Type-2 fuzzy set :
Explore the definition of type-2 fuzzy set and complex operations of type-2 fuzzy set. We discuss the footprint of uncertainty (FOU) and provide a type-2 fuzzy set with example to explain three-dimensional membership grades.
What is a Type-2 Fuzzy Set (T2FS)?
A Type-2 Fuzzy Set is a “Fuzzy Fuzzy Set.” In a Type-1 set, the membership of an element is a single value between 0 and 1. In Type-2, the membership is itself a fuzzy set. This allows us to model situations where we aren’t even sure about the membership grade itself.
The primary memberships of a Type-2 fuzzy set create a shaded 3D area called the FOU. It is bounded by two Type-1 membership functions:
- Upper Membership Function (UMF): The maximum possible membership.
- Lower Membership Function (LMF): The minimum possible membership.
Example Scenario: “Comfortable Temperature”
If you ask 10 people if 25°C is “Comfortable”:
- Some say the membership is 0.8.
- Others say it is 0.6.
A Type-2 set captures this variation by saying the membership is the interval [0.6, 0.8].
Operations of Interval Type-2 Fuzzy Sets (IT2FS)
The most common form of T2FS is the Interval Type-2 Fuzzy Set. Operations are performed by applying Type-1 logic to the Upper and Lower boundaries separately.
1. Union (OR)
The union of two IT2FS is found by taking the maximum of the UMFs and the maximum of the LMFs.
Set B = [0.7, 0.5] (UMF=0.7, LMF=0.5)
Calculation:
New UMF: max(0.8, 0.7) = 0.8
New LMF: max(0.4, 0.5) = 0.5
Result: [0.8, 0.5]
2. Intersection (AND)
The intersection takes the minimum of the UMFs and the minimum of the LMFs.
Set B = [0.7, 0.5]
Calculation:
New UMF: min(0.8, 0.7) = 0.7
New LMF: min(0.4, 0.5) = 0.4
Result: [0.7, 0.4]
Type-2 Fuzzy Numbers (T2FN)
A Triangular Interval Type-2 Fuzzy Number is represented by two triangles (Upper and Lower). It is usually written as:
| Operation | Logic |
|---|---|
| Addition | Add corresponding coordinates of both UMF and LMF. |
| Subtraction | Subtract Upper of B from Upper of A, and Lower of B from Lower of A. |
Arithmetic Step-by-Step Example (Addition)
Let A = [ (2, 4, 6), (3, 4, 5) ] and B = [ (1, 2, 3), (1.5, 2, 2.5) ]
2. Add Lower Triangles: (3+1.5, 4+2, 5+2.5) = (4.5, 6, 7.5)
Resulting T2FN: [ (3, 6, 9), (4.5, 6, 7.5) ]
Complement of T2FS
The complement is calculated by subtracting the membership boundaries from 1 and swapping their positions to maintain logical consistency.
Calculation:
New UMF: 1 – 0.2 = 0.8
New LMF: 1 – 0.9 = 0.1
Resulting Complement: [0.8, 0.1]
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator