Frequency & Safety Stock

1. Frequency & Safety Stock April 12, 2005

2. Frequency and Safety Stock Continuous Review – EOQ Periodic Review – Order-up-to With a forecast – Safety lead time Agenda

3. Assumptions Fixed ordering cost Orders can be placed at any time Relatively constant, but uncertain demand Variable lead time Often called (Q, r) policy Q is order quantity r is the re-order point Continuous Review Model

4. Basic tool to manage risk Continuous Review Basics Safety stock = number of standard deviations in lead time demand Order when inventory hits the Average Lead time demand + safety stock Order the EOQ Stock on hand Lead Time Reorder Point Actual Lead Time Demand Order placed Avg LT Demand Safety Stock Time

5. Lead time demand has std deviation s Safety stock levels Choose z to get correct probability that lead time demand exceeds safety stock zs Safety Stock Basics

6. What is the impact of increasing the frequency of orders in this context? Inventory costs? Ordering costs? Service level if we hold safety stock constant? Question

7. Basic tool to manage risk Continuous Review Basics Experience risk MORE often!!! Should you increase your safety stock? Stock on hand Lead Time Reorder Point Actual Lead Time Demand Order placed Avg LT Demand Safety Stock Time

8. Standing contract with carrier or supplier E.g., weekly shipment Daily replenishment Every 4 hours replenishment Order-up-to Policies Bring the inventory up to a target level with each shipment Periodic Review Models

9. Order Quantity Actual Lead Time Demand Actual Lead Time Demand Actual Lead Time Demand How much stock is available to cover demand in this period? Order Up To Policy Target pipeline inventory level Reorder Point Reorder Point Actual Lead Time Demand Stock on hand Lead Time Order placed Time

10. Order Quantity Order Up To Policy: Inventory Reorder Point Reorder Point On Average this is the Expected demand between orders On Average this is the safety stock Stock on hand Time So average on-hand inventory is DT/2+ss

11. Order Quantity Order Up To Policy: Inventory Reorder Point Reorder Point After an order is placed, it is the Order up to level Before an order is placed it is smaller by the demand in the period Stock on hand Time So average Pipeline inventory is OUL – DT/2

12. Probability of stock out is the probability demand in T+L exceed the order up to level, S Set a time unit, e.g., days T = Time between orders (fixed) L = Lead time, mean E[L], std dev sL Demand per time unit has mean D, std dev sD Assume demands in different periods are independent Let sDdenote the standard deviation in demand per unit time Let sLdenote the standard deviation in the lead time. Safety Stock in Periodic Review

13. Probability of stock out is the probability demand in T+L exceed the order up to level, S Expected Demand in T + L D(T+E[L]) Variance in Demand in T+L (T+E[L])sD2 +D2sL2 Order Up to Level: S= D(T+E[L]) + safety stock Question: What happens to service level if we hold safety stock constant, but increase frequency? Safety Stock in Periodic Review

14. What if we double frequency, but hold safety stock constant? Expected Demand in T/2 + L D(T/2+E[L]) Variance in Demand in T/2+L (T/2+E[L])sD2 +D2sL2 Order Up to Level: S = D(T/2+E[L]) + safety stock But now we face the risk of failure twice as often Impact of Frequency This is reduced byTsD2/2

15. Time period is a day Frequency is once per week T = 7 Daily demand Average 105 Std Dev 67 Lead time Average 2 days Std Dev 1 days Expected Demand in T+L D (T + E[L]) = 105 (7 + 2) = 945 Variance in Demand in T+L (T+E[L])sD2 +D2sL2 = (7+2)*672 + (1052)*22 =40,401 + 44,100 = 84,501 Std Deviation = 291 Example

16. Expected Demand in T+L D (T + E[L]) = 105 (7 + 2) = 945 If we ship twice a week this drops to 578 If we ship thrice a week this drops to 456 Variance in Demand in T+L (T+E[L])sD2 +D2sL2 = (7+28)*672 + (1052)*22 =40,401 + 44,100 = 84,501 Std Deviation = 291 If we ship twice a week this drops to 262 If we ship thrice a week this drops to 252 Example Cont’d

17. With weekly shipments: To have a 98% chance of no stockouts in a year, we need .9996 chance of no stockouts in a week .999652 ~ .98 With twice a week shipments, we need .9998 chance of no stockouts between two shipments .9998104 ~ .98 With thrice a week shipments, we need .9999 chance of no stockouts between two shipments .9999156 ~ .98 Note that if Lead time is greater than T this is very conservative. Example Cont’d

18. Overlapping Risk T+ L

19. Assume Demand in L+T is Normal Hold risk constant 98% chance of no shortages all year Example Cont’d

20. When lead time is long relative to T Safety stock is less clear (Intervals of L+T overlap) Very Conservative Estimate Lead time = 28

21. When lead time is long relative to T Safety stock is less clear (Intervals of L+T overlap) Aggressive Estimate: Hold safety stock constant Lead time = 28

22. A forecast of day-to-day or week-to-week requirements Two sources of error Forecast error (from demand variability) Lead time variability Safety Lead Time replaces/augments Safety Stock Example 6 days Safety Lead Time Safety Lead Time translates into a quantity through the forecast, e.g., the next 6 days of forecasted requirements (remember the forecast changes) Periodic Review against a Forecast

23. Safety Lead Time as a quantity Safety Lead Time: The next X days of forecasted demand

24. Periodic shipments every T days Safety lead time of S days Each shipment is planned so that after it arrives we should have S + T days of coverage. The Ship-to-Forecast Policy

25. If all goes as planned Ship to this level Planned Inventory Safety Lead Time: The next X days of forecasted demand

26. Ship-to-Average Average usage or longer term forecast Motivation: flex inventory, keep mfg and shipments constant Effects: Relatively constant demands on production and transport capacity Reduced reliance on forecasts Reduced complexity Have to manage inventories New Strategy