1. 1 Part II: When to Order? Inventory Management Under Uncertainty u Demand or Lead Time or both uncertain u Even “good” managers are likely to run out once in a while (a firm must start by choosing a service level/fill rate) u When can you run out? –Only during the Lead Time if you monitor the system. u Solution: build a standard ROP system based on the probability distribution on demand during the lead time (DDLT), which is a r.v. (collecting statistics on lead times is a good starting point!)

2. 2 The Typical ROP System ROP set as demand that accumulates during lead time Lead Time Average Demand ROP = ReOrder Point

3. 3 The Self-Correcting Effect- A Benign Demand Rate after ROP ROP Lead Time Average Demand Lead Time Hypothetical Demand

4. 4 What if Demand is “brisk” after hitting the ROP? ROP > Lead Time Average Demand Hypothetical Demand Safety Stock EDDLT ROP = EDDLT + SS

5. When to Order u The basic EOQ models address how much to order: Q u Now, we address when to order. u Re-Order point (ROP) occurs when the inventory level drops to a predetermined amount, which includes expected demand during lead time (EDDLT) and a safety stock (SS): ROP = EDDLT + SS.

6. 6 When to Order u SS is additional inventory carried to reduce the risk of a stockout during the lead time interval (think of it as slush fund that we dip into when demand after ROP (DDLT) is more brisk than average) u ROP depends on: –Demand rate (forecast based). –Length of the lead time. –Demand and lead time variability. –Degree of stockout risk acceptable to management (fill rate, order cycle Service Level) DDLT, EDDLT & Std. Dev.

7. 7 The Order Cycle Service Level,(SL) u The percent of the demand during the lead time (% of DDLT) the firm wishes to satisfy. This is a probability. u This is not the same as the annual service level, since that averages over all time periods and will be a larger number than SL. u SL should not be 100% for most firms. (90%? 95%? 98%?) u SL rises with the Safety Stock to a point.

8. 8 Safety Stock LT Time Expected demand during lead time (EDDLT) Maximum probable demand during lead time (in excess of EDDLT) defines SS ROP Quantity Safety stock (SS)

9. 9 Variability in DDLT and SS u Variability in demand during lead time (DDLT) means that stockouts can occur. –Variations in demand rates can result in a temporary surge in demand, which can drain inventory more quickly than expected. –Variations in delivery times can lengthen the time a given supply must cover. u We will emphasize Normal (continuous) distributions to model variable DDLT, but discrete distributions are common as well. u SS buffers against stockout during lead time.

10. 10 Service Level and Stockout Risk u Target service level (SL) determines how much SS should be held. –Remember, holding stock costs money. u SL = probability that demand will not exceed supply during lead time (i.e. there is no stockout then). u Service level + stockout risk = 100%.

11. 11 Computing SS from SL for Normal DDLT u Example 10.5 on p. 374 of Gaither & Frazier. u DDLT is normally distributed a mean of 693. and a standard deviation of 139.: –EDDLT = 693. –s.d. (std dev) of DDLT =  = 139.. –As computational aid, we need to relate this to Z = standard Normal with mean=0, s.d. = 1 »Z = (DDLT - EDDLT) / 

12. 12 Reorder Point (ROP) ROP Risk of a stockout Service level Probability of no stockout Expected demand Safety stock 0z Quantity z-scale

13. 13 Area under standard Normal pdf from -  to +z Standard Normal(0,1) 0z z-scale P(Z <z) Z = standard Normal with mean=0, s.d. = 1 Z = (X -  ) /  See G&F Appendix A See Stevenson, second from last page

14. 14 Computing SS from SL for Normal DDLT to provide SL = 95%. u ROP = EDDLT + SS = EDDLT + z (  ). z is the number of standard deviations SS is set above EDDLT, which is the mean of DDLT. u z is read from Appendix B Table B2. Of Stevenson - OR- Appendix A (p. 768) of Gaither & Frazier: –Locate.95 (area to the left of ROP) inside the table (or as close as you can get), and read off the z value from the margins: z = 1.64. Example: ROP = 693 + 1.64(139) = 921 SS = ROP - EDDLT = 921 - 693. = 1.64(139) = 228 u If we double the s.d. to about 278, SS would double! u Lead time variability reduction can same a lot of inventory and $ (perhaps more than lead time itself!)

15. 15 Summary View ROP > Lead Time Holding Cost = C[ Q/2 + SS] (1)Order trigger by crossing ROP (2)Order quantity up to (SS + Q) Safety Stock EDDLT ROP = EDDLT + SS Q+SS = Target Not full due to brisk Demand after trigger

16. 16 Part III: Single-Period Model: Newsvendor u Used to order perishables or other items with limited useful lives. –Fruits and vegetables, Seafood, Cut flowers. –Blood (certain blood products in a blood bank) –Newspapers, magazines, … u Unsold or unused goods are not typically carried over from one period to the next; rather they are salvaged or disposed of. u Model can be used to allocate time-perishable service capacity. u Two costs: shortage (short) and excess (long).

17. 17 Single-Period Model u Shortage or stockout cost may be a charge for loss of customer goodwill, or the opportunity cost of lost sales (or customer!): C s = Revenue per unit - Cost per unit. u Excess (Long) cost applies to the items left over at end of the period, which need salvaging C e = Original cost per unit - Salvage value per unit. (insert smoke, mirrors, and the magic of Leibnitz’s Rule here…)

18. 18 The Single-Period Model: Newsvendor u How do I know what service level is the best one, based upon my costs? u Answer: Assuming my goal is to maximize profit (at least for the purposes of this analysis!) I should satisfy SL fraction of demand during the next period (DDLT) u If C s is shortage cost/unit, and C e is excess cost/unit, then

19. 19 Single-Period Model for Normally Distributed Demand u Computing the optimal stocking level differs slightly depending on whether demand is continuous (e.g. normal) or discrete. We begin with continuous case. u Suppose demand for apple cider at a downtown street stand varies continuously according to a normal distribution with a mean of 200 liters per week and a standard deviation of 100 liters per week: –Revenue per unit = $ 1 per liter –Cost per unit = $ 0.40 per liter –Salvage value = $ 0.20 per liter.

20. 20 Single-Period Model for Normally Distributed Demand u C s = 60 cents per liter u C e = 20 cents per liter. u SL = C s /(C s + C e ) = 60/(60 + 20) = 0.75 u To maximize profit, we should stock enough product to satisfy 75% of the demand (on average!), while we intentionally plan NOT to serve 25% of the demand. u The folks in marketing could get worried! If this is a business where stockouts lose long-term customers, then we must increase C s to reflect the actual cost of lost customer due to stockout.

21. 21 Single-Period Model for Continuous Demand u demand is Normal(200 liters per week, variance = 10,000 liters 2 /wk) … so  = 100 liters per week u Continuous example continued: –75% of the area under the normal curve must be to the left of the stocking level. –Appendix shows a z of 0.67 corresponds to a “left area” of 0.749 –Optimal stocking level = mean + z (  ) = 200 + (0.67)(100) = 267. liters.

22. 22 Single-Period & Discrete Demand: Lively Lobsters u Lively Lobsters (L.L.) receives a supply of fresh, live lobsters from Maine every day. Lively earns a profit of $7.50 for every lobster sold, but a day-old lobster is worth only $8.50. Each lobster costs L.L. $14.50. u (a) what is the unit cost of a L.L. stockout? C s = 7.50 = lost profit u (b) unit cost of having a left-over lobster? Ce = 14.50 - 8.50 = cost – salvage value = 6.  (c) What should the L.L. service level be? SL = C s /(C s + C e ) = 7.5 / (7.5 + 6) =.56 (larger C s leads to SL >.50) u Demand follows a discrete (relative frequency) distribution as given on next page.

23. 23 Lively Lobsters: SL = C s /(C s + C e ) =.56 Demand follows a discrete (relative frequency) distribution: Result: order 25 Lobsters, because that is the smallest amount that will serve at least 56% of the demand on a given night.