  All intuitionistic fuzzy TOPSIS methods contain two key elements: (1) the
order structure, which can affect the choices of positive ideal-points and
negative ideal-points, and construction of admissible distance/similarity
measures; (2) the distance/similarity measure, which is closely related to the
values of the relative closeness degrees and determines the accuracy and
rationality of decision-making. For the order structure, many efforts are
devoted to constructing some score functions, which can strictly distinguish
different intuitionistic fuzzy values (IFVs) and preserve the natural partial
order for IFVs.This paper proves that such a score function does not exist,
namely the application of a single monotonous and continuous function does not
distinguish all IFVs. For the distance or similarity measure, some examples are
given to show that classical similarity measures based on the normalized
Euclidean distance and normalized Minkowski distance do not meet the axiomatic
definition of intuitionistic fuzzy similarity measures. Moreover,some
illustrative examples are given to show that classical intuitionistic fuzzy
TOPSIS methods do not ensure the monotonicity with the natural partial order or
linear orders, which may yield some counter-intuitive results. To overcome the
limitation of non-monotonicity, we propose a novel intuitionistic fuzzy TOPSIS
method,using three new admissible distances with the linear orders measured by
a score degree/similarity function and accuracy degree, or two aggregation
functions, and prove that the proposed TOPSIS method is monotonous under these
three linear orders.} This is the first result with a strict mathematical proof
on the monotonicity with the linear orders for the intuitionistic fuzzy TOPSIS
method.
