Linear Diophantine Neutrosophic Sets and Their

Abstract

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 ï­T  x and falsity ï€¨ï§ F  x membership values and
indeterminacy ï€¨ï® I  x membership value with the
restriction         
2 2 2 0 2 T F I ï‚£   ï‚£ ï­ ï§ ï® x x x . If we consider a neutrosophic triplet
as 0.9,0.9,0.9 then 222 0.9 0.9 0.9   will 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.