The first (R,Q) model in this section assumes that the total shortage cost is proportional to the amount of demand that cannot be met from on-hand inventory. Similarly, the second model assumes that the service level constraint is in terms of the fill rate, the fraction of all customer demand that can be met with on-hand inventory. Consider the following variations of these models. The first, labeled model 3, assumes that a shortage cost is incurred on every order cycle that experiences a stockout. This cost is independent of the size of the stockout. The second model, labeled model 4, prescribes a service level constraint but now on the fraction of order cycles that experience no stockouts.
a. In each of these new models, you need to calculate the probability of a stockout during an order cycle. This is the probability that the demand during lead time is greater than the safety stock. Assuming that demand during lead time is normally distributed, how can this probability be calculated?
b. Given your method in part a, solve the model from Example 12.7 when the cost of having a shortage in any cycle is $100, and all other parameters are as before. What are the optimal reorder point and the safety stock level?
c. Continuing part b, what model 4 service level constraint is this $100 stockout cost equivalent to?
Example 12.7
ORDERING CAMERAS WITH UNCERTAIN DEMAND AT MACHEY'S
I n Example 12.1, we considered Machey's department store, which sells, on average, 1200 cameras per year. The store pays a setup cost of $125 per order, and the holding cost is $8 per camera per year. It takes one week for an order to arrive after it is placed. In that example, the optimal order quantity Q was found to be 194 cameras. Now we assume that the annual demand is normally distributed with mean 1200 and standard deviation 70. Machey's wants to know when to order and how many cameras to order at each ordering opportunity.
Objective To find the (R,Q) policy that minimizes the company's expected annual cost.
Example 12.1
ORDERING CAMERAS AT MACHEY'S
Machey's Department Store sells 1200 cameras per year, and the demand pattern throughout the year is very steady. The store orders its cameras from a regional warehouse, and it usually takes one week for the cameras to arrive after an order has been placed. Each time an order is placed, an ordering cost of $125 is incurred. The store pays $100 for each camera and sells them for $130 apiece. There is no physical storage cost, but the store's annual cost of capital is estimated at 8% per year-that is, it can earn 8% on any excess cash it invests. The store wants to determine how often it should order cameras, when it should place orders, and how many cameras it should order in each order.
Objective To determine when to order and how much to order so that the store never runs out of cameras and profit is maximized.