## 3 Methodology
### Some Fuzzy Collaborative Forecasting Models for the Unit Cost Forecasting  
An example is given in Table 1. The data of the first 7 periods were used as training data, and the remaining data are left for testing. \begin{table}
\begin{tabular}{c c} \hline \hline
**t** & **c\text{ (US$)}** \\ \hline
1 & 2.57 \\
2 & 1.61 \\
3 & 1.76 \\
4 & 1.28 \\
5 & 1.53 \\
6 & 1.19 \\
7 & 1.32 \\
8 & 1.32 \\
9 & 1.61 \\
10 & 1.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An example. Some linear fuzzy collaboration forecasting models for used to predict the unit cost. **Model 1. FCF(WT(\(s_{1}\)), WT(\(s_{2}\)))**  
In this model, the two objects use the same fuzzy linear regression method (WT(\(s\))), but with different values of \(s\) to predict the unit cost. In WT(\(s\)), the most precise forecast is associated with the minimum value of \(s\) that satisfies the constraints. In addition, the results when \(s\) is large often contain the results when \(s\) is relatively small, which makes it less effective for their collaboration. Nevertheless, the fuzzy collaborative forecasting method, to a certain extent, improves the precision of WT(\(s\)). In the previous example, assuming the \(s\) values specified by the objects are 0.3 and 0.6, respectively:  
\(Prec_{\text{AR}}(\text{WT}(0.3))=0.56\)  
\(Prec_{\text{AR}}(\text{WT}(0.6))=1.14\)  
\(Prec_{\text{AR}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=0.53\)  
Therefore, the quality of collaboration with respect to the forecasting precision can be evaluated as  
\(QoC_{\text{MPLAR}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=\text{max}((1. 14-0.53)/1.14,\,(0.56-0.53)/0.56)=54\%\). \(QoC_{\text{API,AR}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=((1.14-0.53) /1.14+(0.56-0.53)/0.56)/2=29\%\). In order to evaluate the forecasting accuracy, the forecasts by the two objects are defuzzified using the center of gravity (COG) method, and then are compared with the actual values:  
\(Accu_{\text{MAE}}(\text{WT}(0.3))=0.16\)  
\(Accu_{\text{MAE}}(\text{WT}(0.6))=0.24\)  
\(Accu_{\text{MAPE}}(\text{WT}(0.3))=10\%\)  
\(Accu_{\text{MAPE}}(\text{WT}(0.6))=15\%\)  
\(Accu_{\text{RMSE}}(\text{WT}(0.3))=0.19\)  
\(Accu_{\text{RMSE}}(\text{WT}(0.6))=0.31\)  
while in the fuzzy collaborative forecasting method, the forecasts by the two objects are aggregated using the fuzzy intersection and back propagation network approach to generate a single crisp value:  
\(Accu_{\text{MAE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=0.06\)  
\(Accu_{\text{MAPE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=4\%\)  
\(Accu_{\text{RMSE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=0.10\)  
Therefore, the quality of collaboration with respect tothe forecasting accuracy can be evaluated as  
\(QoC_{\text{AMPL,MAE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=\text{max} ((0.16-0.07)/0.16,\,(0.24-0.07)/0.24)=71\%\). \(QoC_{\text{API,MAE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=((0.16-0.0 7)/0.16+(0.24-0.07)/0.24)/2=64\%\). \(QoC_{\text{AMPL,MAPE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=\text{max} ((10\%-5\%)/10\%,\,(15\%-5\%)/15\%)=67\%\). \(QoC_{\text{API,MAPE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=((10\%-5 \%)/10\%+(15\%-5\%)/15\%)/2=58\%\). \(QoC_{\text{AMPL,RMSE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=\text{max} ((0.19-0.15)/0.19,\,(0.31-0.15)/0.31)=52\%\). \(QoC_{\text{API,RMSE}}(\text{FCF}(\text{WT}(0.3),\,\text{WT}(0.6)))=((0.19-0.15 )/0.19+(0.31-0.15)/0.31)/2=36\%\). **Model 2. FCF(Peters(\(d_{1}\)), Peters(\(d_{2}\)))**  
In this model, both objects use Peters(\(d\)), but with different \(d\) values to predict the unit cost. In Peters(\(d\)), the most precise forecast is associated with the minimum value of \(d\) that satisfies the constraints. In addition, the results when \(d\) is large often contain the results when \(d\) is relatively small. As a result, the benefits of collaboration are not obvious. In the previous example, assuming the \(d\) values specified by the objects are 0.3 and 0.5, respectively.