intuitionistic fuzzy soft set :
This article covers the definition of intuitionistic fuzzy soft set and common operations of intuitionistic fuzzy soft set. Learn about IFSS membership values, IFSS non-membership values, and study an intuitionistic fuzzy soft set with example for hybrid decision-making.
What is an Intuitionistic Fuzzy Soft Set (IFSS)?
An Intuitionistic Fuzzy Soft Set is a hybrid model. It uses Soft Sets to categorize data by parameters, and Intuitionistic Fuzzy Sets to describe the membership (yes) and non-membership (no) of each item.
• μ (mu): Degree of Membership (0 to 1).
• ν (nu): Degree of Non-membership (0 to 1).
• Constraint: μ + ν ≤ 1.
Example Scenario (The Setup)
Imagine a company interviewing two candidates {c1, c2} based on the parameter “Experience”.
Meaning: 70% experienced, 20% not experienced, 10% uncertain.
Candidate 2 (c2): (0.4, 0.5)
Meaning: 40% experienced, 50% not experienced, 10% uncertain.
Operations of IFSS with Clear Examples
Let’s assume two experts, Expert A and Expert B, give their intuitionistic fuzzy opinions on a candidate for a single parameter.
1. Union (OR Operation)
The Union takes the highest membership and the lowest non-membership.
Expert B: (0.5, 0.4)
Calculation:
Max membership: max(0.6, 0.5) = 0.6
Min non-membership: min(0.3, 0.4) = 0.3
Result: (0.6, 0.3)
2. Intersection (AND Operation)
The Intersection takes the lowest membership and the highest non-membership.
Expert B: (0.5, 0.4)
Calculation:
Min membership: min(0.6, 0.5) = 0.5
Max non-membership: max(0.3, 0.4) = 0.4
Result: (0.5, 0.4)
3. Complement (NOT Operation)
The complement simply swaps the membership and non-membership degrees.
Complement Calculation: Swapping the values…
Result: (0.2, 0.7)
Operations of Intuitionistic Fuzzy Soft Numbers (IFSN)
When IFSS values are treated as numbers for calculation, we use specific arithmetic formulas for Addition and Multiplication.
| Operation | Formula for (μ1, ν1) and (μ2, ν2) |
|---|---|
| Addition (⊕) | ( μ1 + μ2 – μ1μ2 , ν1ν2 ) |
| Multiplication (⊗) | ( μ1μ2 , ν1 + ν2 – ν1ν2 ) |
Arithmetic Step-by-Step Example
Let A = (0.5, 0.3) and B = (0.4, 0.2).
ν Result: 0.3 × 0.2 = 0.06
Sum Result: (0.7, 0.06)
ν Result: 0.3 + 0.2 – (0.3 × 0.2) = 0.5 – 0.06 = 0.44
Product Result: (0.2, 0.44)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator