Supply Disruptions and the Reverse Bullwhip Effect
Lawrence V. Snyder1 Ying Rong1 Zuo-Jun Max Shen2
1Department of Industrial & Systems EngineeringCenter for Value Chain Research
Lehigh University
2Department of Industrial Engineering and Operations ResearchUniversity of California, Berkeley
University of Florida SCM Research Workshop — February 15–16, 2008
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 1 / 51
Outline
1 Motivation
2 Capacity/Price/Demand Model
3 Rationing Game
4 Conclusions
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 2 / 51
Motivation
Outline
1 Motivation
2 Capacity/Price/Demand Model
3 Rationing Game
4 Conclusions
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 3 / 51
Motivation
Katrina Crippled U.S. Oil Drilling and Refining Capacity
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 4 / 51
Motivation
Consumers Panicked, Gas Lines Occured
Even though there were few actual supply shortages.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 5 / 51
Motivation
Gas Prices Rose Sharply
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 6 / 51
Motivation
Gas Prices Rose Sharply
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 6 / 51
Motivation
Gas Prices Rose Sharply
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 6 / 51
Motivation
The Reverse Bullwhip Effect
Demand for gasoline after Katrina was very volatile
But production was stable (capacity maxed out)
The classical bullwhip effect (BWE):
Demand volatility increases as we move upstream
We conjecture that the reverse bullwhip effect (RBWE) occurredafter Katrina and Rita:
Demand volatility increases as we move downstream
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 7 / 51
Motivation
The Reverse Bullwhip Effect
Demand for gasoline after Katrina was very volatile
But production was stable (capacity maxed out)
The classical bullwhip effect (BWE):
Demand volatility increases as we move upstream
We conjecture that the reverse bullwhip effect (RBWE) occurredafter Katrina and Rita:
Demand volatility increases as we move downstream
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 7 / 51
Motivation
Movie
0 10 20 30 40 50 6020
30
40
50
60
70
80
90
100
110
Time Periods
Uni
t
CapacityDemand
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 8 / 51
Motivation
Empirical Evidence
Recent empirical studyMany industries do not exhibit BWEOrder variance smaller at ends of supply chain, larger in middleCachon, et al. (2007)
Behavioral studies using beer gameMany find a significant portion of players not exhibiting BWECroson et al. (2003, 2004, 2006), Kaminski and Simchi-Levi (2000),Wu and Katok (2005)
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 9 / 51
Motivation
String-Vibration Analogy
No amplification [no BWE/RBWE]
Demand shock [BWE]
Fixed point upstream [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 10 / 51
Motivation
String-Vibration Analogy
No amplification [no BWE/RBWE]
Demand shock [BWE]
Fixed point upstream [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 10 / 51
Motivation
String-Vibration Analogy
No amplification [no BWE/RBWE]
Demand shock [BWE]
Fixed point upstream [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 10 / 51
Motivation
String-Vibration Analogy
No amplification [no BWE/RBWE]
Demand shock [BWE]
Fixed point upstream [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 10 / 51
Motivation
String-Vibration Analogy, cont’d.
Supply shock [RBWE]
Supply and demand shock [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 11 / 51
Motivation
String-Vibration Analogy, cont’d.
Supply shock [RBWE]
Supply and demand shock [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 11 / 51
Motivation
String-Vibration Analogy, cont’d.
Supply shock [RBWE]
Supply and demand shock [“umbrella”]
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 11 / 51
Capacity/Price/Demand Model
Outline
1 Motivation
2 Capacity/Price/Demand ModelIntroduction and NotationCapacity Shift =⇒ Demand ShiftExistence of RBWE
3 Rationing Game
4 Conclusions
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 12 / 51
Capacity/Price/Demand Model Introduction and Notation
Capacity/Price/Demand Model
Model describes relationship between random capacity and resultingprice and demand
Use it to demonstrate that capacity shocks create RBWE
2 stages, supplier and buyer
Supplier’s capacity follows process {ct}∞t=1
Capacity changes produce price changes
Buyer anticipates future price changes and sets demand accordingly
Linear, downward-sloping demand curve
We assume capacity is always tight
i.e., supplier’s production quantity always equals capacity
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 13 / 51
Capacity/Price/Demand Model Introduction and Notation
Capacity/Price/Demand Model
Model describes relationship between random capacity and resultingprice and demand
Use it to demonstrate that capacity shocks create RBWE
2 stages, supplier and buyer
Supplier’s capacity follows process {ct}∞t=1
Capacity changes produce price changes
Buyer anticipates future price changes and sets demand accordingly
Linear, downward-sloping demand curve
We assume capacity is always tight
i.e., supplier’s production quantity always equals capacity
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 13 / 51
Capacity/Price/Demand Model Introduction and Notation
Notation
c = supplier’s production capacity = production quantity
p = equilibrium price
Q = quantity demanded by buyer
All state variables are indexed by t (time)
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 14 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Capacity Process
For now, we assume a deterministic process for ct
t
ct
Our results can also be proven for iid random ct (yield uncertainty)
Or linear recovery + iid random error
We are currently extending to more general capacity processes
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 15 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Capacity =⇒ Price
Q
P
ct
pt
bt-1
m
For each capacity ct we determine market-clearing price pt
pt = mct + bt−1
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 16 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Price =⇒ Demand Curve Shift
Q
P
pt–1
bt–1
ct
pt
Buyer observes price pt and change in price from last period
Adjusts demand curve based on change in price
Assumes price trend will continueReplaces pt with pt − r(pt − pt−1)r ∈ [0, 1) is a “reaction factor”
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 17 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Demand Curve =⇒ Order Quantity
Q
P
pt–1
bt–1
ct
pt
New curve (and current price) =⇒ demand:
Qt =(1− r)pt + rpt−1 − b
m
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 18 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Demand Curve =⇒ Order Quantity
Q
P
pt–1
bt–1
ct
pt
Qt
New curve (and current price) =⇒ demand:
Qt =(1− r)pt + rpt−1 − b
m
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 18 / 51
Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift
Perceived Demand Curve
Q
P
pt–1
bt–1
ct
pt
Qt
bt
Buyer observes order quantity Qt and updates “perceived demandcurve”
And the process repeats
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 19 / 51
Capacity/Price/Demand Model Existence of RBWE
Plot of Demand vs. Supply (Capacity)
0 10 20 30 40 50 6030
40
50
60
70
80
90
100
110
120
Time Periods
Uni
ts
How demand chases supply
CapacityDemand
Demand is more variable than supply =⇒ RBWE
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 20 / 51
Capacity/Price/Demand Model Existence of RBWE
Approximation of Demand Variance
pt is polynomial function of r
(Recall: r = shift in demand curve)
Let pt = first-order approximation of pt with respect to r
And Qt the resulting demand
Then we calculate the variance of Qt
Key Question: Is
Var(Qt) > Var(ct) [RBWE] orVar(Qt) < Var(ct) [BWE]?
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 21 / 51
Capacity/Price/Demand Model Existence of RBWE
BWE or RBWE?
Theorem
There exists a unique r∗ > 0 such that:
(a) If r = 0 or r = r∗, then Var(Qt) = Var(ct) [no BWE or RBWE].
(b) If r ∈ (0, r∗), then Var(Qt) < Var(ct) [BWE].
(c) If r ∈ (r∗,∞), then Var(Qt) > Var(ct) [RBWE].
r
V(Qt) – V(ct)
›
r *
RB
WE
BW
E
We know that r∗ ∈ (0, 0.2547) and are working on narrowing thisrange further.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 22 / 51
Capacity/Price/Demand Model Existence of RBWE
BWE or RBWE?
Theorem
There exists a unique r∗ > 0 such that:
(a) If r = 0 or r = r∗, then Var(Qt) = Var(ct) [no BWE or RBWE].
(b) If r ∈ (0, r∗), then Var(Qt) < Var(ct) [BWE].
(c) If r ∈ (r∗,∞), then Var(Qt) > Var(ct) [RBWE].
r
V(Qt) – V(ct)
›
r *
RB
WE
BW
E
We know that r∗ ∈ (0, 0.2547) and are working on narrowing thisrange further.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 22 / 51
Capacity/Price/Demand Model Existence of RBWE
Numerical Study: Difference in Variance vs. r
We conjecture that Var(Qt) always underestimates Var(Qt)
Then RBWE is more frequent and more exaggerated than suggested bythe Theorem.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
0
1
2
3
4
5
6
Ratio of Demand Shift to Price Change (r)
(Var
(Dem
and)
− V
ar(S
uppl
y))/
Var
(Sup
ply)
% Increase in Variance vs. r
Actual differenceApproximate difference
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 23 / 51
Capacity/Price/Demand Model Existence of RBWE
Severity of RBWE
Proposition
The magnitude of RBWE (Var(Qt)− Var(ct)):
(a) increases with ∆c (drop in capacity)
(b) increases with T (time to recovery)
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 24 / 51
Rationing Game
Outline
1 Motivation
2 Capacity/Price/Demand Model
3 Rationing GameIntroductionLPW97 ModelRevised ModelReservation Costs
4 Conclusions
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 25 / 51
Rationing Game Introduction
Rationing Game
Lee, Padmanabhan, and Whang (1997) [LPW97] discuss therationing game.
They conclude that...?
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 26 / 51
Rationing Game Introduction
LPW97’s Main Result
Theorem (LPW97)
z∗ ≥ z , where
z∗ is the Nash equilibrium base-stock level for the problem withstochastic capacity
z is the optimal base-stock level for the newsvendor problem
“This in turn implies the bullwhip effect when the mean demand changesover time.”
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 27 / 51
Rationing Game Introduction
LPW97’s Main Result
Theorem (LPW97)
z∗ ≥ z , where
z∗ is the Nash equilibrium base-stock level for the problem withstochastic capacity
z is the optimal base-stock level for the newsvendor problem
“This in turn implies the bullwhip effect when the mean demand changesover time.”
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 27 / 51
Rationing Game Introduction
Bullwhip Effect from Changes in Demand Mean
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time Period
Nash Eq z
Newsboy z
Demand
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 28 / 51
Rationing Game Introduction
Bullwhip Effect from Changes in Demand Mean?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time Period
Nash Eq z
Newsboy z
Demand
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 29 / 51
Rationing Game Introduction
BWE and RBWE in Rationing Game
Lingering Questions:
Does a Nash equilibrium exist?
Does the BWE really occur, even if the demand mean changes?
If BWE occurs, can RBWE occur too?
Retailer Customer
BWE
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 30 / 51
Rationing Game Introduction
BWE and RBWE in Rationing Game
Lingering Questions:
Does a Nash equilibrium exist?
Does the BWE really occur, even if the demand mean changes?
If BWE occurs, can RBWE occur too?
Supplier
RBWE
Retailer Customer
BWE
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 30 / 51
Rationing Game LPW97 Model
Assumptions and Notation
Same assumptions as LPW:
N identical retailers
Single-period model
Stochastic iid demand, D ∼ f ,F
Stochastic supplier capacity, V ∼ g ,G
Holding cost h, backorder cost p
Order quantity zi for retailer i
If capacity < total orders =⇒ proportional allocation
Retailers order before capacity is realized
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 31 / 51
Rationing Game LPW97 Model
Assumptions and Notation
Same assumptions as LPW:
N identical retailers
Single-period model
Stochastic iid demand, D ∼ f ,F
Stochastic supplier capacity, V ∼ g ,G
Holding cost h, backorder cost p
Order quantity zi for retailer i
If capacity < total orders =⇒ proportional allocation
Retailers order before capacity is realized
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 31 / 51
Rationing Game LPW97 Model
Assumptions and Notation
Same assumptions as LPW:
N identical retailers
Single-period model
Stochastic iid demand, D ∼ f ,F
Stochastic supplier capacity, V ∼ g ,G
Holding cost h, backorder cost p
Order quantity zi for retailer i
If capacity < total orders =⇒ proportional allocation
Retailers order before capacity is realized
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 31 / 51
Rationing Game LPW97 Model
Assumptions and Notation
Same assumptions as LPW:
N identical retailers
Single-period model
Stochastic iid demand, D ∼ f ,F
Stochastic supplier capacity, V ∼ g ,G
Holding cost h, backorder cost p
Order quantity zi for retailer i
If capacity < total orders =⇒ proportional allocation
Retailers order before capacity is realized
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 31 / 51
Rationing Game LPW97 Model
Expected Cost Function
Focus on retailer i
Assume all other order quantities are known: z¬i (a vector)
Ci (zi , z¬i ) = expected cost when i orders zi , others order z¬i
Expected Cost Function
Ci (zi , z¬i ) =
∫ y
v=0
[h
∫ vziy
0
(vzi
y− x
)dF (x) + p
∫ ∞
vziy
(x − vzi
y
)dF (x)
]dG (v)
+(1− G (y))
[h
∫ zi
0
(zi − x)dF (x) + p
∫ ∞
zi
(x − zi )dF (x)
]where y =
∑j zj .
z∗i minimizes Ci (zi , z¬i )
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 32 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
Recall: z = optimal newsboy order quantity
z∗i = z if capacity is always sufficient
LPW97’s theorem (z∗ ≥ z) assumes z∗ is NE
Proposition
If V is bounded above such that V < Nz (i.e., capacity is always tight),then there is no NE of retailers’ order quantities.
Intuition: There’s nothing to prevent z from growing to ∞
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 33 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
Recall: z = optimal newsboy order quantity
z∗i = z if capacity is always sufficient
LPW97’s theorem (z∗ ≥ z) assumes z∗ is NE
Proposition
If V is bounded above such that V < Nz (i.e., capacity is always tight),then there is no NE of retailers’ order quantities.
Intuition: There’s nothing to prevent z from growing to ∞
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 33 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
Recall: z = optimal newsboy order quantity
z∗i = z if capacity is always sufficient
LPW97’s theorem (z∗ ≥ z) assumes z∗ is NE
Proposition
If V is bounded above such that V < Nz (i.e., capacity is always tight),then there is no NE of retailers’ order quantities.
Intuition: There’s nothing to prevent z from growing to ∞
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 33 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
What if capacity may be sufficient?
i.e., V is unbounded, or bounded by something > Nz
Difficult to prove existence or non-existence of NE in this case
LPW97 assume NE but do not prove
We are unable to prove existence of BWE or RBWE in this case
What to do?
Adjust LPW97’s sequence of events slightly...
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 34 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
What if capacity may be sufficient?
i.e., V is unbounded, or bounded by something > Nz
Difficult to prove existence or non-existence of NE in this case
LPW97 assume NE but do not prove
We are unable to prove existence of BWE or RBWE in this case
What to do?
Adjust LPW97’s sequence of events slightly...
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 34 / 51
Rationing Game LPW97 Model
Non-Existence of Nash Equilibrium
What if capacity may be sufficient?
i.e., V is unbounded, or bounded by something > Nz
Difficult to prove existence or non-existence of NE in this case
LPW97 assume NE but do not prove
We are unable to prove existence of BWE or RBWE in this case
What to do?
Adjust LPW97’s sequence of events slightly...
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 34 / 51
Rationing Game Revised Model
Revised Sequence of Events
Suppose instead that retailers order after capacity is realized
If v ≥ Nz , each retailer orders z
Proposition
If v < Nz (i.e., capacity is tight), then there is no NE of retailers’ orderquantities.
What to do?
Assume each retailer thinks others order z
Retailer i thinks it’s the only one gamingRetailer i minimizes Ci (zi , z)
Let z0 = optimal order quantity under this assumption.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 35 / 51
Rationing Game Revised Model
Revised Sequence of Events
Suppose instead that retailers order after capacity is realized
If v ≥ Nz , each retailer orders z
Proposition
If v < Nz (i.e., capacity is tight), then there is no NE of retailers’ orderquantities.
What to do?
Assume each retailer thinks others order z
Retailer i thinks it’s the only one gamingRetailer i minimizes Ci (zi , z)
Let z0 = optimal order quantity under this assumption.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 35 / 51
Rationing Game Revised Model
Revised Sequence of Events
Suppose instead that retailers order after capacity is realized
If v ≥ Nz , each retailer orders z
Proposition
If v < Nz (i.e., capacity is tight), then there is no NE of retailers’ orderquantities.
What to do?
Assume each retailer thinks others order z
Retailer i thinks it’s the only one gamingRetailer i minimizes Ci (zi , z)
Let z0 = optimal order quantity under this assumption.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 35 / 51
Rationing Game Revised Model
Optimal Order Quantity
Proposition
z0 satisfies
F
(v
z0 + (N − 1)zz0
)=
p
p + h.
Therefore,
z0 =N − 1
v − zz2 ≥ z
provided that z < v ≤ Nz .
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 36 / 51
Rationing Game Revised Model
Existence of BWE
Suppose demand mean µ is a random variable
µ = µ0 + X ,
where X is a r.v. representing the demand shift
μ0
Retailer knows µ before placing order
New newsvendor quantity: z + X
Multiple copies of single-period model (no inventory carryover)
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 37 / 51
Rationing Game Revised Model
Existence of BWE
Suppose demand mean µ is a random variable
µ = µ0 + X ,
where X is a r.v. representing the demand shift
μ0
Retailer knows µ before placing order
New newsvendor quantity: z + X
Multiple copies of single-period model (no inventory carryover)
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 37 / 51
Rationing Game Revised Model
Existence of BWE
We examine Var(X ) as a proxy for Var(D)
i.e., ignore short-term demand variability
Order quantity Z is now a r.v., too
We are interested in Var(NX ) vs. Var(NZ )
Theorem
If z + X < v ≤ N(z + X ) (i.e., capacity is always tight but notextremely tight),
then Var(NZ ) > Var(NX ), i.e., the BWE occurs between retailers andcustomers.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 38 / 51
Rationing Game Revised Model
Existence of BWE
We examine Var(X ) as a proxy for Var(D)
i.e., ignore short-term demand variability
Order quantity Z is now a r.v., too
We are interested in Var(NX ) vs. Var(NZ )
Theorem
If z + X < v ≤ N(z + X ) (i.e., capacity is always tight but notextremely tight),
then Var(NZ ) > Var(NX ), i.e., the BWE occurs between retailers andcustomers.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 38 / 51
Rationing Game Revised Model
BWE Simulation
10 20 30 40 50 60 70 80 90 100−15
−10
−5
0
5
10
15
Time Periods
Uni
t
Demand MeanCapacity RealizationRetailer Orders
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 39 / 51
Rationing Game Revised Model
BWE: Done, RBWE: To Do
Supplier
RBWE
Retailer Customer
BWE
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 40 / 51
Rationing Game Revised Model
Existence of RBWE
Suppose now that capacity V is a random variable
Demand mean is deterministic
Theorem
If z < V ≤ Nz (i.e., capacity is always tight but not extremely tight),
then Var(V ) < Var(NZ ), i.e., the RBWE occurs between supplier andretailers.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 41 / 51
Rationing Game Revised Model
RBWE Simulation
10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10
15
Time Periods
Uni
t
Demand MeanCapacity RealizationRetailer Orders
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 42 / 51
Rationing Game Revised Model
Existence of “Umbrella”
Finally, suppose capacity and demand mean are stochastic
Corollary
If z + X+ < V ≤ N(z − X−) (i.e., capacity is always tight but notextremely tight),
then Var(V ) < Var(NZ ) > Var(NX ), i.e., the “umbrella shape”occurs.
Supplier
RBWE
Retailer Customer
BWE
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 43 / 51
Rationing Game Revised Model
“Umbrella” Simulation
10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10
15
20
25
30
Time Periods
Uni
t
Demand MeanCapacity RealizationRetailer Orders
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 44 / 51
Rationing Game Reservation Costs
Reservation Costs
There is no NE in previous models because there is no penalty forover-ordering
Suppose retailer pays r per unit ordered
And an additional cost per unit received
As before, assume V ≤ Nz
Capacity and demand mean are random
Retailer orders after capacity and demand mean are realized
Key Questions:
Are reservation costs sufficient to ensure NE?
Do BWE, RBWE occur?
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 45 / 51
Rationing Game Reservation Costs
Reservation Costs
There is no NE in previous models because there is no penalty forover-ordering
Suppose retailer pays r per unit ordered
And an additional cost per unit received
As before, assume V ≤ Nz
Capacity and demand mean are random
Retailer orders after capacity and demand mean are realized
Key Questions:
Are reservation costs sufficient to ensure NE?
Do BWE, RBWE occur?
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 45 / 51
Rationing Game Reservation Costs
Reservation Costs
There is no NE in previous models because there is no penalty forover-ordering
Suppose retailer pays r per unit ordered
And an additional cost per unit received
As before, assume V ≤ Nz
Capacity and demand mean are random
Retailer orders after capacity and demand mean are realized
Key Questions:
Are reservation costs sufficient to ensure NE?
Do BWE, RBWE occur?
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 45 / 51
Rationing Game Reservation Costs
Nash Equilibrium of Order Quantities
Let
v0 = NF−1
(p
p + h− r
p + h
1
N − 1
)< Nz
Proposition
1 There exists a symmetric NE of order quantities.
2 If v < v0, then
Nz∗ = −v
r
(1− 1
N
) [−(p + r) + (p + h)F
( v
N
)]≥ v .
3 If v0 ≤ v ≤ Nz , then Nz∗ = v .
Corollary
If v < v0, then as r → 0, z∗ →∞.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 46 / 51
Rationing Game Reservation Costs
Nash Equilibrium of Order Quantities
Let
v0 = NF−1
(p
p + h− r
p + h
1
N − 1
)< Nz
Proposition
1 There exists a symmetric NE of order quantities.
2 If v < v0, then
Nz∗ = −v
r
(1− 1
N
) [−(p + r) + (p + h)F
( v
N
)]≥ v .
3 If v0 ≤ v ≤ Nz , then Nz∗ = v .
Corollary
If v < v0, then as r → 0, z∗ →∞.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 46 / 51
Rationing Game Reservation Costs
Existence of BWE and RBWE
Theorem
If X (demand shift) is restricted to be in a certain interval, then BWEoccurs between retailers and customers.
If X is restricted to be outside this interval, then RBWE occursbetween retailers and customers.
Theorem
If V (capacity) is bounded above by a certain threshold, then RBWE occursbetween supplier and retailers.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 47 / 51
Rationing Game Reservation Costs
Existence of BWE and RBWE
Theorem
If X (demand shift) is restricted to be in a certain interval, then BWEoccurs between retailers and customers.
If X is restricted to be outside this interval, then RBWE occursbetween retailers and customers.
Theorem
If V (capacity) is bounded above by a certain threshold, then RBWE occursbetween supplier and retailers.
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 47 / 51
Conclusions
Outline
1 Motivation
2 Capacity/Price/Demand Model
3 Rationing Game
4 Conclusions
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 48 / 51
Conclusions
Conclusions
Supply disruptions can cause the RBWE
At least two mechanisms:
Pricing: capacity changes =⇒ demand volatility since buyers worryabout future price increasesRationing: capacity changes =⇒ demand volatility since buyersworry about future availability
In both cases, RBWE may propagate through several stages of supplychain, as long as buyers adjust policies to anticipate shortages
When demand stops reacting to shortages, BWE may occur
e.g., customer in LPW97’s rationing gameor gasoline usage (not consumption) in Katrina example
Disruptions may also cause RBWE in centralized supply chains
Buyers reduce order quantity during disruptions to prevent upstreamstockoutsWe have confirmed this empirically using a variant of the beer game
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 49 / 51
Conclusions
Conclusions
Supply disruptions can cause the RBWE
At least two mechanisms:
Pricing: capacity changes =⇒ demand volatility since buyers worryabout future price increasesRationing: capacity changes =⇒ demand volatility since buyersworry about future availability
In both cases, RBWE may propagate through several stages of supplychain, as long as buyers adjust policies to anticipate shortages
When demand stops reacting to shortages, BWE may occur
e.g., customer in LPW97’s rationing gameor gasoline usage (not consumption) in Katrina example
Disruptions may also cause RBWE in centralized supply chains
Buyers reduce order quantity during disruptions to prevent upstreamstockoutsWe have confirmed this empirically using a variant of the beer game
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 49 / 51
Conclusions
Conclusions
Supply disruptions can cause the RBWE
At least two mechanisms:
Pricing: capacity changes =⇒ demand volatility since buyers worryabout future price increasesRationing: capacity changes =⇒ demand volatility since buyersworry about future availability
In both cases, RBWE may propagate through several stages of supplychain, as long as buyers adjust policies to anticipate shortages
When demand stops reacting to shortages, BWE may occur
e.g., customer in LPW97’s rationing gameor gasoline usage (not consumption) in Katrina example
Disruptions may also cause RBWE in centralized supply chains
Buyers reduce order quantity during disruptions to prevent upstreamstockoutsWe have confirmed this empirically using a variant of the beer game
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 49 / 51
Conclusions
Future Directions
More general capacity process in pricing model
Empirical calibration for demand-curve shift
Sharper results for R/BWE in rationing game
Strategies for mitigating RBWE
Supply information sharing: RFID
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 50 / 51
Conclusions
Questions?
[email protected]/∼lvs2
Research supported by NSF grant #CMMI-0726822
Snyder (Lehigh/Berkeley) The Reverse Bullwhip Effect UF Workshop Feb 08 51 / 51
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