$[-/20$ Points$]$

CAMMIMS$167.$E$\mathrm{.}011.$

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.

$($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

s$.$t$.$

Department $A$

Department $B$

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$)?$

$

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 $3$ 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?

$(P_1,P_2,P_3,y_1,y_2,y_3)=()$ with profit $

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