Binary mixed$-$integer linear program $(70$ points in total$)$

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

$\backslash$table$[[$Department$,$Product $1,$Product $2,$Product $3],[$A$,1\mathrm{.}50,3\mathrm{.}00,2\mathrm{.}00],[$B$,2\mathrm{.}00,1\mathrm{.}00,2\mathrm{.}50],[$C$,0\mathrm{.}25,0\mathrm{.}25,0\mathrm{.}25]]$

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 $3.$

The company also realizes that, in order to produce a specific type of product, it must set up a corresponding production facility which is associated with a setup cost and a maximum production capacity. It estimates that setup costs are $$400$ for product $1,$$$550$ for product $2,$ and $$600$ for product $3.$ The company also states that the facility's maximum production capacity for product $1$ is $175$ units, for product $2$ is $150$ units, and for product $3$ is $140$ units.

The company wants to determine how much of each product should be produced in order to maximize total profit contribution considering the possible setup costs.

$2\mathrm{.}1$ What is the mixed$-$integer linear program for this problem expressed in the mathematical form? Write down the entire mathematical model in the following space including the explicit and implicit constraints. Let $P_i=$ units of product