optimum parameters for optimum parameters for...
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Optimum Parameters for Optimum Parameters for CrostonCroston
Intermittent Demand MethodsIntermittent Demand Methods
The 32The 32ndnd Annual International Symposium on ForecastingAnnual International Symposium on Forecasting
www.lancs.ac.uk
NikolaosNikolaos KourentzesKourentzesLancaster University Management School
The 32The 32ndnd Annual International Symposium on ForecastingAnnual International Symposium on Forecasting
► Production & inventory management require knowledge of future demand
� forecasting!
► Conventional forecasting methods & techniques perform poorly for intermittent
(or lumpy) demand.
► Intermittent demand
Intermittent Demand Time SeriesIntermittent Demand Time Series
6
MotivationMotivationCroston’s Method
► Intermittent demand
o Frequent intervals with no demand
o Large variation in demand levels when it occurs
► Observed in both manufacturing and service environments
o Heavy machinery & spare parts [Willemain et al., 05], automotive industry parts
[Syntetos & Boylan 01, 05], durable goods parts [Kalchschmidt et al., 03], aircraft maintenance
[Ghobbar and Friend, 03], telecommunications, large compressors textile machines
[Bartezzaghi et al., 99], RAF spare parts [Teunter & Babangida Sani, 09], etc
1 2 3 4 5 6 7 8 9 10 11 12 13 140
2
4
6
Period
Dem
and
► Croston’s method widely used for intermittent demand forecasting [Syntetos & Boylan
2005]
o Based on exponential smoothing [Croston, 72]
o Extract non-zero demand dt and demand intervals pt from observed demand.
Forecasting intermittent time seriesForecasting intermittent time series
MotivationMotivationCroston’s Method
o Extract non-zero demand dt and demand intervals pt from observed demand.
o Use exponential smoothing to predict both.
o Demand of future period(s) is given by ct+1 = dt+1/pt+1.
► Outperforms conventional forecasting methods [Croston, 72, Syntetos, 01]
► Later corrected for positive bias [Syntetos & Boylan, 01, Leven Segerstedt, 04]
► Bias introduced due to division � Quantified analytically for any exponential
smoothing parameter.
MotivationMotivationCroston’s Method
► Based on exponential smoothing [Croston, 72]
Forecasting intermittent time seriesForecasting intermittent time series
tttFAF )1(
1αα −+=
+
Previous
forecast
Smoothing
► Exponential smoothing for demand and exponential smoothing for intervals
► Single smoothing parameter α � same for nominator (demand forecast) and
denominator (interval forecast) [Croston, 72]
► How to select best smoothing parameter?
� Literature rather cryptic � Manually select between [0.05, 0.3]
Current actualsSmoothing
parameter [0,1]
Intermittent DemandIntermittent DemandCroston’s Method
Conventional model optimisationConventional model optimisation
100
110
120
►Minimise in-sample MSE � Choose model parameter
►What is the problem of this approach for intermittent demand?
tttFAF )1(
1αα −+=
+ ∑ −=2
)(1
ttFA
nMSE
5 10 15 20 25 30 35 4080
90
Measuring PerformanceMeasuring Performance
How can we best measure method performance?
►Most error measures depend on different aggregations of four basic error units
o Squared Error SEt = (Actualt – Foreacastt)2.
o Absolute Error AEt = |Actualt – Foreacastt|.
10
Data
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
2
4
6
8
10
Period
De
ma
nd
Data
Croston's
Error measures are problematic ���� Need a different way to measure method performance
Demand “Rate” =
Demand/Interval
Demand
Zero?
Using error measures for intermittent demand is misleading ���� Consider zero forecasts!
Optimising Optimising Croston’sCroston’s
A novel optimization method on inventory metrics
► Simulate inventory
o Track total stock
o Track total backlog
oo Track realised service levels
► Consider lead times, inventory policy and target service levels
► For each item (time series) run a simulation based on the in-sample data
Optimising Optimising Croston’sCroston’s
► Different model parameters � different inventory performance
Minimise holding and backlog volume ���� Best model
Multi-objective optimisation; low holding ���� high backlog, low backlog ���� high holding
Need to optimise both!
Optimising Optimising Croston’sCroston’s
► Convert multi-objective optimisation to weight-of-sum linear optimisation.
Holding volume for a given
smoothing parameter &
service level
Smoothing
parameter
Service level
Total cost (to be minimised)
Backlog volume for a given
smoothing parameter &
service levelScalarisation factor �
Convert multiobjective to
weight of sums
� Scalarisation λ must be between [0, 1]
Optimising Optimising Croston’sCroston’s
Identification of λ and α parameters
► For each λ find optimum smoothing parameter α.
► Find combination of λ and α that gives minimum total cost.
► Select that α as the result of the optimisation
► λ is not related to underage or overage cost � That is the service level!
► Several combinations of λ & α may have same cost � Indifferent � Paretto optimal
Dual Parameter Dual Parameter Croston’sCroston’s
Relax constrain on smoothing parameter of demand size & interval
► Conventional method uses same smoothing parameter for demand and intervals
►Why? � Convenient � Distributions point that parameters should be different
► If optimisation is possible no need to constrain parameters to be equal
oo New Croston’s variant � More flexible
tXtX
tZtZ
t
XX
ZZF
ˆ)1(
ˆ)1(1
αα
αα
−+
−+=
+
Demand
Intervals
Empirical EvaluationEmpirical Evaluation
Experimental Setup
► Simulate inventory for lead times L = 1, 3, 6
► Simulate service levels 80%, 90%, 95%, 99%
► Data: Empirical distributions from Syntetos and Boylan (2005) � Automotive spare
parts.
► To test data conditions multiply empirical distributions by 1, 3, 5 � More
intermittency
► Each of 9 simulations (3 lead times X 3 multipliers) has 1000 time series
► Each time series: 36 observations in-sample
100 observations burn-in and 100 observations out-of-sample
► Track service levels, holding volume and backlog volume
ResultsResults
Results – Lead time 1 (similar for lead time = 3 & 6)Best manual α
OptD: Dual α
P-Opt is
conventional
optimization
With SB With SB
modification
ResultsResults
Results – Comparison between Croston’s and de-biased Croston’s SB
Using separate smoothing parameters is beneficial ���� Extend SB modification for
two parameters?
ConclusionsConclusions
►Optimisation on inventory metrics performs well � Bootstrapping in-sample
provided no gains
►Better than manually presetting parameters or conventional optimisation on
accuracy metrics
►►Results hold for different lead times and different intermittency levels
►Using different smoothing parameter for the non-zero demand and the inter-
demand intervals better than standard approach in the literature (using
single parameter for both)
Nikolaos KourentzesLancaster University Management School
Centre for Forecasting
Lancaster, LA1 4YX, UK
Tel. +44 (0) 7960271368
email [email protected]
Optimising Optimising Croston’sCroston’s
► Forecasts are used to replenish stocks
Order-up-to (T,S) policy
60
70
80
Demand
S
Orders
Stock
Stock < S
Initiate orders
S = Demand over lead
time + k*standard
deviation over lead time
40 50 60 70 80 90 100 110 120 1300
10
20
30
40
50
60
Stockout
Initiate orders