## 2 Membership-Fusing Model of Sludge-Treatment Process
### Membership-Fusing Model  
Based on the three-stage sludge-treatment process and the quantity ratio \(x\), the membership-fusing model is established by combing fuzzy sets and DS evidence theory. First, four fuzzy sets \(S_{\text{I}}\), \(S_{\text{II}}\), \(S_{\text{III}}\), and \(S_{\text{un}}\) are put forward to elucidate the affiliation of the processing stage of a sludge sample being Stage I, Stage II, Stage III, and stage uncertain. \(S_{\text{I}}\), \(S_{\text{II}}\), and \(S_{\text{III}}\) contain \(x\) values belonging to Stage I, II, and III, respectively. \(S_{\text{un}}\) clarifies the indetermination of the processing stage of a sludge sample, considering of both the vague boundary of the methanogen community structure between two processing stages, and the absence of \(x\) values around 10. The data error generated among paralleling samples caused by sample contamination and incidental and instrumental errors occurring in data measurement and analysis processes would lead to the appearance of \(x\) values outside the fuzzy sets representing the three stages. The degree of the probability of a sludge sample from Stage I, II, III, and uncertain is described by the corresponding basic membership function of each fuzzy set, \(A(S_{\text{I}})\), \(A(S_{\text{II}})\), \(A(S_{\text{III}})\), and \(A(S_{\text{un}})\), respectively, established in Equation (2), referring to the traditional normally distributed membership function:  
\[\left\{ \begin{array}{l}A(S_{\text{I}})=e^{\frac{-(x-x_{\text{I}})^{2}}{2 \alpha_{1}}^{2}}\\ A(S_{\text{II}})=(1+e^{\tau_{1}(x-c_{1})})^{-1}\\ A(S_{\text{III}})=(1+e^{-\tau_{2}(x-c_{2})})^{-1}\\ A(S_{\text{un}})=e^{\frac{-(x-x_{\text{II}})^{2}}{2\alpha_{2}}^{2}}\end{array} \right. \tag{2}\]  
where parameters \(\tau_{1}\), \(c_{1}\) and \(\tau_{2}\), \(c_{2}\) reflect the trends of \(A(S_{\text{II}})\) and \(A(S_{\text{III}})\), respectively. Parameters \(\tau_{1}\) and \(\tau_{2}\) indicate the changing gradients of the uncertainties of the two basic membership functions, while \(c_{1}\) and \(c_{2}\) clarify the limits of the \(x\) value where the two basic membership functions become uncertainly. Values of \(c_{1}\) and \(c_{2}\) are equal to the values of \(x\) when \(A(S_{\text{II}})\) and \(A(S_{\text{III}})\) equal 0.5. Parameters \(u_{1}\) and \(u_{2}\) are the average values of \(S_{\text{I}}\) and \(S_{\text{un}}\) and describe the most probable \(x\) values belonging to Stage I and stage uncertain. Parameters \(\sigma_{1}\) and \(\sigma_{2}\) are the standard deviation of \(S_{\text{I}}\) and \(S_{\text{un}}\) and reveal the degree of the deviation of \(x\) values away from the average \(x\) value. Parameters mentioned above should satisfy condition (3) and are determined by statistics method elucidated in Section 3.3:\[\left\{ \begin{array}{l}u_{1}=x_{\text{taver}}\\ \sigma_{1}=x_{\text{lsd}}\\ c_{1}=x_{\text{lImax}}\\ c_{2}=x_{\text{IImin}}\\ (1+e^{\tau_{1}(x_{\text{lImin}}-c_{1})})^{-1}>0.999\\ (1+e^{-\tau_{2}(x_{\text{IImax}}-c_{2})})^{-1}>0.999\\ u_{2}-2\sigma_{2}=x_{\text{taver}}\\ u_{2}+2\sigma_{2}=x_{\text{IIlayer}}\end{array} \right. \tag{3}\]  
where \(x_{\text{javer}}\), \(x_{\text{jsd}}\), \(x_{\text{jmax}}\), and \(x_{\text{jmin}}\) are the average value, standard deviation, maximum and minimum of \(x\) of \(Sj\) (\(j\) = I, II, III). \(2\sigma\) principle [40] means the probability of the value of the normally distributed function is 0.95443 when the independent variable distributes in (\(u\)-\(2\sigma\), \(u\)+\(2\sigma\)). In other words, \(x\) outside (\(u\)-\(2\sigma\), \(u\)+\(2\sigma\)) is most unlikely to exist, which is applied to determine the values of \(u_{2}\) and \(\sigma_{2}\). Considering of the vagueness of the critical state of methanogen community structure between two sludge-processing stages, the average value of \(x\) is chosen to be the boundary value. Then the basic membership functions are normalized in Equation (4):  
\[\left\{