The multiple criteria decision making (MCDM) problems indicate the alternatives which have more or less resemblance to each other. An important mathematical tool used by decision-makers (DMs) to quantify these resemblances is the similarity measure (SM). SM is a powerful tool that measures the resemblance more accurately. Mostly, fuzzy sets (FSs) and its extensions handle the vague and uncertain information by considering the membership, non-membership, and indeterminacy degrees whose sum always lies in the interval [0, 1]. However, single-valued neutrosophic sets (SVNSs) and interval-valued neutrosophic sets (IVNSs) have information whose sum is bounded in [0, 3]. In the present work, we extended the SM presented by William and Steel for SVNSs and IVNSs by using the concept of Euclidean distance. The weights of criteria indicate much influence for the selection of the best alternative, sometimes DMs feel hesitation to allocate the weights to the criteria. We applied the linear programming (LP) model to evaluate the weights of the criteria to reduce the hesitancy. Later on, SM is utilized to establish an MCDM model for the selection of the best option. Moreover, the Spearman’s rank correlation coefficient is implemented to analyze the ranking order. Finally, a medical diagnosis example is illustrated for the feasibility and effectiveness of the proposed model.