# Multi-Strategy Improved Flamingo Search Algorithm for Global Optimization
## 3 Improving Flamingo Search Algorithm
### Information Feedback Model  
[PERSON] et al. [23] found that in most metaheuristic algorithms, the update process fails to utilize the information available to the individuals in previous iterations. Therefore, an information feedback model was proposed to incorporate useful information from previous iterations into the update process, resulting in a significant improvement in the solution quality. The information feedback model essentially generates a new individual by combining the information from several previous generations of individuals through weighted summation. There are two operational modes for the information feedback model: randommode and fixed mode [24; 25]. To prevent a significant increase in algorithm complexity resulting from the retention of too many generations of population information, the number of previous individuals selected is usually no more than three generations [23]. In this study, we only focus on the case where the case when the number of predecessors is 1. In this case, the information feedback model is expressed according to the following equation. \[x_{i}^{t+1}=\alpha x_{i}^{t}+\beta y_{i}^{t+1} \tag{10}\]  
\[\alpha=\frac{f_{i}^{t+1}}{f_{i}^{t+1}+f_{j}^{t}},\beta=\frac{f_{j}^{t}}{f_{i}^ {t+1}+f_{j}^{t}} \tag{11}\]  
Assume the current generation is \(t+1\); then, \(x_{j}^{t}\) refers to the position of the \(j\)th flamingo of the previous generation, \(y_{i}^{t+1}\) is the intermediate individual derived using the FSA algorithm, \(x_{i}^{t+1}\) is the position of the \(i\)th flamingo of the same generation, and \(f\) denotes the fitness of the corresponding individual. \(x_{i}^{t+1}\) is derived from the weighted sum of intermediate and predecessor individuals, and the weight coefficients \(\alpha\) and \(\beta\) are influenced by the fitness value. There are two choices for the value of \(j\). The first one is when \(j=i\), which is a fixed approach at this time. The second is when \(j=rand(1,2,\cdots,N)\), which is a randomized approach. Different ways of \(j\) have different facilitation effects on the information feedback model. Different choices of the parameter \(j\) will result in different types of information feedback models. Each of the two strategies for selecting \(j\) has its own advantages and disadvantages, and one approach is not necessarily superior to the other. The fixed model in the information feedback model can effectively improve the convergence speed of the algorithm when the algorithm is not trapped in a local optimum. However, the information feedback model in fixed mode has the disadvantage of insufficient ability to jump out of the local optimum. The random mode in the information feedback model has good randomness and can increase the learning probability for excellent individuals in the population, thereby enhancing the global search capability of the algorithm. Nevertheless, in contrast to the facilitation effect of the fixed mode, the random mode can result in weakened convergence speed and precision before the algorithm reaches a local optimum. Based on the analysis above, this study proposes an information feedback model based on adaptive fitness adjustment and applies it to the FSA algorithm. The model dynamically selects the appropriate information feedback mode based on the current fitness value. By combining the strengths of two different models, the performance of the algorithm is jointly improved. The following formula is designed as a threshold for dynamically adjusting the model based on fitness. \[\sigma=\left|\frac{f_{i}^{t+1}-f_{i}^{t}}{f^{t}\text{max}-f^{t}\text{min}}\right| \tag{12}\]  
Assume the current generation is \(t+1\), where \(f_{i}^{t+1}\) represents position of the \(i\)th flamingo of the current generation, \(f_{i}^{t}\) represents the fitness value of the \(i\)th flamingo of the previous generation, and \(f^{t}\)min and \(f^{t}\)max represent the minimum and maximum fitness values of the previous generation, respectively. In the beginning stage of the algorithm, individuals tend to have a dispersed distribution.