*3.1. Fuzzification*

At this stage , first, each linguistic input *A* is mapped to the linguistic terms set *T*(*A*) = {*T*1*A*, *<sup>T</sup>*2*A*, ··· , *<sup>T</sup>kA*}, and then, the meaning of each linguistic term *TjA*(*j* = 1, *k*) is represented by the membership function *μjA*(*<sup>j</sup>* = 1, *k*).

According to the needs, considering the computational efficiency, different types of membership functions can be used. The most used are *fuzzy numbers* (FNs).

By a FN we understand a mapping *f* : R → [0, 1], such that:


(∀)*x* ∈ <sup>R</sup>,(∀)*<sup>α</sup>* ∈ (0, 1] : *f*(*x*) < *<sup>α</sup>*,(∃)*<sup>δ</sup>* > 0 such that |*y* − *x*| < *δ* ⇒ *f*(*y*) < *α* .

Among the various types of fuzzy sets or their generalizations, the most common (see [6,23,24]) are:

(1) *Triangular FNs* which have membership function

$$f(x) = \begin{cases} 0 & \text{if} \quad x < a\\ \frac{\frac{x-a}{b-a}}{\frac{c-x}{c-b}} & \text{if} \quad a \le x < b\\ \frac{\frac{c-x}{c-b}}{0} & \text{if} \quad b \le x < c\\ 0 & \text{if} \quad x > c \end{cases}, \text{where } a \le b \le c \text{ (}. $$

and they are denoted by *f* = (*a*, *b*, *c*).

> (2) *Trapezoidal FNs* defined by membership function

˜

$$f(\mathbf{x}) = \begin{cases} 0 & \text{if } \quad \mathbf{x} < a \\ \frac{\mathbf{x} - a}{b - a} & \text{if } \quad a \le \mathbf{x} \le b \\\ 1 & \text{if } \quad b < \mathbf{x} < c \\\ \frac{d - \mathbf{x}}{d - c} & \text{if } \quad c \le \mathbf{x} \le d \\\ 0 & \text{if } \quad \mathbf{x} > d \end{cases}, \text{where } a \le b \le c \le d \ne \mathbf{x}$$

and expressed as ˜ *f* = (*a*, *b*, *c*, *d*).

*x*∈*X*

(3) *Gaussian FNs* defined by *f*(*x*) = *e*<sup>−</sup> (*<sup>x</sup>*−*<sup>m</sup>*)<sup>2</sup> <sup>2</sup>*σ*<sup>2</sup> .

*x*∈*X*

(4) *Interval-valued fuzzy sets*, defined by the membership mapping *f* : *X* → I([0, <sup>1</sup>]), where I([0, 1]) represents the set of all closed subintervals of [0, 1].

(5) *Intuitionistic fuzzy sets*, defined by a membership function *f* and also by a nonmembership function *g* such that 0 ≤ *f*(*x*) + *g*(*x*) ≤ 1,(∀)*x* ∈ *X*.

(6) *Interval-valued intuitionistic fuzzy sets*, defined by two functions *f* , *g* : *X* → I([0, 1]) such that 0 ≤ sup *f*(*x*) + sup *g*(*x*) ≤ 1.

 Our basic references for interval-valued fuzzy sets, intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are [25–32].

(7) F. Smarandache [33] proposed in 1999 the concept of neutrosophic set. A *Neutrosophic set* in *X* is defined as

$$A = \{ < \ge, T\_A(\mathfrak{x}), I\_A(\mathfrak{x}), F\_A(\mathfrak{x}) > \colon \mathfrak{x} \in X \}$$

where *TA*(*x*), *IA*(*x*), *FA*(*x*) are subsets of ]<sup>0</sup><sup>−</sup>, 1+[ and represent the truth-membership function, indeterminacy-membership function and falsity-membership function such that

$$0^- \le \sup\_{\mathbf{x} \in X} T\_A(\mathbf{x}) + \sup\_{\mathbf{x} \in X} I\_A(\mathbf{x}) + \sup\_{\mathbf{x} \in X} F\_A(\mathbf{x}) \le \mathbf{3}^+,$$

where

> *a* − = {*a* − : ∈ R<sup>∗</sup>, is infinitesimal} *b*<sup>+</sup> ={*b*+:∈R<sup>∗</sup>, isinfinitesimal}.

 For applications we can consider that *TA*(*x*), *IA*(*x*), *FA*(*x*) are subsets of [0, 1].

(8) *Pythagorean fuzzy sets* were proposed by R.R. Yager in 2013 [34]. A *Pythagorean fuzzy set* is defined by the functions *f* , *g* : *X* → [0, 1] which give us the degree of membership and degree of non-membership, respectively such that

$$0 \le (f(x))^2 + (g(x))^2 \le 1,\\
(\forall) x \in X.$$

The function *h* : *X* → [0, 1] defined by

$$h(\mathbf{x}) = \sqrt{1 - \left[ (f(\mathbf{x}))^2 + (g(\mathbf{x}))^2 \right]}$$

is called the degree of indeterminacy.

## *3.2. Fuzzy Rules*

Fuzzy rules allow the logic fuzzy system to take rational decisions. Fuzzy rules offer a frame in which human knowledge is integrated. In the fuzzy process the relation between input variables and output variables are described through schemes like: "If . . . then". Generally we have a number *N* of fuzzy rules, each of them with the form [24,35]:

If *A*1 is *TjA*1 and *A*2 is *TjA*2 , then *B*0 is *TjB*0 , where *B*0 is the consequent (output) linguistic variable of the rule.

## *3.3. Inference Engine*

The mission of the inference engine is to obtain output variables from input variables, based on fuzzy logic rules.

For example, let us assume there are two rules: 

If *A*1 is *T*1*A*1and *A*2 is *<sup>T</sup>*1*A*2, then *B*0 is *T*1*B*0,

If *A*1 is *T*2*A*1and *A*2 is *<sup>T</sup>*2*A*2, then *B*0 is *T*2*B*0.

 The firing strengths of the two rules are: *f*1 = *<sup>μ</sup>*<sup>1</sup>*A*1 ∧ *<sup>μ</sup>*<sup>1</sup>*A*2 and *f*2 = *<sup>μ</sup>*<sup>2</sup>*A*1 ∧ *<sup>μ</sup>*<sup>2</sup>*A*2 , where ∧ represents AND operation in fuzzy logic, this being the minimum most often, namely

*<sup>μ</sup>*<sup>1</sup>*A*1 ∧ *<sup>μ</sup>*<sup>1</sup>*A*2 = min{*μ*<sup>1</sup>*A*1 , *<sup>μ</sup>*<sup>1</sup>*A*2 } and *<sup>μ</sup>*<sup>2</sup>*A*1 ∧ *<sup>μ</sup>*<sup>2</sup>*A*2 = min{*μ*<sup>2</sup>*A*1 , *<sup>μ</sup>*<sup>2</sup>*A*2 }, but there can be used other t-norms as well.

We also consider *μ*ˆ1*B*0and *μ*ˆ2*B*0defined by: *μ*ˆ1*B*0= *f*1 ∧ *μ*1*B*0and *μ*ˆ2*B*0= *f*2 ∧ *μ*2*B*0

 Finally, the membership degree of the output is obtained using OR operation (denoted ∨), namely *μ* = *μ*ˆ1*B*0∨ *μ*ˆ2*B*0.

.

For OR operation most often we meet: *μ*ˆ1*B*0 <sup>∨</sup>*μ*ˆ2*B*0 = max{*μ*ˆ1*B*0 , *μ*ˆ2*B*0 }, but other t-conorms can be used.
