$3.$ Lupin, a construction company, are evaluating the selection of five projects over a $3-$year planning horizon. For this purpose, their total available budget is $300$ million and they decide the equal allocation of the budget over $3$ years. The following table gives the expected profits for each project and the associated yearly expenditures.

Projects Expenditure $($in millions$)/$ year Profits $($in millions$)$

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

$12043280$

$2162840160$

$31236880$

$42816460$

$5322440120$

Now Lupin wants to maximize its profit. Based on the information given above, answer the following questions.

a$)$ Write down the objective function of the Integer programming consistent with the objective of Lupin $[2]$

b$)$ Write down the constraints of Integer programming consistent with the objective of Lupin $[6]$

c$)$ Which projects should be selected? What will be the maximum return for Lupin? $($You can use either excel or QM for windows to solve the problem$)[2]$