Abstract
A max-algebraic approach is applied in this study to assess the inconsistency of pairwise comparisons (PC) matrices. An input PC matrix is flexible: It can be nonreciprocal, inconsistent, and incomplete. Contrary to previous studies, inconsistency is examined for all polyads with varying cycle lengths, while typical assessment methods are triad based. Central is max-plus algebra, also known as tropical geometry. An input PC matrix is converted by a logarithmic mapping into linear space, an eigenvalue problem in max-plus algebra is then solved to obtain the most significant inconsistent cycle across the associated graph. The max-algebraic eigenvalue reflects the extent of inconsistency, and its exponential inversion signifies the maximum geometric mean of cycle inconsistencies. The new measure can thus be comprehended in both geometric mean and polyad contexts. First-time users of max-plus algebra can compute an approximate value with an off-the-shelf computational environment that provides matrix functionalities. The resulting measure passes all property requirements stipulated in three related categories of literature. Analytical results highlight the significance of our method.
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Acknowledgments
We would like to acknowledge Professor Waldemar W. Koczkodaj at Laurentian University, Canada, for his useful advice. Additionally, we would like to thank Professor Nikolai Krivulin at St. Petersburg State University, Russia, for his valuable comments regarding mathematical validation. We look forward to collaborating with the abovementioned professors in the future. Hiroyuki Goto is financially supported in part by two research Grants, JSPS KAKENHI Grant Numbers JP18K01126 and JP18K04628.
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Goto, H., Wang, S. Polyad inconsistency measure for pairwise comparisons matrices: max-plus algebraic approach. Oper Res Int J 22, 401–422 (2022). https://doi.org/10.1007/s12351-020-00547-9
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DOI: https://doi.org/10.1007/s12351-020-00547-9