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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. Formulate the linear programming model for this problem.

A cargo airplane operated by 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
Capacity
(tons) Space
Capacity
(cu.ft.)
Front
Center
Back 12
13
10 7,000
9,000
5,000

To ensure proper weight and balance, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity.

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

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

Any portion of these cargoes can be accepted. 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. Formulate the linear programming model for this problem.

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