Hyperfuzzy and SuperHyperfuzzy Extensions of Linear Programming: Modelsand Mathematical Foundations - Fujita, Takaaki (Optimality, 2025)
A fuzzy set assigns to each element of a universe a membership degree within the interval [0, 1], thereby modeling imprecision and vagueness. A hyperfuzzy set extends this concept by associating each element with a nonempty subset of [0, 1], capturing both uncertainty and variability through a range of possible membership degrees. Building on this, a superhyperfuzzy set generalizes the framework further by assigning to each nonempty element in the nth power-set hierarchy a nonempty subset of [0, 1], thus enabling the representation of recursively structured and hierarchical uncertainty. Linear programming is an optimization technique that aims to maximize or minimize a linear objective function subject to a set of linear equality and inequality constraints. Fuzzy linear programming generalizes this framework by incorporating fuzzy numbers into the objective coefficients and constraints, allowing for uncertainty in both parameters and feasible regions. In this paper, we propose mathematical models for Hyperfuzzy Linear Programming and Superhyperfuzzy Linear Programming, and briefly examine their theoretical properties. We hope that these models will provide a foundation for further validation, development, and refinement in future research