## 4 Proposed Transfer Functions Based on DFs  
### Integration of DFs into Transfer Functions  
DF \(d(y)\) scales the possible response (\(y\)) value to the range of [0, 1], where \(d(y)=0\) and \(d(y)=1\) represent completely undesirable and desirable values, respectively. Among the three DFs proposed by [PERSON] and [PERSON] [40] for three situations of a particular response, the \"nominal is best\" and \"smaller is better\" DFs are utilized as new transfer functions in the hidden NN layers in order to estimate the process mean and standard deviation functions. Assume that there are \(q\) runs, and hence, the summation at each hidden neuron represents vector \(\mathbf{y}_{j}\) consisting of \(\{y_{1},\ldots,y_{p},\ \ldots,y_{q}\}\) values, where the input of each hidden neuron can then be calculated as \(y_{p}(\mathbf{x})=\sum_{i=1}^{k}x_{i}w_{ij}+a_{j}\) at each run order. Therefore, the associated DF output can be represented as \(\mathbf{d}_{j}=\left\{d_{1},\ldots,d_{p},\ \ldots,d_{q}\right\}\). When a response belongs to the \"nominal is best\" case, the individual DF is defined as  
\[d_{p}(y_{p})=\left\{ \begin{array}{c}0\quad if\ y_{p}(\mathbf{x})<L_{p}\\ \left(\frac{y_{p}(\mathbf{x})-L_{p}}{T_{p}-L_{p}}\right)^{s}if\ L_{p}\leq y_{p}( \mathbf{x})\leq T_{p}\\ \left(\frac{y_{p}(\mathbf{x})-U_{p}}{T_{p}-U_{p}}\right)^{t}if\ T_{p}\leq y_{p}( \mathbf{x})\leq U_{p}\\ 0\quad\quad\ if\ y_{p}(\mathbf{x})>U_{p}\end{array} \right. \tag{8}\]  
When a response needs to be minimized, the individual DF is  
\[d_{p}(y_{p})=\left\{ \begin{array}{c}1\quad if\ y_{p}(\mathbf{x})<L_{p}\\ \left(\frac{y_{p}(\mathbf{x})-U_{p}}{L_{p}-U_{p}}\right)^{r}if\ L_{p}\leq y_{p}( \mathbf{x})\leq U_{p}\\ 0\quad\quad\ if\ y_{p}(\mathbf{x})>U_{p}\end{array} \right. \tag{9}\]  
where \(L_{p}\), \(T_{p}\), and \(U_{p}\) denote the lower specification, target, and upper specification values, respectively. Superscripts \(r\) and \(s\) in Equation (8) and superscript \(t\) in Equation (9) represent weight parameters of the \"nominal is best\" and the \"smaller is better\" DFs, respectively. When the values of \(s\), \(t\), and \(r\) are set to 1, the DFs become linear. For \(s<1\), \(t<1\), and \(r<1\) the functions are convex, and for \(s>1\), \(t>1\), and \(r>1\) the functions become concave. The integration of the \"nominal is best\" and \"smaller is better\" DFs as new transfer functions in the hidden layers to estimate the process mean and standard deviation is depicted in Figure 3.  
From Equation (5), the proposed estimated process mean and standard deviation functions (namely, \(\hat{\mu}_{pro}(\mathbf{x})\) and \(\hat{\sigma}_{pro}(\mathbf{x})\)) are defined, respectively, as  
\[\hat{\mu}_{pro}(\mathbf{x})=\sum_{j=1}^{h_{mean\_pro}}\left\{v_{j}^{\text{ mean\_pro}}\left[d_{mean}^{*}\left(\sum_{i=1}^{k}w_{ij}^{\text{mean\_pro}}x_{i}+a_{j}^{ \text{mean\_pro}}\right)\right]\right\}+b_{mean}^{pro}, \tag{10}\]  
\[\hat{\sigma}_{pro}(\mathbf{x})=\sum_{j=1}^{h_{std\_pro}}\left\{v_{j}^{\text{ std\_pro}}\left[d_{std}^{*}\left(\sum_{i=1}^{k}w_{ij}^{\text{std\_pro}}x_{i}+a_{j}^{ \text{std\_pro}}\right)\right]\right\}+b_{std}^{pro}, \tag{11}\]  
where \(h\_mean\_pro\), \(h\_std\_pro\), \(a_{j}^{\text{mean\_pro}}\), \(a_{j}^{\text{std\_pro}}\), \(b_{mean}^{pro}\), \(b_{std}^{\text{mean\_pro}}\), \(v_{j}^{\text{mean\_pro}}\), \(v_{j}^{\text{std\_pro}}\), \(w_{ij}^{\text{mean\_pro}}\), \(w_{ij}^{\text{std\_pro}}\), \(d_{mean}^{*}\), and \(d_{std}^{*}\) denote the number of hidden neurons, bias at the hidden node \(j\), bias at the output neuron, weights connecting hidden node \(j\) to the output responses, weights connecting input factors \(x_{i}\) to hidden node \(j\) of the proposed feed-forward NN, and the optimal DFs for the process mean and standard deviation functions, respectively. In addition, the log-sigmoid and linear functions are usually used as conventional transfer  
Figure 3: Integration of DFs as transfer functions in hidden layers: (**a**) “nominal is best” transfer function and (**b**) “smaller is better” transfer function.