Chapter 5. Probabilistic Inventory Control

1. Chapter 5. Probabilistic Inventory Control

2. Production Management 2 Probabilistic Inventory Control Demand is expressed by Definitions Lead Time Time interval between the time when an order is placed and the time when it is actually received. Constant, Deterministic, and/or Probabilistic Reorder Level (R) The inventory level at which orders are placed for replenishing the inventory. Function of the lead time Safety Stock (SS) The inventory that is carried to prevent stockout when there is uncertainty in demand.

3. Production Management 3 Probabilistic Inventory Control

4. Production Management 4 Probabilistic Inventory Control Customer Service Level Type 1 : Probability of no stockout in a cycle Type 2 : Proportion of demand that are met from stock (Fill rate) Example Type 1 Service Level = 0.2 Type 2 Service Level (Fill Rate) = (1450-55)/1450 = 0.96

5. Production Management 5 Probabilistic Inventory Control

6. Production Management 6 Probabilistic Inventory Control Newsboy Problem Very important model, which can be applied for fashion items, seasonal items, and so on. At the beginning of the season, company must decide how many units they have to purchase. Notations Expected Average Annual Cost

7. Production Management 7 Probabilistic Inventory Control Example Suppose the demand for Christmas trees during Christmas season are exponentially distributed with a mean of 100. The trees are purchased once. The sales revenue and purchasing cost are $200 and $100 per tree respectively. Any unsold trees must be disposed at the end of the season at a cost of $2/tree for waste treatment. A shortage of trees results in lost sales at a cost of $10/tree. How many trees do you purchase?

8. Production Management 8 Probabilistic Inventory Control Lot Size – Reorder Point Systems, (Q,R) System (s,Q), (s,S), or (r,Q) system Continuous Review : Demands are recorded as they occur Demand is random and stationary, and expected demand is D. There is a fixed positive lead time for placing an order. Notations

9. Production Management 9 Probabilistic Inventory Control Expected Average Annual Cost Solution Procedure

10. Production Management 10 Probabilistic Inventory Control Example Harvey Specialty Shop is a popular spot that specializes in international gourmet foods. One of the items that Harvey sells is a popular mustard that he purchases from an English company. The mustard costs Harvey $10 a jar and requires six-month lead time for replenishment of stock. Harvey uses a 20 percent annual interest rate to compute holding costs and estimates that if a customer requests the mustard when he is out of stock, the loss-of-goodwill cost is $25 a jar. Book keeping expenses for placing an order amount to about $50. During the six-month lead time, Harvey estimates that he sells an average of 100 jars, but there is substantial variation from one six-month period to the next. He estimates that the standard deviation of demand during each six-month period is 25. Assume that demand is described by a normal distribution. a) How should Harvey control the replenishment of the mustard? b) Determine safety stock, the average annual holding, setup, and shortage costs associated with the inventory control of the mustard, the average time between placement of orders, the proportion of order cycles in which no stock-outs occur, the proportion of demands that are not met.

11. Production Management 11 Probabilistic Inventory Control Service Levels in (Q,R) System The penalty cost is difficult to estimate. It is common practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are: Type 1 Service Level : Choose R so that the probability of not stocking out in the lead time is equal to a specified value. Type 2 service Level : Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value. Example Consider again Harvey’s Specialty Shop, Harvey feels uncomfortable with the assumption that the stock-out cost is $25 and decides to us a service level criterion instead. Suppose that he chooses to use a 98 % service objective.

12. Production Management 12 Probabilistic Inventory Control Example A company control its ordering policy using a continuous review policy. Basic information is as follows; K=$200, i=0.2, C=$10/unit, E(D)=20,000 units/year. Demand during lead time is normally distributed with mean 400 units and variance 2500 units. All demand out of stock is backordered at a cost of $8/unit. a) Find the EOQ without considering the shortage. b) What should be the reorder point to minimize average annual cost if a lot of obtained in a) is to be used? c) The present forecasting system costs $100/year. If a consultant offers you a new forecasting system which can reduce the variance of demand during lead time to 625 units, how much would you willing to pay for new forecasting system?

13. Production Management 13 Probabilistic Inventory Control Example The company stocks lumbers for construction. The demand rate is forecasted to be normally distributed wit mean 200 lumbers/week and the standard deviation is 100 lumbers/week. The purchasing cost is $30/lumber, the storage cost is $2/unit.year, and the annual inventory carrying cost rate is 0.20. the orders are delivered by truck at a cost of $100/delivery. All demand when out of stock would be backordered. The company would like to maintain a fraction of demand satisfied without backorder not less than 0.95. the company controls the inventory by a continuous review system, and the EOQ is used to determine the order quantity. The ordering lead time from the supplier is 2 weeks. (Assume 50 weeks per year.) a) Determine the EOQ, the safety stock, the reorder point and the average annual total cost. b) the supplier offers to deliver fixed quantity every week and charge $3,000 per year which include the trucking cost. Does the company accept the new offer? Why?

14. Production Management 14 Probabilistic Inventory Control Periodic Review System (S,T), (R,T), or (s,S,T) system Inventory levels are reviewed every T period, and place an order upto S.

15. Production Management 15 Probabilistic Inventory Control Expected Average Annual cost Solution Service Levels in (S,T) System Type 1 Service Level : Type 2 Service Level :

16. Production Management 16 Probabilistic Inventory Control Example Harvey Specialty Shop is a popular spot that specializes in international gourmet foods. One of the items that Harvey sells is a popular mustard that he purchases from an English company, and its inventory is reviewed every six months at a cost of $50. The mustard costs Harvey $10 a jar and requires six-month lead time for replenishment of stock. Harvey uses a 20 percent annual interest rate to compute holding costs and estimates that if a customer requests the mustard when he is out of stock, the loss-of-goodwill cost is $25 a jar. Book keeping expense s for placing an order amount to about $50. During the six-month lead time, Harvey estimates that he sells an average of 100 jars, but there is substantial variation from one six-month period to the next. He estimates that the standard deviation of demand during each six-month period is 25. Assume that demand is described by a normal distribution. a) Find the optimal order up to level. b) Determine safety stock, the average annual holding, setup and review, and shortage costs associated with the inventory control of the mustard, the proportion of order cycles in which no stock-outs occur, the proportion of demands that are not met.

17. Production Management 17 Probabilistic Inventory Control ABC Analysis ABC analysis is based on the Pareto Curve. Pareto discovered that the distribution of wealth follows an increasing exponential curve. A similar curve describes the distribution of the value of inventory items in a multi-item system. The value of a Pareto curve analysis in this context is that one can identify the items accounting for most of the dollar volume of sales. Rough guidelines are that the first 20% of the items account for 80% of the sales, the next 30% of the items account for 15% of the sales, and the last 50% of the items only account for 5% of the sales.