#### Question

# Linear Program help needed. Absolutely clueless.?

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.

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?

Data for problem below:

Leg

Atlanta-Los Angeles

Los Angeles-Houston

Houston-New Orleans

New Orleans-Atlanta

Minimum Fuel Required (1,000 Gal.)

24

15

9

11

Maximum Fuel Allowed (1,000 Gal.)

36

23

17

20

Regular Fuel Consumption (1,000 Gal.)

12

7

3

5

Fuel Price Per Gallon

$1.15

$1.25

$1.10

$1.18

#### Answer

Atlanta to LA to Houston to NO to Atlanta

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. (This is clearly an approximation since the percentage loss would depend on the flight length, the weight of the aircraft, the wind, etc. and I'm thinking a more accurate formula wouldn't be linear.)

Leg 1: Min 24,000 Max 36,000 Consumption 12,000 Price 1.15 Initial fuel : X1 Purchase fuel: P1 = ?

Leg 2: Min 15,000 Max 23,000 Consumption 7,000 Price 1.25 X2 = ? P1=?

Leg 3: Min 9,000 Max 17,000 Consumption 3,000 Price 1.10 X3 = ? P2 = ?

Leg 4: Min 11,000 Max 20,000 Consumption 5,000 Price 1.18 X4 = ? P3 = ?

X1, X2, X3, X4 = initial fuel when plane is ready to be fueled up

P1, P2, P3, P4 = amount of fuel purchased

Formulate an LP problem:

This means:

Identify the limits for each variable

Write an equation for the quantity to be optimized (total cost)

The question says to account for the fuel load in the airplane before adding fuel in Atlanta

Let X1 be the initial amount in Atlanta. Then the amount it will carry is X1 + P1. The cost will be

C1 = P1 x (1.15) (I apologize I used P for the amount purchased, I should have used something else since P is usually price.)

The limits are

24000 < X1 + P1 < 36000

The amount left when it gets to LA will be, assuming regular consumption,

X2 = X1 + P1 - 12000

The limits for P2, P3, and P4 are

15000 < X1 + P1 - 12000 + P2 < 23000

9000 < X1 + P1 + P2 + P3 - 12000 - 7000 < 17000

11000 < X1 + P1 + P2 + P3 + P4 - 12000 - 7000 - 3000 < 20000

Then to solve it you use linear programming techniques. I think it would make my answer too long if I wrote it, so it will be simpler if you repost that LP problem now that I stated it for you! Teamwork is good.