# Neutrosophic Linear Equations and Application in Traffic Flow Problems
## 1 Introduction  
The condensed traffic flow along with the increasing number of vehicles is increasingly complex and uncertain in actual road traffic situations [1]. Since fuzzy logic and fuzzy algebra were introduced by [PERSON] [2] in 1965, fuzzy theory has been successfully applied in a wide range of subject fields [1; 3; 4]. The main reason for this widespread application is that it can process various vague or uncertain data and provide results that are suitable for decision making. In a fuzzy set, fuzzy numbers are proven to be very suitable for expressing vague, imprecise, uncertain values [3; 4]. Then, fuzzy numbers can be defined by a membership/truth function, which specifies the membership/truth degree for each element \(x\) from the universe \(X\). In many real situations (e.g., traffic flow), the information obtained by decision makers may be partial determinacy and/or partial indeterminacy due to a lack of data, time pressure, measurement errors, or the decision makers' limited attention and knowledge. However, the fuzzy sets/numbers cannot represent data with both determinate and indeterminate information. To express it, [PERSON] [5; 6; 7] first proposed the concept of a neutrosophic number (NN) as a subclass of neutrosophy. NN is composed of the determinate part \(a\) and the indeterminate part \(ul\), denoted by \(z=a+ul\) for \(a\), \(u\in R\) (\(R\) is all real numbers), where \(I\) is indeterminacy. Hence, NN is much more suitable for representing and handling real problems with both determinate and indeterminate information. Recently, NNs have been applied to decision making [8; 9] and fault diagnoses [10; 11]. Then, [PERSON] [12] further proposed the concept of the neutrosophic function, neutrosophic precalculus, and neutrosophic calculus in 2015 as another subclass of neutrosophy to deal with various indeterminate problems in the real world. In general, a neutrosophic function (thick function) \(f\): \(R\)\(\rightarrow\)\(F(R)\) can be defined as the forms of a closed interval function \(f(x)=[f_{1}(x),f_{2}(x)]\), an open interval function \(f(x)=(f_{1}(x),f_{2}(x))\), and semi-open/semi-closed interval functions \(f(x)=[f_{1}(x),f_{2}(x))/f(x)\)\(=(f_{1}(x),f_{2}(x)]\). Then, [PERSON] et al. [13] applied the neutrosophic function to the expression of the jointroughness coefficient (JRC) and the shear strength in rock mechanics. However, the neutrosophic functions introduced in [12; 13] are interval functions (thick function), but they cannot express and handle actual problems containing NN information. Furthermore, NNs were also applied to expressions and analyses of rock joint roughness coefficient in rock mechanics [14; 15; 16]. Further, [PERSON] [17] put forth an NN linear programming method to obtain optimal solutions of NN linear programming problems under NN environments. Unfortunately, there is no study on neutrosophic linear equations and their solution methods in the currently existing literature. It is obvious that existing fuzzy sets/numbers cannot express neutrosophic linear equations with both determinate and indeterminate information (NNs). In this paper, based on NNs we propose for the first time the new concepts of neutrosophic linear equations and the neutrosophic matrix, and the solving methods of the system of neutrosophic linear equations by using the substitution method, the addition method, and the inverse matrix method. To do so, the rest of this paper is organized as follows. Section 2 describes some basic concepts and operational laws of NNs. In Section 3, we propose some new concepts in neutrosophic linear equations and neutrosophic matrices and introduce some neutrosophic matrix operations. Section 4 introduces the solving methods of the system of neutrosophic linear equations or the neutrosophic matrix equation by using the substitution, addition, and inverse matrix methods.