*4.2. Case Study*

The previous procedure was applied to a case study which has been widely employed in the literature [11,12,24,25]: three decision makers (DM1, DM2 and DM3) must compare 5 alternatives (A1, ... , A5). The individual pairwise comparison matrices are given in Table 1. The decision makers were given different weights (*<sup>π</sup>*1 = 5; *π*2 = 4; and *π*3 = 2).

**Table 1.** Pairwise comparison matrices for the three decision makers.


Table 2 gives the resulting priorities using the RGM for each of the three individual matrices and their corresponding rankings.


**Table 2.** Individual priority vectors, rankings and GCIs.

The PCCM matrix that is obtained by applying the procedure explained in [11,12] is shown in Table 3.

**Table 3.** Precise Consistency Consensus Matrix (PCCM).


Two other AHP-GDM procedures have been applied: the AIJ that was explained in Section 2, and the Dong procedure [23]. Table 4 shows the priority vectors obtained with the three AHP-GDM procedures. It can be observed that the ranking of the alternatives is the same for the three procedures.


**Table 4.** Priority vectors and rankings for the AHP-GDM procedures.

Table 5 shows the consistency and compatibility indicator values for the three AHP-GDM procedures.

**Table 5.** Consistency and compatibility indicator values (the best value of the methods is in bold, for each indicator).


With respect to the indicators that measure consistency (GCI and CVN), the values obtained with the PCCM are considerably better than those obtained with the other two approaches. The values of the GCI for the AIJ procedure (0.122) and for the Dong procedure (0.069) are, respectively, more than five times (535.7%) and three times (304.2%) greater than that of the PCCM (0.023). The behaviour of the CVN is also better for the PCCM (CVN(PCCM) = 0 while CVN(AIJ) = CVN(Dong) = 0.018).

With respect to the compatibility, the value of the GCOMPI for the AIJ procedure (0.464) is better than those of the PCCM (the value 0.529 is 14% greater than the AIJ) and the Dong procedure (the value 0.472 is 1.7% greater). Finally, in the analysis of the number of violations (ordinal compatibility), the three methods gave the same result (0.136).

Having observed that the PCCM is the procedure (among the three being compared) that achieves the highest value for the GCOMPI indicator, the iterative procedure proposed at Section 4.1 is applied with the aim of detecting an improvement in the compatibility of the PCCM.

The iterative procedure was applied with different values of θ (θ = 0.75; θ = 0.5; θ = 0.25; and θ = 0); the PCCM corresponds to θ = 1. In order to compare the results obtained for the combinations considered, the focus is on the two cardinal indicators—the GCI for consistency and the GCOMPI for compatibility.

Tables 6–9 show the sequence of iterations followed when applying the procedure (each column) and the values obtained for the two indicators for each iteration. The second row specifies the judgement that is modified in the corresponding iteration. The values for the original PCCM are shown in the first column as it corresponds to the starting point of the iterative procedure (t = 0). The modified values for each entry can be seen in Table 10. The values of the GCI and GCOMPI for the judgment (1,4), t = 8, are empty because modifying this judgement will lead to a figure out of the matrix GCIJA. The initial value is maintained and the procedure continues, selecting the following judgement.

**Table 6.** Results for the iterative procedure with θ = 0.75 (the best value of the methods is in bold, for each indicator).


**Table 7.** Results for the iterative procedure with θ = 0.5 (the best value of the methods is in bold, for each indicator).


**Table 8.** Results for the iterative procedure with θ = 0.25 (the best value of the methods is in bold, for each indicator).


**Table 9.** Results for the iterative procedure with θ = 0 (the best value of the methods is in bold, for each indicator).


From Tables 6–9, it is possible to make the following observations:


Figures 1 and 2 give the information provided in Tables 6–9 in the form of graphs; they illustrate the relationship between the two indicator values, GCI and GCOMPI. Figure 1 shows the paths that these values follow in the iterative procedure (as new judgements are modified) for each value of θ separately, while Figure 2 shows all the paths in the same graphic presentation. The graphic visualisations help us to understand the relationship between the two indicators and to identify the steps of the algorithm that provide the biggest changes.

**Figure 1.** GCI and GCOMPI values for the iterative procedure for different values of θ. Red points correspond to the initial values of the iterative process.

**Figure 2.** GCI and GCOMPI values for the iterative procedure (all the values of θ).

It can be observed that for all the values of θ, there is a turning point where the value of the GCI begins to decrease significantly. It corresponds to iterationt=5 when judgment (1,5) is modified. Moreover, when θ decreases, the value of the GCOMPI also decreases and the variability of the GCI increases (Figure 2).

Table 10 includes the modified PCCMs corresponding to the last iteration for each value of θ. Table 11 gives the priorities associated to these matrices; all the priority vectors have the same ranking and their range increases when the value of θ decreases.


**Table 10.** Modified PCCMs for different values of θ.

**Table 11.** Priority vectors and rankings for the modified PCCMs with different values of θ.

