*3.2. Kriging*

The Kriging metamodel was originally created by Dannie Gerhardus Krige, later much research contributed to its development and finally, Matheron formalized the approach in 1963 [28]. The main idea of the kriging metamodel is that the deterministic response *y*(*x*) can be described by

$$y(\mathbf{x}) = (\mathbf{x}) + Z(\mathbf{x})\tag{2}$$

where the approximation function is known is called *f*(*x*), and *Z(x)* is an execution of a stochastic process with a mean zero, variance σ*2* and a non-zero covariance. The first term of the equation, *f(x)*, offers a global approach to the design space that is similar to a regression model (Equation (3)). The second term, *Z(x)*, generates local deviations to interpolate the initial sample points using the kriging model (Equation (4)).

$$f(\mathbf{x}) = \sum\_{i=1}^{n} \beta\_i \cdot f\_i(\mathbf{x}) \tag{3}$$

$$\text{cov}\left[Z(\mathbf{x}\_i), Z(\mathbf{x}\_j)\right] = \sigma^2 \cdot \text{R}\left(\mathbf{x}\_i, \mathbf{x}\_j\right) \tag{4}$$

where the process variance σ2 scales the spatial correlation function *R(xi,xj)* between two data points. The Gaussian correlation function (Equation (5)) is widely used in engineering design [29]. It can be defined with a single parameter (θ) that determines the area of influence of the adjacent points [30]. When the sample points have a high correlation, then θ is low, thus *Z(x)* will be similar throughout the design space. As the θ grows, the closest points will have the greatest correlation, thus *Z(x)* will vary according to the point in the design space:

$$R\left(\mathbf{x}\_{i\prime},\mathbf{x}\_{j}\right) = \boldsymbol{\sigma}^{-\sum\_{k=1}^{\text{II}} \theta \left|\mathbf{x}\_{k}^{i} - \mathbf{x}\_{k}^{j}\right|^{2}} \tag{5}$$

#### *3.3. The Fitted Model*

The search for the Best Linear Unbiased Predictor (BLUP) is used by the formulation of kriging. Simpson et al. [31] have discussed fitting methods for most widely-used metamodels.

#### *3.4. Mean and Variance*

The mean (μ) and standard deviation (σ) of the responses of the objectives measure the robust optimum design. These statistical parameters have been obtained for four different levels of uncertainty (10%, 20%, 30%, and 40%). The value of the uncertain initial parameter has been calculated according to a uniform distribution depending on the level of uncertainty. In this way, the mean refers to the optimum design, and the standard variation refers to the robust design.
