**3. Fuzzy sets, fuzzy numbers, alpha-cuts and interval arithmetic, L–R fuzzy numbers, arithmetic of L–R fuzzy numbers, triangular fuzzy numbers**

#### **3.1 Fuzzy sets**

**Definition 2**: Let *E* be a classical set or a universe. A fuzzy subset *A*~ (or a fuzzy set *<sup>A</sup>*~) in *<sup>E</sup>* is defined by the function *<sup>η</sup>A*<sup>~</sup> , called membership function of *<sup>A</sup>*~, from *<sup>E</sup>* to the real unit interval [0,1]. *ηA*<sup>~</sup> ð Þ *x* is called the grade or the membership degree of *x*, ∀*x*∈ *A*~ (cf [11]).

**Definition 3**: Let *<sup>A</sup>*<sup>~</sup> fuzzy set on *<sup>E</sup>*. The *<sup>α</sup>*�cut of *<sup>A</sup>*<sup>~</sup> denoted *<sup>A</sup>*<sup>~</sup> *<sup>α</sup>* the support sup *<sup>A</sup>*<sup>~</sup> , the height *h A*~ and the core *A*~ are crisp sets defined as follows:

$$\tilde{A}\_a = \left\{ \mathbf{x} \in E \, : \, \eta\_{\vec{A}}(\mathbf{x}) \ge a \right\} \tag{4}$$

$$\sup(\tilde{A}) = \left\{ \mathbf{x} \in E : \eta\_{\vec{A}}(\mathbf{x}) > \mathbf{0} \right\} \tag{5}$$

$$h(\vec{A}) = \max \{ \eta\_{\vec{A}}(\mathbf{x}) : \mathbf{x} \in E \} \tag{6}$$

$$\text{core}(\tilde{A}) = \{ \mathfrak{x} \in E : \eta\_{\vec{A}}(\mathfrak{x}) = \mathbf{1} \}\tag{7}$$

**Definition 4**: A fuzzy set on a universe is said to be *normal* if:

$$h(\tilde{A}) = \mathbf{1} \tag{8}$$

that is, <sup>∃</sup>*<sup>m</sup>* <sup>∈</sup> *<sup>A</sup>*<sup>~</sup> : *<sup>η</sup>A*<sup>~</sup> ð Þ¼ *<sup>m</sup>* 1. In these conditions, *<sup>m</sup>* is called *modal value* of the fuzzy set *A*~.

**Definition 5**: A fuzzy set *<sup>A</sup>*<sup>~</sup> on the universe *<sup>E</sup>* <sup>¼</sup> is said to be *convex* iff:

$$\forall \boldsymbol{\aleph}, \boldsymbol{y} \in \tilde{\boldsymbol{A}}, \forall \boldsymbol{\aleph} \in [\mathbf{0}, \mathbf{1}] : \boldsymbol{\eta}\_{\tilde{\boldsymbol{A}}}(\boldsymbol{\aleph} + (\mathbf{1} - \boldsymbol{\lambda})\boldsymbol{y}) \ge \min \left\{ \boldsymbol{\eta}\_{\tilde{\boldsymbol{A}}}(\boldsymbol{x}), \boldsymbol{\eta}\_{\tilde{\boldsymbol{A}}}(\boldsymbol{y}) \right\} \tag{9}$$

#### **3.2 Fuzzy numbers**

**Definition 6**: A fuzzy set *A*~ on a universe *E* is called a fuzzy number if it satisfies the following conditions:

1.*E* ¼

2.*A*~ is normal

3.*A*~ is convex

4.The membership function *ηA*<sup>~</sup> is piecewise continuous

#### **3.3 Fuzzy numbers of type L: R of L: R type**

**Definition 7**: A fuzzy set *A*~ is said to be of L–R type if there exists three reals *m*, *a*> 0, *b*> 0 and two continuous and decreasing positive functions L and R from in [0,1] such that: L 0ð Þ¼ R 0ð Þ¼1

$$L(\mathbf{1}) = \mathbf{0}, \quad \text{or} \quad L(\mathbf{x}) > \mathbf{0}, \text{with} \lim\_{\mathbf{x} \to \mathbf{0}} L(\mathbf{x}) = \mathbf{0} \tag{10}$$

$$R(\mathbf{1}) = \mathbf{0}, \quad \text{or} \quad R(\mathbf{x}) > \mathbf{0},\\\text{with}\\\lim\_{\mathbf{x} \to \mathbf{s}} R(\mathbf{x}) = \mathbf{0} \tag{11}$$

$$\eta\_A(\varkappa) = \begin{cases} L\left(\frac{m-\varkappa}{a}\right) & \text{if } \varkappa \in [m-a, m] \\ R\left(\frac{m-\varkappa}{b}\right) & \text{if } \varkappa \in [m, m+b] \\ 0 & \text{otherwise} \end{cases} \tag{12}$$

The L–R representation of a fuzzy number *<sup>A</sup>*<sup>~</sup> is *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> h i *<sup>m</sup>*, *<sup>a</sup>*, *<sup>b</sup> <sup>L</sup>*�*<sup>R</sup>*, *<sup>m</sup>* is called the modal value of *A*~. *a* and *b* are called respectively the *left spread* and *right spread* of *A*~.

By convention, h i *<sup>m</sup>*,0,0 *<sup>L</sup>*�*<sup>R</sup>* is the ordinary real number *<sup>m</sup>*; called also fuzzy singleton. The support of *A*~ is the open interval:

sup *A*~ � � ¼�*<sup>m</sup>* � *<sup>a</sup>*, *<sup>m</sup>*�∪½*m*, *<sup>m</sup>* <sup>þ</sup> *<sup>b</sup>*½ � <sup>¼</sup> *<sup>m</sup>* � *<sup>a</sup>*, *<sup>m</sup>* <sup>þ</sup> *<sup>b</sup>*½.

From the Definition (8) and the expression (12) of *<sup>η</sup>A*<sup>~</sup> , the support of *<sup>A</sup>*<sup>~</sup> is determined by the open following interval:

$$\sup(\tilde{A}) = [m - a, m] \cup [m, m + b] = ]m - a, m + b[ \tag{13}$$

*Computing the Performance Parameters of the Markovian Queueing System FM/FM/1… DOI: http://dx.doi.org/10.5772/intechopen.110388*

#### **3.4 Arithmetic of fuzzy numbers of L: R Type**

#### *3.4.1 Addition and subtraction of fuzzy numbers of L: R type*

According to [10], if there exists two fuzzy numbers of the same L–R type. *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> h i *<sup>m</sup>*, *<sup>a</sup>*, *<sup>b</sup> <sup>L</sup>*�*<sup>R</sup>* and *<sup>B</sup>*<sup>~</sup> <sup>¼</sup> h i *<sup>n</sup>*,*c*, *<sup>d</sup> <sup>L</sup>*�*R*; then their sum and their difference are also fuzzy numbers of L–R type given respectively by:

$$
\tilde{A} \oplus \tilde{B} = \langle m+n, a+c, b+d \rangle\_{L-R} \tag{14}
$$

$$
\tilde{A} \ominus \tilde{B} = \langle m - n, a + c, b + d \rangle\_{L-R} \tag{15}
$$

#### *3.4.2 Multiplication and division*

According to [12], if there exist two fuzzy numbers of the same L–R type. *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> h i *<sup>m</sup>*, *<sup>a</sup>*, *<sup>b</sup> <sup>L</sup>*�*<sup>R</sup>* and *<sup>B</sup>*<sup>~</sup> <sup>¼</sup> h i *<sup>n</sup>*,*c*, *<sup>d</sup> <sup>L</sup>*�*<sup>R</sup>*; then:

$$
\tilde{A} \odot \tilde{B} \approx \langle mn, mc, + na - ac, md + nb + bd \rangle\_{L-R} \tag{16}
$$

$$\frac{\tilde{A}}{\tilde{B}} = \frac{\langle m, a, b \rangle\_{L-R}}{\langle n, c, d \rangle\_{L-R}} \approx \left\langle \frac{m}{n}, \frac{m d}{n(n+d)} + \frac{a}{n} - \frac{a d}{n(n+d)}, \frac{m c}{n(n-c)} + \frac{b}{n} + \frac{b c}{n(n-c)} \right\rangle\_{L-R} \tag{17}$$

The product and the quotient of two numbers of the same type L–R are obtained by the secant approximation of Hanss [14], whose kernel and the support, for the quotient are given by:

$$\ker\left(\frac{\tilde{A}}{\tilde{B}}\right) = \frac{m}{n} \tag{18}$$

$$\text{supp}\left(\frac{\tilde{A}}{\tilde{B}}\right) = \left[\frac{m-a}{n+d}, \frac{m+b}{n-c}\right] \tag{19}$$

#### **3.5 Fuzzy triangular numbers**

**Definition 8**: A fuzzy number *A*~ is said to be a *fuzzy triangular number* iff there exists three real numbers *a*<*b*<*c* such that:

$$\eta\_A(\mathbf{x}) = \begin{cases} \left(\frac{\mathbf{x} - a}{b - a}\right) & \text{if } a \le \mathbf{x} \le b \\\left(\frac{\mathbf{c} - \mathbf{x}}{c - b}\right) & \text{if } b \le \mathbf{x} \le c \\\ 0 & \text{otherwise} \end{cases} \tag{20}$$

### **Remark 1**.


$$\tilde{A} = (a, b, c) = \langle b, b - a, c - b \rangle\_{L - R} \tag{21}$$
