## 3 Methodology
### Some Fuzzy Collaborative Forecasting Models for the Unit Cost Forecasting
After forecasting the unit cost, the performances of the two objects are evaluated as  
\(Prec_{\text{AR}}\)(CL1(3, 0.3)) = 0.56  
\(Prec_{\text{AR}}\)(CL1(2, 0.6)) = 1.12  
\(Accu_{\text{MAE}}\)(CL1(3, 0.3)) = 0.16  
\(Accu_{\text{MAE}}\)(CL1(2, 0.6)) = 0.23  
\(Accu_{\text{MAPE}}\)(CL1(3, 0.3)) = 10%  
\(Accu_{\text{MAPE}}\)(CL1(2, 0.6))) = 14%  
\(Accu_{\text{RMSE}}\)(CL1(3, 0.3)) = 0.19  
\(Accu_{\text{RMSE}}\)(CL1(2, 0.6)) = 0.30  
Then, the forecasting performances are compared with those of the linear methods WT(0.3) and WT(0.6). The results are summarized in \[Accu_{\text{MAE}}(\text{[PERSON]}(0.3))=0.16\]1  
Obviously, the use of a nonlinear objective function may change the optimal solution. The quality of collaboration is evaluated as follows: \begin{table}
\begin{tabular}{c c c c c} \hline  & \(Prec_{\text{AR}}\) & \(Accu_{\text{MAE}}\) & \(Accu_{\text{MAPE}}\) & \(Accu_{\text{RMSE}}\) \\ \hline WT(0.3) & 0.56 & 0.16 & 0.1 & 0.19 \\ CL1(3, 0.3) & 0.56 & 0.16 & 0.1 & 0.19 \\ WT(0.6) & 1.14 & 0.24 & 0.15 & 0.31 \\ CL1(2, 0.6) & 1.12 & 0.23 & 0.14 & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the performances of CL1(\(o\), \(s\)) and WT(\(s\)). \[QoC_{\text{MPI,AR}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6)))=50 \%.\] \[QoC_{\text{API,AR}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6)) )=25\%.\] \[QoC_{\text{AMPI,MAE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=71\%.\] \[QoC_{\text{AMPI,MAE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=65\%.\] \[QoC_{\text{AMPI,MAPE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=67\%.\] \[QoC_{\text{AMPI,MAPE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=60\%.\] \[QoC_{\text{AMPI,RMSE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=66\%.\] \[QoC_{\text{AMPI,RMSE}}(\text{FCF}(\text{CL1}(3,\,0.3),\,\text{CL1}(2,\,0.6 )))=57\%.\]  
Then, the quality of collaboration in FCF(CL1(3, 0.3), CL1(2, 0.6)) is compared with that in FCF(WT(0.3), WT(0.6)). The results are shown in Table 3.  
As can be seen from this table, the use of a nonlinear model CL1(\(o\), \(s\)) instead of the linear model WT(\(s\)) can indeed achieve a better quality of collaboration, especially in the forecasting accuracy. **Model 5. FCF(CL2(\(o_{1}\), \(d_{1}\), \(m_{1}\)), CL2(\(o_{2}\), \(d_{2}\), \(m_{2}\)))**  
This model assumes that both of the two objects use CL2(\(o\), \(d\), \(m\)), but with different values to predict the unit cost  
\((o_{1}\), \(d_{1}\), \(m_{1})=(3\), 0.3, 2)  
\((o_{2}\), \(d_{2}\), \(m_{2})=(2\), 0.5, 3)  
The performances of the two objects are evaluated as  
\(Prec_{\text{AR}}(\text{CL2}(3,\,0.3,\,2))=0.48\)  
\(Prec_{\text{AR}}(\text{CL2}(2,\,0.5,\,3))=0.66\)  
\(Accu_{\text{MAE}}(\text{CL2}(3,\,0.3,\,2))=0.16\)  
\(Accu_{\text{MAE}}(\text{CL2}(2,\,0.5,\,3))=0.16\)  
\(Accu_{\text{MAPE}}(\text{CL2}(3,\,0.3,\,2))=10\%\)  
\(Accu_{\text{MAPE}}(\text{CL2}(2,\,0.5,\,3))=10\%\)  
\(Accu_{\text{RMSE}}(\text{CL2}(3,\,0.3,\,2))=0.19\)  
\(Accu_{\text{RMSE}}(\text{CL2}(2,\,0.5,\,3))=0.20\)  
Through the collaboration of the two objects, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) achieves a better forecasting performance:  
\(Prec_{\text{AR}}(\text{FCF}(\text{CL2}(3,\,0.3,\,2),\,\text{CL2}(2,\,0.5,\,3 )))=0.35\)  
\(Accu_{\text{MAE}}(\text{FCF}(\text{CL2}(3,\,0.3,\,2),\,\text{CL2}(2,\,0.5,\,3 )))=0.05\) \begin{table}
\begin{tabular}{c c c} \hline \hline  & **FCF(WT(0.3), WT(0.6))** & **FCF(CL1(3, 0.3), CL1(2, 0.6))** \\ \hline \(QoC_{\text{MPI,AR}}\) & **54\%** & 50\% \\ \(QoC_{\text{API,AR}}\) & **29\%** & 25\% \\ \(QoC_{\text{AMPI,MAE}}\) & 71\% & 71\% \\ \(QoC_{\text{API,MAE}}\) & 64\% & **65\%** \\ \(QoC_{\text{AMPI,MAPE}}\) & 67\% & 67\% \\ \(QoC_{\text{AMPI,MAPE}}\) & 58\% & **60\%** \\ \(QoC_{\text{AMPI,RMSE}}\) & 52\% & **66\%** \\ \(QoC_{\text{AMPI,RMSE}}\) & 36\% & **57\%** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of FCF(WT(0.3), WT(0.6)) and FCF(CL1(3, 0.3), CL1(2, 0.6)).