## 4 Numerical Experiments
### Relay Node Deployment Experiments
In order to verify the stability of the pdABC algorithm in the process of solving the deployment problem of relay nodes in TT-WSN, the box plots shown in Figures 11 and 12 demonstrate the distribution of the solution. Figure 10: Lifetime optimization trend with different RN numbers in 200 \(\times\) 200 terrain. The box plot evaluates the symmetry and distribution of the solution distribution using five statistical values in the data (the minimum, first quartile, median, third quartile, and maximum), which can be intuitive. The above shows that the solution distribution is stable. The 6 RNs and 22 RNs in the two terrains have the most changes, respectively; thus, they are used to evaluate the stability of the algorithms under the two terrains. It can be seen from Figures 11 and 12 that the distribution of the GA solutions is the most concentrated, that is, its stability is the best. The SA algorithm has the worst performance in the first terrain, although it performs better in the other terrain; in general, the overall health value of the solution is the lowest. The distribution of the solutions of the pdABC algorithm is better than that of ABC algorithm. Although the stability of the pdABC solution distribution is worse than that of GA, the median and average values of the pdABC solutions compared to other algorithms in the two terrains are the largest. In addition, the maximum value found by pdABC is the largest, which means that the algorithm has stronger global optimization performance. In addition to comparison with the abilities of the SA, GA, and ABC algorithms in solving the problem of deploying RNs in TT-WSN, we compared the dpABC algorithm with other state-of-the-art algorithms, using the same 100 \(\times\) 100 terrain with 6 RNs and 200 \(\times\) 200 terrain with 22 RNs for the simulation. The algorithms compared include WOA [32], SCA [33], and GWO [34]. These three algorithms have been widely studied and applied in recent years. Thus, experiments were conducted to verify the performance of dpABC algorithm. A preliminary experimental study aimed at ensuring the fairness of algorithm comparison was conducted to ensure that all the algorithms had the same  
Figure 11: Optimal lifetime distribution box plot with six RNs in 100 \(\times\) 100 terrain. Figure 12: Optimal lifetime distribution box plot with twenty-two RNs in 200 \(\times\) 200 terrain. number of problem evaluations in one iteration. The results can be comprehensively evaluated through iteration numbers and network lifetime. Thus, the parameter of each algorithm's population size was set. To study the other parameters of these algorithms, see the related references [32; 33; 34]. The lifetime optimization results of the different algorithms are shown in Table 2, in which Ave.NL represents the average network lifetime of the TT-WSN and Ave.CT denotes the number of iterations when the algorithm reaches the convergence state. Both parameters were obtained by averaging them over several experiments. However, for simplicity, both were taken directly as integers. As can be seen from the table, while WOA has the best coverage ratio, it requires more iterations due to its slow convergence speed. The dpABC algorithm has the second-highest average network lifetime compared to other algorithms, and its average iteration times are reduced nearly by 15% compared with WOA. It can be concluded that the dpABC algorithm not only achieves a higher average network lifetime, it has a faster convergence rate as well.