Problem $11-11$

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:

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 $$25$ for product $1,$ $$28$ for product $2,$ and $$30$ for product $3.$

Formulate a linear programming model for maximizing total profit contribution. If required, round your answers to two decimal places. For those boxes in which you must enter subtractive or negative numbers use a minus sign. $($Example: $-300)$

Let Pi $=$ units of product i produced

Max fill in the blank $1$

P$1+$ fill in the blank $2$

P$2+$ fill in the blank $3$

P$3$

s$.$t$.$

fill in the blank $4$

P$1+$ fill in the blank $5$

P$2+2$P fill in the blank $6$

$2$P$1+$ fill in the blank $7$

P$2+$ fill in the blank $8$

P fill in the blank $9$

fill in the blank $10$

P$1+\mathrm{.}25$P$2+$ fill in the blank $11$

P fill in the blank $12$

P$1,$ P$2,$ P$3>=0$Problem $11-11$

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:

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 $$25$ for product $1,$$$28$ for product $2,$ and $$30$ for product $3.$

a$.$ Formulate a linear programming model for maximizing total profit contribution. If required, round your answers to

two decimal places. For those boxes in which you must enter subtractive or negative numbers use a minus sign.

$($Example: $-300)$

Let $P_i=$ units of product i produced

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?

$P_1=$

$P_2=$

$P_3=$

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 $$400$ for product $1,$$$550$ for product $2,$ and

$$600$ for product $3.$ If the solution developed in part $($b$)$ is to be used, what is the total profit contribution after

taking into account the setup costs?

Profit $=$$

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 $175$ units of product $1,150$ units of product

$2,$ or $140$ units of product $3.$ If required, round your answers to two decimal places. For those boxes in which you

must enter subtractive or negative numbers use a minus sign. $($Example: $-300)$ Here introduce a $0-1$ variable $y_i$ that

is one if any quantity of product $i$ is produced and zero otherwise.

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? Compare this profit contribution to that obtained in part $($c$).$

$P_1=$

$P_2=$

$P_3=$

Profit $=$$

The profit is