## 3 Simulation Results
### Case When Forecasting Error Follow A Log-Normal Distribution
Nevertheless, the performance of the GARCH-based methods can be shown by choosing particular values of the autoregressive parameter \(\phi\) and plotting the tradeoff curves and tick loss value for each quantile of interest. Table 2 gives the tick loss values of each safety stock measurement method for the four CSL levels and for an LT of one week and four weeks assuming demand that follow the AR (1) process with log-normal distribution and \(\phi=0.70\) or \(\phi=-0.70\). Values in blood in Table 2 indicate the minimum tick loss for a given CSL. The tick loss values in the table are also plotted in Figure 3. For example, the value of tick loss in the case of \(\phi=-0.70\) and an LT of four weeks is 3.08 at an 85% level when we apply the CEVT method. This value decreases to 0.48 when the CSL becomes at 99%. Choosing an LT of one week implies a minimum value of tick loss of 1.58 only for the 95% level. In the same case and at a 90% level, the FHS method gives better results. Additionally, in the case of \(\phi=0.70\) and LT of one week, FHS is more important compared with other methods given a minimum of tick loss for all CSL. This result is the same for four weeks. It appears that the FHS method provides the most performance in the majority of cases followed by the CEVT method and GARCH. Traditional methods cannot provide good results in comparison to GARCH-based methods. Figure 2: Average tick loss values for lead times of four weeks, when forecasting errors follow a lognormal distribution. Figure 3 shows the tick loss values in the case of log-normal distribution for \(\phi=0.70\) (**a**,**c**) and \(\phi=-0.70\) (**b**,**d**) and lead times of one week (**a**,**b**) and four weeks (**c**,**d**). Figure 3 shows the tick loss values in the case of log-normal distribution for \(\phi=0.70\) (on the left) and \(\phi=-0.70\) (on the right) and lead times of one week (upper panel) and four weeks (lower panel). We can remark for the four panels when the CEVT provides a higher tick loss value we find that the FHS provides the lowest loss, followed by the GARCH method and vice versa. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Safety Stock Estimation Method**} & \multirow{3}{*}{**LT = 1 Week**} & \multicolumn{6}{c}{\(\phi\) = 0.70} & \multicolumn{6}{c}{\(\phi\) = \(-\)0.70} \\ \cline{3-10}  & & \multicolumn{3}{c}{**CSL**} & \multicolumn{3}{c}{**CSL**} \\ \cline{3-10}  & & **85\%** & **90\%** & **95\%** & **99\%** & **85\%** & **90\%** & **95\%** & **99\%** \\ \hline \multirow{3}{*}{Simple methods} & SES & 3.68 & 3.02 & 2.12 & 1.01 & 2.88 & 2.38 & 1.66 & 0.76 \\  & HS & 3.12 & 2.63 & 1.87 & 0.74 & 2.65 & 1 & 2.24 & 1.61 & 0.70 \\  & KDE & 3.14 & 2.66 & 1.90 & 0.80 & 2.67 & 2.26 & 1.64 & 0.74 \\  & GARCH & 2.46 & 2.05 & 1.46 & 0.69 & 2.66 & 2.20 & 1.64 & 0.71 \\ \hline \multirow{3}{*}{Combined methods} & FHS & 2.42 1 & 1.99 1 & 1.44 1 & 0.62 1 & 2.67 & 2.08 1 & 1.63 & 0.72 \\  & CEVT & 3.10 & 2.60 & 1.84 & 0.64 & 3.53 & 2.99 & 1.58 1 & 0.74 \\ \hline \multirow{3}{*}{
\begin{tabular}{c} LTD = 4 Weeks \\ \end{tabular} } & \multicolumn{6}{c}{\(\phi\) = 0.70} & \multicolumn{6}{c}{\(\phi\) = \(-\)0.70} \\ \cline{3-10}  & LT = 4 Weeks & \multicolumn{6}{c}{CSL} & \multicolumn{6}{c}{CSL} \\ \cline{3-10}  & & 85\% & 90\% & 95\% & 99\% & 85\% & 90\% & 95\% & 99\% \\ \hline \multirow{3}{*}{Simple methods} & SES & 12.00 & 9.77 & 6.68 & 2.98 & 3.38 & 2.83 & 2.03 & 1.04 \\  & HS & 10.55 & 8.77 & 6.13 & 2.30 & 3.23 & 2.77 1 & 2.01 & 0.92 \\  & KDE & 10.62 & 8.86 & 6.32 & 2.60 & 3.28 & 2.81 & 2.11 & 0.96 \\  & GARCH & 10.36 & 8.44 & 5.70 & 2.29 & 3.38 & 2.80 & 1.97 & 0.92 \\ \hline \multirow{3}{*}{Combined methods} & FHS & **10.20 1** & **8.40 1** & **5.65 1** & **1.90 1** & 3.27 & 2.78 & 1.97 & 0.77 \\  & CEVT & 11.42 & 9.69 & 6.27 & 2.53 & **3.08 1** & 2.78 & **1.97 1** & **0.48 1** \\ \hline \hline \end{tabular}  
* \({}^{1}\) Minimum values of tick loss shown in bold.  
\end{table}