A family of fuzzy multi-criteria sorting models, FTOPSIS-Sort, as a fuzzy extension of Multi-Criteria Decision Analysis (MCDA) ordinary method TOPSIS, is introduced and analyzed. Models from this family differ by approaches to determining functions of fuzzy numbers (approximate computations, standard fuzzy arithmetic, and transformation method) and by methods for ranking of fuzzy numbers (two defuzzification based ranking methods are considered). The features of developing and adjusting Fuzzy TOPSIS (FTOPSIS) models to sorting problematic are presented. The developed models are implemented in the case study on a healthcare supply chain alternative selection problem. For exploring distinctions in sorting alternatives by FTOPSIS-Sort models, the special algorithms have been developed along with their integrating with Monte Carlo simulation of a large number of input scenarios, each of which is a separate (and independent of the others) multicriteria problem on sorting alternatives. The results of such an analysis demonstrate a significant distinction in sorting alternatives by different FTOPSIS-Sort models. The latter has theoretical, methodological, and applied significance within the use of Fuzzy TOPSIS (Fuzzy MCDA) sorting models.