In this study, the multi-objective optimisation of a solid oxide fuel cell (SOFC) system by defining the objective functions to maximise the power output, energy efficiency and exergy efficiency, and minimise the cost under various constraints is conducted. In this regard, energy, exergy and exergoeconomic analyses are performed. Some specific cases are considered and studied parametrically by varying practical operating conditions, namely temperature, pressure, current density and stack assembly thickness. An exergoeconomic model is developed for the system and incorporated into the developed computer program MULOP (multi-objective optimiser) which is based on a genetic algorithm to investigate the system parametrically, depending on the multi-objective optimisation of the objective function ratios. The best result obtained for each objective function is 1.65 W for the power produced, 0.242 and 0.269 for both exergy and energy efficiencies, respectively, and 0.0017 $/W for the cost generated.