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
Economics of
Implementing a
Supply chain
The golden
The future
approaches the
Solutions tolaw
Square root to
in
effect the
thebullwhip
bullwhip
smoothing rule
for
strategies for taming
of bullwhip rule
solving
bullwhip problem
replenishment
effect
in Tesco
chains
supply problem
the bullwhip effect
bullwhip
The bullwhip effect in supply chains
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Measures of the bullwhip effect
Stochastic measures Deterministic measures
Orders
2
Var(Orders)
Demand Var( Demand)
2
Orders Stdev(Orders)
Demand Stdev( Demand)
Orders
Orders COV Orders
Demand COV Demand
Demand
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The bullwhip effect is important
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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 +
+ Lead-time
Variance +
+
+ +
Capacity
+
Overtime / + +
+
Agency work / +
Stock-outs Stock
Subcontracting -
Utilisation
- + +
+
+ Costs + Obsolescence
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Methodological
approaches to
solving the
bullwhip problem
Representations of time
Discrete Time v Continuous Time
Inventory positions are assessed Inventory positions are assessed
and orders are placed at discrete and order rates are adjusted at all
moments in time moments of time
- At the end of every day, or the end - May be suitable for a petrol refinery
of every week, for example or in a chemical plant
- May be suitable of the way a - The system states are known at
supermarket operates, or a every moment of time
distribution company
- 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 Aleksandr Mikhailovich
Lyapunov 1857-1918
Leonhard Euler Pierre-Simon Laplace
1707 - 1783 1749 - 1827
0
xt f t , xt , xt d
d
dt
Differential equations
f (W ) We W
Johann Heinrich Lambert
1728 – 1777 Lambert W functions
Discrete time approaches
Stochastic processes / ARIMA
p d q
D D D
t i t j k t k
t
i 1
i j 1 k 1
t
George Box
White
Auto Regressive term s Moving Averageterm s noise
Integrative term s
The ARMA(1,1) demand process for 16
P&G products in their Homecare range
Discrete time approaches
Stochastic processes / ARIMA
p d q
D D D
t i t j k t k
t
i 1
i j 1 k 1
t
George Box
White
Auto Regressive term s Moving Averageterm s noise
Integrative term s
F ( z ) Z f (t ) f (t ) z t
E ft 1 f1,..., ft ft t 0
Martingales z-transforms
Yakov Zalmanovitch Tsypkin
Joseph Leo Doob
1920-2004
1919-1997
State space
methods
Xt 1 Ax[t ] Bu (t )
Y[t ] Cx[t ] Du (t )
Rudolfl Kalman
1930-
Table of transforms and their properties
Other useful approaches
F k f x e 2ikx 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
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
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….
WIP
Desired
Target net stock
Net stock Tp
Actual WIP
Ot ˆ
Dt:t T p 1 ( TNS NS t ) ( Dt:t i WIPt )
ˆ
i 1
Ordersat tim e t
Forecastof dem and Inventorydiscrepancy
m adeat tim e t of dem and WIP discrenpancy
in period t T p 1
Set via the newsboy
Forecasts
approach to achieve
the critical fractile
Generating forecasts inside the OUT policy
• Exponential smoothing ˆ ˆ
Dt Dt 1
D D
t
ˆ
t 1
1 Ta
Tm
D t i
• Moving average ˆ
Dt i 1
Tm
• Conditional expectation Dt ,t x EDt x Dt i , i 0..
ˆ
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 Tp 1 Dt
Demandat time t
Previousordersat
time t T p 1
WIP WIP1 Ot 1 Ot Tp 1
t t
The replenishment lead-time, Tp
The influence of the
replenishment policy
The inventory balance equation….
NSt NSt 1 Ot Tp 1 Dt
Demandat time t
Previousordersat
time t T p 1
….shows us that the replenishment policy
influences both the orders and the net stock.
Therefore, when studying bullwhip we should also
consider
Net Stock Var( Net Stock)
2
NSAmp 2
Demand Var(Demand)
The impact of forecasting on net
stock variance amplification
NSAmp
2
NetStock
1 Tp
1 T
p
2
2
Demand 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
2 2
Bullwhip
Orders a p p a p
2
Demand 1 Ta 1 2Ta
23 2Ta Tp
2
5 2Ta 2Tp
1 2Ta 1 3Ta 2Ta 1 3Ta 2Ta 2
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
5 2Ta 23 2Ta Tp
2
2Tp
Bullwhip
1 2Ta 1 3Ta 2Ta
2
1 3Ta 2Ta
2
• However, reducing lead-times will always
reduce the inventory variance
NSAmp 1 Tp
1 T
p
2
1 2Ta
The OUT policy through the eyes of a
control engineer… Desired
Target
Target
WIP
WIP
Desired
1 Net stock
net stock
Net stock 1 T pT p
net stock
WIP
Actual
Actual WIP
* TNS NS ( (ˆ tˆt t i WIP
OtOt DtDtT:tpT1p 1 1*((TNS NS t t )) 1* * DD:ti WIPt t ) )
ˆ ˆ
:t
Ti Tw i:
11
Ordersat tim e t e t
Ordersat tim Forecastof dem andand
Forecastof dem y
Inventorydiscrepanc y
Inventorydiscrepanc
i
m at tim tim e of dem
m adeadeat e t of tdem andand WIP discrenpan
WIP discrenpan cy
cy
in period 1
in period t TtpT p 1
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
Matched feedback controllers
Tp
Ot Dt:t Tp 1 TNS NS t Dt:t i
ˆ 1 1 ˆ WIP
t
Ti Ti i 1
• When Tw=Ti the maths becomes very much
simpler
• With MMSE forecasting ( Ta ) we have…
Orders
2
1
Bullwhip 2
Demand 2Ti 1
NetStock
NSAmp 2
2
1 Tp
Ti 12 T Ti 2
Demand 2Ti 1 2Ti 1
p
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
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…
1 P N
S O
*
2 erf
N P
…for a given set of costs (H, B, N, P) and lead-time, (Tp)
2 2
1 2B 1 2N
erf B H 1 erf 1 N P
NS B H e
O N P e
Total costs
2 2
thus linearly related
Total costs are Constants Bullwhip costs
Inventory costs
to the standard deviations
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
Lead-time
Tp=1 Tp=3
TNS*=1.81 TNS*=2.56
Inventory feedback
Ti=1
S*=0.25 S*=0.25
£T=6.34 £T=7.37
gain
TNS*=2.27 TNS*=2.99
Ti=Ti*
S*=0.1001 S*=0.091
Ti*=3.69 Ti*=4.32
£T=4.63 £T=5.49
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Square root law
for bullwhip
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 2 3 4 6 12
Demand process faced by each DC N (60 , 12 )
Factory demand
N (60 , 12 )
… for 2 DC’s…
Each customer‟s demand = N(5,1)
DC demand=
N (30 , 6 )
DC demand=
N (30 , 6 ) Factory demand=N (60 , 12 )
Number of DC‟s (n) 1 2 3 4 6 12
Demand process faced by each DC N (60 , 12 ) N (30 , 6 )
Factory demand
N (60 , 12 ) N (60 , 12 )
… for 3 DC’s…
Each customer‟s demand = N(5,1)
Each DC‟s demand=
N ( 20 , 4 ) Factory demand=N (60 , 12 )
Number of DC‟s (n) 1 2 3 4 6 12
Demand process faced by each DC N (60 , 12 ) N (30 , 6 ) N ( 20 , 4 )
Factory demand
N (60 , 12 ) N (60 , 12 ) N (60 , 12 )
… for 4 DC’s…
Each customer‟s demand = N(5,1)
Each DC‟s demand=
Factory demand=N (60 , 12 )
N (15 , 3 )
Number of DC‟s (n) 1 2 3 4 6 12
Demand process faced by each DC N (60 , 12 ) N (30 , 6 ) N ( 20 , 4 ) N (15 , 3 )
Factory demand
N (60 , 12 ) N (60 , 12 ) N (60 , 12 ) N (60 , 12 )
Inventory
n The Square Root Law
Bullwhip
n
Number of DC‟s, n
1 2 3 4 6 12
Inventory
£8.59 £12.15 £14.89 £17.20 £21.06 £29.78
cost
Inventory cost
£8.59 £8.59 £8.60 £8.60 £8.60 £8.60
n
“If the inventories of a single product (or stock keeping unit) are originally
maintained at a number (n) of field locations (refereed to as the
Number of DC‟s, n
but
decentralised system) 1 are then consolidated into one central inventory
2 3 4 6 12
Capacity decentralised systeminventory
then the ratio £13.38 £18.93 £23.18 £26.77 n
£32.78 £46.36
cost
centralised systeminventory
exists”,Capacity cost £13.38
Maister, (1976). £13.39 £13.38 £13.39 £13.38 £13.38
n
Bullwhip
Proof of “the Square Root
n Law for bullwhip” 2
2N
erf 1 1
O N P e
N P
The bullwhip (capacity) costs are given by C£ OY
2
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,
decentralised bullwhip costs n c Y
2
centralise d bullwhip costs
n c Y
2
n
which is the “Square Root Law for Bullwhip”.
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Implementing a
smoothing rule
in Tesco
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
Peak weekly workload workload amplified
smoothed by modified by existing system
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 Tescoworking environment in the distribution
stable case study will be discussed in
system. more detail this afternoon in
the President’s Medal presentation
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The future
of bullwhip
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
Auto-regressive process with interaction with the other product
Demand for product 2 at time t Auto-regressiveitself Random processes
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
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Replenishment orders
Production Lead time 1000
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Retailer
600
orders 400
Manufacturer 200
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Consumer Demand
RetailerManufacturer his orders (with a onto the manufacturer
uses an OUT policy and places queuing model.
If the retailer smoothesis represented by aorders proportional controller)
Operates on a make to order principle
Processes can replenish first served basis
then the manufacturerorders on a first comethe retailers orders quicker.
Thus there is an interaction effect between bullwhip and lead-
times 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 www.cardiff.ac.uk
Steve@bullwhip.co.uk DisneySM@cardiff.ac.uk
The IOBPCS family
Stability issues (Tp=1)
Stability issues (Tp=2)