**3. Type-2 fuzzy numbers: Two concrete examples**

#### **3.1 Motivation for type-2 fuzzy theory**

The key to fuzzy theory is the concept of membership grades, and it is represented by our individual degrees of confidence. However, membership grades we set may be *Review of Type-1 and Type-2 Fuzzy Numbers DOI: http://dx.doi.org/10.5772/intechopen.110495*

ambiguous. Put another way, determining membership grades means determining the degrees of confidence, but there is also the degree of confidence for the degree of confidence *α*. For example, let us say that we are willing to accept the sentiment that we have determined *α* (100*α* %) with a confidence level of 0.8 (80 %). These are called fuzzy membership grades. In summary, type-2 fuzzy theory is a theory of fuzzy sets with fuzzy membership grades.

A type-1 fuzzy set *A* is characterized by its membership function *A* : *X* ! ½ � 0, 1 for the universal (crisp) set *X*, whereas a type-2 fuzzy set *A*~ is characterized by its membership function *<sup>A</sup>*<sup>~</sup> : *<sup>X</sup>* ! ½ � 0, 1 ½ � 0,1 . Here, *<sup>U</sup><sup>V</sup>* denotes the set of mappings *<sup>U</sup>* ! *<sup>V</sup>* for crisp sets *U*,*V*.

There are other advantages of type-2 fuzzy theory. See Section 3.5 for that.

#### **3.2 Type-2 fuzzy sets and those associated with them**

We begin with the definition of a type-2 fuzzy set. Type-2 fuzzy theory has many concepts and their terms, but we prepare them required in this chapter. (There is a slight change from the traditional definition and notation.) For example, the membership grade of a type-2 fuzzy set is called the fuzzy membership grade of it. Unlike type-1 fuzzy sets, type-2 fuzzy sets can be three-dimensional figures and are generally difficult to depict.

**Definition 3.1.** If *A*~ is characterized by the membership function

$$
\mu\_{\vec{A}} : I \times J\_{\mathfrak{x}} \mathfrak{z}(\mathfrak{x}, \mathfrak{u}) \mapsto \mu\_{\vec{A}}(\mathfrak{x}, \mathfrak{u}) \in [0, 1], \tag{5}
$$

*A*~ is called a type-2 fuzzy set on *X*. Here, *I* ⊂*X* is the universe for the primary variable *x*∈ *X*, and *Jx* ⊂½ � 0, 1 is the interval determined for each *x*∈ *I*. Then, *I* and *Jx* are called the primary and secondary domains of *A*~, respectively.

There are other representations of *A*~:

$$\tilde{A} = \left\{ (\mathbf{x}, u; \mu\_{\hat{A}}(\mathbf{x}, u)) : \mathbf{x} \in I, \ u \in I\_x \right\}, \quad \int\_{\mathbf{x} \in I} \int\_{\mathbf{u} \in I\_x} \mu\_{\hat{A}}(\mathbf{x}, u) / (\mathbf{x}, u), \quad \text{etc.} \tag{6}$$

Remark that " Ð " in the representation on the right side of Eq. (6) means a continuous union for sets, not an integral. Moreover, "*=*" means a marker, not a division, and " Ð " is rewritten as " P" if *A*~ is discrete.

**Definition 3.2.** Let *A*~ be a type-2 fuzzy set on *X*. The type-1 fuzzy set for *A*~ appears if *x*∈*I* is fixed arbitrarily. It is called the vertical slice of *A*~, and its membership function

$$\nu\_{\vec{A}}^{\mathfrak{x}} : J\_{\mathfrak{x}} \to [\mathbf{0}, \mathbf{1}].$$

is called the secondary membership function of *A*~ at *x*. *A*~ can be said to be characterized by *ν<sup>x</sup> <sup>A</sup>*<sup>~</sup> as follows:

$$\bar{A} = \int\_{x \in X} \left( \int\_{u \in J\_x} \nu\_{\bar{A}}^{\times}(u)/u \right) / \infty,$$

where *ν<sup>x</sup> <sup>A</sup>*<sup>~</sup> ð Þ *u* is the value of the secondary membership function at *u*, that is, the secondary membership grade of *A*~.

**Note:** The concepts of vertical slices and secondary membership functions are often treated in the same sense. Because of this, "secondary membership functions" are sometimes also called "vertical slices."

There are two kinds of cutting with respect to *β* and what is important is what to cut with respect to *β*. First, the following is the cutting of vertical slices.

**Definition 3.3.** Let *A*~ be a type-2 fuzzy set on *X*. The crisp set

$$S\_{\hat{A}}(\mathfrak{x}|\beta) = \begin{cases} \left\{ u \in \mathcal{J}\_{\mathfrak{x}} : \nu\_{\hat{A}}^{\mathfrak{x}}(u) \ge \beta \right\} & (\beta \in (\mathbf{0}, \mathbf{1}]), \\\mathbf{cl}\left(\left\{ u \in \mathcal{J}\_{\mathfrak{x}} : \nu\_{\hat{A}}^{\mathfrak{x}}(u) > \mathbf{0} \right\} \right) & (\beta = \mathbf{0}), \end{cases}$$

is called the *β*-cut set of the vertical slice of *A*~ (**Figure 7**). Next, the following is the cutting of type-2 fuzzy sets. **Definition 3.4.** Let *<sup>A</sup>*<sup>~</sup> be a type-2 fuzzy set on *<sup>X</sup>*. For *<sup>β</sup>* <sup>∈</sup> ½ � 0, 1 ,

$$\tilde{A}\_{\beta} = \bigcup\_{\mathbf{x} \in I} \mathbb{S}\_{\vec{A}}(\mathbf{x}|\beta)$$

is called the *β*-plane of *A*~. In particular, *A*~<sup>1</sup> and *A*~ <sup>0</sup> are called the principal (or, principle) set and footprint (set) of *A*~, respectively.

Roughly speaking, a type-2 fuzzy set can be characterized by a membership function in the form of two mountains. We use the following notation for the *β*-plane of *A*~ because we want to make it geometrically easy to see what a type-2 fuzzy set looks like. That is, it brings in the notion of "left- and right-sided type-1 fuzzy sets."

**Definition 3.5.** If the *β*-plane of *A*~ is the interval-valued fuzzy set, there exist type-1 fuzzy sets *A<sup>β</sup>* and *Aβ*. Then, we denote

$$
\tilde{\mathbf{A}}\_{\beta} = \left\langle \underline{\mathbf{A}\_{\beta}}, \ \overline{\mathbf{A}\_{\beta}} \right\rangle.
$$

Here, *A<sup>β</sup>* and *A<sup>β</sup>* are called the lower membership function (briefly, LMF) and upper membership function (briefly, UMF) on *A*~, respectively.

**Definition 3.6.** Let *<sup>A</sup>*<sup>~</sup> be a type-2 fuzzy set on *<sup>X</sup>*. For each *<sup>β</sup>* <sup>∈</sup>½ � 0, 1 , the coupling of *α*-cut sets of *A<sup>β</sup>* and *A<sup>β</sup>* is written as

$$\left[\tilde{\mathbf{A}}\right]\_{\beta}^{a} = \left\langle \left[\underline{\mathbf{A}\_{\beta}}\right]\_{a}, \left[\overline{\mathbf{A}\_{\beta}}\right]\_{a} \right\rangle, \quad a \in [0, 1], \tag{7}$$

and is called the ð Þ *<sup>α</sup>*, *<sup>β</sup>* -cut set of *<sup>A</sup>*~.

**Figure 7.**

*0-cut set of the vertical slice of "about 1" as the meaning of type-2 fuzzy numbers; Figure 1 [11].*

*Review of Type-1 and Type-2 Fuzzy Numbers DOI: http://dx.doi.org/10.5772/intechopen.110495*

As with type-1, we can discuss type-2 fuzzy sets as their *β*-planes or ð Þ *α*, *β* -cut sets. Indeed, Hamrawi found the formula that is a type-2 version of Eq. (3) as follows; we leave the details to Ref. [20] for more information on the contents of this neighborhood.

**Proposition 3.7.** Any type-2 fuzzy set *A*~ on *X* satisfies

$$\tilde{A} = \bigcup\_{\beta \in [0,1]} \beta \bigcup\_{a \in [0,1]} a \left[ \tilde{A} \right]\_{\beta}^{a},$$

where *α A*~ � �*<sup>α</sup> <sup>β</sup>* : *X* ! f g 0, *α* is a type-1 fuzzy set.

#### **3.3 Perfect quasi-type-2 fuzzy numbers**

We hereafter set *X* ¼ .

Hamrawi introduced the following type-2 fuzzy number, which we can call a "triangular type-2 fuzzy number."

**Definition 3.8.** ([20], Section 3.4)**.** Let A be a type-2 fuzzy set on . A is a perfect type-2 fuzzy number if and only if

i. UMF and LMF of FPð Þ A are equal as type-1 fuzzy numbers, and

ii. UMF and LMF of Pð Þ A are equal as type-1 fuzzy numbers.

Moreover, if a perfect type-2 fuzzy number A satisfies that

iii. A can be completely determined by using its FPð Þ A and Pð Þ A ,

such a A is called the perfect quasi-type-2 fuzzy number (briefly, PQT2FN) on . **Definition 3.9.** A PQT2FN <sup>A</sup> is triangular if and only if ½ � <sup>A</sup> *<sup>α</sup> <sup>β</sup>* has the *α*-cut set of LMF on A:

$$\begin{aligned} \left[\underline{A}\_{\beta}\right]\_{a} &= \left[L^{a}\_{\underline{A}\_{\beta}}, \ R^{a}\_{\underline{A}\_{\beta}}\right]; \\ L^{a}\_{\underline{A}\_{\beta}} &= X^{a}\_{A\_{1}} - (1 - \beta) \left(X^{a}\_{A\_{1}} - L^{a}\_{\underline{A}\_{\beta}}\right), \\ R^{a}\_{\underline{A}\_{\beta}} &= Y^{a}\_{A\_{1}} + (1 - \beta) \left(R^{a}\_{\underline{A}\_{0}} - Y^{a}\_{A\_{1}}\right), \\ L^{a}\_{\underline{A}\_{0}} &= C\_{\mathcal{A}} - (1 - a) \left(C\_{\mathcal{A}} - L\_{\underline{A}\_{0}}\right), \\ R^{a}\_{\underline{A}\_{0}} &= C\_{\mathcal{A}} + (1 - a) \left(R\_{\underline{A}\_{0}} - C\_{\mathcal{A}}\right) \end{aligned}$$

and the *α*-cut set of UMF on A:

$$\begin{aligned} \left[\overline{A}\_{\beta}\right]\_a &= \left[L^{\underline{a}}\_{\underline{A}\_{\beta}}, \ R^{\underline{a}}\_{\overline{A}\_{\beta}}\right]; \\ L^{\underline{a}}\_{\underline{A}\_{\beta}} &= X^{a}\_{A\_1} - (1 - \beta) \left(X^{a}\_{A\_1} - L^{\underline{a}}\_{\underline{A}\_0}\right), \\ R^{\underline{a}}\_{\overline{A}\_{\beta}} &= Y^{a}\_{A\_1} + (1 - \beta) \left(R^{\underline{a}}\_{\overline{A}\_0} - Y^{a}\_{A\_1}\right), \\ L^{\underline{a}}\_{\overline{A}\_0} &= C\_{\mathcal{A}} - (1 - a) \left(C\_{\mathcal{A}} - L\_{\overline{A}\_0}\right), \\ R^{\underline{a}}\_{\overline{A}\_0} &= C\_{\mathcal{A}} + (1 - a) \left(R\_{\overline{A}\_0} - C\_{\mathcal{A}}\right), \end{aligned}$$

where

$$\begin{aligned} X\_{A\_1}^a &= \mathcal{C}\_{\mathcal{A}} - (\mathbf{1} - a)(\mathcal{C}\_{\mathcal{A}} - X\_{A\_1}), \\ Y\_{A\_1}^a &= \mathcal{C}\_{\mathcal{A}} + (\mathbf{1} - a)(Y\_{A\_1} - \mathcal{C}\_{\mathcal{A}}). \end{aligned}$$

They are called the left principle number and right principle number of A, respectively. *<sup>C</sup>*<sup>A</sup> denotes the core of <sup>A</sup>, that is, the crisp number ½ � <sup>A</sup> <sup>1</sup> <sup>1</sup>. A triangular perfect quasi-type-2 fuzzy number is abbreviated as TPQT2FN.

**Figure 8** shows

$$L^{\underline{a}}\_{\overline{A}\_0} \le X^{\underline{a}}\_{A\_1} \le L^{\underline{a}}\_{\underline{A}\_0} \le C\_{\mathcal{A}} \le R^{\underline{a}}\_{\underline{A}\_0} \le Y^{\underline{a}}\_{A\_1} \le R^{\underline{a}}\_{\overline{A}\_0} \dots$$

In particular, the supports of A are represented by the *α*-cut sets of LMF and UMF of FPð Þ A :

$$[\underline{\mathbf{A}}\_0]\_a = \left[ L^a\_{\underline{\mathbf{A}}\_0}, R^a\_{\underline{\mathbf{A}}\_0} \right], \quad [\overline{\mathbf{A}}\_0]\_a = \left[ L^a\_{\overline{\mathbf{A}}\_0}, R^a\_{\overline{\mathbf{A}}\_0} \right].$$

Also, the *α*-cut set of Pð Þ A is given as

$$[A\_1]\_a = \left[X^a\_{A\_1}, Y^a\_{A\_1}\right].$$

Now, recall that the triangular type-1 fuzzy number *u* is determined by three information, that is, its left end *l*, core *c* and right end *r*:

$$
\mu = \langle \langle l; c; r \rangle \rangle.
$$

In contrast, TPQT2FN A is determined by seven information, that is, its upper left end *LA*<sup>0</sup> , left principle number *XA*<sup>1</sup> , lower left end *LA*<sup>0</sup> , core *C*A, lower right end *RA*<sup>0</sup> , right principle number *YA*<sup>1</sup> and upper right end *RA*<sup>0</sup> . We then write

$$\mathcal{A} = \left\langle \left\langle L\_{\overline{A}\_0}, X\_{A\_1}, L\_{\underline{A}\_0}; C\_{\mathcal{A}}; R\_{\underline{A}\_0}, Y\_{A\_1}, R\_{\overline{A}\_0} \right\rangle \right\rangle.$$

In general, any type-2 fuzzy set/number satisfies that both the principal set and the vertical slice are type-1 fuzzy numbers as shown in **Figure 9**.

**Figure 8.** *A view of a TPQT2FN* A *from directly above.*

*Review of Type-1 and Type-2 Fuzzy Numbers DOI: http://dx.doi.org/10.5772/intechopen.110495*

**Figure 9.** *Membership function of a perfect quasi-type-2 fuzzy number.*
