Please compute the optimal traffic flow along each of the 4 roads marked x, y, z and w in the diagram of the city block below.?

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answers:
ted s say: there is one parameter.....x = 10 - y ; z = y - 4 ; w = 15 - y , y = y ...for y in [ 4,10]

thus there is no " optimal " flow....set up flow at each corner and rref a 4 by 5 matrix
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Steve4Physics say: You have not defined your criteria for ‘optimum’ but this might help.

Assume x, y, z and w all flow clockwise around the central square (each will be negative if its direction turns out to be clockwise).

At each node, total entering/hr = total leaving/hr. Apply this at each node.

At the top left node:
900 + 100 + y = x
x = y +1000 (equation 1)

At the top right node:
x = z + 200 + 400
. .= z + 600 (equation 2)

At the bottom right node:
z + 700 + 400 = w
w = z + 1100 (equation 3)

At the bottom left node:
w = y + 1200 + 300
w = y + 1500 (equation 4)
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The above equations provide the constraints for an optimisation. Suppose the marginal cost per vehicle is fixed, is the same for each of the 4 roads and is independent of flow direction. Then we have linear programming problem to minimise the objective function F = |x| + |y| + |z| + |w|.
So our constraints are equations 1 to 4 plus x≥0, y≥0, z≥0, w≥0.

Could be done manually but I’d use some software.
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Steve A say: Let the top two be A,B; the four on the side be C,D,E,F; the bottom two be G,H.
A+F+H = E
C = B+D+G

Figure from there.
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WHANKING-WILLY say: 42
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