Question

Case 1 (Due: 7/11/2019) Delta Express is an overnight package delivery company based in Minneapolis, Minnesota. Jet fuel is one of the largest operating costs incurred by the company and they want your assistance in managing this cost. The price of jet fuel varies considerably at different airports around the country. As a result, it seems that it might be wise to 'fill up’ on jet fuel at airports where it is least expensive. However, the amount of fuel an airliner burns depends, in part, on the weight of the plane-and excess fuel makes an airplane heavier and, therefore, less fuel-efficient. The following table summarizes the flight schedule (or rotation) flown nightly by one of the company's planes. For each flight segment, the table summarizes the minimum required and maximum allowable amount of fuel on board at takeoff and the cost of fuel at each point of departure. The final column provides a linear' function relating fuel consumption to the amount of fuel on board at takeoff. Segment Depart Arrive Minimum Fuel level At takeoff Maximum Fuel level At takeoff Cost per Gallon Fuel used in flight G gallons On board at takeoff 1 Minneapolis Chicago 10,000 35,000 $ 50 2,000 + 0.6*(G) 2 Chicago Boston 12,000 45,000 $ 45 2,800 + 0.4*(G) 3 Boston Denver 18,000 36,000 $ 35 1,500 + 0.8*(G) 4 Denver Minneapolis 15,000 27,000 $ 37 2,300 + 0.7* (G) For instance, if the plane leaves Minneapolis for Chicago with 25,000 gallons on board, it should arrive in Chicago with approximately 25,000 - (2000 + 0.6 x 25000) = 8,000 gallons of fuel. The company has many other planes that fly different schedules each night, so the potential cost savings from efficient fuel purchasing is quite significant. But before turning you loose on all of their flight schedules, the company wants you to create a LP model to determine the most economical fuel purchasing plan for the previous schedule. (Hint: Keep in mind that the most fuel you would purchase at any departure point is the maximum allowable fuel level for takeoff at that point. Also, assume that whatever fuel is on board when the plane returns to Minneapolis at the end of the rotation will still be, on board when the plane leaves Minneapolis the next evening.) 1. Define the decision variables Pm= Pc= Pb= Pd= = Minneapolis Chicago Boston Denver Beginning fuel Purchase Pm Pc Pb Pd Use Ending fuel 2. Define the objective function Minimize 50*Pm+ 45*Pc + 35*Pb+ 37*Pd 3. Define the constraints 4. How much fuel should US Express purchase at each departure point and what the cost is of this purchasing plan?

Answer 1

1) Decision variables:

D1, D2, D3, D4 = Fuel (gallons) on-board at landing at Minneapolis, Chicago, Boston, Denver respectively

P1, P2, P3, P4 = Fuel (gallons) purchased at Minneapolis, Chicago, Boston, Denver respectively

2) Objective function: Min 50P1 + 45P2 + 35P3 + 37P4

3) Constraints:

D1+P1 >= 10000

D1+P1 <= 35000

D2 = D1+P1-(2000+0.6*(D1+P1))

D2+P2 >= 12000

D2+P2 <= 45000

D3 = D2+P2-(2800+0.4*(D2+P2))

D3+P3 >= 18000

D3+P3 <= 36000

D4 = D3+P3-(1500+0.8*(D3+P3))

D4+P4 >= 15000

D4+P4 <= 27000

D1 = D4+P4-(2300+0.7*(D4+P4))

P1, P2, P3, P4, D1 >= 0

Solution using Excel Solver is as follows

Formulas:

Cell	Formula	Copy to
F14	=D14+E14	F14:F17
H14	=F14-G14	H14:H17
G14	=2000+0.6*F14
G15	=2800+0.4*F14
G16	=1500+0.8*F14
G17	=2300+0.7*F14
E19	=SUMPRODUCT(E14:E17,F4:F7)

4) Fuel to be purchased at each departure point:

Segment	Depart	Fuel purchased (gallons)
1	Minneapolis	7,800
2	Chicago	10,000
3	Boston	13,600
4	Denver	12,900

Cost of this purchasing plan = $ 1,793,300