# Optimizing Aircraft Routes in Dynamic Conditions Utilizing Multi-Criteria Parameters
## 3 Materials and Methods
Summarizing the analysis of existing solutions and research on optimizing the flight path of an aircraft (as well as the corresponding algorithms), we can see that they all work in a two-dimensional coordinate plane and do not take into account the construction of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm Name** & **Description** & **Application in Aviation** & **Advantages** & **Disadvantages** \\ \hline \multirow{4}{*}{
\begin{tabular}{c} D* \\ (basic algorithm) \\ \end{tabular} } & Dynamic planning with & & & \\  & updated path cost & & & \\  & estimation: & & & \\  & \(f^{\prime}(n)=g^{\prime}(n)+l^{\prime}(n)\) & & & & \\  & with adaptation. & & & \\ \hline \multirow{4}{*}{D\_Lite*} & A simplified algorithm for & & & \\  & faster real-time work: & & & \\  & \(f(n)=g(n)+h(n)\), but with & & & \\  & optimized computation. & & & \\ \hline \multirow{4}{*}{D-Focused*} & A specialized algorithm & & & \\  & with optimization for & & & \\  & specific types of problems: & & & \\  & \(f(n)=g(n)+h(n)\) & & & \\  & with a focus on & & & \\  & certain properties. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of modern improvements of the D* algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm Name** & **Description** & **Application in Aviation** & **Advantages** & **Disadvantages** \\ \hline \multirow{4}{*}{Bidirectional Dijkstra} & Simultaneous search & & & \\  & from both ends to & & & \\  & reduce search time: & Reduced search time & & \\  & [PERSON]’s two-way & & in static conditions. & Faster route search. & \\  & algorithm. & & & \\ \hline \multirow{4}{*}{Dijkstra + Heuristics} & Adding heuristics to & & & \\  & speed up the search: & & & \\  & \(f(n)=g(n)+h(n)\), & & & \\  & where \(h(n)\)—a & & & \\  & heuristic similar to & & \\  & the A* algorithm. & & & \\ \hline \multirow{4}{*}{Lazy Dijkstra} & Avoids calculations & & & \\  & for unlikely paths, & & \\ \cline{1-1}  & saving computing resources: \(f(n)=g(n)\) & & & \\ \cline{1-1}  & \(+h(n)\) using filtering. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Analysis of modern improvements of the Dijkstra algorithm. an optimal route when changing the flight altitude (or factor in the impact of weather conditions on fuel consumption or the impact on aircraft speed). Therefore, it is necessary to modify algorithms by imposing additional restrictions and adding criteria when calculating distances between points in space. Despite all the algorithms that work in a two-dimensional plane, the Dijkstra algorithm is best suited for the needs of this study because it determines the set of all possible paths from point A to point B. This makes it possible, by imposing various constraints, to provide more accurate information about the optimal paths according to different optimality criteria. One of the determining factors in building an optimal route is the number of points in the graph, which affects the speed of route finding and memory usage. Therefore, it is also necessary to consider the main methods of limiting the search area of the optimal route [4,17]. Currently, the most well-known methods for restricting the sample are those based on rectangular or elliptical regions. It should be noted that the use of rectangular areas is not correct and relevant from the point of view that the route is built in the air and the flight path will always have the form of an arc. Therefore, when constructing this technical solution, it is advisable to use the method of elliptical domains. Paper [4] gives an example of limiting the area of sampling points for route finding using an ellipse. Another area of important optimization research in flight route planning is the study of the impact of weather conditions (especially winds). Consider the impact on flight planning of major weather phenomena such as thunderstorms, snow and ice, and jet streams.