## 4 Discussion  
To validate the superiority of the two proposed SS estimation methods, it is necessary to study the significance of the difference between the tick loss of each combined SS estimation method. For these reasons, the Analysis of Variance (ANOVA) technique is used to test the difference between the six studied methods means and between other factor means, such as the value of CSL with four levels (85%, 90%, 95%, and 99%), the value of \(\phi\) with 10 levels (range from \(-\)0.90 to 0.90), LT with two levels (1 week, and 4 weeks), and FED with two levels (gamma distribution, log normal distribution).  
To perform the ANOVA test, a full design of experiments with 960 runs is used with only one response (output): TLF and five input factors: Method (six levels), CSL (four levels), \(\phi\) (10 levels), LT (two levels), and FED (two levels). Using Minitab Software, the ANOVA results are presented in Figure 12.  
Figure 11: Backorders of each method for real demand that follows ARMA-GARCH model for one week (**a**) and for four weeks (**b**).  
For each input factor, this test has two hypotheses (H\({}_{0}\): all means are the same, vs. H\({}_{1}\): not all the means are equal). To the right of the ANOVA table (Figure 12), under the column headed P, is a \(p\)-value. If the \(p\)-value is less than 0.05 then H\({}_{0}\) is rejected and the result provides sufficient evidence to conclude that the TLF is not all the same for all levels.  
Through the ANOVA test, we can conclude that the used estimation method, the CSL, the autoregressive parameter (\(\phi\)) and the lead time have a significant influence on TLF and consequently on the estimation of the safety stock. However, the forecast error distribution (FED) has no influence on the decision since the related \(p\)-value = 0.356 is higher than 0.05. This indicates that the TLFs are the same when the forecast errors follow a log-normal or a gamma distribution.  
Although an ANOVA test does tell whether different groups or treatments were statistically significantly different, it does not tell (1) if that difference is practically significant and (2) it does not use a grouping method that can help separate the differences in performance between each of the different safety stock estimation methods used in this manuscript. Consequently, it is necessary to look at confidence intervals or run post hoc tests to determine that. However, in this research, pair-wise comparison between methods is not applied because the ANOVA test used in this manuscript is not one-way ANOVA. Extending the current research in this direction is one of our interesting perspectives.  
ANOVA results (effect and interactions) and the main effect plot shown in Figure 13, validate the simulation results and confirm that the two combined method for safety stock estimation FHS and CEV gives the minimum tick loss values and they are better than the classical methods, such as the SES method, Simulation method, KDE method, and GARCH method. We can verify also in Figure 13, that the tick loss is better when CLS = 99% and when the lead time is equal to one week. The value of TLF decreases when the CSL increases from 85% to 99% levels. This indicates that gradually as the target probability of no stock out over the LT (CSL) increases the actual demands and the forecast demands values are getting closer, and therefore, the TLF values decrease. Similarly, as the LT increases, the uncertainty increases implying the growth of TLF values. We can conclude also that when the autoregressive parameter increases implying a strong autocorrelation between the past and actual demands, the TLF values also increases.  
Figure 12: ANOVA results applied to TLF as output response and Method, CSL, \(\phi\), LT, and FED as inputs.