## 5 Results and Analysis
### Test Performance
#### 5.1.2 Program Size  
In scheduling processes, it is crucial to provide concise, simply interpretable dispatching rules. Rule size, often described as the sum of all nodes included inside a rule, is an essential aspect that affects interpretability. The advantages of compact dispatching rules include a reduced computational complexity and greater generality. This paper employed the numerical reduction technique, as proposed by [PERSON] [18], to simplify the rules and enhance our understanding of their complexity and interpretability. The changes in the mean rule size across generations for the MT, MVT, and MFT scenarios are illustrated in Figure 4a-c, respectively. The observation of Figure 5 indicates that, in all scenarios, the IGEP and GEP algorithms exhibited a tendency to generate rules of significantly larger size than those produced with the FS-IGEP algorithm. As we expected, the designed rule length of the FS-IGEP was significantly different from those generated with IGEP and GEP algorithms. This was because more compact feature sets can help GEP to obtain simpler and shorter rules. In addition, we can observe that the rule length of the IGEP was not too much different from the GEP. This shows that despite the improvements, GEP was unable to generate simplified rules without a feature selection mechanism. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Test Sets** & **TWKR** & **SPT** & **PT + WINQ + AT** & **GEP** & **IGEP** & **FS-IGEP** \\ \hline  
mathbf{-14.24\pm 0.30}\) \\ \hline \end{tabular}
\end{table} 
Table 4: The test performance of the rules in MS scenarios (% relative to that of SPT/TWKR). In order to further understand the impact of feature selection on the structure of dispatching rules, we compared and analyzed the structure of the best-evolved rules using GEP, IGEP, and FS-IGEP algorithms. Specifically, our study took the best dispatching rule (its objective value being 1213.25) achieved using the GEP algorithm, the best rule (its objective value being 1199.34) obtained using the IGEP algorithm, and the best rule designed using the FS-IGEP algorithm (its objective value being 1186.61) to be found in the MWT scenario <4, 15, 95%>. This was because the mean weighted tardiness in scenario <4, 15, 95%> was more difficult to optimize than other scenarios. We noted that the smaller the values calculated using the rules, the more priority the operation had. Figure 4: The mean rule size of the GEP algorithms in the three scenarios: (**a**) mean rule size across generations in the MS scenario; (**b**) mean rule size across generations in the MFT scenario; (**c**) mean rule size across generations in the MWT scenario. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test Sets**} & **OPFSLK/PT;** & \multirow{2}{*}{**2 PT + LWKR + FDD**} & \multirow{2}{*}{**2 PT + WINQ + NPT**} & \multirow{2}{*}{**GEP**} & \multirow{2}{*}{**IGEP**} & \multirow{2}{*}{**FS-IGEP**} \\  & **FDD** & & & & & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 5: The test performance of the rules in MFT scenarios (% relative to that of PT + WINQ). For the GEP algorithm, in the examined run, the terminal set consisted of NOIQ, NOINQ, WIQ, MWT, PT, NPT, OWT, WKR, NOR, MRT, and W (i.e., eleven features). For the dispatching rules obtained using the IGEP algorithm, all the features in the terminal set were used. There were seven features (NOIQ, WINQ, NPT, PT, W, WKR and NOR) selected as terminal set. Fewer features were used by the FS-IGEP algorithm compared to IGEP and GEP. However, the evolved dispatching rules obtained using the FS-IGEP algorithm had better performance than that of IGEP and GEP. According to the numerical reduction technique, the dispatching rule achieved using the FS-IGEP algorithm could be simplified, as shown in Equation (16). It is noted that for all the operations in the queue of a machine, the machine-related features, such as the workload of the current queue (WIQ), was the same for all operations. This means that it was not a vitally important feature.