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

Max$27P_1+29P_2+32P_3$

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 $C$

$P_1,P_2,P_{3}0$

$($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 $$,|.$

$($in dollars$)$ after taking into account the setup costs?

$

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

Max

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

$P_{1^{\text{'}}}{P}_{2^{\text{'}}}{P}_{3}0$;$y_{1^{\text{'}}}{y}_{2^{\text{'}}}{y}_3=0,1$

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

$(P_1,P_2,P_{3^{\text{'}}},y_1,y_2,y_3)=!$ with profit $