## 6 Conclusions and Future Studies  
In this study, a solution was presented for 2D path optimization problems that can be encountered, particularly in AMC applications and other similar areas. The structure in plane, which is in fact a single layer of a 3D form, consists of vertices and edges between them. The principal constraint of the problem is that some edges will be traveled at least once, and during that travel a manufacturing process will occur. In other edges, travel may occur or not, and the travels will be idle passages in which no manufacturing will be performed. It is assumed that the process will begin from a predefined vertex and progress through to a predetermined edge, and the rest of the path will be optimized by the algorithm defined. Analyses on the problems solved in this study showed that distances covered by idle movements can be as much as 18% of the total distances traveled when a flat or structural component is concreted. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Movement of the Nozzle**} & \multirow{2}{*}{**Path**} & \multicolumn{10}{c}{**Order**} \\ \cline{3-14}  & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** \\ \hline \multirow{4}{*}{R} & 1 & 1 & 2 & 10 & 6 & -11 & -3 & 16 & 12 & 7 & -13 & -5 & -14 & 8 & 4 & -9 & 15 & 17 \\  & 2 & 1 & 14 & 4 & -9 & 2 & 16 & -5 & -8 & 15 & 12 & 7 & -13 & -10 & 3 & 11 & -6 & 17 \\  & 3 & 1 & 15 & 6 & -11 & -3 & 16 & 17 & -13 & -5 & -4 & -8 & 9 & -14 & 2 & 10 & 12 & 7 \\ \hline \multirow{4}{*}{D} & 1 & 1 & 9 & -14 & 8 & 4 & 5 & 6 & -11 & -3 & -2 & 15 & 17 & -13 & -16 & 10 & 12 & 7 \\  & 2 & 1 & 9 & -14 & 8 & 4 & 5 & -16 & -3 & 11 & -6 & 12 & 7 & -13 & -15 & 2 & 10 & 17 \\  & 3 & 1 & 15 & -10 & 3 & 16 & -5 & -4 & -8 & 9 & -14 & 2 & 11 & 6 & 13 & -17 & 12 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Best solutions obtained for the 17-wall component. T = 12. Figure 13: Comparison of idle to total distances. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Example** & **Number of Walls** & **Non-Idle Distance** & **Idle Distance** & **Total Distance** & **Idle/Total Distance (\%)** \\ \hline \(1\) & 7 & 185 & 25 & 210 & 11.9 \\ \(2\) & 8 & 195 & 35 & 230 & 15.2 \\ \(3\) & 14 & 39 & 6 & 45 & 13.3 \\ \(4\) & 31 & 85 & 18.3 & 103.3 & 17.7 \\ \(5\) & 17 & 65 & 12 & 77 & 15.6 \\ \hline \hline \end{tabular}
\end{table} 
TableThis is a critical percentage and indicates that the optimization process considered in this study is essential to these operations. Clearly, if this type of optimization work is not conducted, more than 20% of the nozzle movements will be idle, which represents a major waste of time and energy. Analyses showed that the number of possible paths in this type of problem increased rapidly as the number of edges increased, making the problem a nondeterministic polynomial (NP) one. For example, when the number of edges was 30, indicating that the problem being considered was common, the number of possible paths was \(4.747\times 10^{39}\). It is also interesting to note that if the number of edges increased by 1 (i.e., from 30 to 31), the path possibilities increased to \(2.801\times 10^{41}\), making the problem unsolvable by many algorithms. Metaheuristic algorithms represent a group of solutions that can be used for NP problems. In this study, the HS algorithm was chosen from among them. Computations showed that acceptable results were obtained with this choice. The study also revealed that 45,000 runs of HS were necessary to arrive at acceptable results for a problem with 31 walls. One research task for the future will be to identify a metaheuristic or hybrid algorithm to reduce this number. In the 3D case, if the problem involves concrete pouring or a similar operation, constraints will exist that will make the problem much more difficult. Two of these types of constraints can be explained as follows.