*Article* **The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices**

#### **Juan Aguarón, María Teresa Escobar, José María Moreno-Jiménez \* and Alberto Turón**

Grupo Decisión Multicriterio Zaragoza (GDMZ), Facultad de Economía y Empresa, Universidad de Zaragoza, Gran Vía 2, 50005 Zaragoza, Spain; aguaron@unizar.es (J.A.); mescobar@unizar.es (M.T.E.); turon@unizar.es (A.T.)

**\*** Correspondence: moreno@unizar.es

Received: 5 May 2020; Accepted: 3 June 2020; Published: 6 June 2020

**Abstract:** The paper presents the Triads Geometric Consistency Index (*T*-*GCI*), 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*-*GCI* 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 (*GCI*) and this allows the utilisation of the inconsistency thresholds as well as the properties of the *GCI* when using the *T*-*GCI*. In addition, the decision tools developed for the *GCI* can be used when working with triads (*T*-*GCI*), 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*-*GCI* and *GCI*) 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*-*GCI*. 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.

**Keywords:** Analytic Hierarchy Process (AHP); consistency; Geometric Consistency Index (GCI); triads; cycles
