On Generalized Picture Fuzzy Numbers with Gaussian Membership Functions

Abstract

The Gaussian membership function (GMF) is widely used in fuzzy logic for its smoothness, symmetry, continuity, differentiability, intuitive interpretation, and minimal parameter requirements. Classical fuzzy set theory becomes insufficient in addressing decision-making scenarios where uncertainty stems from a lack of confidence in expert assessments, or when experts exhibit hesitation, neutrality, or refrain from providing evaluations. This study aims to present novel generalized types of picture fuzzy numbers that are characterized by GMFs. The new class includes both fuzzy numbers (FNs) and intuitionistic fuzzy numbers with GMFs, whether normal or subnormal. We introduce symmetric triangular approximations for Gaussian FNs, designed to preserve their core, value, and ambiguity characteristics, and achieving a structural similarity of at least 86% with the original functions. These approximations are then employed to construct a consistent fuzzy arithmetic framework. Unlike most of the literature, which conservatively employs the minimum or a t-norm operator in all arithmetic operations to compute the overall height, our approach adopts a compensatory mechanism to better reflect the reliability of the information. We propose a preference index for generalized FNs with GMFs, incorporating value and standard deviation as proxies for central tendency and variability. The proposed index produces rankings that are consistent with established behavioral indices, such as the total integral value and those based on value and ambiguity. To extend the applicability of this approach to certain non-standard FNs, we develop a degeneration process grounded in ordered weighted arithmetic operators and behavioral risk parameters. A formal linguistic scale is constructed by mapping the identified risk parameters onto risk behaviors. The method achieves linear computational complexity, making it suitable for large-scale problems. The efficacy of the proposed technique is tested through its application to multi-attribute decision-making problems. The resulting rankings are consistent with those reported in the literature. Furthermore, sensitivity analyses with respect to variations in the risk parameters are conducted to quantitatively assess the robustness of the proposed approach.

Keywords

Gaussian membership function / generalized fuzzy number / intuitionistic fuzzy number / picture fuzzy number / multiple attribute decision-making / triangular approximation

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