## 3 Methodology
### Some Fuzzy Collaborative Forecasting Models for the Unit Cost Forecasting
The forecasting performances of the two objects are evaluated as \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\] \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\] \[Accu_{\text{MAE}}(\text{[PERSON]}(0.3))=0.16\] \[Accu_{\text{MAE}}(\text{[PERSON]}(0.5))=0.20\] \[Accu_{\text{MAPE}}(\text{[PERSON]}(0.3))=10\%\] \[Accu_{\text{MAPE}}(\text{[PERSON]}(0.5))=12\%\] \[Accu_{\text{RMSE}}(\text{[PERSON]}(0.3))=0.19\] \[Accu_{\text{RMSE}}(\text{[PERSON]}(0.5))=0.25\]  
After collaboration, the forecasting precision and accuracy are both improved:  
\[Prec_{\text{AR}}(\text{FCF}(\text{[PERSON]}(0.3),\text{[PERSON]}(0.5)))=0.48\] \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]0 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]1 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]2  
The quality of collaboration in the two aspects can be evaluated as  
\[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]3 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]4  
and  
\[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]5 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]6 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]7 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]8 \[Prec_{\text{AR}}(\text{[PERSON]}(0.3))=0.48\]9 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]0  
respectively. **Model 3. FCF(Donoso(\(k_{11}\), \(k_{21}\), \(s_{1}\)), Donoso(\(k_{12}\), \(k_{22}\), \(s_{2}\)))**  
In this model, both objects use Donoso(\(k_{1}\), \(k_{2}\), \(s\)), but with different parameter values to predict the unit cost. This method has more parameters that can be adjusted, so there is a greater degree of freedom, which provides a space for coordination. Assuming in the previous example, the parameter values specified by the two objects are  
\[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]1 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]2  
Then their forecasting performances are  
\[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]3 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]4 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]5 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]6 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]7 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]8 \[Prec_{\text{AR}}(\text{[PERSON]}(0.5))=0.68\]9 \[Accu_{\text{MAE}}(\text{[PERSON]}(0.3))=0.16\]0  
Comparatively, the forecasting performance of the fuzzy collaborative forecasting method is \(Prec_{\text{AR}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 0.48  
\(Accu_{\text{MAE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 0.07  
\(Accu_{\text{MAPE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 5%  
\(Accu_{\text{RMSE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 0.10  
Therefore, the quality of collaboration can be evaluated as  
\(QoC_{\text{MPLAR}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 16%. \(QoC_{\text{API,AR}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 8%. \(QoC_{\text{AMPL,MAE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 53%. \(QoC_{\text{API,MAE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 52%. \(QoC_{\text{AMPL,MAPE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 44%. \(QoC_{\text{API,MAPE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 44%. \(QoC_{\text{AMPL,RMSE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 44%. \(QoC_{\text{API,RMSE}}\)(FCF([PERSON](0.2, 0.8, 0.2), [PERSON](0.7, 0.3, 0.3))) = 43%. It is worth noting that the performance of this collaboration model is not as good as expected. **Model 4. **\(\mathbf{FCF}\)(CL1(\(o_{1},s_{1}\)), CL1(\(o_{2},s_{2}\)))  
In this model, the two objects use the same method CL1(\(o\), \(s\)), but with different values of \(o\) and \(s\) to predict the unit cost. CL1(\(o\), \(s\)) is an extension of WT(\(s\)) by considering a nonlinear objective function instead. To make a comparison, the parameter values of the two objects are set to  
\((o_{1},s_{1})=(3,\,0.3)\)  
\((o_{2},s_{2})=(2,\,0.6)\)  
The \(s\) values are the same with those in the original WT(\(s\)) methods.