## 5 Case Study  
This case study was performed using the printing data example proposed by [PERSON] and Myers [6] and [PERSON] [7]. The impacts of three input factors, that is, speed (\(x_{1}\)), pressure (\(x_{2}\)), and distance (\(x_{3}\)), on the capabilities of a printing machine that applies colored inks to package levels (\(y\)) were investigated. Each input factor had three levels, resulting in a total of 27 runs. Three replications were performed for each data combination. In comparative studies between methods, the expected quality loss (EQL) is usually applied as a critical optimization criterion in RD. The expectation of the loss function is defined as  
\[\text{EQL}=\boldsymbol{\theta}\Big{[}(\hat{\mu}(\mathbf{x})-\tau)^{2}+\hat{ \sigma}^{2}(\mathbf{x})\Big{]}. \tag{41}\]  
where \(\boldsymbol{\theta}\) represents a positive loss coefficient. \(\hat{\mu}(\mathbf{x})\), \(\tau\), and \(\hat{\sigma}(\mathbf{x})\) are the estimated process mean function, desirable target value, and estimated standard deviation function, respectively. In this case study, the target value was \(\tau=500\). Using MATLAB, the process-estimated mean and standard deviation functions obtained using the LSM based on RSM were as follows:  
\[\hat{\mu}_{\text{LSM}}(\mathbf{x})=327.6296+\mathbf{x}^{T}\mathbf{m}_{LSM}+ \mathbf{x}^{T}\mathbf{M}_{LSM}\mathbf{x}\,, \tag{42}\]  
\[\hat{\sigma}_{\text{LSM}}(\mathbf{x})=34.8832+\mathbf{x}^{T}\mathbf{n}_{LSM}+ \mathbf{x}^{T}\mathbf{N}_{LSM}\mathbf{x} \tag{43}\]where  
\[\mathbf{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right],\ \mathbf{m}_{LSM}=\left[ \begin{array}{c}177.000\\ 109.426\\ 131.463\end{array} \right],\ \mathbf{M}_{LSM}=\left[ \begin{array}{ccc}32.000&66.028&75.472\\ 66.028&-22.389&43.583\\ 75.472&43.583&-29.056\end{array} \right],\] \[\mathbf{n}_{LSM}=\left[ \begin{array}{c}11.527\\ 15.323\\ 29.190\end{array} \right],\ \text{and}\ \mathbf{N}_{LSM}=\left[ \begin{array}{ccc}4.204&7.7120&5.1093\\ 7.720&-1.316&14.082\\ 5.109&14.082&16.778\end{array} \right].\]  
For comparison, the criteria used in the NN (mainly, the number of hidden neurons, training algorithms, and the number of epochs of NNs) for the conventional log-sigmoid transfer function in the hidden layer were the same as those of the proposed NNs. The NN information for the conventional log-sigmoid transfer function and the proposed DR estimation method are presented in Table 1. The NN architecture consists of several input factors, hidden neurons, and output responses. The weight and bias matrix values of the NNs using the conventional log-sigmoid transfer function for the process mean and standard deviation functions are presented in Table 2. Based on the structures, training algorithms, transfer functions, and the number of epochs of the proposed feed-forward NNs used to estimate the functions of process mean and standard deviation, optimal weight parameters \(\hat{s}\), \(\hat{t}\), and \(\hat{r}\) were calculated using Equations (20), (22), and (24), respectively. As shown in Table 3, twenty hidden neurons in the hidden layer are used to estimate mean function, and twenty hidden neurons in the hidden layer are used to estimate standard deviation function. For each hidden neuron, a desirability function is proposed as a transfer function.