Global warming continues to pose significant challenges to our planet, with chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs) used in conventional refrigeration cycles being major contributors. A promising alternative to conventional refrigeration is magnetocaloric cooling technology. Which exploits the magnetocaloric effect, where the temperature of a magnetocaloric material (MCM) changes upon magnetization or demagnetization in response to an applied magnetic field. A crucial component of magnetocaloric cooling devices is the active magnetic regenerator (AMR), which facilitates heat transfer between the MCM and the refrigerant. By cyclically exciting a MCM, a refrigeration cycle can be established. In a rotary magnetocaloric cooling device, when a refrigerant, such as water, flows through the AMRs, its temperature increases or decreases when the MCM is magnetized or demagnetized. Numerical simulations provide a powerful tool for predicting the performance of such devices and for minimizing material usage and manufacturing costs, in addition to improve the overall performance when optimization techniques are applied. Recent studies incorporating shape and topology optimization have demonstrated substantial performance improvements for magnetocaloric systems \cite{Wiesheu2023}. However , multiphysics modeling of the AMR or multi-objective optimization strategies have not been included. In this work, we employ the isogeometric finite element method to optimize a rotary magnet assembly in 2D in terms of its shape and parameters, while including the influence of the AMRs. Our simulation includes a one-way coupling to a thermal simulation of the AMR and a nonlinear magnetic model of the MCM. The optimization considers multiple objectives, including the magnetic field profile, cost, coefficient of performance, and torque. We propose a comprehensive numerical study to determine the optimal shape and dimensions for the permanent magnets, soft iron components, and AMRs, as well as the optimal number of AMRs. This work aims to enhance the overall performance and efficiency of magnetocaloric cooling systems.