Question

4. A network of railway lines connects the main lines entering and leaving a city. Speed limits, track reconstruction, and train length restrictions lead to the flow diagram below where the numbers represent how many cars can pass per hour Formulate and solve an LP to find the maximal flow in cars per hour from Node 1 to Node F 400 600 300 100 500 200 (+04) 500 500

Answer 1

Let Xij be the flow rate from node-i to node-j.

Maximize Z = X12 + X16 + X15 (Or, we can also take X4F + X8F + X9F)

Subject to,

Zero net flow at the intermediate nodes

X12 - X23 - X24 = 0 (node 2)
X23 - X34 = 0 (node 3)
X24 + X34 + X84 - X48 - X4F = 0 (node 4)
X15 - X57 = 0 (node 5)
X16 + X76 - X67 - X69 = 0 (node 6)
X57 + X67 - X76 - X79 = 0 (node 7)
X48 + X98 - X84 - X89 - X8F = 0 (node 8)
X79 + X69 + X89 - X98 - X9F = 0 (node 9)

Capacity constriants

X12 <= 500; X16 <= 600; X15 <= 300;
X23 <= 300; X24 <= 400;
X34 <= 150;
X48 <= 800; X4F <= 600;
X57 <= 400;
X67 <= 300; X69 <= 500;
X76 <= 200; X79 <= 350;
X84 <= 200; X89 <= 300; X8F <= 450;
X98 <= 300; X9F <= 500

Xij >= 0

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LINDO Code

So, the maximum possible flow = 1300 cars per hour