This paper reports a modified axiomatic foundation of the analytic hierarchy process (AHP), where the reciprocal property of paired comparisons is broken. The novel concept of reciprocal symmetry breaking is proposed to characterize the considered situation without reciprocal property. It is found that the uncertainty experienced by the decision maker can be naturally incorporated into the modified axioms. Some results are derived from the new axioms involving the new concept of approximate consistency and the method of eliciting priorities. The phenomenon of ranking reversal is revisited from a theoretical viewpoint under the modified axiomatic foundation. The situations without ranking reversal are addressed and called ranking equilibrium. The likelihood of ranking reversal is captured by introducing a possibility degree index based on the Kendall's coefficient of concordance. The modified axioms and the derived facts form a novel operational basis of the AHP choice model under some uncertainty. The observations reveal that a more flexible expression of decision information could be accepted as compared to the judgments with reciprocal property.

中文翻译：

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层次分析法的修正公理基础

本文报告了层次分析法 (AHP) 的修改公理基础，其中配对比较的互惠性质被打破。提出了互易对称性破缺的新概念来表征所考虑的没有互易性质的情况。发现决策者所经历的不确定性可以自然地纳入修改后的公理中。一些结果是从新公理中推导出来的，这些新公理涉及近似一致性的新概念和引出优先级的方法。在修正公理基础下，从理论角度重新审视排名反转现象。没有排名反转的情况被处理并称为排名均衡。通过引入基于 Kendall 一致性系数的可能性度指数来捕捉排名反转的可能性。修改后的公理和推导出的事实在某些不确定性下形成了层次分析法选择模型的新操作基础。观察结果表明，与具有互惠属性的判断相比，可以接受更灵活的决策信息表达。