The Hellinger distance in Multicriteria Decision Making: An illustration to the TOPSIS and TODIM methods

Due to the difficulty in some situations of expressing the ratings of alternatives as exact real numbers, many well-known methods to support Multicriteria Decision Making (MCDM) have been extended to compute with many types of information. This paper focuses on the information represented as probability distribution. Many of the methods that deal with probability distribution use the concept of stochastic dominance, which imposes very strong restrictions to differentiate two probability distributions, or uses the probability distributions to obtain a quantity that will be used to rank the alternatives. This paper brings the Hellinger distance concept to the MCDM context to assist the models to deal with probability distributions in a direct way without any transformation. Transformations in the data or summary quantities may miss represent the original information. For direct comparisons among probability distributions we use the stochastic dominance degree (SDD). We illustrate how simple it can be to adapt the existing methods to deal with probability distributions through the Hellinger distance and SDD by adapting the TOPSIS and TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) methods.