Methods to improve the ordinal and multiplicative consistency for reciprocal preference relations

This paper investigates both the ordinal and multiplicative consistencies for reciprocal preference relations simultaneously. First, the relationship among ordinal consistency, acceptable cardinal consistency and perfect cardinal consistency of reciprocal preference relations are analyzed. Then, based on the basic forms of all ordinal inconsistencies, i.e. 3-cycle cases, a new ordinal consistency index, called OCI(R),is introduced to measure the degree of ordinal consistency for reciprocal preference relations. An algorithm to identify and modify ordinal inconsistent elements in reciprocal preference relations is further developed. Afterwards, a multiplicative inconsistency identification and modification method is developed, in which an induced matrix is constructed on the basis of multiplicative consistency property, to improve the consistency of reciprocal preference relations. Meanwhile, some numerical examples are illustrated to show the effectiveness and efficiency of the developed methods Compared with the existing methods, our proposed methods not only significantly improve both the ordinal and multiplicative consistency but also are appropriate for both strict and non-strict reciprocal preference relations, meanwhile retaining the original preference information as much as possible.