Question

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Problem 6-29 (Algorithmic) The north-south highway system passing through Albany, New York, can accommodate the capacities shown. If the constant is "1" it must be entered in the box. If your answer is zero enter "O For negative values enter "minus" sign () Flow capacity per hour 6,000vchicles Entering Albany (north) Leaving Albany (south) Max s.t. Flow Out Flow In 1 x121 x13 -1 | X x14 1 x25 Node 2 0 1 X X13+ 1 X X43 0 Node 3 X36 -1 x24 + x42 -1x34 + 1x54 0 Node 4 -1 | X x46 X45+ 1x56 -1X45 0 Node 5 1x36+ -1X46+ Node 6 X12 S X24 S x34 S x42 S x54 S 2 X13 S X25 S X36 S x43 S x56 S 6 X 3 X X14 S 3 X x45 S 6 xij 2 0 for all i and j Can the highway system accommodate a north-south flow of 9,000 vehicles per hour? Yes X Maximum flow of vehicles per hour9,000 X

Answer 1

The LP model is following:

Max 1 X61

s.t.

Node 1: 1 X12 +1 X13 +1 X14 -1 X61 = 0

Node 2: 1 X24 +1 X25 -1 X12 -1 X42 = 0

Node 3: 1 X34 +1 X36 -1 X13 -1 X43 = 0

Node 4: 1 X42 +1 X43 +1 X45 +1 X46 -1 X14 -1 X24 -1 X34 -1 X54 = 0

Node 5: 1 X54 +1 X56 -1 X25 -1 X45 = 0

Node 6: 1 X61 -1 X36 -1 X46 -1 X56 = 0

X12 <= 2

X13 <= 5

X14 <= 2

X24 <= 1

X25 <= 4

X34 <= 3

X36 <= 2

X42 <= 1

X43 <= 3

X45 <= 1

X46 <= 2

X54 <= 1

X56 <= 6

Xij >= 0

Solution using LINGO is following:

Can highway system accommodate a north-south flow of 9000 vehicles per hour ?

No

Maximum flow of vehicles per hour = 8,000

Optimal solution (maximum flow from node to node is following)

Variable	Value
X61	8
X12	2
X13	5
X14	1
X24	0
X25	3
X42	1
X34	3
X36	2
X43	0
X45	1
X46	2
X54	0
X56	4