With the emergence of drone-based delivery systems like Prime Air, unmanned aerial vehicles have revolutionized logistics operations. Delivery drones offer significant advantages over traditional methods through enhanced efficiency and cost reduction, positioning them as vital tools for urban air mobility. As cargo volumes and distribution complexity increase, single delivery UAVs become inadequate, necessitating optimal allocation of distribution tasks across multiple heterogeneous delivery drones to achieve intelligent logistics operations.
We establish a multi-objective optimization model for delivery drone task allocation in urban environments. Consider a distribution center deploying $n$ heterogeneous delivery UAVs $U = \{U_1, U_2, \dots, U_n\}$ to service $m$ delivery points $R = \{R_1, R_2, \dots, R_m\}$. Each delivery UAV must return to the distribution center after completing tasks, represented as task $R_{m+1}$. The decision variable $x_{ij}$ determines task assignment:
$$x_{ij} =
\begin{cases}
1 & \text{if task } R_j \text{ is assigned to delivery UAV } U_i \\
0 & \text{otherwise}
\end{cases}$$
The objective function minimizes total costs comprising transportation, management, and time penalty components:
$$\min C = \omega_1 C_1 + \omega_2 C_2 + \omega_3 C_3$$
where $\omega_1 + \omega_2 + \omega_3 = 1$.
Cost Components
Transportation Cost: Includes energy consumption, depreciation, and maintenance
$$C_1 = \sum_{i=1}^{n} \sum_{j=1}^{m+1} \alpha_i x_{ij} L_{ij} w_j$$
where $\alpha_i$ is the cost coefficient per kg-km, $L_{ij}$ is flight distance, and $w_j$ is cargo weight.
Management Cost: Administrative expenses per delivery UAV
$$C_2 = \sum_{i=1}^{n} \sum_{j=1}^{m+1} \beta_i x_{ij}$$
Time Penalty Cost: For late deliveries beyond time window $[ET_j, LT_j]$
$$C_3 = \sum_{i=1}^{n} \sum_{j=1}^{m+1} \gamma (t^f_{ij} – LT_j) x_{ij}$$
Penalty coefficient $\gamma$ varies with tardiness:
$$\gamma =
\begin{cases}
0 & t^f_{ij} \leq LT_j \\
\gamma_{\max} \frac{(t^f_{ij} – t^s_{ij})}{\partial b_i} & LT_j < t^f_{ij} < MLT \\
\gamma_{\max} & t^f_{ij} \geq MLT
\end{cases}$$
Operational Constraints
| Constraint Type | Mathematical Formulation |
|---|---|
| Flight Range | $\sum_{j=1}^{m} x_{ij} L_{ij} \leq L_i^{\max} \quad \forall i \in U$ |
| Payload Capacity | $\sum_{j=1}^{m} x_{ij} w_j \leq W_i^{\max} \quad \forall i \in U$ |
| Task Completion | $\sum_{i=1}^{n} \sum_{j=1}^{m} x_{ij} = m$ |
| Unique Assignment | $\sum_{i=1}^{n} x_{ij} = 1 \quad \forall j \in \{1,2,\dots,m\}$ |
| Time Window | $t^f_{ij} \in [ET_j, LT_j]$ |
| Safety Intervals | $x_{ij}(t^s_{i+1,j+1} – t^s_{ij}) \geq \Delta t_{\text{safe}}$ |
Enhanced Harris Hawk Optimization Algorithm
We propose an Improved Harris Hawk Optimization (IHHO) algorithm to solve this NP-hard delivery drone allocation problem. Key enhancements include:
Hénon Chaotic Mapping for population initialization:
$$\begin{cases}
x_{n+1} = 1 – ax_n^2 + |y_n| \\
y_{n+1} = bx_n
\end{cases} \quad (a=1.4, b=0.3)$$
Nonlinear Escape Energy Update balancing global/local search:
$$E = 2E_1 \times E_0 \quad \text{where} \quad E_1 = \left(1 – \frac{t}{t_{\max}}\right)^{\frac{1}{2}}$$
Sine-Cosine Optimization integration:
$$X(t+1) =
\begin{cases}
X(t) + r_1 \sin r_2 \cdot |r_3 X^* – X(t)| & r_4 < 0.5 \\
X(t) + r_1 \cos r_2 \cdot |r_3 X^* – X(t)| & r_4 \geq 0.5
\end{cases}$$
Adaptive t-Distribution Mutation with iteration-dependent degrees of freedom:
$$X_i^t = X_i + X_i \cdot t(\text{iter})$$
Simulation Results
Experimental setup for 25 delivery points and 8 delivery drones:
| Drone Type | Cost Coefficient ($\alpha_i$) | Management Cost ($\beta_i$) | Speed (km/h) | Range (km) | Payload (kg) |
|---|---|---|---|---|---|
| U1-U3 | 0.6 | 0.8 | 53 | 10 | 2.5 |
| U4-U6 | 0.8 | 1.2 | 35 | 20 | 4.5 |
| U7-U8 | 0.5 | 1.0 | 31 | 20 | 3.6 |
Algorithm performance comparison (20 runs):
| Algorithm | Best Cost | Average Cost | Worst Cost |
|---|---|---|---|
| IHHO | 16.29 | 17.57 | 18.81 |
| Standard HHO | 18.64 | 20.16 | 27.52 |
Optimal delivery drone allocation scheme:
| Drone ID | Route | Distance (km) | Utilization (%) |
|---|---|---|---|
| U1 | 0→19→3→0 | 2.48 | 68 |
| U4 | 0→18→13→25→7→22→0 | 8.31 | 100 |
| U5 | 0→15→1→9→16→0 | 4.86 | 100 |
| U6 | 0→11→10→20→8→12→0 | 5.26 | 97.8 |
| U7 | 0→21→2→17→4→0 | 5.44 | 91.7 |
The IHHO algorithm demonstrates superior convergence and stability for delivery UAV task allocation, achieving a 17.8% cost reduction versus standard HHO. Our model efficiently coordinates multiple delivery drones while respecting operational constraints, significantly enhancing urban logistics efficiency. Future research will address dynamic airspace constraints and weather impacts on delivery drone operations.