department, are as follows:

During the next production period the labor$-$hours available are $450$ in department A$,350$ in department $B,$ and $50$ in department C$.$ The profit contributions per

unit are $$30$ for product $1,$$$25$ for product $2,$ and $$28$ for product $3.$

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

If the constant is $"1"$ it must be entered in the box. If required, round your answers to two decimal places.

Let $P_i=$ units of product i produced

$($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?

Profit $

$($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 provided in part $($c$)$ 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 $3.$ What are the new objective function and additional equation constraints?

If the constant is $"1"$ it must be entered in the box.

Let $Y_i$ is one if any quantity of product $i$ is produced and zero otherwise.

$($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$).$

If required, round your answers to nearest whole number. If your answer is zero enter $"0".$

Updated Profit $