## 3 Simulation Results
### Case When Forecasting Error Follow to Gamma Distribution
Figure 7 shows the tick loss values in the case of gamma 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 7 shows the tick loss values in the case of gamma 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. Figure 8 shows the tradeoff curves in the case of gamma \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Safety Stock** \\ **Estimation Method** \\ \end{tabular} } & \multirow{2}{*}{**LT= 1 Week**} & \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}  & & **0.85** & **0.90** & **0.95** & **0.99** & **0.85** & **0.90** & **0.95** & **0.99** \\ \hline \multirow{3}{*}{Simple methods} & SES & 3.87 & 3.10 & 2.07 & 0.84 & 3.10 & 2.50 & 1.66 & 0.64 \\  & HS & 3.37 & 2.73 & 1.79 & 0.57 & 3.04 & 2.48 & 1.65 & 0.54 \\  & KDE & 3.42 & 2.77 & 1.83 & 0.59 & 3.05 & 2.50 & 1.67 & 0.56 \\  & GARCH & 2.75 & 2.25 & 1.54 1 & 0.65 & 2.97 & 2.42 & 1.63 & 0.67 \\ \hline \multirow{3}{*}{Combined methods} & FHS & 2.74 1 & 2.24 1 & 1.54 1 & 0.55 & 2.89 & 2.38 & 1.64 & 0.56 \\  & CEVT & 3.76 & 2.95 & 1.84 & 0.521 & 2.51 1 & 2.04 1 & 1.48 1 & 0.44 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} \\ \cline{3-10}  & & 0.85 & 0.90 & 0.95 & 0.99 & 0.85 & 0.90 & 0.95 & 0.99 \\ \hline \multirow{3}{*}{Simple methods} & SES & 11.88 & 12.27 & 6.17 & 2.38 & 3.49 & 2.83 & 1.92 & 0.79 \\  & HS & 10.93 & 9.43 & 5.62 & 1.82 & 3.48 & 2.83 & 1.87 & 0.62 \\  & KDE & 11.14 & 8.73 & 5.73 & 1.84 & 3.50 & 2.85 & 1.90 & 0.67 \\  & GARCH & **10.66 1** & 8.88 & 5.34 & 1.80 & 3.46 & 2.79 & 1.85 & 0.70 \\ \hline \multirow{3}{*}{Combined methods} & FHS & 10.72 & **8.40 1** & **5.28 1** & 1.58 & 3.47 & 2.80 & 1.82 & **0.56 1** \\  & CEVT & 12.27 & 8.41 & 5.56 & **1.37 1** & **2.79 1** & **2.32 1** & **1.62 1** & 0.59 \\ \hline \hline \end{tabular} \({}^{1}\) Minimum values of Tick loss shown in bold.  
\end{table} 
Table 3: Tick loss values corresponding to \(\phi\) = 0.70 and \(\phi\) = \(-\)0.70 when forecasting errors follow a gamma distribution. Figure 7: Tick loss values in the case of gamma 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**). 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). Figure 7 plotted the backorders versus inventory investment. Each curve for each panel contained four specific points corresponding to the four targets CSL (85%, 90%, 95% and 99%). We can see in the case of \(\phi\) = \(-\)0.70 that the CEVT method gives good results in terms of inventory investment and backorders for CSL targets 85%, 90% and 95%. The 99% level provides a lower level of backorders but does not give the minimum inventory investment. Therefore, it is difficult to determine the best method. We can also see in the case of \(\phi\) = 0.70 that the GARCH method provides results with the FHS, notably for lead times of one week.