The network in Figure 2 shows the traffic flow (in vehicles per hour) over several one-way streets in downtown Baltimore during a typical early afternoon. Determine the general flow pattern for the network.

Question

Accepted Answer

Write equations that describe the flow, and then find the general solution of the system. Label the street intersections (junctions) and the unknown flows in the branches, as shown in Figure 2. At each intersection, set the flow in equal to the flow out.

Intersection	Flow in	Flow out
A	300 + 500 =	[latex] x_{1} + x_{2} [/latex]
B	[latex] x_{2} + x_{4} [/latex]=	[latex] 300 + x_{3} [/latex]
C	100 + 400 =	[latex] x_{4} + x_{5} [/latex]
D	[latex] x_{1} + x_{5} [/latex]=	600

Also, the total flow into the network (500 + 300 + 100 + 400) equals the total flow out of the network (300 + [latex] x_{3} [/latex] + 600), which simplifies to [latex] x_{3} [/latex] = 400. Combine this equation with a rearrangement of the first four equations to obtain the following system of equations:

[latex] \begin{matrix} x_{1} & + x_{2} & & & &= 800 \ & x_{2} & - x_{3} & + x_{4} & & = 300 \ & & &x_{4} &+ x_{5} &= 500 \ x_{1} & & & &+ x_{5} & = 600 \& & x_{3} & & & = 400 \end{matrix}[/latex]

Row reduction of the associated augmented matrix leads to

[latex]\begin{matrix} x_{1} & & & & + x_{5} & = 600 \ & x_{2} & & & - x_{5} & = 200 \ & & x_{3} & & & = 400 \ & & & x_{4} & + x_{5} & = 500 \end{matrix} [/latex]

The general flow pattern for the network is described by

[latex]\begin{cases} x_{1} = 600 - x_{5} \ x_{2} = 200 + x_{5} \ x_{3} = 400 \ x_{4} = 500 - x_{5}\ x_{5} is free \end{cases} [/latex]

A negative flow in a network branch corresponds to flow in the direction opposite to that shown on the model. Since the streets in this problem are one-way, none of the variables here can be negative. This fact leads to certain limitations on the possible values of the variables. For instance, [latex] x_{5} [/latex] ≤ 500 because [latex] x_{4} [/latex] cannot be negative. Other constraints on the variables are considered in Practice Problem 2.

Question 1.6.2: The network in Figure 2 shows the traffic flow (in vehicles ...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions