Unit 5: Fuzzy Arithmetic & Solution of Fuzzy Equations

1. Addition of Fuzzy Numbers ($A + B$)

Let $A$ and $B$ be two fuzzy numbers on $\mathbb{R}$. By Zadeh's Extension Principle, their sum $C = A + B$ has membership function:

Verification via $\alpha$-Cuts:

For every $\alpha \in (0, 1]$, let $A_\alpha = [a_1(\alpha), a_2(\alpha)]$ and $B_\alpha = [b_1(\alpha), b_2(\alpha)]$. By Nguyen's theorem:

Addition of Triangular Fuzzy Numbers (TFNs):

If $A = (a_1, a_2, a_3)$ and $B = (b_1, b_2, b_3)$ are TFNs:

Both endpoints remain linear in $\alpha$. Therefore:

Theorem 5.1 (TFN Addition): The sum of two Triangular Fuzzy Numbers is strictly a Triangular Fuzzy Number:

2. Subtraction of Fuzzy Numbers ($A - B$)

By the Extension Principle, $C = A - B$ has membership function:

In terms of $\alpha$-cuts:

Subtraction of TFNs:

For $A = (a_1, a_2, a_3)$ and $B = (b_1, b_2, b_3)$:

Therefore:

Theorem 5.2 (TFN Subtraction):

Notice that the left bound is $a_1 - b_3$ and the right bound is $a_3 - b_1$!

1. Multiplication of Fuzzy Numbers ($A \cdot B$)

By the Extension Principle:

In terms of $\alpha$-cuts, for positive fuzzy numbers ($A_\alpha, B_\alpha > 0$):

The Quadratic Shape Distortion:

For two TFNs $A = (a_1, a_2, a_3)$ and $B = (b_1, b_2, b_3)$ with positive vertices:

This expression contains a quadratic term in $\alpha$ ($\alpha^2$)! Consequently:

Crucial Observation: The product of two Triangular Fuzzy Numbers is NOT a Triangular Fuzzy Number! Its left and right membership branches become non-linear (parabolic curves). However, for practical approximations, engineers often approximate the product as a TFN: $(a_1 b_1, a_2 b_2, a_3 b_3)$.

2. Reciprocal and Division ($A / B$)

For a strictly positive fuzzy number $B$ ($b_1 > 0$):

Then the division $A / B = A \cdot B^{-1}$ on $\alpha$-cuts is:

The branches of $A / B$ are rational functions of $\alpha$, exhibiting hyperbolic curvature.

1. Extended Min and Max

Beyond standard arithmetic, one can apply the Extension Principle to the crisp binary functions $\min(x, y)$ and $\max(x, y)$.

Let $A$ and $B$ be fuzzy numbers. We define:

2. $\alpha$-Cut Characterization

For any intervals $I = [a_1, a_2]$ and $J = [b_1, b_2]$:

Therefore:

Both $\text{MIN}(A, B)$ and $\text{MAX}(A, B)$ are valid fuzzy numbers. Together with the fuzzy number ordering $A \le B \iff a_1(\alpha) \le b_1(\alpha) \text{ and } a_2(\alpha) \le b_2(\alpha)$, the set of fuzzy numbers forms a distributive lattice.

1. The Fundamental Problem of Fuzzy Equations

Consider the simple linear algebraic equation where $A$ and $B$ are known fuzzy numbers, and $X$ is an unknown fuzzy number to be determined:

In classical algebra, one simply subtracts $A$ from both sides: $X = B - A$. In fuzzy mathematics, setting $X = B - A$ generally FAILS to solve $A + X = B$!

Why $B - A$ Fails:

Compute $A + (B - A)$:

The width of $A + (B - A)$ is $(b_2 - b_1) + 2(a_2 - a_1)$, which is strictly wider than $B_\alpha$! Thus $A + (B - A) \ne B$ whenever $A$ is non-crisp.

2. Exact Solvability Criterion for $A + X = B$

Theorem 5.3 (Solvability of Linear Fuzzy Equation $A + X = B$): Let $A$ and $B$ be fuzzy numbers with $\alpha$-cuts $A_\alpha = [a_1(\alpha), a_2(\alpha)]$ and $B_\alpha = [b_1(\alpha), b_2(\alpha)]$. The equation $A + X = B$ has an exact fuzzy number solution $X$ if and only if:

1. For all $\alpha \in (0, 1]$, $x_1(\alpha) \equiv b_1(\alpha) - a_1(\alpha)$ is non-decreasing in $\alpha$.
2. For all $\alpha \in (0, 1]$, $x_2(\alpha) \equiv b_2(\alpha) - a_2(\alpha)$ is non-increasing in $\alpha$.
3. $x_1(1) \le x_2(1)$.

In terms of spreads (uncertainty widths $\Delta A_\alpha = a_2(\alpha) - a_1(\alpha)$ and $\Delta B_\alpha = b_2(\alpha) - b_1(\alpha)$): An exact solution exists if and only if the uncertainty of $B$ is greater than or equal to the uncertainty of $A$ at every level:

When this condition holds, the unique solution $X$ has $\alpha$-cuts:

For Triangular Fuzzy Numbers $A = (a_1, a_2, a_3)$ and $B = (b_1, b_2, b_3)$: The candidate solution is $X = (b_1 - a_1, \; b_2 - a_2, \; b_3 - a_3)$. It is a valid TFN if and only if:

1. Multiplicative Fuzzy Equations ($A \cdot X = B$)

Consider the multiplicative equation for strictly positive fuzzy numbers $A, B > 0$:

In terms of $\alpha$-cuts: $[A \cdot X]_\alpha = [a_1(\alpha) x_1(\alpha), \; a_2(\alpha) x_2(\alpha)] = [b_1(\alpha), \; b_2(\alpha)]$

Theorem 5.4 (Solvability of Multiplicative Equation $A \cdot X = B$): For strictly positive fuzzy numbers $A, B$, the equation $A \cdot X = B$ possesses an exact fuzzy number solution $X$ if and only if:

1. $x_1(\alpha) = \frac{b_1(\alpha)}{a_1(\alpha)}$ is non-decreasing with $\alpha \in (0, 1]$.
2. $x_2(\alpha) = \frac{b_2(\alpha)}{a_2(\alpha)}$ is non-increasing with $\alpha \in (0, 1]$.
3. $x_1(1) \le x_2(1)$.

When these conditions hold, the unique solution is given by:

2. Fuzzy Polynomial Equations

For general equations such as $A X^2 + B X = C$, solutions are obtained by solving the coupled non-linear interval equations at each $\alpha$-cut level:

and verifying monotonicity to ensure the resulting family of intervals $[x_1(\alpha), x_2(\alpha)]$ defines a legitimate fuzzy number.