## 4 Algorithm for Determining Critical Combinations of Parameters  
Due to the large number of combinations required, the vertex method is not effective, but the combination of parameters obtained using this method allows one to calculate the bounds of the objective function. Using Taylor series expansion does not always result in determining the same combination of parameters as in the vertex method, especially if uncertainties are large. The algorithm presented below uses the Taylor series expansion, but even in the case of a large number of parameters or their variability, it gives exactly the same combination of parameter values as the vertex method. In [25], this algorithm was used to analyze structures with viscoelastic dampers; however, in this study, it is proved that this algorithm is general and can be used for other structures with viscoelastic damping elements. In addition, this algorithm is more efficient than the vertex method. In the first step, the first- and second-order sensitivities with respect to the change in each design parameter, which was assumed as an interval value, are calculated. Then, according to Equation (37), increments in the limit values of design parameters are performed, and the largest among them is selected. In the second step, the value of the parameter with the largest is updated, and first- and second-order sensitivities are recalculated with respect to the change in design parameters. The parameter whose value was set in the previous step is not taken into further consideration. As mentioned earlier, the largest increment is selected, and the appropriate parameter value is updated. The calculations are repeated until the limit values of all parameters are determined. In the last step, the upper bound of the objective function for the set parameter values is calculated. The same analysis is conducted to determine the parameter values, thus allowing the calculation of the lower bound of the objective function. The analyzed examples show that this algorithm allows one to determine the same critical combination of limit values of parameters as the vertex method, but it is less labor intensive. The pseudocode for the presented algorithm is as follows (see Algorithm 1):  
```
1\({}^{\circ}\) Input data:
2\(r\)-number of uncertain parameters
3\(p\)-vector of lower bounds of uncertain parameters
4\(\overline{p}\)-vector of upper bounds of uncertain parameters
5\(p^{c}\) set \(n\):= \(r\) set \(p^{crit}\):= \(p^{c}\) (\(p^{crit}\) denotes parameter vector for the critical combination)
6\({}^{\circ}\) for \(i\) =1,2,..., \(n\) calculate sensitivities of the response function with respect to change in parameter \(p_{i}^{c}\) \(\frac{\partial f(p^{crit})}{\partial p_{i}^{c}}\) (sensitivities of the first order) and \(\frac{\partial^{2}f(p^{crit})}{\left(2\mu_{i}\right)^{2}}\) (sensitivities of the second order)
7\(4\) for \(i\) =1,2,..., \(n\) calculate the increment in response function \(\Delta f\big{(}p^{crit}\big{)}\) according to Equation (37) for \(\underline{p}_{i}\) and \(\overline{p}_{i}\) \(5^{\circ}\) select the maximum value of \(\Delta f\big{(}p^{crit}\big{)}\) and corresponding parameter \(p_{k}\) \(\overline{\Delta f}(p_{k})=max\big{(}\Delta f\big{(}p^{crit}\big{)}\big{)}\) \(6^{\circ}\) substitute \(p^{crit}(k)=p_{k}\) \(7^{\circ}\) exclude parameter \(p_{k}\) and its lower \(\underline{p}_{k}\) and upper \(\overline{p}_{k}\) bounds from futher calculations
8\({}^{\circ}\) update \(n\):= \(n-1\) \(9^{\circ}\) repeat steps \(3^{\circ}\) to \(8^{\circ}\) until \(n=0\) \(10^{\circ}\) calculate the maximum value of the response function for the critical combination of parameters \(\overline{f}=f\big{(}p^{crit}\big{)}\). ```  
**Algorithm 1** Determining the critical combination of parameters.