RESULTS

In Table 7.10 I show the performance of the WGP combined forecasting models. I explore the combinations of all eight individual forecasting methods and report the overall average MAPE and the average MAPE on different types of time series.

Table 7. 10 Weighted goal programming combinations - MAPE PERFORMANCE OF COMBINED FORECASTS

MAPE WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 7.27 7.30 7.28 7.23 7.25 7.25

Smooth 3.13 3.14 3.13 3.13 3.14 3.15

Hard 23.85 24.03 23.87 23.55 23.56 23.48

Seasonal 5.41 5.42 5.44 5.45 5.49 5.52

As we can see, the WGP model results are very similar to the single objective LP (Table 7.4). We can observe a small improvement for weights 4, 5 and 6 with the best results weight 4. The only approach that outperforms WGP is the average LP combination technique, nevertheless, WGP is a bit simpler because we need to solve only one linear program compared to the latter where we need to solve three (MinSAD, MinSAPE and MinMaxAD) and calculate the average of them. However, the differences are very small.

Table 7.11 shows the performance of the WGP according to the sMAPE. The results are similar. The best approaches are with weight 4, 6 and 5. Weight 6 is the best on the hard series, weights 1, 2, 3 and 4 are the best on the smooth series and weights 1 and 2 on the seasonal series. WGP outperforms simple LP overall, but the latter is better on the smooth and seasonal series. However, the differences are small.

159 Table 7. 11 Weighted goal programming combinations - sMAPE

PERFORMANCE OF COMBINED FORECASTS

sMAPE WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 7.08 7.11 7.09 7.03 7.05 7.03

Smooth 3.03 3.03 3.03 3.03 3.04 3.05

Hard 23.16 23.31 23.13 22.76 22.79 22.59

Seasonal 5.36 5.36 5.38 5.39 5.44 5.46

The performance according to MASE is found in Table 7.12. All the alternative weights perform the same overall and on the smooth series. On the seasonal series the best are weights 1, 2,3 and 4 and on the hard series weights 4, 5 and 6. WGP outperforms simple LP overall and performs as good as MinMaxAD on the hard series and as good as MinSAD and MinSAPE on the seasonal series. On the other hand, it performs slightly worse than MinSAD on the smooth series.

Table 7. 12 Weighted goal programming combinations - MASE PERFORMANCE OF COMBINED FORECASTS

MASE WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 0.81 0.81 0.81 0.81 0.81 0.81

Smooth 0.96 0.96 0.96 0.96 0.96 0.96

Hard 0.85 0.86 0.86 0.84 0.84 0.84

Seasonal 0.52 0.52 0.52 0.52 0.53 0.53

In Table 7.13 we can see the percentage difference between the MAD of the combined forecasting technique and this of the best individual technique.

Table 7. 13 Weighted goal programming combinations - Difference between the best PERFORMANCE OF COMBINED FORECASTS

%BEST WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 4.91 5.31 5.42 5.20 5.54 5.64

Smooth 6.40 6.72 6.90 6.95 7.14 7.34

Hard 10.48 11.54 10.89 9.24 9.54 8.95

Seasonal -0.84 -0.65 -0.24 -0.15 0.46 0.78

The results are very similar to the single objective LP models (Table 7.7). Weight 1 gives the smallest average percentage difference between MAD; weights 6 and 4 give the smallest average percentage difference on the hard series.

Table 7.14 shows the percentage difference between the MAD of the worst technique on each series and this of the WGP combinations.

160 Table 7. 14 Weighted goal programming combinations - Difference between the worst

All Techniques PERFROMANCE OF COMBINED FORECASTS

%WORST WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 43.60 43.33 43.29 43.47 43.30 43.26

Smooth 42.72 42.50 42.39 42.35 42.20 42.05

Hard 22.23 21.49 22.01 23.29 23.08 23.49

Seasonal 57.01 56.92 56.72 56.67 56.48 56.40

By comparison with Table 7.8, the difference between the WGP approach and the single objective MinSAD, MinSAPE and average LP are insignificant.

Table 7. 15 Weighted goal programming combinations - Difference between the average All Techniques PERFORMANCE OF COMBINED FORECASTS

%AVERAGE WEIGHTED GOAL PROGRAMMING

a2 1 2 3 4 5 6

All 26.27 25.94 25.89 26.09 25.88 25.84

Smooth 20.67 20.40 20.27 20.22 20.06 19.88

Hard 7.06 6.17 6.73 8.14 7.88 8.36

Seasonal 46.89 46.78 46.53 46.48 46.24 46.14

Table 7.15 shows the percentage difference between the average MAD of all the individual techniques on each of the series and the MAD of the WGP combinations. The results are similar with Table 7.13. There WGP approaches perform as good as the single objective LP and the differences are small.

A general conclusion about the WGP approaches is that they slightly improve the overall results of the three LP approaches as well as for each subgroup of the series separately. WGP performs slightly worse than the average LP model according to the MAPE, but it is better according to the other accuracy indices. In addition, the first approach requires solving one program only. WGP seem to be stable and have the same good performance according to all the performance measurement indices. Nevertheless, the differences between all the LP approaches are small.

7.6 CONCLUSION

The aim of the chapter was to examine the applicability of LP as a tool to combine forecasts. For this reason four simple linear programs and one weighted goal program was formulated (the latter was tested with alternative weights). The performance of these approaches was compared with traditional combination approaches that are found in the literature. The analysis shows that linear

161 programming is a very good tool for combined forecasting. The simple LP methods outperform the traditional combined forecasting techniques that are found in the literature. On the other hand, the WGP methods outperform the former. Moreover, it seems that the distribution of the more and less accurate individual forecasts does affect the accuracy of traditional combination techniques, but not this of the LP based combinations. This is because while traditional techniques tend to distribute the weights over all the individual forecasts, LP approaches tend to select only the most accurate approaches.

The conclusions of this chapter are summarised as follows:

1. LP models outperform the traditional combination techniques in general.

2. LP is the only method for combination that outperforms all the individual techniques.

3. The inverse proportion to MSE and the weights based on the absolute error are the most accurate traditional combination techniques.

4. WGP performs slightly better than single objective LP.

5. All LP models, except MinMaxAD, are better than the traditional techniques on the smooth and seasonal series, but they give similar results on the hard series.

6. MinMaxAD gives the best results on the hard series.

LP is a very good method for combing forecast. The last step of the study is to test LP as a tool to minimise forecasting cost.

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