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The prevailing concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs) and q-rung orthopair fuzzy sets (q-ROFSs) have numerous applications in various fields from real life. Unfortunately, these theories have their own limitations related to the membership and non-membership grades. To eradicate these restrictions, we introduce the novel concept of linear Diophantine fuzzy set (LDFS) with the addition of reference parameters. This idea removes the restrictions of existing methodologies and the decision maker (DM) can freely choose the grades without any limitations. This structure also categorizes the problem by changing the physical sense of reference parameters. We present some fundamental operations on linear Diophantine fuzzy sets (LDFSs). We present geometrical interpretation for different operations of LDFSs. We also introduce the novel concepts of linear Diophantine fuzzy topological space (LDFTS) and linear Diophantine fuzzy weighted geometric aggregation (LDFWGA) operator. We discuss several properties of LDFTS with the help of examples. We introduce score functions and accuracy functions with different orders for the comparison of linear Diophantine fuzzy numbers (LDFNs). We propose two algorithms for solving multi-attribute decision-making (MADM) problem accompanied by an interesting application employing LDFTSs and LDFWGA operator.
Riaz et al. (Fri,) studied this question.