# Suitability Evaluation of Urban Underground Space Development: A Case Study of Qingdao City
## 3 Research Method
### Weight Assignment of Evaluation Indicators
Considering the tendency of the suitability evaluation of urban underground space development to generate neutral effects, the variable weight model constructed in this study dynamically adjusts the weights of various decision factors by introducing a penalty mechanism. This is achieved by constructing the variable weight model based on the idea of penalizing factors with poor performance. The variable weight model \(W(X)=(w_{1}(x),w_{2}(x),\ldots,w_{n}(x))\) indicates that \(W(X)\) possesses \(n\) dimensions, \(w_{i}(i=1,2,\ldots,n)\), \(W_{i}\): \([0,1]^{n}\rightarrow[0,1]\), \((x_{1},x_{2},\ldots,x_{n})|\to w_{i}\)\((x_{1},x_{2},\ldots,x_{n})\). The main idea of the Variable-Weight Analytic Hierarchy Process (VWAHP) is to introduce a variable weight model into the traditional Analytic Hierarchy Process (AHP). Initially, the AHP is used to determine the initial weights \(W^{0}\) of evaluation indicators. Then, based on the specific research conditions, a local state variable weight function \(Y(X)\) is constructed. The normalized product of these two vectors yields the variable weight vector \(W(X)\). This approach adjusts the weights of evaluation factors according to different evaluation areas. \[W(X)=\frac{\big{(}w_{1}^{0}\cdot Y_{1}(X),w_{2}^{0}\cdot Y_{2}(X),\ldots,w_{n} ^{0}\cdot Y_{n}(X)\big{)}}{\sum_{i=1}^{n}\big{(}w_{i}^{0}\cdot Y_{i}(X)\big{)} }=\frac{W^{0}\cdot Y_{i}(X)}{\sum_{i=1}^{n}\big{(}w_{i}^{0}\cdot Y_{i}(X)\big{)}} \tag{1}\]  
To ensure more objective and rational weight allocation, this study systematically investigated the principle of local variable weighting based on previous experiences. Combining decision-makers' preferences, an improved local indicator state variable weight function \(Y(X)\) was proposed. The core idea behind constructing the variable weight function in this study is to increase the weight of indicators with lower suitability evaluation grades while keeping the weights of indicators with higher suitability evaluation grades unchanged. The goal is to amplify the impact of adverse factors on the evaluation area, reduce the scores in areas with lower-grade indicators, and thereby establish a penalty-based variable weight model that exceeds the incentive, making the evaluation results more scientifically reasonable. The axis \(X\) of the constructed state variable weight function represents the magnitude of a certain indicator value within the evaluation unit, ranging from 0 to 1. In the first interval \([0,a]\), due to the very low rating of the indicator, it requires extremely severe penalties, and the growth rate of the state variable weight function increases as the indicator value decreases. In the interval \([a,b]\), the penalty is significantly reduced compared to the previous interval, and in the interval \([b,c]\), the rate of change of the state variable weight function decreases as the indicator value increases. When the indicator value is greater than \(c\), the value of the state variable weight function remains unchanged. The constructed function graph is shown in Figure 5.  
Among them, \(a\), \(b\), and \(c\) represent the indicator values at the node, while \(d\), \(e\), and \(f\) represent the function values of the state variable weight function. Based on the characteristics of the state variable weight function mentioned above, the specific functions obtained are as follows:  
\[Y(X)= \begin{cases}\frac{f-c}{b-a}a\ln\frac{a}{x_{i}}+c,&x_{i}e(0,a)\\ \frac{e-f}{b-a}x_{i}+\frac{f_{i}b-c-a}{b-a},&x_{i}e(a,b)\\ \frac{c-d}{(c-d)^{2}}(c-x_{i})^{2}+d,&x_{i}e(b,c)\\ d,&x_{i}e(c,1)\end{cases}  \tag{2}\]  
The determination of parameter values for the variable weight status function is a critical factor in assessing the scientific validity of weight calculation results. The segmentation of interval ranges is initially based on the grading score system established in the previous section for the degree of influence of factors. Specifically, we set \(a=0.25\), \(b=0.5\), \(c=0.75\), dividing the indicator values into four intervals.