## 4 Numerical Experiments
### Relay Node Deployment Experiments  
When the network is only composed of SNs, the lifetime of a network with 30 sensor nodes deployed in 100 \(\times\) 100 terrain is 480 min, while the lifetime of a network of 114 sensor routers in 200 \(\times\) 200 terrain is 110 min. On this basis, RNs can be added to the network in a gradually increasing manner to observe the network life optimization curve in the \(100\times 100\) and \(200\times 200\) terrains with different numbers of RNs added, as shown in Figures 7 and 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Parameters** & **Value Selected** & **Range** \\ \hline GA & Mutation & 0.2 & \([0.05,0.1,\ldots,0.95]\) \\  & Crossover & 0.95 & \([0.05,0.1,\ldots,0.95]\) \\ SA & \(T_{0}\) & 4 & \([1,2,\ldots,20]\) \\  & \(\lambda\) & 0.85 & \([0.05,0.1,\ldots,0.95]\) \\ dpABC & C & 1.5 & \([0.5,1.0,\ldots,5.0]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter selection of GA, SA, and dpABC. Figure 7: WSN lifetime optimization with different number of RNs in 100 \(\times\) 100 terrain. When an RN is added, the optimized lifetime results of \(100\times 100\) and \(200\times 200\) deployed WSNs are 570 min and 170 min, respectively, 18.75% and 54.54% longer than the WSN without an RN node. This means that adding an RN to an ordinary WSN has a great effect in improving network life. When an RN node is added, all four algorithms compared here obtain the same optimization results. However, as the number of deployed RN nodes continues to increase, the differences between the different algorithms increase. In the \(100\times 100\) network, the pdABC algorithm performs slightly better than the ABC algorithm for lifetime optimization; in turn, the ABC algorithm is better than GA algorithm, while the SA algorithm has the worst performance when the number of RNs exceeds three. In the \(200\times 200\) terrain, the optimization effect of the different algorithms is more different, and the optimization effect of the pdABC algorithm is further reflected. In order to verify the convergence of the algorithm in solving the TT-WSN relay node deployment problem, the relationship between the different algorithms in terms of lifetime optimization based on the different numbers of RNs and number of iterations is shown in Figures 9 and 10. Figure 8: WSN lifetime optimization with different number of RNs in \(200\times 200\) terrain. Figure 9: Lifetime optimization trend with different RN numbers in \(100\times 100\) terrain. When the number of RNs is small, that is, when the problem is low-dimensional, the performance of the algorithms is the same or similar. In order to fully demonstrate the convergence speed of the different algorithms in the different dimensions, the number of RNs is 3 and 6 in the 100 \(\times\) 100 terrain, and the number of RNs is 3, 9, 15, and 22 in the 200 \(\times\) 200 terrain. It can be seen from Figures 9 and 10 that the GA algorithm has the fastest convergence speed; however, compared to ABC and pdABC, it can easily fall into a local optimum. The SA algorithm is more likely to fall into local optima in the 200 \(\times\) 200 terrain than in the 100 \(\times\) 100 terrain, and it has the worst performance. The convergence speed of pdABC is faster than that of ABC, though weaker than that of GA. As the number of RNs increases and the number of iterations increases, however, pdABC is able to explore better solutions and show stronger global optimization. Compared with the other algorithms, the pdABC algorithm has better ability to avoid falling into local optima. SA, GA, ABC, and pdABC are all metaheuristic algorithms, which prevents them from falling into a local optima by increasing the random change of the solution. Randomness provides the algorithms with the ability to explore better solutions, although it makes them more volatile.