qdm
DESCRIPTION
Linear ProgrammingTRANSCRIPT
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Coast-to-Coast Airlines is investigating the possibility of reducing the cost of fuel purchases
by taking advantage of lower fuel costs in certain cities. Since fuel purchases represent a
substantial portion of operating expenses for an airline, it is important that these costs be
carefully monitored. However, fuel adds weight to an airplane, and consequently, excess fuel
raises the cost of getting from one city to another. In evaluating one particular flight rotation,
a plane begins in Atlanta, flies from Atlanta to Los Angeles, from Los Angeles to Houston,
from Houston to New Orleans, and from New Orleans to Atlanta. When the plane arrives in
Atlanta, the flight rotation is said to have been completed, and then it starts again. Thus, the
fuel on board when the flight arrived in Atlanta must be taken into consideration when the
flight begins. Along each leg of this route, there is a minimum and a maximum amount of fuel
that may be carried. This and additional information is provided in the table on this page. The
regular fuel consumption is based on the plane carrying the minimum amount of fuel. If more
than this is carried, the amount of fuel consumed is higher. Specifically, for each 1,000
gallons of fuel above the minimum, 5% (or 50 gallons per 1,000 gallons of extra fuel) is lost
due to excess fuel consumption. For example, if 25,000 gallons of fuel were on board when
the plane takes off from Atlanta, the fuel consumed on this route would be 12 + 0.05 = 12.05
thousand gallons. If 26 thousand gallons were on board, the fuel consumed would be
increased by another 0.05 thousand, for a total of 12.1 thousand gallons.
Formulate this as an LP problem to minimize the cost. How many gallons should be
purchased in each city? What is the total cost of this?
.:.
SOLUTION:
Let
A = 1,000 gallons of fuel to purchase in Atlanta
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L = 1,000 gallons of fuel to purchase in Los Angeles
H = 1,000 gallons of fuel to purchase in Houston
N = 1,000 gallons of fuel to purchase in New Orleans
FA = fuel remaining when plane lands in Atlanta
FL = fuel remaining when plane lands in Los Angeles
FH = fuel remaining when plane lands in Houston
FN = fuel remaining when plane lands in New Orleans
Minimize cost = 4.15A + 4.25L + 4.10H + 4.18N
subject to
A + FA 24 minimum amount of fuel on board when leaving Atlanta
A + FA 36 maximum amount of fuel on board when leaving Atlanta
L + FL 15 minimum amount of fuel on board when leaving Los Angeles
L + FL 23 maximum amount of fuel on board when leaving Los Angeles
H + FH 9 minimum amount of fuel on board when leaving Houston
H + FH 17 maximum amount of fuel on board when leaving Houston
N + FN 11 minimum amount of fuel on board when leaving New Orleans
N + FN 20 maximum amount of fuel on board when leaving New Orleans
FL = A + FA – (12 + 0.05(A + FA – 24))
This says that the fuel on board when the plane lands in Los Angeles will equal the amount on
board at take-off minus the fuel consumed on that flight. The fuel consumed is 12 (thousand
gallons) plus 5% of the excess above 24 (thousand gallons). This simplifies to:
0.95A + 0.95 FA – FL = 10.8
Similarly,
FH = L + FL – (7 + 0.05(L + FL – 15)) becomes 0.95L + 0.95FL – FH = 6.25
FN = H + FH – (3 + 0.05(H + FH – 9)) becomes 0.95H + 0.95FH – FN = 2.55
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FA = N + FN – (5 + 0.05(N + FN – 11)) becomes 0.95N + 0.95FN – FA = 4.45
All variables 0
The optimal solution is
A = 18 (1,000 gallons of fuel to purchase in Atlanta)
FA = 6 (1,000 gallons of fuel remaining when plane lands in Atlanta)
L = 3 (1,000 gallons of fuel to purchase in Los Angeles)
FL = 12 (1,000 gallons of fuel remaining when plane lands in Los Angeles)
H = 1 (1,000 gallons of fuel to purchase in Houston)
FH = 8 (1,000 gallons of fuel remaining when plane lands in Houston)
N = 5 (1,000 gallons of fuel to purchase in New Orleans)
FN = 6 (1,000 gallons of fuel remaining when plane lands in New Orleans)
Total cost = 112.45 ( 1,000)