Inventory Analysis under Uncertainty: Lecture 6
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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 Point
Inventory level
Q
Usage
rate R
Average inventory = Q/2
Reorde
r
point
Receive
order
Place
order
Receive
order
Place
order
Receive
order
Time
2
When to Order?
ROP (reorder point):
inventory level which triggers
the placing of a new order
Example:
R = 20 units/day with
certainty
• Average inventory = cycle
inventory
L (days)

ROP
0
2
Q*= 200 units
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L =
leadtime with certainty
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μ=
LR = leadtime demand
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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?
Inventory on hand
order
ROP1
ROP2
L
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ROP under Uncertainty
Random Variable
Demand
Mean
std
Leadtime
Leadtime demand (DL)
• 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
–
R
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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
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Average weekly demand
Demand SD
ROP = 1200
Safety stock
Safety factor
µ = 1000
 = 250
• 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
Safety Inventory
NormSInv ( 0.99)·200
NormSInv ( 0.95)·200
NormSInv ( 0.9)·200
0.5
NormSInv ( 0.8)·200
NormSInv ( 0.7)·200
NormSInv ( 0.6)·200
NormSInv ( 0.5)·200
0.6
0.7
0.8
0.9
1.0
Service Level
Spreadsheet
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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
Inventory on hand
Q
order
order
order
ROP
mean demand
during supply
lead time
safety stock
Time t
L
Leadtime
L
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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   L R2  R2 L2
• 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)
I sa  z  n
• Average inventory
= Q/2 + zβσ
= Q/2 + zβσ+RL
Do not own pipeline
Own pipeline
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