*4.1. Iterative Procedure*

The PCCM decisional tool has been applied to decisional problems [11,12] and the values of consistency are significantly better than those obtained with other GDM approaches (AIJ, Dong procedure [23]), but they are slightly worse in terms of compatibility.

This paper suggests an iterative procedure to improve compatibility without significantly worsening consistency (keeping it below a preset threshold). If the PCCM is constructed by sequentially considering the judgements from the least to the greatest variance, the proposed improvement of compatibility will sequentially consider the judgments with the greatest contribution (participation) to the global compatibility measure employed (4). This value corresponds to the entry *prs* of the PCCM for which:

$$\max\_{k=1}^{r} \sum\_{k=1}^{r} \pi\_k \log^2(a\_{ij}^{(k)} \frac{\upsilon\_j^{(G)}}{\upsilon\_i^{(G)}}) \tag{10}$$

where *v*(G) is the priority vector derived from the PCCM using the RGM method.

The procedure will modify the selected judgement *prs* approaching it to the ratio *w*(*G*) *r w*(*G*) *s* of the priorities derived for the AIJ matrix; following a similar idea of that employed in the Dong procedure [23].

$$p\_{rs}^{\prime} = \left(p\_{rs}\right)^{\theta} \cdot \frac{w\_r^{(G)}}{w\_s^{(G)}}\Big|^{1-\theta}, \; \theta \in [0,1] \tag{11}$$

In any case, the modified value would never exceed the limits of the consistency stability intervals for this judgment, guaranteeing that the level of inconsistency for each decision maker is acceptable.

In what follows, the new iterative procedure for improving compatibility is explained in detail. It is described for any judgement matrix P; it will be applied to P = PCCM, as following Algorithm 1.
