(a) Set up a system of equations that represents traffic flow for the network shown in Fig. 1.11. (The numbers give the average flows into and out of the network at peak traffic hours.)

(b) Solve the system of equations. What is the traffic flow if $x_{6}=300$ and $x_{7}=1300$ vehicles per hour?

Question

(a) Set up a system of equations that represents traffic flow for the network shown in Fig. 1.11. (The numbers give the average flows into and out of the network at peak traffic hours.)

(b) Solve the system of equations. What is the traffic flow if [latex]x_{6}=300[/latex] and [latex]x_{7}=1300[/latex] vehicles per hour?

Accepted Answer

(a) Since the flow into a node is equal to the flow out, we obtain the following system of equations:
[latex]800=x_{1}+x_{5} \quad( ext { Node } A)[/latex]
[latex]x_{1}+x_{4}=400+x_{2} \quad( ext { Node } B)[/latex]
[latex]x_{2}=600+x_{3} \quad( ext { Node } C)[/latex]
[latex]1600+x_{3}=400+x_{7} \quad( ext { Node } D)[/latex]
[latex]x_{7}=x_{4}+x_{6} \quad( ext { Node } E)[/latex]
[latex]x_{5}+x_{6}=1000. \quad( ext { Node } F)[/latex]
(b) The augmented matrix for the system above is
[latex]\left[\begin{array}{rrrrrrrr}1 & 0 & 0 & 0 & 1 & 0 & 0 & 800 \1 & -1 & 0 & 1 & 0 & 0 & 0 & 400 \0 & 1 & -1 & 0 & 0 & 0 & 0 & 600 \0 & 0 & 1 & 0 & 0 & 0 & -1 & -1200 \0 & 0 & 0 & 1 & 0 & 1 & -1 & 0 \0 & 0 & 0 & 0 & 1 & 1 & 0 & 1000\end{array} ight] .[/latex]
Some calculations show that this matrix is row equivalent to
[latex]\left[\begin{array}{rrrrrrrr} 1 & 0 & 0 & 0 & 0 & -1 & 0 & -200 \0 & 1 & 0 & 0 & 0 & 0 & -1 & -600 \0 & 0 & 1 & 0 & 0 & 0 & -1 & -1200 \0 & 0 & 0 & 1 & 0 & 1 & -1 & 0 \0 & 0 & 0 & 0 & 1 & 1 & 0 & 1000 \0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array} ight].[/latex]
Therefore, the solution is
[latex]x_{1}=x_{6}-200[/latex]
[latex]x_{2}=x_{7}-600[/latex]
[latex]x_{3}=x_{7}-1200[/latex]
[latex]x_{4}=x_{7}-x_{6}[/latex]
[latex]x_{5}=1000-x_{6}.[/latex]
If [latex]x_{6}=300 ext { and } x_{7}=1300,[/latex] then (in vehicles per hour)
[latex]x_{1}=100, \quad x_{2}=700, \quad x_{3}=100, \quad x_{4}=1000, \quad x_{5}=700[/latex].

Question 1.4.1: (a) Set up a system of equations that represents traffic flo...

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