58 2010 10 25879.15 26432 552.8517
59 2010 11 23942.08 24471 528.9158
60 2010 12 23827.88 25337 1509.125
61 2011 1 21859.07 62 2011 2 20144.56 63 2011 3 24452.56 64 2011 4 25922.16 65 2011 5 28587.76 66 2011 6 28752.58 67 2011 7 30742.26 68 2011 8 30484.26 69 2011 9 29140.02 70 2011 10 28983.88 71 2011 11 26300.12 72 2011 12 25675.7
4.4 Results of Forecasting for the Monthly Sales of Air Conditioner 4.4.1 Validation
The monthly market demand from January 2006 to December 2009 (48 months) is used to find the parameters of the forecasting models and validate the projections for the period from January 2010 to December 2010(12 months). In this research, the above sales data was ran using Number Cruncher Statistical System (NCSS) based on two models—winters multiplicative trend seasonal model and the decomposition model.
A. Forecast (Validation) Based on Winters Multiplicative Trend Seasonal Model The result was obtained at5:38:23 PM on 2011/6/30 using NCSS software and
summarized as in Table 20. A search is conducted to find the values of the smoothing constants that minimize MSE. The Mean Squared Error is 9.854667E+07and Pseudo R-Squared is 0.874496. A value near zero indicates a poorly fitting model, while a value near one indicates a well fitting one. Thus, this model is well fitted. 150
iterations were needed to find the best values for the smoothing constants. The values of the smoothing constants, α, β, and γ are 6.259737E-07, 2.238804E-02, and
0.5738754, respectively. In the current month, Intercept(A) is 4.686174 and Slope(B) is 5.427435E-10.
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The seasonal ratios for the next 12 months are also listed from Seasonal 1 Factor to Seasonal 12 Factor.
Figure 8 shows the monthly market sales forecast for air conditioner in Taiwan.
Table 20: Summary of the values of the smoothing constants, α, β, γ, and the 14 coefficients for validation based on Winters Multiplicative Trend Seasonal Model (air conditioner)
Log10(Variable) Sales Number of Rows 48
Mean 54961.98
Pseudo R-Squared 0.874496 Mean Square Error 9.854667E+07
Forecast Method Winter's with multiplicative seasonal adjustment.
Search Iterations 150
Search Criterion Mean Square Error
Alpha 6.259737E-07
Beta 2.238804E-02
Gamma 0.5738754
Intercept (A) 4.686174
Slope (B) 5.427435E-10
Season 1 Factor 0.9569405 Season 2 Factor 1.026979 Season 3 Factor 1.043513 Season 4 Factor 1.062904 Season 5 Factor 1.050147 Season 6 Factor 1.032684 Season 7 Factor 1.03014 Season 8 Factor 0.9768854 Season 9 Factor 0.9463719 Season 10 Factor 0.947819 Season 11 Factor 0.9456754 Season 12 Factor 0.9799403
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—Forecast; … Actual Sales
Figure 8: The monthly market sales and forecasts for air conditioner in Taiwan(ton) for validation based on WintersMultiplicativeTrend Seasonal Model
Table 21 shows the values of the forecasts, the dates, the actual values, and the
residuals. The residual is the difference between the actual sales and the forecast sales (actual – forecast).
Table 21: The monthly market sales for air conditioner in Taiwan (forecasts vs.
actual) for validation based on WintersMultiplicativeTrend Seasonal Model Row
1 2006 1 32145.6 27486 -4659.604
2 2006 2 53538.58 52957 -581.5829
3 2006 3 71706.66 78204 6497.341
4 2006 4 95442.55 116473 21030.45
5 2006 5 86570.91 93163 6592.089
6 2006 6 67374.88 63238 -4136.874
7 2006 7 86912.03 96299 9386.967
8 2006 8 38444.5 35477 -2967.498
9 2006 9 21929.44 18649 -3280.439
10 2006 10 24307.1 21219 -3088.102
11 2006 11 26477.6 25793 -684.6022
0.0 30000.0 60000.0 90000.0 120000.0
2005.9 2007.1 2008.4 2009.6 2010.9
Sales Forecast Plot
Time
Sales
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12 2006 12 31841.59 31280 -561.5923
13 2007 1 29382.73 40219 10836.27
14 2007 2 53204.05 62879 9674.949
15 2007 3 75366.29 78855 3488.714
16 2007 4 106997.3 94385 -12612.26
17 2007 5 90294.69 93782 3487.314
18 2007 6 64968.82 69081 4112.182
19 2007 7 92180.98 91320 -860.9882
20 2007 8 36712.44 43093 6380.564
21 2007 9 19982.18 19727 -255.1779
22 2007 10 22483.81 23511 1027.192
23 2007 11 26082.54 24506 -1576.542
24 2007 12 31518.1 34369 2850.905
25 2008 1 35183.09 45931 10747.91
26 2008 2 58557.93 67051 8493.065
27 2008 3 77349.08 87890 10540.93
28 2008 4 99566.69 109988 10421.32
29 2008 5 92279.84 92893 613.1529
30 2008 6 67297.83 69084 1786.17
31 2008 7 91685.95 55706 -35979.95
32 2008 8 40248.65 23724 -16524.65
33 2008 9 19835.33 19859 23.66716
34 2008 10 23067.67 23324 256.3333
35 2008 11 25165.8 21921 -3244.799
36 2008 12 33123.91 32745 -378.9044
37 2009 1 40998.92 24494 -16504.92
38 2009 2 63290.77 66216 2925.226
39 2009 3 83233.04 73731 -9502.042
40 2009 4 105420 89082 -16337.96
41 2009 5 92631.09 77147 -15484.09
42 2009 6 68317.04 69648 1330.959
43 2009 7 68883.48 65989 -2894.482
44 2009 8 29717.35 45259 15541.65
45 2009 9 19848.9 34410 14561.1
46 2009 10 23214.42 31477 8262.582
47 2009 11 23249.13 30200 6950.872
48 2009 12 32905.93 44441 11535.07
49 2010 1 30506.29
50 2010 2 64953.36
51 2010 3 77639.74
52 2010 4 95709.3
53 2010 5 83400.74
54 2010 6 69077.78
55 2010 7 67207.31
56 2010 8 37831.59
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57 2010 9 27218.44
58 2010 10 27646.79
59 2010 11 27014.64
60 2010 12 39099.44
B. Forecast (Validation) Based on Decomposition Model
The result was obtained at 5:42:16 PM on 2011/6/30 using NCSS software and summarized as in Table 22. The Mean Squared Error (MSE) is4.677556. Pseudo R-Squared is 0.9183424. A value near zero indicates a poorly fitting model, while a value near one indicates a well-fitting one. Thus, this model is also well fitted.
The equation used to predict the trend is:
Trend = a + bt where
a is the intercept b is the slope t is the time period
Table 22 shows thatais0.999897 and b is -0.000064. Note that the trend value obtained from this equation will be a ratio type value that will be multiplied by the mean to obtain the actual forecast.
Table 22: Summary of the output report for validation based on the Decomposition Model (air conditioner)
Forecast 10^[(Mean) x (Trend) x (Cycle) x (Season)]
Variable Sales
Number of Rows 48
Mean 4.677556
Mean Square Error 64,118,246.6 Pseudo R-Squared 0.9183424 Forecast Std. Error 8007.387
Trend Equation Trend = (0.999897) + (-0.000064) * (Time Season Number) Number of Seasons 12
First Year 2006
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First Season 1
Table 23 shows the seasonal component ratios in the Decomposition model. The ratios used to adjust for each season (month or quarter). For example, the last ratio in this example is 0.975439. This indicates that the December correction factor is a 2.4561% decrease in the forecast.
Table 23: Seasonal component ratios for validation based on the Decomposition Model (air conditioner)
No. Ratio No. Ratio No. Ratio No. Ratio
1 0.970635 2 1.028424 3 1.050456 4 1.072341
5 1.059104 6 1.033179 7 1.043449 8 0.974671
9 0.931187 10 0.940598 11 0.944458 12 0.975439
Figure 9 shows the monthly market sales forecast for air conditioner in Taiwan for validation based on the Decomposition Model. The data plot allows us to analyze how closely the forecasts track the data. The plot also shows the forecasts at the end of the data series.
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—Forecast; … Actual Sales
Figure 9: The monthly market sales and forecasts for air conditioner in Taiwan(ton) for validation based on the Decomposition Model
Table 24 shows the values of the forecasts, the dates, the actual values, and the residuals for validation based on the Decomposition Model. The residual is the difference between the actual sales and the forecast sales. A value of one is used for all future cycle components. This ignores the cycle in the forecasts, and the random factor is assumed to be one.
Table 24: The monthly sales forecast for air conditioner in Taiwan (forecast vs.
actual) for validation based on the Decomposition Model Row 1 2006 1 32435.32 27486 -4949.324 0.9998 0.9937 0.9706 0.9841 2 2006 2 60565.01 52957 -7608.007 0.9998 0.9943 1.0284 0.9878 3 2006 3 77524.3 78204 679.6945 0.9997 0.9954 1.0505 1.0008
0.0 30000.0 60000.0 90000.0 120000.0
2005.9 2007.4 2008.8 2010.3 2011.8
Sales Chart
Time
Sales
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4 2006 4 98714.84 116473 17758.15 0.9996 0.9961 1.0723 1.0144 5 2006 5 85845.14 93163 7317.856 0.9996 0.9963 1.0591 1.0072 6 2006 6 65125.73 63238 -1887.734 0.9995 0.9966 1.0332 0.9973 7 2006 7 74228.03 96299 22070.97 0.9994 0.9985 1.0434 1.0232 8 2006 8 36246.94 35477 -769.9415 0.9994 1.0007 0.9747 0.9980 9 2006 9 22850.44 18649 -4201.438 0.9993 1.0014 0.9312 0.9798 10 2006 10 25090.58 21219 -3871.58 0.9993 1.0007 0.9406 0.9835 11 2006 11 25946.83 25793 -153.8295 0.9992 1.0000 0.9445 0.9994 12 2006 12 36354.46 31280 -5074.455 0.9991 1.0004 0.9754 0.9857 13 2007 1 34571.55 40219 5647.455 0.9991 1.0006 0.9706 1.0145 14 2007 2 64799.48 62879 -1920.481 0.9990 1.0012 1.0284 0.9973 15 2007 3 83064.68 78855 -4209.683 0.9989 1.0023 1.0505 0.9954 16 2007 4 105921.9 94385 -11536.85 0.9989 1.0029 1.0723 0.9900 17 2007 5 92032.46 93782 1749.537 0.9988 1.0032 1.0591 1.0016 18 2007 6 69700.85 69081 -619.8524 0.9987 1.0034 1.0332 0.9992 19 2007 7 78644.15 91320 12675.85 0.9987 1.0044 1.0434 1.0133 20 2007 8 37709.26 43093 5383.738 0.9986 1.0052 0.9747 1.0127 21 2007 9 23723.66 19727 -3996.665 0.9985 1.0059 0.9312 0.9817 22 2007 10 26536.94 23511 -3025.938 0.9985 1.0070 0.9406 0.9881 23 2007 11 27826.38 24506 -3320.383 0.9984 1.0076 0.9445 0.9876 24 2007 12 38911.88 34369 -4542.884 0.9984 1.0077 0.9754 0.9883 25 2008 1 36207.4 45931 9723.602 0.9983 1.0058 0.9706 1.0227 26 2008 2 64551.73 67051 2499.269 0.9982 1.0017 1.0284 1.0034 27 2008 3 79750.47 87890 8139.535 0.9982 0.9994 1.0505 1.0086 28 2008 4 100885.2 109988 9102.766 0.9981 0.9995 1.0723 1.0075 29 2008 5 87050.25 92893 5842.753 0.9980 0.9991 1.0591 1.0057 30 2008 6 65443.4 69084 3640.596 0.9980 0.9985 1.0332 1.0049 31 2008 7 70949.8 55706 -15243.8 0.9979 0.9960 1.0434 0.9783 32 2008 8 33105.11 23724 -9381.114 0.9978 0.9936 0.9747 0.9680 33 2008 9 20657.15 19859 -798.1517 0.9978 0.9929 0.9312 0.9960 34 2008 10 22495.91 23324 828.0914 0.9977 0.9915 0.9406 1.0036 35 2008 11 23077.35 21921 -1156.35 0.9976 0.9900 0.9445 0.9949 36 2008 12 31855.02 32745 889.978 0.9976 0.9894 0.9754 1.0027 37 2009 1 30487.38 24494 -5993.376 0.9975 0.9901 0.9706 0.9788 38 2009 2 58379.97 66216 7836.027 0.9975 0.9933 1.0284 1.0115 39 2009 3 77821.58 73731 -4090.575 0.9974 0.9980 1.0505 0.9952 40 2009 4 102207.3 89082 -13125.31 0.9973 1.0014 1.0723 0.9881 41 2009 5 91102.37 77147 -13955.37 0.9973 1.0039 1.0591 0.9854 42 2009 6 70767.05 69648 -1119.046 0.9972 1.0064 1.0332 0.9986 43 2009 7 77971.28 65989 -11982.28 0.9971 1.0052 1.0434 0.9852 44 2009 8 36155.74 45259 9103.263 0.9971 1.0027 0.9747 1.0214 45 2009 9 22472.15 34410 11937.85 0.9970 1.0021 0.9312 1.0425 46 2009 10 24493.3 31477 6983.696 0.9969 1.0006 0.9406 1.0248 47 2009 11 25135.14 30200 5064.856 0.9969 0.9992 0.9445 1.0181 48 2009 12 34792.87 44441 9648.134 0.9968 0.9985 0.9754 1.0234
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49 2010 1 33530.56 0.9967 1.0000 0.9706 1.0000
50 2010 2 62310.73 0.9967 1.0000 1.0284 1.0000
51 2010 3 78879.24 0.9966 1.0000 1.0505 1.0000
52 2010 4 99692.17 0.9966 1.0000 1.0723 1.0000
53 2010 5 86424.8 0.9965 1.0000 1.0591 1.0000
54 2010 6 65386.72 0.9964 1.0000 1.0332 1.0000
55 2010 7 72952.94 0.9964 1.0000 1.0434 1.0000
56 2010 8 34850.31 0.9963 1.0000 0.9747 1.0000
57 2010 9 21841.27 0.9962 1.0000 0.9312 1.0000
58 2010 10 24146.33 0.9962 1.0000 0.9406 1.0000
59 2010 11 25150.88 0.9961 1.0000 0.9445 1.0000
60 2010 12 35043.92 0.9960 1.0000 0.9754 1.0000
Note: This section shows the values of the forecasts, the dates, the actual values, the residuals, and the forecast ratios. A value of one is used for all future cycle
components. This ignores the cycle in the forecasts. And the random factor is assumed to be one.
From Table 25 and Table 26, we compare the MSEs for forecasts based on two models—the Winters Multiplicative Trend Seasonal Model and the Decomposition Model. It shows that the Decomposition Model generates better forecasts than those from Winters Multiplicative Trend Seasonal Model. Therefore, we complete the validation procedure.
Table 25: MSE of forecast for validation based on Winters Multiplicative Trend Seasonal Model (air conditioner)
Year
season Winter's Trend Seasonal (Forecast Sales)
Actual
Monthly Sales (Actual-Forecast)Squared
2010 1 30506.29 46198 8208.628
2010 2 64953.36 64560 -24259.54
2010 3 77639.74 96523 -1168.432
2010 4 95709.3 95333 -25495.7
2010 5 83400.74 97656 -3786.486
2010 6 69077.78 87720 3746.408
2010 7 67207.31 112769 31021.96
2010 8 37831.59 52779 3713.022
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2010 9 27218.44 35877 1518.764
2010 10 27646.79 32687 -711.8906
2010 11 27014.64 38619 6795.067
2010 12 39099.44 69289 21443.42
MSE (winter's trend) 383,298,846
Table 26: MSE of Forecast for validation based on the Decomposition Model (air conditioner)
2010 1 33530.56 46198 57322.94197
2010 2 62310.73 64560 2912.491056
2010 3 78879.24 96523 373857.6507
2010 4 99692.17 95333 241654.8291
2010 5 86424.8 97656 95932.05344
2010 6 65386.72 87720 13774.54322
2010 7 72952.94 112769 500073.8513
2010 8 34850.31 52779 274599.0565
2010 9 21841.27 35877 388097.8506
2010 10 24146.33 32687 416681.8691
2010 11 25150.88 38619 53005.30035
2010 12 35043.92 69289 12658.5226
MSE (decomposition) 387,630,017.6
4.4.2 Forecasts
After the validation procedures on the monthly market sales from January 2006 to December 2009 (48 months) of air conditioner,the Winters multiplicative trend seasonal model is selected as the forecasting model to project the sales of next 12 months. In this section, we will use
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theWinters model to generate forecasts for the period from January 2011 to December 2011(12 months).
The result was obtained at 12:25:13 PM on 2011/6/30 using NCSS software and summarized as in Table 27. A search is conducted to find the values of the smoothing constants that minimize MSE. The mean square error is 1.416344E+08 and Pseudo R-Squared is 0.824081. A value near zero indicates a poorly fitting model, while a value near one indicates a well-fitting one. Thus, this model is well-fitted. 86 iterations were required to find the best values for the smoothing constants. The values of the smoothing constants, α, β, γ, are 0.191321, 1.564809E-02, 0.696428, respectively. In the current month, Intercept(A) is 4.707559 and Slope(B) is 2.590202E-03.
Figure 10 shows the monthly market sales forecasts for air conditioner in Taiwan.
Table 27: Summary of the values of the smoothing constants, α, β, γ, and the 14 coefficients for forecasting based on Winters Multiplicative Trend Seasonal Model (air conditioner)
Forecast Method Winter's with multiplicative seasonal adjustment.
Forecasts are made for the period from January 2011 to December 2011(12 months)
Forecast Summary Section Log10(Variable) Sales Number of Rows 60
Mean 57803.08
Pseudo R-Squared 0.824081 Mean Square Error 1.416344E+08
Mean |Error| 7717.047
Mean |Percent Error| 14.11755 Search Iterations 86
Search Criterion Mean Square Error
Alpha 0.191321
Beta 1.564809E-02
Gamma 0.696428
Intercept (A) 4.707559
Slope (B) 2.590202E-03
Season 1 Factor 0.9661455 Season 2 Factor 1.012807 Season 3 Factor 1.042347 Season 4 Factor 1.049959 Season 5 Factor 1.048082 Season 6 Factor 1.035443
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Season 7 Factor 1.046123 Season 8 Factor 0.9812949 Season 9 Factor 0.9459578 Season 10 Factor 0.9389964 Season 11 Factor 0.9455498 Season 12 Factor 0.9872943
—Forecasts; … Actual Sales
Figure 10: The monthly market sales forecast for air conditioner in Taiwan(ton) based on Winters Multiplicative Trend Seasonal Model
Table 28 shows the values of the forecasts, the dates, the actual values, and the residuals. The residual is the difference between the actual sales and the forecast sales (actual-forecast).
Table 28: The monthly market sales for air conditioner in Taiwan (forecast vs actual) based on Winters Multiplicative Trend Seasonal Model
Row
1 2006 1 28879.4 27486 -1393.397
2 2006 2 51101.12 52957 1855.884
3 2006 3 72953.38 78204 5250.621
4 2006 4 106764.5 116473 9708.522
5 2006 5 97479.74 93163 -4316.741
6 2006 6 68512.74 63238 -5274.741
7 2006 7 89102.81 96299 7196.19
8 2006 8 36805.44 35477 -1328.441
9 2006 9 18674.17 18649 -25.17243
0.0 35000.0 70000.0 105000.0 140000.0
2005.9 2007.4 2008.9 2010.4 2011.9
Sales Forecast Plot
Time
Sales
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10 2006 10 19640.96 21219 1578.037
11 2006 11 21584.89 25793 4208.114
12 2006 12 28029.33 31280 3250.667
13 2007 1 30720.46 40219 9498.541
14 2007 2 61304.61 62879 1574.392
15 2007 3 89720.34 78855 -10865.33
16 2007 4 128445.6 94385 -34060.64
17 2007 5 100895.6 93782 -7113.563
18 2007 6 69081.13 69081 -0.1217377
19 2007 7 100010.5 91320 -8690.451
20 2007 8 37389.23 43093 5703.765
21 2007 9 19953.33 19727 -226.3326
22 2007 10 21923.02 23511 1587.981
23 2007 11 25497.24 24506 -991.2367
24 2007 12 30571.29 34369 3797.715
25 2008 1 36701.99 45931 9229.012
26 2008 2 63336.18 67051 3714.819
27 2008 3 85532.13 87890 2357.87
28 2008 4 114179.7 109988 -4191.672
29 2008 5 107908.7 92893 -15015.71
30 2008 6 75745.3 69084 -6661.298
31 2008 7 102658.5 55706 -46952.46
32 2008 8 39639.38 23724 -15915.38
33 2008 9 17250.72 19859 2608.282
34 2008 10 20353.51 23324 2970.493
35 2008 11 22479.87 21921 -558.8668
36 2008 12 29382.22 32745 3362.784
37 2009 1 37231.77 24494 -12737.77
38 2009 2 51036.69 66216 15179.31
39 2009 3 69861.54 73731 3869.461
40 2009 4 89662.89 89082 -580.8927
41 2009 5 79895.7 77147 -2748.704
42 2009 6 59340.33 69648 10307.67
43 2009 7 62621.05 65989 3367.95
44 2009 8 28972.63 45259 16286.37
45 2009 9 21821.97 34410 12588.03
46 2009 10 27445.29 31477 4031.705
47 2009 11 27768.33 30200 2431.672
48 2009 12 40452.3 44441 3988.704
49 2010 1 37989.37 46198 8208.628
50 2010 2 88819.54 64560 -24259.54
51 2010 3 97691.43 96523 -1168.432
52 2010 4 120828.7 95333 -25495.7
53 2010 5 101442.5 97656 -3786.486
54 2010 6 83973.59 87720 3746.408
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55 2010 7 81747.04 112769 31021.96
56 2010 8 49065.98 52779 3713.022
57 2010 9 34358.23 35877 1518.764
58 2010 10 33398.89 32687 -711.8906
59 2010 11 31823.93 38619 6795.067
60 2010 12 47845.58 69289 21443.42
61 2011 1 50215.78 62 2011 2 85211.59 63 2011 3 119399.9 64 2011 4 130858.1 65 2011 5 128934.2 66 2011 6 112569.2 67 2011 7 127712.1 68 2011 8 61993.04 69 2011 9 41900.33 70 2011 10 38961.39 71 2011 11 42181.54 72 2011 12 67900.21
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