Lancaster University Management School.

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1 Optimum Parameters for Croston Intermittent Demand Methods The 32 nd Annual International Symposium on Forecasting Nikolaos Kourentzes Lancaster University Management School

2 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 Series o Frequent intervals with no demand o Large variation in demand levels when it occurs Observed in both manufacturing and service environments Motivation Croston s Method Period 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 Demand

3 Motivation Croston s Method Forecasting intermittent time series Croston s method widely used for intermittent demand forecasting [Syntetos & Boylan 2005] o Based on exponential smoothing [Croston, 72] o Extract non-zero demandd t and demand intervalsp t from observed demand. o Use exponential smoothing to predict both. o Demand of future period(s) is given byc t+1 =d t+1 /p t+1. Outperforms conventional forecasting methods [Croston, 72, Syntetos, 01] Later corrected for positive bias [Syntetos & Boylan, 01, Leven Segerstedt, 04] t Bias introduced due to division Quantified analytically for any exponential smoothing parameter. t

4 Motivation Croston s Method Forecasting intermittent time series Based on exponential smoothing [Croston, 72] F = A + (1 ) F t+ 1 α t α t Previous forecast Smoothing parameter [0,1] Current actuals 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]

5 Intermittent Demand Croston s Method Conventional model optimisation F = A + (1 ) F t 1 α t α + t MSE = ( Minimise in-sample MSE Choose model parameter What is the problem of this approach for intermittent demand? 1 n A t F t 2 )

6 Measuring Performance How can we best measure method performance? Most error measures depend on different aggregations of four basic error units o Squared Error SE t = (Actual t Foreacast t ) 2. o Absolute Error AE t = Actual t Foreacast t. 10 Data Demand Demand Zero? Croston's Demand Rate = Demand/Interval Period Using error measures for intermittent demand is misleading Consider zero forecasts! Error measures are problematic Need a different way to measure method performance

7 Optimising Croston s A novel optimization method on inventory metrics Simulate inventory o Track total stock o Track total backlog o 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

8 Optimising Croston 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!

9 Optimising Croston s Convert multi-objective optimisation to weight-of-sum linear optimisation. Smoothing parameter Service level Holding volume for a given smoothing parameter & service level Total cost (to be minimised) Scalarisation factor Convert multiobjective to weight of sums Backlog volume for a given smoothing parameter & service level Scalarisation λ must be between [0, 1]

10 Optimising Croston 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 costthat is the service level! Several combinations of λ & α may have same costindifferentparetto optimal

11 Dual Parameter Croston 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 o New Croston s variant More flexible F t+ 1 = α α X Z Z X t t + (1 + (1 α α Z X ) Zˆ t ) Xˆ t Demand Intervals

12 Empirical 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

13 Experimental Setup - Data Empirical Evaluation

14 Results Results Lead time 1 (similar for lead time = 3 & 6) Best manual α OptD: Dual α P-Opt is conventional optimization With SB modification

15 Results Results Comparison between Croston s and de-biased Croston s SB Using separate smoothing parameters is beneficial Extend SB modification for two parameters?

16 Conclusions 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 interdemand intervals better than standard approach in the literature (using single parameter for both)

17 Nikolaos Kourentzes Lancaster University Management School Centre for Forecasting Lancaster, LA1 4YX, UK Tel. +44 (0)

18 Optimising Croston s Forecasts are used to replenish stocks Order-up-to (T,S) policy Stock < S Initiate orders S = Demand over lead time + k*standard deviation over lead time Demand S Orders Stock Stockout