NEED HELP WITH E Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: A$,$ B$,$ and C$.$ The labor$-$hour requirements,

by 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 $

$5,540$

$($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.

The supervisor noted that setup costs are $$550$ for product $1,$$$400$ 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 $