Hart manufacturing makes three products. Each product requites manufactoring operations in three departments. A$,$B$,$C$.$ The labor hours requiremts by department are as follows. Department A product $1-1\mathrm{.}5$ poduct $2-3$ and profuct $3-2.$

Department B product $1-2,$ product $2-1$ and product $3-2\mathrm{.}5.$ Department C product $1-\mathrm{.}25,$ product $2-\mathrm{.}25$ and produuct $3-\mathrm{.}25$

During the next production. The labor hours available are $450$ in department a $350$ in department B and $50$ in department C$.$

The profit contributions per unit are $27$ for product $128$ for product two and $32$ for product three.

$($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 in dollars?

$($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 $$410$ for product $1,$$$600$ for product $2,$ and $$610$ for product $3.$ If the solution developed in part $($b$)$ is to be used, what is the total profit contribution $($in dollars$)$ 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 mixedinteger linear program that takes setup costs into account. Management also stated that we should not consider making more than $155$ units of product $1,160$ units of product $2,$ or $180$ units of product $3.($Let $P_i=$ units of product i produced and $y_i$ be the $0-1$ variable that is one if any quantity of product $i$ is produced and zero otherwise, for $i=1,2,3.)$

What is the objective function of the mixed$-$integer linear program?

Max$27P_1+28P_2+32P_3-$

In addition to the constraints from part $($a$),$ what other constraints should be added to the mixed$-$integer linear program?

s$.$t$.$

units of Product $1$ produced

units of Product $2$ produced

units of Product $3$ produced

$P_1,P_2,P_{3}0$;$y_1,y_2,y_3=0,1$

$($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 $($in dollars$)$ contribution?