Definition of q-rung orthopair fuzzy set :
Understand the definition of q-rung orthopair fuzzy set and the operations of q-rung orthopair fuzzy set. We discuss the q-rung parameter and provide a q-rung orthopair fuzzy set with example for flexible data representation.
q-Rung Orthopair Fuzzy Set (q-ROFS)
Introduction
The q-Rung Orthopair Fuzzy Set (q-ROFS), introduced by Yager (2017), is a powerful tool for handling high levels of uncertainty. As the value of the rung q increases, the space of acceptable membership grades expands. This means that if a piece of information cannot be expressed as an Intuitionistic or Pythagorean fuzzy set, it can almost always be expressed as a q-ROFS by simply increasing q.
Definition of q-ROFS
A q-Rung Orthopair Fuzzy Set A in a universe X is defined as:
Subject to the fundamental constraint:
Where q ≥ 1. The hesitancy degree is defined as:
Algebraic Operational Laws
Let α₁ = (μ₁, ν₁) and α₂ = (μ₂, ν₂) be two q-ROFNs. For any scalar λ > 0, the operations are:
Let α₁ = (0.7, 0.4) and α₂ = (0.5, 0.6).
Check Constraint: 0.7³ + 0.4³ = 0.343 + 0.064 = 0.407 (≤ 1) ✔
Addition (α₁ ⊕ α₂):
• μres = (0.7³ + 0.5³ – 0.7³ × 0.5³)1/3
• μres = (0.343 + 0.125 – 0.0428)1/3 = (0.4252)1/3 ≈ 0.752
• νres = 0.4 × 0.6 = 0.24
Result: (0.752, 0.24)
Multiplication (α₁ ⊗ α₂):
• μres = 0.7 × 0.5 = 0.35
• νres = (0.4³ + 0.6³ – 0.4³ × 0.6³)1/3
• νres = (0.064 + 0.216 – 0.0138)1/3 = (0.2662)1/3 ≈ 0.643
Result: (0.35, 0.643)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator