Abstract
The paper presents the Triads Geometric Consistency Index ( T - G C I ), a measure for evaluating the inconsistency of the pairwise comparison matrices employed in the Analytic Hierarchy Process (AHP). Based on the Saaty’s definition of consistency for AHP, the new measure works directly with triads of the initial judgements, without having to previously calculate the priority vector, and therefore is valid for any prioritisation procedure used in AHP. The T - G C I is an intuitive indicator defined as the average of the log quadratic deviations from the unit of the intensities of all the cycles of length three. Its value coincides with that of the Geometric Consistency Index ( G C I ) and this allows the utilisation of the inconsistency thresholds as well as the properties of the G C I when using the T - G C I . In addition, the decision tools developed for the G C I can be used when working with triads ( T - G C I ), especially the procedure for improving the inconsistency and the consistency stability intervals of the judgements used in group decision making. The paper further includes a study of the computational complexity of both measures ( T - G C I and G C I ) which allows selecting the most appropriate expression, depending on the size of the matrix. Finally, it is proved that the generalisation of the proposed measure to cycles of any length coincides with the T - G C I . It is not therefore necessary to consider cycles of length greater than three, as they are more complex to obtain and the calculation of their associated measure is more difficult.
Highlights
Since the origin of the species, humans have used pairwise comparisons to choose between tangible elements
The current paper presents two new inconsistency measures based, respectively, on triads and cycles, and proves that, under certain conditions, both coincide with the Geometric Consistency Index (GCI) [10]
The evaluation of the consistency of decision makers when incorporating their preferences in the Analytic Hierarchy Process (AHP), in other words, when eliciting their judgements in pairwise comparison matrices, is one of the most outstanding characteristics of this multi-criteria technique
Summary
Since the origin of the species, humans have used pairwise comparisons to choose between tangible elements This has generally involved taking the lesser element as a reference unit, and indicating how many times the greater element “includes” or “dominates” the lesser. Following the substantive rationality that was prevalent in the traditional scientific method, pairwise comparisons were generally utilised to order tangible elements, based on an objective attribute with a known unit of reference or measurement scale. Consistency in AHP is defined in terms of triads of the elements of the matrix A = ( aij ), its evaluation, following Saaty’s proposal (and the majority of methods that are traditionally followed for the evaluation of inconsistency), depends on the prioritisation procedure that is employed, that is to say, the measurement of inconsistency is linked to the prioritisation procedure.
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