## 2 Materials and Methods
### Bio-Inspired Metaheuristic Optimization Method
The solution of a stochastic differential equation is a random process, the trajectories of which have sections of continuous change, interrupted by jumps of a given intensity. It describes the movement of the tomitt, accompanied by relatively short jumps. This solution can be found by numerical integration with a step size \(h\). If any coordinate value hits the boundary of the search area or goes beyond it, then it is taken equal to the value on this boundary. The best position achieved during the current iteration is chosen as the new position \(x^{j,k,search}\) of the tomitt. This process is shown in Figure 1a. Among all the new positions of tomitts, the best one is selected. It is recorded in the memory matrix and identified with the final position of the leader of the flock at the current iteration, after which the next iteration begins with the procedure for finding a new position of the leader of the flock and the initial positions of the members of the flock relative to it. This procedure is shown in Figure 1b. The method terminates when the maximum number of passes \(P\) is reached. The proposed hybrid method uses the ideas of evolutionary methods to create the initial population [15; 16; 17; 18], the method of imitating the behavior of cuckoo to simulate the jump of the flock leader based on Levy flights [29], the application of a modified numerical Euler-Maruyama method for solving a stochastic differential equation describing the movement of individuals in a population [31]; the idea of particle swarm optimization technique in a flock for describing the interaction of individuals with each other, the modified method of artificial immune systems for updating the population, the Luus-Jaakola method for updating the search set at the end of the next pass [1]. Figure 2 below illustrates the block diagram of the algorithm. Figure 1: Movement of tomtits; (**a**) stochastic movement of tomtits, choosing the best solution and adding into the memory matrix, (**b**) new position of the leader and flight results of other tomtits. Figure 2: A block diagram of the proposed TFO algorithm. Below is a detailed description of the algorithm. **Step 1**. Creation of the initial tomitt population:  
Step 1.1. Set parameters of method:  
* Number of tomitts in population \(NP\);
* Flock leader movement step \(\alpha\);
* Reduction parameter of search set \(\gamma\);
* Reconstruction parameter of search set \(\eta\);
* Levy distribution parameter \(\lambda\);
* Tomtit's neighborhood radius size \(p\);
* Parameters \(c_{1}\), \(c_{2}\), \(c_{3}\), which describe drift vector and diffusion matrix in the stochastic differential equation;
* Number of maximum records in memory matrix \(K\);
* Integration step size \(h\);
* Number of maximum discrete steps \(L\);
* Time required for searching by tomitts \(T_{t}=Lh\);
* Number of maximum passes \(P\);
* Jump intensity parameter \(\mu\). Set the value \(p=0\) (the number of passes), \(r^{0}=1\). Step 1.2. Create initial population \(I=\left\{x^{j}={(x_{1}^{j},x_{2}^{j},\ldots,x_{n}^{j})}^{T},j=1,2,\ldots,NP\right\}\subset D\) of \(NP\) solutions (tomtits) with randomly generated coordinates \(x_{i}\) from the segment \([a_{i},b_{i}]\) using a uniform distribution:  
\[x_{i}^{j}=a_{i}+rand_{i}[0,\ 1]\cdot(b_{i}-a_{i}),i=1,\ldots,n;\ j=1,\ldots,NP,\]  
where \(rand_{i}[0,1]\) is the uniform distribution law on the segment \([0;1]\). **Step 2**. Movement of flock members. Implementation of the diffusion search procedure with jumps. Step 2.1. Set the value: \(k=0\) (iteration counter). Step 2.2. For each member of flock calculate the value of objective function: \(f(x^{1,k})\), \(\ldots\), \(f(x^{NPk})\). Order flock members in increasing (nondecreasing) objective function values. The solution \(x^{1,k}\) corresponds to the best value, i.e., the position of the leader. Step 2.3. Process current information about flock members. For \(j=1\) set the following. The best position is \(x^{best}=x^{1,k}\) and the corresponding value of objective function \(f(x^{best})=f(x^{1,k})\) in population.