Model 11: A packing problem

A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space, as summarized below:

Compartment Weight
(cu ft)
Front 12 7,000
Center 18 9,000
Back 10 5,000

Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the airplane.

The following four cargoes have been offered for shipment on an upcoming flight as space is available:

Cargo Weight
(cu ft/ton)
1 20 500 280
2 16 700 360
3 25 600 320
4 13 400 250

Any portion of these cargoes can be accepted. The object is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.