Supply Disruptions and the Reverse Bullwhip EffectBehavioral studies using beer game Many ﬁnd a...

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

Motivation

Outline

1 Motivation

3 Rationing Game

4 Conclusions

Motivation

Katrina Crippled U.S. Oil Drilling and Refining Capacity

Motivation

Consumers Panicked, Gas Lines Occured

Even though there were few actual supply shortages.

Motivation

Gas Prices Rose Sharply

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

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

Motivation

0 10 20 30 40 50 6020

Time Periods

CapacityDemand

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)

Motivation

String-Vibration Analogy

No amplification [no BWE/RBWE]

Demand shock [BWE]

Fixed point upstream [“umbrella”]

Motivation

Demand shock [BWE]

Motivation

Demand shock [BWE]

Motivation

Demand shock [BWE]

Motivation

String-Vibration Analogy, cont’d.

Supply shock [RBWE]

Supply and demand shock [“umbrella”]

Motivation

Supply shock [RBWE]

Motivation

Supply shock [RBWE]

Capacity/Price/Demand Model

Outline

1 Motivation

2 Capacity/Price/Demand ModelIntroduction and NotationCapacity Shift =⇒ Demand ShiftExistence of RBWE

3 Rationing Game

4 Conclusions

Capacity/Price/Demand Model Introduction and Notation

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

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

Notation

c = supplier’s production capacity = production quantity

p = equilibrium price

Q = quantity demanded by buyer

All state variables are indexed by t (time)

Capacity/Price/Demand Model Capacity Shift =⇒ Demand Shift

Capacity Process

For now, we assume a deterministic process for 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

Capacity =⇒ Price

For each capacity ct we determine market-clearing price pt

pt = mct + bt−1

Price =⇒ Demand Curve Shift

pt–1

bt–1

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”

Demand Curve =⇒ Order Quantity

pt–1

bt–1

New curve (and current price) =⇒ demand:

Qt =(1− r)pt + rpt−1 − b

Demand Curve =⇒ Order Quantity

pt–1

bt–1

New curve (and current price) =⇒ demand:

Qt =(1− r)pt + rpt−1 − b

Perceived Demand Curve

pt–1

bt–1

Buyer observes order quantity Qt and updates “perceived demandcurve”

And the process repeats

Capacity/Price/Demand Model Existence of RBWE

Plot of Demand vs. Supply (Capacity)

0 10 20 30 40 50 6030

Time Periods

How demand chases supply

CapacityDemand

Demand is more variable than supply =⇒ 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]?

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].

V(Qt) – V(ct)

We know that r∗ ∈ (0, 0.2547) and are working on narrowing thisrange further.

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].

V(Qt) – V(ct)

We know that r∗ ∈ (0, 0.2547) and are working on narrowing thisrange further.

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

Ratio of Demand Shift to Price Change (r)

% Increase in Variance vs. r

Actual differenceApproximate difference

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)

Rationing Game

Outline

1 Motivation

3 Rationing GameIntroductionLPW97 ModelRevised ModelReservation Costs

4 Conclusions

Rationing Game Introduction

Rationing Game

Lee, Padmanabhan, and Whang (1997) [LPW97] discuss therationing game.

They conclude that...?

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.”

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.”

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

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

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 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

Retailer Customer

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

Single-period 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 ) =

∫ vziy

y− x

)dF (x) + p

∫ ∞

(x − vzi

)dF (x)

]dG (v)

+(1− G (y))

∫ zi

(zi − x)dF (x) + p

∫ ∞

(x − zi )dF (x)

]where y =

∑j zj .

z∗i minimizes Ci (zi , z¬i )

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 ∞

Proposition

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...

What to do?

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.

Proposition

What to do?

Proposition

What to do?

Optimal Order Quantity

Proposition

z0 satisfies

z0 + (N − 1)zz0

p + h.

Therefore,

z0 =N − 1

v − zz2 ≥ z

provided that z < v ≤ Nz .

Existence of BWE

Suppose demand mean µ is a random variable

µ = µ0 + X ,

where X is a r.v. representing the demand shift

Retailer knows µ before placing order

New newsvendor quantity: z + X

Multiple copies of single-period model (no inventory carryover)

Existence of BWE

Suppose demand mean µ is a random variable

µ = µ0 + X ,

where X is a r.v. representing the demand shift

Retailer knows µ before placing order

New newsvendor quantity: z + X

Multiple copies of single-period model (no inventory carryover)

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.

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.

BWE Simulation

10 20 30 40 50 60 70 80 90 100−15

Time Periods

Demand MeanCapacity RealizationRetailer Orders

BWE: Done, RBWE: To Do

Supplier

Retailer Customer

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.

RBWE Simulation

10 20 30 40 50 60 70 80 90 100−10

Time Periods

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

Retailer Customer

“Umbrella” Simulation

10 20 30 40 50 60 70 80 90 100−10

Time Periods

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?

Reservation Costs

Key Questions:

Do BWE, RBWE occur?

Reservation Costs

Key Questions:

Do BWE, RBWE occur?

Nash Equilibrium of Order Quantities

v0 = NF−1

p + h− r

N − 1

Proposition

1 There exists a symmetric NE of order quantities.

2 If v < v0, then

Nz∗ = −v

(1− 1

) [−(p + r) + (p + h)F

)]≥ v .

3 If v0 ≤ v ≤ Nz , then Nz∗ = v .

Corollary

If v < v0, then as r → 0, z∗ →∞.

Nash Equilibrium of Order Quantities

v0 = NF−1

p + h− r

N − 1

Proposition

1 There exists a symmetric NE of order quantities.

2 If v < v0, then

Nz∗ = −v

(1− 1

) [−(p + r) + (p + h)F

)]≥ v .

3 If v0 ≤ v ≤ Nz , then Nz∗ = v .

Corollary

If v < v0, then as r → 0, z∗ →∞.

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.

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.

Conclusions

Outline

1 Motivation

3 Rationing Game

4 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

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

Conclusions

Questions?

[email protected]/∼lvs2

Research supported by NSF grant #CMMI-0726822

Supply Disruptions and the Reverse Bullwhip EffectBehavioral studies using beer game Many ﬁnd a...

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Transcript of Supply Disruptions and the Reverse Bullwhip EffectBehavioral studies using beer game Many ﬁnd a...