The helium cryo-plant is an indispensable subsystem for the application of low-temperature superconductors in large-scale scientific facilities. However, it is important to note that the cryo-plant requires stable operation and consumes a substantial amount of electrical power for its operation. Additionally, the construction of the cryo-plant incurs significant economic costs. To achieve the necessary cooling capacity while reducing power consumption and ensuring stability and economic feasibility, constrained multi-objective optimization is performed using the interior point method in this work. The Collins cycle, which uses liquid nitrogen precooling, is selected as the representative helium liquefaction cycle for optimization. The discharge pressure of the compressor, flow ratio of turbines, and effectiveness of heat exchangers are taken as decision parameters. Two objective parameters, cycle exergy efficiency (ηEx,cycle) and liquefaction rate (m˙L), are chosen, and the wheel tip speed of turbines and UA of heat exchangers are selected as stability and economic cost constraints, respectively. The technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) is utilized to select the final optimal solution from the Pareto frontier of constrained multi-objective optimization. Compared to the constrained optimization of ηEx,cycle, the TOPSIS result increases the m˙L by 23.674%, but there is an 8.162% reduction in ηEx,cycle. Similarly, compared to the constrained optimization of m˙L, the TOPSIS result increases the ηEx,cycle by 57.333%, but a 10.821% reduction in m˙L is observed. This approach enables the design of helium cryo-plants with considerations for cooling capacity, exergy efficiency, economic cost, and stability. Furthermore, the wheel tip speed and UA of heat exchangers of the solutions in the Pareto frontier are also studied.