Delta Express is an overnight package delivery company based in Minneapolis, Minnesota. Jet fuel is one of the most significant operating costs incurred by the company, and they want your assistance in managing this cost. In addition, jet fuel prices vary considerably at airports around the country. As a result, it seems that it might be wise to 'fill up jet fuel at airports where it is least expensive. However, the amount of fuel an airliner burn depends partly on the plane's weight, 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 plans. For each flight segment, the table summarizes the minimum required and maximum allowable amount of fuel onboard at takeoff and the fuel cost at each point of departure. The final column provides a linear' function relating fuel consumption to the amount of fuel onboard at takeoff.

Segment	Depart	Arrive	Minimum Fuel level At takeoff	Maximum Fuel level At takeoff	Cost per Gallon	Fuel used in flight G gallons Onboard at takeoff
1	Minneapolis	Chicago	8,000	26,000	$ 158	3,100 + 0.6*(G)
2	Chicago	Boston	10,000	30,000	$ 170	3,600 + 0.4*(G)
3	Boston	Denver	9,000	40,000	$ 165	2,900 + 0.5*(G)
4	Denver	Minneapolis	6,000	27,000	$ 167	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 - (3100 + 0.6 x 25000) = 6,900 gallons of fuel.

The company has many other planes that fly different schedules each night, so the potential cost savings from efficient fuel purchasing are significant. But before turning you loose on all of their flight schedules, the company wants you to make an LP model to determine the previous schedule's most economical fuel purchasing plan. (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 following evening, and assume that 600 gallons of fuel remaining when the plane return from Denver to Minneapolis)

1. Define the decision variables, objective function, and constraints.
2. How much fuel should allow the Delta Express purchase at each departure airport, and what is the total cost of this purchasing plan?