# Integration of Functional Link Neural Networks into a Parameter Estimation Methodology
###### Abstract  
In the field of robust design, most estimation methods for output responses of input factors are based on the response surface methodology (RSM), which makes several assumptions regarding the input data. However, these assumptions may not consistently hold in real-world industrial problems. Recent studies using artificial neural networks (ANNs) indicate that input-output relationships can be effectively estimated without the assumptions mentioned above. The primary objective of this research is to generate a new, robust design dual-response estimation method based on ANNs. First, a second-order functional-link-NN-based robust design estimation approach has been proposed for the process mean and standard deviation (i.e., the dual-response model). Second, the optimal structure of the proposed network is defined based on the Bayesian information criterion. Finally, the estimated response functions of the proposed functional-link-NN-based estimation method are applied and compared with that obtained using the conventional least squares method (LSM)-based RSM. The numerical example results imply that the proposed functional-link-NN-based dual-response robust design estimation model can provide more effective optimal solutions than the LSM-based RSM, according to the expected quality loss criteria. response surface methodology; estimation; functional link neural network +
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mean is assigned equal to the target value. However, the dual-response approach in [PERSON] and [PERSON] [8] may not always provide efficient optimal RD solutions, which have been discussed in [PERSON] and [PERSON] [9] and [PERSON] and [PERSON] [10]. Instead, they employed the standard nonlinear programming techniques of the generalized reduced gradient method and the Nelder-Mead simplex method to provide better RD solutions. Subsequently, [PERSON] and [PERSON] [11] identified a drawback in the dual-response model approach whereby the process bias and variance are not simultaneously minimized. To overcome this issue, they proposed a mean square error (MSE) model. The RSM comprises statistical and mathematical techniques to develop, improve, and optimize processes. It helps design, develop, and formulate new products, as well as improve the existing product designs [12]. The unidentified relationship between input factors and output responses can be investigated using the RSM. To define the input-output functional relationship, the conventional LSM is used to estimate unknown model coefficients. The LSM-based RSM assumes that the sample data follow a normal distribution, and the error terms hold a fixed variance with zero mean. Unfortunately, the Gauss-Markov theorem is not applicable in several practical situations, which implies that those assumptions are not valid. Therefore, weighted least squares, maximum likelihood estimation (MLE), and Bayesian estimation methods can be used as alternatives to determine model parameters. Pertaining to MLE, the unknown parameters are considered as constant, and the observed data are treated as random variables [13]. The MLE approach with abnormal distributed data was implemented in [PERSON] and [PERSON] [14], [PERSON] et al. [15], and [PERSON] and [PERSON] [16], whereas [PERSON] [17] and [PERSON] and [PERSON] [18] proposed the weighted least squares methods to estimate the model coefficients in the case of unbalanced data. Most estimation methods based on the RSM consider several assumptions or require specific data types to determine functions between the process mean or variance and input factors. Over the past two decades, artificial neural networks (ANNs), often known as neural networks (NNs), have been widely used to classify, cluster, approximate, forecast, and optimize datasets in the fields of biology, medicine, industrial engineering, control engineering, software engineering, environmental science, economics, and sociology. An ANN is a quantitative numerical model that originates from the organization and operation of the neural networks of the biological brain.