Problem $1.$ A division in a large company has been allocated $$160$ million for capital projects this year. Managers have proposed five projects for the budgeting committee to consider. The committee would like to maximize the total NPV$.$ What projects should be selected?

$$ There is one project of each type, each project has an estimated NPV $($net present value, a measurement of profit$)$ and each requires a capital expenditure $($this comes out of the budget above$).$

$$ At least $2$ projects must be done.

$$ If Project $5$ is chosen, Project $1$ must be done as well.

Project $1$ Project $2$ Project $3$ Project $4$ Project $5$

NPV $($in millions of $$)($Project $1)-10,($Project $2)-17,($Project $3)-16,($Project $4)-8,($Project $5)-14$

Expenditure $($millions of $$)($Project $1)-48,($Project $2)-96,($Project $3)-80,($Project $4)-32,($Project $5)-64$

a$.$ What are your decision variables $($x$)?$ State the units for each.

b$.$ Write down your objective function $($include min$/$max with the equation$)$ and constraints below, include descriptions$/$explanations$/$calculations on the right. Delete$/$add rows as needed.