## 3 Solution Method
### Metamodel Construction  
Various techniques are available to approximate the relationships between the inputs and outputs of a system. The purpose of metamodels is to approximate the predictions of the underlying model as accurately as possible and to be interpretable at the same time. There have been a various number of studies concerning distinct metamodels for particular applications, such as support vector regression [17], multivariate adaptive regression splines [18], kriging [19], artificial neural network [20], and RMTS [15]. Among them, RMTS adopts minimal energy and regularization to improve accuracy for low-dimensional problems with large unstructured datasets. Energy minimization has been used for specific applications such as extending a curve or surface with additional points. In RMTS, an energy function is treated as a cost minimization to determine the coefficients of the multivariate spline. RMTS has been successfully applied to curve fitting, function approximation, prediction, forecasting and classification [15]. The advantages of RMTS include fast prediction time, unsusceptible to training failure and scalability to the variability in parameters. Thus, a metamodel based on the RMTS is utilized to quickly estimate the fitness of a solution. The output variable of RMTS may be represented by a linear combination of a set of basis functions, where the coefficients are precomputed during training. The training data patterns are \((\mathbf{x}_{i},F_{a}(\mathbf{x}_{i}))\), \(i=1,\ldots,M\), where \(\mathbf{x}_{i}\) and \(F_{a}(\mathbf{x}_{i})\) denote the solution vector and objective value assessed using accurate evaluation, respectively. \(M\) represents the number of training data patterns, which can be determined from the sample size formula with a finite population correction factor. The framework of RMTS is displayed in Figure 1[21] and the general model of RMTS can be stated formally as below. \[f(\mathbf{x}|\boldsymbol{\omega})=\sum_{i=1}^{P}\omega_{i}\psi_{i}(\mathbf{x}) \tag{7}\]  
where \(P\) denotes the number of basis functions, \(\mathbf{x}\) denotes an input vector, \(\boldsymbol{\omega}=\left[\omega_{1},\ldots,\omega_{P}\right]^{T}\) is the vector of spline coefficients, \(\omega_{i}\) represents the coefficient of the multivariate spline, which provides a connection between the predicted \(f(\mathbf{x}|\boldsymbol{\omega})\) and basis functions \(\psi_{i}(\bullet)\). Two types of splines are commonly used as the basis functions: B-splines and cubic Hermite splines. The B-splines function is expressed as  
\[\psi(\mathbf{x})=Q^{T}B_{d}(\mathbf{x}) \tag{8}\]  
where \(Q\) is a vector of \(n\) coefficients and \(B_{d}(\mathbf{x})\) is a vector of \(n\) B-spline piecewise functions. The piecewise functions in \(B_{d}(\mathbf{x})\) are \(d\)th degree polynomial pieces in the variable \(\mathbf{x}\). Typically, third- or fourth-order degrees are used. The purpose of RMTS is to make each of its outputs \(f(\mathbf{x}|\boldsymbol{\omega})\) much closer to the actual expected values \(F_{a}(\mathbf{x})\). RMTS solves an energy minimization function where the splines pass through the training points to obtain the coefficients \(\omega_{i}\) of the splines. The energy minimization function (9) comprises three terms, the second derivatives of the splines, regularization and approximation error related to the training points [21].