Explore the definition of dual hesitant fuzzy set and the operations of dual hesitant fuzzy set. This guide covers DHFS membership and non-membership sets with a clear dual hesitant fuzzy set with example.
Dual Hesitant Fuzzy Set (DHFS)
Introduction
The Dual Hesitant Fuzzy Set (DHFS) was introduced by Zhu et al. (2012) to overcome the limitations of standard hesitant sets. While a hesitant fuzzy set only considers a set of membership degrees, the DHFS allows for a set of both membership and non-membership degrees.
Definition of DHFS
A Dual Hesitant Fuzzy Set \(D\) on a fixed set \(X\) is represented as:
\( D = \{ \langle x, h_D(x), g_D(x) \rangle \mid x \in X \} \)
Where:
- \(h_D(x)\) is a set of values in \([0, 1]\) (Membership hesitancy).
- \(g_D(x)\) is a set of values in \([0, 1]\) (Non-membership hesitancy).
Condition: For all \(\gamma \in h\) and \(\eta \in g\), the following must hold:
\( 0 \le \gamma, \eta \le 1 \quad \text{and} \quad \sup(h) + \sup(g) \le 1 \)
Dual Hesitant Fuzzy Element (DHFE)
An element \(d = (h, g)\) is called a Dual Hesitant Fuzzy Element (DHFE).
Detailed Mathematical Operations
Let \(d_1 = (h_1, g_1)\), \(d_2 = (h_2, g_2)\) be two DHFEs and \(\lambda > 0\). The operational laws are:
1. Addition (\(\oplus\))
\( d_1 \oplus d_2 = \bigcup_{\gamma_1 \in h_1, \gamma_2 \in h_2, \eta_1 \in g_1, \eta_2 \in g_2} \{ (\gamma_1 + \gamma_2 – \gamma_1\gamma_2, \eta_1\eta_2) \} \)
2. Multiplication (\(\otimes\))
\( d_1 \otimes d_2 = \bigcup_{\gamma_1 \in h_1, \gamma_2 \in h_2, \eta_1 \in g_1, \eta_2 \in g_2} \{ (\gamma_1\gamma_2, \eta_1 + \eta_2 – \eta_1\eta_2) \} \)
3. Scalar Multiplication (\(\lambda d\))
\( \lambda d = \bigcup_{\gamma \in h, \eta \in g} \{ (1 – (1 – \gamma)^\lambda, \eta^\lambda) \} \)
4. Power (\(d^\lambda\))
\( d^\lambda = \bigcup_{\gamma \in h, \eta \in g} \{ (\gamma^\lambda, 1 – (1 – \eta)^\lambda) \} \)
5. Score Function (\(s\))
\( s(d) = \frac{1}{|h|} \sum_{\gamma \in h} \gamma – \frac{1}{|g|} \sum_{\eta \in g} \eta \)
Used to rank DHFEs. Higher scores indicate superior elements.
Numerical Example:
Let \(d_1 = (\{0.4, 0.5\}, \{0.2\})\) and \(d_2 = (\{0.3\}, \{0.4\})\).
Addition (\(d_1 \oplus d_2\)):
• Combination 1: \((0.4 + 0.3 – 0.4 \times 0.3, 0.2 \times 0.4) = (0.58, 0.08)\)
• Combination 2: \((0.5 + 0.3 – 0.5 \times 0.3, 0.2 \times 0.4) = (0.65, 0.08)\)
Result: \(d_1 \oplus d_2 = (\{0.58, 0.65\}, \{0.08\})\)
Let \(d_1 = (\{0.4, 0.5\}, \{0.2\})\) and \(d_2 = (\{0.3\}, \{0.4\})\).
Addition (\(d_1 \oplus d_2\)):
• Combination 1: \((0.4 + 0.3 – 0.4 \times 0.3, 0.2 \times 0.4) = (0.58, 0.08)\)
• Combination 2: \((0.5 + 0.3 – 0.5 \times 0.3, 0.2 \times 0.4) = (0.65, 0.08)\)
Result: \(d_1 \oplus d_2 = (\{0.58, 0.65\}, \{0.08\})\)
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