Public transport is the enabler of social and economic development, as it allows the movement of people and provides access to opportunities that otherwise might have been unattainable. Access to public transport is a key aspect of social equity, with step-free access improving the inclusivity of the transport network in particular for mobility impaired population groups. Thus, this study develops a two-step algorithm for determining the optimal allocation of resources for the refurbishment of public transit metro stations to provide step-free accessibility in pubic transport networks. The first step consists of k-shortest path finding algorithm between every origin-destination pair in the network. The non step-free shortest paths are then fed into the second step of the algorithm, a mixed-integer linear optimization problem that selects the station to be refurbished considering inequality as well as population, costs, budget and demand constraints, whilst considering the impact of construction times and disruptions on the journeys. The developed methodology is applied to enhance the accessibility of the London Underground. In doing so, several demographic components, including population in each borough or districts, economic background, and disability reported, are parameterised and factored into the determination of the optimal solution. Our analysis shows in average above 5% increase in step-free trips per borough and 8% increase overall in the London boroughs. Thanks to the refurbishments over 850,000 daily step-free journeys can be made through the London Underground transport system and at least additional 3% of the paths will be made step-free and helping each existing step-free journey to take around 4% shorter journey time.