QUESTION $22(10$ points$)-$ Fixed$-$Quantity Inventory Systems $($FQS$)$

ComfyShoes is an online retailer which carries various shoe lines. For simplicity, we are only

interested in the number of pairs of shoes $($the unit is a pair of shoes$),$ not in differences

amongst lines and sizes. The company uses a fixed$-$quantity inventory system $($FQS$)$ to manage

their stock. Following is key information either current or averaged over years of operations:

Average demand $840$ pairs per week

Lead time $2$ weeks

Order cost $$210$ per order

Unit cost $$16$ for a pair of shoes

Annual inventory holding cost $27\%$ for the whole year

Standard deviation of daily demand $14$ pairs per day

Desired service level with safety stock $95\%$

Current on$-$hand inventory $1762$ pairs of shoes

Current scheduled receipts $0$ pairs of shoes

Current backorders $13$ pairs of shoes

Weeks of operation $51$ weeks

Days of operation $7$ days in a week

Backorders happen because of shortages in certain sizes. A restock order would include all sizes.

You would need to show equations, steps, and the final results with units for full credits. Time

units should be carefully treated. Goods quantities should be rounded up to whole numbers.

$($a$)[2]$ Find the Economic Order Quantity $($EOQ$)($rounded up to a whole number$).$ Find the

average total annual cost $($TAC$)$ for the EOQ $(2$ decimal places$).$

$($b$)[1]$ Given the EOQ, find the average number of orders made in a year $(2$ decimal places$).$

What is the average time between orders $($in days with $2$ decimal places$)?$

$($c$)[1]$ Find the current inventory position $($IP$).$

$($d$)[3]$ Find the reorder point without safety stock R$($AD$).$ State the ordering rule. Based on the

current IP$,$ determine if the company would make an order and state the order quantity. Sketch

the normal demand distribution in the lead time to illustrate the service level. Then, loosely

sketch how the system would operate over time $($movements of IP over time$).$

$($e$)[3]$ Find the reorder point with safety stock R$($ST$).$ State the ordering rule. Based on the

current IP$,$ determine if the company would make an order and state the order quantity. Sketch

the normal demand distribution in the lead time to illustrate the service level. Then, loosely

sketch how the system would operate over time $($movements of IP over time$).$

Hint: Use the following table to find the z$-$value corresponding to the desired service level.

$\backslash$alpha v $0\mathrm{.}800\mathrm{.}850\mathrm{.}900\mathrm{.}950\mathrm{.}99$

z $=$ NORM.S$.$INV$(\backslash$alpha v$)0\mathrm{.}841621\mathrm{.}036431\mathrm{.}281551\mathrm{.}644852\mathrm{.}32635$

$2$