The dynamics of material
flows in supply chains
Dr Stephen Disney
Logistics Systems Dynamics Group
Cardiff Business School
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Methodological
The
bullwhip
Implementing
Economics
ofa
Supply
chain
The
future
Solutions
The golden
tolaw
the
Square
root
approaches
to
smoothing
the
bullwhip
rule
effect
in
strategies
for
taming
of
bullwhip
for
bullwhip
solving
the rule
replenishment
bullwhip
problem
ineffect
Tesco
the
bullwhip
effect
supply
chains
bullwhip problem
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The bullwhip effect in supply chains
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Measures of the bullwhip effect
Stochastic measures
2
 Orders
Var (Orders )

2
 Demand
Var ( Demand )
 Orders
Stdev(Orders )

 Demand Stdev( Demand )
 Orders
 Demand
Orders
 Demand

COVOrders
COVDemand
Deterministic measures
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The bullwhip effect is important
because it causes
Unstable production schedules
Insufficient or excessive capacities
Increased lead-times
Poor customer service
due to unavailable products
Runaway transportation and
warehousing costs
Excessive labour and learning costs
Up to 30% of costs are due to the bullwhip
effect!
How the bullwhip effect
creates unnecessary costs
+
Demand
Variance
+
Overtime /
Agency work /
Subcontracting
+
+
+
Capacity
Lead-time
+
+
+
+
Stock-outs
-
+
+
Stock
Utilisation
- +
+
+
Costs +
+
Obsolescence
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Methodological
approaches to
solving the
bullwhip problem
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Representations of time
Discrete Time
Inventory positions are assessed
and orders are placed at discrete
moments in time
v
Continuous Time
Inventory positions are assessed
and order rates are adjusted at all
moments of time
- At the end of every day, or the end
of every week, for example
- May be suitable for a petrol refinery
or in a chemical plant
- May be suitable of the way a
supermarket operates, or a
distribution company
- The system states are known at
every moment of time
- In between the discrete moments
of time nothing is known about the
system
Continuous time approaches
L f (t )(s)   f (t )e st dt
t
0
Laplace transforms
Leonhard Euler
1707 - 1783
Pierre-Simon Laplace
1749 - 1827
0


d
xt   f  t , xt ,  xt   d  
dt



Differential equations
f (W )  WeW
Johann Heinrich Lambert
1728 – 1777
Lambert W functions
Aleksandr Mikhailovich
Lyapunov 1857-1918
Discrete time approaches
Stochastic processes / ARIMA
D   D
 D
  
 

p
t
George Box
q
d
i
t i
t j
k t k
i 1



j 1


k 1



Auto Regressiveterms
Integrative terms
Moving Average terms
t
White
noise
The ARMA(1,1) demand process for 16
P&G products in their Homecare range
Discrete time approaches
Stochastic processes / ARIMA
D   D
 D
  
 

p
t
George Box
q
d
i
t i
t j
k t k
i 1



j 1


k 1



Auto Regressiveterms
Integrative terms
Moving Average terms
E ft 1 f1 ,..., ft   ft
Martingales
t
White
noise

F ( z )  Z  f (t )   f (t ) z t
t 0
z-transforms
Yakov Zalmanovitch Tsypkin
1919-1997
Joseph Leo Doob
1920-2004
State space
methods
Rudolfl Kalman
1930-
Xt  1  Ax[t ]  Bu(t )
Y[t ]  Cx[t ]  Du(t )
Table of transforms and their properties
Other useful approaches
F k  


f  x e  2ikx dx

Jean Baptiste Joseph
Fourier (1768-1830)
Fourier transforms
The beer game
John Sterman
Jay Forrester
(1918-)
System dynamics / simulation
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Supply chain
strategies for taming
the bullwhip effect
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Traditional supply chains
Definition: ‘Traditional’ means that each level in the supply chain issues production
orders and replenishes stock without considering the situation at either up- or
downstream tiers of the supply chain. This is how most supply chains still operate; no
formal collaboration between the retailer and supplier.
Bullwhip increases geometrically in a traditional supply chain
Supply chains with information sharing
Definition: Information exchange (or information sharing) means that retailer and
supplier still order independently, yet exchange demand information in order to align
their replenishment orders and forecasts for capacity and long-term planning.
Bullwhip increases linearly in supply chains with information sharing
Synchronised Supply (VMI)
Definition: Synchronized supply eliminates one decision point and merges the
replenishment decision with the production and materials planning of the supplier. Here,
the supplier takes charge of the customer’s inventory replenishment on the operational
level, and uses this visibility in planning his own supply operations.
Bullwhip may not increase at all in VMI supply chains
Integrating internal and external decision in
supply chains with long lead-times
RFID technologies now allow
us to monitor the distribution leg
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Solutions to the
bullwhip problem
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Replenishment rules and the
bullwhip problem
•
Replenishment decisions influence both inventory levels &
production rates.
•
A common replenishment decision is the “Order-Up-To” (OUT)
policy….
Net
stock
Actual
WIP
Tp



 ( TNS  NSt )  ( Dˆ t:t i  WIPt )

 i 1



Inventorydiscrepancy
Target net stock
Ot

Orders at time t

Dˆ t:t T p 1

Forecast of demand
made at time t of demand
in period t T p 1
Desired WIP



WIP discrenpancy
Set via the newsboy
Forecasts
approach
to achieve
the critical fractile
Generating forecasts inside the OUT policy
Dˆ t  Dˆ t 1
• Exponential smoothing
Tm
• Moving average
Dˆ t 

D  Dˆ 

t
t 1
1  Ta
D
i 1
t i
Tm
• Conditional expectation Dˆ t ,t  x  EDt  x Dt i , i  0..
We will assume normally distributed i.i.d.
demand & exponential smoothing
forecasting from now on
The inventory and
WIP balance equations
NSt  NSt 1  Ot T p 1

Previousorders at
time t T p 1

Dt

Demand at time t
WIPt  WIPt 1  Ot 1  Ot Tp 1
The replenishment lead-time, Tp
The influence of the
replenishment policy
The inventory balance equation….
NSt  NSt 1  Ot T p 1

Previousorders at
time t T p 1

Dt

Demand at time t
….shows us that the replenishment policy
influences both the orders and the net stock.
Therefore, when studying bullwhip we should also
consider
2
 Net
Var ( Net Stock )
Stock
NSAmp  2

 Demand
Var (Demand )
The impact of forecasting on net
stock variance amplification

NSAmp 

2
NetStock
2
Demand

1 T 

2
 1  Tp
p
1  2Ta 
• As Ta   then NSAmp approaches 1+Tp.
• Minimising the Mean Squared Error between the
forecast of demand over the lead-time and review
period and its realisation will result in the minimum
inventory variance.
• This holds in a single echelon (Vassian 1954) and
across a complete supply chain (Hosoda and Disney,
2006) when the traditional OUT policy is used
The impact of forecasting
on bullwhip
5  2T  2T 3  T   T 7  4T 

Bullwhip 


2
Orders
2
Demand
2
a
p
p
a
1  Ta 1  2Ta 
2
23  2Ta Tp
2Tp
p
5  2Ta



2
1  2Ta 1  3Ta  2Ta 1  3Ta  2Ta 2
As Ta   then bullwhip approaches unity.
Thus, we can see that as we make more
accurate forecasts the bullwhip problem is
reduced (but is not eliminated in this scenario)
Reducing lead-times
• Reducing lead-times usually (but not always)
reduces bullwhip
2
2Tp
5  2Ta 23  2Ta Tp
Bullwhip 
1  2Ta

1  3Ta  2Ta
2

1  3Ta  2Ta
2
• However, reducing lead-times will always
reduce the inventory variance

1 T 

2
NSAmp  1  Tp
p
1  2Ta 
The OUT policy through the eyes of a
control engineer…
Target
Target
net
stock
netstock
Desired
WIP
WIP
Desired









Net
stock
Actual
WIP
WIP
Actual
stock
Net
T pT p






1
1
ˆ
ˆDˆ t:t i WIP
ˆ
OtOt   DtD

*
(
TNS

NS
)

*
(
WIP
)
D
(
*
1

)
NS

TNS
(
*
1


:
t

T

1
t

t t)
t:t  i
t
1

T

t
:
t
p
p








 Tw i 1i 1

T






i
Orders
at
time
t
Orders at time t Forecast of demand


Inventory
yy









discrepanc
Inventorydiscrepanc
Forecast of demand
made
at time
t of tdemand
of demand
at time
made
in period
t

T
in period tpT1p 1
WIP
discrenpan
cycy
discrenpan
WIP
Unity feedback gains!
Inventory feedback gain (Ti)
WIP feedback gain (Tw)
• A control engineer would not be at all surprised
that the OUT policy generates bullwhip as there
are unit gains in the two feedback loops
• Let’s add in a couple of proportional feedback
controllers….
The first proportional controller:
The Maxwell Governor
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The golden
replenishment rule
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Matched feedback controllers
Tp


1
1
ˆ
ˆ
Ot  Dt:t T p 1  TNS  NSt     Dt:t i  WIPt 
Ti
Ti  i 1

• When Tw=Ti the maths becomes very much
simpler
• With MMSE forecasting ( Ta   ) we have…
2
 Orders
1
Bullwhip  2

 Demand 2Ti  1
2
2
2

 NetStock
Ti  1
Ti
NSAmp  2
 1  Tp 
 Tp 
 Demand
2Ti  1
2Ti  1
The golden ratio in supply chains
For i.i.d. demand, matched feedback controllers, MMSE forecasting
The golden ratio
1 5

 1.618034
2
1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072...…
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Economics of
the bullwhip
effect
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Economics of inventory
Inventory costs are governed by the safety stock (TNS)
The Target Net Stock (TNS*) is an investment decision to be optimized
In each period, when the
inventory is positive a holding
cost is incurred of £H per unit.
In each period, if a backlog
occurs (inventory is negative),
a backlog cost of £B per unit
is incurred
The economics of capacity
Capacity per period = Average demand +/- slack capacity
The amount of slack capacity (S*) is an investment decision to be optimized
Production above capacity
results in some over-time
working (or sub-contracting).
The cost of this type of
capacity is £P per unit of
over-time.
Production below capacity
results in some lost capacity
cost of £N per unit lost.
Costs are a linear function of the
standard deviation
Setting the amount of safety stock we need via the
newsboy…
 1  B  H  
*
TNS   NS 2  erf 


B  H 

… and the amount of capacity to invest in…
S O
*
 1  P  N  
2  erf 


 N  P

…for a given set of costs (H, B, N, P) and lead-time, (Tp)
erf
Total costs 
 NS B  H e
2
1 
2B

 B  H 1


2

 O N  P e
erf
1 
2N 
1 N  P 


2
Inventory
costs
Bullwhip
costs
Total
costs
are Constants
thus linearly
related
to the standard deviations
2
Sample designs for the 4
different scenarios
Assuming the costs are;
Holding cost, H=£1,
Backlog cost, B=£9
Lost capacity cost, N=£4,
Over-time cost, P=£6
Tp=1
Tp=3
Ti=1
TNS*=1.81
S*=0.25
£T=6.34
TNS*=2.56
S*=0.25
£T=7.37
Ti=Ti*
Inventory feedback
gain
Lead-time
TNS*=2.27
S*=0.1001
Ti*=3.69
£T=4.63
TNS*=2.99
S*=0.091
Ti*=4.32
£T=5.49
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Square root law
for bullwhip
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Distribution Network
Design: Bullwhip costs
12 customers
…
n DC’s
One manufacturer
Each customer produces an i.i.d. demand,
normally distributed with a mean of 5 and unit variance
All lead-times in the system are one period long
…and it all depends on how many
distribution centres we have…
Each customer’s demand = N(5,1)
DC demand= N (60, 12 )
Factory demand=N (60, 12 )
Number of DC’s (n)
1
Demand process faced by each DC
N (60, 12 )
Factory demand
N (60, 12 )
2
3
4
6
12
… for 2 DC’s…
Each customer’s demand = N(5,1)
DC demand=
N (30, 6 )
DC demand=
Factory demand=N (60, 12 )
N (30, 6 )
Number of DC’s (n)
1
2
Demand process faced by each DC
N (60, 12 )
N (30, 6 )
N (60, 12 )
N (60, 12 )
Factory demand
3
4
6
12
… for 3 DC’s…
Each customer’s demand = N(5,1)
Each DC’s demand=
Factory demand=N (60, 12 )
N (20, 4 )
Number of DC’s (n)
1
2
3
Demand process faced by each DC
N (60, 12 )
N (30, 6 )
N (20, 4 )
N (60, 12 )
N (60, 12 )
N (60, 12 )
Factory demand
4
6
12
… for 4 DC’s…
Each customer’s demand = N(5,1)
Each DC’s demand=
N (15, 3 )
Factory demand=N (60, 12 )
Number of DC’s (n)
1
2
3
4
Demand process faced by each DC
N (60, 12 )
N (30, 6 )
N (20, 4 )
N (15, 3 )
N (60, 12 )
N (60, 12 )
N (60, 12 ) N (60, 12 )
Factory demand
6
12
Inventory
n
Bullwhip
n
The Square Root Law
Number of DC’s, n
1
2
3
4
6
12
Inventory
cost
£8.59
£12.15
£14.89
£17.20
£21.06
£29.78
Inventory cost
n
£8.59
£8.59
£8.60
£8.60
£8.60
£8.60
“If the inventories of a single product (or stock keeping unit) are originally
maintained at a number (n) of field locations
to as the
Number of(refereed
DC’s, n
decentralised system) 1but are then
consolidated
2
3
4 into one
6 central
12inventory
then the
Capacity
ratio
cost
decentrali sed system inventory
£13.38 £18.93 £23.18 £26.77 
£32.78
n
centralise d system inventory
exists”,Capacity
Maister,cost
(1976).
£13.38
n
£13.39
£13.38
£13.39
£13.38
£46.36
£13.38
Bullwhip
n
Proof of “the Square Root
Law for bullwhip”
The bullwhip (capacity) costs are given by C£ 
 O N  P e
 2N 
erf 1 1

 N P 
2
2
  OY
In the decentralised supply chain the standard deviation of the orders is ,
 O  n  c2
In the centralised supply chain the standard deviation of the orders is
 O  n c2
Thus,




n  c2 Y
decentrali sed bullwhip costs

 n
2
centralise d bullwhip costs
n c Y
which is the “Square Root Law for Bullwhip”.
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Implementing a
smoothing rule
in Tesco
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10
20
30
40
50
60
70
80
Tesco project brief
• Tesco’s store replenishment algorithms were
generating a variable workload on the physical
delivery process
– this generated unnecessary costs
• The purpose of the project was to;
– investigate the store replenishment rules to evaluate their
dynamic performance
– to identify if they generated bullwhip
– offer solutions to any bullwhip problems
Inventory replenishment approaches
High volume products
• Account for 65% of sales volume and 35% of product lines
• Deliveries occur up to 3 times a day
The simulation approach
Weekly workload profile:
Before and after
Peak weekly workload
smoothed by modified
system
Peak weekly
workload amplified
by existing system
Summary
• Tesco’s replenishment system was found to increase the daily
variability of workload by 185% in the distribution centres
• A small change to the replenishment algorithms was
recommended that smoothed daily variability to
approximately 75% of the sales variability
• The solution was applied to 3 of the 7 order calculations. This
accounted for 65% of the total sales value of Tesco UK
• This created a The
stable
in the distribution
Tescoworking
case study environment
will be discussed in
system.
more detail this afternoon in
the President’s Medal presentation
Orders
1000
1000
800
800
600
600
400
400
200
200
0
0
0
The future
of bullwhip
10
20
30
40
50
60
70
80
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
Multiple products with interacting
demand
Demand for product 1 at time t
D1,t  1,1D1,t 1  1, 2 D2,t 1  1,t
D2,t  2, 2 D2,t 1  2,1D1,t 1   2,t
Demand for product
2 at time t Auto-regressive
interaction with the
Random
other product
processes
Auto-regressive
process with itself
The Inventory Routing and
Joint Replenishment Problem
• In a multiple product or multiple customer scenario
• Place an order to bring inventory up-to S,
– if inventory is below a reorder point
– OR if inventory is below a “can-deliver” level AND another product (or retailer)
has reached its reorder point
Consolidation of orders/ deliveries can
generate significant savings
The interaction between bullwhip
inventory variance & lead-times
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
80
Replenishment orders
Production Lead time
1000
800
orders
Retailer
Manufacturer
600
400
200
0
0
10
20
30
40
50
60
70
80
Consumer Demand
Retailersmoothes
Manufacturer
uses an OUTishis
policy
represented
and places
by
aorders
queuing
onto
model.
the manufacturer
If the retailer
orders
(with
a proportional
controller)
Operates on a make to order principle
then the manufacturer
can on
replenish
orders quicker.
Processes orders
a first comethe
first retailers
served basis
Thus there is an interaction effect between bullwhip and leadtimes that allows supply chains to break the inventory / order
variance trade-off!
Multi-echelon supply
chain policies
The impact of errors
•
•
•
•
•
•
•
Demand parameter mis-identification
Demand model mis-identification
Lead-time mis-identification
Information delays
Random errors in information
Non-linear, time-varying systems
…
Thank you
The dynamics of material flows in supply chains
Dr Stephen Disney
Logistics Systems Dynamics Group
Cardiff Business School
www.bullwhip.co.uk
Steve@bullwhip.co.uk
www.cardiff.ac.uk
DisneySM@cardiff.ac.uk
The IOBPCS family
Stability issues (Tp=1)
Stability issues (Tp=2)