$1.$ A mechanical department has back log mechanical work of $500$ man$-$hours. The mechanical work is classified in three types. Priority $1,$ Priority $2$ and Priority $3.$ The classification is based on the importance of the work and its impact on the plant operation. Out of the $500$ manhours $100$ are priority $1$ and $150$ priority $2$ and $250$ priority $3.$ There are three types of sources who can provide workers to do the work. The sources are in house hired workers, over time workers who are in house hired but allowed to work overtime. The in house hired are of two type skilled and average workers. The third source is outsource workers who are classified as skilled workers. The man hour for skilled in house hired worker is $50$ Saudi Riyals $($SR$)$ and for average craft costs $30$ SR$.$ The over time hour from the in house workers costs $30\%$ more from the salary they get when worked in regular time. The over time workers only work on priority $1$ and $2$ work only. The outsourced workers hour costs $80$ SR$.$ The outsourcing places have only skilled workers and they do not work over$-$time. In addition they are hired to work on priority $1$ and $2$ work only. The productivity of in house skilled workers are $0\mathrm{.}70,0\mathrm{.}75$ and $\mathrm{.}85$ for priority $1,$ priority $2$ and priority $3$ work respectively. The productivity of in house average worker are $0\mathrm{.}55,0\mathrm{.}65$ and $0\mathrm{.}80$ for priority $1,$ priority $2$ and priority $3$ work respectively. The productivity of the outsources workers productivity are $0\mathrm{.}75,0\mathrm{.}85$ and $0\mathrm{.}87$ for priority $1,$ priority $2$ and priority $3$ work respectively. Formulate a linear programming problem to find the optimal mix of workers to accomplish the work that minimizes the total cost. $[20].$