In the last decade, opinion mining and sentiment analysis have been the subject of fascinating interdisciplinary research. Alongside the evolution of social media networks, the sheer volume of social media text available for sentiment analysis has increased multi-fold, leading to a formidable corpus. Sentiment analysis of tweets have been carried out to gauge public opinion on breaking news, various policies, legislations, personalities and social movements. Fuzzy logic has been used in the sentiment analysis of twitter data, whereas neutrosophy which factors in the concept of indeterminacy has not been used to analyse tweets. In this paper, the concept of multi refined neutrosophic set (MRNS) with two positive, three indeterminate and two negative memberships is proposed. Single valued neutrosophic set (SVNS), triple refined indeterminate neutrosophic set (TRINS) and MRNS have been used in the sentiment analysis of tweets on ten different topics. Eight of these topics chosen for sentiment analysis are related to Indian scenario and two topics to international scenario. A comparative analysis of the methods show that the approach with MRNS provides better refinement to the indeterminacy present in the data. © 2019 Elsevier B.V.