Α-CUT NEUTROSOPHIC Z-NUMBERS WITH SHAPLEY NORMALIZED WEIGHTED BONFERRONI MEAN AND ITS ALGEBRAIC PROPERTIES

Abstract

Previous research has demonstrated that the Bonferroni Mean (BM) and its extensions possess valuable properties for capturing interrelationships between criteria in fuzzy multi-criteria decision-making (MCDM). However, most existing BM-based models are restricted to handling pairwise interactions, limiting their ability to represent complex dependencies among multiple criteria. To overcome the limitation mentioned, this research introduces a new Shapley-based Normalized Weighted Bonferroni Mean aggregation operator within the framework of ????-cut Neutrosophic Z-numbers (α-NZNs). The proposed method integrates the α-cut technique to transform continuous neutrosophic Z-number information into interval representations at a given confidence level, thereby simplifying computations while retaining uncertainty characteristics. The Shapley measure is then employed to capture the overall interaction among all criteria, not only in pairs, by considering their cooperative importance. The developed α-cut NZN Shapley Normalized Weighted Bonferroni Mean (α-NZN-SNWBM) exhibits key mathematical properties including reducibility, monotonicity, commutativity, boundedness, and idempotency, reflecting its robustness in modelling complex relationships. Furthermore, a structured decision-making procedure is proposed, incorporating the α-NZN-SNWBM as the core aggregation step. A detailed numerical case study is presented to demonstrate the applicability of the approach, and sensitivity analysis confirms the stability and consistency of the ranking results across varying parameter values. The findings indicate that the proposed approach effectively captures higher-order interdependencies among criteria in uncertain and imprecise decision-making environments, offering a more comprehensive tool for real-world MCDM problems.