## 3 Capacity Optimization Model
### Establishment of Multi-Objective Optimization Model  
This study develops a multi-objective function \(F(X)\) based on the combination of economy, energy saving and environmental protection, which gives a certain weight to each evaluation index to reflect its contribution to the whole [28].  
\[F(X)=min[\omega_{1}\cdot ATC+\omega_{2}\cdot TEC+\omega_{3}\cdot CDE] \tag{15}\]In this paper, the AHP algorithm is adopted to determine the value of the weight coefficient for the allocation of capacity of the CCHP system. The procedure is described as follows.  
1. Establish the hierarchical analysis structure of capacity optimization. According to the composition of the multi-objective function, the target is the capacity optimization, where the criterion layer includes economy, energy saving and environmental protection, as shown in Figure 2.
2. Construct judgment matrix. \(A(a_{ij})_{n\times n}\) is the judgment matrix. \(a_{ij}\) is the comparison weight of relative importance obtained by pairwise comparison of standard layer indicators. Twenty experts were consulted to complete the questionnaire [29]. Experts rate the relative importance (between two factors) of the criterion layer. The scale of relative importance is 1-9 [30]. Calculate the arithmetic mean of the comparison weight column of relative importance between each two factors, and then round to get \(a_{ij}\). Finally, \(A(a_{ij})_{n\times n}\) is obtained. \[A=\left[\begin{array}{ccc}1&3&5\\ 1/3&1&3\\ 1/5&1/3&1\end{array}\right]\] (16)
3. Weight sorting. The weight ordering refers to the ordering of the importance of each element in the standard layer for the target layer. First, calculate the element product of each row of the judgment matrix, and then find the \(n\)-th square root of the element product of each row to obtain the eigenvector matrix. At last, normalize the sum of the eigenvectors to be 1, and then obtain the weight vector \(W\). The specific steps are expressed as follows. \[\left\{\begin{array}{c}M_{i}=\prod_{j=1}^{n}a_{ij}(i=1,2,\ldots,n)\\ \overline{W_{i}}=\sqrt[n]{M_{i}}\\ W_{i}=\overline{W_{i}}/\sum_{j=1}^{n}\overline{W_{j}}\end{array}\right.\] (17) where \(M_{i}\) is the score of elements in each row of the judgment matrix; \(\overline{W_{i}}\) is the \(n\)-th root of \(M_{i}\); \(W_{i}\) is the \(i\)-th element of the matrix \(W\). The calculated results are: \(W\) = [0.637, 0.258, 0.105]T.
4. Conduct theconsistency test for the judgment matrix. Consistency index (\(CI\)), Randomconsistency index (\(RI\)) and Consistency ratio (\(CR\)) are calculated as follows. \[\left\{\begin{array}{c}\lambda_{max}=\frac{1}{n}\sum\limits_{i=1}^{n}\frac{ (AW)_{i}}{W_{i}}\\ CI=(\lambda_{max}-n)/(n-1)\end{array}\right.\] (18) where \(\lambda_{max}\) is the maximum characteristic root of matrix \(A\). \((AW)_{i}\) is the \(i\)-th element of the matrix \(AW\). Substitute the \(W\) matrix obtained in step (3) into Equation (18): \(\lambda_{max}\) = 3.038, \(CI\) = 0.019. According to the \(RI\) value table with a sample size of 100-500 given by \([PERSON]\), it can be obtained: \(RI\) = 0.58. \(CR\) = \(CI\)/\(RI\) = 0.033 \(<\) 0.10 [31]. This indicates that the judgment matrix has a satisfactory consistency, and its normalized feature vector can be used as the weight vector. The final result from the above steps is obtained as \(\omega_{1}=0.637,\ \omega_{2}=0.258,\ \omega_{3}=0.105\).  
Figure 2: The structure diagram of AHP.