Eigen Neutrosophic Z- Set and Neutrosophic Z- Relation

This paper introduces an innovative framework for computing the Greatest Eigen Neutrosophic Z-set and the Least Eigen Neutrosophic Z-set using the composition operators, namely max-min-min and min-max-max. The proposed Eigen Neutrosophic Z-set, along with the Neutrosophic Z-relation, remains constant across different computational perspectives. This study addresses the limitation of existing neutrosophic and fuzzy models that fail to effectively capture eigen-based relationships under uncertainty by introducing the Eigen Neutrosophic Z-set framework for more consistent and interpretable decision analysis. Furthermore, Neutrosophic Z-matrices are developed, and their properties are examined in relation to Neutrosophic Z-relations. In this paper several similarity relations among Neutrosophic Z-matrices are presented, along with discussions on their permutations and the invertibility characteristics. Two distinct algorithms are formulated to establish the Greatest Eigen Neutrosophic Z-set and the Least Eigen Neutrosophic Z-set, accompanied by a numerical example. Additionally, a practical application is provided to demonstrate the enhancement of score value while addressing both effectiveness and uncertainty for future advancements of hotel management decision-making systems.