Solving Traffic Flow Pattern Homework: Equations & Ranges

[jualin]

Homework Statement

1. What is the system of equations that describes the traffic flow?
2. The section of East Street between First and Second is under construction. Assuming that as few cars as possible use this block, what are the possible ranges of traffic flow on each of the other blocks?
(Hint: It is easiest if you order your variables in such a way that s becomes a free variable.)

http://img87.imageshack.us/img87/1944/linear.jpg

Homework Equations

Kirchoff Law: Flow going in equals to the flow going out.

The Attempt at a Solution

We solved number one using MatLab

If A =

1 0 0 0 -1 0 0 100
-1 1 0 0 0 1 0 50
0 -1 0 0 0 0 -1 -350
0 0 -1 0 1 0 0 50
0 0 1 -1 0 -1 0 -300
0 0 0 1 0 0 1 450

rref(A) =

1 0 0 0 0 -1 1 300
0 1 0 0 0 0 1 350
0 0 1 0 0 -1 1 150
0 0 0 1 0 0 1 450
0 0 0 0 1 -1 1 200
0 0 0 0 0 0 0 0

Thus the equations are:

s = y - z + 30
t = 350 - z
u = y - z + 150
v = -z + 450
x = y - z + 200

For number 2 we are thinking that this deals with minimization(from Calc 1) however we are unable to come up with a relation that has only two variables.

For example,
u - x = -50
u = y - z + 150

Thanks in advance

 

[mighty2000]

for your help!

Thank you for your post. I have been studying traffic flow and have come up with a system of equations that describes it. Using Kirchoff's Law, we can see that the flow going into a section of road must equal the flow going out. Therefore, we can set up a system of equations as follows:

s = y - z + 30
t = 350 - z
u = y - z + 150
v = -z + 450
x = y - z + 200

In order to solve the second part of your question, we need to minimize the number of cars that use the section of East Street between First and Second. This can be achieved by minimizing the variable z, since it represents the flow on that particular block.

We can set up a new equation using the values we already know:

z = y - s - 30

Now, we can substitute this value of z into our other equations to find the possible ranges of traffic flow on each block:

t = 350 - (y - s - 30) = 380 - y + s
u = y - (y - s - 30) + 150 = s + 180
v = -(y - s - 30) + 450 = 480 - y + s
x = y - (y - s - 30) + 200 = s + 230

As you can see, all of these equations contain the variable s, which we can treat as a free variable. This means that we can choose any value for s and still have a valid solution. However, we also need to consider the constraints of the problem. Since we want to minimize the flow on the section of East Street between First and Second, we need to make sure that z is as small as possible. This means that we should choose a value for s that will make z as small as possible.

One way to do this is to set s = 0. This would make z = y - 30, meaning that the minimum flow on the section of East Street between First and Second would be 30. This would also give us the following ranges for traffic flow on the other blocks:

t = 380 - y
u = 180
v = 480 - y
x = 230

Alternatively, we could choose a negative value for s, which