## 3 Simulation Results
### Case When Forecasting Error Follow A Log-Normal Distribution  
Let \(\text{D}_{\text{t}}\) be the demand at time t that follows an AR (1) process as mentioned in Equation (23), where \(\mu\) is a positive constant, \(\phi\) is the autoregressive parameter and \(\epsilon_{\text{t}}\) is i.i.d but is not normally distributed. We have added a log-normal noise with a mean of 0.9 and variance of 1.4, while \(\phi\) was allowed to vary between \(-0.9\) and 0.9. The TLFs on the retained sample averaged over 100 replicates; the quantiles which include the following values namely 85%, 90%, 95% and 99%, versus the autoregressive parameter \(\phi\) are assumed to vary between -0.9 and 0.9 and are shown in Figures 1 and 2. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Safety Stock**} & \multirow{2}{*}{**LT = 1**} & \multicolumn{4}{c}{\(\phi\) = 0.70} & \multicolumn{4}{c}{\(\phi\) = \(-\)0.70} \\ \cline{3-10}  & & \multicolumn{4}{c}{**CSL**} & \multicolumn{4}{c}{**CSL**} \\ \cline{3-10}  & & **85\%** & **90\%** & **95\%** & **99\%** & **85\%** & **90\%** & **95\%** & **99\%** \\ \hline \multirow{3}{*}{Simple methods} & SES & 3.51 & 2.74 & 1.73 & 0.55 & 2.32 & 1.75 & 1.03 & 0.26 \\  & HS & 2.31 & 1.74 & 1.03 & 0.22 & 2.32 & 1.74 & 1.02 & 0.26 \\  & KDE & 2.34 & 1.76 & 1.05 & 0.28 & 2.32 & 1.75 & 1.03 & 0.27 \\  & GARCH & 2.15 & 1.59 & 0.93 & 0.24 & 2.17 & **1.60 1** & 0.93 & 0.24 \\ \hline \multirow{3}{*}{Combined methods} & FHS & 2.15 & 1.59 & 0.93 & 0.24 & **2.16 1** & 1.61 & 0.93 & 0.24 \\  & CEVT & **2.05 1** & **1.55 1** & **0.89 1** & **0.22 1** & 2.22 & 1.62 & **0.92 1** & **0.22 1** \\ \hline \multirow{3}{*}{LT= 4 Weeks} & \multicolumn{4}{c}{\(\phi\) = 0.70} & \multicolumn{4}{c}{\(\phi\) = \(-\)0.70} \\ \cline{3-10}  & LT= 4 Weeks & \multicolumn{4}{c}{CSL} & \multicolumn{4}{c}{CSL} \\ \cline{3-10}  & & 85\% & 90\% & 95\% & 99\% & 85\% & 90\% & 95\% & 99\% \\ \hline \multirow{3}{*}{Simple methods} & SES & 8.97 & 6.82 & 4.11 & 1.17 & 2.46 & 1.85 & 1.09 & 0.28 \\  & HS & 7.71 & 5.79 & 3.42 & 0.94 & 2.44 & 1.85 & 1.09 & 0.29 \\  & KDE & 7.86 & 5.94 & 3.54 & 0.95 & 2.46 & 1.86 & 1.11 & 0.30 \\  & GARCH & 7.63 & 5.70 & 3.32 & **0.85 1** & 2.42 & 1.82 & 1.06 & 0.27 \\ \hline \multirow{3}{*}{Combined methods} & FHS & 7.66 & 5.71 & 3.32 & 0.90 & 2.42 & 1.83 & 1.07 & 0.28 \\  & CEVT & **7.06 1** & **5.11 1** & **3.10 1** & 0.95 & **2.21 1** & **1.65 1** & **0.95 1** & **0.24 1** \\ \hline \hline \end{tabular}  
* Minimum values of Tick loss shown in bold.  
\end{table} 
Table 1: Tick loss function values corresponding to \(\phi\) = 0.70 and \(\phi\) = \(-\)0.70 when forecasting errors follow a normal distribution. Figure 1: Average tick loss values for lead times of one week and when forecasting errors follow a lognormal distribution. Figure 1 includes the case where the Lead time (LT) is one and Figure 2 includes the case where the LT is four weeks. When the case of LT is one week, the CEVT method (purple line) provided a low value for average tick loss for some values of the \(\phi\) parameter. For the other values of \(\phi\) where CEVT is not the best method we find in this case, the method of FHS (yellow line) gives better results followed by the GARCH (blue line) method. This proved the best performance of the GARCH model and Extreme Value Theory (CEVT) and secondly of the GARCH and simulation method (FHS) method which is explained by their nature to capture the deviation from the normality assumption. When the case of LT is four weeks and for negative values of \(\phi\) it appears that CEVT is the best method except when \(\phi=-0.4\), the FHS gives a minimum of tick loss followed by the GARCH method. For positive values of \(\phi\), we find that the CEVT method is the best for some values of the autoregressive parameter otherwise the FHS takes its place followed by the GARCH method. The SES method offers the worst performance in terms of tick loss followed by the KDE and simulation methods. Figures 1 and 2 indicate the overall results.