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

. 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.


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][7][8][9][10][11][12][13][14] and fuzzy integrodifferential equations [15][16][17][18][19][20][21][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][29][30][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][38][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 2 Complexity 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][41][42][43][44][45][46][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 widerange 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][52][53][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 risk In category one, Yager [55] and Lee and Li [56] first borrowed statistical center-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 an areadriven 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 a deviation-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 this deviation-driven ordering index have somewhat reignited the development of category 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 fuzzynumber 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 (, , , ) becomes either (, , , ) or (, , , ) 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.
Ostensibly, as opposed to the prolific ranking indices to date that have been presented in category one, the established ranking indices related to category 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-andtrapezoidal 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.Section 2 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.

Preliminaries
The following definitions and remarks are mainly adopted from Zimmermann [69] and Lee [70].
Definition 1 (fuzzy subset).Let R be a nonempty set.The fuzzy subset Ã of R is defined by a function  Ã : R → [0, 1]. Ã is called a membership function.
A fuzzy number has the following properties: (i) Ã is normal if there exists an  ∈ R such that  Ã() = 1; that is,  = 1.
Since Ã ⊂ Ã0 for each  ∈ (0, 1], condition (iv) shows that the -level sets Ã are bounded subsets of R for all  ∈ (0, 1].It is well known that condition (ii) is satisfied if and only if the -level set Ã is a convex subset of R. Therefore, from conditions (i)-(iv), it is implied that if Ã is a fuzzy number, then the -level set of Ã is a closed, bounded, and convex subset of R, that is, a closed interval in R, denoted by Ã = [ Ã  , Ã  ].
Remark 5. Let Ã be a fuzzy number.Then, the following statements hold true: Remark 6.Let Ã be a fuzzy number such that its membership function is strictly increasing on interval [, ] and strictly decreasing on interval [, ].From the fact of strict monotonicity,   Ã() and   Ã() are continuous functions on [0, 1].This implies that Ã is also a real fuzzy number.

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 leftand-right area (weighted area), is proposed in this section.
Definition 9 (left-and-right areas (an area-driven measure that belongs to category one)).Left-and-right areas of a fuzzy number Ã for  = 1, , denoted by    and    , are given by AA Complexity where  ∈ [0, 1] is level of optimism reflecting a data-revelation optimism degree of a decision-maker, where the larger the  set by the decision-maker is, the more optimistic attitude the decision-maker has on the data revelation.Two extreme cases are  = 0, meaning the decision-maker is completely pessimistic, and  = 1, meaning the decisionmaker is completely optimistic.Case  = 1/2 reflects a neutral decision attitude.From the mathematical viewpoint, (4) can be seen as a weighted-area value of Ã .
For boosting the fuzzy-number discrimination power, let us consider an index named UI   by multiplying two sizediscriminatory values of a fuzzy number; that is, UI   is called unified index.And,   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, CV  = 0.It is used to provide consistent ranking power when CV  = 0. Particularly, this paper suggests using   =   × 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, Ã and Ã .Given the data-optimistic level , from (5), we obtain their realized unified indices, UI   and UI   .Then, the following decisions can be made: Now, we will prove the unified index's consistency property when ranking fuzzy numbers and their images.Without loss of generality, CV  ̸ = 0 is considered in the following.
Based on (3), According to (4) and with the above results,     =    and     =    , we further have Similarly, Finally, regarding (5) and the aforementioned outcomes, we can simply obtain We complete the proof.

Comparative Studies
In this section, several fuzzy-number 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 up-todate representative indices from the publications.To make it easier to follow the whole discussion of comparison, Table 1 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 Propositions 11 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 imageranking results is not mentioned or shown on the result tables.

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 numbers Ã1 = (1, 4, 5) and Ã2 = (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, Ã1 ≺ Ã2 ( Ã 1 ≻ Ã 2 ) at any arbitrary level-of-optimism attitude of data revelation from the decision-maker,  ∈ [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][59][60][61][62].
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  = 1, their approach leads to a disparate ranking, Ã1 ≃ Ã2 ≃ Ã3 ( Ã 1 ≃ Ã 2 ≃ Ã 3 ), indicating that their index as a whole somewhat lacks reliability.
Example 17.Again, examine three fuzzy numbers, Ã1 = (1, 3, 5), Ã2 = (2,3,4), and Ã3 = (1,4,6), as shown in Figure 5. Visibly, Ã3 = (1,4,6) is right way out Ã1 and Ã2 , so there is no dispute that a capable index should rate Ã3 ( Ã 3 ) as the largest (smallest).The challenging one is to distinguish Ã1 and Ã2 ( Ã 1 and Ã 2 ) due to their symmetry with respect to  = 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 Ã1 ≃ Ã2 , 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, 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  value manifests the decision-maker's optimism towards revelation of left-and right-area data. ∈ (0.5, 1] implies that the right-area data is more preferred by the decision-maker;  ∈ [0, 0.5) represents the notion that the decision-maker is more optimistic regarding the left-area data;  = 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  = 0, where it suggests that Ã3 does not dominate Ã2 ; that is, Ã2 ≃ Ã3 ( Ã 2 ≃ Ã 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.

Ranking for Normal Triangular-and-Trapezoid Mixed
Fuzzy Numbers.Here, the proposed unified index is used to broaden the ranking comparisons to normal triangular-andtrapezoid 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 number Ã1 = (1, 5, 5) overlapping with a trapezoidal fuzzy number Ã2 = (2, 3, 5, 5), as shown in Figure 6.Of ten existing measures that have been studied in this case, three (30%) support Ã1 ≺ Ã2 [30,66,86] and seven (70%) stand for Ã1 ≻ Ã2 [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.
Table 5 is the ranking results of using the unified index, where  ∈ [0, 0.8], Ã1 ≻ Ã2 and  ∈ [0.9, 1], Ã1 ≺ Ã2 .Once more, the chosen  value manifests the decision-maker's optimism towards revelation of the left-and-right area of fuzzy data.From the -probability point of view, around 80% support Ã1 ≻ Ã2 and 20% favor Ã1 ≺ Ã2 .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 Ã1 ≻ Ã2 with a confidence degree of 73% and Ã1 ≺ Ã2 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 computationcomplexity problem, but also lacks capacity for ranking the fuzzy-number image.
Example 19.Taken from [38] and shown in Figure 7, one trapezoid fuzzy number, Ã3 = (0, 2, 4, 6), mingled with two triangular fuzzy numbers, Ã1 = (0, 3, 6) and Ã2 = (−1, 0, 2), is considered in this example.Noticeably, Ã2 left distances away from Ã1 and Ã3 , so there is no argument that a reliable  index should discriminate Ã2 ( Ã 2 ) as the smallest (largest).The question is the rating result of the triangular fuzzy number Ã1 and the trapezoid fuzzy number Ã3 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.
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, " Ã1 ≻ Ã2 " and " Ã3 ≻ Ã2 ," leaving undecided ranking for Ã1 and Ã3 .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 Ã1 and Ã3 that are overlapping and each has axis-of-symmetry property.

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.
In this nonlinear case, by using the unified index, the conclusions in Table 8, Ã1 ≻ Ã2 ( Ã 1 ≺ Ã 2 ) for  ∈ [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.

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.
and   Ã () stand for inverse functions of the left-and-right membership functions,   Ã () and   Ã (), respectively, and visual views of    and    are shown in Figure 2 [72].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-andright area, signified by AA   .

Figure 2 :
Figure 2: Left area    and right area    .

Table 1 :
The ranking performance assessments for some representative indices as opposed to the unified index.Figure 3: Fuzzy numbers Ã1 and Ã2 in Example 15.

Table 2 :
Ranking results for Example 15.

Table 3 :
Ranking results for Example 16.

Table 4 :
Ranking results at different optimism levels in Example 17.

Table 5 :
Ranking results at different optimism levels in Example 18.

Table 6 :
Ranking results of the three fuzzy numbers in Example 19.

Table 7 :
Ranking results of the three fuzzy numbers in Example 20.Figure 9: Fuzzy numbers Ã1 and Ã2 in Example 22.

Table 8 :
Ranking results at different optimism levels in Example 22.