**1. Introduction**

One of the multicriteria decision making techniques that best responds to the challenges and needs of the Knowledge Society [1], especially the consideration of intangible aspects and decision-making with multiple actors, is the Analytical Hierarchy Process (AHP). AHP was proposed by Thomas L. Saaty in the early 1970s (20th century) [2]. This multicriteria technique incorporates the intangible aspects associated with the human factor through the use of pairwise comparisons. In group decision-making, where all the actors work as a single unit, AHP usually follows one of the two most traditional approaches [3–5]: the Aggregation of Individual Judgements (AIJ) and the Aggregation of Individual Priorities (AIP).

Both methods present two important limitations that have been addressed in some of the most recent proposals: the certainty of the data and the use of the geometric mean as the synthesising procedure of the considered values (judgments in AIJ and priorities in AIP). Escobar and Moreno-Jiménez [6] consider the principle of certainty and incorporate the context effect through the procedure called the Aggregation of Individual Preference Structures (AIPS). Altuzarra et al. [7] advance a Bayesian approach as a prioritisation procedure and a group decision-making aggregation procedure.

The concept of consistency [2] is one of the characteristics that distinguishes AHP from the other multicriteria techniques and gives coherence to the method; Moreno-Jiménez et al. [8–10] used this to design a new procedure for group decision making: the Consistency Consensus Matrix (CCM). Under certain conditions, the CCM automatically provides an interval judgement matrix where each entry reflects the range of values in which all decision makers would simultaneously be consistent in their initial matrices.

One limitation of this new decision-making tool is that the CCM is sometimes incomplete. The Precise Consistency Consensus Matrix (PCCM) has been proposed [11,12] to respond to this limitation by including more judgments in the group consensus matrix and allowing decision makers to have different weights assigned in the resolution of the problem. This new consensus matrix has, by construction, demonstrated good behaviour with respect to consistency, but it can be improved with respect to compatibility, understood as the discrepancy between the individual positions and the collective position that synthesises them.

This work presents a procedure to improve the compatibility of the PCCM guaranteeing that the consistency does not exceed a predetermined level. Compatibility is improved by modifying those judgments of the PCCM that most contribute to the global compatibility, with the idea of reducing this contribution. The combination of what happens to the consistency and the compatibility will allow selecting, as the preferred option, the one that most improves the cumulative relative changes of the two criteria (consistency and compatibility).

The paper is structured as follows: Section 2 gives the background to the developments; Section 3 describes the PCCM and the algorithm that solves the optimisation problem that aims to find the precise value that maximises the slack of consistency that remains free for the following steps when the actors have different weights; Section 4 explains the proposal for improving the compatibility of the PCCM and applies it to a case study; Section 5 highlights the most important conclusions of the study.
