Data acquisition preparation
This study establishes a 12-indicator framework spanning exposure, sensitivity, and coping capacity to evaluate social vulnerability in Changchun City, informed by expert insights and comprehensive literature review26. In the process of indicator selection, a study was first conducted on significant historical earthquakes in China to filter key factors that have a considerable impact on social vulnerability in the context of earthquake disasters. Research indicates that population vulnerability is a primary determinant of social vulnerability during seismic events, as factors such as health, education levels, and social security capacity directly influence post-disaster response and recovery capabilities. Building vulnerability directly threatens post-disaster life safety and infrastructure functionality, thereby increasing societal risks. Moreover, it exacerbates social inequality, particularly among low-income groups, as they tend to reside in buildings with weaker disaster resilience, increasing the risk of life and property loss, and further deepening social vulnerability27,28. The exposure of economic assets and infrastructure further intensifies the impact of earthquakes29, as it hampers recovery efforts, disrupts essential services, and increases economic losses. Moreover, enhanced emergency response capacity can significantly mitigate the negative consequences of earthquakes. Based on this analysis, this paper selects the following 12 indicators: per capita GDP, average household size, fixed asset investment, road network density, the proportion of the population under 14 years and over 60 years of age, the proportion of unprotected buildings, the proportion of buildings with cracks, proportion of buildings in use for over 30 years, the proportion of illiterate population, the number of medical beds, the number of shelters, and the number of personnel engaged in emergency response. These indicators comprehensively consider characteristics related to population, buildings, economy, infrastructure and response capacity. Table 1 summarizes the data sources of the selected indicators.
Weight calculation
Subjective weight
The only reference comparison judgment method, namely the G2 method43,44, is a subjective weighting method. Its basic idea is as follows: first, experts select an indicator from the evaluation indicator set \(\{{x}_{1},{x}_{2},\cdots ,{x}_{m}\}\) that they consider the least important, denoted as \({x}_{jm}\). At this point, the evaluation indicator set can be redefined as \(\{{x}_{1},{x}_{2},\cdots ,{x}_{jm}\}\). Then, the least important indicator \({x}_{jm}\) is used as the sole reference, and experts assign relative importance ratios to the other indicators \({x}_{km}(k=\text{1,2},\dots ,\text{m}-1)\) compared to \({x}_{jm}\). Finally, the weights of each evaluation indicator are calculated based on the importance ratios. Let \({r}_{km}(k=\text{1,2},\dots ,\text{m}-1)\) represent the ratio of the importance of \({x}_{jk}\) to \({x}_{jm}\). Suppose experts assign a specific value to \({r}_{k}\) according to a certain criterion: \({r}_{km}={a}_{k},k=\text{1,2},\dots ,\text{m}-1\). At this point, the weight of the evaluation indicator \({x}_{jm}\):
$$w_{k} = \frac{{a_{k} }}{{\sum\limits_{i = 1}^{m} {a_{i} } }},k = 1,2, \ldots ,m$$
(1)
In this study, we invited twenty experts in the field of earthquake risk, independent from the research team, to participate in the assessment. First, the experts selected the least important indicator from the list and assigned it a value of 1. Subsequently, the experts rated the other indicators based on their relative importance compared to this least important indicator, with a scoring range from 1 to 5. Finally, we consolidated the scores from twenty experts and calculated the mean as the subjective weight.
Objective weight
The CRITIC method is a technique employed to calculate the objective weights of evaluation criteria45,46,47. It assesses the importance of multiple evaluation criteria on decision outcomes by calculating weights based on the degree of interrelation between criteria, aiming to more accurately determine their influence in the decision-making process. The method expresses the comparative strength between evaluation criteria using standard deviation, where a larger standard deviation typically indicates greater dispersion and, consequently, a higher amount of information, leading to larger weights for evaluation criteria. Conflict is represented in this method using correlation coefficients; the greater the correlation, the smaller the difference in information expressed between criteria, resulting in smaller weights. The specific operations are detailed as follows:
Firstly, the formula for standardizing the data is as follows:
$$x_{ij}{\prime} = \left\{ {\begin{array}{*{20}l} {\frac{{x_{ij} – \min x_{ij} }}{{\max x_{ij} – \min x_{ij} }},} \hfill & {for \, positive \, indicators} \hfill \\ {\frac{{\max x_{ij} – x_{ij} }}{{\max x_{ij} – \min x_{ij} }},} \hfill & {for \, negative \, indicators} \hfill \\ \end{array} } \right.$$
(2)
Establishing an evaluation matrix for n samples with m indicators, the matrix is as follows:
$$\left( {\begin{array}{*{20}c} {x_{11} } & {x_{12} } & \cdots & {x_{1n} } \\ {x_{21} } & {x_{22} } & \cdots & {x_{2n} } \\ \vdots & \vdots & \vdots & \vdots \\ {x_{m1} } & {x_{m2} } & \cdots & {x_{mn} } \\ \end{array} } \right)$$
(3)
The conflict of indicators is represented using correlation coefficients, with the formula as follows:
$$R_{i} = \sum\limits_{i = 1}^{n} {1 – r_{ij} }$$
(4)
where \({R}_{j}\) represents the conflictiveness of indicator, and \({r}_{ij}\) represents the correlation coefficient between evaluation indicators \(i\) and \(j\).
The formula for calculating information entropy is as follows:
$$C_{j} = \sigma_{j} \times R_{j}$$
(5)
where \({C}_{j}\) represents the information content contained in the \(j\)th indicator, and \({\sigma }_{j}\) represents the standard deviation of the \(j\)th indicator.
The formula for calculating objective weights is as follows:
$$W_{1j} = \frac{{C_{j} }}{{\sum\limits_{j = 1}^{n} {C_{j} } }}$$
(6)
where \({W}_{j}\) represents the objective weight of the \(j\)th indicator.
The entropy weighting method is a technique used in multi-criteria decision-making where the weights of different indicators are determined based on their entropy48. One of the advantages of the entropy weighting method is its ability to fully consider the correlation and importance of various indicators, thereby enhancing the accuracy and reliability of the evaluation results. The core idea of this method is that the higher the entropy, the greater the uncertainty of the indicator, hence it should be assigned a lower weight, and vice versa. The specific steps are as follows:
The evaluation comprises \(m\) samples and \(n\) indicators, with the measurement value of the \(j\)th sample for the \(i\)th indicator recorded as \({r}_{ij}\)
The first step involves standardizing the measurement values. The calculation method is as follows:
$$p_{ij} = \frac{{r_{ij} }}{{\sum\limits_{j = 1}^{m} {r_{ij} } }}$$
(7)
To calculate the entropy value \(e_{i}\) for the \(i\) th indicator:
$$e_{j} = – k\sum\limits_{i = 1}^{m} {p_{ij} } \ln p_{ij}$$
(8)
The final weights are determined as follows:
$$w_{2j} = \frac{{1 – e_{j} }}{{\sum\limits_{j = 1}^{n} {d_{j} } }}$$
(9)
The CRITIC method cannot measure the degree of indicator discreteness, while the entropy weight method determines weights based on the discreteness of indicators. Therefore, combining the CRITIC weight method and the entropy weight method can more objectively reflect the weights of indicators. The calculation of the final weight \(w_{j}\) is as follows:
$$w_{j} = \alpha w_{1j} + (1 – \alpha )w_{2j}$$
(10)
Game theory combined weighting
The limitations of subjective and objective weighting methods are evident. Subjective weighting primarily relies on experts’ experience, age, and professional knowledge, while objective weighting imposes strict requirements on data formats, often affecting evaluation results due to data format issues. To address these limitations, we propose a game theory combined weighting approach49,50. This method combines weights obtained from different methods linearly to seek the most accurate indicator weights, thus avoiding the shortcomings of using each method individually and maximizing the accuracy of the estimation process.
Utilizing properties of matrix differentiation, the system of linear equations can be equivalently transformed into the optimal first-order derivative conditions, as illustrated below:
$$\left( {\begin{array}{*{20}c} {\omega_{1} \omega_{1}^{T} } & {\omega_{1} \omega_{2}^{T} } \\ {\omega_{2} \omega_{1}^{T} } & {\omega_{2} \omega_{2}^{T} } \\ \end{array} } \right)\left[ {\begin{array}{*{20}c} {\alpha_{1} } \\ {\alpha_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\omega_{1} \omega_{1}^{T} } \\ {\omega_{2} \omega_{2}^{T} } \\ \end{array} } \right]$$
(11)
After obtaining the optimal linear combination coefficients from the formula and normalizing them, the comprehensive weights of the game theory combined weighting are finally obtained as follows:
$$W = \alpha_{1}^{*} \omega_{1}^{T} + \alpha_{2}^{*} \omega_{2}^{T} ,\alpha_{1}^{*} = \frac{{\alpha_{1} }}{{\alpha_{1} + \alpha_{2} }};\alpha_{2}^{*} = \frac{{\alpha_{2} }}{{\alpha_{1} + \alpha_{2} }}$$
(12)
TOPSIS model
The TOPSIS model is a multi-attribute decision-making method commonly used for selecting optimal solutions or evaluating the merits of alternatives51,52. For each attribute, there exists an ideal best solution and an ideal worst solution. The ideal best solution typically maximizes the value of the attribute, while the ideal worst solution typically minimizes it. Similarity between each alternative and the ideal best and worst solutions is computed, often using Euclidean distance or other similarity metrics. By comparing the similarity of each alternative to the ideal best solution with its similarity to the ideal worst solution, a comprehensive score is calculated for each alternative.
The evaluation method is as follows:
Establishing the evaluation matrix, assuming there are \(m\) indicators for \(n\) samples, the evaluation matrix is as shown in Eq. (3). After standardization, the normalized matrix is obtained \(p = \left[ {p_{ij} } \right]_{m \times n}\).
After considering the weights of each evaluation criterion, the weighted normalized matrix is as follows:
$$V = P \times W = \left[ {v_{ij} } \right]_{m \times n}$$
(13)
\(W\) is the calculated comprehensive weight.
Determine the positive and negative ideal solutions:
$$\begin{aligned} V^{ + } = & (V_{1}^{ + } ,V_{2}^{ + } , \cdots V_{n}^{ + } ) \\ = & \left( {\max \left\{ {v_{11} ,v_{12} ,v_{1n} } \right\},\max \left\{ {v_{12} ,v_{22} ,v_{m2} } \right\},\max \left\{ {v_{1n} ,v_{2n} ,v_{mn} } \right\}} \right) \\ \end{aligned}$$
(14)
$$\begin{aligned} V^{ – } = & (V_{1}^{ – } ,V_{2}^{ – } , \cdots V_{n}^{ – } ) \\ = & \left( {\min \left\{ {v_{11} ,v_{12} ,v_{1n} } \right\},\min \left\{ {v_{12} ,v_{22} ,v_{m2} } \right\},\min \left\{ {v_{1n} ,v_{2n} ,v_{mn} } \right\}} \right) \\ \end{aligned}$$
(15)
Calculate the distance of each sample from the positive and negative ideal solutions:
$$D_{i}^{ – } = \sqrt {\sum\limits_{j = 1}^{n} {v_{ij} – v_{j}^{ – } } } \, i = 1,2, \cdots m$$
(16)
$$D_{i}^{ + } = \sqrt {\sum\limits_{j = 1}^{n} {v_{j}^{ + } – v_{ij} } } \, i = 1,2, \cdots m$$
(17)
Calculate the proximity of each evaluated object to the optimal solution and perform ranking:
$$C_{i} = \frac{{D_{i}^{ – } }}{{(D_{i}^{ + } + D_{i}^{ – } )i}} = 1,2, \cdots m$$
(18)
Social vulnerability index
The calculated \(SoVI_{i}\) represents the calculation formula for the \(i\) th county’s \(SVSD_{i}\) as follows:
$$SoVI_{i} = \sum\limits_{i = 1}^{n} {x_{ij}{\prime} \times w_{j} } = SoVI_{A} + SoVI_{B} + SoVI_{C}$$
(19)
where \(x_{ij}{\prime}\) represents the standardized indicator, \(SoVI_{A}\), \(SoVI_{B}\), and \(SoVI_{C}\) respectively represent exposure index, sensitivity index, and coping capacity index.