Question $1(45$ points$)$

Boris Milkem's financial firm owns six assets. The expected sales price $($in millions of dollars$)$ for each asset at the beginning of each year is given in the following table. For example, if asset $1$ is sold at the beginning of year $2,$ the firm receives $$20$ million.

$\backslash$table$[[,$Sold ln$],[$Asset$,$Year $1,$Year $2,$Year $3,$Ycar $4],[1,15,20,24,27],[2,16,18,21,22],[3,22,30,36,40],[4,10,20,30,36],[5,17,19,22,25],[6,19,25,29,35]]$

Milkem has to make the following payments in the next three years:

$$20$ million in year $1,$$$30$ million in year $2,$$$35$ million in year $3$

The firm currently has $$5$ million cash available and has to sell some or all of its assets over four years to meet the payment requirements. Cash from an asset sold at the beginning of a given year can be immediately used to make payments in that year. During any year, excess funds can be invested in a money market account whose yearly return is $5\%.$ Milkem wants to maximize the fotal cash on hand at the beginning of year $4.$ Assume that all the transactions $($paymentsand receipts$)$ occur at the beginning of a year.

$($a$)(20$ points$)$ Set up an IP that Milkem can use to determine which assets to sell and when to maximize total cash on hand at the beginning of year $4.$ Define your decision variables clearly: $($Hint: Obviously, Milkem wants to sell all of the assets by the beginning of year $4$ to maximize total cash.$)$

$($b$)(10$ points$)$ Solve the model you developed in part $($a$)$ using OpenSolver and report the optimal solution. $($Hint: Optimal objective function value is $$83,815,250)$

$($c$)(5$ points$)$ Assume that Milkem is not allowed to sell assets $1$ and $2$ in the same year. How would you modify the formulation in part $($a$)$ to handle this? You do not need to submit the optimal solution here but you may want to test your model to see if your constraint$($s$)$ really works.

$($d$)(5$ points$)$ Assume that asset $3$ can only be sold if asset $6$ was sold in a prior year. How would you modify the formulation in part $($a$)$ to handle this? You do not need to submit the optimal solution here but you may want to test your model to see if your consiraint$($s$)$ really works.

$1$

$($e$)(5$ points$)$ Assume that asset $5$ can only be sold if both assets $3$ and $4$ were sold in a prior year. How would you modify the formulation in part $($a$)$ to handle this? You do not need to submit the optimal solution here but you may want to test your model to see if your constraint$($s$)$ really works.