Managing Uncertainty with Inventory I. John H. Vande Vate Spring, 2007. Topics. Integrate Obermeyer (wholesaler) with the Retail Game (retail pricing) Continuous Review Inventory Management Periodic Review Inventory Management Safety Lead Time. The Retail Game Revisited.

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Managing Uncertainty with Inventory I

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Topics • Integrate Obermeyer (wholesaler) with the Retail Game (retail pricing) • Continuous Review Inventory Management • Periodic Review Inventory Management • Safety Lead Time 2

The Retail Game Revisited • How much inventory to bring to the market? 2000? • What will demand be? • How to estimate it? That’s not demand! It’s supply • How to estimate demand for this item? 3

Average 209/week Estimating Demand • How fast was it selling? • So an estimate of season demand for this item is • 2473 = 2000 – 154 + 3*209 4

New Estimate • Should we order 1664? • What are the issues? • If salvage value exceeds our cost? • If salvage value is less than our cost? 5

Risk & Return • Will Demand be 1664? • How to measure our uncertainty about demand? • Method 1: Standard deviation of diverse forecasts • Method 2: Historical A/F ratios + Point forecast • Trade off Risks (out of stock and overstock) vs Return (sales) 6

Measuring Risk and Return • Profit from the last item • $profit if demand is greater, $0 otherwise • Expected Profit • $profit*Probability demand is greater than our choice • Risk posed by last item • $risk if demand is smaller, $0 otherwise • Expected Risk • $risk*Probability demand is smaller than our choice Example: risk = Salvage Value - Cost What if Salvage Value > Cost? 7

Balancing Risk and Return • Expected Profit • $profit*Probability demand is greater than our choice • Expected Risk • $risk*Probability demand is smaller than our choice • How are probabilities related? 8

Prob. Outcome is smaller Prob. Outcome is larger Risk & Reward Our choice How are they related? 9

Balance • Expected Revenue • $profit*(1- Probability demand is smaller than our choice) • Expected Risk • $risk*Probability demand is smaller than our choice • Set these equal • profit*(1-P) = risk*P • profit = (profit+risk)*P • profit/(profit + risk) = P = Probability demand is smaller than our choice 10

Prob. Demand is smaller Making the Choice Our choice Cumulative Probability profit/(profit - risk) 11

Swimsuit Case p 49 • Fixed Production Cost $100K • Variable Production Cost $80 • Selling Price $125 • Salvage Value $20 • Profit is $125 - $80 = $45 • Risk is $80 - $20 = $60 • Profit + Risk is $125 - $20 = $105 • Order to an expected stock out probability 57% = 1-$45/$105 = 1-43% • Several Sales Forecasts 12

43% Inferred Cum. Probability 57% stockout: 11,490 units 11490=10000+ 2000* [43%-Pr(10000)]/ [Pr(12000)-Pr(10000)] 14

Net Profit as a function of Quantity Gross Profits from sales Net Profits = Gr. Profits from sales – Cost of liquidation-fixed cost Costs of liquidations

Retail Game What to order? • So, we want P to be (Selling Price – Cost) (Selling Price– Salvage) • Assume Cost = $30, • But what’s the selling price? • In a wholesale environment this is easier. In a retail environment, it is messierSome protection from vendor some times 16

Extend Idea • Ship too little, you have to EXPEDITE the rest • Ship Q • If demand < Q • We sell demand and salvage (Q – demand) • If demand > Q • We sell demand and expedite (demand – Q) • What’s the strategy? 20

Same idea • Ignore profit from sales – that’s independent of Q • Focus on salvage and expedite costs • Look at last item • Chance we salvage it is P • Chance we expedite it is (1-P) • Balance these costs • Unit salvage cost * P = Unit expedite cost (1-P) • P = expedite/(expedite + salvage) 21

Safety Stock • Protection against variability • Variability in demand and • Variability in lead time • Typically described as days of supply • Should be described as standard deviations in lead time demand • Example: BMW safety stock • For axles only protects against lead time variability • For option parts protects against usage variability too 22

Continuous Review Basics Order Up to Level On Hand Position If lead time is long, … EOQ Lead Time Inventory Reorder Point Actual Lead Time Demand Order placed Avg LT Demand Safety Stock Time 24

Assumptions • Fixed Order Cost • Constant average demand • Typically assume Normally distributed lead time demand 25

Safety Stock Basics • Lead time demand N(m, s) • Typically Normal with • Average lead time demand m • Standard Deviation in lead time demand s • Setting Safety Stock • Choose z from N(0,1) to get correct probability that lead time demand exceeds z, • Safety stock is zs 26

Only Variability in Demand Sq. Root because we are adding up L independent (daily) demands. • If Lead Times are reliable • Average Lead Time Demand L * D • Standard Deviation in lead time demand sL = LsD • Sqrt of Lead time * Standard Deviation in demand • Units (Example) • L is the Lead Time in days, • sD is the standard deviation in daily demand 27

Implementation • Inventory On-hand • Inventory Position: On-hand and on-order • When Inventory Position reaches a re-order point (ROP), order the EOQ • This takes the Inventory Position to the Order-Up-To Level: EOQ + ROP • That’s because review is continuous – we always re-order at the ROP • Often called a (Q,r) policy (when inventory reaches r, order Q) 28

Example 3-7 page 61 Model assumes constant average monthly sales with variability around that average: no seasonality or growth in our sales NormInv(0.97)√(L) D ss+ L* AvgD ROP+EOQ [ROP+(EOQ+ROP)]/2 ss+EOQ/2 29

Lead Time Variability If Lead Times are variable • D = Average (daily) demand • sD = Std. Dev. in (daily) demand • L = Average lead time (days) • sL = Std. Dev. in lead time (days) • Average lead time demand • DL • Std. Dev. in lead time demand • sL = Ls2D + D2 s2L • Remember: Std. Dev. in lead time demand drives safety stock 30

Levers to Pull • Std. Dev. in lead time demand • sL = Ls2D + D2 s2L Reduce Variability in Lead Time Reduce Lead Time Reduce Variability in Demand 31

Periodic Review • Orders can only be triggered at certain times • Examples • Batched transmissions (e.g., every night, week, …) • Imposed by transportation (e.g., weekly vessel) • Examples of Continuous Review? 32

No Ordering Cost • Example? • Cost typically viewed as • Inventory cost • Service Level seen as a constraint • Probability of stock out in an order cycle • Key Assumption: NO COST TO CHANGE ORDER SIZE • Is this typically the case? 33

Order-Up-To Policy • Order-up-to Policy: At each period place an order to bring inventory position up to a level S • What problem might we encounter? 34

(S,s) Policy • To avoid small orders • In each period, if the inventory position is below s, place an order to bring it up to S. 35

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 Inventory Position Reorder Point Reorder Point Actual Lead Time Demand Stock on hand Lead Time Order placed Time 36

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 37

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 S – DT/2 38

Safety Stock in Periodic Review • Probability of stock out is the probability demand in T+L exceeds 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. 39

Safety Stock in Periodic Review • Probability of stock out is the probability demand in T+L exceeds 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? 40

Impact of Frequency • 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 This is reduced byTsD2/2 41

Example • 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 2 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 42

Example Cont’d 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 43

Example Cont’d • 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 44

Example Cont’d • Assume Demand in L+T is Normal • Hold risk constant 98% chance of no shortages all year S-D(T+E[L]) NormInv(0.9996) DT/2+ss OHI+DE[L] 45

Lead time = 28 • When lead time is long relative to T • Safety stock is less clear (Intervals of L+T overlap) • Very Conservative Estimate Assume independence 46

Lead time = 28 • When lead time is long relative to T • Safety stock is less clear (Intervals of L+T overlap) • Aggressive Estimate: Hold safety stock constant 47

Periodic Review against a Forecast • 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) 48

The Ship-to-Forecast Policy • 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. • Coverage: Inventory on hand should meet S+T days of forecasted demand 50