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Inventory Analysis under Uncertainty: Lecture 6

• Leadtime and reorder point

• Uncertainty and its impact

• Safety stock and service level

• Cycle inventory, safety inventory, and pipeline inventory

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Leadtime and Reorder PointIn

vent

ory

leve

lQ

Receive order

Placeorder

Receive order

Placeorder

Receive order

Reorderpoint

Usage rate R

Time

Average inventory = Q/2

3

When to Order?

ROP (reorder point): inventory level which triggers the placing of a new order

Example:

R = 20 units/day with certainty

Q*= 200 units

L = leadtime with certainty

μ = LR = leadtime demand

• Average inventory = cycle inventory

L (days)

ROP

0

2

7

14

22

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Uncertain Leadtime Demands

• Sandy is in charge of inventory control and ordering at Broadway Electronics

• The leadtime for its best-sales battery is one week fixed

• Sandy needs to decide when to order, i.e., with how many boxes of batteries left on-hand, should he place an order for another batch of new stock

• How different is this from Mr. Chan’s task at Motorola?

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Forecast and Leadtime Demand• Often we forecast demand and stock goods

accordingly so that customers can be satisfied from on-hand stock on their arrivals

• But it is impossible to forecast accurately, especially for short time periods, i.e., we may have a good estimate for the total demand in a year, but the leadtime (2 weeks) demand can be highly uncertain

• A further problem is the uncertainty of the length of the leadtime

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Stockout Risk

• When you place an order, you expect the remaining stock to cover all leadtime demands

• Any new order can only be used to satisfy demands after L

• When to order?

L

order

Inventory on hand

ROP1

ROP2

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ROP under Uncertainty

R• When DL is uncertain, it always makes sense to order a little earlier, i.e., with more on-hand stock

• ROP = + IS where – IS = safety stock = extra inventory–

Random Variable Mean std

Demand

Leadtime

Leadtime demand (DL)

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Safety Stock and Service Level

• Determining ROP is equivalent to determining the safety stock

• Service level SL or β

Service level is a measure of the degree of stockout protection provided by a given amount of safety inventory

• Or the probability that all customer demands in the leadtime are satisfied immediately

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Example, Broadway

• The weekly demand for batteries at Broadway varies. The average demand is estimated to be 1000 units per week with a standard deviation of 250 units

• The replenishment leadtime from the suppliers is 1 week and Broadway orders a 2-week supply whenever the inventory level drops to 1200 units.

• What is the service level provided with this ROP?• What is the average inventory level?

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Solution Using the Normal Table• Average weekly demand µ = 1000• Demand SD = 250• ROP = 1200• Safety stock• Safety factor

• Service Level:

β = SL = Prob.(LD ≤ 1200) Use normal table

=

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Computing the Service Level

Mean: µ = 1000

SL = Pr (LD ROP) = probability of meeting all demand(no stocking out in a cycle)

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Safety Stock for Target SL

• If Sandy wants to provide an 85% service level to the store, what should be the reorder point and safety stock?

• Solution: from the normal table

z0.85 =

ROP =

Safety stock = Is =

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Using Excel• Solve Pr(DL ROP) = SL for ROP

– If DL is normally distributed– zβ = NormSInv(SL),

ROP = + zβσ = + NormSInv( SL)·σ=

Or = NormInv( SL, ,σ) = • For given ROP

SL = Pr(LT Demand ROP) = NormDist( ROP, , σ, True)=

Spreadsheet

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Price of High Service Level

0.5 0.6 0.7 0.8 0.9 1.0

Saf

ety

Inve

nto

ry

Service Level

NormSInv ( 0.5)·200

NormSInv ( 0.6)·200

NormSInv ( 0.7)·200

NormSInv ( 0.8)·200

NormSInv ( 0.9)·200NormSInv ( 0.95)·200

Spreadsheet

NormSInv ( 0.99)·200

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Reducing Safety Stock

Levers to reduce safety stock

- Reduce demand variability

- Reduce delivery leadtime

- Reduce variability in delivery leadtime

- Risk pooling

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Demand Aggregation• By probability theory

Var(D1 + …+ Dn) = Var(D1) + …+ Var(Dn) = nσ2

• As a result, the standard deviation of the aggregated demand

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The Square Root Rule Again• We call (3) 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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Cost of Safety Stock at Each Outlet

• The safety stock level at each outlet is

Is =

• The monthly safety stock holding cost

TC(Is) =

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Safety Inventory Level

Q

Time t

ROP

L L

order order order

mean demand during supply lead time

safety stock

Inventory on hand

Leadtime

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Saving in Safety Stock from Pooling

• System-wide safety stock holding cost without pooling

• System-wide safety stock holding cost with pooling

Annual saving =

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Pipeline Inventory• If you own the goods in transit from the supplier

to you (FOB or pay at order), you have a pipeline inventory

• On average, it equals the demand rate times the transit time or leadtime by Little’s Law

• Your average inventory includes three parts

Average Inventory =

=

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Examples

• Sandy’s average inventory with SL=0.85: Q=2000, L =1 week, R = 1000/week

Average inventory:

• BMW’s consolidated average inventory with SL = 0.9: L = 2, Q = 36 (using EOQ), R=100/month Average inventory:

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Takeaways• ROP = + IS = RL + zβσ

• Leadtime demand: = RL and std • Assuming demand is normally distributed:

– For given target SL

ROP = + zβσ = NormInv(SL, ,σ) = +NormSInv(SL)·σ– For given ROP

SL = Pr(DL ROP) = NormDist(ROP, , σ, True)

• Safety stock pooling (of n identical locations)

• Average inventory= Q/2 + zβσ Do not own pipeline

= Q/2 + zβσ+RL Own pipeline

nzI sa

2 2 2R LL R