# Suitability Evaluation of Urban Underground Space Development: A Case Study of Qingdao City
## 4 Comprehensive Evaluation Model
### Fuzzy Comprehensive Evaluation Model  
Fuzzy comprehensive evaluation is a comprehensive assessment method grounded in fuzzy mathematics that is primarily used to address decision-making problems with uncertainty and strong fuzziness in evaluation factors. This method can effectively handle evaluation issues that combine qualitative and quantitative aspects, particularly suitable for complex system evaluations that are challenging to describe using precise mathematical models. Its core principle involves synthesizing fuzzy relations to combine multiple factors or evaluation criteria influencing the evaluation object. Through fuzzy transformation, these factors are integrated into a fuzzy set, reflecting the overall evaluation level of the evaluation object. The specific construction process is as follows:  
1. Determine the factor set: Based on the constructed evaluation system, the secondary evaluation indicators in the system are taken as the factor set. U = {Terrain and landforms, Geotechnical engineering geology, Hydrogeology, Adverse geological effects and disasters, Existing construction conditions, Urban planning requirements}. 2. The evaluation set is established according to the specific circumstances of each criterion or indicator, and in this experiment, the suitability evaluation is divided into four levels. Therefore, the constructed evaluation set is V = {Highly Suitable for Development, Moderately Suitable for Development, Marginally Suitable for Development, Not Suitable for Development}. 3. Weight Allocation: Based on the importance of each criterion, assign a weight to each criterion or indicator, forming a weight set \(W=\{w_{1},w_{2},\ldots,w_{n}\}\), where \(w_{i}(i=1,2,\ldots,n)\) represents the weight value of the \(i\)-th evaluation criterion. The specific weight values have been obtained in the previous chapter. 4. Membership Function Construction: Membership functions are one of the core tools in fuzzy comprehensive evaluation, used to convert qualitative evaluations into quantitative values. The construction process of membership functions involves transforming actual evaluation data into membership degrees ranging from 0 to 1. In this study, trapezoidal membership functions are selected for calculating membership degrees. The specific functional expression is as follows: \[y_{1}= \begin{cases}1&x\leq 0.2\\ 3-10x&0.2<x\leq 0.3\\ 0&x>0.3\end{cases} \] (4) \[y_{2}= \begin{cases}0&x\leq 0.2\\ 10x-2&0.2<x\leq 0.3\\ 1&0.3<x\leq 0.5\\ 6-10x&0.5<x\leq 0.6\\ 0&x>0.6\end{cases} \] (5)\[y_{3}= \begin{cases}0&x\leq 0.5\\ 10x-5&0.5<x\leq 0.6\\ 1&0.6<x\leq 0.8\\ 9-10x&0.8<x\leq 0.9\\ 0&x>0.9\end{cases}  \tag{6}\]  
\[y_{4}= \begin{cases}0&x\leq 0.8\\ 10x-8&0.8<x\leq 0.9\\ 1&x>0.9\end{cases}  \tag{7}\]  
In the equation, \(x\) represents the standardized value of the influencing factor, and \(y_{1},y_{2},y_{3},\) and \(y_{4}\) represent the four levels of suitability development evaluation results: \"Not Suitable for Development\", \"Marginally Suitable for Development\", \"Moderately Suitable for Development\", and \"Highly Suitable for Development\", respectively. Based on this membership degree function, the membership degree set for each evaluation indicator can be calculated. 5. The fuzzy matrix R is a matrix representing the fuzzy relationships between the factor set \(U_{i}(i=1,2,\ldots,n)\) and the evaluation set \(V_{j}(j=1,2,\ldots,m)\). The fuzzy matrix \(R\) is an \(n\times m\) matrix, where n represents the number of evaluation indicators and m represents the number of evaluation levels. Taking shallow underground space evaluation as an example, there are a total of 13 evaluation indicators, so \(n=13\), and there are four evaluation levels, so \(m=4\). The element \(r_{ij}\) in the fuzzy matrix R (\(n\times m\)) represents the degree of membership of the \(i\)-th evaluation indicator to the \(j\)-th evaluation level. \[R= \begin{bmatrix}r_{11}&r_{12}&\cdots&r_{1m}\\ r_{21}&r_{22}&\cdots&r_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\cdots&r_{nm}\end{bmatrix}  \tag{8}\]
6.