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

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

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

he 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 $($manually or automatically$)$ due to quality targets, and the amount of skilled labor time

used must be at 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.

Consider the following solution:

$600$ units are produced manually,

$1100$ units are produced automatically,

$300$ units are outsourced.

Is this solution feasible? If yes, determine how much skilled labor, unskilled labor, and

machine time are unused.