## 3 Proposed Algorithm
### Population Initialization Based on Tent Mapping
The concrete implementation steps of CDPIO are shown in Algorithm 3. Step 1: Initialize the parameters in CDPIO, including the number of pigeons \(N_{p}\), the maximum iteration times of compass part \(N_{\text{C1}}\), and the landmark part \(N_{\text{C2}}\) (\(N_{\text{C1}}<N_{\text{C2}}\)), and the state of each pigeon, the position \(X_{i}\), and the velocity \(V_{i}\). Step 2: The population is initialized according to Algorithm 1. Step 3: Calculate the fitness \(f_{i}\) of each pigeon, select the current best fitness \(f_{g}\) and its corresponding position \(x_{b}\), along with the current worst fitness \(f_{w}\) and its corresponding position. Step 4: Update \(X_{i}\) and \(V_{i}\) according to Equations \((1)\) and \((2)\). Step 5: If \(n>N_{\text{C1}}\), jump to Step 6, or \(n=n+1\), and return Step 4. Step 6: Update \(N_{p},\ X_{c},\ X_{i}\) according to Equations \((3)\)-\((5)\), and update and preserve \(X_{g}\). Step 7: After the completion of one iteration, it is judged as to whether the pigeons need to be disturbed according to Algorithm 3. Step 8: According to the current status of the pigeons, the optimal position \(x_{b}\) and its fitness \(f_{g}\) experienced by the whole population are updated, as well as the current worst fitness \(f_{w}\) and its corresponding position. Step 9: If \(n>N_{\text{C2}}\), jump to Step 10; or if \(n=n+1\), return to Step 6. Step 10: According to Equation \((8)\), \(X_{p}\) is obtained and retained. The loop ends and the optimization result is output as the output of the algorithm. **Algorithm 3** Chaotic Disturbance Pigeeon-inspired Optimization Algorithm

## 4 Performance Evaluation of CDPIO

### Simulation Setup

This section is mainly to verify the feasibility and superiority of CDPIO algorithm. In order for the performance and characteristics of the algorithm to be fully tested, many standard functions with different characteristics are usually introduced. Here, nine complex functions [25], as shown in Table 1, are used as experimental objects. The functions selected involve multiple complex functions such as single peak, multi-peak, low dimension, and high dimension. The optimization ability of CDPIO algorithm can be fully investigated through various types of reference functions. ### Performance of CDPIO

The population size of each algorithm and the total number of iterations were the same, and each individual was evaluated for individual fitness only once during each iteration. The average value, standard deviation, and optimal value of each algorithm were used as the final evaluation index for nine different types of benchmark functions. The correlation performance of each test function was tested and compared with that of PIO algorithm. The optimization results are shown in the table below. The best value and average value in Table 3 can be expressed as the convergence precision and optimization ability of the algorithm. For f1, f4, f5, f6, and f7 functions, the convergence accuracy was very high, while for f5, f6, and f8 functions, the average value was very close to the optimal value, and compared with PIO algorithm, the ability of CDPIO has been significantly improved. The standard deviation in Table 3 reflects the robustness of the algorithm. Low standard deviation proves that there is little difference in the optimization data and the optimization process is more stable. For f5, f6, f8, and f9, the standard deviation is 0, which is a great improvement compared with the standard deviation of PIO algorithm, indicating that CDPIO has better robustness. As shown in Figure 1, the four pictures (a-d) represent the low-dimensional unimodal and low-dimensional multimodal, high-dimensional unimodal, and iterative graph of high-dimensional multimodal function. The blue curve shows the test curve of the benchmark functions using the PIO algorithm, and the red curve represents the test curve of functions using CDPIO algorithm.