that it is both technically valid and practically useful, the method does have its critics. In the early 1990s a series of debates between critics and proponents of AHP was published in Management Science and The Journal of the Operational Research Society, two prestigious journals where Saaty and his colleagues had considerable influence. These debates seem to have been settled in favor of AHP: An in-depth paper discussing and rebutting the academic criticisms of AHP was published in Operations Research in 2001. A 2008 Management Science paper reviewing 15 years of progress in all areas of Multicriteria Decision Making showed that AHP publications have far outnumbered those in any other area, characterizing their growth as "enormous." Also in 2008, the major society for operations research, the Institute for Operations Research and the Management Sciences formally recognized AHP's broad impact on its fields. Occasional criticisms still appear. A 1997 paper examined possible flaws in the verbal (vs. numerical) scale often used in AHP pairwise comparisons. Another from the same year claimed that innocuous changes to the AHP model can introduce order where no order exists. A 2006 paper found that the addition of criteria for which all alternatives perform equally can alter the priorities of alternatives. In 2021, the first comprehensive evaluation of the AHP was published in a book authored by two academics from Technical University of Valencia and Universidad Politécnica de Cartagena, and published by Springer Nature. Based on an empirical investigation and objective testimonies by 101 researchers, the study found at least 30 flaws in the AHP and found it unsuitable for complex problems, and in certain situations even for small problems. Rank reversal Decision making involves ranking alternatives in terms of criteria or attributes of those alternatives. It is an axiom of some decision theories that when new alternatives are added to a decision problem, the ranking of the old alternatives must not change — that "rank reversal" must not occur. There are two schools of thought about rank reversal. One maintains that new alternatives that introduce no additional attributes should not cause rank reversal under any circumstances. The other maintains that there are some situations in which rank reversal can reasonably be expected. The original formulation of AHP allowed rank reversals. In 1993, Forman introduced a second AHP synthesis mode, called the ideal synthesis mode, to address choice situations in which the addition or removal of