Problem 1: Airline fuel problem: 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, files 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?
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LEG
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Minimum fuel required (1,000 Gal)
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Maximum fuel allowed (1,000 Gal)
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Required fuel consumption (1,000 Gal)
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Fuel price per gallon
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Atlanta-Los Angeles
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24
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36
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12
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$4.15
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Los Angeles-Houston
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15
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23
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7
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$4.25
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Houston-New Orleans
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9
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17
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3
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$4.10
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New Orleans-Atlanta
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11
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20
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5
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$4.18
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Problem 2: A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows:
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Hours/Unit
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Product
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Line 1
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Line 2
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A
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12
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4
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B
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4
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8
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Total hours
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60
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40
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a. Formulate a linear programming model to determine the optimal product mix that will maximize profit.
b. Transform this model into standard form.
Problem 3: a. Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points.
b. What would be the effect on the optimal solution is the production time on line 1 was reduced to 40 hours?
c. What would be the effect on the optimal solution is the profit for product B was increased form $7 to $15? To $20?