This document discusses various methods for defuzzification, which is the process of converting a fuzzy quantity into a crisp quantity. It describes seven common defuzzification methods: 1) max membership principle, 2) centroid method, 3) weighted average method, 4) mean max membership, 5) center of sums, 6) centre of largest area, and 7) first of maxima, last of maxima. For each method, it provides details on the calculation approach and formulas used to determine the defuzzified crisp value. The centroid method is noted as the most commonly used defuzzification technique.

1. DEFUZZIFICATION METHODS

2. DEFUZZIFICATION
 Defuzzification is the process of converting a fuzzified output into a single
crisp value with respect to a fuzzy set.
 Defuzzification is the process of conversion of fuzzy quantity into a Crisp
quantity.
 Defuzzification technique is required in order to find a crisp value for whether
it is a fuzzy set or it is a fuzzy relation or it is a fuzzy rule.
 The defuzzified value in FLC (Fuzzy Logic Controller) represents the action to
be taken in controlling the process.

3. DEFUZZIFICATION METHODS
[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.

4. [1] MAX MEMBERSHIP PRINCIPLE
 This method is also known as height method and is limited to peak output
functions.
 This method is given by the algebraic expression:

5. [2] CENTROID METHOD
 This method is also known as the centre of mass, centre of area or centre of
gravity.
 It is the most commonly used defuzzification method.
 The defuzzified output x* is given by

7. [3] WEIGHTED AVERAGE METHOD
 This method is valid for symmetrical output membership functions only.
 Each membership function is weighted by its maximum membership value.
 The output in the case is given by
where xi
1 is the maximum value of the membership function

8. From the figure the output is

9. [4] MEAN MAX MEMBERSHIP
 This method is also known as the middle of the maxima.
 This is closely related to the max-membership method, except that the
locations of the maximum membership can be nonunique.
 The output here is given by

10. From the figure the output is

11. [5] CENTER OF SUMS
 This method employs the algebraic sum of the individual fuzzy subsets instead
of their union.
 The calculations here are very fast, but the main drawback is that the
intersecting areas are added twice.
 The defuzzified value x* is given by

13. [6] CENTRE OF LARGEST AREA
 This method can be adopted when the output of at least two convex fuzzy
subsets which are not overlapping.
 The output, in this case, is biased towards a side of one membership function.
 When output fuzzy set has at least two convex regions, then the centre of
gravity of the convex fuzzy subregion having the largest are is used to obtain the
defuzzified value x*.
 The value is given by

15. [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.

16.  The steps used for obtaining x* as fallows,
1) Initially, the maximum height in the union is found,
where “SUP” is supremum i.e., least upper bound
2) The first of maximum is found,
Where “inf” is the infimum i.e., greatest lower bound
3) The last maximum is found,