Introduction

Spherical Fuzzy Sets (SFS) are a powerful extension of Picture Fuzzy Sets and Pythagorean Fuzzy Sets. They allow decision-makers to define a broader membership domain by mapping the levels of satisfaction, dissatisfaction, and hesitation on a three-dimensional spherical surface. This provides more flexibility in handling uncertainty and imprecision in complex decision-making environments.

Definition of Spherical Fuzzy Set

A Spherical Fuzzy Set S in a universe U is defined by three functions: membership (μ), non-membership (ν), and hesitancy (π). It is expressed as:

Subject to the spherical constraint:

Spherical Fuzzy Number (SFN)

A Spherical Fuzzy Number is represented as a triple α = (μ, ν, π). Unlike traditional fuzzy sets, the sum of the squares of these three parameters must not exceed 1, allowing the values to occupy a larger space (a sphere) compared to Intuitionistic or Pythagorean sets.

Operations & Examples

Let α₁ = (μ₁, ν₁, π₁) and α₂ = (μ₂, ν₂, π₂) be two SFNs. Basic operations include:

• 1. Addition (⊕):
  α₁ ⊕ α₂ = [√(μ₁² + μ₂² – μ₁²μ₂²), ν₁ν₂, √((1-μ₂²)π₁² + (1-μ₁²)π₂² – π₁²π₂²)]
• 2. Multiplication (⊗):
  α₁ ⊗ α₂ = [μ₁μ₂, √(ν₁² + ν₂² – ν₁²ν₂²), √((1-ν₂²)π₁² + (1-ν₁²)π₂² – π₁²π₂²)]

Introduction

Spherical Fuzzy Sets (SFS) are a powerful extension of Picture Fuzzy Sets and Pythagorean Fuzzy Sets. They allow decision-makers to define a broader membership domain by mapping the levels of satisfaction, dissatisfaction, and hesitation on a three-dimensional spherical surface. This provides more flexibility in handling uncertainty and imprecision in complex decision-making environments.

Definition of Spherical Fuzzy Set

A Spherical Fuzzy Set S in a universe U is defined by three functions: membership (μ), non-membership (ν), and hesitancy (π). It is expressed as:

Subject to the spherical constraint:

Spherical Fuzzy Number (SFN)

A Spherical Fuzzy Number is represented as a triple α = (μ, ν, π). Unlike traditional fuzzy sets, the sum of the squares of these three parameters must not exceed 1, allowing the values to occupy a larger space (a sphere) compared to Intuitionistic or Pythagorean sets.

Operations & Examples

Let α₁ = (μ₁, ν₁, π₁) and α₂ = (μ₂, ν₂, π₂) be two SFNs. Basic operations include:

• 1. Addition (⊕):
  α₁ ⊕ α₂ = [√(μ₁² + μ₂² – μ₁²μ₂²), ν₁ν₂, √((1-μ₂²)π₁² + (1-μ₁²)π₂² – π₁²π₂²)]
• 2. Multiplication (⊗):
  α₁ ⊗ α₂ = [μ₁μ₂, √(ν₁² + ν₂² – ν₁²ν₂²), √((1-ν₂²)π₁² + (1-ν₁²)π₂² – π₁²π₂²)]