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Linear System - Network Flow Matrices

  1. Nov 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the network of streets with intersections a,b,c,d and e below. The arrows indicate the direction of traffic flow along the oneway streets, and the numbers refer to the exact number of cars observed to enter or leave a,b,c,d and e during one minute. Each xi denotes the unknown number of cars which passed along the indicated streets during the same period.
    See Attached Image
    Linear System:
    x1+x6-x5=55
    x5-x4=35
    x6+x3-x4=60
    x3-x2=40
    x1-x2=70

    Reduced row-echelon form
    1 0 0 0 -1 1 | 55
    0 1 0 0 -1 1 |-15
    0 0 1 0 -1 1 |25
    0 0 0 1 -1 0 |-35
    0 0 0 0 0 0 |0
    s t
    General Solution:
    x1=55+s-t
    x2=-15+s-t
    x3=25+s-t
    x4=-35-s
    x5=s
    x6=t

    The question is: if ED were closed due to roadwork, find the minimum flow along AC, using your results in the genral solution.
    Note: x2 = ED therefor x2=0, and AC is x6

    2. Relevant equations



    3. The attempt at a solution

    I know x2=0 and no idea what to do with S and T...
    what I did is set x2=0 so -15+s-t=0 so S=t+15
    And sub it into the general solution you get:
    x1=70
    x2=0
    x3=40
    x4=-20+t
    x5=15+t
    x6=t

    so x4≥0 when t≥20
    x6≥0 when t≥0
    So minflow is 0?
     

    Attached Files:

  2. jcsd
  3. Nov 2, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No, you are not thinking it through! You need ALL xi >= 0, so you need x4 >= 0 and so you need t >= 20. Just having t >= 0 is not good enough.

    RGV
     
  4. Nov 2, 2012 #3
    I am trying to find the MINIMUM flow so....
     
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