This paper presents a multi-objective optimal sizing of battery storage system (BSS) and rooftop solar photovoltaic (PV) for a grid-connected household. The objective functions are selected as cost of electricity (COE) and grid dependency (GD). COE is the sum of annual cost over total electricity usage in a year. GD is the ratio of annual purchased electricity from the main grid over the yearly electricity demand of the house. The main aim of this study is to render insights for the electricity consumers to select the optimal capacity of PV and BSS by a compromise between cost and dependency on the grid. The non-dominated sorting genetic algorithm II (NSGA-II) is conducted to solve the multi-objective problem. A grid-connected home in Australia is studied by incorporating actual data of weather and electricity consumption. It is found that by decreasing the GD, the COE increases and vice versa. The minimum COE in the system results in a GD of 0.57 p.u. To verify the optimization with NSGA-II, the obtained results are compared with those of multi-objective particle swarm algorithm.