## 4 Modified Primary Compression Index Model  
### Model Creation  
The assumptions of the model in this article are as follows: (1) the Modified Primary Compression Index (MPCI) of MSW is mainly influenced by four factors: dry unit weight (\(\gamma_{d}\)), organic matter content (\(OW\)), moisture content on a wet basis (\(\omega_{w}\)), and temperature (\(T\)); (2) the dry unit weight, organic matter content, moisture content on a wet basis, and temperature are independent and do not affect each other; (3) and the product of exponential operation on the four parameters above is linearly related to the Modified Primary Compression Index.  
Based on the assumptions above, the dimensionless factor called the waste compressibility index (\(WCI\)), which is used to represent the comprehensive effect of four parameters (dry unit weight, organic matter content, moisture content on a wet basis, and temperature) above, is adopted. The expression of the model is as follows:  
\[\left\{\begin{array}{l}C_{c}^{\prime}=a\cdot WCI+b\\ WCI=\omega_{w}^{m}\cdot\left(\frac{\gamma_{d}}{\gamma_{w}}\right)^{n}\cdot OW^ {k}\cdot\left(\frac{T}{T_{0}}\right)^{i}\end{array}\right. \tag{2}\]  
where \(\omega_{w}\) is moisture content on a wet basis (%); \(\gamma_{w}\) is the weight of water, which is taken as 10 kN/m\({}^{3}\); \(\gamma_{d}\) is dry unit weight (kN/m\({}^{3}\)); \(OW\) is organic matter content (%); \(T\) is the initial temperature of MSW, which is taken as 20 \({}^{\circ}\)C; and \(a\), \(b\), \(m\), \(n\), \(k\), and \(l\) are the parameters of the model.  
Figure 9: Change rule of MPCI for moisture content on a wet basis at different temperatures.  
Regression fitting analysis was conducted based on the summary of the literature presented in Table 4 and the results of the consolidation test considering the temperature effects in this article. The parameters \(a\), \(b\), \(m\), \(n\), \(k\), and \(l\) in the model were determined to be \(0.133\), \(-0.107\), \(0.951\), \(0.664\), \(0.227\), and \(0.078\), respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Reference** & **Age** & \(\mathbf{t}\) (\(\mathbf{\mathrm{{}^{\circ}}}\)**C)** & **Size (cm)** & \(\mathbf{C}_{\mathbf{c}}^{\prime}\) & \(\mathbf{\omega}_{\mathbf{w}}\) & \(\mathbf{\gamma}_{\mathbf{d}}\) & \(\mathbf{OW}\) \\ \hline \multirow{4}{*}{Xu [24]} & fresh & 20 & \(\varnothing\)500 & 0.381 & 70.2 & 2.44 & 84.45 \\  & & & \(\varnothing\)500 & 0.323 & 62.31 & 2.87 & 83.1 \\ \cline{2-8}  & \multirow{4}{*}{fresh} & & \multirow{4}{*}{20} & \(\varnothing\)30 & 0.314 & 68.8 & 3.02 & 83.15 \\  & & & & 0.229 & 57.39 & 3.85 & 57.45 \\  & & & & 0.175 & 53.1 & 4.33 & 18.5 \\ \hline \multirow{4}{*}{Liu [25]} & \multirow{4}{*}{fresh} & & \multirow{4}{*}{20} & \(\varnothing\)6.18 & 0.287 & 37.5 & 7.61 & 52.5 \\  & & & & 0.194 & 37.5 & 7.61 & 52.5 \\  & & & & 0.178 & 37.5 & 7.61 & 26.25 \\  & & & & 0.164 & 37.5 & 7.61 & 26.25 \\ \hline \multirow{4}{*}{[PERSON] [13]} & \multirow{4}{*}{20} & \multirow{4}{*}{20} & \(\varnothing\)6.4 & 0.091 & 23.84 & 5.6 & \multirow{4}{*}{32.6} \\ \cline{3-4}  & & & \(\varnothing\)10 & 0.187 & 23.84 & 5.71 & \\ \cline{3-4}  & & & \(\varnothing\)30.5 & 0.162 & 23.84 & 5.61 & \\ \cline{3-4}  & &  
The model of the MPCI can be expressed as follows:  
\[C^{\prime}_{c}\ =\ 0.227\cdot\omega_{w}^{0.133}.\left(\frac{\gamma_{d}}{\gamma_{w}} \right)^{-0.107}\cdot OW^{0.951}.\left(\frac{T}{T_{0}}\right)^{0.664}+0.078 \tag{3}\]  
The fitting results of the MPCI model of the test and the literature are shown in Figure 10. In the figure, the two dashed lines have the same regression coefficients for the regression line and correspond to \(\pm\)2 SD (\(\sigma\)) from the regression line. It can be seen that for the given WCI, \(C^{\prime}_{c}\) can be calculated to be \(\pm\)0.0719. The regression of \(C^{\prime}_{c}\) and the WCI include all the data, and the fitting degree of the model (\(R^{2}\)) is 0.6554 in the figure. The results of the test are distributed near the curve which reflect that this model has a good fitting effect.