1. 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

2. Intermittent Demand

3. Accessed on 05 Aug 2010

4. 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)

5. 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

6. SBA : Syntetos & Boylan Approximation

7. 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

8. 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.

9. 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.

10. 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).

11. 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).

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

13. 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.)

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

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

16. 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.

17. 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.

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

19. 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 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

20. 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

21. STOCK CONTROL performance

22. 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.

23. 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.

24.
Thank you ? 24

25.
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