Question

An online clothing retailer sells men’s white undershirts. In each week, the demand for these undershirts has a normal distribution with mean 2,500 and standard deviation 600. At the end of each week, the retailer observes its inventory position and decides whether or not to place an order, and if so, for how many units (so this is a periodic review system). When the retailer orders more shirts from its supplier, it receives them after a lead time of 3 weeks (don’t forget to add a week to get the effective lead time!). If the retailer doesn’t have enough inventory on hand to fulfill a customer’s order, the order is backlogged and then fulfilled once the retailer has sufficient stock on hand. When an order is backlogged, the retailer compensates each affected customer with a discount of $1 per unit ordered (so, if a customer ordered 3 shirts, they would receive a discount of $3 on their order). The inventory holding/carrying cost for one unit is $5 per year.

(a) (1 point) What should be the target fill rate for these undershirts? Remember to account for and/or convert to matching time scales when doing your calculations.

(b) (1 point) What is the average lead time demand?

(c) (1 point) What is the standard deviation of lead time demand?

(d) (2 points) What is the optimal base-stock level s ∗ ?

Answer 1

Weekly demand = 2500

Annual Demand = 2500*52

Standard Deviation , = 600

Lead Time = 3 weeks

Backorder cost, B_c = $ 1

Annual holding cost, H = $5 per unit

Q* = 606.32

a)

Max probability of Stockout = (H/(H+B_c) = ( 2.5/2.5+1) = 0.71428

Target fill rate = 71.428%

b)

Average lead time demand = 3 weeks * 2500 = 7500

c)

Standard deviation of lead time demand = 600 units

d)

Optimal base stock = Average lead time demand + zscore * Standard deviation

Z score for fill rate 71.428% is 0.565949

Optimal Base stock = 7500 + (0.565949*600) = 7839.569