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The interval-valued Pythagorean fuzzy set (IVPFS) presents a novel approach to tackling vagueness and uncertainty, while neutrosophic sets, a broader concept encompassing of fuzzy sets and intuitionistic fuzzy sets, are tailored to depict real-world data characterized by uncertainty, imprecision, inconsistency, and incompleteness. Additionally, the development of Interval Value Neutrosophic Sets (IVNS) enhances precision in handling problems involving a range of numbers within the real unit interval, rather than focusing solely on a single value. However, despite these advancements, there is a deficiency in research addressing practical implementation challenges, conducting comparative analyses with existing methods, and applying these concepts across various fields. This study aims to bridge this research gap by proposing a novel concept based on the Interval Valued Pythagorean Neutrosophic Set (IVPNS), which is a generalization of the IVPFS and INS. The development of IVPNS provides a more comprehensive framework for handling uncertainty, ambiguity, and incomplete information in various fields, leading to more robust decision-making processes, improved problem-solving capabilities, and better management of complex systems. Furthermore, this research introduces the algebraic operations for IVPNS, including addition, multiplication, scalar multiplication, and exponentiation and provides a comparative analysis with IVPFS and IVNS. The study incorporates illustrative numerical examples to demonstrate these operations in practice. Additionally, this study provides and rigorously proves the algebraic properties of IVPNS, specifically discussing their commutative and associative properties. This validation ensures compliance with established conditions for IVPNS, reinforcing their theoretical soundness and practical applicability.
Razak et al. (Tue,) studied this question.