## 1 Introduction
Then, they developed a new algorithm for the SMA wires and combined it with a bilinear kinematic hardening model, which was considered for the LRB. They verified their results by the finite element model, which was validated through experimental tests. [PERSON] et al. [8] implemented their proposed constitutive model for simulating the behavior of SMA-LRBs in Opensees software to assess the accuracy of the model. The results showed that the new model could precisely predict the shear hysteretic response of smart SMA-LRBs compared with other existing models. [PERSON] and [PERSON] [9] used four different retrofitting devices to enhance the seismic performance of an LRB-isolated bridge, including yielding steel cables (YSCs), viscous dampers (VDs), friction dampers (FDs), and super-elastic shape memory alloy cables. The results indicated that the SMA device was better than the other measures regarding its self-centering capability and showed the least residual superstructure displacement. Many other researchers also investigated the application of SMAs in mitigating the seismic response of bridges, including but not limited to [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [12], [PERSON] and [PERSON] [13], [PERSON] et al. [14], [PERSON] et al. [15], [PERSON] and [PERSON] [16], and [PERSON] et al. [17]. Researchers have suggested control devices and applied them to different bridges. For this reason, it was hard to compare their results. To overcome this difficulty, a benchmark highway bridge was developed in two phases. Many researchers have implemented their control devices on phase I of the benchmark bridge, in which the deck of the bridge was rigidly connected to the central pier ([PERSON] et al. [18]). [PERSON] and [PERSON] [19] presented sample designs for three control systems. [PERSON] et al. explored the use of different passive and semi-active devices on the phase I bridge [20; 21; 22]. [PERSON] et al. also applied passive control devices on this bridge [23; 24; 25]. Some other research was done on the benchmark bridge in phase II, in which the deck was completely isolated from the central pier ([PERSON] et al. [26]). [PERSON] et al. [27] suggested a damper with SMA for controlling the displacements of the second phase of the benchmark bridge. Also, these researchers studied the effect of the temperature changes on the behavior of SMA and their presented device. They observed that the effect of temperature was negligible, and the suggested device was successful in reducing the displacements and residual deformations. [PERSON] et al. [28] investigated the seismic performance of a new SMA device in the second phase of the benchmark bridge. Their passive SMA damper considerably reduced the maximum displacement of the base isolator and displacements. However, this system increased the maximum base shear and overturning moment. Recently, [PERSON] et al. [29] proposed a novel negative stiffness device (NSD) and a damper system for seismic control of the benchmark bridge in phase II and compared its effectiveness with sample passive and semi-active controllers. In their system, passive fluid dampers were used in parallel to the NSD to alleviate excessive displacement. The results showed that the proposed approach was more successful than the two other systems in the reduction of the base shear and mid-span acceleration. However, the semi-active sample controller was slightly more effective in controlling mid-span displacement and bearing deformation. In the literature, the effectiveness of SMA isolators in reducing displacements and residual deformations of bridges is widely studied; however, it usually causes an increase in the base shear and the deck acceleration or may not be effective in large shear deformations. Therefore, to address these issues, in this study, the radius of the wire was optimized using a multi-objective non-dominated sorting genetic algorithm (NSGA-II) to minimize the base shear and the mid-span displacement simultaneously.