# Routing Design for Pipe System Considering Natural Frequency
## 5 Path Planning for Multi-Pipe System by Using Novel Collaborative Evolution  
Previous studies on path planning have primarily focused on optimizing individual pipes, which may not guarantee a global optimal solution for the entire pipe system. To address this limitation and achieve an optimal solution for the entire system, this paper proposes a co-evolutionary approach for pipe routing. In this approach, each pipe in the system generates its own population, resulting in a multi-population system. By promoting cooperation and competition among the pipes, a global optimal solution is established. In the collaborative evolutionary computation of pipes, each pipe population is composed of individuals representing various paths. Heuristic algorithms like genetic algorithms are utilized to drive the evolutionary process for each pipe. During the evaluation of individuals, the evaluation process considers both coordination between different pipes and the natural frequency. This approach assigns lower fitness to individuals that positively impact coordination. Conversely, individuals that hinder coordination are assigned higher fitness. By incorporating coordination, the evolutionary process encourages the development of pipe paths that promote mutual coordination and overall system performance. Previous works have used multi-ant colony collaborative evolutionary algorithms to solve multi-pipe routing and branch pipe routing problems [5]. However, these approaches heavily rely on the control parameters of the ant colony optimization algorithm. Another  
Figure 13: The natural frequency with different diameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Diameter** & \(A_{0}\) & \(t\) & \(y_{0}\) \\ \hline
5 mm & \(1.44\times 10^{7}\) & 8.01 & \(5.4\times 10^{3}\) \\
10 mm & \(7.02\times 10^{6}\) & 18.1 & \(8.28\times 10^{3}\) \\
20 mm & \(7.08\times 10^{4}\) & \(1.08\times 10^{2}\) & \(4.03\times 10^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The fitting constant parameters for different diameters. pioneering work [27] was the first to utilize genetic algorithm co-evolution to address multi-pipe routing problems. Nevertheless, these methods lacked consideration for engineering constraints and the natural frequency of pipes. Moreover, they solely assessed the coordination relationship between optimal individuals from different populations, without achieving true co-evolution. To overcome these limitations, this paper proposes a novel collaborative evolution method (See Figure 14). In Figure 14, the distinctive aspect of this method is highlighted in red, which sets it apart from other collaborative evolutionary algorithms. The approach involves N populations, each containing m initial paths obtained through the RRT algorithm. After acquiring the initial fitness values of each individual, the evolution proceeds sequentially based on the order of the initial populations. During population evolution, a path is randomly selected from each population, forming the temporary solution for the current system. The fitness value of this temporary solution is evaluated using Equation (3). This process is repeated k times to obtain the average fitness value and update the fitness values of each individual. The above evolutionary process continues until the maximum number of generations is reached. Figure 15 is the detailed diagram of two pipes in the red part of Figure 14. Figure 15 is a pipe system with two pipes. The population number of each pipeline is 4. First, record its current fitness function and the number of times selected. Then, randomly select a path from each population and repeat it for k times (k = 3 in Figure 15) to form a temporary solution of the pipe, evaluate the temporary solution of the system according to Equation (3) (such as k1 in Figure 15), obtain the average fitness value of the pipe, and start the next update. Figure 14: Path planning for multi-pipe system by using novel collaborative evolution. Figure 15: The detailed diagram of two pipes in the red part of Figure 14.