Changes in the energy sector, associated with the move away from fossil fuels, pose a challenge for appropriate thermal energy management in residential buildings. The important element to deal with the variability of renewable energy in thermal systems is latent heat thermal energy storage. Due to the low thermal conductivity of phase change materials, a number of techniques are proposed to enhance the heat transfer process. In this research, the global sensitivity of fin geometrical parameters on the melting and solidification times and energy efficiency of these processes was investigated. The computational model of the phase change was developed using the finite volume method with the enthalpy-porosity model and Boussinesq approximation. Numerical simulations were carried out according to the design of experiments technique. The multi-dimensional response surface was developed, and the multi-objective optimisation was done. The research shows that the melting process is most influenced by the position of the top fin (α angle) and the solidification process by the position of the bottom fin (γ angle). The angle of the tree fin (β) has a different effect on both processes, with the energy efficiency decreasing during melting and increasing during solidification. Maximum values for the energy efficiencies of melting (ηm=0.973) and solidification (ηs=0.988) were obtained for α=18.2°, β=89.0°, L=10.7 mm and γ=21.0°.