Modeling Directional Uncertainty for Sustainability: IndetermSoft and IndetermHyperSoft Multi-Directed Sets

Abstract

Uncertainty-Management frameworks—Fuzzy Sets, Intuitionistic Fuzzy Sets, Hyperfuzzy Sets, Neutrosophic Sets, Soft Sets, Rough Sets, and Plithogenic Sets—are widely used to model imprecision. For managerial decision-making under ambiguity and incomplete data, Soft Sets are especially practical: they associate each decision parameter with a subset of the universe, yielding a transparent basis for screening options and tracing rationales. To address rising problem complexity, extensions such as Hypersoft Sets, SuperHypersoft Sets, IndetermSoft Sets, IndetermHyperSoft Sets, and TreeSoft Sets add expressiveness and control over parameter interactions. However, despite progress on oriented structures like Multi-Directed Sets, there is no framework that jointly encodes indeterminacy and directional relations—nor tested applications in management areas such as sustainability portfolios aligned with the SDGs, supply-chain resilience, or project prioritization. To close this gap, we introduce two constructs: the IndetermSoft Multi-Directed Set and the IndetermHyperSoft Multi-Directed Set, which combine the indeterminacy handling of IndetermSoft models with the relational directionality of Multi-Directed Sets. We further define the SuperHyperSoft Multi-Directed Set as a natural extension of SuperHypersoft Sets. We then illustrate how these models support multi-criteria evaluation, policy design, resource allocation, and risk propagation with direction-labeled links (e.g., upstream→downstream effects, feedback loops). These contributions provide actionable tools for governance dashboards, scenario planning, and performance management, and we expect them to stimulate further set-theoretic advances for managing uncertain systems.