Wild West produces two types of cowboy hats. A Type $1$ hat requires twice as much labor time as a Type $2.$ If all the available labor time is dedicated to Type $2$ alone, the company can produce a total of $400$ Type $2$ hats a day. The respective market limits for the two types are $150$ and $200$ hats per day. The revenue is $$8$ per Type $1$ hat and $$5$ per Type $2$ hat.

$($a$)$ Use the graphical solution to determine the number of bats of each type that maximizes revenue.

$($b$)$ Determine the dual price of the production capacity $($in terms of the Type $2$ hat$)$ and the range for which it is applicable.

$($c$)$ Determine the optimality range for the unit revenue ratio of the two types of hats that will keep the current optimum unchanged.