**2. Preliminaries**

In classical set theory, a set is defined as any well-defined collection of objects. This theory highlights the fact that there can never be any doubt at all as to whether any object belongs to the set or not. If the object belongs to it, it is said to belong to the set in degree 1. If the object does not belong to it, it is said to belong to the set in degree 0, there is no other option. With the birth of fuzzy set theory [2], the membership constraint is relaxed, accepting that objects can partially belong to the set. Thus, for example, a bitten apple may belong to the set of apples in grade 0.5 [17,18]. The sets created with this approach are called fuzzy sets and make up the basic elements of Zadeh's theory. In order to establish the nomenclature that will be used in the present paper, we proceed to define the different concepts related to fuzzy sets.

**Definition 1.** *Fuzzy subset, membership function and support*.

*We consider E an ordinary set that we take as referential set. A fuzzy subset A of E is a set of ordered pairs:*


$$\bar{A} = \left\{ \left( \mathbf{x} \; , \; \mu\_{\bar{A}}(\mathbf{x}) \right) / \mathbf{x} \in E \right\}$$

*where μA*- *is a function: μA*- : *E* → [0, <sup>1</sup>].


*Function μA*- *is called membership function of fuzzy subset A , and given an element x* ∈ *E, the value μA*-(*x*) *is called membership degree, compatibility degree or truth degree of element x in fuzzy subset A* - .

*We call Support of A, and we indicate by Supp* - *A*-*at the ordinary set:* 

$$\operatorname{Supp}\left(\tilde{A}\right) = \left\{\mathbf{x} \in \operatorname{E}/\mu\_{\tilde{A}}(\mathbf{x}) > 0\right\},$$

## **Definition 2.** *Fuzzy number*

*A fuzzy number A is defined as a fuzzy subset of the referential R of the set of real numbers such that the membership function fulfils the following conditions:*

*1a. A minimum of value x exists so that μA*-(*x*) = 1 *(normality)*

*1b. μA*-(*x*) ≥ *min <sup>μ</sup>A*-(*<sup>x</sup>*1, *μA*-(*<sup>x</sup>*2)) ∀ *x*1*, x*2 ∈ *R and* ∀ *x* ∈ *[x*<sup>1</sup>*, <sup>x</sup>*2*] (convexity) 1c. μA*-(*x*) *is a piecewise continuous function in all the points of* R*, this is, it's a continuous function except perhaps at a finite number of points in its domain.*

The fuzzy number is in accordance with a prediction made by a human being. For example, convexity guarantees that as we approach a value *x* the closer *x*0 with *μA*-(*<sup>x</sup>*0) = 1, then its possibility value of this *x* is higher. It is important that readers are aware of this restrictive assumption.

Further, note that any real number "*m*" (which may be referred to as a crisp number) can be considered a particular case of fuzzy number whose membership function is *μ*(*x*) = ! 1 *if x* = *m*

0 *if x* = *m*

It is important to know that, by virtue of the representation theorem established by Zadeh [19], every fuzzy number *A* - can be expressed via its α-cuts, and for this reason we can express in a simplified way:

$$\tilde{A} = \{A\_a \; , \; 0 \le a \le 1\} \tag{1}$$


where *Aα* = *x* ∈ *<sup>R</sup>*/*μA*-(*x*) ≥ *α* <sup>⊆</sup> *R*.

.

By virtue of convexity, *Aα* is reduced to a closed interval of *R*, which shall be represented thus:

$$A\_a = \begin{bmatrix} \underline{A}(a), \ \overline{A}(a) \end{bmatrix} \tag{2}$$

where *<sup>A</sup>*(*α*) = *min<sup>x</sup>*/*μA*-(*x*) ≥ *α* and *<sup>A</sup>*(*α*) = *max<sup>x</sup>*/*μA*-(*x*) ≥ *<sup>α</sup>* .

Note that the representation theorem is justified in its formal form trough the following equality:

$$\widetilde{A} = \bigcup\_{\alpha \in [0,1]} \alpha \cdot A\_{\alpha}$$

understanding the union through the maximum operator, because the membership function verifies:

$$\mu\_{\bar{A}}(\mathbf{x}) = \max\_{\boldsymbol{\alpha} \in [0,1]} \{ \boldsymbol{\alpha} \cdot \mu\_{A\_{\boldsymbol{\alpha}}}(\mathbf{x}) \},$$

We must also take into account that *<sup>A</sup>*(*α*) is an increasing function with regard to *α*, and that *<sup>A</sup>*(*α*) is a decreasing function with regard to α due to convexity.

However, just as algebraic operations can be conducted using real numbers, here our interest lies in determining how common operations on real numbers can incorporate the use of fuzzy numbers.

This can be resolved if we apply the extension principle of operations or composition laws, initially introduced by Zadeh [19], and subsequently modified by other authors [20–23], which considers a general method for extending the usual operations of arithmetic to the case in which uncertain amounts are represented through fuzzy subsets or fuzzy numbers.

In the case that we have a unary operation *f*(*a*) or a binary operation *f*(*<sup>a</sup>*, *b*) between one or two quantities respectively, which are the cases we will study in our model, we will have that the principle of extension of operations to uncertain magnitudes expressed by fuzzy subsets will represent the degree of possibility of each possible solution from the following membership functions:

1. If *A* is a fuzzy subset of *E*, the extension of a unary operation from a set *E* to another set *F*:

$$f: E \to F$$

will be given by the fuzzy subset *C* - = *f A*-with the membership function:

$$\mu\_{\tilde{C}}(z) = \left\{ \bigvee\_{\mathbf{x} \in f^{-1}(z)} \mu\_{\tilde{A}}(\mathbf{x}) \qquad \text{if} \qquad f^{-1}(z) \neq \mathcal{Q} \right. \tag{3}$$

where ∨ represents the supremum operator.




2. For the most common case, whereby a binary operation or internal composition law between the elements of *E* is defined by:

$$f: E \times E \to E$$

$$(x, y) \to f(x, y) = x \ast y$$

if we consider *A* - and *B* - as fuzzy subsets of *E*, then *C* - = *f*(*A* - , *B* ) = *A* - (∗)*<sup>B</sup>* - is expressed by:

$$\mu\_{\vec{C}}(z) = \left\{ \bigvee\_{\{(\mathbf{x},\mathbf{y})/x\*y=z\}} \left[ \mu\_{\vec{A}}(\mathbf{x}) \wedge \mu\_{\vec{B}}(\mathbf{y}) \right] \text{ if } f^{-1}(z) \neq \mathcal{Q} \right. \tag{4}$$

#

where ∨ represents the supremum and ∧ represents the infimum, which in this case coincides with the minimum due to there being a finite number of values.

The support of *C* contains the possible results of the operation obtained from the elements of the respective supports of *A* - and *B* - , and using the max-min convolution we determine the degree of membership of each possible result of the operation. In fact, when *A* - and *B* - are fuzzy numbers, and, therefore, fuzzy convex subsets of *R*, the degree of membership *μC*-*(z)* of a possible solution is the maximum for the values of membership *μAx*- *B*-(*<sup>x</sup>*, *y*) among all elements (*<sup>x</sup>*, *y*), which following the operation give us the result z; that is to say that *f*(*<sup>x</sup>*, *y*) = *z*.

More symbolically, if it is being understood that if *f* <sup>−</sup><sup>1</sup>(*z*) = ∅ then *μC*-(*z*) = 0 we simply write:

$$\mu\_{\vec{C}}(z) = \bigvee\_{z = x \ast y} \left[ \mu\_{\vec{A}}(x) \wedge \mu\_{\vec{B}}(y) \right] \tag{5}$$

When we write *C* - = *A* - (∗)*<sup>B</sup>* - , observe that the extension to the fuzzy subsets *A* - and *B* - from the binary operation ∗ defined in *E* are denoted by means of the (∗) symbol.

In particular, if *A* and *B* are fuzzy numbers with continuous membership functions *μA*- and *μB*-, respectively, and ∗ is a binary operation in *R*, then the membership function of the extension *A* - (∗)*<sup>B</sup>* - is given by:

$$\mu\_{\bar{A}(\*)\bar{B}}(z) = \bigvee\_{z=x\*y} [\mu\_{\bar{A}}(x) \wedge \mu\_{\bar{B}}(y)] \tag{6}$$


If the function *f*(*<sup>x</sup>*, *y*) = *x* ∗ *y* which defines the binary operation is continuous, the supremum ∨ coincides with the maximum and ∧ is the minimum. Adapting the extension principle to the Equation (6) is therefore also referred to as a max-min convolution operation.

**Remark 1.** *Compatibility of the extension principle with the α-cuts*.



*In the case that A and B are fuzzy numbers, and the binary operation f is a continuous function, Nyugen [24] states that as the supremum of Equation (4) is achieved by an (x,y), then the supremum coincides with the maximum, resulting in the following property being fulfilled:*

$$\mathcal{C}\_{a} = \left[ f \left( \tilde{A} \; \; \; \; \tilde{B} \right) \right]\_{a} = f(A\_{a}, B\_{a}) \tag{7}$$

*where Aα and Bα are the respective α-cuts of A* - *and B , and f*(*<sup>A</sup><sup>α</sup>*, *<sup>B</sup>α*) *indicates, in this case, the image set.*


**Remark 2** (Dubois and Prade [22])**.** *If A and B* - *are fuzzy numbers with continuous membership functions μA*- *and μB*-*, respectively, and* ∗ *is an increasing monotonous (or decreasing monotonous) binary operation in R, then A* - (∗)*<sup>B</sup> is a fuzzy number with a continuous membership function.* -


**Remark 3** (Buckley [25])**.** *If* ∗ *is a binary operation in R defined by x* ∗ *y* = *f*(*<sup>x</sup>*, *y*)*, where f is a continuous function, and A* - *and B* - *are fuzzy numbers whose membership functions are continuous, then, if we consider C* - = *A* - (∗)*B, we have:* -

$$\mathbb{C}\_{\mathbb{A}} = \{ z = \mathbf{x} \* y / \mathbf{x} \in A\_{\mathbb{A}}, y \in B\_{\mathbb{A}} \}\tag{8}$$

**Remark 4** (Moore [26])**.** *If* ∗ *is a binary operation in R defined by x* ∗ *y* = *f*(*<sup>x</sup>*, *y*)*, where f is a continuous rational function in its domain and each variable appears only once as a maximum and is elevated to the first power, and A* - *and B* - *are fuzzy numbers whose membership functions are continuous, then, if we consider C* - = *A* - (∗)*B, the* - *α-cuts of the result are:*

$$C\_{\mathfrak{a}} = A\_{\mathfrak{a}}(\*)B\_{\mathfrak{a}} \tag{9}$$

*where <sup>A</sup>α*(∗)*<sup>B</sup>α in this case represents the corresponding operation induced by the function f through the elementary operations of the confidence intervals.*
