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,

Equation to compute crisp value using CoS method

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

center of sums

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:

Computing area of trapezoidal
Computing area of trapezoidal

Example – 1:

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

Output fuzzy set - 1
Output fuzzy set – 1

x1 = (1 + 5) / 2 = 3

A1 = 1/2 (2 + 4) × 0.2

A1 = 0.6

 Output fuzzy set - 2
Output fuzzy set – 2

x2 = (3 + 7) / 2 = 5

A2 = 1/2 (2 + 4) × 1.0

A2 = 3

 Output fuzzy set - 3
Output fuzzy set – 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

Center of sums

Example – 2:

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

cos - output 1

A1 = 1/2 (3 + 5) × 0.3 = 1.2

x1 = (0 + 5) / 2 = 2.5

cos - output 2

A2 = 1/2 (2 + 4) × 0.5 = 1.5

x2 = (3 + 7) / 2 = 5

cos - output 3

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.

Please post your answer / query / feedback in comment section below !