# Optimizing Aircraft Routes in Dynamic Conditions Utilizing Multi-Criteria Parameters
## 3 Materials and Methods
For a three-dimensional space, the number of vertices checked can grow exponentially relative to the search depth, making it less efficient for large problems. * The computational complexity can reach \(O(t^{d})\), where \(b\)--branching factor and \(d\)--solution depth. 4. Dependence of the algorithm on the heuristic function: if the heuristic is inadequate or imperfect, the algorithm may run slower or even find a suboptimal route. Let us consider a possible modification of the A* algorithm. The modified evaluation function \(f(n)\) is given by the following expression:  
\[f(n)=g(n)+h(n)+WeatherPenalty(n)+FuelCost(n), \tag{2}\]where \(g(n)\)--the real cost of the path from the starting point to the current vertex, \(h(n)\)--a heuristic estimate of the distance from the current vertex to the target vertex (for example, the Euclidean distance), and \(\textit{WeatherPenalty}(n)\)--a penalty for weather conditions in the area, which can be determined as follows:  
\[\textit{WeatherPenalty}(n)=\lambda_{\omega}\cdot\omega(n), \tag{3}\]  
where \(\omega(n)\)--a function for assessing weather conditions, which may include factors such as rain, wind, snow, etc. ; \(\lambda_{\omega}\)--a weather influence factor; and \(\textit{FuelCost}(n)\)--an estimate of fuel consumption that depends on the length of the route and the speed of movement thus:  
\[\textit{FuelCost}(n)=\lambda_{f}\times d(n)\times\frac{v_{opt}\left(n\right)} {v_{real}\left(n\right)}, \tag{4}\]  
where \(d(n)\)--the length of the route section, \(v_{opt}(n)\)--the optimal speed for minimum fuel consumption, and \(v_{real}(n)\)--the real speed. Advantages of the algorithm: The A* algorithm becomes more accurate when taking into account real-world conditions such as weather and fuel consumption. This makes it suitable for practical applications where not only distance but also other factors are important. Disadvantages of the algorithm: Additional factors complicate the calculations and can lead to a significant increase in computational complexity, especially if you need to take into account changing weather conditions in real time. _[PERSON]'s algorithm_ is used to find the shortest path from the initial vertex to all other vertices in the graph [15]. A snapshot of [PERSON]'s algorithm can be found in [16]. The minimum distance according to the [PERSON] algorithm is calculated using the following expression:  
\[d(v)=\min\{d(u)+\omega(u,v)\},\forall u\in N(v), \tag{5}\]  
where \(d(v)\)--the minimum distance from the initial vertex to vertex \(u\), \(d(u)\)--the minimum distance to the adjacent vertex \(u\), \(\omega(u,v)\)--the weight of the edge between vertices u and \(v\), and \(N(u)\)--the set of neighbors of vertex \(v\). Explanation: [PERSON]'s algorithm works on graphs with positive edge weights and finds the shortest path by checking all possible options and gradually updating the distances for each vertex. Advantages of the algorithm: ease of implementation; it guarantees finding the optimal route if the weights are non-negative. Disadvantages of the algorithm: not suitable for graphs with negative weights; it has high memory and time consumption for large graphs. Influence of weather conditions: weather conditions can affect the weights of edges (for example, delays due to poor visibility), which may require the algorithm to adapt to dynamic changes. Let us consider a modification of the [PERSON] algorithm (modified path cost function):  
\[d^{\prime}(v)=\min\{d(u)+\omega(u,v)+\textit{WeatherPenalty}(u,v)+\textit{ FuelCost}(u,v)\},\forall u\in N(v), \tag{6}\]  
where \(\omega(u,v)\)--the weight of the edge between vertices \(u\) and \(v\); \(\textit{WeatherPenalty}(n,v)\)--the penalty for weather conditions on the section \(u\) to \(v\), which is calculated similarly to the A* algorithm; and \(\textit{FuelCost}(n,v)\)--the fuel cost on the section \(u\) to \(v\), which takes into account the length of the path and the speed of movement.