Production Planning $(20$ points$)|$

The Bogard Corporation produces three types of bookcases, which it sells to large office supply

companies. The production of each bookcase requires two machine operations, trimming and shaping,

followed by assembly, which includes inspection and packaging. All three types require $0\mathrm{.}4$ hours of

assembly time, but the machining operations have different processing times, as shown in the following

table, in hours per unit:

Each machine is available $150$ hours per month, and the current size of the assembly department provides

capacity of $600$ hours. Each bookcase produced yields a unit profit contribution as follows:

$(1)$ What is the optimal production quantity for "Standard" the company? $(10$ points$)$

$(2)$ What is the shadow price on the Trimmer? $(5$ points$)$

$(3)$ What is the price you are willing to pay for one more hour of the Shaper? $(5$ points$)$