Fuzzy Numbers and Fuzzy Operations

A fuzzy number can be thought of as a weighted interval of real numbers, where each point of the interval has a degree of membership, ranging from 0 to 1 [7]. The higher this degree, the higher the confidence that the point belongs to the fuzzy number. The function $F_A(x)$, called membership function, gives us the degree of membership for $x$ in the fuzzy number $A$.

Before defining the arithmetic with fuzzy numbers, we have to introduce the concept of $\alpha$-cut. The $\alpha$-cut ( $\alpha \in [0,1]$) of a fuzzy number $A$, is the interval $\{A\}_\alpha=$ $[a_1,a_2]$ such that $F_A(x)>=\alpha, \forall x\in[a_1,a_2]$.

Let $A$ and $B$ be fuzzy numbers, and $\{A\}_\alpha$ and $\{B\}_\alpha$ $\alpha$-cuts. The fuzzy arithmetic operations are defined as follows,

$A+B = C, s.t. \{C\}_\alpha = \{A\}_\alpha\oplus\{B\}_\alpha \ \forall \alpha$

$A-B = C, s.t. \{C\}_\alpha = \{A\}_\alpha\ominus\{B\}_\alpha \ \forall \alpha$

$A \times B = C, s.t. \{C\}_\alpha = \{A\}_\alpha\otimes\{B\}_\alpha \forall \alpha$

$A \div B = C, s.t. \{C\}_\alpha = \{A\}_\alpha\oslash\{B\}_\alpha \forall \alpha$
where the operations $\oplus, \ominus, \otimes$ and $\oslash$ are performed on intervals and are defined as

$[a_1,a_2]\oplus[b_1,b_2] = [a_1+b_1,a_2+b_2]$

$[a_1,a_2]\ominus[b_1,b_2] = [a_1-b_2,a_2-b_1]$

$[a_1,a_2]\otimes[b_1,b_2] = [min(a_1b_1,a_1b_2,a_2b_1,a_2b_2),$ $max(a_1b_1,a_1b_2,a_2b_1,a_2b_2)]$

$[a_1,a_2]\oslash[b_1,b_2] = [a_1,a_2]\otimes[\frac{1}{b_2},\frac{1}{b_1}], 0 \notin [b_1,b_2]$