Ranking Alternatives Based on the Dominance Degree. an Investigation of Rank Reversal

Qian et al. (2008) and Song et al. (2012) proposed an ordinal pairwise comparison method for ranking alternatives based on the dominance degree. They consider alternatives characterized by interval assessments with respect to several criteria. The dominance degree is a score computed by taking into account all other alternatives. Therefore it is to be expected that the resulting ranking may change when adding or removing alternatives. In other words, rank reversal may occur.
In this paper, we study the occurrence of rank reversal (RR) phenomena when an alternative is removed from or added to the initial set of alternatives. For any pair of alternatives, we study the possibility of RR depending on the position of the removed/added alternative with respect to the pair. We do this not only for the dominance degree introduced by the above authors but also for the dominated degree and the net dominance degree. These methods bear similarities with the PROMETHEE method: a valued binary relation is used to obtain a ranking by computing the outgoing, the entering or the net flow issued from each alternative.
Various aspects of the RR issue in PROMETHEE were studied e.g. by Mareschal et al. (2008), Verly and De Smet (2013), Eppe and De Smet (2017). In the simple setting of the dominance degree score, it is possible to analytically determine whether RR may occur or not depending on the position of the removed/added alternative. We also made extensive simulations in order to assess the frequency of RR in the different positions of the removed/added alternative. It turns out that the net dominance degree is less prone to RR than the dominance or the dominated degrees.
References.
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S. Eppe and Y. De Smet. On the influence of altering the action set on PROMETHEE II's relative ranks. In Evolutionary Multi-Criterion Optimization - 9th International Conference, EMO 2017, Münster, Germany, March 19-22, 2017, Proceedings, pp. 206-220, 2017.
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