This document reviews recent advancements in multi-objective optimization algorithms for balancing mechanisms, categorizing them based on the type of algorithm employed, with particular emphasis on whether they are inherently multi-objective or rely on a weighted combination of objectives. Furthermore, it examines the mathematical processes involved in deriving balancing equations. Most optimization studies have implemented these algorithms using Cartesian coordinates incorporating angular parameters. However, as demonstrated by [PERSON] et al. [3], it is possible to formulate these equations using _FCC_--eliminating angles and associated trigonometric functions--resulting in more computationally efficient optimization processes. In 1972, [PERSON] et al. [4] presented one of the earliest studies on dynamic balance optimization. They investigated a four-bar mechanism, formulating the objective function as a weighted sum of **Sh**Fs, _ShM_s, and joint loads. Similarly, [PERSON] and [PERSON] [5] analyzed a four-bar mechanism with counterweights, modeling the counterweights as point masses with no predefined upper limits. Their approach involved optimizing a single performance criterion while enforcing constraints on another. In 1976, [PERSON] et al. [6; 7] developed a computer-based system called KOGEOP to optimize linkage dynamics. Their research focused on minimizing an objective function's mean square and peak values, integrating **Sh**Fs, _ShM_s, and driving torque. Additionally, they explored the performance of various search algorithms in the optimization process. [PERSON] et al. [8] applied optimization techniques using a nonlinear programming approach combined with the Simplex algorithm. They developed the MEDIC program, which iteratively adjusts counterweight parameters to achieve optimal balance. Their methodology employs an objective function that assigns weights to multiple criteria. [PERSON] et al. [9] applied multi-objective optimization techniques, including goal programming, goal attainment, the lexicographic method, and the bounded objective function method, to balance a four-bar linkage mechanism. These approaches aim to simultaneously minimize **Sh**Fs and moments by employing weighted-sum methods to construct the objective function. [PERSON] et al. [10] introduced a numerically efficient technique for optimizing the dynamic balance of linkages. Their method integrates [PERSON]'s equations with the virtual work principle to decouple the link equilibrium equations, facilitating an efficient dynamic analysis. Optimization is achieved through a weighted-sum approach to construct the objective function. Multi-objective optimization techniques, which utilize methods not inherently designed for multi-objective problems in combination with mathematical approaches like weighted factors, are widely employed to explore a range of potential solutions and identify the optimal configuration that simultaneously minimizes both the **Sh**F and _ShM_. These techniques are particularly advantageous due to their ability to address the complex, non-linear nature of the balancing problem, offering a set of Pareto-optimal solutions, the most suitable of which can be selected based on specific performance criteria. Such advanced optimization methods have been shown to significantly enhance the dynamic performance of mechanisms, leading to improved stability and extended operational lifespan. Recent studies have introduced various approaches for optimizing the balancing of planar mechanisms using these methods. Key contributions in this field include [11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. All these optimization approaches employ traditional Cartesian coordinates methods to formulate the objective function. [PERSON] et al. [3; 21] introduced a methodology for optimizing the balance of a four-bar mechanism by formulating the objective functions exclusively in _FCC_. Subsequently, they extended this approach to optimize a six-bar Watt linkage [22], employing a similar technique that relies on \(FCC\).