data is more preferred by the decision-maker; λ∈[0,0.5) represents the notion that the decision-maker is more optimistic regarding the left-area data; λ=0.5 indicates that the decision-maker is neutral towards preference of data location.
Then, we evaluate the indices proposed by Yu and Dat [48] and Liou and Wang [67]. While Yu and Dat’s work confirms most of the results in Table 4, it does exhibit an apparent counterintuition issue at λ=0, where it suggests that A~3 does not dominate A~2; that is, A~2≃A~3 (A~2′≃A~3′). Moreover, Liou and Wang’s index [67] not only afflicts the same shortage of Yu and Dat’s index, but also has shown inconsistent results for ranking the fuzzy numbers and their images due to the index’s limited definition and generalization.Table 4
Ranking results at different optimism levels in Example17.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
6.000
7.500
9.167
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.1
6.600
7.800
10.083
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.2
7.200
8.100
11.000
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.3
7.800
8.400
11.917
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.4
8.400
8.700
12.833
A
~
1
≺
A
~
2
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.5
9.000
9.000
13.750
A
~
1
≃
A
~
2
≺
A
~
3 and A~1′≃A~2′≻A~3′
0.6
9.600
9.300
14.667
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.7
10.200
9.600
15.583
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.8
10.800
9.900
16.500
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
0.9
11.400
10.200
17.417
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′
1.0
12.000
10.500
18.333
A
~
2
≺
A
~
1
≺
A
~
3 and A~1′≻A~2′≻A~3′Figure 5
Fuzzy numbersA~1, A~2, and A~3 in Example 17.


## 4.2. Ranking for Normal Triangular-and-Trapezoid Mixed Fuzzy Numbers

Here, the proposed unified index is used to broaden the ranking comparisons to normal triangular-and-trapezoid mixed fuzzy numbers. The cases from the literature that have one trapezoid mixed with one triangular fuzzy number, followed by two examples with two triangular fuzzy numbers, are investigated.Example 18.
Compare a triangular fuzzy numberA~1=(1,5,5) overlapping with a trapezoidal fuzzy number A~2=(2,3,5,5), as shown in Figure 6. Of ten existing measures that have been studied in this case, three (30%) support A~1≺A~2 [30, 66, 86] and seven (70%) stand for A~1≻A~2 [47, 53, 63, 73, 74, 83, 87]. Clearly, this stark contrast outcome is intriguing for further investigation. Therefore, in this example, we first attempt to explain the predecessors’ conflicting consequence by using the unified index. Then, the index itself will be compared with the recent work proposed by Zhang et al. in 2014 [73] to lay out their result similarity as well as their performance with regard to computation easiness and image consistency.
Table5 is the ranking results of using the unified index, where λ∈[0,0.8], A~1≻A~2 and λ∈0.9,1, A~1≺A~2. Once more, the chosen λ value manifests the decision-maker’s optimism towards revelation of the left-and-right area of fuzzy data. From the λ-probability point of view, around 80% support A~1≻A~2 and 20% favor A~1≺A~2. In fact, this result, providing a level-of-optimism attitude-based explanation for conflicts among the comparison, is interesting to be approximate with aforementioned percentages obtained from the literature conclusions. Moreover, it is also similar to Zhang et al.’s [73] result who uses a preference-probability relation to explain the uncertainty level of the comparison; with seven intricate and somewhat complicated steps, they concluded A~1≻A~2 with a confidence degree of 73% and A~1≺A~2 with 27%.
Finally, it is worth mentioning that as opposed to the unified index, Zhang et al.’s [73] seven-step algorithm for ranking fuzzy numbers not only suffers a computation-complexity problem, but also lacks capacity for ranking the fuzzy-number image.Table 5
Ranking results at different optimism levels in Example18.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
11.000
9.333
A
~
1
≻
A
~
2
0.1
11.733
10.267
A
~
1
≻
A
~
2
0.2
12.467
11.200
A
~
1
≻
A
~
2
0.3
13.200
12.133
A
~
1
≻
A
~
2
0.4
13.933
13.067
A
~
1
≻
A
~
2
0.5
14.667
14.000
A
~
1
≻
A
~
2
0.6
15.400
14.933
A
~
1
≻
A
~
2
0.7
16.133
15.867
A
~
1
≻
A
~
2
0.8
16.867
16.800
A
~
1
≻
A
~
2
0.9
17.600
17.733
A
~
1
≺
A
~
2
1.0
18.333
18.667
A
~
1
≺
A
~
2Figure 6
Fuzzy numbersA~1 and A~2 in Example 18.Example 19.
Taken from [38] and shown in Figure 7, one trapezoid fuzzy number, A~3=(0,2,4,6), mingled with two triangular fuzzy numbers, A~1=(0,3,6) and A~2=(-1,0,2), is considered in this example. Noticeably, A~2 left distances away from A~1 and A~3, so there is no argument that a reliable index should discriminate A~2 (A~2′) as the smallest (largest). The question is the rating result of the triangular fuzzy number A~1 and the trapezoid fuzzy number A~3 and their images. Therefore, this example is to compare the unified index with the recent works of Asady in 2010 and Ky Phuc et al. [38] in 2012 who proposed a deviation-degree ranking measure.
First, we check the unified index’s results in Table6. Regardless of λ∈[0,1], A~2 (A~2′) is always the smallest (largest), which confirms human intuition. For the ranking of A~1 and A~3, dividing from λ=0.5, A~1≃A~3 (A~1′≃A~3′); the upper part λ∈[0,0.5), A~1≻A~3 (A~1′≺A~3′); the lower part λ∈(0.5,1], A~1≺A~3 (A~1′≻A~3′). Literally, this finding (refer to Theorem 14) is consistent with two fuzzy numbers with the same attitude and symmetry, shown in Example 17.
Then, we evaluate Ky Phuc et al.’s [38] and Asady’s [46] deviation-degree index. Despite the exhausted computation, its capability can only provide the partial result, “A~1≻A~2” and “A~3≻A~2,” leaving undecided ranking for A~1 and A~3. Actually, as mentioned in Section 1, the deviation-degree index, belonging to the category one ranking measure, has the limitation for ranking the fuzzy numbers akin to A~1 and A~3 that are overlapping and each has axis-of-symmetry property.Table 6
Ranking results of the three fuzzy numbers in Example19.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
4.500
0.167
3.000
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.1
5.400
0.183
4.200
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.2
6.300
0.200
5.400
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.3
7.200
0.217
6.600
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.4
8.100
0.233
7.800
A
~
1
≻
A
~
3
≻
A
~
2 and A~1′≺A~3′≺A~2′
0.5
9.000
0.250
9.000
A
~
1
≃
A
~
3
≻
A
~
2 and A~1′≃A~3′≺A~2′
0.6
9.900
0.267
10.200
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.7
10.800
0.283
11.400
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.8
11.700
0.300
12.600
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
0.9
12.600
0.317
13.800
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′
1.0
13.500
0.333
15.000
A
~
3
≻
A
~
1
≻
A
~
2 and A~3′≺A~1′≺A~2′Figure 7
Fuzzy numbersA~1, A~2, and A~3 in Example 19.Example 20.
Additionally, let us consider one trapezoidal fuzzy number,A~3=(-6,0,1,1), blended with two triangular fuzzy numbers, A~1=(-12,1,2) and A~2=(-23/12,1/12,13/12), which are adapted from [66] and shown in Figure 8. Unlike the previous challenging one that is with a symmetric and triangle-embedded trapezoid shape, they are all left-skewed fuzzy numbers and easy to be distinguished by human perception; that is, A~1≺A~3≺A~2. Hence, for this subsection, this example is capable of judging the indices’ performance if intuition-satisfied.
The result in Table7, obtained with the unified index, clearly shows that A~1≺A~3≺A~2, which is identical to previous works in [46, 47, 63, 64, 66, 87]. However, counter results are claimed by Abbasbandy and Hajjari [74] who ranked them as A~1≃A~2≃A~3 and Nasseri and Sohrabi [75] who suggested A~2≺A~3≺A~1. Both works’ counterintuition is obvious.Table 7
Ranking results of the three fuzzy numbers in Example20.
λ
U
I
1
λ
U
I
2
λ
U
I
3
λ
Ranking result
0.0
-
18.333
-
0.229
-
4.125
A
~
1
≺
A
~
3
≺
A
~
2
0.1
-
16.833
-
0.221
-
3.850
A
~
1
≺
A
~
3
≺
A
~
2
0.2
-
15.333
-
0.213
-
3.575
A
~
1
≺
A
~
3
≺
A
~
2
0.3
-
13.833
-
0.204
-
3.300
A
~
1
≺
A
~
3
≺
A
~
2
0.4
-
12.333
-
0.196
-
3.025
A
~
1
≺
A
~
3
≺
A
~
2
0.5
-
10.833
-
0.188
-
2.750
A
~
1
≺
A
~
3
≺
A
~
2
0.6
-
9.333
-
0.179
-
2.475
A
~
1
≺
A
~
3
≺
A
~
2
0.7
-
7.833
-
0.171
-
2.200
A
~
1
≺
A
~
3
≺
A
~
2
0.8
-
6.333
-
0.163
-
1.925
A
~
1
≺
A
~
3
≺
A
~
2
0.9
-
4.833
-
0.154
-
1.650
A
~
1
≺
A
~
3
≺
A
~
2
1.0
-
3.333
-
0.146
-
1.375
A
~
1
≺
A
~
3
≺
A
~
2Figure 8
Fuzzy numbersA~1, A~2, and A~3 in Example 20.Example 21.
Now, two special cases taken from R. Chutia and B. Chutia [81] are considered. The first set includes A~1=(0.1,0.1,0.1,0.1;0.8) and A~2=(-0.1,-0.1,-0.1,-0.1;1.0) which were ranked as A~1>A~2; and the second one includes B~1=(1,1,1,1;0.5) and B~2=(1,1,1,1;1.0) which were ranked as B~1<B~2. The proposed unified index also leads to similar conclusions as in [37, 78, 81, 88–90], indicating that the index can effectively work with crisp numbers as well.


## 4.3. Ranking for Nonlinear Fuzzy Numbers

Finally, although empirical phenomenon and human perception are rather unlikely to gather the nonlinear fuzzy numbers, this more general type can be justifiable for investigating the index’s computation easiness as well as adaptability.Example 22.
Let us consider two fuzzy numbers shown in Figure9, adapted from Liou and Wang [67]: A~1=(1,2,5) and A~2=(1,2,2,4) with a nonlinear membership function(14)fA~2x=1-x-221/2,1≤x≤2,1-14x-221/2,2≤x≤4,0,otherwise.
In this nonlinear case, by using the unified index, the conclusions in Table8, A~1≻A~2 (A~1′≺A~2′) for λ∈[0,1], do not add much complexity for the computation. Obviously, previous proposed measures in [53, 63, 66, 67, 76] possess the same conclusion and computation easiness. However, in recent works, Ky Phuc et al. [38], Asady [46], and Zhang et al. [73], their indices become more complicated and elaborate for ranking the nonlinear fuzzy numbers as well as their images.Table 8
Ranking results at different optimism levels in Example22.
λ
U
I
1
λ
U
I
2
λ
Ranking result
0.0
6.750
2.945
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.1
7.650
3.516
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.2
8.550
4.087
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.3
9.450
4.658
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.4
10.350
5.230
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.5
11.250
5.801
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.6
12.150
6.372
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.7
13.050
6.943
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.8
13.950
7.515
A
~
1
≻
A
~
2 and A~1′≺A~2′
0.9
14.850
8.086
A
~
1
≻
A
~
2 and A~1′≺A~2′
1.0
15.750
8.657
A
~
1
≻
A
~
2 and A~1′≺A~2′Figure 9
Fuzzy numbersA~1 and A~2 in Example 22.


## 5. Conclusions

Numerous indices for fuzzy-data comparisons and rankings have been widely implemented to resolve decisive problems that are evaluated under resources constraint and with to some extent linguistic preference of multiattribute, realized from users’ perspectives, as well as subjective quantification of multiple characteristics, assessed from decision-makers. However, counterintuition, computation complexity, less reliability, and/or inconsistency on their fuzzy-number rating outcomes have hampered their comprehensive implementation. To lessen their exhibited ranking weaknesses, this paper develops a unified index that multiplies weighted-mean and weighted-area discriminatory components of a fuzzy number, respectively, called centroid value (a measure that values the fuzzy number itself) and attitude-incorporated left-and-right area (a fuzzy-number measure that also reflects on the decision-maker’s attitude as regards data revelation). From theoretical proofs and comparative studies, this unified index has demonstrated four advantages for ranking fuzzy numbers.First, ranking results of the unified index support the human-intuition judgement. Secondly, it shows computation easiness regardless of different types of fuzzy numbers. It can be noted that this computation simplicity becomes crucial for multiagents-multicriteria decision-making problems, which normally involve numerous comparisons and analyses of fuzzy numbers. Thirdly, the unified index can provide a level-of-optimism attitude-based explanation for ranking conflicts among the literature. Most importantly, the unified index possesses the consistency property for ranking fuzzy numbers and their images as well as for symmetric fuzzy numbers with an identical altitude. Literally, in fields of computer vision and image pattern recognition, this property has been a rather critical one for accurate matching and/or retrieval of information.



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*Source: 3083745-2017-07-13.xml*