## 3 Simulation Results
### Case Study Results
#### 3.4.2 Case Study Safety Stock Estimation  
Two simulations are performed with the real data for one and four weeks of lead times. Figure 9 shows the weekly demand for the item over time. Figure 10 presents the tick loss values of each method for real demand that follows the ARMA-GARCH model for one week (on the right) and for four weeks (on the left). When the case of LT is one week, the CEVT approach reduced the loss to a 85% level. This is valid when the LT is four weeks. For the other CSL, almost all methods give the same value of tick loss.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Min** & **Max** & **Median** & **Mean** & **Skewness** & **Kurtosis** & \
\begin{tabular}{c} **Jarque** \\ **Bera Test** \\ \end{tabular} &
\begin{tabular}{c} **Augmented Dickey** \\ **Fuller Test** \\ \end{tabular} & **Q (12)** & **Q \({}^{2}\) (12)** \\ \hline
0 & 928.9 & 463.7 & 458 & \(-\)0.003 & 45.365 & 790.3 & \(-\)9.0986 & 30.47 & 28.707 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Descriptive statistics and preliminary tests.  
Figure 10: Tick loss values of each method for real demand that follows ARMA-GARCH model for one week (**a**) and for four weeks (**b**).  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Estimate** & **Std. Error** & **t-Value** & _p-Value_ \\ \hline
**mu** & 458.45 & 6.31 & 72.64 & 0.000 \\
**ar1** & 0.17 & 0.04 & 4.76 & 0.000 \\
**omega** & 31.56 & 77.84 & 0.41 & 0.685 \\
**alpha1** & 0.00 & 0.00 & 0.81 & 0.420 \\
**beta1** & 1.00 & 0.00 & 1688.02 & 0.000 \\
**Q (12)** & 6.68 & & & 0.878 \\
**Q (12)** & 18.42 & & & 0.110 \\
**ARCH test** & 19.15 & & & 0.090 \\ \hline
**Log Likelihood** & & & \(-\)4962.29 & & \\
**AIC** & & & 12.82 & & \\
**BIC** & & & 12.85 & & \\
**Shibata** & & & 12.82 & & \\
**H-Q** & & & 12.83 & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: ARMA-GARCH parameters estimation.  
Figure 9: Case study weekly demands over time.  
Figure 11 shows the backorders of each method for real demand that follows the ARMA-GARCH model for one week (on the left) and for four weeks (on the right).  
In the case of shorter LT and at an 85% level we can remark that the GARCH method provides a lower level of inventory investment but gives higher backorders. Contrary the CEVT method gives lower backorders but provides a higher level of inventory investment. The result of the FHS method is the compromise between such methods. When the CSL is equal to 90% the combination of FHS and CEVT approaches provide almost the same result. Concerning backorders, the two techniques outperform the GARCH method but in terms of inventory investment this approach works better. At the 95% level, the GARCH-based methods yield the lowest backorders and inventory investment with CEVT improving on FHS and GARCH slightly in terms of backorders. We can see for CSL= 99% that the GARCH approach outperforms slightly the rest of the methods.  
In the case of longer lead times for CSL= 85% the CEVT performs better regarding backorders and there is a slight difference regarding inventory investment in favor of FHS followed by the GARCH technique. Note, that these three approaches always give better results for 90% and 95% CSL with a slight difference regarding backorders or inventory investment. The combination technique FHS attains the best results with the lowest level of backorders and lowest inventory investment for CSL= 99%. Generally, such a figure indicates the good performance of GARCH-based methods compared with the traditional approaches for every CSL target for shorter lead times except for the 99% level where GARCH performs better. For longer lead times that achievement remains except for the 99% level where FHS yields the best results. These results obtained with real data match those with the simulated data carried out in the previous section. That coincidence is in the sense that the GARCH method and their combination with the simulation and EVT always perform better.