## 3 Proposed Algorithm
### Population Initialization Based on Tent Mapping
When \(f_{i}\geq f_{avg}\), it indicates that the pigeons are in the \"divergence\" trend, and tent chaotic disturbance should be added to optimize the pigeons. Firstly, the chaotic variable \(Z_{d}\) carrier generated in the initial stage of the population in Section 3.3.2 is added to the solution space to be solved, which is defined as follows. \[newX_{d}=min_{d}+(max_{d}-min_{d})\cdot Z_{d} \tag{12}\]

where \(max_{d}\) and \(min_{d}\) are the maximum and minimum values of the d-dimensional variable \(newX_{d}\), respectively. If it is necessary to disturb the current sequence of pigeons, the individuals will be disturbed according to the following disturbance rules. \[newX^{\prime}=\left(X^{\prime}+newX\right)/2 \tag{13}\]

where \(X^{\prime}\) is the individual requiring chaotic disturbance, \(newX\) is the amount of chaotic disturbance generated, and \(newX^{\prime}\) is the new individual after chaotic disturbance. If the performance of the disturbed individual is better, the new individual is used to replace the original individual; otherwise, the original individual remains unchanged. By introducing chaotic disturbance, the algorithm can avoid falling into local optimum and improve the global searching ability and searching precision. Implementation steps of Tent chaotic disturbance mechanism are shown in Algorithm 2. ```
Step1: The chaos variable \(Z_{d}\) is generated by Equation (11). Step2: Carrier the chaotic variable into the solution space of the problem to be solved. Step3: According to Equation (12), the chaotic disturbance is performed on the individual, and the replacement and update operation is decided according to the performance comparison between the new and the old individual. ```

**Algorithm 2** Tent Chaotic Disturbance

### Cdpio

The PIO algorithm is well known for its fast convergence speed, and the optimal solution is stable and satisfactory in the search process. However, too high search speed will reduce the traversal ability of PIO algorithm in the whole search space, leading to premature convergence of the algorithm. Therefore, on the basis of the above two aspects of chaotic mapping and chaotic disturbance mechanism, we supplemented and optimized the PIO algorithm. #### 3.3.1 Population Initialization

In the initialization stage, initial values are set by the initialization mechanism in Section 3.1 to set the parameters required by the algorithm, and the improved tent chaotic mapping expression is used to generate tent chaotic sequence in the feasible domain; then, the fitness \(f_{i}\) of each pigeon is calculated, and the current optimal fitness \(f_{g}\) and its corresponding position \(x_{b}\) are selected, as well as the current worst fitness \(f_{w}\) and its corresponding position. The population initialization is thus completed. #### 3.3.2 Iteration Updating Stage

After the initial stage, the algorithm will enter the iterative update stage. In this section, each pigeon will be updated according to the existing speed and position update strategy. When the algorithm completes an iteration, it will judge whether the pigeons are in a \"divergent\" trend. If so, the individuals will be disturbed according to the tent disturbance mechanism described in Section 3.2 of this paper to generate new individuals. After the disturbance, the optimal position \(x_{b}\) and its fitness \(f_{g}\) experienced by the whole population are updated according to the current status of the flock, as well as the current worst fitness \(f_{w}\) and its corresponding position. After updating, the algorithm will judge whether the number of iterations has reached the maximum number of iterations. If not, the cycle of updating iteration will continue. Otherwise, the cycle will end, and the optimal sequence of pigeons will be output as the optimal result of the algorithm. CDPIO algorithm introduces tent chaotic mapping, which increases the population diversity, improves the performance of the algorithm, and avoids falling into local optimum.