Question

A music store is restocked weekly. The weekly demand for a best-selling CD is normal, with mean of 200 and standard deviation of 50. These CDs cost $8.00 per piece, and the store earns $4.50 per CD sold. Excess demand is lost (salvage value is $0); Customers go to a competitor's store rather than wait for resupply.

Suppose that exactly 10% of the start-of-week inventory of this CD disappears from the store without being sold; this is euphemistically known as a 10% shrinkage rate. Now, what order quantity maximizes expected profit?

Answer 1

Since 10% of the start of the week inventory is lost , cost of 100 CDs are built on 90 CDs

Therefore , effective cost of each CD =Cost of each CD / 0.9 = $8/0.9 = $ 8.89 ( rounded to 2 decimal places )

From the given data :

Effective Cost of each CD = C = $8.89

Selling price of each CD = P = $ 8 + $4.50 = $12.50

Salvage value of each CD = S = 0

Therefore ,

Underage cost , Cu = P – C = $12.50 - $8.89 = $3.61

Overage cost , Co = C – S = $8.89 – 0 = $8.89

Therefore , critical ratio = Cu/Cu + Co= 3.61/ ( 3.61 + 8.89 ) = 3.61 / 12.5 = 0.2888

Critical ratio is the in stock probability of the optimum order quantity

Therefore, in stock probability = 0.2888

Corresponding Z value = NORMSINV ( 0.2888) = - 0.5568

Optimum order quantity which maximizes profit

= Mean demand + Zvalue x standard deviation of demand

= 200 - 0.5568 x 50

= 200 – 27.84

= 172.16 ( 172 rounded to nearest whole number )

ORDER QUANTITY THAT WILL MAXIMIZE PROFIT = 172