ABSTRACT Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision-making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually yield nonunique solutions; therefore, an approach blending ordinal and cardinal information is needed. In this work, we consider two cascading problems: first, we compute ordinal preferences, maximizing an index that combines ordinal and cardinal information; then, we obtain a cardinal ranking by enforcing ordinal constraints. Notably, we provide a sufficient condition (that is likely to be satisfied in practical cases) for the first problem to admit a unique solution and we develop a provably polynomial-time algorithm to compute it. The effectiveness of the proposed method is analyzed and compared with respect to other approaches and criteria at the state of the art.

中文翻译：

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具有最小加权序数违规的不完整层次分析过程

摘要 不完全的成对比较矩阵提供了一种在决策过程中表达偏好的自然方式。尽管序数信息至关重要，但文献中存在偏见：基数模型占主导地位。序数模型通常会产生非唯一解；因此，需要一种混合序数和基数信息的方法。在这项工作中，我们考虑两个级联问题：首先，我们计算序数偏好，最大化结合序数和基数信息的索引；然后，我们通过执行序数约束来获得基数排名。值得注意的是，我们为第一个问题提供了一个充分条件（在实际情况下可能会得到满足），以允许唯一解决方案，并且我们开发了一个可证明的多项式时间算法来计算它。