Set-Valued Fuzzy Normed Linear Spaces: A Mathematical Framework for Multi-Alternative Decision-Making Under Uncertainty

This paper introduces Set-Valued Fuzzy Normed Linear Spaces, a novel mathematical framework that generalizes fuzzy norms from single points to sets. This extension addresses a critical gap in the literature by providing a rigorous foundation for modeling complex business scenarios involving uncertainty and multi-alternative decision-making, such as evaluating portfolios of strategies, bundles of resources, and segments of customers. We establish the complete mathematical structure of Set-Valued Fuzzy Normed Linear Spaces, including a rigorous axiomatic definition and key topological properties like convergence and completeness, leading to the concept of a Set-Valued Fuzzy Banach Space. The practical utility of this framework is demonstrated through its direct application to fundamental challenges in management and marketing, including robust customer segmentation, optimal resource allocation, and strategic portfolio optimization under imprecise information. By bridging advanced functional analysis with pressing business needs, this work offers a powerful and unified analytical tool for decision-makers to navigate fuzzy, complex environments effectively, moving beyond traditional point-based evaluations to a more realistic set-based paradigm.