ONLY PART E$)$ NEEDED. FILL OU THE MISSING GAPS

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.

$\backslash$table$[[$Department$,$Product $1,$Product $2,$Product $3],[$A$,1\mathrm{.}50,3\mathrm{.}00,2\mathrm{.}00],[$B$,2\mathrm{.}00,1\mathrm{.}00,2\mathrm{.}50],[$C$,0\mathrm{.}25,0\mathrm{.}25,0\mathrm{.}25]]$

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 $$25$ for product $1,$$$27$ for product $2,$ and $$29$ for product $3.$

$($a$)$ Formulate a linear programming model for maximizing total profit contribution. $($Let $P_i=$ units of product i produced, for $i=1,2,3.)$

Ma$)$

s$.$t$.$

Department A

Department B

Department $C0\mathrm{.}25P_1+0\mathrm{.}25P_2+0\mathrm{.}25P_{3}50$

$P_1,P_2,P_{3}0$

$($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$)?$

$(P_1,P_2,P_3)=(,)$ with 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 $$370$ for product $1,$$$560$ for product $2,$ and $$580$ 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 mixed$-$integer linear program that takes setup costs into account. Management also stated that we should not consider making more than $145$ units of product $1,155$ units of product $2,$ or $160$ 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?

Ma$)$

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 $3$ producec

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

$($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?