$($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 $\backslash (\backslash$mathrm$\{$P$\}\_\{\backslash$mathrm$\{$i$\}\}=\backslash )$ 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 $\backslash$$ the total profit contribution after taking into account the setup costs?

$\backslash$$ 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 $\backslash ($ Y$\_\{$i$\}\backslash )$ is one if any quantity of product $\backslash ($ i $\backslash )$ 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 $"0".$