37 2009 1 809.8107 905 95.18927
38 2009 2 909.4203 1038 128.5797
39 2009 3 1356.912 1350 -6.91211
40 2009 4 1558.833 1565 6.166496
41 2009 5 1851.291 1903 51.7087
42 2009 6 2273.668 2475 201.3322
43 2009 7 3212.855 4454 1241.145
44 2009 8 3488.607 4005 516.3934
45 2009 9 2766.078 3647 880.9222
46 2009 10 1974.475 1359 -615.4746
47 2009 11 936.9445 820 -116.9445
48 2009 12 778.7621 735 -43.76217
49 2010 1 982.3508 742 -240.3508
50 2010 2 906.9282 936 29.07175
51 2010 3 1283.419 1864 580.5806
52 2010 4 1786.146 1959 172.8543
53 2010 5 2226.52 2096 -130.5196
54 2010 6 2626.334 2324 -302.3345
55 2010 7 3351.405 3600 248.5954
56 2010 8 3205.901 3226 20.09929
57 2010 9 2392.863 2603 210.1373
58 2010 10 1573.596 1871 297.4036
59 2010 11 978.0168 964 -14.01678
60 2010 12 861.1707 780 -81.17062
61 2011 1 1067.558 62 2011 2 1136.387 63 2011 3 1599.269 64 2011 4 1848.076 65 2011 5 2199.113 66 2011 6 2674.779 67 2011 7 3639.806 68 2011 8 3357.914 69 2011 9 2494.585 70 2011 10 1572.923 71 2011 11 902.4263 72 2011 12 801.3745
4.3 Results of Forecasting for the Monthly Sales of Fresh Milk 4.3.1 Validation
The monthly market demand from January 2006 to December 2009 (48 months) of fresh milk is used to find the parameters of the forecasting models and validate the projections for the period
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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 the same two models that was also used for ice cream—winters multiplicative trend seasonal model and the decomposition model.
A. Forecast (Validation) Based on Winters Multiplicative Trend Seasonal Model The result was obtained at4:52:29 PM on 2011/6/11 using NCSS software and
summarized as in Table 11. A search is conducted to find the values of the smoothing constants that minimize MSE. The Mean Squared Error is 790666.8 and Pseudo R-Squared is 0.914516. 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. 151 iterations were needed to find the best values for the smoothing constants. The values of the smoothing constants, α, β, and γ are 0.8145158, 5.716994E-08, and 2.286691E-03, respectively. In the current month, Intercept(A) is 4.380449 and Slope(B) is 1.021276E-04.
The seasonal ratios for the next 12 months are also listed from Seasonal 1 Factor to Seasonal 12 Factor.
Figure 5 shows the monthly market sales forecast for fresh milk in Taiwan.
Table 11: Summary of the values of the smoothing constants, α, β, γ, and the 14 coefficients for validation based on Winters Multiplicative Trend Seasonal Model (fresh milk)
Forecast Method Winter's with
multiplicative seasonal adjustment.
Log10(Variable) Sales Number of Rows 48
Mean 22284.81
Pseudo R-Squared 0.914516 Mean Square Error 790666.8
Mean |Error| 723.2148
Mean |Percent Error| 3.293337 Search Iterations 151
Search Criterion Mean Square Error
Alpha 0.8145158
Beta 5.716994E-08
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Gamma 2.286691E-03
Intercept (A) 4.380449
Slope (B) 1.021276E-04
Season 1 Factor 0.9797085 Season 2 Factor 0.9726461 Season 3 Factor 0.9911656 Season 4 Factor 0.9972439 Season 5 Factor 1.007274 Season 6 Factor 1.00772 Season 7 Factor 1.015774 Season 8 Factor 1.015128 Season 9 Factor 1.01006 Season 10 Factor 1.009061 Season 11 Factor 0.9990233 Season 12 Factor 0.9951959
—Forecasts; … Actual Sales
Figure 5: The monthly sales and forecasts for fresh milk in Taiwan(ton) for validation based on Winters Multiplicative Trend Seasonal Model.
Table 12 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).
14000.0 18000.0 22000.0 26000.0 30000.0
2005.9 2007.1 2008.4 2009.6 2010.9
Sales Forecast Plot
Time
Sales
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Table 12: The monthly market sales for fresh milk in Taiwan (Forecasts vs.
Actual) for validation based on Winters Multiplicative Trend Seasonal Model Row
1 2006 1 18524.37 17263 -1261.367
2 2006 2 16304.96 16452 147.0355
3 2006 3 19763.45 19355 -408.4469
4 2006 4 20647.93 21954 1306.069
5 2006 5 24004.4 24242 237.5971
6 2006 6 24311.96 24164 -147.9632
7 2006 7 26230.05 27492 1261.954
8 2006 8 27083.58 29099 2015.417
9 2006 9 27286.1 26740 -546.0951
10 2006 10 26577.32 25574 -1003.32
11 2006 11 23287.58 23704 416.4235
12 2006 12 22737.37 20779 -1958.372
13 2007 1 18098.91 18703 604.0874
14 2007 2 17322.6 16645 -677.6018
15 2007 3 20185.11 20433 247.8888
16 2007 4 21671.71 21322 -349.7108
17 2007 5 23647.33 24406 758.6683
18 2007 6 24377.85 24066 -311.8484
19 2007 7 26156.49 26041 -115.4931
20 2007 8 25900.96 24348 -1552.966
21 2007 9 23421 23167 -253.9982
22 2007 10 22989.44 23505 515.5574
23 2007 11 21184.72 20084 -1100.717
24 2007 12 19531.2 19439 -92.19598
25 2008 1 16688.12 17357 668.8781
26 2008 2 16064.96 15271 -793.9627
27 2008 3 18526.66 19081 554.3356
28 2008 4 20163.89 20514 350.1097
29 2008 5 22600.62 22044 -556.6193
30 2008 6 22249.29 22121 -128.2872
31 2008 7 23994.6 23875 -119.5966
32 2008 8 23749.86 24697 947.1389
33 2008 9 23317.1 23972 654.9006
34 2008 10 23618.25 23762 143.7509
35 2008 11 21476.79 20720 -756.7885
36 2008 12 20082.71 20820 737.2847
37 2009 1 17722.46 18686 963.5344
38 2009 2 17241.8 18911 1669.195
39 2009 3 22422.36 21995 -427.355
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40 2009 4 23475.92 22062 -1413.921
41 2009 5 24687.64 24287 -400.6447
42 2009 6 24475.45 25094 618.5548
43 2009 7 27090.71 26091 -999.7139
44 2009 8 26110.6 24859 -1251.595
45 2009 9 23854.44 23934 79.55725
46 2009 10 23687.36 23928 240.6438
47 2009 11 21608.9 23225 1616.096
48 2009 12 22056.95 23388 1331.054
49 2010 1 19790.93
50 2010 2 18432.91
51 2010 3 22228.67
52 2010 4 23641.38
53 2010 5 26167.59
54 2010 6 26292.02
55 2010 7 28526.69
56 2010 8 28348.08
57 2010 9 26939.96
58 2010 10 26675.74 59 2010 11 24109.64 60 2010 12 23200.82
B. Forecast (Validation) Based on Decomposition Model
The result was obtained at 4:50:35 PM on 2011/6/11 using NCSS software and summarized as in Table 13. The Mean Squared Error (MSE) is 1006689.149. Pseudo R-Squared is 0.8911608. 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 13 shows that a is0.999829 and b is -0.000019. 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.
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Table 13: Summary of the output report for validation based on the Decomposition Model (fresh milk)
Forecast 10^[(Mean) x (Trend) x (Cycle) x (Season)]
Variable Sales
Number of Rows 48
Mean 4.343749
Mean Square Error 1006689.149 Pseudo R-Squared 0.8911608 Forecast Std. Error 1003.339
Trend Equation Trend = (0.999829) + (-0.000019) * (Time Season Number) Number of Seasons 12
First Year 2006
First Season 1
Table 14 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 0.993180. This indicates that the December correction factor is a 0.682% decrease in the forecast.
Table 14: Seasonal Component Ratios for validation based on the Decomposition Model (fresh milk)
No. Ratio No. Ratio No. Ratio No. Ratio
1 0.977703 2 0.971128 3 0.990244 4 0.996752
5 1.007150 6 1.007762 7 1.015624 8 1.014469
9 1.008775 10 1.007440 11 0.997255 12 0.993180
Figure 6 shows the monthly market sales forecast for fresh milk 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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—Forecasts; … Actual Sales
Figure 6: The monthly market sales and forecasts for fresh milk in Taiwan(ton) for validation based on the Decomposition Model
Table 15 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 15: The monthly sales forecast for fresh milk in Taiwan (forecast vs. actual) for validation based on the Decomposition Model.
Row
No. Year
Season Forecast
Sales Actual
Sales Residual Trend
Factor Cycle
Factor Season
Factor Error Factor 1 2006 1 19189.66 17263 -1926.658 0.9998 1.0087 0.9777 0.9893
14000.0 18000.0 22000.0 26000.0 30000.0
2005.9 2007.4 2008.8 2010.3 2011.8
Sales Chart
Time
Sales
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2 2006 2 17790.11 16452 -1338.109 0.9998 1.0078 0.9711 0.9920 3 2006 3 21285.06 19355 -1930.065 0.9998 1.0064 0.9902 0.9905 4 2006 4 22511.74 21954 -557.7376 0.9998 1.0055 0.9968 0.9975 5 2006 5 24731.82 24242 -489.8178 0.9997 1.0045 1.0072 0.9980 6 2006 6 24642.65 24164 -478.6522 0.9997 1.0035 1.0078 0.9981 7 2006 7 26680.74 27492 811.259 0.9997 1.0036 1.0156 1.0029 8 2006 8 26475.63 29099 2623.374 0.9997 1.0040 1.0145 1.0093 9 2006 9 25074.18 26740 1665.821 0.9997 1.0043 1.0088 1.0063 10 2006 10 24766.27 25574 807.7272 0.9996 1.0044 1.0074 1.0032 11 2006 11 22337.32 23704 1366.683 0.9996 1.0043 0.9973 1.0059 12 2006 12 21444.19 20779 -665.1892 0.9996 1.0044 0.9932 0.9968 13 2007 1 18314.07 18703 388.9309 0.9996 1.0042 0.9777 1.0021 14 2007 2 16983.71 16645 -338.7123 0.9996 1.0032 0.9711 0.9979 15 2007 3 20301.7 20433 131.2979 0.9995 1.0019 0.9902 1.0006 16 2007 4 21465.03 21322 -143.0292 0.9995 1.0010 0.9968 0.9993 17 2007 5 23570.17 24406 835.8257 0.9995 0.9999 1.0072 1.0035 18 2007 6 23484.51 24066 581.4897 0.9995 0.9990 1.0078 1.0024 19 2007 7 25251.3 26041 789.699 0.9995 0.9984 1.0156 1.0030 20 2007 8 24792.58 24348 -444.5848 0.9995 0.9978 1.0145 0.9982 21 2007 9 23272.21 23167 -105.215 0.9994 0.9971 1.0088 0.9995 22 2007 10 22861.6 23505 643.3984 0.9994 0.9967 1.0074 1.0028 23 2007 11 20535.24 20084 -451.2422 0.9994 0.9961 0.9973 0.9978 24 2007 12 19567.51 19439 -128.5135 0.9994 0.9954 0.9932 0.9993 25 2008 1 16658.45 17357 698.5532 0.9994 0.9947 0.9777 1.0042 26 2008 2 15558.52 15271 -287.5247 0.9993 0.9944 0.9711 0.9981 27 2008 3 18851.7 19081 229.3056 0.9993 0.9946 0.9902 1.0012 28 2008 4 20149.25 20514 364.751 0.9993 0.9948 0.9968 1.0018 29 2008 5 22383.48 22044 -339.4853 0.9993 0.9950 1.0072 0.9985 30 2008 6 22614.69 22121 -493.6901 0.9993 0.9955 1.0078 0.9978 31 2008 7 24602.55 23875 -727.5468 0.9993 0.9961 1.0156 0.9970
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32 2008 8 24618.56 24697 78.44176 0.9992 0.9973 1.0145 1.0003 33 2008 9 23611.03 23972 360.9675 0.9992 0.9988 1.0088 1.0015 34 2008 10 23509.59 23762 252.4112 0.9992 0.9997 1.0074 1.0011 35 2008 11 21385.09 20720 -665.0854 0.9992 1.0004 0.9973 0.9968 36 2008 12 20721.76 20820 98.24145 0.9992 1.0014 0.9932 1.0005 37 2009 1 17904.67 18686 781.3312 0.9991 1.0023 0.9777 1.0044 38 2009 2 16828.29 18911 2082.705 0.9991 1.0027 0.9711 1.0120 39 2009 3 20385.17 21995 1609.828 0.9991 1.0027 0.9902 1.0077 40 2009 4 21763.6 22062 298.3979 0.9991 1.0028 0.9968 1.0014 41 2009 5 24276.68 24287 10.31796 0.9991 1.0033 1.0072 1.0000 42 2009 6 24663.57 25094 430.4343 0.9990 1.0043 1.0078 1.0017 43 2009 7 26903.91 26091 -812.9073 0.9990 1.0051 1.0156 0.9970 44 2009 8 26918.68 24859 -2059.677 0.9990 1.0063 1.0145 0.9922 45 2009 9 25804.08 23934 -1870.08 0.9990 1.0078 1.0088 0.9926 46 2009 10 25690.19 23928 -1762.194 0.9990 1.0087 1.0074 0.9930 47 2009 11 23347.69 23225 -122.6869 0.9990 1.0095 0.9973 0.9995 48 2009 12 22615.37 23388 772.6312 0.9989 1.0104 0.9932 1.0034
49 2010 1 17470.36 0.9989 1.0000 0.9777 1.0000
50 2010 2 16356.65 0.9989 1.0000 0.9711 1.0000
51 2010 3 19795.06 0.9989 1.0000 0.9902 1.0000
52 2010 4 21120.88 0.9989 1.0000 0.9968 1.0000
53 2010 5 23428.6 0.9988 1.0000 1.0072 1.0000
54 2010 6 23567.85 0.9988 1.0000 1.0078 1.0000
55 2010 7 25488.83 0.9988 1.0000 1.0156 1.0000
56 2010 8 25191.5 0.9988 1.0000 1.0145 1.0000
57 2010 9 23794.24 0.9988 1.0000 1.0088 1.0000
58 2010 10 23474.64 0.9988 1.0000 1.0074 1.0000
59 2010 11 21199.69 0.9987 1.0000 0.9973 1.0000
60 2010 12 20350.38 0.9987 1.0000 0.9932 1.0000
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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 16 and Table 17, 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 16: MSE of Forecast for validation based on Winters Multiplicative Trend Seasonal Model (fresh milk)
Year season Winters
Forecast Actual
Monthly Sales (Actual-Forecast)Squared
2010 1 19790.93 21513 2965525.085
2010 2 18432.91 19069 404610.4881
2010 3 22228.67 23576 1815298.129
2010 4 23641.38 24613 944045.4244
2010 5 26167.59 26646 228876.1281
2010 6 26292.02 26905 375744.4804
2010 7 28526.69 27054 2168815.836
2010 8 28348.08 26610 3020922.086
2010 9 26939.96 26092 719036.1616
2010 10 26675.74 26432 59409.1876
2010 11 24109.64 24471 130581.0496
2010 12 23200.82 25337 4563264.992
MSE (winter's trend) 1,449,677.42
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Table 17: MSE of Forecast for validation based on the Decomposition Model (fresh milk)
2010 1 17470.36 21513 16342938.17
2010 2 16356.65 19069 7356842.523
2010 3 19795.06 23576 14295507.28
2010 4 21120.88 24613 12194902.09
2010 5 23428.6 26646 10351662.76
2010 6 23567.85 26905 11136570.12
2010 7 25488.83 27054 2449757.129
2010 8 25191.5 26610 2012142.25
2010 9 23794.24 26092 5279701.018
2010 10 23474.64 26432 8745978.17
2010 11 21199.69 24471 10701469.12
2010 12 20350.38 25337 24866379.02
MSE (decomposition) 10,477,820.80
4.3.2 Forecasts
After the validation procedures on the monthly market sales from January 2006 to December 2009 (48 months) of fresh milk, 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 the Winters model to generate forecasts for the period from January 2011 to December 2011(12 months).
The result was obtained at 4:56:48 PM on 2011/6/11 using NCSS software and summarized as in Table 18. A search is conducted to find the values of the smoothing constants that minimize MSE. The mean square error is 797011.9 and Pseudo R-Squared is 0.916643. 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. 136 iterations were required to find the best values for the smoothing constants. The values of the smoothing constants, α, β, γ, are 0.8730332, 3.756879E-08,
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3.947698E-03, respectively. In the current month, Intercept(A) is 4.373165 and Slope(B) is 7.648958E-04.
Figure 6 shows the monthly market sales forecasts for fresh milk in Taiwan.
Table 18: Summary of the values of the smoothing constants, α, β, γ, and the 14
coefficients for forecasting of fresh milk based on Winters Multiplicative Trend Seasonal Model
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 22799.82
Pseudo R-Squared 0.916643 Mean Square Error 797011.9 Mean |Error| 712.8579 Mean |Percent Error| 3.172555 Search Iterations 136
Search Criterion Mean Square Error
Alpha 0.8730332
Beta 3.756879E-08
Gamma 3.947698E-03
Intercept (A) 4.373165
Slope (B) 7.648958E-04
Season 1 Factor 0.9818563 Season 2 Factor 0.9736617 Season 3 Factor 0.9925297 Season 4 Factor 0.9980898 Season 5 Factor 1.007528 Season 6 Factor 1.007919 Season 7 Factor 1.014312 Season 8 Factor 1.01331 Season 9 Factor 1.008709 Season 10 Factor 1.008008 Season 11 Factor 0.9983027 Season 12 Factor 0.9957737
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—Forecast; … Actual Sales
Figure 7: The monthly market sales forecast for fresh milk in Taiwan(ton) based on Winters Multiplicative Trend Seasonal Model
Table 19 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 19: The monthly market sales for fresh milk in Taiwan (Forecast vs Actual) based on Winters Multiplicative trend seasonal model
Row
1 2006 1 18786.5 17263 -1523.498
2 2006 2 16111.33 16452 340.6747
3 2006 3 19838.37 19355 -483.3698
4 2006 4 20556.02 21954 1397.978
5 2006 5 23970.55 24242 271.4522
6 2006 6 24345.36 24164 -181.3639
7 2006 7 25832.35 27492 1659.653
8 2006 8 27049.8 29099 2049.197
9 2006 9 27566.1 26740 -826.1008
10 2006 10 26701.19 25574 -1127.194
11 2006 11 23360.32 23704 343.6839
12 2006 12 23104.55 20779 -2325.548
15000.0 20000.0 25000.0 30000.0 35000.0
2005.9 2007.4 2008.9 2010.4 2011.9
Sales Forecast Plot
Time
Sales
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13 2007 1 18355.65 18703 347.346
14 2007 2 17219.05 16645 -574.0549
15 2007 3 20218.14 20433 214.8584
16 2007 4 21611.04 21322 -289.041
17 2007 5 23511.05 24406 894.9453
18 2007 6 24428.79 24066 -362.7858
19 2007 7 25752.43 26041 288.5715
20 2007 8 25790.35 24348 -1442.348
21 2007 9 23466.98 23167 -299.9824
22 2007 10 23084.05 23505 420.9515
23 2007 11 21323.73 20084 -1239.727
24 2007 12 19768.79 19439 -329.7863
25 2008 1 16997.98 17357 359.0195
26 2008 2 15985.05 15271 -714.049
27 2008 3 18546.95 19081 534.0519
28 2008 4 20127.27 20514 386.7318
29 2008 5 22518.43 22044 -474.4281
30 2008 6 22228.12 22121 -107.1171
31 2008 7 23628.18 23875 246.8245
32 2008 8 23648.67 24697 1048.332
33 2008 9 23500.65 23972 471.3493
34 2008 10 23787 23762 -25.00126
35 2008 11 21605.67 20720 -885.6664
36 2008 12 20347.09 20820 472.9115
37 2009 1 18098.05 18686 587.9476
38 2009 2 17173.88 18911 1737.116
39 2009 3 22643.87 21995 -648.8704
40 2009 4 23390.41 22062 -1328.408
41 2009 5 24475.77 24287 -188.7715
42 2009 6 24448.85 25094 645.1514
43 2009 7 26719.79 26091 -628.7867
44 2009 8 25954.18 24859 -1095.18
45 2009 9 23913.81 23934 20.19368
46 2009 10 23806.44 23928 121.5641
47 2009 11 21737.89 23225 1487.106
48 2009 12 22491.32 23388 896.6757
49 2010 1 20256.48 21513 1256.518
50 2010 2 19679.5 19069 -610.4973
51 2010 3 23216.34 23576 359.662
52 2010 4 24939.16 24613 -326.1627
53 2010 5 27176.58 26646 -530.5823
54 2010 6 26865.81 26905 39.18773
55 2010 7 28750.25 27054 -1696.253
56 2010 8 27037.49 26610 -427.4862
57 2010 9 25503.13 26092 588.8686
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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