Achieving optimal allocation of urban flood control funds: integrating hydrodynamic model-based risk assessment with diverse multi-objective optimization models

Against the backdrop of global climate change, the increasing frequency of flooding events is imposing greater demands on urban flood prevention systems. Traditional approaches to allocating flood control funds, which rely primarily on historical city-level data, often lead to inefficient and misaligned distribution. This inefficiency stems from an inability to account for complex, high-dimensional flood risk scenarios, as well as a lack of systematic comparison of algorithmic performance in terms of adaptability, convergence to optimal solutions, and handling of high-dimensional discrete search spaces. To address this issue, this study first developed a refined flood risk assessment system by integrating a coupled hydrological-hydrodynamic model with socioeconomic and infrastructure data. Subsequently, several multi-objective optimization algorithms were applied to this system to identify the most cost-effective funding allocation strategy under high-dimensional scenarios. The model accurately identified localized high-risk areas, such as river bends and zones where steep slopes meet plains, thereby shifting the optimal allocation strategy from “regional coverage” to “targeted risk-based precision.” Notable differences were observed among the optimization algorithms. Specifically, the SPEA2 algorithm achieved optimal overall benefits while reducing the proportion of extremely high-risk areas to 0.02%. This study highlights the mechanistic advantages of hydrological-hydrodynamic models in pinpointing flood risks and clarifies how algorithmic features influence funding allocation efficiency. The findings provide actionable insights for enhancing urban flood resilience and supporting sustainable development.

1 Introduction

Global climate change is intensifying urban flood risks, thereby creating serious challenges for urban development and public safety (Khosravi et al., 2018). As one of the countries most severely affected by flooding, China has recently experienced a worrying trend of “small floods causing major disasters,” underscoring the vulnerability of existing infrastructure. For example, in 2023 alone, catastrophic floods across four provincial-level regions—Beijing, Hebei, Heilongjiang, and Henan—caused direct economic losses exceeding 25.71 billion dollars, accounting for a staggering 76.2% of the nation’s total disaster losses. At the same time, flood disasters in China are exhibiting new characteristics: major losses are increasingly caused by extreme weather events in small watersheds (Xu et al., 2024), often triggering significant chain effects of disasters. The rapid urbanization process in developing regions has also led to inadequate disaster prevention infrastructure (Tin et al., 2024), while increased surface runoff has extended post-disaster recovery cycles. These phenomena highlight systemic weaknesses in current urban flood prevention systems. In response, China has invested heavily in traditional flood control infrastructure such as levees and drainage channels, as well as in “sponge city” initiatives—urban systems designed to absorb, store, infiltrate, and utilize rainwater like a sponge during rainfall (Qi et al., 2021), exemplified by cities like Beijing and Shanghai. However, a fundamental problem remains: traditional funding allocation tends to rely on historical disaster data or coarse risk assessments at the city scale, lacking the spatial precision needed to address modern climate extremes. Inefficient use of funds itself represents a major risk. Therefore, maximizing urban flood resilience through efficient and targeted allocation of limited resources has become an urgent priority.

Conventional approaches to allocating flood control resources have largely been confined to administrative boundaries, relying on macro-level historical data or coarse risk indicators such as annual average rainfall or past flood records (Ma et al., 2019). While these methods offer some reference value, they often fail to capture the spatial heterogeneity of flood hazards and disaster propagation mechanisms under extreme rainfall (Darji et al., 2024). Allocating funds solely by administrative unit risks overlooking the actual distribution of high-risk zones and critical infrastructure, resulting in a “mismatch between funding and risk” and reducing the efficiency of resource utilization (Zbigniew et al., 2010; Geng et al., 2024). In the current context of climate change, with frequent extreme events in small watersheds and rapidly changing urban surfaces, such static, low-resolution assessment methods can no longer meet the demands of precise disaster prevention planning (Xu et al., 2024). To achieve refined optimization of flood control funding, high-resolution risk assessment methods are essential. Early studies often relied on historical disaster data or remote sensing, which could identify general risk patterns but struggled to capture localized hazards and exposed elements under complex terrain (Areas with significant topographic changes, such as steep slopes, narrow valleys, and transitional zones between mountainous regions and plains) (Tufano et al., 2023). By contrast, coupled hydrological-hydrodynamic models can simulate flood propagation under various scenarios and have been widely used for high-precision hazard mapping (Manh Xuan and Molkenthin, 2021). This study employs such a model to simulate flood dynamics, including water depth and flow velocity at grid points, under different rainfall return periods. Combined with socioeconomic and infrastructure exposure data, it enables a comprehensive risk assessment (Bertsch et al., 2022). This approach mechanistically reflects flood evolution and disaster formation processes, supporting granular risk identification at street and grid scales, and providing reliable data support for the spatial optimization of flood control funding.

Advances in flood risk assessment have laid a crucial foundation for improving the cost-effectiveness of flood control investments. Multi-objective optimization techniques offer a scientific framework for flood risk mitigation through mathematical modeling (Lu et al., 2022). As extreme precipitation events become more frequent, governments worldwide are increasing disaster prevention investments (Coetzee et al., 2023). In recent years, some researchers have begun applying these methods to optimize flood control funding allocation. For instance, Yang et al. (2021) and Zhong et al. (2025) conducted preliminary explorations using genetic algorithms, though the potential of other algorithms remains largely untested. In related domains such as reservoir operation, multi-objective algorithms like NSGA-II (Yao et al., 2019), MOEA/D (Xu and Bai, 2019), and MOPSO (Dabral et al., 2025) have been widely applied, with comparative studies analyzing their strengths and limitations. By contrast, research on flood control funding allocation remains largely confined to genetic algorithms and their variants. Given that resource allocation problems exhibit high-dimensional (simultaneously determining numerous decision variables) and discrete (the algorithmic search domain consists of discrete points rather than continuous space) characteristics, different algorithms demonstrate unique global search capabilities and convergence properties (Azamathulla et al., 2008). A mismatch between algorithm selection and problem structure may lead to suboptimal resource distribution (Qiao et al., 2024), severely limiting improvements in funding efficiency and disaster resilience. Therefore, building on the aforementioned precise risk assessment framework, this study addresses the complex, high-dimensional nature of flood control funding allocation by applying and comparing several multi-objective optimization algorithms. This approach is essential for establishing a precise and efficient funding allocation methodology and comprehensively enhancing urban flood control capacity.

In summary, this study employs a high-precision coupled hydrodynamic model for refined risk assessment and systematically compares the performance of several advanced multi-objective optimization algorithms, NSGA-II, MOEA/D, MOPSO, SPEA2, and MOCOA, in resource allocation. The specific aims are: (1) to develop a comprehensive flood risk index based on detailed hydrodynamic simulations for accurate identification of high-risk areas; and (2) to systematically compare the performance of these multi-objective optimization algorithms in the high-dimensional funding allocation scenario, thereby identifying the most effective algorithm and determining the optimal fund allocation strategy based on the Pareto-optimal solution. This research provides a scientific pathway for transitioning flood control funding decisions from experience-based to data-driven approaches, ultimately enhancing urban flood resilience and promoting sustainable development.

2 Methodology

2.1 Study area

This study focuses on the Aojiang River Basin (27°46′–27°68′N, 120°34′–120°60′E), located in southeastern Zhejiang Province, China (Figure 1). The basin spans Pingyang and Cangnan counties, covering a total area of 326.11 km², and represents one of Zhejiang’s most flood-prone regions. The southwestern and northeastern regions of the study area are highly urbanized, with a total population of 865,000 and an average population density of approximately 20 people per square kilometer. Urban areas exhibit a population density of about 200 people per square kilometer, creating significant exposure risks within flood-prone watersheds. Topographically, the area is characterized by higher elevations in the west and lower in the east, with an elevation range of 626 m (−7 to 619 m) and complex geomorphology. The western, northern, and southern parts of the basin consist of hilly terrain with steep slopes, where only limited terraces are distributed along river valleys. These areas exhibit rapid rainfall-runoff response and a high risk of flash flooding. The south-central part of the basin is an alluvial plain with low-lying terrain, serving as a concentrated zone for urban development and agricultural production. The region experiences a central subtropical oceanic monsoon climate, with an average annual precipitation of approximately 1700 mm, showing a decreasing trend from west to east. Due to its unique geological structure and geographical conditions, the basin is highly susceptible to multiple hazards under typhoon and heavy rainfall conditions, including flash floods, landslides, waterlogging, and tidal flooding. According to hydrological records, on 26 July 2024, influenced by Typhoon No. 3 “Gaemi,” the study area received an average rainfall of 131 mm. The water level at the key hydrological station (Daitou Station) in the main stream of the Aojiang River rose to 14.03 m, exceeding the warning level by 0.03 m and resulting in the first numbered flood event of the year. This event clearly demonstrates the region’s high sensitivity and vulnerability to heavy rainfall and extreme weather.

Figure 1

Figure 1. Multi-scale geographic characteristics of the study area: (a) Location map of Zhejiang Province, China; (b) The study area is located in the southeastern zone of Zhejiang Province; (c) Digital elevation model of the study area; (d) Land-use types are dominated by forested and cultivated land.

2.2 Research framework

This study aims to achieve high-precision flood hazard risk assessment and systematically optimize the allocation of flood control funds. As shown in Figure 2, the research consists of two main components: (1) High-precision flood modeling through coupled hydrological and hydrodynamic simulations, integrated with socioeconomic and infrastructure exposure indicators, followed by a comparative analysis with non-coupled models; (2) Application of several multi-objective optimization algorithms, based on the flood risk assessment results, to identify optimal benefit solutions within high-dimensional and complex flood control funding allocation scenarios (the high-dimensional complexity discussed in this paper stems from the enormous number of decision variables in flood control funding allocation, specifically, the product of spatial unit counts and measure types. The decision space consists of discrete points, resulting in an optimization algorithm search domain that is a set of discrete points rather than a continuous space).

Figure 2

Figure 2. Research framework.

2.3 Hydrological-hydrodynamic model

In the study area, an external coupling mode was selected for hydrological-hydrodynamic simulations. Its core mechanism is reflected in the unidirectional data transfer relationship, where the hydrological model independently drives the hydrodynamic model. Under this mode, the two types of models maintain their respective independence and do not interfere with each other’s internal computational logic during the calculation process, making it the most widely used coupling form in engineering practice (the technical workflow is detailed in Figure 3). External coupling first extracts runoff hydrograph data at key nodes of the hydrological model (such as river confluences and reservoir inflow points); then, this hydrograph is imported as an input boundary condition into the hydrodynamic model; finally, relying on the fluid dynamics equations, it generates the spatial distribution of flood evolution parameters (including key physical quantities such as inundation depth, flow velocity vectors, and water level elevations), significantly reducing system development complexity.

Figure 3

Figure 3. Overview of hydrological-hydrodynamic coupling.

Under the implementation framework of the external coupling architecture, DEM was used as the topographic foundation for both models to ensure spatial consistency. Furthermore, by synchronizing model inputs and outputs, the time steps of both models were set to be consistent to maintain temporal consistency (the time step for both is 1 min). The core implementation path of this coupled system is shown in Figure 3. Key nodes J5, J7, J18, and J24 of the hydrological model serve as hubs for hydrodynamic boundary transfer. The flood hydrographs for these four nodes are generated through hydrological model calculations, and this dynamic runoff sequence is then imported into the hydrodynamic model as the driving source for basin-wide flood evolution computation. The blue area (sub-basin S1) is the key study area of the hydrodynamic model, receiving runoff inputs from the four upstream hydrological nodes. Finally, key disaster characteristic parameters such as inundation depth, flow velocity distribution, and water level elevation are output through hydrodynamic calculations. The integration of HEC-HMS and HEC-RAS enables comprehensive flood simulation, from rainfall-runoff processes at the watershed scale to flood evolution in river channels (Peker et al., 2024). Coupling hydrological (HEC-HMS) and hydrodynamic (HEC-RAS) modeling significantly enhances the accuracy and completeness of flood risk assessment. The model utilizes standard rainfall scenarios representing return periods of 20, 50, and 100 years. These scenarios are based on Zhejiang Province’s local Intensity-Duration-Frequency (IDF) curve derived from long-term historical rainfall statistics. The Manning’s roughness coefficient (N) values for channels and floodplains were determined using land use data, adhering to standard values recommended in the HEC-RAS Hydraulic Reference Manual. Specific values are presented in Table 1. The model employs a 12.5-m resolution DEM) from the ALOS PALSAR satellite mission as the terrain foundation. Inherent uncertainties influence simulation results in key input parameters. Therefore, subsequent validation through river discharge flow verification and inundation area comparison ensures simulation reliability, mitigating the impact of parameter uncertainties on the model.

Table 1

Table 1. Regression results for Manning’s N value.

River flow calibration employs stations with relatively abundant data as validation targets. From the rainfall dataset provided by the Water Resources Department of Wenzhou City, Zhejiang Province, six major rainfall events were selected as input for duration-based rainfall data. Each event lasted 24 h with 1-h intervals. The initial model’s simulated data were compared with measured runoff data. Model applicability and reliability were validated using peak runoff relative error and Nash coefficient as evaluation metrics. The validation results indicate that the maximum relative error in peak flow was −12.14%, with a minimum of 6.34% (Table 2). The maximum Nash coefficient was 0.85, and the minimum was 0.72. Overall, the validation results were satisfactory. The absolute values of the peak relative errors ranged between 6% and 13%, meeting the validation requirements for error tolerance.

Table 2

Table 2. Verification of rainfall event simulation results.

To validate the hydrodynamic model’s flood inundation results, a comparative analysis was conducted between the simulated inundation areas and historical inundation areas from ten floods obtained from the Global Flood Database (https://global-flood-database.cloudtostreet.ai). The overlap ratio between simulated and historical inundation areas reached 79.58% (as shown in Figure 4), indicating a high degree of agreement between the simulation results and actual historical inundation events.

Figure 4

Figure 4. Overlay of historical flood zones and simulated flood zones.

2.4 Coupled water-hydrodynamic flood risk computational model

Based on the outputs from the hydrological-hydrodynamic model, a hazard index, defined as the product of water depth and velocity (h × v), is calculated for return periods of 20, 50, and 100 years to quantify the physical intensity of flooding (Zhao et al., 2024).

Although coupled hydrological-hydrodynamic models provide a physical foundation for flood risk assessment by precisely depicting flood intensity, establishing a comprehensive flood risk assessment system still requires integrating multidimensional indicators to reflect the actual impact of disasters. Therefore, building upon the disaster risk index, this study incorporates indicators reflecting exposure characteristics, regional socioeconomic conditions, and disaster adaptation and resilience, following established indicator selection principles. The existing framework primarily assesses human exposure risk through population density, while acknowledging that a comprehensive evaluation of human impacts requires detailed vulnerability data (e.g., building structure types, resident age distribution). However, these elements fall outside the scope of this study. GDP serves as an alternative indicator for economic exposure, primarily used to prevent flood control funding from being allocated to areas with high flood risk but no economic value (such as uninhabited mountainous regions without economic industries). The density of hydrological monitoring stations and the accessibility of emergency rescue services represent regional pre-disaster monitoring and response capabilities and post-disaster adaptive recovery capabilities, respectively. Specific indicators are detailed in Table 3. The integration of physical flood simulations with socioeconomic and infrastructure data was achieved through a static composite index method. Specifically, all indicators were first standardized, then aggregated into a single composite risk value using weights derived from PCA analysis. This approach is used because the physical flood simulation employs the maximum peak flow conditions to determine the maximum inundation depth, aiming to characterize the vulnerability of exposed facilities and the socio-economic population within the study area under the maximum flood intensity.

Table 3

Table 3. Indicators for flood risk assessment.

Principal component analysis (PCA) was employed exclusively for the flood risk assessment stage to determine the objective weights of the eight indicators listed in Table 3. The percentage of total variance explained by each principal component determines its weight in the PCA. The comprehensive weight of each evaluation indicator was then calculated based on the proportion of the squared factor loadings after rotation. Finally, all weighted indicators were integrated using the indicator aggregation formula to compute the comprehensive risk value and generate a risk map. The risk calculation is expressed in Equation 1, where $W_{ij}$ is the value of the $j$-th indicator in the $i$-th unit, ${In}_i$ is the weight of the $j$-th indicator derived from PCA, and the resulting ${Risk}_i$ is the comprehensive risk value for the $i$-th analysis unit.

${Risk}_i={\displaystyle \sum _{i=1}^n}{W}_{ij}*{In}_j(1)$

2.5 Flood control funds allocation model

To enhance the efficiency of flood control fund allocation, this study formulates the problem as a multi-objective optimization model based on FAOM and SO-FAOM (Zhong et al., 2025; Yang et al., 2021), supported by quantified flood risk data. Risk assessment results can identify areas with higher potential flood hazards, dense populations, vulnerable infrastructure, inadequate emergency response capabilities, and greater potential for economic losses. These regions are most vulnerable during disasters. By focusing on risk assessment findings to determine priority support areas, funding can be directed precisely to regions with the most urgent needs. Therefore, areas with medium or higher risk levels are designated as priority support regions. Practical and readily implementable disaster prevention measures were chosen from the risk evaluation system to quantify funding benefits. The benefit coefficients of these measures were determined by analyzing the quantitative relationship between historical flood events and flood control interventions using the geodetector method. This approach is justified by the fact that historical flood events have historically driven the construction of flood control infrastructure in the region (Welsch et al., 2022). A stronger correlation implies higher potential effectiveness of the measures. These coefficients were incorporated into a tiered quantification mechanism to reflect varying urgency across risk levels and differentiate benefits according to regional priorities.

To achieve refined and spatially explicit fund allocation, the problem was abstracted into two objective functions and one constraint condition. This formulation aims to maximize funding benefits from both global and local perspectives under a total planned investment, thereby transforming the fund allocation problem into a multi-objective optimization problem.

Objective Function 1: Minimize the sum of risk values within priority support areas (aiming to reduce overall flood risk from a global perspective) (calculation process as per Equations 2–7).

$x_{ij}=\frac{P_{ij}}{I_j}(2)$

$X_{ij}=\frac{x_{ij}-x_{ijmin}}{x_{ijmax}-x_{ijmin}}(3)$

${DRisk}_i={\displaystyle \sum _{j=1}^k}{X}_{ij}*w_{nj}*W_j(4)$

${PRisk}_i={Risk}_i-{DRisk}_i(5)$

$P{Risk}_{total}={\displaystyle \sum _{i=1}^n}{PRisk}_i(6)$

$f_1=\mathit{min}\left(P{Risk}_{total}\right)(7)$

Where $P_{ij}$ is the decision variable representing the capital investment for measure $j$ in analysis unit $i$; $I_j$ denotes the cost of measure j; $x_{ij}$ is the quantity of measure $j$ invested within analysis unit $i$; $X_{ij}$ is the standardized value; $w_{nj}$ is the benefit coefficient of measure $j$ within the nth level priority support region; $W_j$ is the weight of measure $j$; ${DRisk}_i$ is the risk reduction value in analysis unit $i$; ${PRisk}_i$ is the residual risk value in analysis unit $i$ after implementing measures; $P{Risk}_{total}$ is the sum of risk values within the priority support region.

Objective Function 2: Minimize the number of high-risk and above analysis units (reduce the number of areas with extreme flood risk from a local perspective) (calculation process as per Equation 8).

$f_2=\min \left({\displaystyle \sum _{i=1}^n}NUM\left({PRisk}_i>{Risk}_H\right)\right)(8)$

Where ${Risk}_H$ is the threshold for extreme risk units.

The weights of the two objective functions are treated deterministically, with equal weighting assigned to each. This deterministic weighting reflects a balanced policy perspective that neither prioritizes overall risk reduction nor favors the management of local extreme risks. This configuration enables the model to naturally balance these two competing objectives.

Constraint: The total amount of allocated funds shall not exceed the total amount of planned investment funds (calculation process as per Equation 9).

$P={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{i=1}^k}{P}_{ij}(9)$

Where $P$ is the total planned investment fund.

After defining the constraints and objective functions, a multi-objective optimization algorithm is applied to obtain the Pareto optimal solution set. To select the best compromise solution from the Pareto-optimal set obtained by each multi-objective optimization algorithm, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was applied. The input for the TOPSIS method consisted of the normalized values of the two objective functions (f1 and f2) for each solution in the Pareto set. Each objective was assigned a weight of 0.5 in the TOPSIS calculation, reflecting its equal importance in the decision-making process for fund allocation. The solution with the highest relative closeness to the ideal solution was identified as the optimal allocation scheme.

Given the inherently high-dimensional, discrete, and nonlinear nature of flood control fund allocation, as well as the complex trade-off between minimizing overall risk and minimizing localized extreme risks, selecting an appropriate multi-objective optimization algorithm is crucial. Based on the aforementioned requirements and adhering to the principle of representing different mainstream paradigms within the field, five mainstream algorithms were selected through a literature review for comparative analysis to ensure comprehensiveness. Non-dominated Sorting Genetic Algorithm II (NSGA-II) serves as a mainstream representative of algorithms based on the Pareto and congestion distance approaches. The Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D) represents the decomposition-based methodology. Multi-objective Particle Swarm Optimization (MOPSO) stands as a representative of swarm intelligence-based methods. Strength Pareto Evolutionary Algorithm (SPEA2) exemplifies the Pareto algorithm known for its robust archiving strategy. The Multi-Objective Coati Optimization Algorithm (MOCOA) is a recently proposed nature-inspired algorithm. These algorithms represent distinctly different solution characteristics in multi-objective optimization (e.g., Pareto-based, decomposition-based, and swarm intelligence-based approaches). This diverse selection enables to evaluation of the suitability of different solution approaches for high-dimensional, discrete flood control funding allocation problems. Table 4 summarizes the general characteristics of these algorithms. Accurate flood control fund allocation involves addressing a high-dimensional, multi-objective optimization problem characterized by nonlinear conflicts between competing analysis units. Through systematic algorithm comparison, this study identifies methods with superior capability in solving these complex, high-dimensional nonlinear problems. Finally, to illustrate the convergence behavior and solution diversity of various algorithms in flood control funding allocation problems, quantitative evaluation was conducted using hypervolume (HV) (Zhu et al., 2025) and spacing (SP) (Pandya et al., 2024).

Table 4

Table 4. Comparison of the advantages and disadvantages of different multi-objective optimization algorithms.

3 Results

3.1 Spatial distribution of flood risk

Simulated flood extents under the 2020 extreme rainfall scenario reveal a broadly contiguous spatial pattern. The total inundated area reached 78.45 km² under the 20-year return period, increasing to 81.24 km² for the 50-year return period, and peaking at 83.14 km² under the 100-year return period scenario. As shown in the hazard index (depth × velocity) distribution (Figure 5), high-hazard zones expanded by approximately 1.52 km² when moving from the 20- to the 50-year return period, and by a further 1.08 km² under the 100-year scenario. These results indicate that although rainfall intensity increased considerably with longer return periods, the absolute growth in both total inundation area and high-hazard area remained limited.

Figure 5

Figure 5. Hazard index rating map based on the hydrodynamic model. (a) 20-year. (b) 50-year. (c) 100-year.

Risk assessment indicators were derived by coupling hydrological and hydrodynamic processes. The weight of each indicator was determined using principal component analysis (PCA), incorporating both the percentage of variance explained by each principal component and the rotated factor loadings of the indicators, as summarized in Table 5. It is noteworthy that PCA assigned nearly identical weights to the three risk indicators (R₂₀, R₅₀, R₁₀₀) at 0.172, 0.178, and 0.177, respectively. Principal Component Analysis (PCA) assigns weights based on the variance and covariance structure of the data. The high consistency of these weight values indicates a strong positive correlation in the spatial distribution patterns of flood hazards across the three distinct rainfall scenarios. Locations classified as hazardous zones during a 20-year flood event are highly likely to remain core hazard areas during more extreme 50-year and 100-year floods. The primary difference lies in increased water depth and flow velocity, rather than a fundamental shift in spatial distribution patterns. Among these, the slightly lower weight of the 100-year return period relative to the 50-year return period is a direct result of PCA. This minor discrepancy indicates that, compared to the 50-year return period, the additional spatial information provided by the 100-year return period exhibits a slightly diminished uniqueness or influence when calculating the risk index. Therefore, assigning similar weights to these indicators when constructing a comprehensive risk index is justified, as they provide redundant information that collectively enhances the identification of the most vulnerable areas.

Table 5

Table 5. Indicator weights.

Figure 6a illustrates the flood risk assessment results obtained through the coupled hydrological–hydrodynamic modeling approach. Low-risk areas represent the largest proportion of the study area (32.84%), followed by medium-risk (29.93%) and extremely low-risk areas (25.49%). High-risk and extremely high-risk zones cover comparatively smaller proportions, accounting for 8.7% and 3.05% of the total area, respectively. The spatial distribution of flood risk exhibits clear regional characteristics, with elevated risk primarily concentrated in the southern part of the study area, the northwestern mountainous regions, and urban zones along rivers in the northeast, forming distinct striped and clustered spatial agglomeration patterns.

Figure 6

Figure 6. Spatial distribution of flood risk: (a) results from the coupled hydrodynamic model; (b) results from the uncoupled hydrodynamic simulation.

To evaluate the effectiveness of the coupled hydrodynamic model in identifying high-risk areas and key risk drivers, a comparative risk assessment was performed using the same methodology but excluding the hydrodynamic simulation results, as shown in Figure 6b. Under this non-coupled modeling scenario, extremely high-risk areas were predominantly concentrated in northeastern urban regions (1.88%), while high-risk zones were mainly distributed across northeastern and southwestern urban areas. Approximately half of the remaining territory was classified as medium-risk regions.

3.2 Results of flood control fund allocation

Based on the flood risk assessment and considering the principles of disaster prevention resource optimization, particularly the relationship between risk level and loss probability, medium- and higher-risk areas were selected as funding investment zones. This approach reflects the understanding that, under limited budget constraints, investments in low-risk areas yield significantly lower returns than those in higher-risk regions (Zhong et al., 2025). These priority areas were further classified into three support levels according to risk gradation, as illustrated in Figure 7. Figure 7a shows the priority support area map, while Figure 7b depicts the population density distribution and urbanized area distribution. Both exhibit consistent spatial distribution patterns.

Figure 7

Figure 7. Priority support areas and population density, urbanization area map. (a) Priority support area distribution. (b) Population density and urbanized area distribution.

To quantify the risk reduction effect of flood control funding, three disaster prevention measures were selected according to regional risk levels and natural conditions: hydrological stations, emergency rescue stations, and high-standard farmland. The selection of disaster prevention measures must be targeted and relatively easy to implement. For example, investing in hydrological monitoring systems is more feasible than relocating populations or modifying complex terrain. Hydrological stations enhance real-time monitoring and early warning capabilities, addressing the rapid onset of flash floods, particularly in remote areas. Emergency response stations are deployed in high-risk areas to effectively mitigate long-standing delays in emergency response caused by complex terrain (such as hilly areas, ravines, or regions prone to road flooding), thereby enhancing medical accessibility in these regions. High-standard farmland refers to cultivated land that has undergone systematic infrastructure upgrades, characterized by contiguous plots, fertile soil, and strong disaster resilience. Its flood resistance is enhanced through water management facilities such as drainage ditches and flood detention basins. The benefit coefficients of these measures vary across priority support areas (Table 6), with higher-priority areas yielding greater benefits per investment. Among the three measures, hydrological stations consistently generated the highest benefit coefficients at each priority level.

Table 6

Table 6. Benefit coefficients within priority support areas of different levels.

The allocation of flood control funds was formulated as a multi-objective optimization problem, using comprehensive risk values derived from the coupled hydrodynamic model as the primary input. Five multi-objective algorithms, NSGA-II, MOEA/D, MOPSO, SPEA2, and MOCOA. To ensure a fair comparison, all five multi-objective optimization algorithms (NSGA-II, MOEA/D, MOPSO, SPEA2, and MOCOA) were implemented under consistent computational conditions. The population size was set to 100 for all algorithms (where MOPSO denotes the particle swarm size, and MOEA/D is characterized by the number of reference directions), and each was run for 1,000 iterations, which was determined to be sufficient for convergence based on preliminary tests. The specific parameters for each algorithm are pre-configured based on the high-dimensional discrete characteristics of this problem. The configuration parameters are summarized in Table 7 (where n is the number of decision variables).

Table 7

Table 7. Parameter configurations for the multi-objective optimization algorithms.

The resulting Pareto frontiers are shown in Figure 8. Different algorithms generate varying numbers of solutions, primarily due to their inherent characteristics, which may result in differing convergence rates and solution sets in complex problems. With optimal solutions selected by the TOPSIS method based on relative proximity. SPEA2 achieved the best performance on both objective functions, with its Pareto solutions showing only minor trade-off differences. NSGA-II ranked second in performance, though its Pareto solutions were concentrated in a high-performance region. In contrast, MOEA/D and MOCOA each produced only a single solution, indicating poor diversity, while MOPSO generated the most diverse Pareto front but the lowest overall performance. Quantitative performance evaluation (Table 8) reveals significant differences among algorithms. SPEA2 exhibits the most uniform solution distribution (SP = 0.0050) and competitive convergence (HV = 0.8321). NSGA-II achieves the highest hypervolume value (0.8654), indicating it maintains good diversity throughout the optimization process. In contrast, MOPSO generated the largest solution set yet performed worst (HV = 0.4235, SP = 0.1247). This indicates that while MOPSO explores a broad solution space, it struggles to converge to high-quality solutions in this high-dimensional discrete problem. MOEA/D and MOCOA produced single solutions with good convergence but exhibited insufficient diversity (SP = 0).

Figure 8

Figure 8. Pareto fronts for each algorithm.

Table 8

Table 8. Comparative performance of multi-objective optimization algorithms.

The spatial distribution of allocated funds from each algorithm is shown in Figure 9, with funding levels divided into three equal intervals. All algorithms produced allocation patterns generally consistent with the risk distribution: the highest funding was directed to extreme- and high-risk zones, decreasing sequentially toward lower-risk areas. High-performing algorithms such as SPEA2 showed more concentrated funding distributions. For example, under SPEA2, the highest-funded units were densely clustered in the northeastern study area, surrounded by moderately funded zones. In contrast, MOPSO resulted in a scattered distribution of top-funded units, with moderately funded areas forming dense but isolated clusters. The remaining algorithms produced relatively similar allocation patterns: only extreme-risk units received the highest funding, areas near the moderate-risk threshold received minimal investment, and other regions received moderate funding levels.

Figure 9

Figure 9. Spatial distribution of flood control funds.

To visually compare the effectiveness of each algorithm, the risk reduction benefits of each allocation scheme were spatialized, providing an intuitive evaluation of their applicability to the disaster prevention fund allocation problem. After fund allocation, risk levels decreased across all algorithms, with the relationship between risk level changes in priority support regions shown in Figure 10. This diagram illustrates the transformation process of the number of analysis units across pre-investment and post-investment risk levels for different optimization algorithms. The width of each chord represents the proportional flow of units, with the chord color corresponding to the color of the target node (the receiving side). The unchanged chords represent the number of units that remain within the same risk tier after capital injection. Most areas transitioned to low-risk, with the most significant shift observed from medium- to low-risk, followed by high- to low-risk transitions. SPEA2, which achieved the highest overall benefit, retained the fewest high- and extremely high-risk areas after allocation. It also converted more high-risk areas to medium-risk and maintained the largest number of medium-risk areas, demonstrating the strongest performance in mitigating extreme risks. In contrast, the poorest-performing algorithm, MOPSO, distributed funds too uniformly under budget constraints, excessively allocating to medium-risk areas while failing to prioritize extreme-risk zones. These conversion results directly validate the algorithm performance rankings observed in the Pareto frontier analysis (Figure 8), establishing a direct link between optimization outcomes and tangible improvements in field risk levels. The spatial distribution of risk levels after investment is shown in Figure 11. Following fund allocation, most regions were classified as low- or extremely low-risk, with only a minimal proportion remaining at high or extremely high risk. Specifically, after SPEA2 allocation, medium-risk areas still accounted for 12.57% of the region, while extremely high-risk areas constituted only 0.02%. In contrast, other algorithms still showed sporadic extreme-risk areas, with MOPSO displaying a relatively distinct distribution of extreme-risk zones, confirming its inferior performance. The remaining algorithms yielded results between these two extremes.

Figure 10

Figure 10. Changes in risk levels within priority support areas.

Figure 11

Figure 11. Risk space distribution after funds investment.

4 Discussion

4.1 Spatial distribution of flood risk

The hydrodynamic simulation results revealed only limited expansion of high-hazard zones across increasing return periods (Figure 5), a phenomenon primarily attributable to the interplay between topographic constraints and flood propagation dynamics (Kelleher and McPhillips, 2020). The upper reaches of the study area are characterized by mountainous terrain and narrow valleys, which substantially restrict the lateral spreading of floodwaters (Figure 1c). Once the inundation extent for a 20-year flood event is established, further rainfall increases mainly lead to deeper flooding rather than significant aerial expansion, with only marginal increases occurring along the inundation boundaries (Ahmad et al., 2025). Additionally, the mutual constraint between water depth and flow velocity suppresses exponential growth in the composite hazard index (Zhao et al., 2018). Nevertheless, the hydrodynamic simulations consistently identified persistently high hazard indices in specific locales, primarily the upper tributaries and constricted valleys of the western and northern basins (Figure 5). We attribute these high hazards primarily to the compound effect of steep slopes, which generate rapid surface runoff, and narrow valley morphologies, which constrict flow and amplify velocities. This finding underscores the need to shift flood risk management from “regional coverage” toward “targeted interventions” in these topographically sensitive units. The interaction between water depth and flow velocity is critical, as it directly relates to the destructive potential of floodwaters. While water depth alone indicates inundation extent, the product of water depth and flow velocity (h × v) approximates unit velocity or hydraulic power, better reflecting structural damage, erosion capacity, and risks to human safety. A single parameter indicator fails to capture the severe scouring risk of fast shallow flows nor the extreme destructive force of deep high-velocity flows. Compared to any isolated variable, this approach more accurately reveals the fundamental mechanisms of flood hazards.

Comparative analysis between the coupled hydrodynamic model (Figure 6a) and the uncoupled approach (Figure 6b) demonstrates that the latter substantially underestimates risk levels along river corridors and in mountainous zones. This underestimation is particularly pronounced in specific high-hazard geomorphic units, such as sharp bends in river channels (which increase flow velocity along the outer bank) and confluence zones in river valleys (where merging currents cause sudden rises in water levels). The underestimation stems from the decoupling methods’ inability to characterize the dynamic feedback between topography, channel morphology, and hydrodynamics. In contrast, the hydrological-hydrodynamic coupling model framework comprehensively captures the disaster formation process (Saksena et al., 2019). Its key lies in simulating how topography (e.g., slope) forces runoff convergence, how landforms (e.g., narrow valleys) constrain and accelerate water flow, and how these factors ultimately determine the energy carried by floodwaters as they flow and spread across urban (plain) areas. This enables coupled models to precisely identify often-overlooked localized high-risk points (Yu et al., 2022), as they mechanistically represent hazard formation processes rather than merely correlating with static characteristics. Furthermore, integrating such physics-based models with data-driven approaches (Mohammadian et al., 2025) holds promise for pioneering new avenues to enhance the accuracy of flood simulations.

These comparative results emphasize the critical importance of dynamic flood process simulation in producing reliable flood risk maps. Within small watersheds featuring complex topography, the coupled hydrodynamic model provides precise spatial identification of high-risk areas. This capability ensures that optimization algorithms operate on authentic and reliable risk distribution data, directing funds toward locations where they are most needed and effective. Consequently, the model establishes a reliable foundation for subsequent validation of different multi-objective optimization algorithms in evaluating flood control funding allocation effectiveness.

4.2 Adaptability of multi-objective optimization algorithms in flood control funding allocation scenarios

The application of different multi-objective optimization algorithms to flood control funding allocation revealed significant variations in their performance, highlighting distinct levels of adaptability to the problem structure. SPEA2 demonstrated the strongest overall performance, attributable to its sophisticated archiving mechanism that effectively preserves non-dominated solutions and enhances global search capability (Emmerich and Deutz, 2018). This enables SPEA2 to identify optimal solutions within the high-dimensional flood control funding allocation problem, which is characterized by a large number of decision variables. Specifically, the “dimension” refers to the total number of individual funding allocations that must be determined, the amount of funds allocated to each of the three disaster prevention measures across thousands of spatial analysis units. Furthermore, its adaptive grid density estimation maintains solution set diversity (Kharrich et al., 2021), allowing precise targeting of extreme-risk core areas within high-dimensional discrete decision spaces (where the algorithmic search domain consists of discrete points rather than continuous space) (Figure 9). NSGA-II achieved the second-best performance. Its non-dominated sorting and crowding distance mechanisms enable rapid convergence toward a uniformly distributed optimal solution set in complex funding allocation problems (Hashemi, 2021). However, this rapid search capability somewhat dilutes funding intensity in critical areas (Figure 9), resulting in a more dispersed spatial allocation pattern compared to SPEA2. MOEA/D struggled to balance cost and benefit during allocation and was prone to missing global optima (Xie et al., 2022). This limitation stems from a fundamental structural conflict: while MOEA/D relies on neighborhood relationships defined for continuous decision variables (Li et al., 2025), flood control funding allocation constitutes a typical high-dimensional discrete optimization problem. Consequently, neighborhood-based solution updates often replace efficient solutions with inferior alternatives. MOCOA (Multi-objective Coati Optimization Algorithm) proved unsuitable for this application context. Although its ecological community co-optimization paradigm performs well in continuous low-dimensional problems, it conflicts with the discrete competitive nature of funding allocation and overemphasizes global exploration (Shang et al., 2024; Shuo et al., 2023). This leads to fund dispersion and resource wastage. MOPSO delivered the poorest performance due to its rapid initial convergence tendency, resulting in premature convergence (Esfahani et al., 2025). This characteristic produces unstable allocation outcomes and generates suboptimal solutions. In summary, algorithm selection proves critical for flood control funding allocation, as inappropriate choices may lead to either resource waste or suboptimal outcomes. Although these applicability assessments were conducted within specific research regions, the identified problem characteristics, high dimensionality, discreteness, and objective conflicts, represent common structural features across different geographical contexts. Therefore, the findings retain applicability where similar problem structures exist. Ultimately, this rigorous algorithm selection process constitutes an essential prerequisite for optimizing flood control fund benefits, providing a scientific foundation for enhanced flood disaster management.

4.3 Model robustness and implementability

To achieve urban flood control, the proposed precision allocation framework must simultaneously possess the robustness of model outputs and the operability of policy implementation. In this regard, we have analyzed both the methodological design and practical implementation.

Firstly, flood control decision-making is based on a forward-looking perspective, and there is inherent uncertainty in the future development of factors such as rainfall and socio-economic conditions that cannot be avoided. Considering the diversity of future development scenarios, this framework incorporates several intrinsic designs to enhance the robustness of the allocation scheme. First, unlike traditional methods that rely on a single rainfall scenario, the coupled hydrodynamic model simulates flood hazards under three return periods—20-, 50-, and 100-year—and integrates them into a comprehensive risk index. This means the final allocation scheme does not target a single extreme scenario but responds to a series of hazard intensities, significantly reducing dependence on any specific future rainfall event. Second, the structure of the composite risk index itself provides a buffer. The final risk value is a weighted fusion of hydrological-physical hazards and multiple socio-economic exposure indicators, which dilutes potential errors or uncertainties from individual data sources during aggregation, preventing them from having a decisive impact on the overall allocation pattern. This approach of integrating multi-source data to enhance model robustness and reliability is a well-established principle in geohazard assessment (Cemiloglu et al., 2023). Beyond designing for future uncertainties, the application of principal component analysis (PCA) for socioeconomic weighting provides an objective, data-driven approach to reduce subjective bias. However, it remains undeniable that factors such as future land-use changes and short-to-long-term climate variations cannot be fully captured by static socioeconomic data and design rainfall models. Therefore, conducting quantitative sensitivity analyses on all input factors represents a critical direction for future research. By exploring diverse scenario configurations, the model’s robustness can be further enhanced.

Building on the foundation of a stable model, the high-precision simulation and optimization of this framework require several hours of computational time, making it suitable for medium- to long-term decision support. Focusing on pre-disaster preparedness mitigates uncertainties in post-disaster emergency resource allocation, such as avoiding information silos caused by road damage or communication disruptions. In practice, given that fiscal investments typically follow a top-down allocation process, high-resolution funding distribution results (Figure 9) provide transparent, data-driven justification. These results can be aggregated across administrative levels (e.g., villages, subdistricts, townships) to support decision-making. This approach also avoids the issue of overlooking both overall and extreme local needs. For instance, the impact of minor flood events may be confined to a village but diluted in overall district/county statistics, leading to uneven investment. By focusing on pre-disaster fund allocation, more vulnerable areas—particularly in regions with uneven development—can be identified, and resources can be precisely allocated to maximize the efficiency of disaster prevention and mitigation.

This study proposes a data-driven approach to flood control funding allocation, shifting from an experience-based model. By integrating multiple scenarios and fusing diverse data sources, the framework enhances model robustness and provides scientific decision-making support for pre-disaster prevention and mitigation investments. Furthermore, the framework structure is adaptable: it can be customized based on local data availability across different urban environments. However, the framework’s accuracy remains heavily dependent on input data quality, which poses a persistent challenge in many developing-country cities. Future research will therefore focus on enhancing the model’s adaptability to coarse or uncertain data to broaden its applicability. Concurrently, incorporating future land-use and climate change scenarios will enable dynamic, multi-stage investment strategy analysis, thereby continuously strengthening the model’s resilience to evolving urban development patterns.

5 Conclusion

This study addresses the critical challenge of optimizing limited resource allocation in urban flood management. Accurate identification of flood risk spatial distribution forms the fundamental basis for efficient resource allocation. Our assessment confirms that the flood risk to the urbanized lowlands is intrinsically driven by processes in the mountainous upstream areas. To address this, we developed a refined flood risk assessment system integrating hydrological and hydrodynamic models. This system can simulate the entire process of flood formation and evolution with high precision, enabling comprehensive risk assessment that incorporates socioeconomic and exposure data. To prevent algorithmic mismatch from causing resource misallocation, five mainstream multi-objective optimization algorithms were applied to simulate disaster prevention benefits under various funding allocation scenarios, thereby identifying optimal allocation strategies. The key findings are as follows:

1. The coupled hydrodynamic model risk assessment system effectively identifies flood-prone areas, precisely locating high-risk and extremely high-risk zones covering 11.75% of the total region, primarily distributed in the southern and northwestern urban areas. Compared to non-coupled models, this approach fully simulates the chain of events from precipitation to runoff generation, revealing the mechanism by which mountainous runoff flows into urban plains. This significantly enhances the accuracy of identifying localized high-risk points.

2. Significant performance variations were observed among multi-objective optimization algorithms in funding allocation. SPEA2, leveraging its robust archiving mechanism, achieved the best overall performance by reducing extreme-risk zones to just 0.02%. NSGA-II ranked second in performance, while MOPSO suffered from premature convergence, resulting in scattered funding distribution and poor disaster prevention outcomes. Both MOEA/D and MOCOA demonstrated weak problem adaptability, leading to inefficient fund utilization.

This research demonstrates that integrating optimization algorithms with strong global search capabilities, such as SPEA2, with a refined risk assessment system based on coupled models, enables precise targeting of flood prevention funds to extreme-risk zones. This integrated approach not only enhances fund utilization efficiency but also provides an effective pathway for scientific urban flood risk management and sustainable development.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

AZ: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing. JZ: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Software, Supervision, Visualization, Writing – original draft, Writing – review and editing. YX: Conceptualization, Data curation, Methodology, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing. JH: Conceptualization, Data curation, Investigation, Software, Writing – original draft, Writing – review and editing. YM: Data curation, Formal Analysis, Investigation, Visualization, Writing – original draft. ZW: Funding acquisition, Project administration, Resources, Supervision, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This research was funded by Open Fund of Wenzhou Future City Research Institute (Grant No. WL2023011); Sichuan Provincial Science and Technology Key R&D Program (Grant No. 24YFHZ0133); Sichuan Science and Technology Program (Grant No. 2025ZNSFSC0004); Open Subjects of Southwest Mountain Natural Resources Remote Sensing Monitoring Engineering and Technology Innovation Center (Grant No. RSMNRSCM-2024-008).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The authors declare that no Generative AI was used in the creation of this manuscript.

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