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Intuition, expressed as verbal arguments or axiom formulations, has often been used as a guiding principle for fuzzy number ranking (FNR). This chapter adopts the multi-attribute decision making (MADM) framework to analyze such intuition with three results. First, intuition in FNR should have involved multiple attributes, which are often implicated in the existing ranking methods. Then, we suggest three attributes (i.e., representative x-value, x-value range, overall membership ratio), which can be used to characterize the FNR intuition. Second, we decouple two issues in FNR: selection of attributes and aggregation of values, where aggregation is concerned with the trade-off among attributes to determine a single index for FNR. Then, the discount factors are proposed for the attributes of range and membership ratio to model the trade-off and formulate a ranking index. Third, the decoupling of attributes and aggregation reveals a fundamental tension between information content and the satisfaction of the FNR axioms. That is, if we can consider more information (in terms of attributes) as relevant to FNR, the ranking method will likely violate some FNR axioms. However, if we consider less information, the ranking method will be less sensitive to distinguish some fuzzy numbers for ranking. In the end, the proposed multi-attribute approach can provide a practical aspect to analyze and address the FNR problems.
Department of Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Alberta, Canada
*Address all correspondence to: simoli@ucalgary.ca
1. Introduction
Intuition has been a criterion for researchers to evaluate and comment the ranking results from a set of fuzzy numbers. As a pattern described by Wang and Kerre [1], a ranking method can be criticized by yielding “counter-intuitive” results from some examples, and thus it is motivated to develop new ranking methods (e.g., [2, 3]). Despite of its common use, the meaning of “intuition for ranking” is somewhat unclear. It should be related to the ranking of real numbers, which is fundamental in our intuition. However, this alone is not sufficient for fuzzy number ranking (FNR). Why? When we compare two real numbers: 3 and 5, we can state 5 > 3 because these real numbers can be ordered on a single dimension, i.e., the real line. In the context of fuzzy sets, the membership information is added. For example, consider two ordered pairs: (3, 0.9) and (5, 0.4), where the second elements are the membership values. Here, we cannot straightforwardly state (5, 0.4) ≻ (3, 0.9) due to the presence of the second dimension, membership, in the ranking consideration. Notably, in this paper, the symbol “>” is used to compare two real numbers, while the symbol “≻” or “≽” is used to represent the ranking relation.
Consider that a fuzzy number contains a set of such ordered pairs. We argue that the problem structure of FNR should contain multiple dimensions to explain “intuition” properly. Then, we employ the classical framework of multi-attribute decision making (MADM) [4] for the analysis of ranking intuition. The framework of MADM distinguishes the concepts of attributes and aggregation. Attributes are used to evaluate the properties of options, and they are subject to the selection by decision makers, who determine what properties (or information) are deemed relevant to the decision problems. On the other hand, aggregation captures the weighting strategies (e.g., weighted sum) to address the trade-off consideration among the option’s properties (or information).
In FNR, each attribute represents a single dimension for ranking consideration. In literature, numerous attributes have been implied in the formulations of ranking indices. For example, the approach of the maximizing and minimizing sets [3, 5, 6] implicates the attributes that articulate the optimistic and pessimistic aspects of a fuzzy number for ranking. In the centroid-based approach [7, 8], centroid can be interpreted as an attribute that focuses on the “middle” aspect over the geometry of a fuzzy number. Notably, each notion of attribute can be quantified in multiple ways. For example, we may express the notion of “average” via the formulations of “value” by Delgado et al. [9] or “median” by Bodjanova [10]. In addition, new ranking methods have been proposed by adding attributes to the ranking indices. For example, to address some non-distinguishable results from Abbasbandy and Hajjari [2], Asady [11] and Ezzati et al. [12] formulated additional attributes (namely, the epsilon-neighborhood and Mag’(u), respectively) in their ranking indices.
The consideration of multiple attributes for FNR is not new. In literature, some approaches have explicitly considered multiple measures (or attributes) to describe a fuzzy number such as value and ambiguity [9, 13, 14], mean and standard deviation [15], average value and degree of deviation [16], expected value (in transfer coefficient) and deviation degree [17, 18], general concepts of area/mode/spreads/weights [19, 20, 21] and extensions from the centroid concept [21, 22, 23, 24, 25].
Aggregation is a separate issue from the selection of attributes. It aims to handle the given information of attributes for decision making. In literature, different aggregation approaches over the same attributes have been reported. For example, aggregation over the x- and y-coordinates of a centroid can be done via a distance measure [26] or an area measure [8, 27]. The weighted sum approach has been used to aggregate two attributes such as the right/left utility values [3] and the average and deviation values [16]. In addition to closed-form equations, aggregation can also be done by rules and procedures. For example, Asady [11] and Chi and Yu [23] determine the ranking of fuzzy numbers based on the priority of two or three attributes, which basically is a lexicographical ordering procedure ([4], pp. 77–79).
The aggregation approach can influence the ranking results since it controls the trade-off among attributes. To illustrate, consider the earlier ordered pairs (3, 0.9) and (5, 0.4). Suppose that two attributes are considered for ranking: real number and membership value, and we assume “higher value ➔ higher rank” for both attributes. Then, we can have multiple ways to aggregate these two values such as 3 + 0.9 and 3 × 0.9, which are consistent with the “higher-the-better” direction. However, different aggregation functions can lead to different ranking results, e.g., (3 + 0.9) < (5 + 0.4) and (3 × 0.9) > (5 × 0.4). Different results can be explained by the trade-off approach implied in the aggregation functions. For example, in this case, addition tends to give an advantage to real number, whereas multiplication allows more influence from membership value.
Based on the above discussion, the theme of this chapter is to adopt the MADM framework, which purposely decouples attributes and aggregation for FNR. In this way, we can compare ranking methods in view of their selections of attributes and the formulations of aggregation functions independently. In addition, the multi-attribute aspect can help explain the axiomatic properties of ranking methods. To avoid the reliance on the “intuition criterion”, Wang and Kerre [1] suggested seven axioms as reasonable properties to specify the meaning of intuition more clearly. Ban and Coroianu [28] derived a class of ranking functions that can satisfy six of these axioms with literature examples that can belong to this class under some conditions (e.g., [2, 29, 30]).
Despite of the formal work by Ban and Coroianu [28], new ranking methods emerge continually as researchers considered this class of ranking functions did not address two aspects. First, the development by Ban and Coroianu [28] was intended for normalized fuzzy numbers, and some work has been developed for the non-normalized cases (e.g., [31]). Second, their class of ranking functions cannot distinguish two symmetric fuzzy numbers with different spreads (e.g., for cases in Ezzati et al., [12]). In some recent work, Dombi and Jónás [32] applied the probability-based preference intensity index, and Van Hop [33] developed the dominant interval measure (namely relative dominant degree) for fuzzy number ranking. Their approaches basically generalized the numerical techniques of intervals for fuzzy number ranking without decomposing or analyzing the ranking attributes.
More fundamentally, it seems to us that if a ranking function is designed to satisfy the axioms by Wang and Kerre [1], this ranking function will be less sensitive to the distribution of membership values and the spreads of fuzzy numbers to determine the ranking results. In other words, the satisfaction of these axioms is strongly influenced by the type of information (or attributes) that is selected for FNR but it is less relevant to the aggregation approach. The distinction between information selection and aggregation has not been investigated for fuzzy number ranking in literature. This chapter will use the multi-attribute aspect to analyze this issue.
After the preliminaries in Section 2, this chapter will discuss and illustrate our selection of three attributes for FNR in Section 3 and then our aggregation approach using the discount factors in Section 4. Section 5 will suggest some guidance for the application of the proposed multi-attribute ranking method. Section 6 will discuss the relation between the information content for FNR and the axiomatic properties of ranking methods. This chapter is concluded in Section 7.
Fuzzy number is described as a fuzzy subset of the real line R [34]. This work considers trapezoidal fuzzy number (TrFN) as a special case of fuzzy number. Let FA denote a TrFN with a maximum membership equal to hA, as illustrated in Figure 1. Let x be any element of the real line, and its membership according to TrFN, denoted as μFAx, can be expressed in the following formulation where a1, a2, a3, a4 are real numbers to specify FA. As a convenient notation, FA can be expressed as a 5-tuple, where FA = (a1, a2, a3, a4; hA).
Let supp(FA) be the support of FA, and we have supp(FA) = {x ∈ R | a1 ≤ x ≤ a4}. Then we have the infimum and supremum of supp(FA) as inf supp(FA) = a1 and sup supp(FA) = a4, respectively. Also, let IFAα=lFAαrFAα be the α-cut interval of FA. For α ≤ hA, the left and right bounds of the α-cut interval can be formulated as follows.
lFAα=a1+αhAa2−a1E2
rFAα=a4+αhAa3−a4E3
Suppose we have two fuzzy numbers: FA = (a1, a2, a3, a4, hA) and FB = (b1, b2, b3, b4, hB), and a constant, denoted as λ (i.e., λ∈ R). We can have fuzzy number addition and multiplication with a constant as follows [17, 18, 34].
FA⊕FB=a1+b1a2+b2a3+b3a4+b4minhAhBE4
λ·FA=λ·a1λ·a2λ·a3λ·a4hAE5
To describe some reasonable properties of ranking methods, Wang and Kerre [1] have proposed seven axioms. Ban and Coroianu [28] have dropped one axiom by considering a ranking (or an ordering) over a given set of fuzzy numbers. This chapter follows the choice made by Ban and Coroianu [28]. Let F be a set of fuzzy numbers, and a ranking method determines the binary relation ≽ over F. Then, their six axioms are summarized (without elaborating their variants) below.
Axiom 1: FA ≽ FA. for any FA∈ F.
Axiom 2: For any FA, FB∈ F, if FA ≽ FB and FB ≽ FA, then FA ∼ FB.
Axiom 3: For any FA, FB, FC∈ F, if FA ≽ FB and FB ≽ FC, then FA ≽ FC.
Axiom 4: For any FA, FB∈ F, if inf supp(FA) ≥ sup supp(FB), then FA ≽ FB.
Axiom 5: Suppose that FA, FB, FA ⊕ FC, FB ⊕ FC are elements of F. If FA ≽ FB, then Fa ⊕ Fc ≽ Fb ⊕ Fc.
Axiom 6: Suppose thatλ∈ R and FA, FB, λ⋅FA, λ⋅FB are elements of F. If FA ≽ FB and λ ≥ 0, then λ⋅FA ≽λ⋅FB. If FA ≽ FB and λ ≤ 0, then λ⋅FA≼λ⋅FB.
Axioms 1 and 3 are referred to as the reflexive and transitive properties of binary relations, respectively, for a total pre-order on F [28]. Axiom 2 defines the conditions for the equality “∼”. Axiom 4 specifies that FA is larger than or equal to FB if the lower bound of the support of FA is larger than the upper bound of the support of FB. Axioms 5 and 6 generally imply that the ordering of FA ≽ FB should be preserved if they are added by the same fuzzy number FC or multiplied by the same positive quantity λ. Notably, index-based ranking methods will satisfy Axioms 1 to 3 [1], and this chapter will focus more on Axioms 4 to 6.
In this section, we characterize the comparison of fuzzy numbers through one primary attribute and two secondary attributes. The primary measure is concerned with the representative value of a fuzzy number on the real line, which is a common intuition for ranking. One secondary attribute checks the range of real numbers enclosed by a fuzzy number, which information is independent of the representative value but can be relevant for ranking. Another secondary attribute is associated with membership, which is concerned with the shape of a fuzzy number.
3.1 Representative x-value
Since the real line of a fuzzy number is often expressed on the x-axis, we use “x-value” to label the values associated with the real line. As a fuzzy number encloses a range of possible x-values, one common intuition is to identify a representative x-value of a fuzzy number for comparison. There can be several options that are aligned with this intuition such as the expected value [34, 35], the x-coordinate of a centroid [36] and median [10]. In this chapter, we adopt the class of ranking indices derived by Ban and Coroianu [28]. Let rep(FA, w) be the function to evaluate the representative x-value of the fuzzy number FA, and its formulation is given as follows.
repFAw=w·a1+12−wa2+12−wa3+w·a4E6
where w is a weighting constant with 0 ≤ w ≤ 1. As proven by Ban and Coroianu [28] (Theorem 39), if this function is used as a ranking index, it satisfies the six axioms discussed in the preliminaries section. Beyond this theorem result, we can interpret this formulation as a weighted function of a fuzzy number’s core values (a2 and a3) with a weight (1/2-w) and boundary values (a1 and a4) with a weight w. When w = 0, only the core values are considered. Alternately, when w = 1/2, only the boundary values are considered. To emphasize the importance of core values (i.e., a2 and a3) through weighting, we set (1/2-w) ≥ w, and then we have 0 ≤ w ≤ 1/4.
Derived from Eq. (6), we have rep(FA, w) ≥ rep(FB, w) if the following condition is satisfied.
wa1+a4−b1+b4+12−wa2+a3−b2+b3≥0E7
This condition implies a weighted comparison between core and boundary values of FA and FB. Apparently, we cannot guarantee the satisfaction of this condition if FA and FB are partially overlapped (i.e., supp(FA) ∩ supp(FB) ≠ ∅). Then, the value of w can influence the ordering of rep(FA, w) and rep(FB, w). Following the discussion in Ban and Coroianu [28], we consider the presence of w as a generalization of some existing indices, which have implicitly pre-defined weighting factors for core and boundary values of a fuzzy number. For example, the ranking index developed by Abbasbandy and Hajjari [2] is an instance by setting w = 1/12. Given a ranking problem, decision makers can consider some sensitivity analysis (e.g., evaluate the value of w that makes rep(FA, w) = rep(FB, w)) to define the value of w for their ranking problems.
3.2 X-value range
Another attribute is associated with the range of possible x-values of a fuzzy number. Fuzzy numbers can have the same representative x-values with different ranges (e.g., symmetric triangular fuzzy numbers with the same core value but different boundary values). Some argue that the information of range should be considered for ranking (e.g., [11]). There can be several options to quantify this intuition such as ambiguity value [9, 13], standard deviation [15] and deviation degree [16, 17, 18]. In this chapter, we adopt the range (or size) of the α-cut interval (denoted as rng(FA, α)), and it is formulated as follows.
rngFAα=rFAα−lFAαE8
Figure 2 illustrates the α-cut interval of a trapezoidal fuzzy number FA., where the lower (left) and upper (right) bounds of the α-cut interval are denoted as lFAα and rFAα, respectively. The formulations of lFAα and rFAα can be found in Eqs. (2) and (3), respectively. The value of α can be interpreted as the minimum membership value that is deemed relevant for the ranking analysis. For example, if we set α at a lower value, we will receive a wider interval.
Figure 2.
Illustration of the α-cut interval.
Here, we suppose that a large range of possible x-values tends to yield a lower rank because decision makers do not want high uncertainty associated with a large range. This stated intuition of “larger range ➔ lower rank” is aligned with Wang and Luo [6] and Nasseri et al. [37]. Also, we classify range as a secondary attribute because some decision makers may find this attribute not necessary to their ranking problems (e.g., ranking a set of triangular fuzzy numbers with a similar size of support). Then, using the measure of representative x-value only could be sufficient for ranking. In contrast, if decision makers find the information of range relevant to their ranking problems, our suggested approach is to take the range information as a modifier to the representative x-value. This approach will be discussed in Section 5.
3.3 Overall membership ratio
The notion of overall membership is associated with the shape of a fuzzy number, regardless of where this shape is placed on the real line. To illustrate, consider two comparisons in Figure 3. In Figure 3a, while FA and FB have different representative x-values, their overall membership values should be the same due to the common shape. In contrast, FC in Figure 3b should have higher overall membership than FD as FC’s membership values are higher than or equal to those of FD over the common support (note: the common support is not necessary; it just makes the comparison easier to observe).
Figure 3.
Illustration of the concept for overall membership a) FA and FB with same membership b) FC with higher membership than FD.
To capture the above idea of the overall membership of a fuzzy number, we formulate the ratio using two areas: the shape’s area and the full membership area over the same support. Also, we keep the concept of α-cut interval so that the decision maker can identify the minimum level of membership that is relevant for their ranking problem. Figure 4 is used to illustrate the concept of both types of area. First, the shape’s area is considered as the area under the fuzzy number and enclosed by the α-cut interval, as shaded by gray lines in Figure 4. Then, the full membership area is based on the rectangle with the width of the α-cut interval and the height of 1 (i.e., maximum membership). Accordingly, the shape’s area (denoted as areashape) and the full membership area (denoted as areafull) can be formulated as follows.
Figure 4.
Illustration of the shape’s area and full membership area.
areashapeFAα=∫lFAαrFAαμFAxdxE9
areafullFAα=rFAα−lFAα×1E10
The overall membership ratio of a fuzzy number (denoted as mem(FA, α)) can be expressed as follows.
memFAα=areashapeFAαareafullFAαE11
Here, we suppose that higher overall membership ratio tends to yield a higher rank. We classify (overall) membership ratio as another secondary attribute because it may not be necessary for ranking problems with normal fuzzy numbers (e.g., if FA is a normal triangular fuzzy number, mem(FA, 0) is always equal to 0.5). Yet, if this information is considered relevant, Section 5 will suggest one approach to use it as a modifying factor for ranking.
Notably, it is probably more common to apply two measures (instead of three) for FNR in literature (e.g., [value, ambiguity] and [average value, degree of deviation] as mentioned in Introduction). From there, they tend to integrate the information of range and membership ratio into one measure. We choose to handle such information in terms of two separate attributes for two reasons. First, the concepts of range and membership ratio are relatively direct for decision makers to visualize and interpret (thus supporting their intuition) in the comparison of fuzzy numbers. Second, range and membership ratio can indicate independent information. For example, consider two normal fuzzy numbers: one triangle and one trapezoid. While the trapezoid shape always yields a higher membership ratio, the ranges of both shapes can be changed arbitrarily, thus explaining the independence of range and membership ratio.
To demonstrate the evaluation of the three attributes, consider a fuzzy number: FA = (1, 2, 3, 4; 1), which has a lower bound of 1 and an upper bound of 4. Its maximum membership value is 1, which covers the range between 2 and 3 (check Figure 1 for an illustrative reference). Suppose that α = 0 (i.e., we consider the whole fuzzy number) and w = 1/12 (i.e., according to Abbasbandy and Hajjari [2]), we can evaluate the values of the three attributes according to the following:
After defining three attributes, we can rank fuzzy numbers for some cases using the Pareto optimality principle [4]. In a less formal expression, we have FA ≽ FB if rep(FA, w) ≥ rep(FB, w), rng(FA, α) ≤ rng(FB, α) and mem(FA, α) ≥ mem(FB, α). To examine how well these attributes can speak for the ranking intuition, numerical examples are used in the next sub-section to check the following situations.
If two fuzzy numbers can be ranked based on Pareto optimality, this ranking order should be considered “obvious” to the ranking intuition with less room for arguments.
If two fuzzy numbers cannot be ranked based on Pareto optimality, decision makers can effectively use the selected attribute to explain their arguments.
3.5 Numerical examples
The numerical cases from Bortolan and Degani [38] are employed for demonstration, and they can illustrate systematically how the selected attributes are changed with different fuzzy numbers. While we keep the case labels from Bortolan and Degani [38] for cross checking, we classify these cases into five groups for discussion. Also, we follow Abbasbandy and Hajjari [2] by setting w = 1/12 to evaluate rep(FA, w). Also we set α = 0 for rng(FA, α) and mem(FA, α) in this numerical demonstration.
Group 1: Non-overlapping, triangular fuzzy numbers
This group covers the cases of a, b, c, d and e from Bortolan and Degani [38], and the results are shown in Table 1. By examining the Pareto optimality with the three attributes, we can first pass the membership ratio because mem(FA, 0) is always equal to 0.5 if FA is normal and triangular. The rankings of fuzzy numbers in cases a to d are obvious as the fuzzy numbers with higher representative x-values have the same (i.e., cases a, b, d) or smaller (i.e., case c) ranges. In case e, while the fuzzy number FE3 is ranked highest, we cannot immediately rank FE2 higher than FE1 based on Pareto optimality only since FE2 has a larger range. Through these five cases, we want to note that the proposed attributes vary according to our “intuition” to interpret and rank fuzzy numbers (e.g., check how representative x-values and ranges vary independently in these cases).
Table 1.
Results of comparing non-overlapping, triangular fuzzy numbers.
Group 2: Overlapping, triangular fuzzy numbers
This group covers the cases of f, i and l from Bortolan and Degani [38], and the results are shown in Table 2. In case f, while FF2 should be ranked higher than FF1 due to higher representative x-value shown in Table 2, we should note that this ranking is sensitive to the pre-set value of w. If w < 1/6 (i.e., more emphasis to the core values), we have rep(FF2) > rep(FF1). If w ≥ 1/6 (i.e., more emphasis to the boundary values), we have rep(FF1) ≥ rep(FF2).
Table 2.
Results of comparing overlapping, triangular fuzzy numbers.
In contrast, as the fuzzy numbers in case i share the same support, their ranking is not sensitive to the value of w. Finally, the ranking in case l depends on the information of range, and our intuition assumes that smaller range is better. Notably, our intuition here is not universal, and some decision maker can rank a fuzzy number of larger range higher for a positive likelihood of higher x-values. Here we are not arguing which “intuition” (or ranking rule) is right. Instead, we want to keep the intuition more transparent through explicit attributes so that researchers can argue their ranking intuitions on a common ground.
Group 3: Triangular and trapezoidal fuzzy numbers
This group covers the cases of g and h from [38], and the results are shown in Table 3. The trapezoid fuzzy numbers have a large shape, giving higher values of range and membership ratio. The triangular fuzzy numbers in both cases have higher representative x-values. Their triangular shapes are the same, with a shift to the right side by 0.1 in case h. In view of Pareto optimality with three attributes, there is no dominant fuzzy number. Yet, we can note that if FG2 ≽ FG1 in case g, we would have FH2 ≽ FH1 in case h. It is because FH2 ≽ FG2 due to Pareto optimality and FG1 = FH1. This note should make sense when we observe the graphical shift of triangular fuzzy numbers from FG2 to FH2 in Table 3. This demonstrates how the three attributes can characterize some intuitive reasoning in FNR.
Table 3.
Results of comparing triangular and trapezoidal fuzzy numbers.
Group 4: Nested fuzzy numbers.
This group covers the cases of j and k from [38], and the results are shown in Table 4. In case j, FJ2 is created by shifting the lower bound of FJ1 to the left; FJ2 and FJ3 share the same support with a different shape. Fuzzy numbers in case k have a similar pattern in an opposite direction (see Table 4). By checking from the order FJ1 ➔ FJ2 ➔ FJ3 or FK1 ➔ FK2 ➔ FK3, we argue that the three attributes can reasonably capture and quantify the characteristics of these fuzzy numbers.
Table 4.
Results of comparing nested fuzzy numbers.
Group 5: Non-normal fuzzy numbers.
Non-normal fuzzy numbers have their maximum membership less than 1 (i.e., hA < 1). Notably, the literature of FNR often assumes normal fuzzy numbers (e.g., [28]). By inspecting the earlier cases, we should note that the variations of membership ratio of normal fuzzy numbers do not change much (from 0.5 for triangular to 0.7 or 0.75 for trapezoidal). Thus, it is not unreasonable if one chooses not to consider membership ratio for comparing normal fuzzy numbers. Yet, non-normal fuzzy numbers will open other possibilities, where the membership ratio can be an important consideration.
This group covers the cases of n, o, p, q and r from Bortolan and Degani [38], and the results are provided in Table 5. As shown in Table 5, the values of membership ratio vary more significantly as some fuzzy numbers have smaller maximum membership. Consequently, the trade-off consideration can be more challenging. For example, how should we compare FN1 and FN2 in case n with the trade-off of representative x-value and membership ratio (similarly for case o)? While we see FP2 ≽ FP1 in case p and FQ1 ≽ FQ2 in case q due to Pareto optimality, the trade-off consideration is present in case r with different values of range.
Table 5.
Results of comparing non-normal fuzzy numbers.
The main theme of this section is that we need some attributes to characterize our intuition for FNR. Otherwise, it is difficult to get a common ground for constructive arguments. In this section, we choose three attributes to make clear our “intuition” for FNR. Aligned with the note in Keeney and Raiffa [4], we do not claim the uniqueness of this selection of attributes for FNR. Other researchers can propose other sets of attributes to characterize their intuition.
If the Pareto optimality principle cannot rank two fuzzy numbers, trade-off consideration is required to finalize the ranking decision. That is, a fuzzy number of a higher rank must have some “weaker” aspect in terms of the three attributes but its “stronger” aspect is sufficient to bring it to a higher rank overall. This ranking process should involve an aggregation that combines all aspects into an overall evaluation and then determines the ranking result. This section will propose a ranking index for aggregation along with numerical examples.
4.1 Discount factors and ranking index
As discussed in Section 3, representative x-values are used as the primary attribute to rank fuzzy numbers. Then, we view the information of range and membership ratio as secondary attributes that will “discount” the representative x-values. To illustrate, consider a crisp number, 5, which has the representative x-value of 5, range of 0 and membership ratio of 1. If a fuzzy number with the representative x-value of 5 has a range larger than 0 and a membership ratio less than 1, this fuzzy number should be ranked lower than the crisp number 5. The discount factors are intended to capture this idea. Let Irank(FA) be the index as the discounted representative x-value of FA for ranking, and it can be formulated as follows.
IrankFA=drngFA·dmemFA·repFAwE12
where drng(FA) and dmem(FA) are the discount factors associated with range and membership ratio, respectively. To quantify these discount factors, we consider the following conditions:
0 ≤ drng(FA) ≤ 1 and 0 ≤ dmem(FA) ≤ 1
If rng(FA, α) ≥ rng(FB, α), drng(FA) ≤ drng(FB).
If mem(FA, α) ≥ mem(FB, α), dmem(FA) ≥ dmem(FB).
Apparently, many forms of formulations can be used for the discount factors and satisfy these conditions. In this chapter, we use a simple ratio with respect to some reference (or extreme) values. Let rngmin be the minimum reference for range, and memmax be the maximum reference for membership ratio. We also set that rngmin > 0 and 0 < memmax ≤ 1. Then, the discount factors for FA can be formulated as follows.
drngFA=rngminrngFAαE13
dmemFA=memFAαmemmaxE14
With these discount factors, if FA has a range equal to rngmin, its discount factor, drng(FA), is equal to 1 (i.e., no discount). A similar effect is also set for dmem(FA). The selection of the values for rngmin and memmax depends on how decision makers interpret the discount ratio for their ranking problems. One suggestion is to identify the minimum range and the maximum membership ratio from the set of fuzzy numbers to be ranked. That is, suppose that FR = {FA, FB, FC,…} be the set of fuzzy numbers that need to be ranked in a problem. We can select rngmin and memmax according to the following equations. Then, we can interpret the discount ratio with respect to the “best values” among the set of fuzzy numbers in the problem.
rngmin=minrngFAαrngFBαrngFCα…E15
memmax=maxmemFAαmemFBαmemFCα…E16
4.2 Overview of the ranking method
After defining the attributes in Section 3 and the ranking index in Section 4.1, this sub-section will overview our proposed approach to rank fuzzy numbers. The procedure to determine the ranking index is illustrated in Figure 5. Given a fuzzy number FA, we first determine the values of three attributes: representative x-value, x-value range and overall membership ratio. Then, we can evaluate the discount factors for x-value range and overall membership ratio. In the end, we can determine the ranking index for the given fuzzy number.
Figure 5.
Procedure to determine the ranking index.
Suppose that we are tasked to rank a set of fuzzy numbers. We first determine the ranking index for each fuzzy number. Then, we can use the index, Irank, for this ranking task. That is, if IrankFA≥IrankFB, we rank FA higher than FB, symbolically, FA≽FB.
4.3 Numerical examples
As a recall from Section 3, we set w = 1/12 and α = 0 to evaluate representative x-value, range and membership ratio. We use Eqs. (15) and (16) to obtain rngmin and memmax and then calculate the values of the discounts and the ranking index. We reuse the numerical examples from Section 3.5 with the cases where Pareto optimality cannot finalize the ranking. The results are presented in Table 6.
Fuzzy number
Range discount (drng)
Mem. discount (dmem)
Ranking index (Irank)
Case e
FE1 = (0, 0, 0, 0.1; 1)
1
1
0.0083
FE2 = (0.5, 0.6, 0.6, 0.7; 1)
0.5
1
0.3
FE3 = (0.9, 1, 1, 1; 1)
1
1
0.9917
Case g
FG1 = (0, 0.1, 0.5, 1; 1)
0.2
1
0.0667
FG2 = (0.5, 0.6, 0.6, 0.7; 1)
1
0.7143
0.4286
Case h
FH1 = (0, 0.1, 0.5, 1; 1)
0.2
1
0.0667
FH2 = (0.6, 0.7, 0.7, 0.8; 1)
1
0.7143
0.5
Case j
FJ1 = (0.5, 0.7, 0.7, 0.9; 1)
1
0.6667
0.4667
FJ2 = (0.3; 0.7, 0.7, 0.9; 1)
0.6667
0.6667
0.3037
FJ3 = (0.3, 0.4, 0.7, 0.9; 1)
0.6667
1
0.3722
Case k
FK1 = (0.3, 0.5, 0.8, 0.9; 1)
0.6667
1
0.4278
FK2 = (0.3, 0.5, 0.5, 0.9; 1)
0.6667
0.6667
0.2296
FK3 = (0.3, 0.5, 0.5, 0.7; 1)
1
0.6667
0.3333
Case n
FN1 = (0, 0.2, 0.2, 0.4; 1)
1
1
0.2
FN2 = (0.6, 0.8, 0.8, 1; 0.8)
1
0.8
0.64
Case o
FO1 = (0.4, 0.6, 0.6, 0.8; 1)
0.5
1
0.3
FO2 = (0.8, 0.9, 0.9, 1; 0.2)
1
0.2
0.18
Case r
FR1 = (0.6, 1, 1, 1; 1)
0.25
1
0.2417
FR2 = (0.9, 0.95, 0.95, 1; 0.2)
1
0.2
0.19
Table 6.
Ranking index results for cases with trade-off consideration.
Case e comes from Group 1 (see Table 1), where FE3 is ranked on the top per Pareto optimality (same result from the ranking index). Between FE1 and FE2, though FE2 should be ranked higher intuitively, trade-off is involved logically because FE2 has a large range (reflected in its range discount of 0.5 as well). Per the ranking index, we still have FE2≽FE1, which matches the general intuition.
Cases g and h come from Group 3 (see Table 3), where wide trapezoidal fuzzy numbers are compared with narrow triangular fuzzy numbers. Per the ranking index, the triangular fuzzy numbers are ranked higher mainly because of the large difference of the range discount (1 vs. 0.2). In contrast, the difference of the membership ratio discount is less substantial.
Cases j and k come from Group 4 (see Table 4), where two triangular fuzzy numbers are nested in a trapezoidal fuzzy number. In both cases, the wider triangular fuzzy numbers (i.e., FJ2 and FK2) are ranked lowest as they receive both discounts (i.e., drng = dmem = 0.6667). In case j, the narrower triangular fuzzy number (FJ1) is ranked first because its representative x-value and range can “win” over its weaker membership ratio as compared to the trapezoidal fuzzy number (FJ3). In contrast, in case k, the trapezoidal fuzzy number (FK1) “wins” because it has better representative x-value and membership ratio as compared to FK3.
Cases n, o and r come from Group 5 (see Table 5). In case n, we have FN2≽FN1, as FN2 has higher representative x-value despite lower membership ratio (associated with the discount dmem(FN2) = 0.8). In cases o, we have FO1≽FO2 because the membership ratio of FO2 is substantially lower despite its higher representative x-value. In case r, the trade-off between range and membership ratio is relatively close. In the end, we have FR1≽FR2, as FR2 has a lower value of the discount from membership ratio (i.e., dmem(FR2) = 0.2 vs. drng(FR1) = 0.25).
Notably, the judgment for ranking with trade-off can become difficult when the trade-off among the three attributes is getting close. We argue that such difficulty is fundamentally embedded into the problem structure of FNR, which involves multiple dimensions of considerations. Thus, our solution strategy is not about providing the best ranking procedure. Instead, we emphasize the importance of defining attributes to quantify the “intuition”. Then, decision makers can explicitly explain their trade-off considerations in the ranking process.
As we consider that ranking methods should be dependent on a given set of fuzzy numbers to be ranked (i.e., context-dependent), we want to discuss two types of adjustable elements of our proposed method in practice. The first type is the selection of attributes. Among three attributes: representative x-value, range and membership ratio, representative x-value should be a default choice as it intuitively corresponds to the ranking of real numbers (e.g., compare representative x-values of different fuzzy numbers). We suggest the class of ranking indices by Ban and Coroianu [28] (i.e., in Eq. (6)) as it satisfies the six axioms. To determine the weight, w, of this attribute, decision makers may consider sensitivity analysis for their given set of fuzzy numbers (i.e., how sensitive of the value of w can alter the ranking of two fuzzy numbers).
In contrast to representative x-value, the choice of range and membership ratio is optional. The attribute of x-value range is common in literature, and other formulations of this attribute (e.g., ambiguity and deviation degree as mentioned in Section 3.2) can be considered as a choice by decision makers for this attribute. If the ranking problem has fuzzy numbers of similar ranges (e.g., triangular fuzzy numbers with similar supports), we think it is legitimate not to consider range in FNR (in order to preserve some axiomatic properties, to be discussed in Section 6). The attribute of membership ratio is less common, and it should be more relevant for non-normal fuzzy numbers (e.g., normal triangular fuzzy numbers always have the same membership ratio equal to 0.5).
The second type of adjustable elements of our proposed method is the specification of the reference values (i.e., rngmin and memmax) and the formulations of the discount factors. Notably, our formulations of discount factors (i.e., Eqs. (13) and (14)) are only one simple suggestion. One possible disadvantage of our discount factors is that they can be too sensitive to the reference values. For example, if two fuzzy numbers with the range values of 0.5 and 1 are compared, the range discount (drng) can be equal to 0.5 for one fuzzy number, cutting half of its representative x-value. Decision makers can consider adjusting the effects of discount factors through other formulations for their problems (e.g., additional scaling component).
5.2 Applicability to specific forms
The origin of fuzzy numbers can be viewed as a generalization of crisp numbers to describe approximate information. Consider the 5-tuple definition of a fuzzy number, FA = (a1, a2, a3, a4; hA) as the generalized form of fuzzy numbers in this work. Accordingly, three specific forms can be considered as follows.
Interval: if a1 = a2, a3 = a4 and hA = 1;
Ordered pair (an element of a fuzzy set): if a1 = a2 = a3 = a4 and hA < 1;
Crisp number: if a1 = a2 = a3 = a4 and hA = 1.
Then, we want to investigate the reducibility property that whether a ranking method can still be applicable if the above specific forms are considered. Table 7 shows that our proposed ranking method can be still used for these specific forms. First, a general fuzzy number can be evaluated using the equations as listed in the first row of Table 7. When these equations are applied to interval, ordered pair and crisp number, we can obtain the results that match our expectations. For example, the representative x-value of an interval will be the midpoint of a2 and a3, and a crisp number has no discount effect (i.e., drng = 1 and dmem = 1). In this way, our proposed method can be used to compare fuzzy numbers with intervals or crisp numbers in the same methodical framework.
Since the pioneer work by Wang and Kerre [1], researchers have examined the axiomatic properties (i.e., the six axioms listed in Section 2) of fuzzy number ranking methods. The intent of this section is to discuss how the information content for ranking can influence the axiomatic properties in the context of our ranking approach. One key message is that the satisfaction of axioms depends on the selection of information that is deemed relevant to FNR. If more information is selected and considered for ranking, the ranking method is more likely to violate the axioms. This message is aligned with the topic of information basis in the analysis of the Arrow’s Impossibility Theorem [39, 40].
6.1 Analysis of Axiom 4
Axiom 4 somewhat dictates the ranking of non-overlapping fuzzy numbers. If we only consider representative x-values for ranking (i.e., no range, membership ratio and discount factors), our ranking procedure will directly follow the results from Ban and Coroianu [28], and it will thus satisfy Axiom 4. However, if range and membership ratio are considered as relevant information for ranking, Axiom 4 can be violated, and the reason is given below.
Axiom 4 only focuses on the boundary values without considering any distributional information (e.g., range and membership). When multiple attributes are considered for ranking, Axiom 4 can be violated by strengthening the distributional aspect of the inferior fuzzy number (in view of Axiom 4). For example, we have FO2≽FO1 in case o (see Table 5) according to Axiom 4, no matter how small of membership ratio of FO2. However, when we consider range and membership ratio, we obtain FO1≽FO2 (see Table 6), which violates Axiom 4. Notably, the logic of such violation can be held whenever we deem membership ratio as relevant information for FNR, regardless of the details of the ranking procedures.
Notably, this discussion is not about rejecting Axiom 4. Instead, we want to explain one logical tension with Axiom 4. That is, Axiom 4 dictates some ranking of fuzzy numbers based on their boundary values only, and this opens a chance for the information of range and membership ratio to violate Axiom 4. Alternately, if we choose the index class by [28], representative x-values will be the only information considered for ranking, and the information of range (for example) will become irrelevant for FNR. In other words, if we consider that FNR should involve trade-off with multiple attributes in addition to representative x-value, Axiom 4 could be violated in some situations.
6.2 Dependence of rngmin and memmax
As we use reference values (i.e., rngmin and memmax) to evaluate the discount factors (i.e., drng and dmem), the ranking index, Irank, belongs to the second class of ranking indices according to the classification by [1]. In their axiomatic analysis, they have identified five indices of the second class, i.e., JK(), K(), CHK(), W() and KPK(), which all do not satisfy Axioms 5 and 6. Without listing counter-examples, Irank of the same class follows the same conclusion because they share a common feature of these ranking indices, i.e., use of reference values.
Why using reference values could violate Axioms 5 and 6? It is because the index values would depend on the information that is external to the fuzzy numbers themselves. For example, if a fuzzy number with a very small range is added to a set of fuzzy numbers for ranking (i.e., FR), this newly added fuzzy number will decrease the reference value, rngmin, and thus generally decrease the range discount values (drng) for the original set of fuzzy numbers. Then, all values of Irank would change because of adding a new fuzzy number to the set (i.e., FR).
While it seems undesirable by setting rngmin and memmax per individual sets of fuzzy numbers, can we simply set these two values as universal numbers that are applicable to all ranking problems (e.g., simply set memmax = 1)? Theoretically, it is a viable option. However, by doing so, we somehow lose our interpretation of “discount” factors that are relevant to a given set of fuzzy numbers that we want to rank in the problem. For example, it is not easy to interpret if the range of a fuzzy number, say 5, is large or small until we know a reference for comparison (e.g., if rngmin = 1, the range of 5 will quite large). In other words, the reference values, rngmin and memmax, provide a numerical context as relevant information for comparison.
To close this section, we want to make a note about the historical development of the Arrow’s Impossibility Theorem, which proves that no voting method (or social welfare function) can satisfy a set of “reasonable” properties (or axioms) [41]. One famous “escaping route” is the information basis approach, which classifies the information content (or availability) for interpersonal comparisons with different axiomatic results [39, 40]. Back to our context, if representative x-value is taken as the only relevant information for FNR, the results by [28] are sufficient to design a ranking index that satisfies the six axioms in Section 2. However, if additional information is considered for FNR, the axiomatic properties cannot be guaranteed. To us, this tension seems fundamental.
The main theme of this chapter is to use the multi-attribute approach to analyze and address the problems of fuzzy number ranking (FNR) since numerous ranking methods in literature have implicated multiple attributes in their ranking intuition. The multi-attribute approach has two phases: the selection of attributes and the formulation of the aggregation function. The selection of attributes determines what information is deemed relevant for FNR, and the aggregation function controls the trade-off of the attribute values of fuzzy numbers. In this work, we propose three attributes (i.e., representative x-value, range and membership ratio) as three possible dimensions to evaluate a fuzzy number. In aggregation, we formulate the discount factors for range and membership ratio to modify the representative x-value of a fuzzy number for FNR. The proposed method has been illustrated via numerical examples to reveal the rationale of using multiple attributes to articulate the intuition behind FNR.
In future work, there can be two directions to consider: practice and theory. In the practice direction, we can develop more methodical guidance toward the selection and formulations of attributes and the aggregation procedures. In particular, we can formalize the ranking intuition in terms of the selected attributes and the trade-off rationale through the aggregation approach for different FNR problems. Along this effort, we can also compare the ranking results from this multi-attribute approach with other FNR approaches. In the theory direction, while this chapter has initially explored the tension between information content and the satisfaction of the FNR axioms. This tension should call for more mathematical analyses such as classification of information content for FNR and relaxation of axioms for expanded information basis.
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Simon Li
Submitted: September 27th, 2022Reviewed: January 12th, 2023Published: February 16th, 2023
Open access peer-reviewed chapter - ONLINE FIRST
