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.