Traffic Flow, Network Analysis

**Alterah** · September 7th 2009, 11:08 AM

Ok, I am having trouble completing the following problem:

The flow of traffic (in vehicles per hour) through a network of streets is shown in Figure 1.16 (my attachment).

(a) Solve this system for $x_i, i = 1, 2, ..., 5.$
(b) Find the traffic flow when $x_2 = 200 and x_3 = 50.$
(c) Find the traffic flow when $x_2 = 150 and x_3 = 0.$

Ok, I am having trouble with part a. I feel once I get part a, parts b and c should be a fairly straightforward, plug in the values and go from there. Using my figure I get the following Input = Output equations for the labeled junctions in my figure:

$A:x_1 + x_2 = 300$
$B:x_1 + x_3 - x_4 = 150$
$C:x_2 - x_3 - x_5 = -200$
$D:x_4 + x_5 = 350$

Anyhow, when I put it into matrix form and proceed to RREF I get:
$<br /> \left(\begin{array}{cccccc}1&0&1&0&1&500\\0&1&-1&0&-1&-200\\0&0&0&1&1&350\\0&0&0&0&0&0\end{array}\right)<br />$

Because of the zero row I know $x_4$ and $x_5$ are free variables, so I set $x_4 = s$ and $x_5 = t$ . It's at this point I get stuck. I have tried to solve it, but I wind up getting an expression for $x_2$ in both $x_1$ and $x_3$ . Do I need to let another variable be a free variable? Thanks for any and all help.

**Alterah** · September 8th 2009, 03:19 PM

Did anyone have any insight? Thanks.

**aidan** · September 8th 2009, 10:07 PM

Originally Posted by Alterah

Ok, I am having trouble completing the following problem:

The flow of traffic (in vehicles per hour) through a network of streets is shown in Figure 1.16 (my attachment).

(a) Solve this system for $x_i, i = 1, 2, ..., 5.$
(b) Find the traffic flow when $x_2 = 200 and x_3 = 50.$
(c) Find the traffic flow when $x_2 = 150 and x_3 = 0.$

Ok, I am having trouble with part a. I feel once I get part a, parts b and c should be a fairly straightforward, plug in the values and go from there. Using my figure I get the following Input = Output equations for the labeled junctions in my figure:

$A:x_1 + x_2 = 300$
$B:x_1 + x_3 - x_4 = 150$
$C:x_2 - x_3 - x_5 = -200$
$D:x_4 + x_5 = 350$

Anyhow, when I put it into matrix form and proceed to RREF I get:
$<br /> \left(\begin{array}{cccccc}1&0&1&0&1&500\\0&1&-1&0&-1&-200\\0&0&0&1&1&350\\0&0&0&0&0&0\end{array}\right)<br />$

Because of the zero row I know $x_4$ and $x_5$ are free variables, so I set $x_4 = s$ and $x_5 = t$ . It's at this point I get stuck. I have tried to solve it, but I wind up getting an expression for $x_2$ in both $x_1$ and $x_3$ . Do I need to let another variable be a free variable? Thanks for any and all help.

Typically, in a network (such as a piping system), the change occurs by increasing/decreasing traffic through a node.

A quick analysis indicates:
x1 & x2 are inversely related
{x1: 0,1,...,299,300} & {x2: 300,299,...,1,0}

x4 & x5 are inversely related
{x5: 350,...,0}{x4: 0,...,350}

x3 is dependent on x2 & x5 but has a larger range
{x3: 0, ... , 500}

It appears that you will need to restrain two values (UNRELATED VALUES ) to have a unique value for the others.

If x2=200 & x3=50
then the others are set
x1=100, x4=0, x5=350

(a) Solve this system for $x_i, i = 1, 2, ..., 5.$
Would this just be the ranges?
Since there are multiple values possible for the other four variables, when you assign a specifc value to 1 variable.

.

**Alterah** · September 9th 2009, 03:34 AM

Thanks for the reply. I took the question to mean to find the general solution. And then from there I can use part b and part c for a more specific solution.

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