## 2 Methodology
Then activated signals are outputted from the hidden layer as such:  
\[\tilde{h}_{l}=(h_{l1},h_{l2},h_{l3})=\frac{1}{1+e^{-\tilde{h}_{l}^{h}}}=(\frac {1}{1+e^{-h_{l1}^{h}}},\frac{1}{1+e^{-h_{l2}^{h}}},\frac{1}{1+e^{-h_{l3}^{h}}}) \tag{3}\]  
where \(\tilde{h}_{l}\) is the output from hidden-layer node \(l\), and  
\[\tilde{h}_{l}^{h}=(n_{l1}^{h},n_{l2}^{h},n_{l3}^{h})=\tilde{I}_{l}^{h}(-) \tilde{\theta}_{l}^{h}=(I_{l1}^{h}-\theta_{l3}^{h},I_{l2}^{h}-\theta_{l2}^{h},I_{l3}^{h}-\theta_{l1}^{h}) \tag{4}\]  
\[\tilde{I}_{l}^{h} =(I_{l1}^{h},I_{l2}^{h},I_{l3}^{h})=\sum_{all~{}k}\tilde{w}_{kl}^ {h}\cdot x_{k} \tag{5}\] \[=(\sum_{all~{}k}\min(w_{kl1}^{h}x_{k},w_{kl3}^{h}x_{k}),\sum_{all~ {}k}w_{kl2}^{h}x_{k},\sum_{all~{}k}\max(w_{kl1}^{h}x_{k},w_{kl3}^{h}x_{k}))\]  
\(\tilde{\theta}_{l}^{h}\) is the threshold for screening out weak signals by hidden-layer node \(l\); \(\tilde{w}_{kl}^{h}\) is the weight of the connection between input node \(k\) and hidden-layer node \(l\); \(x_{k}\) is the \(k\)-th input; \((\
eg)\) and \((\times)\) denote fuzzy subtraction and multiplication, respectively. \(\tilde{h}_{l}\)'s are also transferred to the output layer with the same procedure. Finally, the output of the FBPN is generated as follows:  
\[\tilde{\mathcal{O}}=(o_{1},o_{2},o_{3})=\frac{1}{1+e^{-\tilde{n}^{o}}}=(\frac {1}{1+e^{-n_{1}^{o}}},\frac{1}{1+e^{-n_{2}^{o}}},\frac{1}{1+e^{-n_{3}^{o}}}) \tag{6}\]  
where  
\[\tilde{n}^{o}=(n_{1}^{o},n_{2}^{o},n_{3}^{o})=\tilde{I}^{o}(-) \tilde{\theta}^{o}=(I_{1}^{o}-\theta_{3}^{o},I_{2}^{o}-\theta_{2 Subsequently, the following goal programming (GP) problem is solved to determine the upper bound of each parameter (e.g., \(w_{kl3}^{h}\), \(\theta_{l3}^{h}\), \(w_{l3}^{o}\), and \(\theta_{3}^{o}\)) so that the actual value will be less than the upper bound of the network output:  
\[\mathrm{Min}\sum_{\mathrm{all}\,t}\pi_{t} \tag{9}\]  
subject to  
\[\mathrm{ln}(\frac{1}{P_{t3}}-1)=\theta_{3}^{o}-\sum_{\mathrm{all}\,t}w_{l3}^{o }h_{l3} \tag{10}\]  
\[\sum_{\mathrm{all}\,l}w_{l3}^{o}h_{l3}-\theta_{3}^{o}=-\mathrm{ln}(1/\,\pi_{t} -1) \tag{11}\]  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]0  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]1  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]2  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]3  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]4  
\[x=\frac{N(x)-N_{{}_{L}}}{N_{{}_{U}}-N_{{}_{L}}}\cdot(x_{\max}-x_{\min})+x_{ \min}\]5  
\(\tilde{P}_{t}\) is the fuzzy price forecast of month \(t\); \(a_{t}\) is the normalized value of the actual price of month \(t\). All actual values will fall within the corresponding fuzzy forecasts. The objective function is to minimize the sum of the half-ranges (\(\pi_{t}\)) of the fuzzy price forecasts, which is calculated according to constraint (11). Constraint (12) forces the half-range to be narrower than the agent's requirement \(\psi_{R}(g)\). The other constraints limit the changes that should be made to the network parameters (\(w_{il3}^{h}\), \(\theta_{l3}^{h}\), \(w_{l3}^{o}\), and \(\theta_{3}^{o}\)) for the same purpose. \(s_{R}(g)\) is the acceptable satisfaction level on the right-hand side. If \(s_{R}(g)\) is high, then \(h_{l3}\) will be large, which leads to a large \(\pi_{t}\). A number of possible values for \(s_{R}(g)\) are enumerated to yield many different solutions to the goal programming problem. Of these optimization results, the one giving the minimum upper bound is chosen.