Linear Diophantine Type-2 Fuzzy Sets with Applications

This article proposes the conventional definition and real life based applications of linear Diophantine type-2 fuzzy sets (LDT2FS), manifesting its supremacy over common fuzzy models. LDT2FS is a new mathematical framework that can address the drawbacks of existing fuzzy sets. As common fuzzy systems, however, these models hold restrictions on acceptance and rejection grades, limiting the flexibility of decision-makers in managing uncertainty. LDT2FS extends the previous study by relaxing these limitations, as decision-makers can specify grades-freely with reference parameters in practice in cases where decision making under uncertainty is necessary when functional relationships are unknown, or when data contain high levels of imprecision. To this end, the less demanding structure of LDT2FS allows for establishing the fit to allow dealing with these uncertainties in an efficient way. It also describes basic arithmetic operations on LDT2FS such as union, intersection, complement, containment, etc. along with their algebraic characteristics. Furthermore, two new operators, the Certainty Operator, and Feasibility Operator, are proposed to convert an LDT2FS into a conventional fuzzy set, simplifying mathematical processing and applications. This article proposes Haming Distance and Euclidean Distance commonly used in pattern recognition, clustering, and classification problems to quantify the differences or similarity between LDT2FS instances.