**5. Multi-attribute ranking method in practice**

#### **5.1 General suggestions for application**

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* = (*a*1, *a*2, *a*3, *a*4; *hA*) as the generalized form of fuzzy numbers in this work. Accordingly, three specific forms can be considered as follows.


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


**Table 7.**

*Overview of the reducibility property.*

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 *a*<sup>2</sup> and *a*3, 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.
