Dr. Kostas Nikolopoulos Professor of Decision Sciences, University of Wales, Bangor, U.K. Adjunct Professor, Korea University, ISC 2009/2010 ADIDA : An Aggregate - Disaggregate Intermittent Demand Approach to Forecasting An Empirical Proposition and Analysis of Forecasting and Stock Control performance

Intermittent Demand

http://www.linkedin.com/answers/business-operations/inventory-management/OPS_IVM/664920-48121704 Accessed on 05 Aug 2010

Background Intermittent Demand stands for 60% of demand in a typical inventory setting Croston’s method is the standard approach in software/business but it is biased (S&B 2001) SBA reduces bias and increases predictability (S&B 2005) Various other approaches more or less successful have been proposed (70+ citations in Croston 1972 in the last 5 years)

Croston’s Approach Demand Forecast = (Volume Forecast) / (Interval Forecast) where: (Interval Forecast) = the exponentially smoothed (or moving average) inter-demand interval, updated only if demand occurs in period, and (Volume Forecast) = the exponentially smoothed (or moving average) size of demand, updated only if demand occurs in period

SBA : Syntetos & Boylan Approximation

ADIDA Forecasting Framework A: Original data (months) B: Aggregate data (quarters) C: A quarterly forecast is produced D: The quarterly forecast id broken down to three equal monthly forecasts

Temporal Aggregation Temporal aggregation refers to aggregation in which a low frequency time series (e.g. quarterly) is derived from a high frequency time series (e.g. monthly) This is ignored in much commercial practice, and there is only a small body of academic research on the subject Nevertheless, temporal aggregation is a promising approach for intermittent demand, as forecasts at higher levels of aggregation are generally more accurate and less variable than those at lower levels of aggregation The level of temporal aggregation may be chosen to mirror that of the forecast horizon (lead-time), or may exceed it, in which case disaggregation mechanisms are required.

Advantages and disadvantages The accumulation of demand observations in lower-frequency ‘time buckets’ reduces the number of zero demands and the resulting series bear a greater resemblance to fast-moving items Variance is expected to be lower Identification of the underlying series’ characteristics such as trend and seasonality Use methods originally designed for fast moving items like Theta An obvious disadvantage related to temporal aggregation is that of losing information since the frequency and number of observations is reduced.

The objectives of our empirical study are as follows To provide for the first time some results on the performance of Temporal Aggregation when used for items with intermittent demands; To empirically determine optimum aggregation levels; To consider appropriate disaggregation mechanisms and link their performance to the statistical properties of the original series; To assess the effects of temporal aggregation in time buckets that equal the lead time (plus review period).

Empirical Investigation 5,000 SKUs (spare parts) from the Royal Air Force (RAF), UK Individual demand histories (monthly data) covering 7 years’ history (84 monthly demand observations) Actual lead time is available Two methods were considered: SBA and Naïve Sliding simulation (rolling evaluation) Results based on Mean Absolute Scaled Error (Hyndman and Koehler, 2006).

Accuracy results for two randomly selected series Optimal aggregation levels (per series) Accuracy results for two randomly selected series

Optimal aggregation levels (per series) Interpretation of results The ADIDA process functions as self-improving mechanism for SBA and Naïve methods Across all series, the benefit is perhaps more marked for the SBA One might have expected more modest (comparative) improvements for the former method, since it has in fact been constructed for application on intermittent series However, the sparseness of data, i.e. the great number of zero observations present in each series renders the Naïve method a very accurate estimation procedure The results also demonstrate that although there is an ‘optimum’ level of aggregation this is not the same across series. (This was expected since, theoretically, such a level relates to the underlying demand structure of the series.)

Optimal aggregation levels (across series) (1 of 2) Extrapolation Method: Naive

Optimal aggregation levels (across series) (2 of 2) Extrapolation Method: SBA

Optimal aggregation levels (across series) Interpretation of results The results indicate that the ADIDA process may lead to substantial improvements in a single method’s application. (In the case of SBA the improvements are also statistically significant at the 5% level.) The validity of the results has been further examined and confirmed through the application of two (2) more error measures: Mean Square Error and Relative Geometric Root Mean Squared Error For the Naïve method, a minimum error is achieved (across series) via an aggregation level of nine periods The empirical minimum relates to the specific dataset used for experimentation purposes and the finding may not be necessarily generalised to other situations However, this is a promising result in terms of potentially introducing operationalised rules for an entire group of SKUs as well as linking temporal aggregation to cross-sectional issues.

Empirical determination of the best disaggregation method Three methods considered: i) Equal Weights (EQW); ii) Previous period weights; iii) Average weights The results indicate the ‘best’ performance of the EQW disaggregation mechanism This superiority was theoretically expected due to the stationary nature of the demand data examined in this research.

Aggregation level = LT+R ... a managerial-driven heuristic

Aggregation level = LT+R We have also examined the performance of ADIDA when the aggregation level matches the forecast horizon required for stock control purposes CumError = Cumulative Demand over LT + 1 - Forecast over LT + 1   4,352 SKUs Forecasts ADIDA Forecasts Naïve SBA Bias ME 2.35 -3.57 -0.39 -2.55 MdE 1.00 -1.59 0.00 -1.37 Scaled Errors MAsE 125.53% 92.13% 99.84% 89.24% MdAsE 13.04% 20.93% 19.56% 19.65% Squared Errors MSE 8147.29 2082.39 3092.99 2084.51

STOCK CONTROL performance Forecasting Method Holding (volume) Backlog (volume) Inventory cost (£) Backlog cost (£) Total cost (£) CSL Classical Approach CROSTON 90% 19.530 0.492 5.019 0.674 5.693 0.852 95% 26.810 0.407 6.615 0.538 7.153 0.885 99% 53.048 0.301 11.584 0.387 11.971 0.920 SBA 18.994 0.502 4.885 0.686 5.571 0.849 26.182 0.415 6.479 0.551 7.030 0.882 52.461 0.307 11.440 0.393 11.833 0.918 SES 20.674 0.497 5.094 0.694 5.787 0.843 28.995 0.424 6.789 0.557 7.346 0.872 54.172 0.332 11.549 0.425 11.974 0.909 NAIVE 55.173 0.846 10.852 1.764 12.616 0.550 58.695 0.842 11.580 1.762 13.342 65.665 0.837 13.009 1.760 14.769 0.552 Aggregation Approach (ADIDA,CR) 23.440 0.444 5.662 0.632 6.294 0.866 33.199 0.343 7.689 0.459 8.147 0.904 62.049 0.220 13.168 0.272 13.440 0.946 (ADIDA,SBA) 22.835 0.452 5.535 0.641 6.176 0.863 32.498 0.349 7.505 0.469 7.974 0.902 61.372 0.224 13.005 0.275 13.281 0.944 (ADIDA,SES) 23.002 0.449 5.635 0.635 6.270 0.865 32.989 0.347 7.617 0.463 8.080 0.903 61.849 0.222 13.142 0.273 13.415 0.945 (ADIDA,Naive) 19.197 0.700 3.694 1.177 0.695 22.986 0.675 4.392 1.147 0.703 33.268 0.643 6.072 1.123 0.709

STOCK CONTROL performance

Conclusions The ADIDA process may lead to substantial improvements in a single method’s application; thus, it may be perceived as a method self-improvement mechanism The empirical results demonstrate that an optimal aggregation level may exist. Setting the aggregation level LT+R, shows very promising results. This simple heuristic would make sense in a practical inventory setting, where cumulative forecasts over that time horizon are required for stock control decision making.

Future Work Application of other methods originally designed for fast moving items Interactions between temporal and cross-sectional aggregation Theoretical underpinnings of the ADIDA process.

kostas@fortank.com Thank you ? 24

www.m4competition.com Spyros Makridakis - Distinguished Research Professor of Decision Sciences, INSEAD, France Vassilis Assimakopoulos - Professor of Forecasting Methods, NTUA, Greece Aris Syntetos - Professor of Operational Research and Operations Management, University of Salford, U.K. Dimitrios Thomakos - Professor of Applied Econometrics, University of Peloponnese, Greece Konstantinos Nikolopoulos - Professor of Decision Sciences, University of Wales, Bangor, U.K. Hosted by the Forecasting & Strategy Unit of the National Technical University of Athens