$6-68.$ Rob Roller has been in charge of inventory policy at Cyclorama, a large retail bicycle shop in Orlando, Florida. He now orders Chrome$-$Moly frames from Frameco, a local frame supplier. Cyclorama builds each bike by adding different Shimano component groups to each frame. Currently, cyclorama's annual demand for frames is $2,000$ per year. The lead time is $10$ days, and the ordering cost per order is $$50.$ The holding cost is estimated to be $25\%$ of the unit cost. Frameco offers the following discounts:

$\backslash$table$[[,\backslash$table$[[$LOWER$],[$QUANTITY$],[$PRICE BREAK$]],\backslash$table$[[$UPPER$],[$QUANTITY$],[$UNITS$]],$UNIT PRICE $($$$)],[1,0,10,220\mathrm{.}00],[2,11,20,219\mathrm{.}99],[3,21,30,219\mathrm{.}98],[4,31,40,219\mathrm{.}97],[5,41,50,219\mathrm{.}96],[6,51,60,219\mathrm{.}95],[7,61,70,219\mathrm{.}94],[8,71,80,219\mathrm{.}93],[9,81,90,219\mathrm{.}92],[10,91,100,219\mathrm{.}91],[11,101,110,219\mathrm{.}90],[12,111,120,219\mathrm{.}89],[13,121,130,219\mathrm{.}88],[14,131,140,219\mathrm{.}87],[15,141,,219\mathrm{.}86]]$

$($a$)$ What are the optimal order quantity and the total inventory cost for Cyclorama given the data?

$($b$)$ Rob is optimistic about future demand. If annual demand becomes $3,000$ frames, what is the impact on the optimal order quantity?

$($c$)$ What is the impact if the annual demand is $4,000$ frames?

$($d$)$ In general, what happens to the order quantity and total inventory cost as demand increases?