**2. Background**

### *2.1. Multiactor Decision Making (MADM)*

As previously mentioned, consistency (coherence of decision makers in eliciting their judgments) and a good behaviour in the decision-making with multiple actors are two of the most important properties for multicriteria decision making techniques. [6,13] distinguish three areas in multi-actor decision-making: (i) Group Decision Making (GDM); (ii) Negotiated Decision Making (NDM); and (iii) Systemic Decision Making (SDM).

In GDM, individuals work together in pursuit of a common goal under the *principle of consensus*. Consensus refers to the approach, model, tools, and procedures for deriving the collective position or final group priority vector.

NDM is based on the *principle of agreement* and the assumption that all the actors follow the same scientific approach. The individuals resolve the problem separately, the zones of agreemen<sup>t</sup> and disagreement between the actors are identified and agreemen<sup>t</sup> paths (sometimes known as consensus paths) are constructed by changing, in a personal, semiautomatic or automatic way, one or several judgements.

SDM follows the *principle of tolerance*: the individual acts independently and the individual preferences, expressed as probability distributions, are aggregated to form a collective one—the tolerance distribution. This new approach integrates all the preferences, even if they are provided from different 'individual theoretical models'; the only requirement is that they must be expressed as some kind of probability distribution.

The systemic situation for dealing with multiactor decision making allows capturing the holistic vision of reality and the subjacent ideas of lateral thinking [14]. The information provided by the tolerance distribution can be used to construct tolerance paths to produce a more democratic and representative final decision. In other words, a decision will be accepted by a greater number of actors or by a number of actors with greater weighting in the decisional process [15,16].

### *2.2. Analytic Hierarchy Process*

The Analytic Hierarchy Process is one of the most widely utilised multicriteria decision making techniques. Its methodology consists of three phases [2]: (a) modelling, (b) valuation, and, (c) prioritisation and synthesis.


In the AHP-group decision making context, the two techniques traditionally used are: (i) the Aggregation of Individual Judgements (AIJ) and (ii) the Aggregation of Individual Priorities (AIP); firstly, it is necessary to specify the notation that will be utilised. Given a local context (one criteria in the hierarchy) with *n* alternatives ( *A*1,..., *An*) and *r* decision makers ( *D*1,...,*Dr*), let *A*(*k*) = (*a* (*k*) *ij* ) be the pairwise comparison matrix of decision-maker *Dk* (*k* = 1,...,*r*; *i*, *j* = 1,...,*n*) and *πk* be the relative importance in the group ( *πk* ≥ 0, *r* ∑ *πk* = 1).

The priorities following the two approaches AIJ and AIP are obtained as follows:

*k*=1


Using the Weighted Geometric Mean Method (WGMM) as the aggregation procedure, the group judgement matrix and the group priority vector are given by:

• A(G) = (*a* (*G*) *ij* ) with *a* (*G*) *ij* = ∏*r k*=1 (*a* (*k*) *ij* ) *<sup>π</sup>k* , *i*, *j* = 1, . . . , *n* •w(G/P) =(*w*(*G*/*P*) )with*w*(*G*/*P*) *i* =∏*r k*=1(*w*(*k*) )*<sup>π</sup>k i*=1,

When the WGMM aggregation procedure is employed and the priorities are obtained using the RGM, the two approaches, AIJ and AIP, provide the same solution [17,18].

,  . . . , *n*

*i*

### *2.3. Consistency and Compatibility in AHP*

*i*

AHP allows for the evaluation of the consistency of the decision-maker when the judgements are introduced into the pairwise comparison matrices. Saaty [2] defined consistency in AHP as the cardinal transitivity of the judgements included in the pairwise comparison matrices, that is to say, the reciprocal pairwise comparison matrix *An*x*n* = (*aij*) is *consistent* if ∀*i*,*j*,*k* = 1,...,n satisfies *aij*·*ajk* = *aik*.

*Consistency* is associated with the (internal) coherence of the decision makers when their judgements are considered in the pairwise comparison matrices. Consistency is usually evaluated —depending on the prioritisation procedure that is used— as the 'representativeness' of the local priorities vector derived from the pairwise comparison matrices (*aij* is an estimation of *wi*/*wj*).

In the case of the EGV and RGM, the inconsistency indicators are given, respectively, by the Consistency Index (CI) and the Geometric Consistency Index (GCI) [19]:

$$CI = \frac{1}{n(n-1)} \sum\_{i,j=1}^{n} (\epsilon\_{ij} - 1) \tag{1a}$$

$$GCI = \frac{2}{(n-1)(n-2)} \sum\_{i$$

where *eij* = *aij*(*wj*/*wi*). Obviously, if the matrix is consistent, both indicators of inconsistency are null, thus errors *eij* =1(*aij* = *wi*/*wj*).

The *Consistency Interval Judgement matrix for the group* (GCIJA) is an interval matrix GCIJA = ([*aij*, *aij*]) where the entries correspond to the range of values for which all the decision makers will not exceed the maximum inconsistency allowed and will belong to the Saaty's fundamental scale range of values [1/9, 9]. 

The values that determine the limits of each entry of the GCIJA are given by *aij* = *Max k a*(*k*) *ij* ; 1/9and *aij*= *Mink a*(*k*) *ij* ; 9, where *a*(*k*) *ij* and *a*(*k*) *ij* are the limits of the individual consistency stability interval for *a*(*k*) *ij* ([*a*(*k*) *ij* , *a*(*k*) *ij* ]) with Δ(k) = GCI\* − GCI(*<sup>k</sup>*), GCI\* being the maximum inconsistency allowed for the problem and GCI(*k*) the Geometric Consistency Index for the individual matrix A(*k*) [20].

*Compatibility* refers to the (internal) coherence of the group when selecting its priority vector (*w*(G) = (*<sup>w</sup>*1(G), ... ,*wn*(G))), that is to say, its representativeness in relation to the individual positions (*w*(*k*) = (*<sup>w</sup>*1(*<sup>k</sup>*), ... ,*wn*(*<sup>k</sup>*))). To evaluate the compatibility of an individual *k* (*w*(*<sup>k</sup>*)), *k* = 1, ... ,*r*, with the collective position or group priority vector(*w*(G)), it is sufficient to adapt the previous expression of the GCI, taking *eij* = *a*(*k*) *ij* (*w*(*G*) *j* /*w*(*G*) *i* ) in local context or *eij* = (*w*(*k*) *i* /*w*(*k*) *j* )(*w*(*G*) *j* /*w*(*G*) *i* ) in a global one. The concept of compatibility reflects the distance between the individual and collective positions and is calculated automatically, without the express intervention of the individual with the exception of the emission of the initial judgements of the pairwise comparison matrices. [21] advanced the *Geometric Compatibility Index* (GCOMPI) in order to evaluate the compatibility of the individual positions with respect of the collective position provided by any of the existing procedures. The expression of the GCOMPI for a decision maker *k* in a local context (one criterion) is given by:

$$\text{GCOMPI}^{(k,G)} = \frac{2}{(n-1)(n-2)} \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \log^2(a\_{ij}^{(k)} \frac{w\_j^{(G)}}{w\_i^{(G)}}) \tag{2}$$

and in a global context (hierarchy) by:

$$\text{GCOMPI}^{(k,G)} = \frac{2}{(n-1)(n-2)} \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \log^2(\frac{w\_i^{(k)}}{w\_j^{(k)}} \frac{w\_j^{(G)}}{w\_i^{(G)}}) \tag{3}$$

The *GCOMPI* for the group is given by:

$$\text{GCOMPI}^{(G)} = \sum\_{k=1,\dots r} \pi\_k \text{GCOMPI}^{(k,G)} = \frac{2}{(n-1)(n-2)} \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \sum\_{k=1}^{r} \pi\_k \log^2(a\_{ij}^{(k)} \frac{w\_j^{(G)}}{w\_j^{(G)}}) \tag{4}$$

In addition to the use of the GCI and the GCOMPI, two more indicators are used in the literature to compare the behaviour of the different procedures with respect to consistency and compatibility [11,12]: the Number of Violations in Consistency (CVN) for the consistency; and the Number of Violations in Priorities (PVN) for the compatibility.

The CVN considers the mean number of entries of the group pairwise comparison matrix that do not belong to the corresponding consistency stability interval judgement of each individual, calculated for the inconsistency threshold considered in the problem. The Consistency Violation Number (CVN) for the group is given by CVN(G) = <sup>Σ</sup>*kπkCVN*(*k*,G), where

$$\text{CVN}^{(k,G)} = \frac{2}{n(n-1)} \sum\_{i$$

and

$$I\_{\vec{i}\vec{j}}(\mathcal{C}I\!/A^{(k)}\!/A^{(G)}) = \begin{cases} 1 & \text{if } a\_{\vec{i}\vec{j}}^{(G)} \notin \left[\underline{a}\_{\vec{i}\vec{j}}^{(k)}, \overline{a}\_{\vec{i}\vec{j}}^{(k)}\right] \\ 0 & \text{otherwise} \end{cases} \tag{6}$$

The PVN measures the ordinal compatibility of each AHP-GDM procedure by means of the minimum number of violations [22].

The Priority Violation Number (PVN) for the group is given by PVN(G) = <sup>Σ</sup>*kπkPVN*(k,G), where

$$\text{PVN}^{(k,\mathbb{G})} = \text{PVN}(A^{(k)}/A^{(\mathbb{G})}) \ = \frac{2}{(n-1)(n-2)} \sum\_{i$$

and

$$I\_{ij}(A^{(k)}/A^{(G)}) = \begin{cases} 1 & \text{if } a\_{ij}^{(k)} > 1 \text{ and } w\_i^{(G)} < w\_j^{(G)} \\ 0.5 & \text{if } a\_{ij}^{(k)} = 1 \text{ and } w\_i^{(G)} \neq w\_j^{(G)} \\ 0.5 & \text{if } a\_{ij}^{(k)} \neq 1 \text{ and } w\_i^{(G)} = w\_j^{(G)} \\ 0 & \text{otherwise} \end{cases} \tag{8}$$

### **3. The Precise Consensus Consistency Matrix (PCCM)**

Moreno-Jiménez et al. [9,10] proposed a decisional tool, the Consistency Consensus Matrix (CCM), which identifies the core of consistency of the group decision using an interval matrix that may not be complete or connected. In [12], the same authors refined this tool and introduced the PCCM, which selects a precise value for each interval judgement in such a way that the quantity of slack that remains free for successive algorithm iterations is the maximum possible.

Escobar et al. [11] extended the PCCM to allow the assignment of different weights to the decision makers and to guarantee that the group consensus values were acceptable to the individuals in terms of inconsistency. In the same work, these authors put forward a number of methods for completing the PCCM matrix if it were incomplete.

The improved version of the algorithm for constructing the PCCM proposed in [11] starts by calculating the variance of the logarithms of the corresponding judgements, taking into account the fact that decision makers may have different weights. It also provides (Step 1) the initial Consistency Stability Intervals [20] for the individuals and for the group (GCIJA). The judgement with least variance (Step 2) that has a non-null intersection for the initial individual consistency stability intervals is selected. The consistency stability intervals for each decision maker are calculated for this judgement (Step 3) and the intersection of all these intervals is obtained (Step 4). In this common interval, it is guaranteed that the individual judgements can oscillate without the GCI exceeding a previously fixed level of inconsistency. The intersection of the previous interval with the range of values [1/9,9] and the initial consistency stability intervals is then calculated (Step 5). This avoids taking a value distanced from the initial judgements of all the decision makers more than the amount allowed for the fixed inconsistency level. The algorithm determines a precise value that belongs to the common interval

(Step 6). Any judgment in this interval will have an acceptable inconsistence. Some of the matrices will be more inconsistent than others and they will therefore admit less slack for the following iterations. In order to address this point, the algorithm selects the value that provides the greatest slack for the most inconsistent matrix (the value that minimises the GCI of the most inconsistent matrix). Finally, the value obtained is included as an entry of the PCCM and serves to update the initial individual judgment matrices (Step 7). The detailed version of the algorithm can be seen in [11].

The consideration of different weights for the decision makers has notably increased the difficulty of the optimisation model (9) solved in Step 6. This non-trivial optimisation problem is solved using an iterative procedure which searches for the intersection points of the parabolas (the second order equations associated with the GCI(A(k)) functions).

$$\text{Min}\_{\mathfrak{a}\_{rs}} \text{Max}\_{k} \pi\_{k} \left( \text{GCI}^{(k)} + \frac{2}{n(n-1)} \left[ \left( a\_{rs} - a\_{rs}^{(k)} \right)^{2} + \frac{2n}{n-2} \left( a\_{rs} - a\_{rs}^{(k)} \right) \varepsilon\_{rs}^{(k)} \right] \right) \tag{9}$$

with *αrs* ∈ )log *<sup>a</sup>trs*, log *<sup>a</sup>trs*\*, where *αrs* = log *ars*, *α*(*k*) *rs* = log *a*(*k*) *rs* and *ε*(*k*) *rs* = log *e*(*k*) *rs*

When all decision makers have the same weight (initial version of the algorithm [12]), all the parabolas have the same 'width' (the same coefficient of the quadratic term). In that situation, the parabolas may intersect in one point or none. But when the decision makers have different weights [11], the parabolas may have different coefficients for their respective quadratic terms. Each pair of parabolas may intersect in one or two points, or none. Moreover, in this case, some parabolas can be tangential. The resolution of the optimisation model (9) should consider all these possibilities and carefully analyse each intersection point. A more detailed explanation of the procedure followed to solve this optimisation model (9) can be seen in Appendix A.

### **4. Improving the PCCM's Compatibility**
