# Center of Sums (CoS) method for defuzzification

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

The crisp value according to 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 CoG method, the overlapping area is counted once, whereas, in CoS , the overlapping is counted twice or so

Area of trapezoidal is computed as shown in this diagram:

Example – 1:

Given 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 CoS method:

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

Example – 2:

Given 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 CoG method.