Five projects are being evaluated over a $3-$year planning horizon. The following table gives the expected returns for each project and the associated yearly expenditures.

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Expenditures $($millions $$)/$year

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Project

$1$

$2$

$3$

Returns $($million $$)$

$1$

$5$

$1$

$8$

$20$

$2$

$4$

$7$

$10$

$40$

$3$

$3$

$9$

$2$

$20$

$4$

$7$

$4$

$1$

$15$

$5$

$8$

$6$

$10$

$30$

Available funds $($million $$)$

$25$

$25$

$25$

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a$)$ Formulate the above as an integer linear programming problem, utilizing the $3-$step LP problem formulation technique described during my lectures. $(5$ points$)$

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Note: Points will be given for each component of the formulation according to the answer submitted by the student in the following manner:

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Decision Variables: $1$ point

Objective Function: $1$ point

Constraints: $3$ points