Table 2 Comparison of different fuzzy set extensions.
From: Smart system for forecasting financial outcomes and supporting strategic choices
Fuzzy set type | Membership components | Main concept/strength | Limitations |
|---|---|---|---|
Fuzzy set (FS) | \(\mu (x)\) | Represents the degree of membership of an element in a set. | Cannot express hesitation or explicit non-membership. |
Intuitionistic fuzzy set (IFS) | \(\mu (x), \nu (x)\) | Adds non-membership degree to model hesitation. | Limited flexibility as \(\mu (x) + \nu (x) \le 1\). |
Pythagorean fuzzy set (PyFS) | \(\mu (x), \nu (x)\) | Extends IFS using the quadratic constraint \(\mu ^2(x) + \nu ^2(x) \le 1\). | May still fail to model strong hesitancy or neutrality. |
Fermatean fuzzy set (FFS) | \(\mu (x), \nu (x)\) | Further generalisation using cubic constraint \(\mu ^3(x) + \nu ^3(x) \le 1\). | Complex aggregation; cannot represent neutral opinion. |
Picture fuzzy set (PFS) | \(\mu (x), \eta (x), \nu (x)\) | Introduces a neutral membership degree to express acceptance, neutrality, and rejection simultaneously. | Limited in handling directional or phase-based hesitation. |
CPFS | \(\mu (x)e^{i\theta _1}, \eta (x)e^{i\theta _2}, \nu (x)e^{i\theta _3}\) | Extends PFS by introducing complex-valued membership, allowing magnitude and phase for richer uncertainty modelling. | Higher computational complexity and phase interpretation difficulty. |
