$1.$ Yorktown manufacturing makes three products. Each product requires manufacturing operations in three departments: X$,$ Y$,$ and Z$.$ The labor hour requirements, by department, are as follows:

Department Product $1$ Product $2$ Product $3$

X $1\mathrm{.}503\mathrm{.}002\mathrm{.}00$

Y $2\mathrm{.}001\mathrm{.}002\mathrm{.}50$

Z $0\mathrm{.}250\mathrm{.}250\mathrm{.}25$

During the next production period, the labor$-$hours available are $465$ in department X$,345$ in department Y$,$ and $50$ in department Z$.$ The profit contributions per unit are $$30$ for product $1,$ $$25$ for product $2,$ and $$35$ for product $3.$

$1.$ Formulate a linear programming model for maximizing total profit contribution.

$2.$ Solve the linear program formulated in part $(1).$ How much of each product should be produced and what is the projected total profit contribution? $($Make sure that the number of products are integer valued$).$

$3.$ After evaluating the solution in part $(2),$ one of the production supervisors noted that production setup costs had not been considered. 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 $(2)$ is to be used, what is the total profit contribution after considering the setup costs?

$4.$ Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part $(2).$ 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 $3.$

$5.$ Solve the mixed$-$integer linear program formulated in part $(4).$ 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 $(3).($Make sure that the number of products are integer valued$).$