## 3 A Normalized Approach for the Throughput Prediction
For the sake of clarity, in the following, we will refer to the normalized throughput as a percentage measure, as it may be viewed as a measure of the throughput percentage that the considered production line achieves in respect to the maximum possible throughput \(r_{b}\). The results for the WIP values and a few of the \(\alpha\) values tested are shown in Figures 4 and 5 and Table 3. In the Figure 4, the registered throughput and the normalized throughput in function of different \(\alpha\) values are shown in different colors. As can be seen, the lowest curve corresponds to the smallest \(\alpha\), i.e., the situation in which processing times are highly variable. On the other hand, the curves rise as the alpha value increases because, as the variability within the production system decreases, the system converges to the best case quickly as the WIP increases. A similar pattern holds true for the normalized throughput, which is clearly scaled down by a factor in comparison to the absolute throughput. On the other hand, the Figure 5 shows 10 throughput curves related to 10 different average processing times (and thus different \(r_{b}\)), with \(\alpha\) set to 0.5. The highest throughput values are clearly obtained when the smallest average processing times are considered since the \(r_{b}\) is in general a scale factor for the line throughput. Nevertheless, the curve of the normalized throughput confirms what we searched for since it practically gets values that are independent of the average processing time considered. Indeed, the different processing time curves cannot be distinguished because they are perfectly superimposed upon one on another. To further confirm this, in Table 3, two different measures for each combination of values are reported: the average value of the normalized throughput (\(E(TH/r_{b})\)) computed over all the observations and an error measure (\(\Delta\%\)) evaluated as:  
\[\Delta\%=\frac{\max(TH/r_{b})-\min(TH/r_{b})}{E(TH/r_{b})}\cdot 100 \tag{6}\]  
Practically, the line throughput for each of the 10 average processing times in [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] was determined by executing 40 runs for each combination of \(\alpha\) in \(\{0.5,1,1.5,2,2.5,3\}\) and WIP in the range \([1,30]\). Then, the resulting throughput values were normalized with respect to the related \(r_{b}\) (i.e., the inverse of the average processing time under consideration). For each combination, the maximum, minimum, and mean values were computed, and the \(\Delta\%\) was calculated using Equation (6). The \(\Delta\%\) represents the percentage difference between the highest and lowest observation, and the closer this value is to zero, the more similar the observations for different processing averages are one to the other, basically confirming what the considerations made about the PWC case still hold when general variability is considered. Indeed, it is useful to note that, regardless of the considered \(\alpha\) value, \(\Delta\%\) values remain close to zero, indicating that all observations \begin{table}
\begin{tabular}{c c c} \hline
**Experimental Factor** & **Level** & **Measure Unit** \\ \hline Mean processing time & \(5\div 50\), step 5 & [min] \\ Shape factor (\(\alpha\)) & \(0.5\div 3\), step 0.1 & [] \\ WIP level & \(1\div 30\), step 1 & [unit] \\ Number of machines & 5 & [machines] \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental factors and level for the simulation. have deviations of less than one percentage point. As a confirmation, it is worth noting that the percentage error for the various \(\alpha\) is of the same order of magnitude as the percentage error for the \(\alpha=1\) case, i.e., when the PWC assumptions are verified. This is significant because, in the case of an exponential distribution, the [PERSON] and [PERSON]'s mathematical model confirms the independence of the normalized throughput value from the average processing time, thus pointing out that the overall percentage deviation is due to the simulation variability and not to an unmodeled systematic error.