Following large-scale emergencies, medical facilities in affected areas face dramatic surges in plasma demand. Traditional ground transportation becomes severely constrained when infrastructure is compromised, creating critical delivery bottlenecks. Police drones and police UAVs offer a transformative solution with their ability to bypass terrestrial obstacles and ensure time-sensitive delivery of life-saving blood products. This research develops a distributionally robust optimization framework to maximize both plasma delivery efficiency and equity in police drone networks under supply-demand uncertainty.
Our two-stage optimization model addresses pre-disaster drone route configuration and post-disaster delivery operations. Let $I$ denote blood centers and $J$ represent hospitals. Pre-disaster decisions establish police drone routes $x_{i,j} \in \{0,1\}$ between $i \in I$ and $j \in J$ under budget $F$ and range constraints $d_{i,j} \leq L$:
$$ \min \sum_{i \in I} \sum_{j \in J} e_{i,j}x_{i,j} + \mathbb{E}[Q(\mathbf{x},\xi)] $$
$$\text{s.t. } \sum_{i \in I} \sum_{j \in J} e_{i,j}x_{i,j} \leq F $$
$$ d_{i,j}x_{i,j} \leq L \quad \forall i \in I, j \in J $$
$$ x_{i,j} \in \{0,1\} \quad \forall i \in I, j \in J $$
Post-disaster decisions optimize plasma allocations $q^{G,s}_{i,j}$, $q^{AB,s}_{i,j}$ under supply $(\alpha^s_i, \beta^s_i)$ and demand $(\eta^s_j, \theta^s_j)$ uncertainty across scenarios $s \in S$:
$$ Q(\mathbf{x},\xi) = \min u $$
$$\text{s.t. } q^{AB,s}_{i,j} + q^{G,s}_{i,j} \leq \hat{R}_{i,j}x_{i,j} \quad \forall i,j,s $$
$$ f^{AB,s}_j + \sum_{i \in I} q^{AB,s}_{i,j} \geq \eta^s_j \quad \forall j,s $$
$$ f^{G,s}_j + \sum_{i \in I} (q^{G,s}_{i,j} + \hat{q}^{AB,s}_{i,j}) \geq \theta^s_j \quad \forall j,s $$
$$ \frac{f^{AB,s}_j + f^{G,s}_j}{\eta^s_j + \theta^s_j} \leq u \quad \forall j,s $$
To address distributional ambiguity in $\xi = (\alpha, \beta, \eta, \theta)$, we implement a distributionally robust optimization (DRO) model with $\phi$-divergence ambiguity set $\mathcal{D}$:
$$ \min_{\mathbf{x}} \sup_{\mathbb{P} \in \mathcal{D}} \mathbb{E}_{\mathbb{P}}[Q(\mathbf{x},\xi)] $$
$$ \mathcal{D} = \left\{ \mathbb{P} \in \mathcal{P}(\Xi) : D_{\phi}(\mathbb{P} \| \mathbb{P}_0) \leq \delta \right\} $$
Through strong duality, we transform the semi-infinite program into a tractable mixed-integer linear formulation solvable by commercial optimization software:
$$ \min \sigma\lambda + \mu + \sum_{s \in S} p^0_s \rho_s $$
$$\text{s.t. } \rho_s \geq Q(\mathbf{x},\xi_s) – \lambda \quad \forall s \in S $$
$$ \lambda \geq 0, \quad \mu \in \mathbb{R} $$
Experimental Validation
We implement the police UAV delivery system using data from urban emergency networks. Three blood centers and seven hospitals form the network nodes, with police drone capabilities calibrated to commercial logistics UAV specifications.
| Blood Center | Hospital Connections | Route Cost (¥10K) |
|---|---|---|
| Center 1 | H1, H2, H3 | 2.8 |
| Center 2 | H4, H5, H6 | 2.6 |
| Center 3 | H1, H6, H7 | 2.9 |
Supply-demand scenarios simulate disaster variability:
| Scenario | Plasma Supply Range (U) | Demand Fluctuation (%) |
|---|---|---|
| S1 (Moderate) | 520-710 | ±15 |
| S2 (Major) | 1,572-1,904 | +80 |
| S3 (Severe) | 403-591 | -40 |
Performance Comparison
The police drone network demonstrates significant advantages over deterministic approaches under uncertainty:
| Model Type | Demand Satisfaction (%) | Max Hospital Disparity (p.p.) |
|---|---|---|
| Deterministic | 82.3-86.1 | 12.7-18.3 |
| DRO Police UAV | 88.5 | ≤10.0 |
Sensitivity Analysis
Police drone endurance critically impacts delivery effectiveness:
$$ \text{Satisfaction} = 0.92 – 0.18e^{-0.25L} \quad (R^2=0.96) $$
where $L$ denotes operational range (km). Range reductions from 15km to 12km decrease satisfaction rates by 11.3 percentage points across scenarios.
Budget variations demonstrate the police UAV system’s equity preservation:
| Budget Level | Avg Satisfaction (%) | Max-Min Spread (p.p.) |
|---|---|---|
| 100% (¥350K) | 92.1 | 8.7 |
| 70% (¥245K) | 86.3 | 9.9 |
Conclusions
This research establishes that police drone networks optimized through distributionally robust methods significantly enhance emergency plasma delivery under uncertainty. Key advantages include:
- Demand Satisfaction: Maintains ≥88.5% delivery rates under severe supply-demand fluctuations
- Operational Equity: Limits inter-facility satisfaction disparities to ≤10 percentage points
- Infrastructure Resilience: Functions effectively at 70% budget levels with minimal performance degradation
The optimization framework enables police UAV systems to dynamically adapt to blood type substitution protocols and distribution constraints. Future work will integrate real-time police drone routing adjustments and expand to multi-period delivery operations.