# Center of Sums (CoS) method for defuzzification

Center of Sums (CoS) method is the most commonly used defuzzification method. In this method, the area of overlapping region is considered multiple times, whereas the Center of Gravity (CoG) method counts it once. The fundamental of computing crisp value is otherwise identical to the CoG method.

The crisp value according to the Center of Sum (CoS) is defined as,

Here, Ai denotes the area of the region bounded by the fuzzy set Ci and xi is the geometric center of that area.

Note: In the CoG method, the overlapping area is counted once, whereas, in CoS, the overlapping is counted twice or so

The area of trapezoidal is computed as shown in this diagram:

Example – 1:

Given the following three fuzzy output sets, find the crisp value corresponding to that.

x1 = (1 + 5) / 2 = 3

A1 = 1/2 (2 + 4) × 0.2

A1 = 0.6

x2 = (3 + 7) / 2 = 5

A2 = 1/2 (2 + 4) × 1.0

A2 = 3

x3 = (2 + 6) / 2 = 4

A3 = 1/2 (2 + 4) × 0.4

A3 = 1.2

We can find the corresponding crisp value using the formula of the CoS method:

## Watch on YouTube: Center of Sums (CoS) method

Example – 2:

Given the following three fuzzy output sets, find the crisp value corresponding to that.

A1 = 1/2 (3 + 5) × 0.3 = 1.2

x1 = (0 + 5) / 2 = 2.5

A2 = 1/2 (2 + 4) × 0.5 = 1.5

x2 = (3 + 7) / 2 = 5

A3 = 1/2 (1 + 3) × 1.0 = 2

x3 = (5 + 8) / 2 = 6.5

## Test Your Knowledge:

Find the solution of Example 2 using the CoG method.