Question

A large toy company Müttel currently allows toy retailers to place orders with delivery in 2 weeks. The Gigantic Pocket Monster (Gipokmon) is a new toy that Müttel has introduced. Demand per week for the toy at one of their stores is estimated to be normally distributed with a mean of 25 units and standard deviation of 5 units. Assume TOYS-are-MINE uses the order-up-to model to plan orders and deliveries to this store.

1. Suppose TOYS-are-MINE uses an order-up-to level of 20. After receiving their delivery for this week, they have 2 units on-hand. Last week’s order was for 5 units. How many units will they order this week?

2. Again, suppose they use an order-up-to level of 20 for this store. On average, how many units will this store have on-order?

3. Suppose an order-up-to level of 90 is established. What is the resulting instock probability?

4. Suppose an order-up-to level of 100 is established. What would be the expected end-of-period on-hand inventory of Gipokmons?

Answer 1

1) Inventory position = On hand quantity + On order quantity = 2 + 5 = 7

Order quantity = Order upto level - Inventory = 20 - 7 = 13

2) Expected On-order quantity = Expected demand in one period * Lead time = 25*2 = 50

3)

Demand over l+1 periods, m = 25*(2+1) = 75

Std dev of demand over l+1 periods, x = 5*SQRT(2+1) = 8.66

Order upto level = m + z*s = 75 + z*8.66 = 90 (given)

Therefore, z = (90-75)/8.66 = 1.7321

P(z) = NORMSDIST(1.7321) = 0.9584  

Therefore, in-stock probability = 95.84 %

4) For order upto level of 100, z = (100-75)/8.66 = 2.89

Lookup L(z) in the standard normal loss function table,

L(z) = 0.000567

Expected backorder = s*L(z) = 8.66*0.000567 = 0.0049

Expected on-hand inventory = Order upto level - Expected demand over (l+1) periods + Expected backorder

= 100 - 25*(2+1) + 0.0049

= 25