The figure below shows a network of one way streets. The direction the traffic i

Question

The figure below shows a network of one way streets. The direction the traffic it flowing is indicated with the arrows and the values represent the average number of vehicles per hour entering or exiting the intersection. a) Set up a system of linear equations that represents this scenario, with unknown flow rates as your variables. b) Solve the system form part a) Write your solutions here attach work. c) Suppose there needs to be construction on the road from A to B and traffic needs to be restricted. What is the minimum flow that is required to keep traffic flowing on all roads?

Explanation / Answer

a. & b.

x2 + 550 = x3 + 350
x3 + 400 = x4 + 300
x4 + 200 = x1 + 600
x1 + 250 = x2 + 150

The above system of linear equations can be simplified as follows
x2 – x3 = -200
x3 – x4 = -100
x1 – x4 = -400
x1 – x2 = -100

The above system of linear equations can be represented in the following matrix form
0   1   -1   0   -200
0   0    1 -1   -100
1   0    0 -1   -400
1 -1    0   0 -100

The reduched echleon form is
   1   0    0    -1   -400
   0   1    0    -1   -300
   0   0    1    -1   -100
   0   0    0     0     0
And clearly, x4 is a free variable.

This system has infinitely many solutions, which are (with parameters ri):
x1 = r1 – 400
x2 = r1 – 300
x3 = r1 – 100
x4 = r1
If we want the traffic to flow in the forward direction, then r1>=0

c.

When we restrict the flow from A to B, we substitute a value of x4 in above equations such that remaining all flows should be positive, so when we substittue (x4 >= 400) we get

x1 >= 0
x2 >= 100
x3 >= 200
The minimum flow between A and B must be 400 to keep the traffic flowing in the forward direction on other roads

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