Problem $1(30$ points$)$

Uzay's Electronics Company needs $to$ produce $2,000$ units $of$ its new "Smart Sound System."

There are three methods $to$ produce the product: manual assembly, automated assembly, and

outsourcing $to$ a third$-$party manufacturer. The requirements for each method are $as$ follows:

Manual assembly requires $2$ hours $of$ skilled labor, $10$ hours $of$ unskilled labor, and $1\mathrm{.}5$ hours $of$

machine time per unit. Automated assembly requires $4$ hours $of$ skilled labor, $5$ hours $of$

unskilled labor, and $1$ hour $of$ machine time per unit. Outsourcing requires $no$ labor $or$ machine

time but costs a flat $$20$ per unit.

The company has available, $6,000$ hours $of$ skilled labor; $15,000$ hours $of$ unskilled labor; $3,000$

hours $of$ machine time.

Additionally, the company's contract specifies that $at$ least $80\%of$ all units must $be$ produced $in-$

house $(manuallyor$ automatically$)$ due $to$ quality targets, and the amount $of$ skilled labor time

used must $beat$ least $25\%of$ the total labor time $(unskilled+$ skilled$).$

The production costs are: Manual assembly: $$18$ per unit, Automated assembly: $$16$ per unit,

Outsourcing: $$20$ per unit.

Formulate a linear programming model $to$ minimize the total cost $of$ producing $2,000$ Smart

Sound Systems, considering all resource and contractual constraints. Define the decision

variables, write the objective function and the constraints, and provide a short expression for

each constraint $to$ best describe $it.Do$ not solve.