An airplane pilot buying fuel. Puddlejumper Airlines (PA, for short) has a regular route f...

An airplane pilot buying fuel. Puddlejumper Airlines (PA, for short) has a regular route from city 1 through six more cities with return to city 1. The number of passengers on board each leg of the trip is known fairly well in advance. PA is just getting off the ground. so to speak, and has to count every dollar. As a consequence, the systems department (Jeff) wants the pilot to purchase fuel at stops along the way rather than try to carry the entire flight’s needs from city 1, Prices along the route vary so the decision of where and how much to buy is a complicated one. The systems department knows linear programming and has an idea on how it may be applied.

The fuel consumed in flying the kth leg of the journey depends not only on the number of passengers on board (really their weight plus the weight of their luggage) but also on the weight of fuel being carried. That is, it costs money to carry excess fuel from one city to the next. The rule is that the plane must carry at least the amount of fuel that is needed to carry its load from city k to city k + l (leg k) plus some reserve amount.

Let:

bk = amount of fuel consumed on leg k if no extra fuel is carried on board, and

rk = the reserve amount of fuel to be carried on leg k

By extra fuel we mean any fuel in excess of (bk + rk), that is, in excess of the minimum requirement on leg k, In addition to bk and rk, we have the additional paramenters:

uk = maximum amount of fuel that the plane can safely carry on leg k (actually at the start of leg k)

fk = fraction of extra fuel on leg k burned on leg k; and

ck = cost of fuel purchased at k before beginning the kth leg (the trip from k to k + 1).

The decision variables for the problem are

Yk = the amount of fuel in the plane’s tanks when it lands at city k, and

Xk = the amount of fuel purchased at city k.

To calculate the fuel consumed on leg k requires that we know the extra fuel on board at takeoff from k. That number is

so that the extra fuel burned on leg k, in addition to bk, is

The total fuel burned is

so that you can calulate the fuel in the tanks when the plane arrives at city k + 1.

The PA systems department needs to determine fuel amounts to buy at each city so that the total cost of fuel purchases is as small as possible.. Write the standard-form linear program. (This problem in another situation is the creation of Norman Waite of the IBM Corporation.)