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

Max$24P_1+27P_2+28P_3$

s$.$t$.$

Department A $1\mathrm{.}5P_1+3P_2+2P_{3}450$

Department B $2P_1+1P_2+2\mathrm{.}5P_{3}350$

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

$P_1,P_2,P_{3}0$

$($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 $$5280$

account the setup costs?

$$3720$

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

Max$24P_1+27P_2+28P_3-350y_1-600y_2-610y_3$

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 $1P_1-150y_{1}0$

units of Product $2$ produced $1P_2-155y_{2}0$

units of Product $3$ produced $1P_3-200y_{3}0$

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

$($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?

$(P_{1^{\text{'}}},P_{2^{\text{'}}},P_{3^{\text{'}}},y_{1^{\text{'}}}{y}_{2^{\text{'}}}{y}_3)=$

$]$