# Neutrosophic Linear Equations and Application in Traffic Flow Problems
## 5 Application of a Traffic Flow Problem  
In some city, the downtown traffic flows of four roads are shown in Figure 1 during the rush hours on a typical weekday. In Figure 1, the arrows indicate the direction of traffic flow on each one-way road, and the average number of vehicles per hour entering and leaving each intersection is shown beside each road. Road 3 and Road 4 can handle up to 2000 vehicles per hour without causing congestion, while the maximum capacity of both Road 1 and Road 2 is 1000 vehicles per hour. The flow of traffic is controlled by the traffic lights, which are installed at each of the four intersections. Then, there exists some indeterminate traffic flow on each one-way road in Figure 1. If the neutrosophic number z = 400 + \(I\) is expressed as the determinate and indeterminate information of Road 1, where the indeterminacy \(I\) will ensure no traffic congestion, then we need to find the ranges of traffic flows for the unknowns \(x_{1}\), \(x_{2}\), \(x_{3}\). To avoid traffic congestion, all traffic entering an intersection must also leave that intersection; i.e., the number of entering vehicles is equal to the number of leaving vehicles in each intersection in Figure 1. According to this condition to each of the four intersections in a clockwise direction beginning from the intersection between Road 1 and Road 3, the four linear equations based on the four intersections in Figure 1 can be obtained as follows:  
\[1500=x_{1}+z\]  
\[1300=x_{1}+x_{2}\]  
\[1800=x_{2}+x_{3}\]  
Figure 1: The downtown traffic flows of four roads. \[2000=x_{3}+z\]  
Thus, the system of the four linear equations in the three variables \((x_{1},x_{2},x_{3})\) can be simplified into the following form:  
\[x_{1}=1500-z\]  
\[x_{1}+x_{2}=1300\]  
\[x_{2}+2x_{3}=3800-z\]  
According to the NN \(z=400+I\), the system can also be represented by the following three neutrosophic linear equations:  
Figure 1.0  
Figure 1.1  
Figure 1.2  
Then, there are the three neutrosophic linear equations:  
Figure 1.3  
Figure 1.4  
Figure 1.5  
Thus, let us write the following neutrosophic matrices:  
\[A=\left[ \begin{array}{ccc}1&0&0\\ 1&1&0\\ 0&1&2\end{array} \right],\;X=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right],\;\text{and}\;B=\left[ \begin{array}{c}1100-I\\ 1300\\ 3400-I\end{array} \right].\]  
Hence, the system of neutrosophic linear equations can be written as the form of the neutrosophic matrix equation:  
\[AX=B. \tag{5}\]  
For the system of neutrosophic linear equations, we can therefore use the inverse matrix method to solve the neutrosophic matrix Equation (5). Firstly, we obtain the inverse matrix of the neutrosophic matrix \(A\) below:  
\[A^{-1}=\left[ \begin{array}{ccc}1&0&0\\ -1&1&0\\ 0.5&-0.5&0.5\end{array} \right].\]  
Then, by multiplying both sides of Equation (5) by \(A^{-1}\) (on the left), we can find the solution of the system:  
\[X=A^{-1}B=\left[ \begin{array}{ccc}1&0&0\\ -1&1&0\\ 0.5&-0.5&0.5\end{array} \right]\left[ \begin{array}{c}1100-I\\ 1300\\ 3400-I\end{array} \right]=\left[ \begin{array}{c}1100-I\\ I+200\\ 1600-I\end{array} \right].\]  
Clearly, the values of \(X\) are NNs. In some real situation and/or requirement, when the indeterminacy \(I\in[0,100]\) is considered as a possible range, the solution of the system is as follows:  
\[\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]=\left[ \begin{array}{c}[1000,1100]\\ [200,300]\\ [1500,1600]\end{array} \right].\]Obviously, corresponding to the possible traffic flow \(z=[400,\,500]\), we can obtain the ranges of the three traffic flows: \(x_{1}=[1000,\,1100]\), \(x_{2}=[200,\,300]\), and \(x_{3}=[1500,\,1600]\). If the indeterminacy \(I\in[200,\,400]\) is considered as a possible range, then the solution of the system is given as follows:  
\[\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]=\left[ \begin{array}{c}[700,900]\\ [400,600]\\ [1200,1400]\end{array}