# Evolution of China's Building Energy Service Industry Based on Synergetic Theory
## 5 Empirical Results  
Several steps exist for performing the DEMATEL-cluster method [33; 45]. **Step 1:** Generate the initial direct relation matrix A. In this research, the weighted average method was adopted to record the influence degree. The effect of factor i on factor j is denoted as aij. If factor i does not directly influence factor j, then the corresponding score in the matrix is recorded as 0. Similarly, 1, 2, and 3 represent weak influence, moderate influence, and strong influence, respectively. Then, the direct relation matrix can be expressed in Table 3:  
**Step 2:** Normalize the overall direct relation matrix B using Equation (1). \[\text{B}=\frac{A}{max\sum_{j=1}^{n}a_{ij}},\text{ where }0\leq\text{a}\leq\text{n} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Code** & **Meaning** & **References** \\ \hline P1 & Estimation and control of risk & [29] \\ P2 & Management of building energy service procedures & [29; 38] \\ P3 & The rasising of funds from banks and social capital & [18; 39; 40] \\ P4 & Audit of the amount of energy savings before and after the project’s implementation & [18; 37; 39] \\ P5 & Interests and responsibility among stakeholders & [17; 37; 40] \\ P6 & Allocatation of reasonable tasks and risk to involved stakeholders & [18; 37; 38] \\ P7 & Timely and efficient communication among stakeholders & [18; 41] \\ P8 & Proportion of companies above designated size & [41] \\ P9 & Sufficient number of professionally trained staff & [39] \\ P10 & Financial support from all financial channels & [39; 41] \\ P11 & Effort towards technological innovation & [37; 39; 41] \\ P12 & Payback of the project & [38] \\ P13 & The ratio of actual output and maximum output & [42] \\ P14 & The level of income compared to other industries & [43] \\ P15 & Credit system constraining corporate behavior & [18; 37; 39; 43] \\ P16 & Official system for measuring energy savings & [44] \\ P17 & Standard contract used as a template for companies & [44] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Meaning of order parameters in BESI system. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline  & **P1** & **P2** & **P3** & **P4** & **P5** & **P6** & **P7** & **P8** & **P9** & **P10** & **P11** & **P12** & **P13** & **P14** & **P15** & **P16** & **P17** \\ \hline P1 & 0 & 3 & 3 & 1 & 1 & 3 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ P2 & 1 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 \\ P3 & 0 & 2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1  
**Step 3:** Develop the total relation matrix, using Equation (2). \[\text{C}=\text{B}(\text{I}-\text{B})^{-1} \tag{2}\]  
I = identity matrix. **Step 4:** Compute the prominence (D + E) and net cause/effect (D \(-\) E) values for each barrier using Equations (3) and (4)  
\[\text{D}=\left[\sum_{l=1}^{n}\text{C}_{ij}\right] \tag{3}\]  
\[\text{E}=\left[\sum_{i=1}^{n}\text{C}_{ij}\right] \tag{4}\]  
\[\omega^{\prime}=\sqrt{\left[\left(D_{i}+R_{i}\right)^{2}+\left(D_{i}-R_{i} \right)^{2}\right]} \tag{5}\]  
\[\omega_{i}=\omega_{i}^{\prime}/\sum_{i=1}^{n}\omega_{i} \tag{6}\]  
**Step 5:** Derive the inner dependence matrix by eliminating the entries less than the threshold value (\(\alpha\)) in the total relation matrix. The threshold value is equal to the average of all entries in the total relation matrix, as given by Equation (7). \[\alpha=\frac{\sum_{j=1}^{n}\sum_{i=1}^{n}c_{ij}}{\text{n}^{2}}=0.11 \tag{7}\]  
**Step 6:** Depict the structural relationship between factors by constructing a prominence-causal relationship diagram (see Figure 2). **Step 7:** Calculate the weight of each order parameter. All the results can be found in Tables 4-6. According to the results presented in Table 5, the DEMATEL prominence-causal relationship of all the order parameters could be drawn and is shown in Figure 7, which provides a convenient representation of the factors. The upper half of the diagram comprises causal factors, and the lower half comprises effect factors.