Picture fuzzy set :
Understand the definition of picture fuzzy set PFS and the various operations of PFS. We explain positive, neutral, and negative membership through a practical PFS with example for human opinion modeling.
Picture Fuzzy Set (PFS)
Introduction
A Picture Fuzzy Set (PFS) is an extension of the traditional Intuitionistic Fuzzy Set. While standard fuzzy sets only consider membership and non-membership, PFS introduces a third dimension: neutral membership (abstain). This makes it ideal for real-world scenarios involving human opinions, such as voting, where people can choose to be “Yes”, “Abstain”, “No”, or “Refusal”.
Definition of Picture Fuzzy Set
A Picture Fuzzy Set A on a universe X is an object of the form:
Where:
- μA(x) ∈ [0, 1] is the Degree of Membership
- ηA(x) ∈ [0, 1] is the Degree of Neutral Membership
- νA(x) ∈ [0, 1] is the Degree of Non-membership
These values must satisfy the following condition:
The degree of refusal is calculated as: πA(x) = 1 – (μA(x) + ηA(x) + νA(x)).
Picture Fuzzy Number (PFN)
For convenience, a single element α = (μ, η, ν) is called a Picture Fuzzy Number (PFN), provided that μ, η, ν ∈ [0, 1] and μ + η + ν ≤ 1.
Basic Operations
Let α₁ = (μ₁, η₁, ν₁) and α₂ = (μ₂, η₂, ν₂) be two PFNs. The operations are defined as:
α₁ ⊕ α₂ = ( μ₁+μ₂ – μ₁μ₂, η₁η₂, ν₁ν₂ )
α₁ ⊗ α₂ = ( μ₁μ₂, η₁+η₂ – η₁η₂, ν₁+ν₂ – ν₁ν₂ )
Let α₁ = (0.5, 0.1, 0.2) and α₂ = (0.3, 0.2, 0.1).
Check Constraint: 0.5 + 0.1 + 0.2 = 0.8 (≤ 1) ✔
Addition:
μres = 0.5 + 0.3 – (0.5 × 0.3) = 0.8 – 0.15 = 0.65
ηres = 0.1 × 0.2 = 0.02
νres = 0.2 × 0.1 = 0.02
Result: α₁ ⊕ α₂ = (0.65, 0.02, 0.02)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator