An item is purchased for resale. Shortages are backordered at a cost of $$4$ per unit per year. The cost of carrying inventory is $$1$ per unit per year, based on average inventory. The annual demand is $10,000$ units per year. The fixed cost of ordering a batch of items is $$50,$ and the unit cost of purchasing an item is $$2$ if $0$ Q $2000,$ and $$1\mathrm{.}97$ if Q $2000.($a$)$ Determine the optimal values for the order quantity Q and the backorder quantity B $.($b$)$ Determine the total relevant annual cost