**3. Three attributes for fuzzy number ranking**

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 "xvalue" 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 xvalue 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.

$$\operatorname{rep}(F\_A, w) = w \cdot a\_1 + \left(\frac{1}{2} - w\right) a\_2 + \left(\frac{1}{2} - w\right) a\_3 + w \cdot a\_4 \tag{6}$$

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 (*a*<sup>2</sup> and *a*3) with a weight (1/2-*w*) and boundary values (*a*<sup>1</sup> and *a*4) 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., *a*<sup>2</sup> and *a*3) 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.

$$w[(a\_1 + a\_4) - (b\_1 + b\_4)] + \left(\frac{1}{2} - w\right)[(a\_2 + a\_3) - (b\_2 + b\_3)] \ge 0\tag{7}$$

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*) 6¼ ∅). 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 [16–18]. 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.

$$r\text{mg}(F\_A, a) = r\_{F\_A}(a) - l\_{F\_A}(a) \tag{8}$$

**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 *lFA* ð Þ *α* and

**Figure 2.** *Illustration of the α-cut interval.*

*rFA* ð Þ *α* , respectively. The formulations of *lFA* ð Þ *α* and *rFA* ð Þ *α* 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.

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).

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 αcut 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 3.**

*Illustration of the concept for overall membership a)* FA *and* FB *with same membership b)* FC *with higher membership than* FD.

**Figure 4.** *Illustration of the shape's area and full membership area.*

$$
area\_{shape}(F\_A, a) = \int\_{l\_{F\_A}(a)}^{r\_{F\_A}(a)} \mu\_{F\_A}(\infty) d\infty \tag{9}
$$

$$area\_{full}(F\_A, a) = [r\_{F\_A}(a) - l\_{F\_A}(a)] \times \mathbf{1} \tag{10}$$

The overall membership ratio of a fuzzy number (denoted as *mem*(*FA*, *α*)) can be expressed as follows.

$$mem(F\_A, a) = \frac{area\_{shape}(F\_A, a)}{area\_{full}(F\_A, a)}\tag{11}$$

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:

*Decoupling of Attributes and Aggregation for Fuzzy Number Ranking DOI: http://dx.doi.org/10.5772/intechopen.109992*

	- From Eq. (2): *lFA* ð Þ *<sup>α</sup>* = 1 + (0/1) � (2–1) = 1
	- From Eq. (3): *rFA* ð Þ *<sup>α</sup>* = 4 + (0/1) � (3–4) = 4
	- From Eq. (9) = *areashape*ð Þ¼ *FA*, *<sup>α</sup>* trapezoid' s area = (1 + 3) � 1/2 = 2
	- From Eq. (10) = *areafull*ð Þ¼ *FA*, *<sup>α</sup>* ½ �� <sup>4</sup>–<sup>1</sup> <sup>1</sup> <sup>¼</sup> <sup>3</sup>

#### **3.4 Pareto optimality**

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.


#### **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 *FE*<sup>3</sup> is ranked highest, we cannot immediately rank *FE*<sup>2</sup> higher than *FE*<sup>1</sup> based on Pareto optimality only since *FE*<sup>2</sup> has a larger range. Through these five cases, we want to note

#### **Table 1.**

*Results of comparing non-overlapping, triangular fuzzy numbers.*

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).

*Decoupling of Attributes and Aggregation for Fuzzy Number Ranking DOI: http://dx.doi.org/10.5772/intechopen.109992*

### *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 *FF*<sup>2</sup> should be ranked higher than *FF*<sup>1</sup> 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*(*FF*2) > *rep*(*FF*1). If *w* ≥ 1/6 (i.e., more emphasis to the boundary values), we have *rep*(*FF*1) ≥ *rep*(*FF*2).

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

**Table 2.** *Results of comparing overlapping, triangular fuzzy numbers.*

**Table 3.** *Results of comparing triangular and trapezoidal fuzzy numbers.*

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 *FG*<sup>2</sup> ≽ *FG*<sup>1</sup> in case g, we would have *FH*<sup>2</sup> ≽ *FH*<sup>1</sup> in case h. It is because *FH*<sup>2</sup> ≽ *FG*<sup>2</sup> due to Pareto optimality and *FG*<sup>1</sup> = *FH*1. This note should make sense when we observe the graphical shift of triangular fuzzy numbers from *FG*<sup>2</sup> to *FH*<sup>2</sup> in **Table 3**. This demonstrates how the three attributes can characterize some intuitive reasoning in FNR.

*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, *FJ*<sup>2</sup> is created by shifting the lower bound of *FJ*<sup>1</sup> to the left; *FJ*<sup>2</sup> and *FJ*<sup>3</sup> 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 *FJ*<sup>1</sup> ➔ *FJ*<sup>2</sup> ➔ *FJ*<sup>3</sup> or *FK*<sup>1</sup> ➔ *FK*<sup>2</sup> ➔ *FK*3, 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 *FN*<sup>1</sup> and *FN*<sup>2</sup> in case n with the trade-off of

**Table 5.** *Results of comparing non-normal fuzzy numbers.*

representative x-value and membership ratio (similarly for case o)? While we see *FP*<sup>2</sup> ≽ *FP*<sup>1</sup> in case p and *FQ*<sup>1</sup> ≽ *FQ*<sup>2</sup> in case q due to Pareto optimality, the trade-off consideration is present in case r with different values of range.

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.
