Sam's Cat Hotel operates $52$ weeks per year, $6$ days per week, and uses a continuous review inventory system. It purchases kitty litter for $$11\mathrm{.}70$ per bag. The following information is available about these bags.

Demand $=90$ bags$/$week

Standard deviation of weekly demand $=15$ bags

Order cost $=$ $$54$ per order

Annual Holding cost $=27\%$ of cost

Desired cycle$-$service level $=80\%$

Lead time $=3$ weeks

$($Note : Exclude costs due to Safety Stock for all the questions in this problem$)$

$1.$What is the EOQ?

$2.$ What is the time between orders in weeks?

$3.$ What is the z$-$score for the desired cycle$-$service level of $80\%?$

$4.$ What is the Reorder Point?

$5.$ The current on$-$hand inventory is $320$ bags, with no open orders or backorders. An inventory withdrawal of $10$ bags was just made. Is it time to reorder?

$6.$ What is the annual total cost if the store currently uses a lot size of $500$ bags?

$7.$ What would be the annual cost saved by shifting from $500-$bag lot size to the EOQ?

$8.$ Suppose that the weekly demand forecast of $90$ bags is incorrect and actual demand averages only $60$ bags per week. How much higher will total costs be owing to the distorted EOQ caused by forecast error?

$9.$ Suppose that actual demand is $60$ bags but that ordering costs are cut to only $$6$ by using the internet to automate order placing. However, the buyer does not tell anyone, and the EOQ is not adjusted to reflect this reduction in $"$S$".$ What is the cost penalty for not updating the EOQ?