Based on the above North Face retail store question, the store manager implemented a new ordering system that reduces the fixed cost of a replenishment order from $$30$ to only $$6.$ Please answer the following questions and show your calculations step by step

$(1)$ What is the new optimal order quantity?

$(2)$ What is the cycle inventory $($or average inventory$)?$

$(3)$ What is the annual ordering and holding costs?

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$54$

Annual demand $(D)=50012=6000$

purchasing cost per unit $(C)=\frac{100}{?}$ unit

fixed ordering cost per order $(S)=\frac{6}{?}$ order

Holding rate $(h)=25\%$

$=\sqrt[\phantom{2}]{2S\frac{D}{h}C}Q=\frac{\sqrt[\phantom{2}]{266000}}{25}\%100$