With rising demands for efficient urban logistics, delivery drones offer transformative solutions for last-mile distribution. This research establishes a comprehensive mathematical model for delivery UAV distribution center location-allocation optimization, considering urban airspace constraints, energy consumption dynamics, and environmental impacts. The model minimizes total operational costs while addressing complex urban logistics challenges.
The multi-objective optimization framework minimizes total cost $C$ comprising:
$$C = C_1 + C_2 + C_3 + C_4$$
where:
$$C_1 = \sum_{j \in J} X_j \cdot f \quad \text{(Distribution center construction cost)}$$
$$C_2 = \sum_{i \in I} \sum_{j \in J} \sum_{k \in K} d_{ji}^k \cdot \lambda_{ij} \cdot Y_{ij}^k \cdot Z_{ij}^k \quad \text{(Delivery cost)}$$
$$C_3 = E \cdot p \quad \text{(Energy consumption cost)}$$
$$C_4 = E \cdot \theta \cdot \alpha \quad \text{(Carbon emission cost)}$$
Energy consumption $E$ for delivery drones incorporates payload-sensitive dynamics:
$$E = \sum_{i \in I} \sum_{j \in J} \sum_{k \in K} \frac{(\phi + W_{ij}^k) \cdot d_{ij} \cdot P}{370 \cdot \eta \cdot \gamma \cdot (P – e)}$$
Key operational constraints include:
| Constraint Type | Mathematical Expression |
|---|---|
| Distribution Center Capacity | $\sum_{i \in I} X_{ji} \leq m, \forall j \in J$ |
| Drone Range Limit | $\sum_{i \in I} \sum_{j \in J} 2 \cdot Y_{ij} \cdot d_{ij} \leq D_{\text{max}}$ |
| Payload Capacity | $W_{ij}^k \leq W_{\text{max}}$ |
| No-Fly Zone Compliance | $Z_{ij}^k = 0$ if obstacle exists |
The hybrid GA-PSO algorithm combines exploration and exploitation capabilities:
- Encoding: Chromosome structure with $m + n$ genes representing demand points and distribution centers
- Population Initialization: 60 particles with random feasible solutions
- Fitness Evaluation: $1/C$ where $C$ is total cost
- GA Operations: Multipoint crossover ($P_c = 0.8$) and uniform mutation ($P_m = 0.05$)
- PSO Velocity Update: $v_{id}^{t+1} = \omega v_{id}^t + c_1 r_1 (p_{id} – x_{id}^t) + c_2 r_2 (p_{gd} – x_{id}^t)$
Simulation parameters for urban delivery drone operations:
| Parameter | Value | Unit |
|---|---|---|
| Distribution center cost ($f$) | 20,000 | CNY |
| Drone range ($D_{\text{max}}$) | 20 | km |
| Payload capacity ($W_{\text{max}}$) | 8 | kg |
| Electricity price ($p$) | 0.76 | CNY/kWh |
| Carbon cost coefficient ($\alpha$) | 0.315 | CNY/kgCO₂ |
Optimization results for 36 demand points and 12 candidate centers:
| Selected Centers | Assigned Demand Points | Total Distance (km) |
|---|---|---|
| 2 (4,16) | 4,5,8,10,14,16 | 313.90 |
| 4 (6,6) | 1,2,3,6,7,9,13,15 | |
| 7 (12,4) | 11,12,20,21,25,26,30,31,35 | |
| 8 (13,8) | 17,18,19,22,23,30,32,34 | |
| 9 (15,18) | 24,27,28,29,33,36 |
Sensitivity analysis reveals critical operational insights:
$$ \Delta C_{\text{obstacles}} = 18.7\% \text{ cost increase with 3 additional no-fly zones} $$
$$ \Delta C_{\text{capacity}} =
\begin{cases}
+22.3\% & \text{when } m=6 \\
-14.1\% & \text{when } m=9
\end{cases} $$
Energy consumption for delivery UAVs demonstrates quadratic relationship with payload weight $W$:
$$E(W) = k_1 \cdot W^2 + k_2 \cdot W + k_3 \quad (R^2 = 0.97)$$
This research establishes a robust framework for urban delivery drone network optimization, demonstrating significant cost savings through integrated location-allocation modeling and hybrid metaheuristic problem-solving. The methodology enables sustainable urban logistics planning that balances economic efficiency with environmental responsibility for delivery UAV operations.