## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Size**} & \multirow{2}{*}{**Dataset**} & \multicolumn{2}{c}{**Method 1**} & \multicolumn{2}{c}{**Method 2**} & \multicolumn{2}{c}{**Proposed**} \\ \cline{3-8}  & & **Step** & **Solver** & **Step** & **Solver** & **Step** & **Solver** \\ \hline \multirow{8}{*}{Small} & S1 & 896,212.77 & 896,212.76 & 886,047.38 & 886,047.36 & 885,070.96 & 885,070.92 \\ \cline{2-8}  & S2 & 1,450,217.80 & 1,450,217.50 & 1,419,280.76 & 1,419,280.72 & 1,418,605.85 & 1,418,605.71 \\ \cline{2-8}  & S3 & 82,963.99 & 82,963.98 & 82,284.51 & 82,284.51 & 81,506.58 & 81,505.96 \\ \cline{2-8}  & S4 & 393,051.81 & 393,051.81 & 393,111.26 & 393,111.21 & 392,958.71 & 392,958.71 \\ \cline{2-8}  & S5 & 121,187.43 & 121,187.42 & 116,543.72 & 116,543.71 & 116,648.83 & 116,648.82 \\ \hline \multirow{8}{*}{Medium} & M1 & 29,069.60 & 29,069.60 & 28,837.52 & 28,837.52 & 28,735.94 & 28,735.94 \\ \cline{2-8}  & M2 & 934,245.09 & 934,244.89 & 756,669.78 & 756,669.75 & 765,244.84 & 765,244.02 \\ \cline{2-8}  & M3 & 1,649,100.03 & 1,649,099.38 & 1,599,938.85 & 1,599,937.30 & 1,571,685.18 & 1,571,685.15 \\ \cline{2-8}  & M4 & 519,353.65 & 519,353.61 & 424,846.72 & 424,846.34 & 434,269.59 & 434,269.50 \\ \cline{2-8}  & M5 & 165,837.93 & 165,837.93 & 165,830.64 & 165,830.63 & 165,816.74 & 165,816.72 \\ \hline \multirow{8}{*}{Large} & L1 & 1,516,519.54 & 1,516,519.40 & 1,479,599.21 & 1,479,599.13 & 1,470,586.75 & 1,470,586.56 \\ \cline{2-8}  & L2 & 1,880,659.84 & 1,880,659.83 & 1,880,401.03 & 1,880,400.76 & 1,877,245.36 & 1,877,244.59 \\ \cline{1-1} \cline{2-8}  & L3 & 555,763.08 & 555,762.96 & 555,644.55 & 555,644.38 & 554,925.61 & 554,925.58 \\ \cline{1-1} \cline{2-8}  & L4 & 28,361.93 & 28,361.93 & 28,278.23 & 28,278.21 & 28,270.66 & 28,270.66 \\ \cline{1-1} \cline{2-8}  & L5 & 53,326.57 & 53,326.57 & 50,839.83 & 50,839.76 & 50,986.27 & 50,986.27 \\ \hline \end{tabular}
\end{table} 
Table 4: The lowest MSE of three initial value setting methods using the optimal value for \(\alpha\) from step search and Solver from Microsoft Excel. Figure 4: Plots of the actual and predicted values when setting the initial value using Method 1, Method 2, and the proposed method; for example, of small, medium, and large time series data with S1, S2, M1, M2, L1, and L2. Figure 6 shows a stacked bar chart of the frequencies of obtaining the lowest MSE value when setting the initial value using the three initial value methods: Method 1, Method 2, and the proposed method for each of the 15 datasets. Notably, Figure 6 shows the MSE values for the proposed method were lower than those for the other two methods, a trend that was the same for all of the datasets. Moreover, in Figure 7, it is worth noting that using an initial value, in Method 1, of \(\alpha\) in the range of 0.1-0.4 provided forecasted values far from the actual values, whereas when it exceeded 0.5, all three methods provided similar results. The proposed method and Method 2 usually provided similar MSE values for the  
Figure 5: The lowest MSE using the step search method and Solver from Microsoft Excel to find the optimal value for \(\alpha\) after setting the initial value using M1 (Method 1), M2 (Method 2), and P (the proposed method) for large (L1–L5), medium (M1–M5), and small (S1–S5) time series datasets. same set of conditions, and their plotted lines tended to overlap each other. Meanwhile, lower MSE values for the three initial value setting methods became more evident as the settings were decreased. These results are consistent with the other findings, and it was concluded that the proposed method performed better than the others. It can be seen that when using values of \(\alpha\) ranging from 0 to 1 (0.001, 0.002,..., and 1) to produce 1000 settings, different average MSE levels were provided (Table 6). For example, for S1, Method 1, Method 2, and the proposed method produced MSE values of 984,214.12, 969,740.55, and 967,322.56, respectively.