**2. Preliminaries**

Fuzzy number is described as a fuzzy subset of the real line [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 *μFA* ð Þ *x* , can be expressed in the following formulation where *a*1, *a*2, *a*3, *a*<sup>4</sup> are real

**Figure 1.** *A trapezoidal fuzzy number.*

numbers to specify *FA*. As a convenient notation, *FA* can be expressed as a 5-tuple, where *FA* = (*a*1, *a*2, *a*3, *a*4; *hA*).

$$\mu\_{F\_A}(\mathbf{x}) = \begin{cases} 0 & \mathbf{x} < a\_1 \\ \left(\frac{\mathbf{x} - a\_1}{a\_2 - a\_1}\right) h\_A & a\_1 \le \mathbf{x} < a\_2 \\ \hline h\_A & a\_2 \le \mathbf{x} < a\_3 \\ \left(\frac{\mathbf{x} - a\_4}{a\_3 - a\_4}\right) h\_A & a\_3 \le \mathbf{x} < a\_4 \\ \mathbf{0} & \mathbf{x} > a\_4 \end{cases} \tag{1}$$

Let *supp*(*FA*) be the support of *FA*, and we have *supp*(*FA*)={*x* ∈ | *a*<sup>1</sup> ≤ *x* ≤ *a*4}. Then we have the infimum and supremum of *supp*(*FA*) as inf *supp*(*FA*) = *a*<sup>1</sup> and sup *supp*(*FA*) = *a*4, respectively. Also, let *IFA* ð Þ¼ *α* ½ � *lFA* ð Þ *α* ,*rFA* ð Þ *α* 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.

$$l\_{F\_A}(a) = a\_1 + \left(\frac{a}{h\_A}\right)(a\_2 - a\_1) \tag{2}$$

$$r\_{F\_4}(a) = a\_4 + \left(\frac{a}{h\_A}\right)(a\_3 - a\_4) \tag{3}$$

Suppose we have two fuzzy numbers: *FA* = (*a*1, *a*2, *a*3, *a*4, *hA*) and *FB* = (*b*1, *b*2, *b*3, *b*4, *hB*), and a constant, denoted as *λ* (i.e., *λ*∈ ). We can have fuzzy number addition and multiplication with a constant as follows [17, 18, 34].

$$F\_A \oplus F\_B = (a\_1 + b\_1, a\_2 + b\_2, a\_3 + b\_3, a\_4 + b\_4; \min\{h\_A, h\_B\})\tag{4}$$

$$
\lambda \cdot F\_A = (\lambda \cdot a\_1, \lambda \cdot a\_2, \lambda \cdot a\_3, \lambda \cdot a\_4; h\_A) \tag{5}
$$

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,

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

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 that*λ*∈ 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.
