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T-norm
Hamacher T-norm - Rational parametric t-norm for fuzzy aggregation
Parametric t-norm family - Hamacher (rational generator)
Hamacher, H.1978
Overview
γ=1 (algebraic product) is the standard default for IFN arithmetic. γ=2 (Einstein) gives more cautious (lower) t-norm values. Run for γ=0.5,1,2 to assess sensitivity.
- Output
- membership value, higher is better
- Data
- Intuitionistic Fuzzy, unit interval values
- Size
- 2+ alternatives
- Used for
- IFN MCDM, PFN MCDM, HF MCDM
Fits when / Look elsewhere when
Fits when
- •Most widely used parametric t-norm in IFN MCDM literature
- •Algebraically simple closed-form
- •Einstein as special case
Look elsewhere when
- •Crisp MCDM
Assumptions to verify
- γ ≥ 0
- IFN constraint μ+ν ≤ 1 preserved
Limitations
- •Requires choosing γ (often set to 1 by default)
Edge cases and pitfalls
Confusing γ=2 (Einstein product) with a separate method - Einstein is a special case of Hamacher.
Forgetting that IFN validity μ+ν ≤ 1 must be checked after aggregation.
Applying Hamacher to non-IFN values without verifying the [0,1] constraint.
Works with
How to cite
Hamacher, H. (1978). Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen. Progress in Cybernetics and Systems Research.
System ID, as it appears in reports and the API
TNORM-HAMACHER