**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 epsilonneighborhood 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 [19–21] and extensions from the centroid concept [21–25].

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

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

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 tradeoff 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 nonnormalized 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.
