## 4 Truck-UAV Collaborative Delivery Planning Algorithm Based on Improved Evolutionary Algorithm
### An Analysis of Algorithmic Complexity  
This subsection examines the time complexity and space complexity associated with the algorithm. Let N represent the population size, D denote the number of distribution centers, K signify the average number of nodes per path, M indicate the number of demand nodes, T represent the number of trucks, L denote the length of a single-truck path, U signify the number of drone tasks, Q represent the number of path segments per task, G be the road network topology graph, V indicate the number of nodes in G, E indicate the number of edges in G, k denote the number of objective functions for non-dominated sorting, MaxLtruck signify the maximum length of the truck path, U\({}_{\text{drone}}\) be the number of drone tasks, and IterMax be the maximum number of iterations of the evolutionary algorithm. The time complexity of this algorithm is primarily influenced by the following core operations:  
1. Initialization of the Population  
A two-layer initialization strategy is employed, which encompasses both random generation and heuristic construction. Specifically, the time complexity associated with random generation in the basic layer is O(N\(\cdot\)D\(\cdot\)K), while the time complexity for heuristic generation utilizing the Clarke-Wright algorithm is O\(\left(\text{M}^{2}\right)\). Consequently, the overall complexity of population initialization is O\(\left(\text{N}\cdot\text{D}\cdot\text{K}+\text{M}^{2}\right)\). 2. Assessment of Physical Fitness  
The time complexity of the demand coverage fitness function is O(N\(\cdot\)T\(\cdot\)L), while the time complexity of the delivery cost fitness function is O\(\left(\text{N}\cdot\text{U}\cdot\left(\text{V}+\text{E}\right)\right)\). Consequently, the overall complexity is O\(\left(\text{N}\cdot\left(\text{T}\cdot\text{L}+\text{U}\cdot\left(\text{V}+\text{E} \right)\right)\right)\). 3. Genetic Operations  
The crossover operation, specifically the mixed strategy that combines order crossover (OX) and uniform crossover, exhibits a time complexity of O\(\left(\text{N}\cdot\left(\text{MaxLtruck}+\text{U}_{\text{drone}}\right)\right)\). Additionally, the mutation operation determines the path based on [PERSON]'s algorithm, with its time complexity being O\(\left(\text{V}^{2}\right)\). Consequently, the overall complexity is \(0\Big{(}\) N\(\cdot\Big{(}\text{MaxLtruck}+\text{V}^{2}\Big{)}\Big{)}\). 4. Maintenance of the Pareto Front  
The time complexity associated with non-dominated sorting is O\(\left(\text{kN}^{2}\right)\), while the time complexity for reference point association is O\(\left(\text{NlogN}\right)\). In the computation of crowding entropy, the primary contribution to time complexity arises from the Monte Carlo approximate sampling component, which has a time complexity of O\(\left(\text{kNlogN}\right)\). 5. Comprehensive Analysis of Time Complexity  
The overall time complexity of the algorithm is O\(\left(\text{T}\cdot\left[\text{N}\left(\text{V}^{2}+\text{E}\right)+\text{ kN}^{2}+\text{NlogN}\right]\right)\). The space complexity associated with the population is O\(\left(\text{N}\cdot\left(\text{T}\cdot\text{L}+\text{U}\cdot\text{Q}\right)\right)\). In terms of graph structure storage, the space complexity remains O\(\left(\text{V}+\text{E}\right)\). Due to the implementation of a hybrid coding mechanism, the space requirements of this algorithm are reduced compared to those of traditional decomposition-based multi-objective algorithms. This information is illustrated in Table 3. The green text in the table indicates the highest performance. The graph illustrates that the algorithm proposed in this study is suboptimal regarding time efficiency when compared to the other two algorithms; however, it demonstrates an optimal performance in terms of space efficiency. The algorithm exhibits significant time complexity, primarily due to its nested loop structure, which results in prolonged iterative computation times.