  Multi-Criteria Decision Analysis (MCDA) is extensively used across diverse
industries to assess and rank alternatives. Among numerous MCDA methods
developed to solve real-world ranking problems, TOPSIS remains one of the most
popular choices in many application areas. TOPSIS calculates distances between
the considered alternatives and two predefined ones, namely the ideal and the
anti-ideal, and creates a ranking of the alternatives according to a chosen
aggregation of these distances. However, the interpretation of the inner
workings of TOPSIS is difficult, especially when the number of criteria is
large. To this end, recent research has shown that TOPSIS aggregations can be
expressed using the means (M) and standard deviations (SD) of alternatives,
creating MSD-space, a tool for visualizing and explaining aggregations. Even
though MSD-space is highly useful, it assumes equally important criteria,
making it less applicable to real-world ranking problems. In this paper, we
generalize the concept of MSD-space to weighted criteria by introducing the
concept of WMSD-space defined by what is referred to as weight-scaled means and
standard deviations. We demonstrate that TOPSIS and similar distance-based
aggregation methods can be successfully illustrated in a plane and interpreted
even when the criteria are weighted, regardless of their number. The proposed
WMSD-space offers a practical method for explaining TOPSIS rankings in
real-world decision problems.
