1. EOQ: How much to order; ROP: When to order Re-Order Point (ROP) in periodic inventory system is the start of the period. Where is ROP in a perpetual system? ROP in Perpetual System: Quantity on hand drops to a predetermined amount. If there were no variations in demand, and demand was really constant; ROP is when inventory on hand is equal to the [average] demand during lead time. Lead Time is the time interval from placing an order until receiving the order. When to re-order with EOQ Ordering

2. The greater the variability of demand during lead time, the greater the need for additional inventory to reduce the risk of shortage (i.e., the greater the safety stock). When variability exists in demand or lead time, actual demand during lead time may exceed average demand during lead time. If ROP is equal to average demand during lead time, there is 50% probability of satisfying all the demand. But we want this probability to be greater than 50%. We prefer 90%, 95%, or 99%. Safety Stock (SS): serves to reduce the probability of a stockout during the lead time. ROP in perpetual inventory control is the inventory level equal to the average demand during the lead time + a safety stock (to cover demand variability) When to re-order with EOQ Ordering

3. Example 1 Average demand for an inventory item is 200 units per day, lead time is three days, there is no variation in demand and lead time and therefore, safety stock is zero. What is the reorder point? In a perpetual inventory system, the ROP is the point at which inventory on hand is equal to the [average] demand in lead time plus safety stock. ROP = 3(200) Whenever inventory level drops to 600 units we place an order. We receive the order in 3 days. Because there is no variations in demand, there is no stock out.

4. Example 2 Average demand for an inventory item is 200 units per day, lead time is three days, and safety stock is 100 units. What is the reorder point In a perpetual inventory system, the ROP is the point at which inventory on hand is equal to the average demand in lead time plus safety stock. ROP = 3(200)+100 Whenever inventory level drops to 700 units we place an order. Since there is variation in demand, during the next three days, we may have a demand of 500, 600, 700, or even more.

5. Time Inventory Demand During Lead Time

6. LT ROP when demand during lead time is fixed

7. Time Inventory Demand During Lead Time

8. Demand During Lead Time is Variable

9. Inventory Time Demand During Lead Time is Variable

10. Average demand during lead time A large demand during lead time ROP Time Quantity Safety stock reduces risk of stockout during lead time Safety Stock Safety stock LT

11. ROP Time Quantity Safety Stock LT

12. When to re-order (ROP) Demand during lead time has Normal distribution. We can accept some risk of being out of stock, but we usually like a risk of less than 50%. If we order when the inventory on hand is equal to the average demand during the lead time; then there is 50% chance that the demand during lead time is less than our inventory. However, there is also 50% chance that the demand during lead time is greater than our inventory, and we will be out of stock for a while. We usually do not like 50% probability of stock out

13. RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Safety Stock and ROP Each Normal variable x is associated with a standard Normal Variable z x is Normal (Average x, Standard Deviation x)  z is Normal (0,1) There is a table for z which tells us Given any probability of not exceeding z. What is the value of z Given any value for z. What is the probability of not exceeding z Average demand

14. Common z Values RiskService levelz value 0.1 0.9 1.28 0.05 0.951.65 0.01 0.99 2.33 RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand

15. Relationship between z and Normal Variable x RO P Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand z = (x-Average x)/(Standard Deviation of x) x = Average x +z (Standard Deviation of x) μ = Average x σ = Standard Deviation of x RiskService z value level 0.1 0.9 1.28 0.05 0.95 1.65 0.01 0.99 2.33  x = μ +z σ

16. Relationship between z and Normal Variable ROP ROP Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale Average demand LTD = Lead Time Demand ROP = Average LTD +z (Standard Deviation of LTD) ROP = Average LTD +ss ss = z (Standard Deviation of LTD)

17. Safety Stock and ROP ROP Risk of a stockout Service level Probability of no stockout Safety stock 0z Quantity z-scale RiskService levelz value 0.1 0.91.28 0.05 0.95 1.65 0.01 0.99 2.33 Average demand ss = z × (standard deviation of demand during lead time)

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