$)$ Botswana Electronics Company $($BEC$),$ is contemplating a research and development

program encompassing eight major projects. The company is constrained from embarking on

all of the projects by the number of available scientists $(40)$ and the budget available for the

projects $($$$300,000).$ In the following are the resource requirements and the estimated profit

for each project.

Project Expense $($$$000)$ Scientists Required Profit $($$$000)$

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$65$

$73$

$85$

$92$

$47$

$53$

$110$

$5$

$8$

$6$

$7$

$4$

$8$

$9$

$41$

$48$

$61$

$56$

$16$

$29$

$82$

$860736$

a$.$ Using binary variables write down an algebraic formulation of an integer $($linear$)$

programming formulation for this problem. Document your algebraic model so that it

is clear what the variables, constraints, and objective function mean.

b$.$ Implement your model in Solver. Document your model appropriately so it is possible

to identify the decision variables, objective function, and constraints without going

into Solver. Make sure to paste the Solver dialog box into your spreadsheet.

c$.$ Based on the solution to your model, what is the maximum profit? Which projects

should be selected to achieve this profit?

d$.$ Suppose projects $3$ and $7$ are mutually exclusive. That is$,$ BEC should not undertake

both. Write down a linear constraint that implements this condition.

e$.$ Add the constraint from part d into your model and reoptimize your model. Do this in

a separate worksheet. $($You can right click the name of the original sheet of your model

and select move or copy. Then make a copy of your original sheet, rename it and then

make changes to the solver model therein$).$

f$.$ What is the revised project portfolio and the revised maximum profit?

g$.$ In addition to the previous restrictions, suppose that projects $1-4$ involve consumer

products and that management decides to undertake at least two of those. Write

down a single linear constraint that implements this restriction.

h$.$ Add the constraint from part g into your model from part e and reoptimize your

model. Do this in a separate worksheet.

i$.$ What is the revised project portfolio and revised profit?