## 2 Membership-Fusing Model of Sludge-Treatment Process
### Membership-Fusing Model
\begin{array}{l}B(S_{\text{I}})=\frac{e^{\frac{-(x-u_{\text{I}})^{2} }{2\sigma_{1}2}}}{e^{\frac{-(x-u_{\text{I}})^{2}}{2\sigma_{1}2}}+(1+e^{\tau_{1 }(x-c_{1})})^{-1}+(1+e^{-\tau_{2}(x-c_{2})})^{-1}+e^{\frac{-(x-u_{\text{I}})^ {2}}{2\sigma_{2}2}}}\\ B(S_{\text{II}})=\frac{\frac{1+e^{\tau_{1}(x-c_{1})})^{-1}}{e^{\frac{-(x-u_{ \text{I}})^{2}}{2\sigma_{1}2}}+(1+e^{\tau_{1}(x-c_{1})})^{-1}+(1+e^{-\tau_{2}( x-c_{2})})^{-1}+e^{\frac{-(x-u_{\text{I}})^{2}}{2\sigma_{2}2}}}}{B(S_{\text{III}})= \frac{\frac{1+e^{\tau_{2}(x-c_{2})})^{-1}}{e^{\frac{-(x-u_{\text{I}})^{2}}{2 \sigma_{1}2}}+(1+e^{\tau_{1}(x-c_{1})})^{-1}+(1+e^{-\tau_{2}(x-c_{2})})^{-1}+e^ {\frac{-(x-u_{\text{I}})^{2}}{2\sigma_{2}2}}}}\\ B(S_{\text{un}})=\frac{e^{\frac{-(x-u_{\text{I}})^{2}}{2\sigma_{2}2}}}{e^{ \frac{-(x-u_{\text{I}})^{2}}{2\sigma_{1}2}}+(1+e^{\tau_{1}(x-c_{1})})^{-1}+(1+ e^{-\tau_{2}(x-c_{2})})^{-1}+e^{\frac{-(x-u_{\text{I}})^{2}}{2\sigma_{2}2}}}\end{array} \right. \tag{4}\]  
Sample contamination might occur, possibly brought from improper operations of sampling persons, or from samples transfer process, or from the inconsistency of the environmental condition of each sampling time. Instrumental and incidental data errors brought by measurement instruments and human operations are inevitable in data measurement and analysis processes [41]. All these circumstances could result in several imprecise and conflict \(x\) values generated from a group of paralleling sludge samples, so that the paralleling normalized memberships of a certain fuzzy set might deviated extremely. DS evidence theory is introduced to address such problem of data error brought by multi-sources. The four types of normalized memberships obtained from a group of paralleling sludge samples are fused together to generate the integrated membership \(C(S)\) of each fuzzy set in Equation (5):  
\[\left\{ \begin{array}{l}S=S_{1}\cap S_{2}\cap\ldots\cap S_{M}\\ C(S)=\frac{\sum B(S_{1})B(S_{2})\ldots B(S_{M})}{1-K}\ \ for\ S\
eq\varnothing\\ K=\sum B(S_{1})B(S_{2})\ldots B(S_{M})\ \ for\ S=\varnothing\end{array} \right. \tag{5}\]  
where \(M\) stands for the number of the paralleling sludge samples. \(S_{1}\), \(S_{2}\),..., \(S_{M}\) represent any one of the four fuzzy sets, either \(S_{1}\), or \(S_{\text{II}}\), or \(S_{\text{III}}\), or \(S_{\text{un}}\), and are obtained from the \(M\) paralleling sludge samples, respectively. \(B(S_{m})\) (\(m\) = 1, 2,..., \(M\)) are the corresponding normalized memberships. \(K\) stands for the total conflict results generated from the \(M\) paralleling samples. \(S\) obeys the intersection operation in condition (6):\[\left\{ \begin{array}{l}S_{j}\cap S_{j}=S_{j}\ (\ j=\text{I, II, III})\\ S_{\text{un}}\cap S_{\text{un}}=S_{\text{un}}\\ S_{j}\cap S_{\text{un}}=S_{j}\ (\ j=\text{I, II, III})\\ S_{j}\cap S_{k}=\varnothing\ (\ j,k=\text{I, II, III};\ j\
eq k)\end{array} \right. \tag{6}\]  
For \(M\) paralleling \(x\) values of a certain fuzzy set, if several data not belonging to this fuzzy set appeared, the integrated membership \(C(S_{\text{un}})\) would be higher. The other three integrated memberships would be more reliable by considering the existence of the null set when two conflict data from two stages occur, and eliminating the influence of \(S_{\text{un}}\) when intersecting with other three fuzzy sets. The unreliability regarding the judgment on the specific processing stage would be weakened more reasonably compared with just averaging the paralleling data [42]. The maximum among \(C(S_{\text{I}})\), \(C(S_{\text{II}})\), \(C(S_{\text{III}})\), and \(C(S_{\text{un}})\) elucidates the most probable state of the examined paralleling sludge samples, either to be Stage I, or II, or III, or uncertain, based on the maximum principle of fuzzy sets.