# Optimizing Aircraft Routes in Dynamic Conditions Utilizing Multi-Criteria Parameters
## 3 Materials and Methods
Advantages of the algorithm: the modified Dijkstra algorithm provides the optimal route, taking into account all the factors entered, which allows for a more accurate assessment of the actual cost of travel, which is especially useful for the transportation of goods or passengers. Disadvantages of the algorithm: as in the previous case (the Dijkstra algorithm itself), the complexity of the algorithm increases when new factors are added, so taking into account weather conditions may require constant updating of weighting factors, which can slow down the algorithm. _The B* (B-star) algorithm_ is a variation of the A* algorithm, but it is more focused on searching for tree-like structures in the solution space. In this algorithm, the weight of an edge and the distance between points play important roles. The above algorithm, according to [12], is quite effective, but it depends on the implementation because it requires storing data on the traversed nodes. The formulas used in the B* algorithm are as follows:  
* Expansion function:  
\[f^{\prime}(n)=g(n)+\omega h(n), \tag{7}\]
* Depth of search:  
\[d_{\max}=\frac{\log(b_{avg})}{\log(b_{\min})}, \tag{8}\]  
where \(\omega\)--weighting factor that affects the heuristic evaluation, \(b_{avg}\)--average branching factor, and \(b_{min}\)--minimum branching factor. The B* algorithm is suitable for cases where the search requires a balance between accuracy and performance, and it uses a more complex evaluation function to control the search. Let us consider a modification of the B* algorithm--a modified evaluation function \(f^{\prime}(n)\):  
\[f^{\prime}(n)=g(n)+\omega h(n)+\mathit{WeatherPenatly}(n)+\mathit{FuelCost}(n), \tag{9}\]  
where the added components are similar to those used in the modification for the A* algorithm. Advantages of the algorithm: the B* algorithm can better adapt to the uneven influence of external factors due to the ability to adjust the weighting factors \(\omega\), which allows for more flexible control over the search process. Disadvantages of the algorithm: the choice of parameter \(\omega\) becomes even more complicated due to the addition of new factors, which can lead to an increase in the complexity of the algorithm's configuration. _The D* (Dynamic A-star) algorithm_ is a dynamic variant of A* that can adapt to changes in the environment in real time. The formulas used in the D* algorithm are as follows:  
* Updating the cost of the path:  
\[d^{\prime}(n)=g^{\prime}(n)+h(n), \tag{10}\]
* Reconstruction of the road:  
\[f(n)=g(n)+h(n)+WeatherPenalty(n)+FuelCost(n), \tag{2}\]0where \(g^{\prime}(n)\)--the updated path cost taking into account the changes, \(m\)--the next node after \(n\), and \(\omega(n,m)\)--the weight between nodes n and m.  
The D* algorithm is able to modify the already calculated path if new obstacles appear on the way or conditions change. This makes it suitable for scenarios with unpredictable changes. Table 1 shows the results of the analysis of key scientific achievements in the field of building optimal air routes for aircraft. Table 1 shows that modern scientific achievements in the field of building optimal air routes for aircraft include the use and modification of classical algorithms (A*, B*, D*, and Dijkstra), taking into account external factors such as weather conditions and fuel consumption. The developed mathematical models and adaptive algorithms make it possible to adjust routes in real time, but there are drawbacks. These disadvantages include limited practical application, lack of universal models for different conditions, and significant computational costs, which complicate the widespread implementation of such solutions in aviation logistics. Table 2 shows the results of the analysis of the key areas of building optimal air routes for aircraft (which need to be supplemented as part of the improvement of modern aviation logistics).