Question # $2(20$p$.)$ : Consider a capital budgeting example with five projects from which to select. Let $x_i=1$ if project $i$ is selected, $0$ if not, for $i=1,2,3,4,5.$ Projects cost $(($:$$1,000\}$ are $100,200,150,75,$ and $300,$ respectively. Suppose that the budget is $550.$ Consider the following requirements: Choose at least $3$ projects, if project $3$ is chosen, project $4$ must not be chosen and if project $1$ is chosen, project $5$ must not be chosen. Estimated profit $($$$1,000)$ for each project are $250,400,300,150,$ and $500,$ respectively. Suppose that we want to maximize the profit. Based on the given information answer the following questions:

a$)$ Formulate the integer programming model.

b$)$ Introduce slack$/$surplus$/$artificial variables and prepare LP model in a standard form.

c$)$ Use the standard form of LP given in part $($b$)$ to constract the initial simplex tableau and fill in all cell values. $($DO NOT SOLVE THE PROBLEM$)$