Problem $2($Cargo Airplane Space Allocation Problem$)(45$ pts$.)$

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 capacity $($tons$)$ Space capacity $($f t$3)$

Front $127000$

Center $189000$

Back $105000$

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

Cargo Weight $($tons$)$ Volume $($f t$3/$ton$)$ Profit $($$$/$ton$)$

$120500320$

$216700400$

$325600360$

$413400290$

Any portion of these cargoes can be accepted. The objective 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.

$1.$ Formulate a linear programming model for this problem in its algebraic form. Solve this model

using AMPL.

$2.$ If$,$ 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. Modify the linear pro$-$

gramming model to include this additional requirement. Solve the updated model using AMPL.

Problem $3($Investment Planning Problem$)(35$ pts$.)$

Maureen Laird is the chief financial officer for the Alva Electric Co$.,$ a major public utility in the

midwest. The company has scheduled the construction of new hydroelectric plants $5,10,$ and $20$ years

from now to meet the needs of the growing population in the region served by the company. To cover

at least the construction costs, Maureen needs to invest some of the company$$s money now to meet

these future cash$-$flow needs. Maureen may purchase only three kinds of financial assets, each of which

costs $$1$ million per unit. Fractional units may be purchased. The assets produce income $5,10,$ and

$20$ years from now, and that income is needed to cover at least minimum cash$-$flow requirements in

those years. $($Any excess income above the minimum requirement for each time period will be used to

increase dividend payments to shareholders rather than saving it to help meet the minimum cash$-$flow

requirement in the next time period.$)$ The following table shows both the amount of income generated

by each unit of each asset and the minimum amount of income needed for each of the future time periods

when a new hydroelectric plant will be constructed.

Income per unit of asset $($million $$)$ Minimum Cash Flow Required

Year Asset $1$ Asset $2$ Asset $3$

$5210\mathrm{.}5400$

$100\mathrm{.}50\mathrm{.}51100$

$2001\mathrm{.}52300$

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