Nowadays, attractions are focused on Stirling engines because of their low noise, external combustion, and the possibility of employing solar energy. These engines can be designed and applied in cases of low or high-temperature differences, as needed. Evidently, the cylinder's layouts and how they are arranged and the used movement mechanism can affect the engine performance. Experts and Engineers are always looking to increase efficiency and also increase the output work of the Stirling engines. In this study, the dimensional synthesis of the kinematic chain using in the Stirling engine will be discussed. For this purpose, kinematic relationships of different layouts of Stirling are extracted, and by optimizing the mechanism based on the maximum output work, the optimal values of the geometrical parameters of the mechanism and the lengths of its links are obtained. Evolutionary optimization Algorithms, including genetic algorithm, particle swarm optimization, and imperialist competition algorithm methods are employed to optimize the problem, and their obtained results are compared. The problem is solved for four different layouts of the Stirling engine. Based on the results, the output work can be increased 9 to 14 times by varying the geometrical parameters of the Stirling engine mechanism without considering changes in thermodynamic parameters, high-temperature, and low-temperature values. Moreover, average improvement (between three optimization algorithms) for output work is about 13.05, 9.14, 10.71 and 14.36 times for » type with slider-crank, Ø type with slider-crank, Ú type with slider-crank and » type with Ross-Yoke, respectively. Therefore, the » type Stirling engine has better advance than Ø and Ú types, for maximizing the output work based on changing the geometrical parameters.