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 fuzzydata 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 fuzzynumber rating outcomes have prohibited their comprehensive implementation. To ameliorate their manifested ranking weaknesses, this paper proposes a unified index that multiplies weightedmean and weightedarea discriminatory components of a fuzzy number, respectively, called centroid value and attitudeincorporated leftandright area. From theoretical proof of consistency property and comparative studies for triangular, triangularandtrapezoidal 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.
It has been well recognized that uncertainty inevitably exists in several realworld phenomena due to the inherent errors or impreciseness of measurement tools, methods, and uncontrollable conditions [
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 [
Jain [
In general, the existing ranking measures can be classified into two main categories:
Indices that value the fuzzy number itself such as center, area, and deviationdriven ordering measures
Indices that not only evaluate the fuzzy number itself, but also gauge decisionmaker’s attitude in regard to specific purposes such as confidence and risk
Emphatically, the aforementioned drawbacks plagued on this
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) 

Section 
Example 

Yu & Dat [ 
More laborious in computation 




Section 
Example 

Chu & Tsao [ 
Counterintuition 

Cheng [ 
Counterintuition  

Yu & Dat [ 
Less reliability  


Section 
Example 

Liou & Wang [ 
Inconsistency 






Section 
Example 

Zhang et al. [ 
Computation complexity 

Inconsistency  


Section 
Example 

Ky Phuc et al. [ 
Computation complexity 






Section 
Example 

Abbasbandy & Hajjari [ 
Counterintuition 






Section 
Example 

Ky Phuc et al. [ 
More elaborate in computation 

Ostensibly, as opposed to the prolific ranking indices to date that have been presented in
Aside from the Introduction, the remainder of this paper is organized into four sections as follows. Section
The following definitions and remarks are mainly adopted from Zimmermann [
Let
The
The
A fuzzy number
A fuzzy number has the following properties:
The
Since
Let
Let
Let
Based on integration of the two aforementioned categories for ranking fuzzy numbers, a unified index, which combines centroid value (weighted mean) and attitudeincorporated leftandright area (weighted area), is proposed in this section.
Centroid value of a fuzzy number
Leftandright areas of a fuzzy number
Left area
Now, a fuzzynumber measure belonging to category two is presented. It also contemplates decisionmaker’s attitude as regards data revelation, called attitudeincorporated leftandright area, signified by
For boosting the fuzzynumber discrimination power, let us consider an index named
Consider the ranking of two fuzzy numbers,
At the dataoptimistic level
At the dataoptimistic level
At the dataoptimistic level
Now, we will prove the unified index’s consistency property when ranking fuzzy numbers and their images. Without loss of generality,
Let
From (
Let a set of fuzzy numbers be
Consider
Let a set of fuzzy numbers be
At the dataoptimistic level
At the dataoptimistic level
At the dataoptimistic level
Finally, the following theory is very useful for ranking “symmetric” fuzzy numbers with an identical altitude.
Consider a set of “symmetric” fuzzy numbers,
(i) Since
Therefore,
(ii) According to (
Due to the symmetry, we have
(iii) From (i), (ii), and (
Finally, according to Remark
In this section, several fuzzynumber examples, which are popular in the literature for a wide range of fuzzynumber comparative studies, are used to compare ranking performance between the unified index and some uptodate representative indices from the publications. To make it easier to follow the whole discussion of comparison, Table
It can be noted that, based on Propositions
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 moveaway 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.
Rank two fuzzy numbers
According to the unified index in (
By the same token, when comparing two normal triangular fuzzy numbers
Ranking results for Example



Ranking result 

0.0  8.333  9.167 

0.1  9.000  9.900 

0.2  9.667  10.633 

0.3  10.333  11.367 

0.4  11.000  12.100 

0.5  11.667  12.833 

0.6  12.333  13.567 

0.7  13.000  14.300 

0.8  13.667  15.033 

0.9  14.333  15.767 

1.0  15.000  16.500 

Fuzzy numbers
Consider three triangle fuzzy numbers,
We first check the unified index. Based on (
In the literature, while many support the intuitive results for ranking the fuzzy numbers [
Moreover, due to scarcity of methods in the literature for consistently ranking their images, a recent work from Yu and Dat [
Ranking results for Example




Ranking result 

0.0  33.000  37.485  38.000 

0.1  33.600  37.831  38.317 

0.2  34.200  38.178  38.633 

0.3  34.800  38.524  38.950 

0.4  35.400  38.871  39.267 

0.5  36.000  39.217  39.583 

0.6  36.600  39.564  39.900 

0.7  37.200  39.910  40.217 

0.8  37.800  40.257  40.533 

0.9  38.400  40.603  40.850 

1.0  39.000  40.950  41.167 

Fuzzy numbers
Again, examine three fuzzy numbers,
First, we check the unified index’s results in Table
First, this finding is consistent with that of Wang and Luo [
Then, we evaluate the indices proposed by Yu and Dat [
Ranking results at different optimism levels in Example




Ranking result 

0.0  6.000  7.500  9.167 

0.1  6.600  7.800  10.083 

0.2  7.200  8.100  11.000 

0.3  7.800  8.400  11.917 

0.4  8.400  8.700  12.833 






0.6  9.600  9.300  14.667 

0.7  10.200  9.600  15.583 

0.8  10.800  9.900  16.500 

0.9  11.400  10.200  17.417 

1.0  12.000  10.500  18.333 

Fuzzy numbers
Here, the proposed unified index is used to broaden the ranking comparisons to normal triangularandtrapezoid 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.
Compare a triangular fuzzy number
Table
Finally, it is worth mentioning that as opposed to the unified index, Zhang et al.’s [
Ranking results at different optimism levels in Example



Ranking result 

0.0  11.000  9.333 

0.1  11.733  10.267 

0.2  12.467  11.200 

0.3  13.200  12.133 

0.4  13.933  13.067 

0.5  14.667  14.000 

0.6  15.400  14.933 

0.7  16.133  15.867 

0.8  16.867  16.800 

0.9  17.600  17.733 

1.0  18.333  18.667 

Fuzzy numbers
Taken from [
First, we check the unified index’s results in Table
Then, we evaluate Ky Phuc et al.’s [
Ranking results of the three fuzzy numbers in Example




Ranking result 

0.0  4.500  0.167  3.000 

0.1  5.400  0.183  4.200 

0.2  6.300  0.200  5.400 

0.3  7.200  0.217  6.600 

0.4  8.100  0.233  7.800 






0.6  9.900  0.267  10.200 

0.7  10.800  0.283  11.400 

0.8  11.700  0.300  12.600 

0.9  12.600  0.317  13.800 

1.0  13.500  0.333  15.000 

Fuzzy numbers
Additionally, let us consider one trapezoidal fuzzy number,
The result in Table
Ranking results of the three fuzzy numbers in Example




Ranking result 

0.0 




0.1 




0.2 




0.3 




0.4 




0.5 




0.6 




0.7 




0.8 




0.9 




1.0 




Fuzzy numbers
Now, two special cases taken from R. Chutia and B. Chutia [
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.
Let us consider two fuzzy numbers shown in Figure
In this nonlinear case, by using the unified index, the conclusions in Table
Ranking results at different optimism levels in Example



Ranking result 

0.0  6.750  2.945 

0.1  7.650  3.516 

0.2  8.550  4.087 

0.3  9.450  4.658 

0.4  10.350  5.230 

0.5  11.250  5.801 

0.6  12.150  6.372 

0.7  13.050  6.943 

0.8  13.950  7.515 

0.9  14.850  8.086 

1.0  15.750  8.657 

Fuzzy numbers
Numerous indices for fuzzydata 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 decisionmakers. However, counterintuition, computation complexity, less reliability, and/or inconsistency on their fuzzynumber rating outcomes have hampered their comprehensive implementation. To lessen their exhibited ranking weaknesses, this paper develops a unified index that multiplies weightedmean and weightedarea discriminatory components of a fuzzy number, respectively, called centroid value (a measure that values the fuzzy number itself) and attitudeincorporated leftandright area (a fuzzynumber measure that also reflects on the decisionmaker’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 humanintuition judgement. Secondly, it shows computation easiness regardless of different types of fuzzy numbers. It can be noted that this computation simplicity becomes crucial for multiagentsmulticriteria decisionmaking problems, which normally involve numerous comparisons and analyses of fuzzy numbers. Thirdly, the unified index can provide a levelofoptimism attitudebased 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.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work was partially supported by Lac Hong University under Decision no. 918/QDDHLH.