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Math Help - Traffic Flow, Network Analysis

  1. #1
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    Traffic Flow, Network Analysis

    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.
    Attached Thumbnails Attached Thumbnails Traffic Flow, Network Analysis-trafficsystem.jpg  
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  2. #2
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    Did anyone have any insight? Thanks.
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  3. #3
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    Quote Originally Posted by Alterah View Post
    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.

    .
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  4. #4
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    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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