$($Inventory Management $-$ Part $1)$ 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 $$1\mathrm{.}97Q2000QB0,$ and $$1\mathrm{.}97ifQ2000.$

$(a)$ Determine the optimal values for the order quantity $Q$ and the backorder quantity $B.$

$(b)$ Determine the total relevant annual cost.