Formulate Linear Programming Problem and solve using QM

Workforce Allocation in Three Shifts:

A manufacturing company operates three shifts: Morning, Afternoon, and Night, and produces two products, $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash ).$ The company needs to meet weekly production targets of at least $1,200$ units of Product $\backslash ($ X $\backslash )$ and $900$ units of Product $\backslash ($ Y $\backslash ).$ The available workforce for each shift is limited$,$ with $180$ workers in the Morning shift, $150$ workers in the Afternoon shift, and $120$ workers in the Night shift. Each worker in the Morning shift can produce $4$ units of Product $\backslash ($ X $\backslash )$ or $3$ units of Product $\backslash ($ Y $\backslash ).$ In the Afternoon shift, a worker can produce $5$ units of Product $\backslash ($ X $\backslash )$ or $4$ units of Product $\backslash ($ Y $\backslash ).$ Workers in the Night shift produce $3$ units of Product $\backslash ($ X $\backslash )$ or $2$ units of Product Y$.$ Additionally, no more than $\backslash (25\backslash \%\backslash )$ of the total workforce, which amounts to a maximum of $112\mathrm{.}5$ workers, can be assigned to the Night shift to ensure balance in workforce distribution. The goal is to allocate workers optimally to both products in each shift to meet production targets while maximizing production efficiency and minimizing unused worker capacity.