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Defuzzification
Type Reduction Defuzzification - Karnik-Mendel algorithm for interval type-2 fuzzy sets
Defuzzification operator - type-2 to type-1 reduction via Karnik-Mendel
Karnik, N.N., Mendel, J.M.2001doi:10.1016/S0020-0255(01)00069-X ↗
Overview
Use centroid mode (c_L+c_R)/2 for standard ranking. Use interval mode [c_L,c_R] to preserve and report type-2 uncertainty. Wider [c_L,c_R] = higher second-order uncertainty. N=100 discretisation points is sufficient for most IT2 TFNs.
- Output
- crisp score or interval, higher is better
- Data
- Type-2 Fuzzy, it2 fuzzy numbers with UMF LMF
- Size
- 1+ alternatives
- Used for
- IT2 fuzzy MCDM, type-2 fuzzy AHP, interval type-2 TOPSIS
Fits when / Look elsewhere when
Fits when
- •Handles full FOU uncertainty
- •Produces an uncertainty interval [c_L,c_R]
- •Proven convergence
Look elsewhere when
- •T1 TFN/TrFN inputs - use DEFUZZ-CENTROID
- •IFN/PFN inputs - use DEFUZZ-SCORE-IFN
Assumptions to verify
- LMF(x) ≤ UMF(x) for all x
- N_disc ≥ 20 for reliable convergence
Limitations
- •Requires discretisation (N evaluations per FN)
- •Two separate KM runs needed
- •More complex than T1 defuzzification
Edge cases and pitfalls
Using TFN centroid (a+b+c)/3 on IT2 inputs - ignores FOU entirely.
Too few discretisation points (N<20) - KM may not converge correctly.
Forgetting that c_L and c_R require separate KM runs with swapped MF assignments.
Works with
How to cite
Karnik, N.N.; Mendel, J.M. (2001). Centroid of a type-2 fuzzy set. Information Sciences. https://doi.org/10.1016/S0020-0255(01)00069-X
System ID, as it appears in reports and the API
DEFUZZ-TYPE-REDUCTION