## 4 Optimal Solution Applied to an Elevator Group Control System  
Each car within the elevator group is associated with a unique array, characterized by the upward and downward calls to each floor for the car. Therefore, the length of each solution is (2 \(\times\) N), where N represents the total number of floors in the structure. For individual cars, the solution represents the allocation of up and down hall calls, as outlined in Tables 4 and 5. The example tables correspond to a 10-floor architectural configuration. The solution uses binary encoding, where \(0^{\prime}\) denotes the absence of hall-call allocation for the respective car, while \(`1^{\prime}\) signifies the assignment of the registered hall call on that floor to the designated car. Table 6 presents upward and downward hall calls for the whole elevator group. All possible solutions that are encoded as stated above are evaluated to determine the quality (or fitness) of such solutions. The fitness estimation procedure relies on the state of the elevator (stationary, ascending, or descending) and is based on four peak values denoted as \(\mathrm{X}_{1}\), \(\mathrm{X}_{2}\), \(\mathrm{X}_{3}\), and \(\mathrm{X}_{4}\). For case (i), where the elevator is stationary or ascending:  
- \(\mathrm{X}_{1}\) represents the current floor;  
- \(\mathrm{X}_{2}\) signifies the highest floor for passengers going up;  
- \(\mathrm{X}_{3}\) denotes the lowest floor for passengers going down;  
- \(\mathrm{X}_{4}\) represents the highest floor lower than \(\mathrm{X}_{1}\) for passengers going up, where \(\mathrm{X}_{4}<\mathrm{X}_{1}\). \[\mathrm{Fitness}=[(\mathrm{X}_{2}-\mathrm{X}_{1})+(\mathrm{X}_{2}-\mathrm{X}_{ 3})+(\mathrm{X}_{4}-\mathrm{X}_{3})]\times\text{estimated interfloor trip time}. \tag{1}\]  
Equation (1) considers the maximum known upward trip, the maximum known downward trip, and the subsequent maximum known unexecuted upward trip in the initial upward traffic, given \(\mathrm{X}_{4}<\mathrm{X}_{1}\). Notably, the mathematical expression excludes passenger destination trips, as their information is unknown until passengers enter the cabin. For case (ii), where the elevator is descending:  
- \(\mathrm{X}_{1}\) represents the current floor;  
- \(\mathrm{X}_{2}\) signifies the lowest floor for passengers going down;  
- \(\mathrm{X}_{3}\) denotes the highest floor for passengers going up;  
- \(\mathrm{X}_{4}\) represents the lowest floor higher than \(\mathrm{X}_{1}\) for passengers going down, where \(\mathrm{X}_{4}>\mathrm{X}_{1}\). \[\mathrm{Fitness}=[(\mathrm{X}_{1}-\mathrm{X}_{2})+(\mathrm{X}_{3}-\mathrm{X} _{2})+(\mathrm{X}_{3}-\mathrm{X}_{4})]\times\text{estimated interfloor trip time}. \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Floors** & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** & **F7** & **F8** & **F9** & **F10** \\ \hline Upward hall calls & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ Downward hall calls & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Upward and downward hall calls for the whole elevator group. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Floors** & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** & **F7** & **F8** & **F9** & **F10** \\ \hline Upward hall calls & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ Downward hall calls & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Upward and downward hall calls for the first car. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Floors** & **F1** & **F2** & **F3** & **F4** & **F5** & **F6** & **F7** & **F8** & **F9** & **F10** \\ \hline Upward hall calls & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ Downward hall calls & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Up and down hall calls for the second car.