Analysis of Inventory Models
with Limited Demand
I f
Information
ti
David Simchi-Levi
Sources
z
The Bullwhip Effect
)Drezner, Ryan, Simchi-Levi, “Quantifying the
Bullwhip Effect: The Impact of Forecasting, Lead
Times and Information
Information.”
)Chen, Ryan, Simchi-Levi, “The Impact of
Exponential Smoothing Forecasts on the Bullwhip
Effect.”
z
Minimax Inventory Models
)Gallego, Ryan, Simchi-Levi, “Minimax Analysis for
Finite Horizon Inventory Models
Models.”
Quantifying the
B ll hi Effect
Bullwhip
Eff t
z
The Impact of Forecasting Methods
)Moving Average
Exponential Smoothing
)Exponential
z
Multi-Stage Supply Chains
)Centralized Information
)Decentralized Information
A Si
Simple
l Supply
S
l Ch
Chain
i
z
Single retailer, single manufacturer.
)Retailer observes customer demand, Dt.
)Retailer
Retailer orders qt from manufacturer.
)Lead time + 1 = L.
Dt
Retailer
qt
L
Manufacturer
A Simple Supply Chain:
O d off Event
Order
E
ts
z
At end of period t:
)Retailer updates forecast based on Dt.
Calculates order
order-up-to
up to point, yt+1.
)Calculates
)Places order qt+1.
)O d arrives
)Order
i
att start
t t off period
i d t+L.
t+L
)Demand is observed and filled.
)Unfilled demand is backlogged.
A Simple Supply Chain:
Policy
I
Inventory
t
li
z
Retailer follows order-up-to policy based
on inventory position.
)Approximates
Approximates the optimal policy under
the assumption of normal demand.
)A policy used frequently in practice
practice.
y t = L μ$ t + z LS t
A Simple Supply Chain:
Average Forecast
M i A
Moving
t z
Mean and standard deviation of
demand are estimated using a
moving
g avera ge of p observations:
p
∑ Dt−i
$μ t = i=1
p
p
St =
∑ ( Dt−i − μ$ t )
i=1
p −1
2
Moving Average Forecasting:
Quantity
Od Q
Order
tit Excess demand is returned at no cost.
z The order quantity for period t is:
z
qt = yt − yt−1 + Dt−1
L⎞
⎛
⎛ L ⎞
⎞
= ⎜ 1+ ⎟ Dt−1 − ⎜ ⎟ Dt− p−1 + z L( S t − S t−1 )
⎝
⎝ p⎠
p⎠
Moving Average Forecasting:
Variability
Th V
The
i bilit off Orders
Od
z
Determine the variance of q relative to
the variance of demand, σ2:
⎛ 2L 2L ⎞ 2
2
Var( qt ) = ⎜ 1+ + 2 ⎟ σ + z LVar( St − St −1 )
⎝
p p ⎠
2
⎛ 2L⎞
+2z L⎜ 1+ ⎟ Cov( Dt−1 ,St ).
⎝
p⎠
Moving Average Forecasting:
Symmettriic Demand
d
Lemma: Let Di, i=1,...,p, be i.i.d. observations
from a symmetric distrib
bution with
h variance σ2.
Then
Cov(Di,S)=0.
Corollary: The increase in variability from the
retailer to the manufacturer is:
2
Var (qt )
2L 2L
≥1+
+ 2 .
2
p
p
σ
Moving Average Forecasting:
N
Normal
lD
Demand
d
Lemma: Let Di, i=1,...,p, be i.i.d. observations
from a normall distrib
d
bution with
h variance σ2.
Var ( S t − S t −1 )
σ
2
≥ 1
p
2
.
Corollary: The increase in variability from the
retailer to the manufacturer is:
is:
Var (qt
)
σ2
2L 2L2 + z 2 L
≥1+
+
.
2
p
p
2
Var(q)/σ :
L
B
d vs. Si
l ti
Lower Bound
Simulation
3
25
2.5
2
1.5
1
0.5
0
0
10
20
Bound
p
30
Simulation
40
2
Var(q)/σ :
F
V i
L d Ti
For Various
Lead
Times
14
L=5
12
10
L=3
8
6
L=1
L=1
4
2
0
0
5
10
15
20
25
30
The Bullwhip Effect:
Manageriiall Insi
I ights
ht z
z
Exists, in part, due to the retailer’s need to
estimate the mean and variance of demand
demand.
The increase in variability is an increasing
function of the lead time and the smoothing
parameter.
)With
With longer lead time need more demand
data to reduce the bullwhip effect.
M lti Stage Supplly Chai
Multi-St
Ch ins
Consider an N stage supply chain:
)Stage i places order qi to stage i+1.
Li is lead time between stage i and ii+1.
1.
)L
qo=D
Retailer
Stage 1
q1
L1
Manufacturer
Stage 2
q2
L2
Supplier
Stage 3
Multi-Stage Supply Chain:
C t lized
Centrali
d IInfformation
ti
At the end of period t-1, stage i:
)Receives order qii-11
t.
)Receives updated forecast from retailer.
)Calculates the order-up-to point, yit.
)Orders qit.
Multi-Stage Supply Chains:
C t lized
Centrali
d IInfformation
ti
Each stage in the supply chain uses:
)The estimate of lead time demand
received from the retailer.
)An order-up-to inventory policy:
y = Li μ$ t
i
t
where μ$ t is received from the retailer.
Multi-Stage Supply Chains:
C t lized
Centrali
d IInfformation
ti
Lemma: When the retailer uses a moving average with p observations,
observations the increase in variability at
at
stage k is:
⎡∑ L ⎤ 2 ⎡∑ L ⎤
2
Var ( q )
⎢⎣i=1 i ⎥⎦
⎢⎣i=1 i ⎥⎦
=1+
+
2
σ
p
p
k
k
k
2
Multi-Stage Supply Chains:
D
Decentralized
t li d IInformation
f
ti
z
The retailer does not provide upstream stages with customer demand data
data. )Stage i estimates the mean demand from the orders it receives from stage ii-1
-1, qti-1, i > 1
1.
p −1
$ =
μ
i
t
∑q
j =
0
i
−1
t −
j
p
i
=
Li
t
y
μ$
i
t
Multi-Stage Supply Chains:
D
Decentralized
t li d IInformation
f
ti
Lemma: When the retailer uses a moving average with
i h p observations,
b
i
the
h increase
i
in
i variability
i bili
at
stage k is:
Var ( q
σ
2
k
) ≥ ∏ ⎡1 + 2 L
k
i=1
⎢
⎣
2 Li ⎤
+ 2 ⎥
p
p ⎦
i
2
The Bullwhip Effect:
Manageriiall Insi
I ights
ht z
z
z
z
Exists, in part, due to the retailer’s need to estimate
the mean and variance of demand
demand.
The increase in variability is an increasing function of
the lead time and the smoothing
g parameter.
)With longer lead time need more demand data
to reduce the bullwhip effect.
The more complicated the demand models and the
forecasting techniques, the greater the increase.
Centralized demand information can reduce the
bullwhip effect, but will not eliminate it.
MIT OpenCourseWare
http://ocw.mit.edu
ESD.273J / 1.270J Logistics and Supply Chain Management
Fall 2009
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