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"vKirchhoff's first Law, the flow into an intersection is equal to the flow out. This gives the followingystem
The augmentedmatrix of the above system is
~
1 0
1 -1
0 10 0
0 0
0 0
0 0 1 0
0 1 0 0
-1 0 0 01 0 0 0
0 1 0 10 0 1 1
0 800
0 400
0 600-1 -1200
-1 0
0 1000
800 - x + v (A)u + x - y + 400 (B)
Y - z + 600 (C)1600 + z . t + 400 (D)
t - u + W (E)v + w - 1000 (F)
Therefore, the solution can be written as
x - w - 200
y- t -600z - t - 1200
" - t - w
" - 1000 - w
w) t elK
OM VIr"J Jt~ . (.J~;;!OO, -t?lo.oo, (iew) , W:((OOV
1000 '7 /.A.! ~.1oo IF,orinstance, ifw:300 and t=1300 (in vehicles per hour), then i ,,(aoo J
~-100, y-700, z =100, U= 1000, V -700. w,tE"d:
Supposenowthatthe streetstrom A to B and trom B to C must be closed(for constructionfor instance),that is x=Oandy=O.Howmightthe traffic be rerouted?
To answer this question, set x=y=O in the above solution, we get W - 200, t - 600.z =-600. U =400and
V =800. Of course, the negative value for zis not normal. In order to avoid negative flow, we mustreversedirectionsonthe streetsconnectingC and D; this changemakes z =600insteadof z =-600.
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