*4.3. Optimization Results*

Table 3 presents the tuning parameters and the processing time for every iteration found for objective functions OA1 and OA2, and mass ratios *μ* = 0.02 and *μ* = 0.05, by applying the WOA methodology proposed herein. Additionally, a comparison is made with the DEM and an ES process with precision to two decimal positions. It should be noted that DEM has proved to be effective in solving the tuning problem of TMDs and TMDIs under actual earthquake excitations [58,59]; thus, it was selected to validate the WOA optimization results. Moreover, the work of Caicedo et al. [58] demonstrated the feasibility of DEM over other conventional tuning methodologies considering harmonic loads [6], white noise processes [7,8], and frequency domain analysis [9]. Similarly, Table 4 shows *ζd* and *f* parameters derived from J1, J2, J3, and J4 approaches.

As expected, the results computed by WOA and the other two comparison methodologies show correspondence. From Tables 3 and 4, it can be observed that some of the design variables approach the limits established previously in Equations (21) and (22). Such results are not realistic from a practical point of view, but they may be attributed to the impulsive nature of some of the seismic records used in the optimization process. However, in all cases the results are in excellent agreemen<sup>t</sup> with those computed through the DEM and the ES process, showing small differences from the third decimal position, which has no influence on the global response of the structural system. Furthermore, the numerical results reported by other works in which actual seismic records were used as excitation inputs [46,58,59,65] exhibited a similar trend, since some of the optimal values are also close to the bound limits defined for the optimization process. On the other hand, WOA exhibits notable advantages against ES and DEM, among them, less computational cost avoiding operations, like mutation or crossover, and reductions up to 45% in the processing time, approximately (see Tables 3 and 4). Figure 4 illustrates the dispersion of optimal design values derived from the three optimization methodologies and the six objective functions used in this investigation.


**Table 3.** TMD design parameters optimized by objective function OA1 and OA2.

**Figure 4.** Scatter graphs for TMD design parameters: (**a**) DE; (**b**) ES; (**c**) WOA.
