$1($Project selection problem$)$: A company wants to invest on some of

projects. $($Either the entire project is selected or not at all.$)$ Each of the six projects

has a cost and yields a revenue as shown below.

Project Cost $($in $million$)$ Revenue $($in $million$)$

$1$ $$10$ $$12$

$2$ $$40$ $$50$

$3$ $$55$ $$56$

$4$ $$60$ $$75$

$5$ $$15$ $$18$

$6$ $$30$ $$33$

The company has a total budget $$150($in $million$)$ that cannot be exceeded. The goal is to invest

the capital in such a way that it maximizes the total revenue. Formulate this situation as a zeroone integer program and find the optimal solution. $(4$ points$)$

$($a$)$ What is the maximum revenue that can be obtained?