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T-norm
Frank T-norm - Frank family of associative t-norms and t-conorms
Parametric t-norm family - Frank (generator: log-based)
Frank, M.J.1979doi:10.1007/BF02189866 ↗
Overview
Choose s based on desired compensation: s≈1 (product) → moderate; s→∞ (Łukasiewicz) → bounded; s→0 (min) → non-compensatory. In IFN MCDM: report results for s=1,2,5 to assess sensitivity.
- Output
- membership value, higher is better
- Data
- Intuitionistic Fuzzy, unit interval values
- Size
- 2+ alternatives
- Used for
- IFN MCDM, PFN MCDM, q-ROFS
Fits when / Look elsewhere when
Fits when
- •Continuously parametric from min to Łukasiewicz
- •Includes product and min as special cases
- •Algebraically elegant (generator: log-based)
Look elsewhere when
- •Non-fuzzy crisp MCDM - unnecessary complexity
Assumptions to verify
- s > 0, s ≠ 1 (or use limiting formulas)
- IFN constraint μ+ν ≤ 1 preserved
Limitations
- •s=1 requires special handling (L'Hôpital)
- •Less computationally transparent than Hamacher
Edge cases and pitfalls
Using s=1 in the formula directly - causes division by zero; apply L'Hôpital limit T=xy.
Using s=0 directly - apply min(x,y) limit.
Forgetting that T_s and S_s are not commutative with Hamacher (different parametric family).
Works with
How to cite
Frank, M.J. (1979). On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Mathematicae. https://doi.org/10.1007/BF02189866
System ID, as it appears in reports and the API
TNORM-FRANK