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1 Managing Inventory under Risks Leadtime and reorder point Uncertainty and its impact Safety stock and service level The lot-size reorder point system Managing system inventory

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2 Leadtime and Reorder Point Inventory level Q Receive order Place order Receive order Place order Receive order Leadtime Reorde r point Usage rate R Time Average inventory = Q/2

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3 When to Order? ROP (reorder point): inventory level that triggers a new order ROP = LR (1) Example: R = 20 units/day Q* = 200 units L = leadtime with certainty μ = LR = leadtime demand L (days) ROP

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4 Motorola Hong Kong Revisited It takes the supplier 3 full working days to deliver the material to Motorola Consumption rate is 90 kg/day At what inventory level should Mr. Chan place an order?

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5 Uncertainty and Its Impact Sandy is in charge of inventory control and ordering at Broadway Electronics. The average demand for their best-selling battery is on average 1,000 units per week with a standard deviation of 250 units With a one-week delivery leadtime from the supplier, Sandy needs to decide when to order, i.e., with how many boxes of batteries left on-hand, she should place an order for another batch of new stock What is the difference between Mr. Chan’s task at Motorola and this one?

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6 Forecast and Leadtime Demand Often we forecast demands and make stocking decisions accordingly trying to satisfy arriving customers from on-hand stock Often, forecasting for a whole year is easier than for a week Leadtime demands usually can not be treated as deterministic

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7 Inventory Decision Under Risk When you place an order, you expect the remaining stock to cover all the leadtime demands Any order now or later can only satisfy demands after the leadtime L When to order? ROP 1 ? L order Inventory on hand ROP 1 ROP 2 L

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8 ROP under Uncertainty When D L is uncertain, it often makes sense to order a little earlier, i.e., at an inventory level higher than the mean ROP = + I S (2) I S = safety stock or extra inventory I S = z β × 3 z β = safety factor

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9 Random Leadtime Demand Random VariableMeanstd Demand Leadtime Leadtime demand ( D L ) R L = LR

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10 Safety Stock Time t ROP L L order mean demand during supply lead time safety stock Inventory on hand Leadtime

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11 Some Relations ROP safety stock safety stock safety factor safety factorservice level Given demand distribution, there is a one-to-one relationship, so we also have ROP I s z β β

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12 Safety Stock and Service Level Service level is a measure of the degree of stockout protection provided by a given amount of safety inventory Cycle service level: the probability that all demands in the leadtime are satisfied immediately SL = Prob.( LT Demand ≤ ROP) =β

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13 Service Level under Normal Demands Mean: µ = 1,000 ROP = 1,200 Service Level: SL = ? (The area of the shaded part under the curve) SL = Pr (LD ROP) = probability of meeting all demand (no stocking out in a cycle) I s = ROP – µ = 200

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14 Compute Cycle Service Level Given I s and σ Use normal table, we find β from z β Use excel : SL= NORMDIST(ROP, ,σ,True)(5) (4)

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15 Example 7.3 (MBPF) ROP = 24,000, µ = 20,000, σ = 5,000 z β = β = or SL = NORMDIST(24,000,20,000, 5,000, True) NT 9-EX1

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16 Compute Safety Stock Given β, we obtain z β from the normal table Use (3), we obtain the safety stock Use (2), we obtain ROP Given β, we can also have z β = NORMSINV (β)(6) ROP = NORMINV(β, µ, σ )(7)

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17 Example 7.4 (MBPF) µ = 20,000, σ = 5,000 β = 85%90%95%99% z β = ROP = NT

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18 Price of High Service Level Safety Stock Service Level NORMSINV ( 0.85)·200 NORMSINV ( 0.90)·200 NORMSINV ( 0.95)·200 NORMSINV ( 0.97)·200 NORMSINV ( 0.99)·200 NORMSINV ( 0.999)·200 9-EX2

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19 Example, Broadway Sandy orders a 2-week supply whenever the inventory level drops to 1,250 units. What is the service level provided with this ROP ? If Sandy wants to provide an 95% service level to the store, what should be the reorder point and safety stock ? Average weekly demand µ = 1,000 Demand SD = 250 Reorder point ROP = 1,250

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20 The Service Level Safety stock I s = Safety factor z β = Service level –By normal table β = –By excel SL= NORMDIST (1250, 1000, 250, True) NT 9-EX1

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21 Safety Stock for Target SL For 95% service rate –By the normal table z 0.95 = ROP= I s = –By excel ROP =NORMINV (0.95, 1000, 250) NT 9-EX1

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22 Lot Size-Reorder Point System Having determined the reorder point, we also need to determine the order quantity Note that we can forecast the annual demand more accurately and hence treat it as deterministic Then, the order quantity can be obtained using the standard EOQ

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23 The Average Inventory Let the order quantity be Q The average inventory level = (Q+I s + I s )/2 = Q/2 +I s The holding cost = HQ/2+HI s The ordering cost = S(R/Q) The optimal inventory cost = HQ * + HI s Time t ROP order mean demand during supply lead time safety stock Inventory on hand Leadtime Q +I s

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24 Example, Broadway R=52000/year (52 weeks) H=$1/unit/year S=$200/order Lot-size Reorder point Order quantity Q * = For 95% service rate I s = 250z β = Inventory cost = Sandy’s current policy µ= 1000, Q = 2000 ROP = 1,250, SL =84% Holding cost = Ordering cost = Inventory cost = 9-EX1

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25 Managing System Inventory There are different stocking points with inventories and at each stocking point, there are inventories for different functions Total average inventory includes three parts: Cycle + Safety + Pipeline inventories Total Average Inventory = Q/2 + I s + RL (8)

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26 Pipeline Inventory If you own the goods in transit from the supplier to you (FOB or pay when order), you have a pipeline inventory Average pipeline inventory equals the demand rate times the transit time or leadtime by Little’s Law Pipeline inventory = RL

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27 Sandy’s Current System Inventory Q=2,000, L =1 week, R = 1,000/week ROP = 1250, Safety stock = I s = 250 Total system average inventory: not own pipeline I = 2000/2+250 = 1250 owns pipeline I = 2000/ = 2250

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28 Managing Safety Stock Levers to reduce safety stock - Reduce demand variability - Reduce delivery leadtime - Reduce variability in delivery leadtime - Risk pooling

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29 Demand Aggregation By probability theory Var(D 1 + …+ D n ) = Var(D 1 ) + …+ Var(D n ) = nσ 2 As a result, the standard deviation of the aggregated demand is (9)

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30 The Square Root Rule Again We call (9) the square root rule: For BMW Guangdong –Monthly demand at each outlet is normal with mean 25 and standard deviation 5 –Replenishment leadtime is 2 months. The service level used at each outlet is 0.90 The SD of the leadtime demand at each outlet of our dealer problem The leadtime demand uncertainty level of the aggregated inventory system

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31 Cost of Safety Stock at Each Outlet The safety stock level at each outlet I s = z 0.9 σ = 1.285×7.07 = 9.08 The monthly holding cost of the safety stock TC(I s )= H×I s = 4,000x9.08 = 36,340RMB/month

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32 Saving in Safety Stock from Pooling System-wide safety stock holding cost without pooling 4 × C(I s ) = 4 × 36,340=145,360 RMB/month System-wide safety stock holding cost with pooling C(I sa ) = 2 × 36,340=72,680 RMB/month Annual saving of 12x(145,360-72,680) = 872,160 RMB!!

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33 BMW’s System Inventory With SL = 0.9: L = 2, Q = 36 (using EOQ), R=100/month z 0.9 =1.285, I s =(1.285)(14.4)= 18.5 ROP = 2x =218.5 Total system average inventory: not own pipeline I = 36/ = 36.5 owns pipeline I = 36/ = 236.5

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34 Takeaways (1) Leadtime demand usually must be treated as random, and hence creates risks for inventory decision We use safety stock to hedge the risk and satisfy a desired service level Together with the EOQ ordering quantity, the lot-size reorder point system provide an effective way to manage inventory under risk Reorder point under normal leadtime demand ROP = + I S = RL + z β σ

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35 Takeaways (2) For given target SL ROP= + z β σ = NORMINV (SL, ,σ) For given ROP SL= Pr(D L ROP) = NORMDIST(ROP, , σ, True) Safety stock pooling (of n identical locations) Total system average inventory = Q/2 + I s not own pipeline = Q/2 + I s +RLowns pipeline

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