1. Statistical Inventory control models I
(Q, r) model

2. Learning objective After this class the students should be able to:
Apply optimization technique to inventory model
calculate the appropriate order quantity in the face of uncertain demand.
Derive the (Q, r) model from the integration of the EOQ model and the EPL model.
analyze the implication of the (Q, r) model

3. Time management
The expected time to deliver this module is 50 minutes. 30 minutes are reserved for team practices and exercises and 20 minutes for lecture.

4. The Base Stock Model
Consider the situation facing an appliance store that sells a particular model of refrigerator. Because space is limited and because the manufacturer makes frequent deliveries of other appliances, the store finds it practical to order replacement refrigerators each time one is sold. In fact, they have a system that places purchase orders automatically whenever a sale is made. But, because the manufacturer is slow to fill replenishment orders, the store must carry some stock in order to meet customer demands promptly. Under these conditions, the key question is how much stock to carry.

5. Assumptions Demands occur one at a time;
Any demand not filled from stock is backordered;
Replenishment lead times are fixed and known;
There is no setup cost associated with placing an order; and
There is no constraint on the number of orders that can be placed per year.
Last two assumptions imply that there is no incentive to replenish stock in anything other than one-at-a-time fashion.

6. Notation

7. The question
We place an order when there are r units in stock and expect to incur demand for θ units while we wait for the replenishment order to arrive. Hence, r - θ is the amount of inventory we expect to have on hand when the order arrives. If s = r - θ > 0, then we call this the safety stock for this system, since it represents inventory that protects it against stockouts due to fluctuations in either demand or deliveries. Since finding r - θ is equivalent to finding r (because θ is a constant), we can view the problem either as finding the optimal base stock level (R = r -1), reorder point (r), or safety stock level (s = r - θ).

8. Solution
Since an order is placed every time a demand occurs, the relationship “on-hand + purchase orders - backorders = R” holds at all times.
Because lead times are constant, we know that all of the other R-1 items in inventory and on order will be available to fill new demand before the order under consideration arrives.
Therefore, the only way the order can arrive after the demand for it has occurred is if demand during the replenishment lead time is greater than or equal to R (i.e., X ≥ R).

9. Solution
Hence, the probability that the order arrives before its demand (i.e., does not result in a backorder) is given by P(X < R). If demand has a continuous distribution then
P(X < R) = P(X ≤ R-1) = G(R).
However, if demand has a discrete distribution (i.e., X can take on only integer values and has a probability mass function instead of a density function), then
P(X < R) = P(X ≤ R-1) = G(R-1)=G(R).

10. Solution
Since all orders are alike with regard to this calculation, the fraction of demands that are filled from stock is equal to the probability that an order arrives before the demand for it has occurred, or
Hence, G(R) or G(r) represents the fraction of demands that will be filled from stock (i.e., the fill rate).

11. The (Q, r) model
Consider the situation where inventory is monitored continuously and demands occur randomly, possibly in batches. When the inventory level reaches (or goes below) r, an order of size Q is placed. After a lead-time of ℓ, during which a stockout might occur, the order is received. The problem is to determine appropriate values of Q and r. The model we use to address this problem is known as the (Q, r) model

12. Example
The manager of a maintenance department must stock spare parts to facilitate equipment repairs.
Demand for parts is a function of machine breakdowns, therefore random.
Unlike the base stock model, the costs incurred in placing a purchase order (for parts obtained from an outside supplier) or the costs associated with setting up the production facility (for parts produced internally) are significant enough to make one-at-a-time replenishment impractical.
Thus, the maintenance manager must determine not only how much stock to carry (as in the base stock model), but also how many to produce/order at a time (as in the EOQ and newsboy models).

13. Assumptions
From a modeling perspective, the (Q, r) model is identical to the base stock model, except that we will assume that either
There is a fixed cost associated with a replenishment order.
There is a constraint on the number of replenishment orders per year.
Therefore, replenishment quantities greater than one may make sense.

14. Basic mechanics of the (Q, r) model
Demands occur randomly, possibly in batches. When the inventory level reaches (or goes below) the reorder point r, a replenishment order for quantity Q is placed. After a (constant) lead time of ℓ, during which a stockout might occur, the order is received. The problem is to determine appropriate values of Q and r.

15. Total cost
In some sense, the (Q, r) model represents the integration of the EOQ model and the base stock model two models. Then, the total cost is
Where n(r) is the expected number of backorders that will be placed during a cycle

16. Optimal replenishment quantity
The optimal replenishment quantity Q*, and reorder point r*, can be found by simultaneously solving the following equations:

17. Reflections
Each team is invited to analyze the following insights, based on the statistical model (10) minutes):
“Cycle stock increase as replenishment frequency decrease”
“Safety stock provide a buffer against stockout”

18. Quantitative insights
Increasing the average demand during a replenishment lead time (θ) will tend to increase the optimal reorder point. Note that either increasing the annual demand D, or increasing the replenishment lead time ℓ, will serve to increase the expected demand during replenishment lead time θ. The implication is that either high demand or long replenishment lead times will tend to require more inventory in stock.

19. Basic insight
Increasing the variability of the demand process will tend to increase the optimal reorder point. The key insight here is that a highly variable demand process will require more safety stock as protection against stockouts than a very stable demand process.

20. Basic insight
Increasing the holding cost will tend to decrease the optimal replenishment quantity and reorder point. The holding cost can be increased by either increasing the cost of the item, the interest rate associated with inventory, or the noninterest holding costs (e.g., handling and spoilage). The point is that the more expensive it is to hold inventory, the less we should hold.

21. Reflections
Suppose you are stocking parts purchased from vendors in a warehouse. How could you use a (Q, r) model to determine whether a vendor of a part with a higher price but a shorter lead time is offering a good deal? What other factors should you consider in deciding to change vendors. (10 minutes)

22. Reflections
What is the key difference between the EOQ model and the (Q, r) model? Between the base stock model and the (Q, r) model?

23. Reference
Factory Physics. Hopp & Spearmen, Irwin, Chapter 2, p