In a resource-constrained society in developing nations like India, the importance of health care infrastructure allocation is often undermined and overlooked. This paper aims to quantify the gap in accessing affordable health care facilities faced by the socio-economically weaker sections of society. Majority of the older cities in India have a central core and surrounding peri-urban areas, which were added later on to minimize the stress of urban expansion and to provide adequate infrastructure. This research states that installation of new facilities for catering to the growing needs of citizens in expanding cities is the need of the hour. We propose a novel technique for maximizing the health care coverage of the peri-urban areas by establishing the minimum number of new public health care facilities. Our aim is to suggest strategies for efficient implementation of policies such as the National Urban Health Mission and the National Health Policy. We considered Kolkata Municipal Corporation, India as a case study to assess our proposed methodology. The southern periphery, which was added in 1981 and has currently expanded again in 2015, is completely devoid of public health care facilities. Our optimization model showed that 13 new public facilities are required in the southern periphery, out of which five are extremely critical. The rise in health care coverage from 76.19 to 90.05 % by taking only the southern periphery into consideration shows the impact of the new facilities placed according to the proposed framework.