## 3 Optimal Island Partition Strategy
### Modified Firefly Algorithm  
Firefly algorithm (FA) is one type of artificially bionic optimization search algorithm proposed by [PERSON] et al. with Cambridge University in 2008 through imitating firefly attraction surrounding others with its own light [18; 19]. In a FA optimization search, fireflies are initially randomly distributed into the solution space of the target function. The more superior the target function where the firefly is located, then the more attractive its luminance is to others. The influence of distance and the propagation medium should be considered in conducting the algorithm iteration, so as to obtain the solution for the target optimization function. The attractiveness between fireflies is in an inverse ratio with the distance and light intensity absorption coefficient, so the attractiveness between fireflies \(X_{m}\) and \(X_{n}\) can be defined as:  
\[\beta(X_{m},X_{n})=\beta_{0}e^{-\gamma\cdot\tau_{nm}} \tag{13}\]  
where \(\beta_{0}\) refers to the maximal attractiveness between fireflies, which is related to the target function value of fireflies \(X_{m}\) and \(X_{n}\); \(\gamma\) refers to light intensity absorption coefficient, which is normally set to be \(\gamma=1\) as a constant; \(\tau_{nm}\) is the distance between fireflies \(X_{m}\) and \(X_{n}\). In the \(t^{th}\) iteration, if firefly \(X_{m}\) is the most attractive object for firefly \(X_{n}\), then the position \(X_{n}(t+1)\) of firefly \(X_{n}\) after movement for being attracted can be expressed as:  
\[X_{n}(t+1)=X_{n}(t)+\beta(X_{m}(t)-X_{n}(t))+K\alpha \tag{14}\]  
where \(\alpha\)-N(0, 1) is the random step factor for satisfying normal distribution of 0-1, and \(K\) is the disturbance level. To further improve the global optimization search ability of traditional FA, the mutually exclusive firefly algorithm (MEFA) was proposed with reference to the influence of territorial repulsion during courtship, to realize the interaction of learning and competition among fireflies. The set of male fireflies was assumed as \(\{X_{a}\}_{a=1}^{M}\) and that of female ones was \(\{Y_{b}\}_{b=1}^{N}\), then the mutually exclusive neighborhood of the former was \(\big{\{}\delta(X_{a},r_{rep})\big{\}}_{a=1}^{M}\). If two male fireflies enter into the mutually exclusive neighborhood in searching for optimization, then the disadvantaged one would escape along the diameter direction as shown in Figure 2. To quickly search the whole solution space, this paper also set the optimal firefly at the mutually exclusive neighborhood as male to avoid slipping into a local optimal solution too early. Based on the above, the concrete steps of MEFA proposed in this paper are as follows:  
Step 1: Randomly select firefly set \(\text{W}=\{W_{b}\}_{b=1}^{M+N}\), which is divided into a male firefly set \(\text{X}=\{X_{a}\}_{a=1}^{M}\) and a female firefly set \(\text{Y}=\{Y_{b}\}_{b=1}^{N}\). Set the maximal absorption \(\beta_{0}\), mutually exclusive neighborhood radius \(r_{rep}\), disturbance level \(H\), random step factor \(\alpha\), convergence threshold \(\varepsilon\), and iteration threshold \(T\). Step 2: for female firefly \(Y_{h}(t)\), search for the most attractive firefly \(W(t)\in\{X_{a}(t)\}_{a=1}^{M}\cup\{Y_{b}(t)\}_{b=1}^{N}/Y_{h}(t)\), and calculate the position of the female firefly \(Y_{h}(t+1)\) after iteration:  
\[Y_{h}(t+1)=Y_{h}(t)+\beta(W(t)-Y_{h}(t))+K\alpha \tag{15}\]Step 3: For the male firefly \(X_{h}(t)\), search for the most attractive firefly \(W(t)\in\left\{X_{a}(t)\right\}_{a=1}^{M}\cup\left\{Y_{b}(t)\right\}_{b=1}^{N}/X_{ h}(t)\), and calculate the position of male firefly \(X_{h}(t+1)\) after iteration:  
\[X_{h}(t+1)=X_{h}(t)+\beta(W(t)-X_{h}(t))+K\alpha \tag{16}\]  
Make a judgment whether the intersection \(\delta(X_{h}(t+1),r_{rep})\cap\mathrm{X}\) between the mutually exclusive neighborhoods of the male firefly \(X_{h}(t+1)\) and the female firefly sets is empty or not.