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Explain Defuzzification and its methods.

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written 23 months ago by |

**Defuzzification:**

Defuzzification is the process of conversion of fuzzy quantity into a precise quantity.

Figure 1. [A] first part of fuzzy output (C1)

[B] Second part of fuzzy output (C2)

[C] Union of part [A] and [B].

The union of two membership function in values the max operator, which is going to be the outer envelope of the two or more shapes.

**Defuzzification methods include:**

[1] max membership principle.

[2] centroid method.

[3] weighted average method.

[4] mean max membership.

[5] center of sums.

[6] centre of largest area.

[7] first of maxima, last of maxima.

**[1] Max – membership principle:**

$M \ c \ (x^*) \ \gt \ M \ c \ (x)$ for all x $\in$ X

**[2] Centroid method:** centre of mall, centre of gravity or area.

$X^A = \frac{\int Ms (x) . xdx}{ \int Mc (x) . dx}$

**[3] Weighted average method:** Valid for symmetrical output membership function.

Each membership function is weighted by its max membership value.

$X^* = \frac{ \sum M c \bar{(x i)} . \bar{xp}}{ \sum M C \bar{(xi)}}$

$\bar{Xi}$ = maximum of with member function.

$\sum$ = algebraic sum.

$x^* = \frac{0.5 a + 0.8 b}{0.5 + 0.8}$

**[4] Mean max membership method:**

This is known as middle of the maxima.

$X^* = \frac{\sum^n_{i = 1} \bar{xp}}{n} $

**[5] Centre of sums:** Algebraic sum of individual fuzzy the union, here, interesting areas are value twice, the defuzzified value $X^+$

$X^* = \frac{\int_x X \sum^n_{iz} M C I (x) dx}{\int_x \sum^n_{iz} M ci (x) dx}$

**[6] Centre of largest area:** When output consists of at least two converse fuzzy subsets which are not overlapping. When o/p fuzzy set has at least two converse regions, then the centre of gravity of converse fuzzy sub region having the largest area is used to obtain defuzzified value.

$X^* = \frac{\int mci (x) . x dx}{\int mci (x) dx}$

**[7] first of maxima (last of maxima)**

This method uses the overall output or union of all individual output fuzzy sets ci for determining the smallest value of the domain maximized membership in ci.

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