Research Article  Open Access
ThanhLam Nguyen, "Methods in Ranking Fuzzy Numbers: A Unified Index and Comparative Reviews", Complexity, vol. 2017, Article ID 3083745, 13 pages, 2017. https://doi.org/10.1155/2017/3083745
Methods in Ranking Fuzzy Numbers: A Unified Index and Comparative Reviews
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 fuzzydata comparisons, exploring the general ranking measure that is capable of consistently differentiating the magnitude of fuzzy numbers has widely captivated academicsâ€™ 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.
1. Introduction
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 [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 decisionmaking. Their subbranches have also been established; for example, fuzzy differential equations [6â€“14] and fuzzy integrodifferential equations [15â€“22] are of fuzzy mathematics while fuzzynumber ranking, the focus of this paper, is of fuzzy decisionmaking. 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, 28â€“31], cutting force prediction [32], firmenvironmental knowledge management [33, 34], green supplychain 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 usersâ€™ perspectives, as well as subjective quantification of multiple characteristics, which is assessed from decisionmakers [2, 3, 37â€“39]. In these cases, fuzzydata 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, 40â€“47], not to mention the ordering for problems of multicriteria decisionmaking, 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 fuzzydata analysis.
Jain [49] in 1977 first launched a fuzzy set rating procedure for multipleaspect decisionmaking. Since then, exploring a general ranking measure, capable of consistently differentiating the magnitude of fuzzy numbers, has widely captivated academicsâ€™ attention [50]. Nowadays, a majority of diverse improved approaches/indices established from widerange perspectives focus on either compensating their predecessorsâ€™ 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, 51â€“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 deviationdriven ordering measures(ii)Indices that not only evaluate the fuzzy number itself, but also gauge decisionmakerâ€™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 centeroriented 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 meanandspread 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 coefficientofvariation 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 Leeâ€™s area index does have its own deficiency of ordering correctness when encountering fuzzy numbers with identical centroid points [65]. By defining fuzzynumber maximal and minimal reference sets, Wang et al. [66] first introduced a deviationdriven ordering index by combining rightandleft 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 numbersâ€™ images [46].
Emphatically, the aforementioned drawbacks plagued on this deviationdriven 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 decisionmakerâ€™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.â€™s deviationdegree index [66], Wang and Luo [39] incorporated decisionmakerâ€™s attitude towards risk into leftandright area between fuzzynumber points and the positiveandnegative ideal points; to improve Liou and Wangâ€™s index [67], Yu and Dat [48] incorporated decisionmakerâ€™s attitude regarding confidence into leftrighttotal integral value subjected to fuzzynumber 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 decisionmaking problems in combination with expertâ€™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 attitudeincorporated leftandright area (the category two measurement). According to comprehensively comparative studies from triangular, triangularandtrapezoidal 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 literatureexemplary 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). Let be a nonempty set. The fuzzy subset of is defined by a function . is called a membership function.
Definition 2 (cut set). The cut set of , denoted by , is defined by for all . The cut set is defined as the closure of the set .
Definition 3 (level set). The level set of , denoted by , is defined by for all .
Definition 4 (fuzzy number). A fuzzy number is described as any fuzzy subset of the real line with the membership function which is given bywhere is a constant and , are continuous functions on .
A fuzzy number has the following properties: (i) is normal if there exists an such that ; that is, .(ii) is fuzzy convex; that is, for .(iii) is upper semicontinuous; that is, is a closed subset of for each .(iv)The level set is a closed and bounded subset of .
Since for each , condition (iv) shows that the level sets are bounded subsets of for all . It is well known that condition (ii) is satisfied if and only if the level set is a convex subset of . Therefore, from conditions (i)â€“(iv), it is implied that if is a fuzzy number, then the level set of is a closed, bounded, and convex subset of , that is, a closed interval in , denoted by .
Remark 5. Let be a fuzzy number. Then, the following statements hold true: (i) for all .(ii) is increasing with respect to ; that is, for .(iii) is decreasing with respect to ; that is, for .
Remark 6. Let 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, and are continuous functions on . This implies that is also a real fuzzy number.
Definition 7 (the image of a fuzzy number [4]). Let fuzzy numbers be . Then, the image of is , as shown in Figure 1.
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 attitudeincorporated leftandright area (weighted area), is proposed in this section.
Definition 8 (centroid value (a centerdriven measure that belongs to category one)). Centroid value of a fuzzy number for , symbolized by , is defined as [3, 4, 38, 63, 65, 71]From the statistical point of view, it is the weighted mean of , meaning that when , we can accordingly have .
Definition 9 (leftandright areas (an areadriven measure that belongs to category one)). Leftandright areas of a fuzzy number for , denoted by and , are given bywhere and stand for inverse functions of the leftandright membership functions, and , respectively, and visual views of and are shown in Figure 2 [72].
Now, a fuzzynumber measure belonging to category two is presented. It also contemplates decisionmakerâ€™s attitude as regards data revelation, called attitudeincorporated leftandright area, signified by .where is level of optimism reflecting a datarevelation optimism degree of a decisionmaker, where the larger the set by the decisionmaker is, the more optimistic attitude the decisionmaker has on the data revelation. Two extreme cases are , meaning the decisionmaker is completely pessimistic, and , meaning the decisionmaker is completely optimistic. Case reflects a neutral decision attitude. From the mathematical viewpoint, (4) can be seen as a weightedarea value of .
For boosting the fuzzynumber discrimination power, let us consider an index named by multiplying two sizediscriminatory values of a fuzzy number; that is,
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, . It is used to provide consistent ranking power when . Particularly, this paper suggests using so that we can efficiently rank fuzzy numbers that have similar centroids but different height.
Remark 10. Consider the ranking of two fuzzy numbers, and . Given the dataoptimistic level , from (5), we obtain their realized unified indices, and . Then, the following decisions can be made: (i)At the dataoptimistic level , if , then .(ii)At the dataoptimistic level , if , then .(iii)At the dataoptimistic level , if , then .
Now, we will prove the unified indexâ€™s consistency property when ranking fuzzy numbers and their images. Without loss of generality, is considered in the following.
Proposition 11. Let be the image of a fuzzy number for . Its centroid value is , leftandright areas are and , attitudeincorporated leftandright area is and , and unified index is and .
Proof. From (2),Based on (3),According to (4) and with the above results, and , we further haveSimilarly,Finally, regarding (5) and the aforementioned outcomes, we can simply obtainWe complete the proof.
Proposition 12. Let a set of fuzzy numbers be and their images , . For a pairwise comparison of and for , two statements hold true: (1) if and only if and (2) if and only if .
Proof. Consider . From Proposition 11, we have the results and . Thus, . On the other hand, consider . According to Proposition 11, and . Hence, . Overall, the proof is completed.
Remark 13. Let a set of fuzzy numbers be and their images , . As regards Remark 10 and Propositions 11 and 12, the following decisions can be made for a pairwise comparison of and , for .(i)At the dataoptimistic level , if , which is equivalent to , then , which is equivalent to .(ii)At the dataoptimistic level , if , which is equivalent to , then , which is equivalent to .(iii)At the dataoptimistic level , if , which is equivalent to , then , which is equivalent to .
Finally, the following theory is very useful for ranking â€śsymmetricâ€ť fuzzy numbers with an identical altitude.
Theorem 14. Consider a set of â€śsymmetricâ€ť fuzzy numbers, , and their images , . By using the unified index, the pairwise comparison of and for is , , , , and , .
Proof. (i) Since and for are symmetric, we have . Moreover, from (2),Therefore, .
(ii) According to (3) and (4), we havewhereDue to the symmetry, we have for , when , and vice versa.
(iii) From (i), (ii), and (5), we have (i), ,(ii), ,(iii), . Finally, according to Remark 13, we complete the proof.
4. Comparative Studies
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 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.
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 indexâ€™s computation easiness; then, the work is extended on three similar fuzzy numbers for contrasting indicesâ€™ 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 indicesâ€™ reliability and consistency.
Example 15. Rank two fuzzy numbers and 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, () at any arbitrary levelofoptimism attitude of data revelation from the decisionmaker, . 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 procedurelaborious in practice as reported by some predecessors [58â€“62].
By the same token, when comparing two normal triangular fuzzy numbers and , taken from [76] and based on the proposed approach, we always have , which is coherent with that in [57, 63, 77â€“80]. However, the approaches by R. Chutia and B. Chutia [81] and Deng [82] lead to a counterintuitive result .

Example 16. Consider three triangle fuzzy numbers, , , and [39], which are similar and covered with the same righthand 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 and . Therefore, this example is capable of judging the indicesâ€™ performance if intuition and consistencysatisfied.
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, and .
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 and Cheng [57] gives , 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 , their approach leads to a disparate ranking, (), indicating that their index as a whole somewhat lacks reliability.

Example 17. Again, examine three fuzzy numbers, , , and , as shown in Figure 5. Visibly, is right way out and , so there is no dispute that a capable index should rate () as the largest (smallest). The challenging one is to distinguish and ( and ) due to their symmetry with respect to , identical centroid value, and their geometric enlargement relationship. Actually, majority of the existing ranking measures in category one (evaluating the fuzzy number itself) rank , 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 decisionmakerâ€™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 indicesâ€™ reliability and consistency.
First, we check the unified indexâ€™s results in Table 4. Regardless of , () is always the largest (smallest), which confirms human intuition. For the ranking of and , dividing from , (); the upper part , (); the lower part , (). 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 value manifests the decisionmakerâ€™s optimism towards revelation of left and rightarea data. implies that the rightarea data is more preferred by the decisionmaker; represents the notion that the decisionmaker is more optimistic regarding the leftarea data; indicates that the decisionmaker 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 Datâ€™s work confirms most of the results in Table 4, it does exhibit an apparent counterintuition issue at , where it suggests that does not dominate ; that is, (). Moreover, Liou and Wangâ€™s index [67] not only afflicts the same shortage of Yu and Datâ€™s index, but also has shown inconsistent results for ranking the fuzzy numbers and their images due to the indexâ€™s limited definition and generalization.

4.2. Ranking for Normal TriangularandTrapezoid Mixed 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.
Example 18. Compare a triangular fuzzy number overlapping with a trapezoidal fuzzy number , as shown in Figure 6. Of ten existing measures that have been studied in this case, three () support [30, 66, 86] and seven () stand for [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 predecessorsâ€™ 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 , and , <