## 1 Introduction
proposed NN-based response function estimation (NRFE) identifies a new screening procedure to obtain the best transfer function in an NN structure using a desirability function family while determining its associated weight parameters [45]. [PERSON] and [PERSON] propose an NN-based estimation method as a RD modeling approach. The modeling method based on a feedback NN approach is first integrated in the RD response functions estimation. Two new feedback NN structures are then proposed. Next, the existing recurrent NNs (i.e., Jordan-type and Elman-type NNs) and the proposed feedback NN approaches are suggested as an alternative RD modeling method [46]. The primary motive of this research is to establish feed-forward NN structure-based estimation methods as alternative RD modeling approaches. First, an NN-based estimation framework is incorporated into the RD modeling procedure. Second, RD modeling methods based on the feed-forward back-propagation neural network (FFNN), cascade-forward back-propagation neural network (CFNN), and radial basis function network (RBFN) are proposed. These are applied to estimate the process mean and standard deviation response functions. Third, the efficiency of the proposed modeling methods is illustrated through simulation studies with a given real function. Fourth, the efficacy of the proposed modeling methods is illustrated through a printing case study. Finally, the results of comparative studies show that the proposed methods obtain better optimal solutions than conventional LSM-based RSM. The proposed estimation methods based on feed-forward NN structures are illustrated in Figure 1. From the experimental data, the optimal numbers of hidden neurons in the FFNN and CFNN structures and the dispersion constant \"spread\" in the RBFN are identified to finalize the optimal structures of the corresponding NNs. The DR functions can be separately estimated using the proposed estimation methods from the optimal NN structures with their control factors and output responses. The statistical estimation method of RSM based on conventional LSM, as introduced by [PERSON] and [PERSON] [47], generates the response surface approximation using linear, quadratic, and other functions, while the coefficients are estimated by minimizing the total error between the actual and estimated values. For a more comprehensive understanding of RSM, [PERSON] [48] and [PERSON] and [PERSON] [49] discuss the various development stages and future directions of RSM. When the exact functional relationship is unknown or very complicated, conventional LSM is typically used to estimate the input-output functional responses in RSM [50; 51]. In general, the output response \(y\) can be identified as a function of input factors \(\mathbf{x}\) as follows:  
\[y=\mathbf{x}\boldsymbol{\beta}+\varepsilon \tag{1}\]  
where \(\mathbf{x}\) is a vector of the control factors, \(\boldsymbol{\beta}\) is a column vector of the estimated parameters, and \(\varepsilon\) is the random error. The estimated second-order models for the process mean \(\hat{\mu}_{LSM}\) and standard deviation \(\partial_{LSM}\) are represented as  
\[\hat{\mu}_{LSM}(\mathbf{x})=\hat{\beta}_{0}+\sum_{i=1}^{p}\hat{\beta}_{i}x_{i }+\sum_{i=1}^{p}\hat{\beta}_{i}x_{i}^{2}+\sum_{ \begin{array}{c}i=1\\ i<j\end{array} }^{p}\sum_{j=1}^{p}\hat{\beta}_{ij}x_{i}x_{j} \tag{2}\]  
\[\partial_{LSM}(\mathbf{x})=\hat{\delta}_{0}+\sum_{i=1}^{p}\hat{\delta}_{i}x_{ i}+\sum_{i=1}^{p}\hat{\delta}_{i}x_{i}^{2}+\sum_{ \begin{array}{c}i=1\\ i<j\end{array} }^{p}\sum_{j=1}^{p}\hat{\delta}_{ij}x_{i}x_{j} \tag{3}\]  
where \(\hat{\beta}\) and \(\hat{\delta}\) are the estimators of unknown parameters in the mean and standard deviation functions, respectively.