A New Structure of Hesitant Fuzzy Relations: An Extension from Fuzzy and Intuitionistic Fuzzy Relations with Applications

Abstract

Accurately assessing academic performance is a persistent challenge due to the uncertain and imprecise nature of evaluative information. While Hesitant Fuzzy Sets (HFS) provide a useful tool for handling uncertainty, existing hesitant fuzzy relation (HFR) structures often fail to preserve original information, leading to incomplete evaluations. The main objective of this research is to develop a new HFR structure that preserves original data more effectively, enabling more accurate, reasonable, and complete assessments. We generalize the Cartesian product and hesitant fuzzy relations by introducing a hesitant fuzzy Cartesian product (HFCP), where each pair $(x,y)$ is associated with hesitant fuzzy elements. This extends classical and intuitionistic fuzzy models using a lattice-based mathematical framework. The analytical approach relies on set theory and lattice structures. An illustrative example using simulated academic data demonstrates the improved representation and accuracy of lecturer–student evaluations. Key findings highlight that the new structure reduces information loss and increases the reliability of performance assessment. This framework supports better-informed decision-making in academic management. We recommend incorporating hesitant fuzzy-based evaluation models in institutional appraisal systems to enhance fairness, clarity, and data completeness in academic performance reviews.