This paper proposes a new Multi-Objective Equilibrium Optimizer (MOEO) to handle complex optimization problems, including real-world engineering design optimization problems. The Equilibrium Optimizer (EO) is a recently reported physics-based metaheuristic algorithm, and it has been inspired by the models used to predict equilibrium state and dynamic state. A similar procedure is utilized in MOEO by combining models in a different target search space. The crowding distance mechanism is employed in the MOEO algorithm to balance exploitation and exploration phases as the search progresses. In addition, a non-dominated sorting strategy is also merged with the MOEO algorithm to preserve the population diversity and it has been considered as a crucial problem in multi-objective metaheuristic algorithms. An archive with an update function is used to uphold and improve the coverage of Pareto with optimal solutions. The performance of MOEO is validated for 33 contextual problems with 6 constrained, 12 unconstrained, and 15 practical constrained engineering design problems, including non-linear problems. The result obtained by the proposed MOEO algorithm is compared with other state-of-the-art multi-objective optimization algorithms. The quantitative and qualitative results indicate that the proposed MOEO provides more competitive outcomes than the different algorithms. From the results obtained for all 33 benchmark optimization problems, the efficiency, robustness, and exploration ability to solve multi-objective problems of the MOEO algorithm are well defined and clarified. The paper is further supported with extra online service and guideline at https://premkumarmanoharan.wixsite.com/mysite.