## 1 Introduction
However, relaxing the error assumptions and increasing the estimation precision would result in further improvements. Artificial neural networks (ANNs) or neural networks (NNs) with nonlinear mapping structures using the human brain function have been widely used as powerful tools in data classification, forecasting, clustering, function approximation, and optimization in recent decades. [PERSON] and [PERSON] [35], [PERSON] et al. [36], [PERSON] [37], and [PERSON] [38] recently addressed the question of approximation using feed-forward NNs and proved that NNs are universal functional approximators with the desired accuracy. ANNs belong to a class of self-adaptive and data-driven techniques, and therefore the undefined relationships between input factors and outputs of a process/product can be ascertained effectively. ANNs can provide a linear or nonlinear relationship between inputs and several outputs without any assumptions based on generalizing capacities of the activation function. Consequently, the functional relationships between input factors and their related output responses in an RD procedure can be estimated without any specific assumptions effectively. Most of the literature on NN function approximation has concentrated on their structure, hidden layers, hidden neurons, and training algorithms. On the contrary, less of them focus on the transfer or activation function, which can strongly affect the complexity and performance of NNs. In addition, most of the transfer functions reported in the literature can provide only a particular configuration to transfer the information from inputs to the associated outputs of a neuron. The desirability function (DF) approach is one of the most commonly utilized techniques for multiple response optimization problems, in which each quality characteristic is converted into an individual DF that takes a value ranging from 0 to 1. The DF concept was introduced by [PERSON] [39] for the simultaneous optimization of multiple response problems in the industry field. This approach can be used to determine those optimal factors settings of input variables that result in the most desirable output values, while none of these response values are beyond the specified boundaries. [PERSON] and [PERSON] [40] implemented a further extension of [PERSON]'s DF. Depending on whether the response will minimize, maximize, or specify the desired target value, there are three associated DFs: STB (smaller is better), NTB (nominal is better), and LTB (larger is better). Based on the weight parameters, the DF family may provide a flexible transfer function configuration. [PERSON] et al. [41] integrated an NN approach into RD and used the NN to perform the DoE. [PERSON] and [PERSON] [42], [PERSON] et al. [43], [PERSON] et al. [44], [PERSON] [45], and [PERSON] and [PERSON] [46] combined an NN approach as an estimation method with a genetic algorithm to determine the optimal process parameter settings or the optimal costs in both static and dynamic characteristics in Taguchi's formula without considering the process mean and variance functions. Recently, [PERSON] and [PERSON] [47] presented a feed-forward NN structure based on RSM in the RD concept. In the robust design (RD) concept, the process mean and standard deviation of an output response in experiment results can be estimated as a function by many different statistical approaches (i.e., LSM, MLE, and so on). These processes mean and standard deviation functions can be optimized simultaneously based on their desired targets using existing optimization methods. The significant objective of this paper is to propose an alternative estimation method to DR functions (i.e., mean and standard deviation functions) by integrating NN approaches for RSM and RD modeling and then to compare this method to the conventional LSM-based estimation method. First, a feed-forward NN is integrated into the RD procedure as a new DR estimation approach to estimate the process mean and standard deviation response functions.