Abstract
Fuzzy sets play a central role in decision-making theory, modeling uncertainty by means of membership grade (MG) and non-membership grade (NMG). Non-linear Diophantine fuzzy sets (N-LDFSs), which are the natural extensions of linear Diophantine fuzzy sets and q-linear Diophantine fuzzy sets, are quite successful in modeling data thanks to their larger domains. However, in an N-LDFS, the MG, NMG and reference parameters (RPs) of an element to a set are given just by a pair of certain numbers from the closed interval [0, 1] that causes a strict modelling. Various types of interval-valued fuzzy sets, multi-fuzzy sets, or circular fuzzy sets change this strict modeling with a sensitive one. In this paper, we propose a theoretical framework for Continuous Non-Linear Diophantine Fuzzy Sets, which introduces continuous functions (CFs) over closed intervals to reduce uncertainty, which represent the MG and NMG functions supported by RPs, which enhance the sensitivity and applicability of decision-making tools. We develop continuous non-linear Diophantine fuzzy algebraic aggregation operators and apply them to a multi-attribute decision-making problem in renewable energy source selection. A case study demonstrates the effectiveness of the proposed CN-LDFS framework using a weighted geometric operator. Comparative analysis with existing methods highlights the superiority and success of our approach in improving decision-making accuracy and reliability.