Disturbance-aware collaborative routing for concurrent emergency response

Emergency vehicle routing and priority control

Emergency vehicle routing and priority control have been widely studied to reduce response time under dynamic urban traffic conditions. Existing studies improve EV passage mainly through real-time route adjustment, intersection signal preemption, and cooperative yielding for surrounding social vehicles^[6-13]. These methods show that EV mobility can be enhanced by adapting routes, reserving intersection capacity, and coordinating nearby traffic participants.

However, their decision scope remains largely local. Dynamic routing optimizes a given EV route using segment-level congestion or travel-time costs, while signal preemption and yielding strategies reduce delays at specific intersections or local areas. In CER scenarios, such local decisions may increase route conflicts or delay other urgent tasks because multiple tasks and EVs compete for feasible routes and response resources. This motivates coupling EV routing with collaborative scheduling, so that candidate routes can be evaluated from a system-level perspective under concurrent incidents and resource competition.

Disturbance-aware traffic prediction

Recent traffic prediction studies use graph neural networks and Transformer-based architectures to capture spatiotemporal dependencies in urban road networks^[14-18]. They improve modeling of road-network diffusion, temporal evolution, adaptive spatial correlations, and long-range spatiotemporal patterns. However, most models target regular traffic conditions and rely mainly on historical patterns, making them insufficient for modeling incident-induced capacity degradation and disturbance propagation across neighboring or downstream segments.

Some studies have incorporated event or incident information into traffic prediction, showing that abnormal events affect future traffic states and spatial dependencies^[19,20]. Nevertheless, they still focus mainly on prediction accuracy, and the predicted incident impact is rarely converted into route-level decision information for emergency response. In CER scenarios, traffic prediction should provide both future traffic states and disturbance-aware information to support candidate route generation and collaborative scheduling.

Emergency scheduling

Emergency scheduling studies have shifted from isolated dispatching to joint dispatching, relocation, and multi-period resource redistribution under uncertain and time-varying demand^[21-23]. These studies improve emergency resource availability and responsiveness, but they mainly focus on vehicle deployment or assignment rather than the joint evaluation of task urgency, route quality, and resource competition under concurrent incidents.

Another research line formulates emergency response as multi-vehicle multi-task allocation, path planning, or matching-based optimization^[24-28]. These formulations provide useful structures for assigning limited emergency resources to multiple demands. Such methods still treat route information as fixed input or evaluate assignment costs mainly by distance, travel time, or matching feasibility. Accordingly, collaborative scheduling requires a unified formulation that integrates dynamic traffic disturbances, concurrent tasks, and limited emergency resources.

Incident disturbance modeling

Incidents are treated as external disturbance sources whose impacts diffuse along the road network to downstream areas. For each incident s, the shortest hop distance from $$ d_{\mathbf{e}}^{(s)} $$ segment e to the incident source is computed to describe the spatial influence range. The disturbance feature of segment e is then constructed as

Here, the first three components represent source membership, normalized topological distance, and distance-based decay, where ω > 0 controls the decay rate. In this study, ω is set as a fixed global hyperparameter. For segments beyond the maximum propagation range, the disturbance value is set to zero. The incident severity is $$ \tilde{L}_{s}=\ell_{s} / \ell_{\max } $$, where $$ \ell_{s} $$ and $$ \ell_{\max } $$ denote the current and maximum incident levels, respectively.

When multiple incidents occur, their disturbance features are stacked into a disturbance tensor $$ I \in \mathbb{R}^{S \times|\varepsilon| \times 4} $$, where S denotes the number of incidents. The incident-aware traffic representation $$ G_{t}^{\text {dis }} $$ is fed into DAGWN, enabling the model to capture disturbance locations, propagation patterns, and dynamic evolution, while reflecting the range and intensity decay of incident impacts.

Disturbance-aware Graph WaveNet

DAGWN is built upon Graph WaveNet^[16] and integrates temporal modeling with incident spatial diffusion to predict traffic capacity and disturbance intensity for route generation. Figure 2 shows the overall architecture of DAGWN.

Figure 2. Disturbance-aware Graph WaveNet framework.

A. Graph convolution layer

For spatial modeling, DAGWN uses diffusion graph convolution to describe traffic feature propagation on a directed graph. Let$$ A \in \mathbb{R}^{N \times N} $$ and$$ X^{d i s} \in \mathbb{R}^{N \times C} $$ denote the adjacency matrix and the disturbance-aware segment feature matrix, respectively. The forward and backward diffusion matrices are P_f = D^-1A and P_b = D^-1A^T, where D is the degree matrix. Based on this diffusion process, graph convolution is formulated as

where K is the diffusion order, and W_k1 and W_k2 are learnable parameters for forward and backward diffusion, respectively^[16].

A single adjacency matrix still cannot simultaneously capture network topology, latent segment correlations, and incident-induced disturbance relationships. Accordingly, diffusion graph convolution is extended with multiple supports $$ \mathcal{A} = \left\{ A, A_{a d p}, A_{e v e n t}\right\} $$ for joint information propagation.

In the support set, the physical adjacency matrix A represents actual road connectivity, while the adaptive adjacency matrix A_adp is learned from node embeddings to capture latent segment correlations:

where H₁ and H₂ denote embeddings of source and target segments^[16].

To model local incident propagation, the disturbance-aware adjacency matrix A_event is constructed. For each segment e_i, its disturbance intensity is computed from $$ \mathbf{I}_{e_{i}} $$ as

The disturbance relationship between two segments is then expressed as

To prevent disturbance information from dominating spatial dependency modeling, a stabilization mechanism reweighs the disturbance-aware adjacency matrix and fuses it with the adaptive adjacency matrix through a residual connection.

where λ_e and μ_a denote the disturbance adjacency weight and residual fusion coefficient, respectively. The three adjacency supports are designed to capture complementary spatial dependencies: the physical adjacency preserves road-topology constraints, the adaptive adjacency captures latent traffic correlations beyond direct connectivity, and the disturbance-aware adjacency emphasizes incident-induced local propagation. The residual fusion is therefore used as a stable and lightweight mechanism to retain general traffic correlations while preventing sparse disturbance signals from excessively dominating spatial dependency learning. Based on the fused supports, graph convolution is written as

where $$ P_{f}^{(l)}=\left(D^{(l)}\right)^{-1} A^{(l)} $$ and $$ P_{b}^{(l)}=\left(D^{(l)}\right)^{-1}\left(A^{(l)}\right)^{T} $$ are the forward and backward diffusion operators of the l-th adjacency matrix.

B. Temporal convolution layer

To capture the temporal evolution of incident-affected traffic states, DAGWN uses dilated causal convolution for temporal feature extraction.

Let the input tensor be $$ \mathbf{U} \in \mathbb{R}^{B \times T \times N \times F} $$, where B, T, N and F denote the batch size, historical window length, number of road segments, and feature dimension per segment and time step, respectively. The input is then mapped to a hidden representation $$ \mathbf{H}_{i n} \in \mathbb{R}^{B \times Q \times T \times N} \\ $$, where O denotes the number of hidden channels.

Temporal dependencies are modeled by gated dilated convolution:

where $$ \mathrm{DConv}_{\delta}(\cdot) $$ denotes a dilated causal convolution with dilation rate δ. $$ \Theta_{f i l t e r} $$ and $$ \Theta_{g a t e} $$ denote the convolution parameters of the filter and gate branches. The tanh(·) function extracts nonlinear temporal features, while σ(·) controls information flow across time steps^[16].

For a one-dimensional temporal sequence of a single channel, dilated causal convolution is written as

where K_t denotes the temporal kernel size, and f_τ denotes the kernel parameter at position τ. When δ > 1, sparse temporal sampling enlarges the receptive field without significantly increasing the parameter size^[16].

C. Dual-head output structure with disturbance awareness

In the output layer, DAGWN adopts a dual-head structure to separately predict traffic states and estimate incident impacts. Since incident disturbances are local and sparse, a single traffic-state regression head may suppress them or treat them as noise, weakening the model’s ability to describe the affected range and intensity. To address this issue, two output heads are introduced to preserve both global traffic evolution and local disturbance effects. The primary head (head-1) predicts future traffic states for route generation, while the disturbance-aware subgraph head (head-2) estimates segment-level disturbance intensity $$ \hat{\delta}(\cdot, t) $$ from shared spatiotemporal features. The disturbance intensity label for head-2 is obtained by comparing the normal traffic state and the post-incident traffic state under the same road network and traffic demand. The normal state is used as the reference, while the post-incident state reflects the traffic degradation caused by incidents. The normalized difference between the two states is used as the segment-level disturbance intensity. In this study, the normal traffic state refers to the incident-free baseline state generated under the same road network and traffic demand. It is used only as an offline reference for constructing the supervision label of the disturbance-aware head during training. Given the current traffic observations and incident features, DAGWN directly predicts future traffic states and segment-level disturbance intensity, which are then used for candidate route generation.

During training, the two heads are jointly optimized with the following overall loss:

where Loss_reg and Loss_mask are the losses of head-1 and head-2, respectively, and λ₁ is the weighting coefficient.

Dynamic weight update-based candidate route generation

Under incident-affected traffic states, static edge weights based on length or average travel time cannot reflect local capacity degradation. As a result, Dijkstra edge weights are dynamically updated using predicted travel time and disturbance intensity. The cost of segment e is defined as cost_e = t_e + λ₂g(e), where t_e is its predicted travel time and λ₂ is the disturbance intensity weight. To provide multiple feasible choices for upper-level scheduling, key segments in the current optimal route are penalized after each search to guide subsequent searches toward more diverse and less locally similar alternatives.

Thus, traditional Dijkstra-based routing is extended to dynamic multi-candidate route construction for complex emergency traffic environments.

Decision unit construction and feasibility screening

Tasks, EVs, and routes are modeled as triplet decision units $$ C_{a}=\left\{(i, j, n) \mid r_{i} \in R, \text { evj } \in E V, p_{i j}^{n} \in P_{i j}\right\} $$, where i, j, and n correspond to tasks, EVs, and routes, respectively. Each triplet (i, j, n) represents a candidate scheme in which vehicle ev_j executes task r_i through the n-th candidate route $$ p_{i j}^{n} $$. The original scheduling problem is therefore transformed into a combinatorial selection problem over C_a, with the final solution obtained by selecting an optimized subset$$ \Omega \subseteq C_{a} $$.

Since C_a may contain infeasible triplets, evaluating all candidates can introduce invalid solutions and increase computational complexity. To avoid this, feasibility screening is applied before optimization. First, type matching is checked to ensure that the EV assigned to each task belongs to the required vehicle type set of that task. Second, route feasibility is verified by removing routes with closed or inaccessible road segments. Third, explicitly invalid triplets caused by routing failure, task invalidation, or unavailable EVs are excluded. After filtering, C_a is reduced to a feasible set $$ C_{a}^{\text {fea }} $$ containing only executable task-EV-route triplets under current traffic and system conditions.

Unified scheduling cost with task urgency

After feasibility screening, each triplet (i, j, n) represents an executable scheduling scheme. The scheduling objective is to balance task urgency and execution cost under limited resources, so high-urgency tasks should be prioritized without sacrificing overall system efficiency. Accordingly, task priority weights are first constructed to characterize task importance, and a unified scheduling cost function is then defined to integrate task urgency and route execution cost.

D. Task urgency weight

To prevent high-risk tasks from being delayed by schemes that minimize only route length or travel time, a task-urgency weight is introduced into the scheduling cost function. For each task i, the estimated arrival times of all feasible schemes are denoted as $$ T_{i}=\left\{T_{i j}^{n} \mid(i, j, n) \in C_{a}^{\text {fea }}\right\} $$. The minimum and maximum arrival times are defined as$$ T_{i}^{\min }=\min _{i j n} T_{i j}^{n} $$ and $$ T_{i}^{\max }=\max _{i j n} T_{i j}^{n} $$, respectively. The average minimum arrival time over all tasks is defined as $$ \bar{T}=\frac{1}{|R|} \sum_{r \in R} T_{r}^{\min } $$.

In emergency response, delay has nonlinear and task-dependent effects. A task with a high emergency level, high waiting risk, and strong worst-case delay pressure should receive sharply increased priority because additional delay may cause more serious consequences. Accordingly, the task urgency weight is defined in exponential form to amplify the combined high-risk factors:

Where α_u, α_h, and α_v control the contributions of the three factors, and u_i denotes the urgency of r_i obtained from the task level.

The life risk term h_i reflects the potential loss caused by waiting time under current traffic and resource conditions and adapts the nonlinear risk definition in^[29] to CER scheduling.

where λ balances base waiting risk and additional delay risk, μ controls the increase rate after the response baseline is exceeded, and [x]₊ = max(x, 0). The first term measures the base risk associated with the minimum achievable arrival time, while the second term is activated only when this time exceeds $$ \bar{T} $$ and then increases rapidly but gradually saturates. Weighted by u_i, h_i assigns higher risk to urgent tasks that are difficult to reach quickly.

The relative time pressure under the worst response condition is defined as

where $$ \max_{r \in R} T_{r}^{\max ^{n}} $$ is the maximum arrival time among all tasks, and ϵ avoids division by zero.

E. Unified scheduling cost

For each feasible triplet (i, j, n), the scheduling cost evaluates the global benefit of assigning EV ev_j to task r_i via route $$ p_{i j}^{n} $$, while balancing response efficiency, task urgency, route conflict, and fairness. It is defined as

Here, $$ C o s t_{i j n}^{\text {time }} $$ measures the response cost and is weighted by $$ \left(1+\widetilde{w}_{i}\right) $$ to emphasize high-urgency tasks. $$ C o s t_{i j n}^{\text {conflict }} $$ penalizes routes with high traffic load or strong overlap, whereas $$ C o s t_{i j n}^{\text {fair }} $$ compensates tasks that may suffer long delays by being subtracted from the total cost. Thus, lower $$ C o s t_{i j n} $$ indicates a more suitable triplet for global scheduling.

The time cost is computed as

where $$ T_{i j}^{n} $$, $$ \bar{T} $$, $$ T_{i}^{\text {min }} $$, and β denote the estimated arrival time of route $$ p_{i j}^{n} $$, the average minimum arrival time over all tasks, the best feasible arrival time for task r_i, and the relative delay penalty coefficient, respectively.

The conflict cost consists of route load and overlap risk:

The load risk is defined as

where $$ \rho_{e}=\frac{t_{e}}{t_{e}^{\text {free }}+\varepsilon} $$), t_e is the predicted travel time, and $$ t_{e}^{\text {free }} $$ is the free-flow travel time.

For a feasible triplet $$ a=(i, j, n) \in C_{a}^{\text {fea }} $$ with route p_a, let $$ \mathcal{P} $$ denote the candidate route set. The average overlap is defined as

where route similarity is computed by the Jaccard index to measure the relative proportion of shared road segments and avoid overestimating conflicts on longer routes.

The fairness compensation term is defined as

Since $$ C o s t_{i j n}^{\text {fair }} $$ is subtracted from $$ C o s t_{i j n} $$, tasks with longer estimated arrival times receive scheduling compensation. The logarithmic numerator limits excessive compensation, while the denominator reduces compensation for high-urgency tasks already emphasized by ($$ \left(1+\widetilde{w}_{i}\right) C o s t_{i j n}^{\text {time }} $$, thereby giving relatively more compensation to low-urgency tasks. The disturbance penalty in the lower layer and the fairness compensation in the upper layer act at different decision stages and therefore do not provide conflicting optimization signals. The lower-layer disturbance penalty is used to generate safer and more reliable candidate routes by avoiding highly disturbed segments when alternatives are available. In contrast, the upper-layer fairness compensation does not encourage vehicles to enter disturbed areas; it only adjusts the relative scheduling priority of feasible task-EV-route triplets after candidate routes have been generated and screened. Moreover, the compensation is logarithmically bounded and moderated by the urgency-related term, which prevents excessive correction for tasks located in highly disturbed areas. Therefore, the proposed design balances disturbance avoidance and service fairness without allowing difficult tasks to dominate the scheduling decision solely because of long arrival times.

Minimum cost flow-based collaborative scheduling

After the unified scheduling cost is computed, the scheduling problem is formulated as a Minimum Cost Flow (MCF) problem, which is suitable for CER scheduling because it optimizes additive costs under capacity constraints and enables global coordination of multiple task-EV-route decisions with limited resources. A directed flow network is constructed to match tasks, EVs, and candidate routes, where edge costs are defined by $$ C o s t_{i j n} $$ and capacities represent scheduling constraints. The source node o_s is connected to task nodes r_i, with total outgoing flow equal to the number of tasks. Each task node sends at least one unit of flow within its vehicle assignment limit to ensure task service. For triplet $$ (i, j, n) \in C_{a}^{\text {fea }} $$, a vehicle-route node represents the scheme where ev_j serves r_i through $$ p_{i j}^{n} $$, and its task-to-scheme edge is assigned $$ C o s t_{i j n} $$. Each vehicle-route node is connected to the sink node o_d with capacity 1, so each scheme can be selected at most once. By solving the MCF model, the optimal flow allocation is obtained. Each activated flow path corresponds to a selected task-EV-route triplet, and the selected set Ω = {(i, j, n)} gives the final scheduling solution.

Implementation details

In DAGWN, the model is trained for 500 epochs using Adam with a learning rate of 0.001, and its loss combines Mean Squared Error for future-state prediction and Smooth L1 Loss for disturbance prediction with adaptive weighting via learnable uncertainty parameters.

To improve the reproducibility of the experiments, the key experimental parameters used in the disturbance-aware traffic prediction evaluation are summarized in Table 2.

Baseline methods

Five representative spatiotemporal forecasting models are adopted as baselines to evaluate the effectiveness of disturbance-aware propagation modeling under incident-affected traffic conditions. These baselines include Diffusion Convolutional Recurrent Neural Network (DCRNN)^[14], Spatio-Temporal Graph Convolutional Network (STGCN)^[15], Graph WaveNet (GWN)^[16], DGCRN^[17], and Spatio-Temporal Adaptive Embedding Transformer (STAEformer)^[18].

Evaluation metrics

The prediction performance is evaluated using the following metrics.

• Mean-based error metrics: Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE) evaluate overall prediction errors from different perspectives. MSE emphasizes large deviations, MAE measures average absolute deviation, and RMSE reports squared-error-based deviation in the original data scale.

• Percentile-based error metrics: P50, P90, and P99 represent the 50th, 90th, and 99th percentiles of absolute errors, reflecting typical prediction performance, high-error cases, and tail robustness, respectively.

Results and analysis

Figure 3 and Table 3 demonstrate that DAGWN achieves the best performance in terms of Loss, MSE, and RMSE under incident-affected traffic conditions, while obtaining the second-best MAE result. Although DAGWN and GWN both reduce errors rapidly and converge in fewer epochs, DAGWN maintains a lower and more stable curve in the later training stages. These results indicate that explicitly modeling disturbance propagation significantly improves the ability to capture abnormal traffic evolution.The similar trends in Figure 3A, B and D are expected. In this figure, the plotted loss is the MSE-based prediction loss used to monitor model convergence, so the Loss curve in Figure 3A is naturally close to the MSE curve in Figure 3B. In addition, RMSE is computed as the square root of MSE; therefore, Figure 3D preserves the same variation trend as Figure 3B, while the value range is compressed after the square-root transformation. These relationships explain why the Loss, MSE, and RMSE curves show highly similar convergence patterns.

Figure 3. Training curves of different traffic prediction models over 500 epochs in terms of (A) training loss; (B) mean squared error (MSE); (C) mean absolute error (MAE); and (D) root mean squared error (RMSE). DAGWN: Disturbance-Aware Graph WaveNet; GWN: Graph WaveNet; DCRNN: Diffusion Convolutional Recurrent Neural Network; DGCRN: Dynamic Graph Convolutional Recurrent Network; STGCN: Spatio-Temporal Graph Convolutional Network; STAEformer: Spatio-Temporal Adaptive Embedding Transformer.

DCRNN also converges, but its training loss remains relatively high in the early stage and decreases notably after about 100 epochs, indicating slower convergence and lower stability than DAGWN. Although DCRNN achieves the lowest MAE, its weaker squared-error metrics indicate lower robustness to large deviations. STGCN converges at a higher error level than DAGWN, GWN, and DCRNN, suggesting that fixed graph convolution is less adaptive to dynamic spatial dependencies under incident disturbances. DGCRN and STAEformer maintain high training errors throughout training, suggesting limited ability to fit disturbed traffic states.

As shown in Figure 4, the performance gap among different models becomes substantially larger at P90 and P99, indicating that disturbance-aware modeling mainly improves robustness under high-impact traffic perturbations rather than regular traffic conditions. DAGWN achieves the best P99 value of 6.67 and the second-best P50 and P90 values of 0.072 and 0.40, respectively, showing stronger control of extreme errors and better robustness to incident-induced traffic variations. In contrast, DGCRN and STAEformer perform acceptably at P50 but deteriorate sharply at higher quantiles, indicating weak tail-error control and suggesting that these models fail to capture the spatial diffusion and temporal persistence of incident-induced disturbances.

Figure 4. Error distribution comparison (P50, P90, P99) across different models. (A) comparison of the median absolute prediction error (P50); (B) comparison of the 90th-percentile absolute prediction error (P90); and (C) comparison of the 99th-percentile absolute prediction error (P99). DAGWN: Disturbance-Aware Graph WaveNet; DCRNN: Diffusion Convolutional Recurrent Neural Network; GWN: Graph WaveNet; STGCN: Spatio-Temporal Graph Convolutional Network; STGCN: Spatio-Temporal Graph Convolutional Network; DGCRN: Dynamic Graph Convolutional Recurrent Network.

Implementation details

In the task urgency weight, α_u, α_h, and α_v are set to 0.5, 0.3, and 0.2, respectively, making the task level the dominant factor while using life risk and relative time pressure as supplementary risk information. For the life risk term, λ = 0.4 balances base risk and additional delay risk, while μ = 2 controls the growth rate after the response baseline is exceeded. The relative delay coefficient β = 0.5 treats relative delay as an auxiliary penalty while keeping absolute response time as the main objective. All reported results are averaged over 200 simulation runs.

The key parameters in the proposed framework have different functional roles across the prediction, route generation, and scheduling layers. The disturbance decay parameter controls the spatial range of incident influence: a larger value leads to wider propagation, whereas a smaller value makes the disturbance more localized. The adjacency fusion weights balance incident-induced propagation and general traffic correlations, preventing the disturbance-aware adjacency from dominating spatial dependency modeling. The disturbance penalty in candidate route generation adjusts the trade-off between travel-time minimization and avoidance of highly disturbed segments. In the upper-layer scheduling cost, the urgency-weight coefficients determine the relative importance of task level, life risk, and time pressure, while the relative delay coefficient controls the penalty for selecting routes that deviate from the best feasible response time. These parameters were selected to maintain the primary role of response efficiency while allowing disturbance risk, route conflict, and fairness to influence the final scheduling decision.

Baseline methods

Since the MVMTA problem is NP-hard, representative heuristic, matching-based, and evolutionary baselines are adopted for comparison, including Random, Greedy, Nearest Neighbor, Hungarian, Multiobjective Evolutionary Algorithm Based on Decomposition Using Simulated Annealing (G-MOEA/D-SA)^[21], and Non-dominated Sorting Genetic Algorithm II for the Integrated Emergency Vehicle Scheduling Model (NSGA-II-IEV)^[25].

Evaluation metrics

Scheduling performance under concurrent emergency scenarios is evaluated from three aspects: overall efficiency, worst-case performance, and critical task responsiveness:

• Overall efficiency: Actual completion time measures the time required to complete all tasks, while average response time reflects the mean response delay across tasks.

• Worst-case performance: Maximum response time captures the largest task delay and reflects scheduling robustness.

• Critical task responsiveness: Average response time of high-priority tasks evaluates whether critical tasks are prioritized effectively.

Performance analysis

As shown in Figure 5, MCF achieves the best results across all three overall metrics. Specifically, Figure 5A-C present the total completion time, average response time, and maximum response time, respectively. This demonstrates that the proposed cost-flow scheduling method coordinates vehicles, tasks, and route resources at the system level. This system-level coordination alleviates resource competition under concurrent tasks and improves response performance. In contrast, Random lacks resource competition awareness and only generates feasible solutions, resulting in the lowest efficiency. Greedy and Nearest reduce response time by relying on local cost or spatial distance, but they may occupy high-quality resources too early, affecting subsequent task assignment. Hungarian, G-MOEA/D-SA, and NSGA-II_IEV improve scheduling performance through matching optimization or multi-objective search, but still struggle to ensure system-level optimality when multiple tasks compete for limited vehicles and high-quality routes. This suggests that the key to CER scheduling lies in global coordination under resource constraints rather than local optimality for individual tasks.

Figure 5. Performance comparison of different methods in terms of time-related metrics: (A) total completion time; (B) average response time; and (C) maximum response time. MCF: Minimum cost flow; G-MOEA/D-SA: Multiobjective Evolutionary Algorithm Based on Decomposition Using Simulated Annealing; NSGA-II-IEV: Non-dominated Sorting Genetic Algorithm II for the Integrated Emergency Vehicle Scheduling Model.

Figure 6 also confirms the adaptability of MCF across different task levels. MCF maintains lower response times at most levels, indicating that global joint scheduling can allocate limited emergency resources according to task demands. Its advantage is more evident for Level 1 tasks, where looser constraints provide a larger scheduling space and reduce unnecessary resource occupation. As the task level increases, high-quality vehicles and low-conflict routes become scarce, causing response times to rise and performance gaps to narrow. This indicates that high-level tasks impose greater scheduling pressure, making local rules or independent matching optimization less effective.

Figure 6. Comparison of different methods across task levels. MCF: Minimum cost flow; G-MOEA/D-SA: Multiobjective Evolutionary Algorithm Based on Decomposition Using Simulated Annealing; NSGA-II-IEV: Non-dominated Sorting Genetic Algorithm II for the Integrated Emergency Vehicle Scheduling Model.

To explain this increase in response time, Figure 7 shows the response-time distributions for Level 4 and Level 5 tasks, with each group based on 200 simulation runs. The boxes indicate the interquartile range (IQR, 25th–75th percentiles), the center lines indicate the median, the whiskers extend to the most extreme data points within 1.5 times the interquartile range below the first quartile and above the third quartile, and the circles denote outliers beyond the whiskers. The distributions shift upward and become wider, with higher upper bounds and more high-value outliers, indicating that resource competition amplifies extreme response delays. As high-quality resources are gradually occupied, later assignments are more likely to rely on suboptimal combinations, which increases response time. Consistent with the previous results, Random shows the highest distribution and largest fluctuations, revealing the risk of uncoordinated assignment under high-load scenarios. Although the other optimization methods improve average performance, their distributions become closer under resource scarcity. In contrast, MCF maintains a lower distribution, showing that it reduces the system-level impact of accumulated suboptimal assignments. Overall, Figure 5 to Figure 7 show that the proposed method improves response efficiency and scheduling stability under high-level tasks and high-load conditions.

Figure 7. Response time distributions under high-level tasks (Levels 4–5). Each group contains n = 200 samples. The boxes represent the interquartile range (IQR, 25th–75th percentiles), the central lines indicate the median, the whiskers extend to the most extreme data points within 1.5 times the IQR from the lower and upper quartiles, and the circles represent outliers beyond the whiskers. MCF: Minimum Cost Flow; G-MOEA/D-SA: Multiobjective Evolutionary Algorithm Based on Decomposition Using Simulated Annealing; NSGA-II-IEV: Non-dominated Sorting Genetic Algorithm II for the Integrated Emergency Vehicle Scheduling Model.

Ablation settings

To evaluate the contribution of each component, an ablation study is conducted with four configurations. For the ablation study, all reported results are averaged over 1,000 independent experimental runs to reduce the influence of randomness in incident generation, traffic evolution, and scheduling decisions.

• I: The full model integrates DAGWN-based route generation with the proposed scheduling method.

• II: DAGWN-based candidate route generation is used with NSGA-II-IEV-based task assignment.

• III: The proposed scheduling method with the designed cost function is applied to static shortest routes.

• IV: The baseline combines static shortest route planning with NSGA-II-IEV-based task assignment.

Results and discussion

The full model achieves the best results in both Figures 8 and 9, demonstrating the complementary effects of disturbance-aware route generation and collaborative scheduling. Method II reduces average response time after introducing DAGWN-based candidate routes, indicating that incident disturbance information improves route quality and supports more effective route selection under affected traffic conditions. Method III improves total completion time and maximum response time by applying the proposed scheduling strategy to static shortest routes, showing that collaborative scheduling optimizes vehicle-task-route assignment and reduces extreme response delays. When used separately, route generation lacks global resource coordination, while scheduling optimization is constrained by static route quality; thus, neither component can achieve the best performance across all metrics. In contrast, the full model combines DAGWN-based disturbance-aware routes with MCF-based task assignment under limited resources, improving both route quality and scheduling performance. These results validate the effectiveness of the proposed hierarchical collaborative optimization framework for system-level CER response optimization.

Figure 8. Time comparison of different ablation settings. (A) comparison of the total completion time under configurations I–IV, reflecting the overall efficiency of completing all emergency tasks; (B) comparison of the average response time, indicating the mean response efficiency across tasks; and (C) comparison of the maximum response time, representing the worst-case response delay.

Figure 9. Performance comparison under different ablation settings across task levels.

These ablation results also support the modular design of the proposed framework. The proposed scheduling method can still operate with static shortest routes, indicating that the upper-layer optimizer is independent of a specific lower-layer predictor. Meanwhile, replacing static routes with DAGWN-based disturbance-aware candidate routes further improves scheduling performance, showing that the lower layer provides beneficial route-level information rather than introducing uncontrolled coupling. Therefore, the hierarchical design enables both independent operability of each layer and complementary performance gains when the two layers are integrated.