## 2 Materials and Methods
### Adaptive Selection of Algorithm Parameters  
As noted in [26; 27; 33; 34; 40; 43], the quality of the solutions obtained using the ACO algorithm strongly depends on the coefficients (parameters) used in it. In the above algorithm, such coefficients are \(\alpha\), \(\beta\), \(\gamma\), \(\rho\), \(\lambda\) (Equations (6)-(9). Since each of the coefficients can take an infinite number of values, the question arises of choosing the coefficients that make it possible to obtain a solution that is closest to the optimal one. Selecting coefficients manually is inefficient because of the large range of their values and the lack of methods for their selection. In this study, it is proposed to select coefficients using their evolutionary selection. The most common method for implementing such a selection is a genetic algorithm. Algorithm 3 presents the GA application for tuning ACO parameters. The ACO parameters selection is carried out according to the scheme described below:  
1. Generation of a random initial state. The first generation is created from randomly selected solutions (chromosomes), where the parameters \(\alpha\), \(\beta\), \(\gamma\), \(\rho\), \(\lambda\) are used as genes (initialization in Algorithm 3). 2. Calculation of the coefficient of survival (fitness). Each solution (chromosome) is assigned a certain numerical value, depending on its proximity to the value of the fitness function (rows 2-12). 3. Reproduction. Chromosomes with greater fitness are more likely to pass to offspring, roulette selection, and a single-point crossover operation is performed (rows 13-25). 4. Mutation. If it is randomly determined that it is necessary to carry out a mutation, then the chromosome is changed to a new random chromosome (rows 21-31). If the specified number of iterations is completed, then the problem is solved. Otherwise, steps 2-4 are repeated. In Algorithm 3, the following designations are introduced: \(I_{g}\)--the number of GA iterations; \(C_{g}\)--the number of chromosomes; and _population_--the population of GA chromosomes. ``` Input:\(N\), \(M\), \(I_{g}\), \(C_{g}\), \(I_{\text{a}ov}\), \(C_{\text{ant},f_{\min}}\), \(f_{\max}\) Output:\(T\), \(\mu\)akespan,



### Improving the Performance of the Software Implementation of the Algorithm

With the approach described above, the solution search time increases dramatically, since the solution of the problem by the ACO algorithm is launched many times. The search speed can be significantly increased by parallelizing calculations by dividing the population into parts and distributing the computational load for working with these parts between processors. The mutation and calculation of the fitness function of individuals can be easily parallelized since they occur independently for each individual. At the same time, data common to all individuals is used only for reading, so there are no difficulties with the need to synchronize these stages and waste time on blocking processes while waiting for resources to be released. Crossover is more difficult to parallelize since during this stage there is an interaction between individuals from different parts of the population. However, there is no need for parallelization, since this stage takes negligible time compared to other calculations. However, the higher the efficiency of parallelization, the higher the level at which it is performed. Since the ACO algorithm is stochastic, it seems reasonable to simply run multiple independent instances of the algorithm at the same time. Since this paper uses meta-optimization based on a genetic algorithm, then parallelization is performed at the level of calculating the GA fitness function. In Algorithm 3, row 4 occupies the vast majority of the running time of the entire algorithm and, at the same time, the loop in rows 2-6 is easily parallelized. Since the algorithm requires multiple traversals of the graph and changing the pheromone on edges, the choice of the graph structure and the mechanism for applying the pheromone is very important.