# Center of Largest Area (CoA) method for defuzzification

Center of Largest Area (CoA) method is simple, computationally effective and widely used defuzzification.

If the fuzzy set has two sub regions, then the ** center of gravity of the sub region with the largest area **can be used to calculate the defuzzified value.

x_{∗}= (∫ μ_{Cm}(x) . x′ dx) / ( ∫ μ_{Cm}(x) dx )

Here, C_{m} is the region with largest area, x′ is the center of gravity of C_{m}

**Example: Center of Largest Area (CoA) method**

Find the crisp value corresponding to given fuzzy output functions

**Solution:**

To compute the crisp value for given fuzzy output sets, we shall place them on one axis and should find the area of nonoverlapping regions. The region with highest value is used to compute the crisp value.

The aggregated fuzzy output functions are shown below. First we will compute the area of region A1.

To find the area of region A_{1}, we need to find the line equations for ab and bc lines

**line ab:**

(y – y_{1}) / (x – x_{1}) = (y_{2} – y_{1}) / (x_{2} – x_{1})

for line ab, (x_{1}, y_{1}) = (0, 0), and (x_{2}, y_{2}) = (1.5, 0.5)

⇒ (y – 0) / (x – 0) = (0.5 – 0) / (1.5 – 0)

⇒ y = 0.33x

Line ranges from 0 to 1.5 on X axis.

**line bc:**

for line bc, (x_{1}, y_{1}) = (1.5, 0.5), and (x_{2}, y_{2}) = (x, 0).

We dont know what is the x coordinate of point c. We shall compute the equation of line cd also. The x coordinate of both the line would be same at intersection point

for line bc:

(y – 0.5) / (x – 1.5) = (0 – 0.5) / (3 – 1.5)

⇒ y = 0.33x + 1

for line cd:

**line cd:**

y = 0.67x – 0.67

By comparing equations of line bc and line cd,

0.33x + 1 = 0.67x – 0.67

x = 1.67 ≈ 1.7

so line bc ranges from 1.5 to 1.7 on X axis.

Area enclosed by line ab and bc:

A_{1} = 0.466

To find the area of region A_{2}, we need to find the line equations for cd and de lines

**line cd:**

(y – y_{1}) / (x – x_{1}) = (y_{2} – y_{1}) / (x_{2} – x_{1})

for line cd, (x_{1}, y_{1}) = (1, 0), and (x_{2}, y_{2}) = (2.5, 1)

⇒ (y – 0) / (x – 1) = (1 – 0) / (2.5 – 1)

⇒ y = 0.67x -0.67 [1.7, 2.5]

Line ranges from 1.7 to 2.5 on X axis.

**line de:**

for line de, (x_{1}, y_{1}) = (2.5, 1), and (x_{2}, y_{2}) = (4, 0).

(y – 1) / (x – 2.5) = (0 – 1) / (4 – 2.5)

⇒ y = -0.67x + 2.67

**line ef:**

y = 0.2x – 0.6

By comparing equations of line de and line ef, we can find the x coordinate of point e

-0.67x + 2.67 = 0.2x – 0.6

x = 3.8

so line de ranges from 2.5 to 3.8 on X axis.

Area enclosed by line cd and de:

A_{2} = 1.32

To find the area of region A_{3}, we need to find the line equations for ef and fg lines

**line ef:**

(y – y_{1}) / (x – x_{1}) = (y_{2} – y_{1}) / (x_{2} – x_{1})

for line cd, (x_{1}, y_{1}) = (0, 3), and (x_{2}, y_{2}) = (4, 0.2)

⇒ (y – 0) / (x – 3) = (0.2 – 0) / (4 – 3)

⇒ y = 0.2x – 0.6

Line ranges from 3.8 to 4 on X axis.

**line fg:**

for line de, (x_{1}, y_{1}) = (4, 0.2), and (x_{2}, y_{2}) = (5, 0).

(y – 0.2) / (x – 4) = (0 – 0.2) / (5 – 4)

⇒ y = -0.2x + 1

so line fg ranges from 4 to 5 on X axis.

Area enclosed by line ef and fg:

A_{3} = 0.136

Thus, the are enclosed by line are summarized in following table:

As we can se, region 2 has the largest area. So center of A_{2} will be used to find the crisp value.

## Watch on YouTube: Center of Largest Area method

All the defuzzification methods are summarized below: