$1.(35$ pts $.)$ A manufacturing company produces three types of products: $\backslash ($ P$\_\{1\},$ P$\_\{2\}\backslash )$ and $\backslash ($ P$\_\{3\}\backslash ).$ The company wants to determine the optimal production levels of each product to maximize revenue, minimize production costs, and fulfill specific production goals. However, due to resource limitations$,$ achieving all goals may not be possible. The company wants to apply goal programming to address these conflicting objectives.

Product Data:

Constraints and Goals:

$1.$ Resource Constraints: Machine hours available per week: $500$ hours.

$2.$ Goal $1($Hlighest Priority$)$: The company wants to produce at least $80$ units of $\backslash ($ P$\_\{$l$\}\backslash ).$

$3.$ Goal $2($Second Priority$)$: Achieve a total weekly revenue of at least $\backslash (\backslash$$ $20,000\backslash ).$

$4.$ Goal $3($Third Priority$)$: Keep total production costs below $\backslash (\backslash$$ $8,000\backslash ).$

$5.$ Goal $4($Fourth Priority$)$: Production levels of $\backslash ($ P$\_\{2\}\backslash )$ should not exceed production levels of $\backslash ($ P$\_\{3\}\backslash )$ by more than $20$ units.

Model the problem as a Goal Programming problem. $($DO NOT plot graph, DO NOT attempt to solve$)$