## 2 Response Surface Model  
A response surface model (RSM) is a collection of statistical and mathematical techniques that are useful for developing, improving, and optimizing processes. The choice of RSM for a given computational model depends on the knowledge of the computational model itself [20; 21]. It is used in the development of an adequate functional relationship between a response of interest, \(\mathbf{y}\), and a number of associated input parameters denoted by \((x_{1},x_{2},\ldots,x_{k})\). In general, such a relationship is unknown but can be approximated by a low-degree polynomial model in the form:  
\[\mathbf{y}=f(\mathbf{x})\mathbf{\beta}+\mathbf{e} \tag{1}\]  
where \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})\), \(f(\mathbf{x})\) is a vector function of \(\mathbf{p}\) elements that consists of powers and cross-products of powers of \(\mathbf{x_{1}},\mathbf{x_{2}},\ldots,x_{k}\) up to a certain degree, denoted by \(\mathbf{d}\) (\(\geq 1\)), \(\mathbf{\beta}\) is a vector of \(\mathbf{p}\) unknown constant coefficients, referred to as parameters, and \(\mathbf{e}\) is a random experimental error assumed to have a zero mean. This is conditioned on the consideration that the model provides an adequate representation of the response. In this case, quantity \(f(\mathbf{x})\beta\) represents the mean response, that is, the expected value of \(\mathbf{y}\), and is denoted by \(\mu\)**(\(\mathbf{x}\))**. Two important models are commonly used in RSM. These are special cases of the model in Equation (1) and include the first-degree model (\(d=1\)):  
\[\mathbf{y}=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}+\mathbf{e} \tag{2}\]  
and the second-degree model (\(\mathbf{d}=2\)):  
\[\mathbf{y}=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}+\sum_{i<j}\ \sum\beta_{ij}x_{i}x_{j}+ \sum_{i=1}^{k}\beta_{ii}x^{2}_{i}+\mathbf{e} \tag{3}\]  
A series of \(n\) experiments should first be carried out, in each of which the response \(y\) is measured (or observed) for specified settings of the control parameters. The totality of these settings constitutes the so-called response surface design, or just \"design\", which can be represented by a matrix, denoted by \(\mathbf{D}\), of order \(n\times k\), called the design matrix,  
\[\mathbf{D}=\left( \begin{array}{ccc}x_{11}&x_{12}&\ldots&x_{1k}\\ x_{21}&x_{22}&\ldots&x_{2k}\\ &&\\ &&\ldots\ldots\\ &&\ldots\ldots\\ x_{n1}&x_{n2}&\ldots&x_{nk}\end{array} \right) \tag{4}\]  
where \(x_{ui}\) denotes the \(u\)th design setting of \(x_{i}\) (\(i=1,2,\ldots,k;\mathbf{u}=1,2,\ldots,n\)). Each row of \(\mathbf{D}\) represents a point, referred to as a design point, in a \(k\)-dimensional Euclidean space. Let\(y_{u}\) denote the response value obtained as a result of applying the \(u\)th setting of \(x\), namely \(x_{u}=(x_{u1},x_{u2},\ldots,x_{uh})\), (\(u=1,2,\ldots,n\)). From Equation (1), we then have:  
\[\mathbf{y}_{u}=f^{\prime}(\mathbf{x}_{u})\mathbf{\beta}+\mathbf{e}_{u}\,\ \mathbf{u}=1,2,\ldots,n \tag{5}\]  
where \(\mathbf{e}_{u}\) denotes the error term at the \(u\)th experimental run. Equation (5) can be expressed in a matrix form as:  
\[\mathbf{y}=\mathbf{X}\mathbf{\beta}\ +\mathbf{\epsilon} \tag{6}\]  
where \(\mathbf{y}=(y_{1},y_{2},\ldots,y_{n})\), \(\mathbf{X}\) is a matrix of order \(n\times p\) of which the \(u\)th row is \(f^{\prime}(\mathbf{x}_{u})\), and \(\mathbf{\epsilon}=(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{n})\). Note that the first column of \(\mathbf{X}\) is the column of ones \(\mathbf{1}_{n}\). Assuming that \(\mathbf{\epsilon}\) has a zero mean, the so-called ordinary least-squares estimator of \(\mathbf{\beta}\) is [22]:  
\[\mathbf{\hat{\beta}}=\left(\mathbf{X}^{\prime}\mathbf{X}\right)^{-1}\mathbf{X}^{\prime}\mathbf{y} \tag{7}\]  
The Box-Behnken experimental method is rotatable second-order designs based on three-level incomplete factorial designs (see Figure 1). The special arrangement of the Box-Behnken design levels allows the number of design points to increase at the same rate as the number of polynomial coefficients.