Question $3(4$ points$)$:

Omega Engineering produces three main components: circuit boards, batteries, and casings. The company aims to optimize the daily production of the three components, The operational constraints and quality targets for a single production day are as follows:

Z$,$ Circuit boards: Target production of $300$ units, with a quality defect rate not exceeding $2\%.$

$2_2$ Batteries: Target production of $200$ units, with a quality defect rate not exceeding $1\mathrm{.}5\%$

Casings: Target production of $100$ units, with a quality defect rate not exceeding $3\%.$

$x_3.$ The total daily production capacity ranges from $500$ to $600$ units due to machine and labor limitations$.$

Inventory space constraints limit the storage of excess components to a maximum of $50$ units across all types.

Use the GP method to optimize the problem taking these penalties into consideration:

Quality deviations carry a penalty of $10$ points per unit of deviation.

Production shortfall carries a penalty of $5$ points per unit of deviation.

Excess inventory carries a penalty of $3$ points per unit of deviation.