$2$ 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.

$21$ 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 induding the explicit and implicit constraints Let $P_1=$ units of product $)=(1,2,3$ produced, and $x$ is a binary decision variable with value $1$ if the production facility for product $($is set up and value $0$ otherwise $(20$ points$)$

Pige $2$ or $3$

$22$ Develop a spreadshet model by completing the missing parts indicated by the bordered cells except the six shaded cells in the provided Hartxlsx file and find the optimal solution using Excel Solver. Generate an answer report on a separate worksheet in the same Excel file. According to the optimal solution, how much of each product should be produced? $($Note: remember to submit the completed Hart xlss filealongside with your answer shiet on Canvas$)(35$ points$)$

$23.$ Among the three types of products, which products are produced as the corresponding production facilities are set up in the optimal solution? $(5$ points$)$

$24$ What is the total profit contribution Hart Manufacturing can earn with the optimal solution? How many hours of actual production time will be scheduled in each department? $(5$ points$)$

$25$ Among the thme time constraints $($hours for department A$,$ hours for department B$,$ and hours for Department C$),$ which constraint$($s$)$ are binding? What is the slack time in each department? $(5$ points$)$