## 2 Materials and Methods  
### Proposed Application of the ACO Algorithm for the JSS Problem  
To solve the JSS problem by using the ACO algorithm, it is necessary to:1. Present the problem as a directed graph;
2. Determine the heuristics of the behavior of ants when constructing a solution;
3. Adjust the algorithm parameters. The iterative ACO algorithm includes building a solution by all ants, improving the solution using the local search method, and updating the pheromone. Building a solution starts with an empty partial solution, which is expanded by adding a new, permissible solution component to it. Based on the algorithm and formulas proposed in [26], in this study, the calculation relations presented below, which are used when adapting the method to the problems of JSS, have been written down. The choice of the solution component is carried out according to the rules of probabilistic choice at each step of constructing the solution in accordance with:  
\[P_{k}=\frac{\left(f_{k}\right)^{\alpha}}{\sum\limits_{i}\left(f_{i}\right)^{ \alpha}} \tag{6}\]  
The coefficient \(\alpha\) determines the influence of the amount of pheromone on the \(k\)-th edges (\(f_{k}\)) on the probability that the ant will choose this edge. The denominator is the sum over all edges accessible from the node. The proposed approach does not use any heuristic information; for example, the duration of the selected stage or the duration of the job to which the selected stage belongs. Preliminary experiments have shown that it does not improve accuracy. For the traveling salesman problem, a route does not include all edges. Therefore, it makes sense to increase the probability of choosing a shorter graph edge for each step. For the scheduling problem, a route must include all stages in any case. Pheromone renewal is necessary to increase it on the best (short) path and to decrease its amount on paths corresponding to bad decisions. Pheromone evaporation is also used in order to avoid the too-fast convergence of the algorithm. If \(F\) is the value of the objective function on the route, then the amount of pheromone applied by the ant to all edges of the route \(\Delta f\) can be determined:  
\[\Delta f=\left(\frac{\gamma}{F}\right)^{\beta} \tag{7}\]  
Here \(\beta\) and \(\gamma\) are the intensity coefficients of pheromone release. The coefficient \(\beta\) was introduced in this work in order to make the dependence of applying the pheromone on the graph more flexible (not necessarily linear). The coefficient \(\rho\) characterizes the pheromone evaporability. Here, it is considered that a certain minimum non-zero amount of pheromone should always remain on the edges. Otherwise, the probability of choosing an edge may be zero and it will be \"ignored\" by the ants. The maximum value is also limited, which prevents the convergence of the algorithm to a solution far from the optimal one. The coefficient takes values from 0 (no evaporation) to 1 (evaporates to a minimum level). \[f^{\prime}=\left\{ \begin{matrix}f(1-\rho),\;f_{\min}<f(1-\rho)<f_{\max}\\ f_{\min},\;f(1-\rho)\leq f_{\min}\\ f_{\max},\;f(1-\rho)\geq f_{\min}\end{matrix} \right. \tag{8}\]  
During the experimental studies, an improvement in results was revealed with an increase in the significance of the current best solution. To do this, on all edges of the path corresponding to the best result at each iteration, a certain amount of pheromone is added, which is determined by the coefficient \(\lambda\):  
\[f_{best}=\left\{ \begin{matrix}f_{best}\cdot\lambda,\;f_{best}\cdot\lambda<f_{ \max}\\ f_{\max},\;f_{best}\cdot\lambda\geq f_{\max}\end{matrix} \right. \tag{9}\]  
Thus, the limit on the maximum amount of pheromone is taken into account here as well. It is possible to present the search for a solution to the JSS problem as follows. In order to completely set the schedule, it is enough to determine which job to load on the device it needs at each \(i\)-th step, \(I=1\), \(2\),..., \(C_{s}\), where \(C_{s}\) is the total number of stages of all jobs from the set \(N\).