The dynamics of material flows in supply chains

transcript

Dr Stephen Disney

Logistics Systems Dynamics Group

Cardiff Business School

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The bullwhip effect in

supply chains

Methodological approaches to

solving the bullwhip problem

Supply chain strategies for taming

the bullwhip effect

The golden replenishment rule

Solutions to the bullwhip problem

Implementing a smoothing rule

in Tesco

Economics of the bullwhip

effect

Square root law for bullwhipThe future of bullwhip

The bullwhip effect in supply chains

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Measures of the bullwhip effect

DemandVar

OrdersVar

Demand

Orders

DemandStdev

OrdersStdev

Demand

Orders

Stochastic measures Deterministic measures

Demand

Orders

Demand

Orders

The bullwhip effect is important because it causes

Up to 30% of costs are due to the bullwhip effect!

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Poor customer service due to unavailable products

Runaway transportation and warehousing costsExcessive labour and learning costs

Unstable production schedules Insufficient or excessive capacitiesIncreased lead-times

How the bullwhip effect creates unnecessary costs

DemandVariance

Overtime / Agency work / Subcontracting

Stock-outs

Costs+

Lead-time+

+Obsolescence

Capacity+

Utilisation

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Methodological approaches to

solving the bullwhip problem

Representations of time

Discrete Time Inventory positions are assessed and orders are placed at discrete moments in time

- At the end of every day, or the end of every week, for example

- May be suitable of the way a supermarket operates, or a distribution company

- In between the discrete moments of time nothing is known about the system

Inventory positions are assessed and order rates are adjusted at all moments of time

- May be suitable for a petrol refinery or in a chemical plant

- The system states are known at every moment of time

v Continuous Time

Continuous time approaches

Lambert W functions

WWeWf )(

Johann Heinrich Lambert1728 – 1777

Leonhard Euler 1707 - 1783

t st dtetfstfL0

)()()(

Laplace transforms

Pierre-Simon Laplace1749 - 1827

Differential equations

,, dtxtxtftxdt

Aleksandr Mikhailovich Lyapunov 1857-1918

Discrete time approaches

Stochastic processes / ARIMA

George Box

noise White

termsAverage Moving

termseIntegrativ

termsRegressive Auto

iitit DDD

The ARMA(1,1) demand process for 16 P&G products in their Homecare range

Discrete time approaches

State space

methods

Joseph Leo Doob1920-2004

Martingales ttt ffffE ,...,11

)(][][

Rudolfl Kalman 1930-

Stochastic processes / ARIMA

George Box

noise White

termsAverage Moving

termseIntegrativ

termsRegressive Auto

iitit DDD

z-transformsYakov Zalmanovitch Tsypkin

1919-1997

)()()(t

tztftfZzF

Table of transforms and their properties

Other useful approaches

Jean Baptiste Joseph Fourier (1768-1830)

Fourier transforms

dxexfkF ikx2

System dynamics / simulation

Jay Forrester (1918-)

The beer gameJohn Sterman

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Supply chain strategies for taming

the bullwhip effect

Traditional supply chainsDefinition: ‘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 sharingDefinition: 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

cydiscrenpan WIP

WIPActual

WIPDesired

ydiscrepancInventory

stockNet stocknet Target

1 periodin demand of at time made

demand ofForecast

at time Orders

)ˆ()(ˆt

t WIPDNSTNSDOp

• Replenishment decisions influence both inventory levels & production rates.

• A common replenishment decision is the “Order-Up-To” (OUT) policy….

Set via the newsboy approach to achieve

the critical fractile

Forecasts

Generating forecasts inside the OUT policy

• Exponential smoothing

• Moving average

• Conditional expectation

We will assume normally distributed i.i.d. demand & exponential smoothing

forecasting from now on

tttt T

ˆˆˆ 1

..0,ˆ, iDDED itxtxtt

The inventory and WIP balance equations

Tttt DONSNS

at time Demand

1 timeat orders Previous

111 pTtttt OOWIPWIP

The replenishment lead-time, Tp

The influence of the replenishment policy

The inventory balance equation….

….shows us that the replenishment policy influences both the orders and the net stock.

Therefore, when studying bullwhip we should also consider

)Demand(

)StockNet (2Demand

2StockNet

VarNSAmp

Tttt DONSNS

at time Demand

1 timeat orders Previous

The impact of forecasting on net stock variance amplification

• As 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

Demand

NetStock

TTNSAmp

The impact of forecasting on bullwhip

As 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)

473225

Demand

Orders

TTTTTBullwhip

Reducing lead-times• Reducing lead-times usually (but not always)

reduces bullwhip

• However, reducing lead-times will always reduce the inventory variance

TBullwhip

TTNSAmp

The OUT policy through the eyes of a control engineer…

cydiscrenpan WIP

WIPActual

WIPDesired

demand ofForecast

at time Orders

)ˆ(*1)(*1ˆt

t WIPDNSTNSDOp

Unity feedback gains!

• 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….

cydiscrenpan WIP

WIPActual

WIPDesired

demand ofForecast

at time Orders

)ˆ(*1

)(*1ˆ

t WIPDT

NSTNST

Inventory feedback gain (Ti) WIP feedback gain (Tw)

The first proportional controller:The Maxwell Governor

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The golden replenishment rule

Matched feedback controllers

• When Tw=Ti the maths becomes very much simpler

• With MMSE forecasting ( ) we have…

iTttt WIPD

TNSTNS

:1:ˆ11ˆ

iDemand

Orders

TBullwhip

Demand

NetStock

TTNSAmp

The golden ratio in supply chains For i.i.d. demand, matched feedback controllers, MMSE forecasting

The golden ratio

1.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072...…

618034.12

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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

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.

The amount of slack capacity (S*) is an investment decision to be optimized

Costs are a linear function of the standard deviation

Setting the amount of safety stock we need via the newsboy…

… and the amount of capacity to invest in…

…for a given set of costs (H, B, N, P) and lead-time, (Tp)

HBerfTNS NS

NPerfS O

costs Total

NS ePNeHB

Inventory costs Bullwhip costsConstantsTotal costs are thus linearly related to the standard deviations

Sample designs for the 4 different scenarios

Lead-time

Tp=1 Tp=3

1 TNS*=1.81 S*=0.25 £T=6.34

TNS*=2.56 S*=0.25 £T=7.37

TNS*=2.27 S*=0.1001 Ti*=3.69 £T=4.63

TNS*=2.99 S*=0.091 Ti*=4.32 £T=5.49

Assuming the costs are; Holding cost, H=£1, Backlog cost, B=£9 Lost capacity cost, N=£4, Over-time cost, P=£6

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Square root law for bullwhip

One manufacturer

Distribution Network Design: Bullwhip costs

12 customers

n DC’s

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

Number of DC’s (n) 1 2 3 4 6 12

Demand process faced by each DC

Factory demand

…and it all depends on how many distribution centres we have…

Each customer’s demand = N(5,1)

DC demand= )12,60(N

Factory demand=

)12,60(N

Number of DC’s (n) 1 2 3 4 6 12

Factory demand

… for 2 DC’s…

DC demand=)6,30(N Factory demand=

)12,60(N

)6,30(N

)12,60(N

DC demand=)6,30(N

Number of DC’s (n) 1 2 3 4 6 12

Factory demand

… for 3 DC’s…

Each DC’s demand=)4,20(N Factory demand=

)12,60(N

)6,30(N

)12,60(N

)4,20(N

)12,60(N

Number of DC’s (n) 1 2 3 4 6 12

Factory demand

… for 4 DC’s…

Each DC’s demand=

)3,15(N

Factory demand=

)12,60(N

)6,30(N

)12,60(N

)4,20(N

)12,60(N )12,60(N

)3,15(N

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

£8.59 £8.59 £8.60 £8.60 £8.60 £8.60

The Square Root Law Inventory nBullwhip n

costInventory

costCapacity

Number of DC’s, n

1 2 3 4 6 12

Capacity cost

£13.38 £18.93 £23.18 £26.77 £32.78 £46.36

£13.38 £13.39 £13.38 £13.39 £13.38 £13.38

“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 decentralised system) but are then consolidated into one central inventory

then the ratio

exists”, Maister, (1976).

ninventory system dcentralise

inventory system seddecentrali

Proof of “the Square Root Law for bullwhip”

The bullwhip (capacity) costs are given by

In the decentralised supply chain the standard deviation of the orders is ,

In the centralised supply chain the standard deviation of the orders is

which is the “Square Root Law for Bullwhip”.

Bullwhip n

ePNC O

costs bullwhip dcentralise

costs bullwhip seddecentrali

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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 workload amplified by existing system

Peak weekly workload smoothed by modified

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 stable working environment in the distribution system.

The Tesco case study will be discussed in more detail this afternoon in

the President’s Medal presentation

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The future of bullwhip

Multiple products with interacting demand

,21,11,21,22,2,2

,11,22,11,11,1,1

Demand for product 2 at time t

Demand for product 1 at time t

Random processesAuto-regressive process with itselfAuto-regressive interaction with the other product

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

Manufacturer

Consumer Demand

Production Lead time

Retailerorders

Replenishment orders

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Retailer uses an OUT policy and places orders onto the manufacturerManufacturer is represented by a queuing model. Operates on a make to order principle

Processes orders on a first come first served basis

If the retailer smoothes his orders (with a proportional controller) then the manufacturer can replenish the 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 DisneyLogistics 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)

The dynamics of material flows in supply chains

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