JL$\mathrm{.}53$ Bob's Bumpers has a repetitive manufacturing facility in Kentucky that makes automobile bumpers and other auto body parts. The facility operates $270$ days per year and has annual demand of $51,000$ bumpers. They can produce up to $430$ bumpers each day. It costs $$59$ to set up the production line to produce bumpers. The cost of each bumper is $$113$ and annual holding costs are $$27$ per unit. Setup labor cost is $$30$ per hour.

What is the optimal size of the production run for bumpers? $($Display your answer to the nearest whole number.$)$

Based on your answer to the previous question, and assuming the manufacturer holds no safety stock, what would be the average inventory for these bumpers? $($Display your answer to the nearest whole number.$)$

Based on your answer two questions back, how many production runs will be required each year to satisfy demand? HINT: As a general rule, whenever calculating a value that is based on previous calculations in Excel, always be sure to use cell references rather than a rounded value as a calculation input. $($Display your answer to the appropriate whole number.$)$

Suppose the customer $($an auto manufacturer$)$ wants to purchase in lots of $350$ and that Bob's Bumpers is able to reduce setup costs to the point where $350$ is now the optimal production run quantity. How much will they save in annual holding costs with this new lower production quantity? $($Display your answer to two decimal places.$)$

How much will they save in annual setup costs with this new lower production quantity? $($Display your answer to two decimal places.$)$