Multi$-$product plem in part $($a$).$

Each prod Production Planning. Hart Manufacturing makes three products.

labor requires manufacturing operations in three departments: A$,$ B$,$ and C$.$ The 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 $3.$

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

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?

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?

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