The document discusses defuzzification in fuzzy rule-based systems, focusing on the conversion of fuzzy outputs into crisp values for decision-making in fuzzy logic controllers. It outlines various defuzzification methods, including the Center of Sums, Center of Gravity, and different Maxima methods, explaining their principles and applications. Examples are provided to illustrate each method's implementation in practice.

1. Defuzzification
Course: Computational Intelligence Engineering (Soft Computing)
Prof. (Dr.) Pravat Kumar Rout
Department of EEE, ITER,
Siksha ‘O’Anusandhan (Deemed to be University),
Bhubaneswar, Odisha, India
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2.  Fuzzy rule based systems evaluate linguistic if-then rules using fuzzification,
inference and composition procedures. They produce fuzzy results which
usually have to be converted into crisp output. To transform the fuzzy results
in to crisp, defuzzification is performed.
 Defuzzification is the process of converting a fuzzified output into a single
crisp value with respect to a fuzzy set.
 The defuzzified value in FLC (Fuzzy Logic Controller) represents the action to be
taken in controlling the process.
2 Defuzzification Methods

3. 3
What is defuzzification?

4. 4 Why Defuzzification?

5. 5 Why Defuzzification?

6. 6 Why Defuzzification?

7. Output of a Fuzzy System
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12. Different Defuzzification Methods
 The following are the known methods of defuzzification.
 Center of Sums Method (COS)
 Center of gravity (COG) / Centroid of Area (COA) Method
 Center of Area / Bisector of Area Method (BOA)
 Weighted Average Method
 Maxima Methods
o First of Maxima Method (FOM)
 o Last of Maxima Method (LOM)
o Mean of Maxima Method (MOM)
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13. Center of Sums (COS) Method
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X1 = (0+5)/2 = 2.5
X2 = (3+7)/2 = 5
X3 = (5+8)/2 = 6.5
e.g. Half of the sum of x coordinates

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18. Example
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19. Center of gravity (COG) / Centroid of Area
(COA) Method
 This method provides a crisp value based on the center of gravity of the fuzzy
set.
 The total area of the membership function distribution used to represent the
combined control action is divided into a number of sub-areas.
 The area and the center of gravity or centroid of each sub-area is calculated
and then the summation of all these sub-areas is taken to find the defuzzified
value for a discrete fuzzy set.
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20. 
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27. 27 Example

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31. Center of Area / Bisector of Area Method
(BOA)
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33. Weighted Average Method
 This method is valid for fuzzy sets with
symmetrical output membership functions
and produces results very close to the COA
method. This method is less computationally
intensive. Each membership function is
weighted by its maximum membership
value. The defuzzified value is defined as :
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36. Example
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37. Maxima Methods
 This method considers values with maximum membership. There are different
maxima methods with different conflict resolution strategies for multiple maxima.
 o First of Maxima Method (FOM)
 o Last of Maxima Method (LOM)
 o Mean of Maxima Method (MOM)
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38. First of Maxima Method (FOM)
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39. Last of Maxima Method (LOM)
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40. Mean of Maxima Method (MOM)
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41. Example
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42. 42 Example

43. 43 Example

44. Exercise-1
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45. Exercise-2
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46. Exercise-3
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47. Exercise-4
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48. Exercise-5
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49. Example-1
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54. Example-2
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59. Example
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Example

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