Polyad inconsistency measure for pairwise comparisons matrices: max-plus algebraic approach
Polyad inconsistency measure for pairwise comparisons matrices: max-plus algebraic approach
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DOI：
10.1007/s12351-020-00547-9
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发表时间：
2020-01
影响因子：
2.7
通讯作者：
H. Goto;Shaohua Wang
中科院分区：
文献类型：
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作者：
H. Goto;Shaohua Wang
在这项研究中，应用最大代数方法来评估成对比较（PC）矩阵的不一致性。输入PC矩阵是灵活的：它可以是不可逆的，不一致的和不完整的。与以往的研究相反，不一致的检查所有polyads不同的周期长度，而典型的评估方法是黑社会为基础的。中心是极大代数，也称为热带几何。首先将输入PC矩阵通过对数映射转换到线性空间，然后求解极大代数中的特征值问题，得到相关图上的最显著不一致圈.最大代数特征值反映了不一致的程度，其指数倒数表示周期不一致的最大几何平均值。因此，新的措施可以理解在几何平均数和polyad上下文中。首次使用max-plus代数的用户可以使用提供矩阵功能的现成计算环境计算近似值。由此产生的措施通过了所有的属性要求，规定在三个相关类别的文献。分析结果突出了我们的方法的意义。
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.