Linear Diophantine Neutrosophic Sets and Their Properties

In 2019, Riaz et al. introduced the notion of linear Diophantine fuzzy set(LDFS) where there is an addition of reference parameters that help to address the issues that cannot be managed by the existing theories such as fuzzy sets(FSs), intuitionistic fuzzy sets(IFSs), Pythagorean fuzzy sets(PFSs), and q-rung orthopair fuzzy sets(q-ROFSs). But all these theories are not capable to describe indeterminacy that exists in numerous real-world problems. For this purpose, neutrosophic sets(NSs), single-valued neutrosophic sets(SVNSs), Pythagorean neutrosophic sets(PNSs) are introduced. In PNS, each object x in the universe is characterized by a dependent truth (Formula Presented)and falsity (Formula Presented)membership values and indeterminacy (Formula Presented)membership value with the restriction(Formula Presented)(Formula Presented). If we consider a neutrosophic triplet as 0.9,0.9,0.9 then0.92+0.92+0.92will give 2.43, which is>2. Such a problem cannot be handled by the decision-makers under the Pythagorean neutrosophic environment. To take care of such an issue there is an urgency to develop another mathematical model. This lead to an introduction of linear Diophantine neutrosophic set(LDNS) as an extension of PNS. Thus, the main purpose of this paper is to introduce the LDNS model with an aid of reference parameters to ensure that through this new model the decision-makers can freely choose the neutrosophic membership values with an extended domain. Therefore, in a broad sense, the LDNSs are a new idea that removes the restrictions present in the existing concepts such as FSs, IFSs, PFSs, q-ROFSs, PNSs, LDFSs, etc. From example 3.1.1, it is quite visible that this new structure helps to classify the problem by changing the physical nature of reference parameters. Moreover, some basic properties and operations on LDNSs are investigated. We also define the score and accuracy function based on linear Diophantine neutrosophic number(LDNN). With the help of a novel linear Diophantine single-valued neutrosophic weighted arithmetic-geometric aggregation (LDSVNWAGA) operator, an algorithm has been developed for decision-making. Finally, the proposed algorithm has been successfully executed with the help of a numerical application. © 2023,Neutrosophic Sets and Systems. All Rights Reserved.