Consistency and transitivity are important and leading research topics in the study of decision-making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than consistency and stronger than transitivity and ensures that vectors associated with a matrix, by means of a strictly increasing synthesis functional, provide a preference order, on the related set of decision elements, equal to the actual ranking. This notion extends, to the case in which the decision-maker can be indifferent between two or more alternatives/criteria, weak consistency introduced in previous papers under constraint of no indifference. Finally, we introduce an order relation on the rows of the matrix, that is, a simple order if and only if weak consistency is satisfied; this simple order allows us to easily determine the actual ranking on the set of decision elements
Weak Consistency for Ensuring Priority Vectors Reliability / Cavallo, Bice; D'Apuzzo, Livia; Basile, Luciano. - In: JOURNAL OF MULTICRITERIA DECISION ANALYSIS. - ISSN 1057-9214. - 23:(2016), pp. 126-138. [10.1002/mcda.1559]
Weak Consistency for Ensuring Priority Vectors Reliability
CAVALLO, BICE
;D'APUZZO, LIVIA;BASILE, LUCIANO
2016
Abstract
Consistency and transitivity are important and leading research topics in the study of decision-making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than consistency and stronger than transitivity and ensures that vectors associated with a matrix, by means of a strictly increasing synthesis functional, provide a preference order, on the related set of decision elements, equal to the actual ranking. This notion extends, to the case in which the decision-maker can be indifferent between two or more alternatives/criteria, weak consistency introduced in previous papers under constraint of no indifference. Finally, we introduce an order relation on the rows of the matrix, that is, a simple order if and only if weak consistency is satisfied; this simple order allows us to easily determine the actual ranking on the set of decision elementsI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.