Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: $A,B_r$ and $C.$ The labor$-$hour requirements, by department, are as follows.

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

$Ma$

s$.$t$.$

Department A

Department $B$

Department C

$P_{1^{\text{'}}}{P}_{2^{\text{'}}}{P}_{3}0$

$($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$)?$ With profit

$(P_1,P_2,P_3)=(,)$ with profit $$,x.$

$($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 $$650$ 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?

$

$3660$

Incorrect: Your answer is incorrect.

$($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 in part $($c$)$ into account. Management also stated that we should not consider making more than $150$ units of product $1,155$ units of product $2,$ or $175$ units of product $3.($Let

Pi $=$

units of product i produced and

yi

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

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 contribution $($in dollars$)?$ With profit

$($P$1,$ P$2,$ P$3,$ y$1,$ y$2,$ y$3)=$