$2-40.$ Fox Enterprises is considering six projects for possible construction over the next

four years. Fox can undertake any of the projects partially or completely. A partial

undertaking of a project will prorate both the return and cash outlays proportionately.

The expected $($present value$)$ returns and cash outlays for the projects are given in the

following table.

Cash outlay $($$$1000)$

Project Year $1$ Year $2$ Year $3$ Year $4$ Return $($$$1000)$

$110\mathrm{.}514\mathrm{.}42\mathrm{.}22\mathrm{.}4324\mathrm{.}00$

$28\mathrm{.}312\mathrm{.}69\mathrm{.}53\mathrm{.}1358\mathrm{.}00$

$310\mathrm{.}214\mathrm{.}25\mathrm{.}64\mathrm{.}2177\mathrm{.}50$

$47\mathrm{.}210\mathrm{.}57\mathrm{.}55\mathrm{.}0148\mathrm{.}00$

$512\mathrm{.}310\mathrm{.}18\mathrm{.}36\mathrm{.}3182\mathrm{.}00$

$69\mathrm{.}27\mathrm{.}86\mathrm{.}95\mathrm{.}1123\mathrm{.}50$

Available funds $($$$1000)60\mathrm{.}070\mathrm{.}035\mathrm{.}020\mathrm{.}0$

$($a$)$ Formulate the problem as a linear program, and determine the optimal project mix

that maximizes the total return using AMPL, Solver, or TORA. Ignore the time

value of money.

$($b$)$ Suppose that if a portion of project $2$ is undertaken, then at least an equal portion

of project $6$ must be undertaken. Modify the formulation of the model, and find the

new optimal solution.

$($c$)$ In the original model, suppose that any funds left at the end of a year are used in

the next year. Find the new optimal solution, and determine how much each year

$$borrows$$ from the preceding year. For simplicity, ignore the time value of money.

$($d$)$ Suppose in the original model the yearly funds available for any year can be exceeded,

if necessary, by borrowing from other financial activities within the company. Ignoring

the time value of money, reformulate the LP model, and find the optimum solution.

Would the new solution require borrowing in any year? If so$,$ what is the rate of return

on borrowed money?