Development of optimization-based budget allocation model for seismic reinforcement based on seismic risk assessment

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Additionally, we derived an algorithm for establishing a reinforcement plan that can minimize earthquake damage within a limited budget. The fragility function for earthquake damage assessment utilized the results of previous studies and an optimization-based budget allocation algorithm for seismic reinforcement was developed by calculating the estimated damage before and after seismic reinforcement based on target damage ratio(TDR) concept and linear approximation (COBYLA) algorithm. To verify the applicability of the developed model, a fictitious city with a 30km x 30km section was set up, and bridges, embankments, and buildings were placed. In addition, the pre- and post-reinforcement damages were evaluated according to two earthquake scenarios, and the optimization-based budget allocation model that can minimize the damages was applied. Physical sciences/Engineering/Civil engineering Earth and environmental sciences/Natural hazards Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction Multiple large-scale earthquakes have occurred in various countries worldwide over the past several years and have been associated with increasing instances of damage to facilities, particularly buildings. In particular, the Turkey earthquake (February 26, 2023) and Sichuan earthquake (December 19, 2023) caused large-scale damage to both property and people. These large earthquakes cause significant damage to structures and socioeconomic damage. Of particular concern are infrastructures, such as bridge collapse, embankment settlement, bridge pier lateral displacement, which are significant social infrastructures that can suffer structural damage 1,2,3,4,5,6 . In the Republic of Korea, the 5.8-magnitude Gyeongju earthquake 7 in 2016 and the 5.6-magnitude Pohang earthquake 8 in 2017 caused damage to the columns of buildings near the epicenter, including school facilities and houses. In particular, 235 school buildings and more than 1,400 houses were affected by half- and full-wave damage during the Pohang earthquake. This level of damage is consistent with the seismic resistance rate of Korean buildings, which is only approximately 20%; thus, research on appropriate seismic reinforcement is required to ensure future seismic stability. In addition, Korean roads and railroad facilities are at risk, with 40% of facilities lacking seismic design owing to the age of their construction, highlighting the necessity for planned seismic reinforcement. Therefore, many preceded researches have been conducted to evaluate dynamic behavior of structure by earthquake. Ji et al 9 evaluated the potential risk of liquefaction caused by the Pohang earthquake. Yun and Han 10 analyzed the dynamic behavior of a retaining wall during an earthquake, and Nguyen et al 11 analyzed the dynamic behavior of a retaining wall structure during an earthquake through numerical analysis. However, there is a lack of research on the development of risk assessment models to predict the expected damage in the event of an earthquake and the calculation of optimal reinforcement measures. Currently, seismic hazard mitigation and response are focused on reactive responses, such as post-earthquake damage restoration. Therefore, FEMA(2022) 12 has developed an earthquake damage estimation model. However, its application has been limited, because suggested vulnerability function evaluated based on actual damage cases, it is difficult to consider various ground conditions, and the function value is determined based only on the data of the strong earthquake that actually caused the damage. Due to limitation of FEMA(2022) 12 earthquake model, a number of earthquake risk assessment research have been conducted for various structures. National Disaster Prevention Research Institute (2008) 13 suggested the earthquake fragility curve for bridge structure based on numerical analysis for Korea bridge design. In addition, Mohseni 14 calculated the seismic vulnerability of a railroad bridge by performing dynamic numerical analysis by varying the bridge extension and pier height. Argyroudis, S., Kaynia, A. M. (2015) 15 presented the earthquake fragility curve for embankment structures constructed on transportation system such as highway or railway. Despite various seismic risk assessment studies, there is a lack of research on rational budgeting and allocation for seismic reinforcement. The resultant degree of damage considerably differed depending on the presence or absence of seismic reinforcement and design. Therefore, to improve efficiency earthquake risk management, it is necessary to develop an efficient management system for assessing expected damage and, in turn, establishing appropriate response budgets and plans. In this study, we developed a technology to predict the economic damage to facilities in a certain area during an earthquake to facilitate the efficient application of performance-based maintenance and seismic reinforcement. Additionally, we derived an algorithm for establishing a reinforcement plan that can minimize earthquake damage within a limited budget. Methods Earthquake risk assessment model In this study, we propose an earthquake risk assessment model to evaluate the potential damage caused by an earthquake in a specific area. The seismic risk assessment model is derived for buildings that are relatively vulnerable to earthquakes and for road transportation networks that are essential for disaster response. Seismic risk assessment models are divided into hazard models for earthquake disasters, inventory models for structures, and vulnerability models encompassing both disasters and structures. For hazard modeling, the peak ground acceleration (PGA) is assumed to be attenuated by the ratio of the distance from the epicenter when an earthquake occurs. The actual PGA is affected by the ground conditions through which seismic waves propagate. However, in this study, the ground is assumed to be hard rock with no ground amplification, and only attenuation with distance is considered. By setting the depth of the earthquake and the PGA at the epicenter, the distance ratio at each location can be calculated, from which the PGA value at each structure location is obtained. In inventory modeling, structures are divided into buildings, embankments, and bridges. Transportation facilities are mainly divided into embankments, bridges, and tunnels; however, as tunnels are classified as structures with relatively low earthquake vulnerability, they were excluded from the risk assessment model in this study. We used inventory and hazard modeling to calculate the properties of the structure according to the geological location and the maximum ground acceleration at that location. Based on this, vulnerability modeling was proposed to calculate the damage that a structure can suffer when the PGA value was applied to the structure. In vulnerability modeling, the seismic fragility curve is frequently utilized to express the exceedance probability of a certain damage state as a function of the PGA. Damage states are categorized as slight, moderate, extensive, and perfect damage. For example, as shown in Fig. 1 , for the seismic vulnerability function of a building, the probability of slight damage is approximately 60% at a PGA level of approximately 0.3 g. The seismic fragility curves applied in this study used the seismic vulnerability function from existing literature, with the seismic fragility curves before and after reinforcement being applied to consider the reinforcement effect. For the seismic fragility curves of buildings, we applied FEMA (2022) 12 without seismic design fragility curves and mid-code seismic design fragility curves (Fig. 1 ); for bridges, the seismic fragility curves before and after seismic reinforcement were applied (Fig. 2 ), based on the National Disaster Prevention Research Institute (2008) 13 . The seismic fragility curves proposed by Argyroudis and Kaynia (2015) 15 for soft and stiff embankments are shown in Fig. 3 . We calculated the probability of exceedance of each damage state for the PGA value for each location and combined the damage ratio for each damage state to determine the expected value of the damage caused by the earthquake, that is, the expected damage ratio. We proposed a model to calculate the expected damage rate according to the type of structure, seismic reinforcement, and earthquake magnitude. The expected damage rate for each damage state utilized data from FEMA (2022) 12 (Table 1 ). Table 1 Damage ratio for each damage state (FEMA 2022) 12 . Damage state Slight Moderate Extensive Complete Bridges 0.03 0.08 0.25 1 Embankments 0.05 0.2 0.7 1 Buildings 0.003 0.017 0.086 0.224 Development of an optimization-based budget allocation algorithm for seismic reinforcement In this study, we proposed an optimization-based budget allocation algorithm to achieve optimal seismic reinforcement within a given budget constraint. An overview of the proposed budget allocation algorithm is presented in Fig. 4 . The proposed algorithm used the risk assessment model described in the previous section to screen infrastructure assets to consider seismic reinforcement and estimate the total budget required to retrofit all screened assets. Subsequently, an optimization task was performed to obtain an optimal budget allocation strategy by minimizing an objective function designed to optimize the reductions in the overall structural damage ratio and replacement cost of the screened assets. Further details on the proposed budget allocation algorithm are described below. In this study, we considered three types of assets: bridges, embankments, and buildings. Before operating the budget allocation algorithm, the following parameters should be set: ( 1 ) the target damage ratio level for each infrastructure asset type, ( 2 ) the individual retrofit cost for each asset, and ( 3 ) the budget constraint. Once the parameters are set, the algorithm is operated as follows. In Step 1, the expected damage ratios (EDRs) for all given infrastructure assets were estimated using the fragility curves of the assets described in the previous section. Given an earthquake scenario with an earthquake location \(({x}_{eqk}, {y}_{eqk})\) , depth \({H}_{eqk}\) , and magnitude at the epicenter \({M}_{epi}\) , the PGAs for all assets were estimated. The computed PGA for the i th asset located at \(({x}_{i}, {y}_{i})\) is given as follows: $${PGA}_{i}=\frac{{M}_{epi}{H}_{eqk}}{{D}_{i}},$$ 1 where \({D}_{i}\) is the distance between the i th asset and the earthquake hypocenter, given as follows: $${D}_{i}=\sqrt{{\left({x}_{eqk}-{x}_{i}\right)}^{2}+{\left({y}_{eqk}-{y}_{i}\right)}^{2}.}$$ 2 The computed PGA values for the assets were then input into the following function to estimate the EDR: $$E{DR}_{i}={P}_{net}^{slight}\left({PGA}_{i}\right){RC}^{slight}+{P}_{net}^{moderate}\left({PGA}_{i}\right){RC}^{moderate}+{P}_{net}^{extensive}\left({PGA}_{i}\right){RC}^{extensive}+{P}_{net}^{complete}\left({PGA}_{i}\right){RC}^{complete},$$ 3 where \({P}_{net}^{slight}\) , \({P}_{net}^{moderate}\) , \({P}_{net}^{extensive}\) , and \({P}_{net}^{complete}\) are the asset-type-specific net cumulative probability distribution functions where damage may occur with slight, moderate, extensive, and complete severity levels, respectively. Each net probability function is a function of \({PGA}_{i}\) and the type of asset. In Eq. ( 3 ), \({RC}^{slight}\) , \({RC}^{moderate}\) , \({RC}^{extensive}\) , and \({RC}^{complete}\) represent the corresponding replacement costs for the damage severity category. In Step 2, asset screening was performed to determine the priority of seismic reinforcement by comparing the EDR and target damage ratio (TDR) levels for each asset. Among the total N infrastructure assets, assets with EDR > TDR were considered in the budget allocation step (Step 3) but assets with EDR ≤ TDR were not. The TDR levels for each infrastructure asset type are shown in Table 2 . In Step 3, the target budget ( \({B}_{target}\) ) required to reinforce all the screened assets in Step 2 was computed, from which we determined whether budget allocation was needed. In this study, we assumed a simple reinforcement scenario in which seismic reinforcement was implemented on an asset with a given cost to reduce the damage ratio of the considered asset. \({B}_{target}\) is computed as follows. $${B}_{target}=\sum _{i=1}^{N}{RC}_{i}{W}_{i},$$ 4 where \({RC}_{i}\) is the reinforcement cost for the i th asset depending on its asset type and \({W}_{i}\) is the binary weight, given as follows: $${W}_{i}=\left\{\begin{array}{c}1 (E{DR}_{i} \ge T{DR}_{i})\\ 0 (E{DR}_{i} <T{DR}_{i})\end{array},\right.$$ 5 where \(T{DR}_{i}\) is the TDR of the i th asset. In this study, we set the TDR threshold values to 5%, 10%, and 3% for bridges, embankments, and buildings, respectively (Table 2 ), considering the nature of each asset type, their earthquake vulnerability, and population. \({B}_{target}\) is then compared with the available budget ( \({B}_{available}\) ). When \({B}_{target}{>B}_{available}\) , budget allocation is required, and optimization is performed in the next step. In the case \({B}_{target}{\le B}_{available}\) , budget allocation is not needed as all the screened assets can be reinforced with \({B}_{available}\) . In Step 4, optimization-based budget allocation was performed in the case \({B}_{target}{>B}_{available}\) by solving the following constrained optimization problem: $$\text{Minimize }Obj\left({W}_{k}^{thr}\right) \text{s}\text{u}\text{b}\text{j}\text{e}\text{c}\text{t} \text{t}\text{o} \sum _{k=1}^{{N}_{thr}}{RC}_{k}{W}_{k}^{thr}\le {B}_{available},$$ 6 where \(Obj\left({W}_{k}^{thr}\right)\) is the objective function to minimize, given as follows: $$Obj\left({W}_{k}^{thr}\right)=\frac{1}{{N}_{thr}}{\sum }_{\text{k}=1}^{{N}_{thr}}\left|{TDR}_{\text{k}}-{EDR}_{k}\left({W}_{k}^{thr}\right)\right| + \lambda \times Reg\left({W}_{k}^{thr}\right)$$ 7 , where \({W}_{k}^{thr}\) is the k th element of the allocation weight vector \({\varvec{W}}^{\varvec{t}\varvec{h}\varvec{r}}\) , the value of which can be one or zero. \({N}_{thr}\) is the number of assets with EDR > TDR. Here, \({W}_{k}^{thr}\) = 1 indicates that the k th asset among \({N}_{thr}\) assets is selected for seismic reinforcement, whereas \({W}_{k}^{thr}\) = 0 the k th asset is unselected. \(\lambda\) in Eq. ( 7 ) is a regularization factor that controls the significance of the regularization function \(Reg\left({W}_{k}^{thr}\right)\) given by: $$Reg\left({W}_{k}^{thr}\right) =\frac{1}{{N}_{thr}}{\sum }_{\text{k}=1}^{{N}_{thr}}{EDR}_{k}\left({W}_{k}^{thr}\right){V}_{k},$$ 8 where \({V}_{k}\) is the value of the k th infrastructure asset in the unit of 1ⅹ10 9 KRW. The first term in the objective function shown in Eq. ( 7 ) plays the role of reducing the overall damage levels of the \({N}_{thr}\) assets, and the second term reducing the total earthquake loss amount. Note that the formulated optimization problem (Eq. 6 ) has a constrain condition to limit the solution space for the total reinforcement cost of the optimally selected assets no to exceed \({B}_{available}\) . The optimal solution for \({\varvec{W}}^{\varvec{t}\varvec{h}\varvec{r}}\) is obtained using an optimization solver. In this study, we used constrained optimization via the linear approximation (COBYLA) algorithm 16 implemented in the Python library SciPy 17 . Once the optimal \({\varvec{W}}^{\varvec{t}\varvec{h}\varvec{r}}\) was obtained by solving the optimization problem, budget allocation was finalized by multiplying \({RC}_{k}\) and \({W}_{k}^{thr}\) . Table 2 Target damage ratio for bridges, embankments, and buildings. Asset type Target damage ratio ( TDR ) (%) Bridge 5 Embankment 10 Building 3 Results: Application to a test-bed region in Korea Earthquake risk assessment model In order to apply the proposed risk assessment model, authors assumed a simplified specific test-bed region including bridges, embankments and buildings. For the seismic risk assessment, 76 simplified structures (20 bridges, 27 embankment and 29 buildings) were assumed to be distributed in an area of 30 km x 30 km, and the asset values of bridges, embankment sections, and buildings were assumed to be 10 billion KRW, 5 billion KRW, and 1 billion KRW, respectively. In the case of actual structures, bridges, embankments, and buildings have different asset values depending on their characteristics, and the vulnerability of the structures is also different. However, to evaluate the applicability of the earthquake risk assessment and optimization-based budget allocation algorithm, this study assumes that all the same structures were constructed, and the asset value was also assumed to be the same in each structure type. Structures were assumed to be distributed in two cases. The first case assumes that the road transportation facilities, bridge and embankment, are linearly constructed, and buildings are placed around them. The second case assumes that the same number of structures are randomly distributed over a 30 km x 30 km area. In addition, we analyzed the location of the earthquake which cause maximum damage. First, in case 1, a PGA of 0.3g, which is the PGA of a 4,800-year return period of earthquake based on Korean seismic design code (MOITT,2023), was assumed to occur at the epicenter, and the earthquake depth of 20km, which is the depth of a recent earthquake in Korea, was applied to calculate the location of the maximum damage among such possible earthquakes. For the earthquake location with maximum damage, an iterative operation was used to generate the same earthquake for each location and calculate the earthquake location. For case 2, a PGA at the epicenter of 0.4g and earthquake depth of 30km were applied to assume that the larger earthquake occurred at a deeper depth. The distribution of structures in each case is shown in Fig. 5 and the calculated earthquake scenarios are summarized in Table 3 . Table 3 Two earthquake scenarios considered in this study. Case 1 Case 2 Location (km) (15, 3) (13, 15) Depth (km) 20 30 PGA at the epicenter (g) 0.3 0.4 The damage cost for each type of structure after and before reinforcement by each earthquake scenario is shown in Table 4 . Bridges with relatively high asset value and embankments with high vulnerability were calculated with large damages. In addition, it can be seen that the estimated damage cost decreases significantly with reinforcement, and the reduction ratio of damage cost is larger in case 1, which has a relatively small earthquake magnitude. This is because the seismic vulnerability function decreases more significantly with reinforcement for smaller PGA value. Table 4 Predicted damage by two earthquake scenarios Case 1 Bridge (×10 6 KRW) Embankment (×10 6 KRW) Building (×10 6 KRW) Sum (×10 6 KRW) without reinforcement 11,397 32,870 980 45,247 with reinforcement 4,696 7,246 533 12,475 Case 2 Bridge ( × 10 6 KRW) Embankment ( × 10 6 KRW) Building ( × 10 6 KRW) Sum ( × 10 6 KRW) without reinforcement 28,133 45,855 2,204 76,192 with reinforcement 14,074 14,300 1,214 29,588 Development of an optimization-based budget allocation algorithm for seismic reinforcement To evaluate the performance of the proposed optimal budget allocation algorithm developed in Methods, we performed numerical experiments considering the two earthquake scenarios listed in Table 3 with regard to different locations, depths, and magnitudes at the epicenter. The reinforcement costs are assumed that 6% of asset value as shown in Table 5 and \({B}_{available}\) was set to 11,000×10 6 KRW for each case. Table 5 Asset value and reinforcement cost for each structure Asset type Asset value (×10 6 KRW) Reinforcement cost (×10 6 KRW) Bridge 10,000 600 Embankment 5,000 300 Building 1,000 60 Figure 6 presents the results of EDR computation showing the assets with EDR > TDR under the earthquake scenarios indicated in Table 3 . These results were used to compute the target budget \({B}_{target}\) for the two numerical experiments. In Case 1, 56 of the total 76 assets exhibited EDR > TDR, indicating the need for earthquake reinforcement (Fig. 6 (a)). The corresponding \({B}_{target}\) was computed to be 16,140×10 6 KRW. In Case 2, all 76 assets had EDR > TDR, resulting in a \({B}_{target}\) of 21,840×10 6 KRW (Fig. 6 (b)). The higher EDR levels in Case 2, compared to those in Case 1, resulted from the relatively high earthquake magnitude at the epicenter and favorable earthquake location (i.e., nearly at the center of the asset placements). Notably, the results of the two cases show that both earthquake scenarios result in a limited reinforcement budget ( \({B}_{available}{ TDR can be reinforced, revealing the necessity of optimal budget allocation for seismic reinforcement. The earthquake reinforcement budget allocation results obtained using the proposed optimization-based algorithm are presented in Fig. 7 and Table 6 . In Case 1, among the 53 assets with EDR, 28 assets were selected for reinforcement by the algorithm > TDR (Fig. 7 (a)). In this case, the total budget allocated to the 28 assets was 10,800×10 6 KRW, which is within and close to \({B}_{available}\) (11,000×10 6 KRW). In comparison, for Case 2, 31 assets were selected for reinforcement by the algorithm (Fig. 7 (b)), resulting in a total allocated budget of 10,500×10 6 KRW. For Case 1 (Fig. 7 (a)), assets close to the earthquake source (indicated by a red star) were selected by the algorithm. In addition, because of the nature of a high EDR and short replacement duration, more embankments were selected than other assets. Buildings were not selected, likely owing to their tendency toward a low EDR and long replacement duration. A similar output tendency of the budget allocation algorithm was observed for Case 2 (Fig. 7 (b)): more embankments were selected than other assets. Overall, the results of the numerical experiments demonstrated the feasibility of the proposed budget allocation algorithm for optimal earthquake reinforcement. Table 6 Budget allocation results. Case 1 Case 2 Target budget (×10 6 KRW) 16,140 21,840 Available budget (×10 6 KRW) 11,000 11,000 Allocated budget (×10 6 KRW) 10,800 10,500 Discussion This research was conducted to provide an appropriate solution for proactive response to earthquakes rather than reactive response after an earthquake occurs. The existing seismic reinforcement in Korea is done by identifying the vulnerable sections after the earthquake and reinforcement the damaged sections, and the budget comes into play after the disaster. As a result of this study, it is possible to identify reinforcement locations that can minimize the damage of the expected design earthquake with a limited budget, which can have a positive effect on government decision-making. This study describes a methodological approach for optimizing the seismic retrofit budget calculation, and the data is significantly simplified. Therefore, this research cannot consider various ground conditions and applies only distance attenuation of earthquake acceleration. However, the actual PGA value can be applied as a simple one-dimensional site response analysis when investigating actual ground conditions, and it is judged that there is no problem in applying this algorithm. In addition, there is a limitation that the same structure is placed for each structure, however this can be applied by adjusting the asset value and vulnerability function of the structure through the investigation of the actual structure. It is expected that more reasonable results can be also obtained by applying newly researched pre- and post-reinforcement vulnerability curves for various structures. The generalizability and applicability of the proposed optimization-based budget allocation model are critical considerations for its broader impact. While the model has been developed with a focus on seismic reinforcement in the context of Korea, it is essential to evaluate its transferability to other regions or countries with distinct seismic characteristics. Variations in geological conditions, seismicity patterns, and structural vulnerabilities may pose challenges to the model's seamless adaptation. Future research could explore the model's robustness and adaptability across diverse contexts, considering the need for region-specific adjustments. Additionally, incorporating local seismic data could enhance the budget allocation model's generalizability and make it a valuable tool for proactive seismic risk management on a global scale. Addressing these aspects will contribute to the model's broader applicability and ensure its effectiveness in aiding decision-making processes in different seismic-prone regions. Declarations Author Contribution J.K.Kim.: writing–original draft, development of risk assessment model. S.J.Kim: Review-original draft. H.M.Song.: Development optimization model. M.Y.: organizing, supervision, project administration, review. The authors confirm that this work has not been published before, and its publication has been approved by all co-authors. Acknowledgements This research was supported by a grant (RS-2023-00238458, Development and Verification of Integrated Management System for High-Risk Disaster Response in Deep Railway Facilities) from the Development of a Disaster Response Complex Training Center for Deep Tunnels (GTX, etc.) Program funded by the Ministry of Land, Infrastructure, and Transport of the Korean government. We greatly appreciate the support. Data availability The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. The data sources, data Repository and data access and usage conditions are summarized below. Data Sources: We utilized fragility curve data from preceded research, assumed and reproduced the inventory data. Data Repository: The data was stored in a local repository within our research laboratory for the duration of the study. 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Hennart), Kluwer Academic Publishers , pp. 51–67 Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau,D., … van der Walt, S. J. (2020). SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods , 17(3), 261–272. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3904718","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":271462495,"identity":"c3259e35-500c-4168-8921-cc8ef2986531","order_by":0,"name":"Jong-Kwan Kim","email":"","orcid":"","institution":"Korea Institute of Civil Engineering and Building Technology (KICT)","correspondingAuthor":false,"prefix":"","firstName":"Jong-Kwan","middleName":"","lastName":"Kim","suffix":""},{"id":271462496,"identity":"e84d54de-7c5d-484b-8a00-d67ad113d83e","order_by":1,"name":"Seok-Jung Kim","email":"","orcid":"","institution":"Korea Institute of Civil Engineering and Building Technology (KICT)","correspondingAuthor":false,"prefix":"","firstName":"Seok-Jung","middleName":"","lastName":"Kim","suffix":""},{"id":271462497,"identity":"7eb3de53-01a7-4385-bed8-a8b9e6ce4df9","order_by":2,"name":"Ho-Min Song","email":"","orcid":"","institution":"Gachon University","correspondingAuthor":false,"prefix":"","firstName":"Ho-Min","middleName":"","lastName":"Song","suffix":""},{"id":271462498,"identity":"29cb4c87-7d85-43bc-9fda-62613310d7fd","order_by":3,"name":"Mintaek Yoo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVRIiWNgGAWjYDACZjCZAGIdABIHoMIHsKlF0nIArIUtgUgtDHAtPAbEadFt5z34+UNFWmI/e8/nFz9q7siZMzA//MBw5h5OLWaH+ZIlDpzJSZzZc3abZc+xZ8aWDWzGEgw3ivFo4TGQONhWkbjhRu42A96Gw4kbDjCYMTB8SMCnxfjHwX8gLTnPDP+CtbB/I6TFTOJgQw5IC/NjiC08QFtu4NdiceZYmvHMnmNmzDLHDhsbHOYplkg4g0fL+TPGNypqkmX72Zsff3xTc1jO4Hj7xg8fjuHWAgOODcC4lAAzQZFLWAMDgz1I7QciFI6CUTAKRsEIBACGr2Gh3ft1HAAAAABJRU5ErkJggg==","orcid":"","institution":"Gachon University","correspondingAuthor":true,"prefix":"","firstName":"Mintaek","middleName":"","lastName":"Yoo","suffix":""}],"badges":[],"createdAt":"2024-01-28 05:32:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3904718/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3904718/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50813589,"identity":"a13cbd91-700f-4340-9353-b6c574217830","added_by":"auto","created_at":"2024-02-07 19:30:36","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":167427,"visible":true,"origin":"","legend":"\u003cp\u003eSeismic fragility curve for buildings (FEMA, 2022)\u003csup\u003e 12\u003c/sup\u003e. (\u003cstrong\u003ea\u003c/strong\u003e) Without reinforcement; (\u003cstrong\u003eb\u003c/strong\u003e) with reinforcement.\u003c/p\u003e","description":"","filename":"F1.png","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/5635cf88fc5eb63145157664.png"},{"id":50813592,"identity":"6836e84d-58d1-4b6c-9667-e6ac09ffc4a2","added_by":"auto","created_at":"2024-02-07 19:30:36","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":141632,"visible":true,"origin":"","legend":"\u003cp\u003eSeismic fragility curve for bridges (Korean Seismic Fragility Function Localization Study, 2008)\u003csup\u003e 13\u003c/sup\u003e. (\u003cstrong\u003ea\u003c/strong\u003e) Without reinforcement; (\u003cstrong\u003eb\u003c/strong\u003e) with reinforcement.\u003c/p\u003e","description":"","filename":"F2.png","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/ac6e221559a4dcd4a83f68bb.png"},{"id":50813595,"identity":"003c0fe9-3aab-42df-9265-a0aefdd2f40b","added_by":"auto","created_at":"2024-02-07 19:30:36","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":163396,"visible":true,"origin":"","legend":"\u003cp\u003eSeismic fragility curve for embankments (Argyroudis and Kaynia, 2015)\u003csup\u003e 15\u003c/sup\u003e. (\u003cstrong\u003ea\u003c/strong\u003e) Without reinforcement; (\u003cstrong\u003eb\u003c/strong\u003e) with reinforcement.\u003c/p\u003e","description":"","filename":"F3.png","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/3d1b448b98da38f294f8f093.png"},{"id":50813593,"identity":"d8b208aa-b3ce-444e-b735-3df164067198","added_by":"auto","created_at":"2024-02-07 19:30:36","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":98175,"visible":true,"origin":"","legend":"\u003cp\u003eAn overview of the proposed seismic reinforcement budget allocation algorithm.\u003c/p\u003e","description":"","filename":"F4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/d2d3362de37cb0bfe3b4f618.jpg"},{"id":50813587,"identity":"f3611d20-6e7d-4dfe-b59a-a3a6f63aba7b","added_by":"auto","created_at":"2024-02-07 19:30:34","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":66584,"visible":true,"origin":"","legend":"\u003cp\u003eTwo earthquake scenarios with different asset placements: (\u003cstrong\u003ea\u003c/strong\u003e) Case 1 and (\u003cstrong\u003eb\u003c/strong\u003e) Case 2.\u003c/p\u003e","description":"","filename":"F5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/36ed111e913bbb6aa830508d.jpg"},{"id":50813591,"identity":"e4e0a6ae-75bd-40b0-a57a-07015074f0b0","added_by":"auto","created_at":"2024-02-07 19:30:36","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":64180,"visible":true,"origin":"","legend":"\u003cp\u003eAssets with expected damage ratio (EDR) greater than the target damage ratio (TDR): (a) Case 1 and (b) Case 2.\u003c/p\u003e","description":"","filename":"F6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/d9d6b144d67c86f70cc7c5f2.jpg"},{"id":50813588,"identity":"b952149d-84a9-4f34-8071-792852743202","added_by":"auto","created_at":"2024-02-07 19:30:35","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":55469,"visible":true,"origin":"","legend":"\u003cp\u003eEarthquake reinforcement budget allocation results: assets selected for reinforcement by the proposed algorithm for (a) Case 1 and (b) Case 2.\u003c/p\u003e","description":"","filename":"F7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/7608bb7e90e76303e4d255c3.jpg"},{"id":57364767,"identity":"f3eebb58-de94-4534-9c2c-9e8c98864a9d","added_by":"auto","created_at":"2024-05-29 16:23:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1258910,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3904718/v1/da31c17c-82d5-4055-b359-5a32e187d1db.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Development of optimization-based budget allocation model for seismic reinforcement based on seismic risk assessment","fulltext":[{"header":"Introduction","content":"\u003cp\u003eMultiple large-scale earthquakes have occurred in various countries worldwide over the past several years and have been associated with increasing instances of damage to facilities, particularly buildings. In particular, the Turkey earthquake (February 26, 2023) and Sichuan earthquake (December 19, 2023) caused large-scale damage to both property and people. These large earthquakes cause significant damage to structures and socioeconomic damage. Of particular concern are infrastructures, such as bridge collapse, embankment settlement, bridge pier lateral displacement, which are significant social infrastructures that can suffer structural damage \u003csup\u003e1,2,3,4,5,6\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn the Republic of Korea, the 5.8-magnitude Gyeongju earthquake\u003csup\u003e7\u003c/sup\u003e in 2016 and the 5.6-magnitude Pohang earthquake\u003csup\u003e8\u003c/sup\u003e in 2017 caused damage to the columns of buildings near the epicenter, including school facilities and houses. In particular, 235 school buildings and more than 1,400 houses were affected by half- and full-wave damage during the Pohang earthquake. This level of damage is consistent with the seismic resistance rate of Korean buildings, which is only approximately 20%; thus, research on appropriate seismic reinforcement is required to ensure future seismic stability. In addition, Korean roads and railroad facilities are at risk, with 40% of facilities lacking seismic design owing to the age of their construction, highlighting the necessity for planned seismic reinforcement.\u003c/p\u003e \u003cp\u003eTherefore, many preceded researches have been conducted to evaluate dynamic behavior of structure by earthquake. Ji et al\u003csup\u003e9\u003c/sup\u003e evaluated the potential risk of liquefaction caused by the Pohang earthquake. Yun and Han\u003csup\u003e10\u003c/sup\u003e analyzed the dynamic behavior of a retaining wall during an earthquake, and Nguyen et al\u003csup\u003e11\u003c/sup\u003e analyzed the dynamic behavior of a retaining wall structure during an earthquake through numerical analysis.\u003c/p\u003e \u003cp\u003eHowever, there is a lack of research on the development of risk assessment models to predict the expected damage in the event of an earthquake and the calculation of optimal reinforcement measures. Currently, seismic hazard mitigation and response are focused on reactive responses, such as post-earthquake damage restoration. Therefore, FEMA(2022)\u003csup\u003e12\u003c/sup\u003e has developed an earthquake damage estimation model. However, its application has been limited, because suggested vulnerability function evaluated based on actual damage cases, it is difficult to consider various ground conditions, and the function value is determined based only on the data of the strong earthquake that actually caused the damage. Due to limitation of FEMA(2022) \u003csup\u003e12\u003c/sup\u003e earthquake model, a number of earthquake risk assessment research have been conducted for various structures. National Disaster Prevention Research Institute (2008) \u003csup\u003e13\u003c/sup\u003e suggested the earthquake fragility curve for bridge structure based on numerical analysis for Korea bridge design. In addition, Mohseni \u003csup\u003e14\u003c/sup\u003e calculated the seismic vulnerability of a railroad bridge by performing dynamic numerical analysis by varying the bridge extension and pier height. Argyroudis, S., Kaynia, A. M. (2015) \u003csup\u003e15\u003c/sup\u003e presented the earthquake fragility curve for embankment structures constructed on transportation system such as highway or railway.\u003c/p\u003e \u003cp\u003eDespite various seismic risk assessment studies, there is a lack of research on rational budgeting and allocation for seismic reinforcement. The resultant degree of damage considerably differed depending on the presence or absence of seismic reinforcement and design. Therefore, to improve efficiency earthquake risk management, it is necessary to develop an efficient management system for assessing expected damage and, in turn, establishing appropriate response budgets and plans.\u003c/p\u003e \u003cp\u003eIn this study, we developed a technology to predict the economic damage to facilities in a certain area during an earthquake to facilitate the efficient application of performance-based maintenance and seismic reinforcement. Additionally, we derived an algorithm for establishing a reinforcement plan that can minimize earthquake damage within a limited budget.\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eEarthquake risk assessment model\u003c/h2\u003e \u003cp\u003eIn this study, we propose an earthquake risk assessment model to evaluate the potential damage caused by an earthquake in a specific area. The seismic risk assessment model is derived for buildings that are relatively vulnerable to earthquakes and for road transportation networks that are essential for disaster response.\u003c/p\u003e \u003cp\u003eSeismic risk assessment models are divided into hazard models for earthquake disasters, inventory models for structures, and vulnerability models encompassing both disasters and structures. For hazard modeling, the peak ground acceleration (PGA) is assumed to be attenuated by the ratio of the distance from the epicenter when an earthquake occurs. The actual PGA is affected by the ground conditions through which seismic waves propagate. However, in this study, the ground is assumed to be hard rock with no ground amplification, and only attenuation with distance is considered. By setting the depth of the earthquake and the PGA at the epicenter, the distance ratio at each location can be calculated, from which the PGA value at each structure location is obtained. In inventory modeling, structures are divided into buildings, embankments, and bridges. Transportation facilities are mainly divided into embankments, bridges, and tunnels; however, as tunnels are classified as structures with relatively low earthquake vulnerability, they were excluded from the risk assessment model in this study.\u003c/p\u003e \u003cp\u003eWe used inventory and hazard modeling to calculate the properties of the structure according to the geological location and the maximum ground acceleration at that location. Based on this, vulnerability modeling was proposed to calculate the damage that a structure can suffer when the PGA value was applied to the structure. In vulnerability modeling, the seismic fragility curve is frequently utilized to express the exceedance probability of a certain damage state as a function of the PGA. Damage states are categorized as slight, moderate, extensive, and perfect damage. For example, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, for the seismic vulnerability function of a building, the probability of slight damage is approximately 60% at a PGA level of approximately 0.3 g. The seismic fragility curves applied in this study used the seismic vulnerability function from existing literature, with the seismic fragility curves before and after reinforcement being applied to consider the reinforcement effect. For the seismic fragility curves of buildings, we applied FEMA (2022) \u003csup\u003e12\u003c/sup\u003e without seismic design fragility curves and mid-code seismic design fragility curves (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e); for bridges, the seismic fragility curves before and after seismic reinforcement were applied (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), based on the National Disaster Prevention Research Institute (2008) \u003csup\u003e13\u003c/sup\u003e. The seismic fragility curves proposed by Argyroudis and Kaynia (2015) \u003csup\u003e15\u003c/sup\u003e for soft and stiff embankments are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eWe calculated the probability of exceedance of each damage state for the PGA value for each location and combined the damage ratio for each damage state to determine the expected value of the damage caused by the earthquake, that is, the expected damage ratio. We proposed a model to calculate the expected damage rate according to the type of structure, seismic reinforcement, and earthquake magnitude. The expected damage rate for each damage state utilized data from FEMA (2022) \u003csup\u003e12\u003c/sup\u003e (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDamage ratio for each damage state (FEMA 2022) \u003csup\u003e12\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDamage state\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSlight\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eModerate\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eExtensive\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eComplete\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBridges\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEmbankments\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBuildings\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.017\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.086\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.224\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eDevelopment of an optimization-based budget allocation algorithm for seismic reinforcement\u003c/h2\u003e \u003cp\u003eIn this study, we proposed an optimization-based budget allocation algorithm to achieve optimal seismic reinforcement within a given budget constraint. An overview of the proposed budget allocation algorithm is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The proposed algorithm used the risk assessment model described in the previous section to screen infrastructure assets to consider seismic reinforcement and estimate the total budget required to retrofit all screened assets. Subsequently, an optimization task was performed to obtain an optimal budget allocation strategy by minimizing an objective function designed to optimize the reductions in the overall structural damage ratio and replacement cost of the screened assets. Further details on the proposed budget allocation algorithm are described below.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn this study, we considered three types of assets: bridges, embankments, and buildings. Before operating the budget allocation algorithm, the following parameters should be set: (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) the target damage ratio level for each infrastructure asset type, (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) the individual retrofit cost for each asset, and (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e) the budget constraint. Once the parameters are set, the algorithm is operated as follows.\u003c/p\u003e \u003cp\u003eIn Step 1, the expected damage ratios (EDRs) for all given infrastructure assets were estimated using the fragility curves of the assets described in the previous section. Given an earthquake scenario with an earthquake location \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(({x}_{eqk}, {y}_{eqk})\\)\u003c/span\u003e\u003c/span\u003e, depth \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({H}_{eqk}\\)\u003c/span\u003e\u003c/span\u003e, and magnitude at the epicenter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({M}_{epi}\\)\u003c/span\u003e\u003c/span\u003e, the PGAs for all assets were estimated. The computed PGA for the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset located at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(({x}_{i}, {y}_{i})\\)\u003c/span\u003e\u003c/span\u003e is given as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${PGA}_{i}=\\frac{{M}_{epi}{H}_{eqk}}{{D}_{i}},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the distance between the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset and the earthquake hypocenter, given as follows:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${D}_{i}=\\sqrt{{\\left({x}_{eqk}-{x}_{i}\\right)}^{2}+{\\left({y}_{eqk}-{y}_{i}\\right)}^{2}.}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe computed PGA values for the assets were then input into the following function to estimate the EDR:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$E{DR}_{i}={P}_{net}^{slight}\\left({PGA}_{i}\\right){RC}^{slight}+{P}_{net}^{moderate}\\left({PGA}_{i}\\right){RC}^{moderate}+{P}_{net}^{extensive}\\left({PGA}_{i}\\right){RC}^{extensive}+{P}_{net}^{complete}\\left({PGA}_{i}\\right){RC}^{complete},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{net}^{slight}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{net}^{moderate}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{net}^{extensive}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{net}^{complete}\\)\u003c/span\u003e\u003c/span\u003e are the asset-type-specific net cumulative probability distribution functions where damage may occur with slight, moderate, extensive, and complete severity levels, respectively. Each net probability function is a function of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({PGA}_{i}\\)\u003c/span\u003e\u003c/span\u003e and the type of asset. In Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}^{slight}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}^{moderate}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}^{extensive}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}^{complete}\\)\u003c/span\u003e\u003c/span\u003e represent the corresponding replacement costs for the damage severity category.\u003c/p\u003e \u003cp\u003eIn Step 2, asset screening was performed to determine the priority of seismic reinforcement by comparing the EDR and target damage ratio (TDR) levels for each asset. Among the total \u003cem\u003eN\u003c/em\u003e infrastructure assets, assets with EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR were considered in the budget allocation step (Step 3) but assets with EDR\u0026thinsp;\u0026le;\u0026thinsp;TDR were not. The TDR levels for each infrastructure asset type are shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn Step 3, the target budget (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e) required to reinforce all the screened assets in Step 2 was computed, from which we determined whether budget allocation was needed. In this study, we assumed a simple reinforcement scenario in which seismic reinforcement was implemented on an asset with a given cost to reduce the damage ratio of the considered asset. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e is computed as follows.\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${B}_{target}=\\sum _{i=1}^{N}{RC}_{i}{W}_{i},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the reinforcement cost for the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset depending on its asset type and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the binary weight, given as follows:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${W}_{i}=\\left\\{\\begin{array}{c}1 (E{DR}_{i} \\ge T{DR}_{i})\\\\ 0 (E{DR}_{i} \u0026lt;T{DR}_{i})\\end{array},\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T{DR}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the TDR of the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset. In this study, we set the TDR threshold values to 5%, 10%, and 3% for bridges, embankments, and buildings, respectively (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), considering the nature of each asset type, their earthquake vulnerability, and population. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e is then compared with the available budget (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}\\)\u003c/span\u003e\u003c/span\u003e). When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}{\u0026gt;B}_{available}\\)\u003c/span\u003e\u003c/span\u003e, budget allocation is required, and optimization is performed in the next step. In the case \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}{\\le B}_{available}\\)\u003c/span\u003e\u003c/span\u003e, budget allocation is not needed as all the screened assets can be reinforced with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn Step 4, optimization-based budget allocation was performed in the case \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}{\u0026gt;B}_{available}\\)\u003c/span\u003e\u003c/span\u003e by solving the following constrained optimization problem:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\text{Minimize }Obj\\left({W}_{k}^{thr}\\right) \\text{s}\\text{u}\\text{b}\\text{j}\\text{e}\\text{c}\\text{t} \\text{t}\\text{o} \\sum _{k=1}^{{N}_{thr}}{RC}_{k}{W}_{k}^{thr}\\le {B}_{available},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Obj\\left({W}_{k}^{thr}\\right)\\)\u003c/span\u003e\u003c/span\u003e is the objective function to minimize, given as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$Obj\\left({W}_{k}^{thr}\\right)=\\frac{1}{{N}_{thr}}{\\sum }_{\\text{k}=1}^{{N}_{thr}}\\left|{TDR}_{\\text{k}}-{EDR}_{k}\\left({W}_{k}^{thr}\\right)\\right| + \\lambda \\times Reg\\left({W}_{k}^{thr}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W}_{k}^{thr}\\)\u003c/span\u003e\u003c/span\u003e is the \u003cem\u003ek\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e element of the allocation weight vector \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{W}}^{\\varvec{t}\\varvec{h}\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e, the value of which can be one or zero. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{thr}\\)\u003c/span\u003e\u003c/span\u003e is the number of assets with EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR. Here, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W}_{k}^{thr}\\)\u003c/span\u003e\u003c/span\u003e = 1 indicates that the \u003cem\u003ek\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset among \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{thr}\\)\u003c/span\u003e\u003c/span\u003e assets is selected for seismic reinforcement, whereas \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W}_{k}^{thr}\\)\u003c/span\u003e\u003c/span\u003e = 0 the \u003cem\u003ek\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e asset is unselected. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) is a regularization factor that controls the significance of the regularization function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Reg\\left({W}_{k}^{thr}\\right)\\)\u003c/span\u003e\u003c/span\u003e given by:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$Reg\\left({W}_{k}^{thr}\\right) =\\frac{1}{{N}_{thr}}{\\sum }_{\\text{k}=1}^{{N}_{thr}}{EDR}_{k}\\left({W}_{k}^{thr}\\right){V}_{k},$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{k}\\)\u003c/span\u003e\u003c/span\u003e is the value of the \u003cem\u003ek\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e infrastructure asset in the unit of 1ⅹ10\u003csup\u003e9\u003c/sup\u003e KRW. The first term in the objective function shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) plays the role of reducing the overall damage levels of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{thr}\\)\u003c/span\u003e\u003c/span\u003e assets, and the second term reducing the total earthquake loss amount. Note that the formulated optimization problem (Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) has a constrain condition to limit the solution space for the total reinforcement cost of the optimally selected assets no to exceed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}\\)\u003c/span\u003e\u003c/span\u003e. The optimal solution for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{W}}^{\\varvec{t}\\varvec{h}\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e is obtained using an optimization solver. In this study, we used constrained optimization via the linear approximation (COBYLA) algorithm\u003csup\u003e16\u003c/sup\u003e implemented in the Python library SciPy\u003csup\u003e17\u003c/sup\u003e. Once the optimal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{W}}^{\\varvec{t}\\varvec{h}\\varvec{r}}\\)\u003c/span\u003e\u003c/span\u003e was obtained by solving the optimization problem, budget allocation was finalized by multiplying \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({RC}_{k}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({W}_{k}^{thr}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTarget damage ratio for bridges, embankments, and buildings.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAsset type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTarget damage ratio (\u003cem\u003eTDR\u003c/em\u003e) (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBridge\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEmbankment\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBuilding\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eResults: Application to a test-bed region in Korea\u003c/h2\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003eEarthquake risk assessment model\u003c/h2\u003e \u003cp\u003eIn order to apply the proposed risk assessment model, authors assumed a simplified specific test-bed region including bridges, embankments and buildings. For the seismic risk assessment, 76 simplified structures (20 bridges, 27 embankment and 29 buildings) were assumed to be distributed in an area of 30 km x 30 km, and the asset values of bridges, embankment sections, and buildings were assumed to be 10\u0026nbsp;billion KRW, 5\u0026nbsp;billion KRW, and 1\u0026nbsp;billion KRW, respectively. In the case of actual structures, bridges, embankments, and buildings have different asset values depending on their characteristics, and the vulnerability of the structures is also different. However, to evaluate the applicability of the earthquake risk assessment and optimization-based budget allocation algorithm, this study assumes that all the same structures were constructed, and the asset value was also assumed to be the same in each structure type. Structures were assumed to be distributed in two cases. The first case assumes that the road transportation facilities, bridge and embankment, are linearly constructed, and buildings are placed around them. The second case assumes that the same number of structures are randomly distributed over a 30 km x 30 km area. In addition, we analyzed the location of the earthquake which cause maximum damage. First, in case 1, a PGA of 0.3g, which is the PGA of a 4,800-year return period of earthquake based on Korean seismic design code (MOITT,2023), was assumed to occur at the epicenter, and the earthquake depth of 20km, which is the depth of a recent earthquake in Korea, was applied to calculate the location of the maximum damage among such possible earthquakes. For the earthquake location with maximum damage, an iterative operation was used to generate the same earthquake for each location and calculate the earthquake location. For case 2, a PGA at the epicenter of 0.4g and earthquake depth of 30km were applied to assume that the larger earthquake occurred at a deeper depth. The distribution of structures in each case is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and the calculated earthquake scenarios are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTwo earthquake scenarios considered in this study.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCase 1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCase 2\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLocation (km)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(15, 3)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(13, 15)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDepth (km)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePGA at the epicenter (g)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe damage cost for each type of structure after and before reinforcement by each earthquake scenario is shown in Table \u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Bridges with relatively high asset value and embankments with high vulnerability were calculated with large damages. In addition, it can be seen that the estimated damage cost decreases significantly with reinforcement, and the reduction ratio of damage cost is larger in case 1, which has a relatively small earthquake magnitude. This is because the seismic vulnerability function decreases more significantly with reinforcement for smaller PGA value.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePredicted damage by two earthquake scenarios\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCase 1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBridge\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEmbankment\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBuilding\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSum\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ewithout reinforcement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e11,397\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e32,870\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e45,247\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ewith reinforcement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4,696\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7,246\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e533\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e12,475\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eCase 2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBridge\u003c/p\u003e \u003cp\u003e\u003cb\u003e(\u003c/b\u003e\u0026times;\u003cb\u003e10\u003c/b\u003e\u003csup\u003e\u003cb\u003e6\u003c/b\u003e\u003c/sup\u003e \u003cb\u003eKRW)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEmbankment\u003c/p\u003e \u003cp\u003e\u003cb\u003e(\u003c/b\u003e\u0026times;\u003cb\u003e10\u003c/b\u003e\u003csup\u003e\u003cb\u003e6\u003c/b\u003e\u003c/sup\u003e \u003cb\u003eKRW)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBuilding\u003c/p\u003e \u003cp\u003e\u003cb\u003e(\u003c/b\u003e\u0026times;\u003cb\u003e10\u003c/b\u003e\u003csup\u003e\u003cb\u003e6\u003c/b\u003e\u003c/sup\u003e \u003cb\u003eKRW)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSum\u003c/p\u003e \u003cp\u003e\u003cb\u003e(\u003c/b\u003e\u0026times;\u003cb\u003e10\u003c/b\u003e\u003csup\u003e\u003cb\u003e6\u003c/b\u003e\u003c/sup\u003e \u003cb\u003eKRW)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ewithout reinforcement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e28,133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e45,855\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2,204\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e76,192\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ewith reinforcement\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e14,074\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e14,300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1,214\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e29,588\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003eDevelopment of an optimization-based budget allocation algorithm for seismic reinforcement\u003c/h2\u003e \u003cp\u003eTo evaluate the performance of the proposed optimal budget allocation algorithm developed in Methods, we performed numerical experiments considering the two earthquake scenarios listed in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e with regard to different locations, depths, and magnitudes at the epicenter. The reinforcement costs are assumed that 6% of asset value as shown in Table \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}\\)\u003c/span\u003e\u003c/span\u003e was set to 11,000\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW for each case.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eAsset value and reinforcement cost for each structure\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAsset type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAsset value\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eReinforcement cost\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBridge\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e10,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e600\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEmbankment\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e300\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBuilding\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e60\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e presents the results of EDR computation showing the assets with EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR under the earthquake scenarios indicated in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. These results were used to compute the target budget \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e for the two numerical experiments. In Case 1, 56 of the total 76 assets exhibited EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR, indicating the need for earthquake reinforcement (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(a)). The corresponding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e was computed to be 16,140\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW. In Case 2, all 76 assets had EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR, resulting in a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{target}\\)\u003c/span\u003e\u003c/span\u003e of 21,840\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(b)). The higher EDR levels in Case 2, compared to those in Case 1, resulted from the relatively high earthquake magnitude at the epicenter and favorable earthquake location (i.e., nearly at the center of the asset placements). Notably, the results of the two cases show that both earthquake scenarios result in a limited reinforcement budget (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}{\u0026lt;B}_{target}\\)\u003c/span\u003e\u003c/span\u003e); hence, not all assets with EDR\u0026thinsp;\u0026gt;\u0026thinsp;TDR can be reinforced, revealing the necessity of optimal budget allocation for seismic reinforcement.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe earthquake reinforcement budget allocation results obtained using the proposed optimization-based algorithm are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. In Case 1, among the 53 assets with EDR, 28 assets were selected for reinforcement by the algorithm\u0026thinsp;\u0026gt;\u0026thinsp;TDR (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a)). In this case, the total budget allocated to the 28 assets was 10,800\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW, which is within and close to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{available}\\)\u003c/span\u003e\u003c/span\u003e(11,000\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW). In comparison, for Case 2, 31 assets were selected for reinforcement by the algorithm (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(b)), resulting in a total allocated budget of 10,500\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW. For Case 1 (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a)), assets close to the earthquake source (indicated by a red star) were selected by the algorithm. In addition, because of the nature of a high EDR and short replacement duration, more embankments were selected than other assets. Buildings were not selected, likely owing to their tendency toward a low EDR and long replacement duration. A similar output tendency of the budget allocation algorithm was observed for Case 2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(b)): more embankments were selected than other assets. Overall, the results of the numerical experiments demonstrated the feasibility of the proposed budget allocation algorithm for optimal earthquake reinforcement.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBudget allocation results.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCase 1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCase 2\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTarget budget\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e16,140\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e21,840\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAvailable budget\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e11,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e11,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAllocated budget\u003c/p\u003e \u003cp\u003e(\u0026times;10\u003csup\u003e6\u003c/sup\u003e KRW)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e10,800\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10,500\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThis research was conducted to provide an appropriate solution for proactive response to earthquakes rather than reactive response after an earthquake occurs. The existing seismic reinforcement in Korea is done by identifying the vulnerable sections after the earthquake and reinforcement the damaged sections, and the budget comes into play after the disaster. As a result of this study, it is possible to identify reinforcement locations that can minimize the damage of the expected design earthquake with a limited budget, which can have a positive effect on government decision-making.\u003c/p\u003e \u003cp\u003eThis study describes a methodological approach for optimizing the seismic retrofit budget calculation, and the data is significantly simplified. Therefore, this research cannot consider various ground conditions and applies only distance attenuation of earthquake acceleration. However, the actual PGA value can be applied as a simple one-dimensional site response analysis when investigating actual ground conditions, and it is judged that there is no problem in applying this algorithm. In addition, there is a limitation that the same structure is placed for each structure, however this can be applied by adjusting the asset value and vulnerability function of the structure through the investigation of the actual structure. It is expected that more reasonable results can be also obtained by applying newly researched pre- and post-reinforcement vulnerability curves for various structures.\u003c/p\u003e \u003cp\u003eThe generalizability and applicability of the proposed optimization-based budget allocation model are critical considerations for its broader impact. While the model has been developed with a focus on seismic reinforcement in the context of Korea, it is essential to evaluate its transferability to other regions or countries with distinct seismic characteristics. Variations in geological conditions, seismicity patterns, and structural vulnerabilities may pose challenges to the model's seamless adaptation. Future research could explore the model's robustness and adaptability across diverse contexts, considering the need for region-specific adjustments. Additionally, incorporating local seismic data could enhance the budget allocation model's generalizability and make it a valuable tool for proactive seismic risk management on a global scale. Addressing these aspects will contribute to the model's broader applicability and ensure its effectiveness in aiding decision-making processes in different seismic-prone regions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eJ.K.Kim.: writing\u0026ndash;original draft, development of risk assessment model. S.J.Kim: Review-original draft. H.M.Song.: Development optimization model. M.Y.: organizing, supervision, project administration, review. The authors confirm that this work has not been published before, and its publication has been approved by all co-authors.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThis research was supported by a grant (RS-2023-00238458, Development and Verification of Integrated Management System for High-Risk Disaster Response in Deep Railway Facilities) from the Development of a Disaster Response Complex Training Center for Deep Tunnels (GTX, etc.) Program funded by the Ministry of Land, Infrastructure, and Transport of the Korean government. We greatly appreciate the support.\u003c/p\u003e\u003ch2\u003eData availability\u003c/h2\u003e \u003cp\u003eThe datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. The data sources, data Repository and data access and usage conditions are summarized below.\u003c/p\u003e\n\u003cp\u003eData Sources:\u0026nbsp;We utilized fragility curve data from preceded research, assumed and reproduced the inventory data.\u003c/p\u003e\n\u003cp\u003eData Repository:\u0026nbsp;The data was stored in a local repository within our research laboratory for the duration of the study.\u003c/p\u003e\n\u003cp\u003eData Access and Usage Conditions: Access to the data and the terms of its utilization are outlined as follows. The data\u0026apos;s accessibility for use, whether open to everyone or subject to limitations, is specified.\u003c/p\u003e\n\u003cp\u003eContact Information: For inquiries, please contact [email protected].\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eOgura, M. The niigata chuetsu earthquake-railway response and reconstruction. Japan Railw. Transp. 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SciPy 1.0: fundamental algorithms for scientific computing in Python. \u003cem\u003eNature Methods\u003c/em\u003e, 17(3), 261\u0026ndash;272.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-3904718/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3904718/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this study, we developed a technology to predict the economic damage to facilities in a certain area during an earthquake to facilitate the efficient application of performance-based maintenance and seismic reinforcement. Additionally, we derived an algorithm for establishing a reinforcement plan that can minimize earthquake damage within a limited budget. The fragility function for earthquake damage assessment utilized the results of previous studies and an optimization-based budget allocation algorithm for seismic reinforcement was developed by calculating the estimated damage before and after seismic reinforcement based on target damage ratio(TDR) concept and linear approximation (COBYLA) algorithm. To verify the applicability of the developed model, a fictitious city with a 30km x 30km section was set up, and bridges, embankments, and buildings were placed. In addition, the pre- and post-reinforcement damages were evaluated according to two earthquake scenarios, and the optimization-based budget allocation model that can minimize the damages was applied.\u003c/p\u003e","manuscriptTitle":"Development of optimization-based budget allocation model for seismic reinforcement based on seismic risk assessment","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-07 19:30:23","doi":"10.21203/rs.3.rs-3904718/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"28650384-417b-407b-8aaa-f422412f0195","owner":[],"postedDate":"February 7th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28618130,"name":"Physical sciences/Engineering/Civil engineering"},{"id":28618131,"name":"Earth and environmental sciences/Natural hazards"}],"tags":[],"updatedAt":"2024-05-29T16:14:58+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-07 19:30:23","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3904718","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3904718","identity":"rs-3904718","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}