$[10]$ Angel Consulting is considering $8$ projects. Each project has a cost and an associated re$-$

turn. The budget at Angel is $$580.$ Angel would like to choose the set of projects that maximize

the total project return. This data below is provided in the workbook that accompanies this exam.

$($a$)[4]$ Formulate this project selection problem as an integer program. Clearly define your decision

variables, objective function, and constraints.

$($b$)[4]($Excel$)$ Solve this integer program in the Excel workbook. What are the optimal set of

projects to select? What is the total return of your solution? Write down the answers here

and save the answer report in Excel.

$($c$)[2]$ Suppose an additional constraint is added to Angel's problem that requires the choice of at

least two projects out of the projects numbered $6,7,$ and $8.$ Write down this additional con$-$

straint in the integer program $($using the decision variables you defined$).$ With this additional

constraint, will the optimal solution value increase or decrease?