fermatean fuzzy set :
Master the definition of fermatean fuzzy set and the operations of fermatean fuzzy set. This tutorial covers FFS membership grade, FFS non-membership grade, and provides a fermatean fuzzy set with example for high-order uncertainty modeling.
What is a Fermatean Fuzzy Set (FFS)?
A Fermatean Fuzzy Set is a powerful tool for handling higher levels of uncertainty in decision-making. It expands the search space further than Intuitionistic (IFS) and Pythagorean (PFS) fuzzy sets by using a cubic constraint.
• μ: Degree of Membership
• ν: Degree of Non-membership
The Rule:
Why do we need FFS?
In some complex decisions, an expert might feel an object has a 0.9 membership and a 0.6 non-membership.
- IFS: 0.9 + 0.6 = 1.5 (Invalid)
- PFS: 0.9² + 0.6² = 0.81 + 0.36 = 1.17 (Invalid)
- FFS: 0.9³ + 0.6³ = 0.729 + 0.216 = 0.945 (Valid!)
Operations of Fermatean Fuzzy Sets
Let A = (μ₁, ν₁) and B = (μ₂, ν₂) be two Fermatean Fuzzy values.
1. Union (OR)
Takes the maximum membership and minimum non-membership.
Result: (0.8, 0.4)
2. Intersection (AND)
Takes the minimum membership and maximum non-membership.
Result: (0.6, 0.5)
Fermatean Fuzzy Number (FFN) Arithmetic
Arithmetic operations for FFNs involve cubic roots and powers. These are essential for aggregating data from multiple experts.
| Operation | Formula |
|---|---|
| Addition (⊕) | ( ∛(μ₁³ + μ₂³ – μ₁³μ₂³) , ν₁ν₂ ) |
| Multiplication (⊗) | ( μ₁μ₂ , ∛(ν₁³ + ν₂³ – ν₁³ν₂³) ) |
Arithmetic Step-by-Step Example
Let A = (0.7, 0.5) and B = (0.6, 0.4).
2. Apply formula: ∛(0.343 + 0.216 – (0.343 × 0.216))
3. ∛(0.559 – 0.074) = ∛(0.485) ≈ 0.785
4. Multiply non-memberships: 0.5 × 0.4 = 0.20
Sum Result: (0.785, 0.20)
2. Cube the non-memberships: 0.5³ = 0.125, 0.4³ = 0.064
3. Apply formula: ∛(0.125 + 0.064 – (0.125 × 0.064))
4. ∛(0.189 – 0.008) = ∛(0.181) ≈ 0.565
Product Result: (0.42, 0.565)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator