Problem $2$: $($Multi$-$Period Knapsack Problem$)$ Consider the following multi$-$period

capital budgeting problem. An investor is considering $8$ independent investment projects.

Each project requires full participation, no partial participation is allowed. If selected, a

project may require cash contributions yearly over the next $3$ years, as in the following table.

There is an external fixed amount of $$1,000,000$ available for these investments in each of

the following $3$ years. Each investment project is expected to have a net return as shown in

the table below. The investor needs to determine the subset of projects to invest in$,$ so as

to maximize the total expected net returns. For simplicity, assume that:

$$ Unutilized funds are not carried forward to the next period and are not available for

investments.

$$ Cash flows are not discounted for the time value of money $($we could consider discounting of cash flows, but the problem becomes considerably more difficult to formulate$).$

Table $2$: Problem $2-$ All figures are in $$10,000$s of dollars

Cash Outflows

P$1$ P$2$ P$3$ P$4$ P$5$ P$6$ P$7$ P$8$ Funds Available

Year $1204050251572313100$

Year $23020302525222328100$

Year $3100103530232315100$

Expected Net Return $1015202015131314$

$($a$)$ Formulate an optimization model for this problem. $(10$ points$)$

$($b$)$ Solve the model using Excel Solver? Explain the solution. $(5$ points$)$

$($c$)$ Suppose now that unused funds can be carried forward to the next period and are

available for cash out$-$flows. Formulate an optimization model for this problem, and

solve it in Excel. Compare your solution to the previous one and explain why the

solution is different.