hesitant fuzzy set :
Read the definition of hesitant fuzzy set and the unique operations of hesitant fuzzy set. This post covers HFS membership elements and provides a hesitant fuzzy set with example for cases where experts provide multiple possible values.
Hesitant Fuzzy Set (HFS)
Introduction
In many decision-making problems, experts often hesitate between several values before assigning a membership degree to an element. Developed by Torra (2010), the Hesitant Fuzzy Set (HFS) handles this by allowing the membership degree to be a set of possible values between 0 and 1, rather than a single number or an interval.
Definition of Hesitant Fuzzy Set
A Hesitant Fuzzy Set E on a fixed set X is defined in terms of a function that returns a subset of [0, 1]:
Where hE(x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E.
Hesitant Fuzzy Element (HFE)
For a specific x, the set h = hE(x) is called a Hesitant Fuzzy Element (HFE). It is typically represented as a collection of values:
Where each γ represents a potential membership degree.
Detailed Mathematical Operations
Let h, h₁, and h₂ be three hesitant fuzzy elements. The following operations are defined:
Subtracts every element in the set from 1.
Combines elements using the probabilistic sum across all possible combinations.
The product of every combination of elements from both sets.
Let h₁ = { 0.2, 0.4 } and h₂ = { 0.5 }.
Addition (h₁ ⊕ h₂):
• Calculation 1: 0.2 + 0.5 – (0.2 × 0.5) = 0.7 – 0.1 = 0.6
• Calculation 2: 0.4 + 0.5 – (0.4 × 0.5) = 0.9 – 0.2 = 0.7
Result: h₁ ⊕ h₂ = { 0.6, 0.7 }
Multiplication (h₁ ⊗ h₂):
• Calculation 1: 0.2 × 0.5 = 0.1
• Calculation 2: 0.4 × 0.5 = 0.2
Result: h₁ ⊗ h₂ = { 0.1, 0.2 }
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator