## 3 Methodology
### Fuzzy Linear Regression Methods for Forecasting the Unit Cost  
According to [PERSON] [25], the yield of a product follows a learning process:  
\[Y_{t}=Y_{0}e^{-b/t+r(t)} \tag{1}\]  
The unit cost can be calculated as  
\[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]  
Obviously, the change in the unti cost is also a learning process, not a usual time-series. After converting to logarithms,  
\[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]  
where \(a=\ln C-\ln Y_{0}-\ln G\). To consider the uncertainty in the unit cost, parameters in equation (3) are given in asymmetric triangular fuzzy numbers as follows [26, 27, 28, 29]:  
\[\tilde{a}=(a_{1},\,\,\,a_{2},\,\,\,a_{3}) \tag{4}\]  
\[\tilde{b}=(b_{1},\,\,\,b_{2},\,\,\,b_{3}) \tag{5}\]  
Therefore,  
\[\ln\tilde{c}_{t}\equiv(\ln c_{t1},\ln c_{t2},\ln c_{t3})=\tilde{a}(+)\tilde{b }/t-r(t)=(a_{1}+b_{1}/t,a_{2}+b_{2}/t,a_{3}+b_{3}/t)-r(t) \tag{6}\]  
where (+) represents fuzzy addition. Equation (6) is obviously a fuzzy linear regression equation. A fuzzy linear regression equation can be fitted in various ways. For example, in Tanaka and Watada [30], a linear programming problem is solved to minimize the fuzziness:\[\text{Min}\;\;Z=\sum_{t=1}^{T}\left(\ln c_{t3}-\ln c_{t1}\right) \tag{7}\]  
subject to  
\[\ln c_{t}\geq \ln c_{t1}+s(\ln c_{t2}-\ln c_{t1}) \tag{8}\] \[\ln c_{t}\leq \ln c_{t3}+s(\ln c_{t2}-\ln c_{t3})\] (9) \[c_{t1}=a_{1}+b_{1}/t\] (10) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]0 (11) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]1 (12) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]2 (13) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]3 (14) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]4  
where \(s\) is the satisfaction level. For the training data, the actual values will fall within the ranges of the fuzzy forecasts. Clearly, the higher value of \(s\) leads to wider fuzzy forecasts. This model is indicated with WT(\(s\)). The second method for fitting a fuzzy linear regression equation is [PERSON] method [31], in which the following linear programming problem is solved, aimed at the maximization of the average satisfaction level:  
\[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]5  
subject to  
\[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]6 \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]7 (18) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]8 (19) \[c_{t}=C/(Y_{t}\cdot G)=C/(Y_{0}e^{-b/t+r(t)}\cdot G)=C/Y_{0}Ge^{b/t-r(t)} \tag{2}\]9 (20) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]0 (21) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]1 (22) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]2 (23) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]3 (24) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]4 (25) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]5 (26) \[\ln c_{t}=\ln C-\ln Y_{0}-\ln G+b/t-r(t)=a+b/t-r(t) \tag{3}\]6where \(d\) is the required range of a fuzzy forecast. Clearly, a larger value of \(d\) results in a higher average satisfaction level. This model is indicated with Peters(\(d\)).