## 3 Simulation Results
### Case When Forecasting Error Follow to Gamma Distribution  
Let D\({}_{\rm t}\) 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 \(\varepsilon_{\rm t}\) is followed by the Gamma distribution with parameters alpha= 0.4 and beta = 0.07 while \(\phi\) was allowed to vary between \(-\)0.9 and 0.9. The TLFs on the retained sample averaged over 100 replicates and then over the quantiles which include the following values namely 85%, 90%, 95% and 99%, versus the autoregressive parameter \(\phi\), assumed to vary between -0.9 and 0.9, are shown in Figures 5 and 6. Figure 5 includes the case when the LT is one and Figure 6 shows the case when the LT is four weeks. When the case of LT is one week and for negative values of \(\phi\), the CEVT method provided a low value for average tick loss except in \(\phi\) = \(-\)0.4 and \(\phi\) = \(-\)0.2, the FHS method gives a minimum tick loss. For positive values of \(\phi\), the CEVT method gives a lower loss for only three values of \(\phi\) and for the remaining values, the FHS is the best method followed by GARCH. We can remark that the GARCH method gives good results especially for negative or positive high values of \(\phi\). This proved the best performance of the combination of GARCH-based methods. When the case of LT is four weeks, for  
Figure 4: Tradeoff curves for AR (1) in the case of lognormal 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**). negative values of \(\phi\) it appears that CEVT is the best method except when \(\phi=-0.4\). The other methods including the traditional and combined approaches give nearly the same minimum of tick loss. For positive values of \(\phi\), we find that the CEVT method is the best for 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 KDE and simulation methods. Figures 5 and 6 indicate the overall results. 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 3 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 a gamma distribution and \(\phi\) = 0.70 or \(\phi\) = \(-0.70\). Values in blood indicate the minimum tick loss for a given CSL. The tick loss values of such a table are also plotted in Figure 7. In the case of \(\phi=-0.70\) and LT of one week it appears that the CEVT method provides the most performance in all targets CSL followed by the FHS method. For example, the value of tick loss is 2.51 at an 85% level when we apply the CEVT method. This value decreases to 0.44 when CSL becomes at 99%. For an LT of four weeks, the CEVT method gives a lower loss except at the 99% level where the FHS method provides the minimum value of tick loss. 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 almost all CSL except at the 99% level where the CEVT is better. Contrary to LT of four weeks, FHS is more suitable only in 90% and 95% levels. The GARCH and CEVT methods are more appropriate, respectively, when the CSL  
Figure 5: Average tick loss values for lead times of one week and when forecasting errors follow a gamma distribution. Figure 6: Average tick loss values for lead times of four weeks and when forecasting errors follow a gamma distribution. is equal to 85% and 99%. Traditional methods (SES, simulation and KDE) cannot provide good results by comparison with GARCH-based methods.