a$.$ Formulate a linear programming model for maximizing total profit contribution.

b$.$ Solve the linear program formulated in part $($a$).$ How much of each product should be produced, and what is the projected total profit contribution?

c$.$ After evaluating the solution obtained in part $($b$),$ one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are $$400$ for product $1,$$$550$ for product $2,$ and $$600$ for product $3.$ If the solution developed in part $($b$)$ is to be used, what is the total profit contribution after taking into account the setup costs?

d$.$ Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part $($b$).$ Formulate a mixed$-$integer linear program that takes setup costs into account. Management also stated that we should not consider making more than $175$ units of product $1,150$ units of product $2,$ or $140$ units of product $3.$

e$.$ Solve the mixed$-$integer linear program formulated in part $($d$).$ How much of each product should be produced, and what is the projected total profit contribution? Compare this profit contribution to that obtained in part $($c$).$