Nonlinear preferences in group decision‐making. Extreme values amplifications and extreme values reductions

Consensus Reaching Processes (CRPs) deal with those group decision‐making situations in which conflicts among experts' opinions make difficult the reaching of an agreed solution. This situation, worsens in large‐scale group decision situations, in which opinions tend to be more polarized, because in problems with extreme opinions it is harder to reach an agreement. Several studies have shown that experts' preferences may not always follow a linear scale, as it has commonly been assumed in previous CRP. Therefore, the main aim of this paper is to study the effect of modeling this nonlinear behavior of experts' preferences (expressed by fuzzy preference relations) in CRPs. To do that, the experts' preferences will be remapped by using nonlinear deformations which amplify or reduce the distance between the extreme values. We introduce such automorphisms to remap the preferences as Extreme Values Amplifications (EVAs) and Extreme Values Reductions (EVRs), study their main properties and propose several families of these EVA and EVR functions. An analysis about the behavior of EVAs and EVRs when are implemented in a generic consensus model is then developed. Finally, an illustrative experiment to study the performance of different families of EVAs in CRPs is provided.

• RQ2: Does the nonlinear approach improve the CRPs in comparison with linear approach?
Without loss of generality, 16 we assume that the preferences will be elicited by fuzzy preference relations (FPRs) and nonlinear deformations will be applied to each value of the preference relation, to adjust the initial experts' preferences to a more realistic nonlinear scale when the (LS)GDM problem faces a situation in which such scales are needed. The impact of the nonlinear remapping procedure in CRPs will be evaluated by comparing its convergence and degree of consensus achieved.
Consequently, we will introduce Extreme Values Amplifications (EVAs) as those functions that increase in a nonlinear way the distance between extreme values of the FPRs. Additionally, Extreme Values Reductions (EVRs) will be defined as those nonlinear deformations which reduce the distance between the extreme values of the FPRs.
Several families of these EVAs and EVRs are proposed. EVAs will be then applied in different CRPs to LSGDM problems to show the effectiveness of this nonlinear preference modeling by using the software AFRYCA. 10 Such EVAs (resp., EVRs) will act as: 1. They remap the original linear-scaled FPRs into nonlinear-scaled FPRs. 2. They amplify (resp., reduce) the distance between the extreme values, and reduce (resp., amplify) the distance between the intermediate ones. 3. They have a concrete geometrical pattern. 4. The amplification (resp., reduction) of distances is greater when preferences are close to the extremes.
Finally, we analyze the performance of EVAs and EVRs in CRPs for LSGDM problems evaluating if EVA/EVR approaches outperform the classic linear approach in CRPs.
The remaining of this paper is set up as follows: In Section 2, a brief review of GDM problems and CRPs is presented. Section 3 introduces an exhaustive study of the properties of those automorphisms on the interval [0, 1] which remaps linear-scaled FPRs into nonlinear-scaled FPRs by increasing or reducing distances between extreme preferences. Section 4 defines the main concepts of this contribution, namely, EVA and EVR, and presents its fundamental characteristics. Section 5 proposes a general method to construct EVAs and introduces several families of EVAs and EVRs. In Section 6 we will discuss the performance of EVAs and EVRs when applied in a generic consensus model. In Section 7, we simulate the performance of EVAs when they are applied in CRPs for LSGDM problems. Finally, Section 8 will conclude the contribution.
• A set X X X X = { , , …, } n 1 2 , n 2  ≤ ∈ , of alternatives or possible solutions to the given problem. , where m 2  ≤ ∈ , of experts who express their opinions about the alternatives in X throughout certain preference structure.
In this study, without loss of generality, 16 we will assume that experts elicit their preferences by using an FPR, which has been proved to be effective in managing the uncertainty. 16,17 To obtain these FPRs, each expert e k m , = 1, …, k will elicit the degree to which she/he prefers the alternative X i over the alternative X j , which will be denoted by p i j k , . The FPR associated with the expert e k will be the matrix P ( 3,4 which demand a large number of experts. LSGDM problems are defined as those decision situations in which 20 or more experts are required to solve the GDM problems. 5

| Consensus reaching processes
GDM solving processes may fail when using classical GDM rules, like, the majority rule, since experts may feel unsatisfied with the solution and think that their opinions have not been sufficiently considered. 10,18 To avoid such disagreements, it is necessary to include in the GDM solving process a CRP to obtain agreed solutions that reflect the opinion of all the experts involved in the GDM problem. 11,19 A CRP is an iterative discussion process 7 usually coordinated by a moderator whose main responsibilities are to evaluate the level of agreement achieved in each round of discussion (and if it is enough), identify those experts' opinions that are far away from the collective opinion and provide some feedback/recommendations to such experts to increase the consensus degree in the next round. 8 A general scheme of a CRP (see Figure 1) is briefly summarized as follows: • Gathering preferences: Each expert elicits her/his preferences through a certain preference structure. 16 • Determining the level of consensus: The moderator computes the level of agreement throughout a certain consensus measure. 10 • Consensus control: The consensus level is compared with a threshold level previously established as acceptable. If either this consensus threshold is reached or the maximum number of rounds is surpassed, the process finishes. Otherwise the consensus progress keeps going. • Consensus progress: To increase the level of consensus, experts should change their preferences according to the moderator's recommendations.
When large-scale contexts are considered, CRPs become more complex 10,18 because there are usually more conflicts and the opinions are more polarized. 6 Additionally, new challenges emerge to deal with a large number of experts in CRPs and several proposals have been presented to cope with them [19][20][21][22] in recent years.

| AMPLIFYING DISTANCES BETWEEN EXTREME PREFERENCES
Our main aim is to study the performance of CRPs when the experts' preferences are modeled by a nonlinear scale. To do this, the preferences elicited from experts with FPRs will be transformed by a nonlinear deformation to obtain more realistic FPRs in which extreme values are deformed so that the distances between them are increased or decreased.
To remap the experts' preferences by using a function D: , that is, must transform the unit interval [0,1] such that the distance between the extreme values increases (or decreases) with respect to their original distance. In this section we will focus on those functions which deform the preferences by increasing the distance between extreme values and decreasing it between the intermediate ones, but similar arguments could be developed to describe those functions which increase the distance between the intermediate values and decrease it between the more extreme values.
For the sake of clarity, the description of these nonlinear deformations will be developed heuristically, by progressively adding requirements to a function D: [0,1] [0, 1] → . Therefore, some mathematical properties will be imposed to the function D due to their practical application and, in other cases, the mathematical properties will lead to useful features of these functions. In the following section all of these properties will be then compiled in the main definitions of our proposal, namely, EVAs and EVRs.

| Regularity
To obtain a proper deformation of the interval [0, 1] the function D: [0,1] [0, 1] → , it must be a bijection. Otherwise, different values of the preferences would be mapped into the same value, which is not reasonable if we want to compare how different the preferences are.
In this context, both the strictly increasing character of D and the values D (0) = 0 and D (1) = 1 are mandatory.
is a strictly increasing bijection which satisfies the boundary conditions D (0) = 0 and D (1) = 1; that is, D is an automorphism on the interval [0, 1].
The following well-known result will assure that a function satisfying this property is also a continuous function.
[ , ] → be a bijection. Then f is strictly monotonous if and only if f is continuous.
Proof. Let us prove first the sufficiency. Suppose f is strictly increasing (the decreasing case is similar) and pick x a b [ , ] 0 ∈ and ϵ > 0. Since f is a bijection we can find In such a case A is an interval containing x 0 and we can find To prove the necessary condition pick x y z , , such that x y z < < . Suppose f x f y ( ) < ( ) and f y f z ( ) > ( ). In that case, by using the Intermediate Value Theorem, we ≠∅, which is impossible because of the bijectivity of f . We can apply the same reasoning to the remaining case (i.e., f x f y ( ) > ( ) and f y f z ( ) < ( )) and conclude it must be either f Since a function satisfying Property 1 is continuous, small changes on the original preferences are mapped into small changes of the deformed values.
As we will see, we will need some extra regularity on D to characterize the amplification of the distances between extreme values according to the value of D′, so we impose now some additional smoothness: is a differentiable function whose derivative D′:

| Symmetry
When using an FPR to represent expert's preferences, it is usual to make the calculations only in the superior triangle due to that triangle and the inferior one are related by the standard negation N: . We have to translate this symmetry into an equivalent property for our function D, that is, the modified distance from 0.8 to 0.85 should be the same that the modified distance from 0.2 to 0.15. This kind of symmetry around the value x = 1 2 will be imposed by the following property. In other words, this property guarantees that D remaps FPRs into FPRs. Furthermore, this property has a clear practical purpose since it allows us to construct these nonlinear deformations by only focusing on one half of the interval [0, 1]: if we manage to obtain D : , 1 , 1 as a piecewise function. Note that, when D 1 is a differentiable function, D 2 is also differentiable and its derivative satisfies These tools allow one to compare how similar are the preferences taking into account the nonlinear approach.

| Distance amplification and derivatives
Here it is studied the relation between the first derivative of an arbitrary automorphism defined in [0,1] and the modification of the distances between elements that it will produce. First, a theorem that characterizes those functions which amplify the distance between the elements of their domain is proposed.

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Proof. (1) Let us choose h > 0 such that z h b + < . In that case the auxiliary function is a continuous function such On the other hand, when z b = , we can consider h > 0 such that b a h − > and use the analogue reasoning for the such that x y < . We can use the Mean Value Theorem to obtain ξ Note that if the derivative of D is between 0 and 1 in some subinterval, as it may occur on the intermediate values of [0,1], then the distance between the deformations of those elements will be lower than the distance between the original ones.
In the following proposition we will use the idea of Theorem 1 to describe the amplification of distances between values close to 1 when we are deforming the interval [0, 1]. The key is to ask for the derivative in x = 1 to be higher than 1 and use the continuity of the derivative to obtain a neighborhood of 1 wherein the derivative is greater or equal than 1. Then, Proof. To prove the first assumption let us use the continuity of f ′ around 1 to get r′ To show the second statement we just have to use Theorem 1. □ Using a similar argument we can show the analogous result for distance amplification in values close to 0: The practical interpretation of these results is simple: to amplify distances close to the extremes we need that the derivative of D is greater than 1 in the extremes. Since D is asked to be a 1  function, the continuity of D′ will give us two neighborhoods (one around 0 and the other around 1) where D′ is always greater than 1 and thus the distance between two values which are inside one of these neighborhoods will increase when we apply D.

| Distance amplification and convexity
Finally, we study the relation between the convexity and the distance amplification on extreme values. First, we compare the deformations D with the identity function on the interval [0, 1].
satisfies g′(0) > 0 and g′(1) > 0 and we can find r ]0, [ This result provides a clear geometrical interpretation: the graph of D is over the diagonal of the square [0, 1] × [0, 1] for values close enough to 0 and it is under the same diagonal for those values close enough to 1 (see Figure 2).
We have already shown that the derivative of D must be greater than 1 close to the extremes. The following proposition, which is an immediate consequence of the previous one, will prove that D′ must also be under 1 in some subinterval of [0,1] and thus the distances decrease between the values in such subinterval. Additionally, we will obtain that D cannot be convex nor concave on its full domain.
In that case, g′ 0 ≥ and g is strictly increasing, which is a contradiction due to g g (0) = (1). That means we can find some where f x ′( ) < 1 and the continuity of f ′ will give us the interval we are looking for.
On the other hand, if f ′ is increasing then g′ is also increasing and due to g′(0) > 0 we . Then g is strictly increasing and g g (0) < (1), which is impossible. □ Propositions 6 and 7 provide a characterization of those functions which amplify the difference between nearby elements when we approach extreme values (0 or 1).
 increasing function. The following statements are equivalent: . On the other hand, the Mean Value Theorem gives us ξ ] , ] ∈ and define g h , which is the convexity of f . □ Using a similar proof, we obtain:  increasing function. The following statements are then equivalent: These propositions show that the convexity is related to an increment of the distances between consecutive values of the preferences when we approach the extremes of the interval [0, 1]. We summarize this in Property 5: should be concave in a neighborhood of 0 and convex in a neighborhood of 1.

| EXTREME VALUES AMPLIFICATIONS AND EXTREME VALUES REDUCTIONS
This section introduces the concept of EVA and its dual concept EVR. First, we present the definition of EVAs as those functions satisfying the properties stated in Section 3.

Definition 1 (Extreme Values Amplification). Let
be a function satisfying: This notation reminds that the main purpose of D is to remap FPRs of a GDM problem in a nonlinear way by amplifying the distance between the extreme values. So the new preferences show a larger distance between extreme elements and a smaller distance between elements close to 1 2 . GARCÍA-ZAMORA ET AL.

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The following theorem, which compiles the main properties of the EVAs, is obtained by using the results discussed in Section 3: be an EVA on [0, 1]. Then, is a restricted dissimilarity 23 and the function S : satisfying that 3. The graph of D is over the diagonal of the square [0, 1] × [0, 1] for values close enough to 0 and it is under the same diagonal for those values close enough to 1, 4. There exist a neighborhood U 0 containing 0 and a neighborhood U 1 containing 1 such that for every and for every x y U x y , , Therefore, EVAs behave exactly as we aimed at: 1. They remap the original linear-scaled FPRs into nonlinear-scaled FPRs. 2. They amplify the distance between the extreme values, and reduce the distance between the intermediate ones. 3. They have a concrete geometrical pattern (see Figure 3A). 4. The amplification of distances is greater close to the extremes.
We can also consider the analogous notion of EVR:  satisfying that 1] for values close enough to 0 and it is over the same diagonal for those values close enough to 1, 4. There exist a neighborhood U 0 containing 0 and a neighborhood U 1 containing 1 such that for every x y U x y , The behavior of EVRs is dual to the EVAs as can be seen below: 1. They remap the original linear-scaled FPRs into nonlinear-scaled FPRs. 2. They reduce the distance between the extreme values, and amplify the distance between the intermediate ones.
3. They have a concrete geometrical pattern (see Figure 3B). 4. The reduction of distances is greater close to the extremes.
Both EVAs and EVRs are valid approaches to model nonlinear preferences, which gives an answer to the first research question.

| GENERATING EVAS AND EVRS
In this section several examples of EVAs and EVRs are provided. First, a generic method to construct EVAs is developed.
Let h α β a b : [ , ] [ , ] → be the standard affine transformation given by given by is an EVA.
Remark 2. The previous result could be adapted for EVRs by requiring concavity instead of convexity and changing the direction of the inequalities, that is, f ′(0) > 1 and f ′(1) < 1.
In the following we will show several families of EVAs and EVRs and study their properties.

| Sin-based EVAs and EVRs
is an EVA (see Figure 4). Note that we obtain a family of EVRs.

| Polynomial EVAs and EVRs
(see Figure 5), whose derivatives are So m α is strictly increasing, concave in 0, . In this case the calculation for the amplification/reduction threshold values is not as easy as before, but we can compute a numeric approximation. For α = 2, we obtain that . Therefore these functions can be used in almost any situation in which a proper EVR can be applied. We aim to use these affine transformations to construct a parametric EVA from the function b : , 1 , . The parameter α > 1 controls how faster the distances between extreme values are amplified.

| Piecewise polynomial-based EVAs
It is clear that the function b 1 is not 1  for any combination of the parameters α r s , , . The parameter ϵ has been introduced to solve this issue. Note that where b : , 1 , 1 is an EVA if and only if the following equality holds where λ is the derivative of h a and indicates how much the intermediate values become closer.
Some useful combinations of these parameters are shown in Table 1 and the respective graphs are included in Figure 6.
Two limit cases are considered below.

| Special case r s
, and g′ > 0 the equation g (ϵ) = 0 has the unique solution ϵ = 1, which is not admissible in this problem.

Note that this assumption implies that the affine transformation h a disappears and
To study the sign of g′ let us consider h: we can conclude that h > 0 in its domain and therefore g′(ϵ) > 0 ϵ ]0, 1[ ∀ ∈ In that case g is increasing and such that g lim (ϵ) = 0 Then, for every λ ]0, 1[ ∈ we can find ϵ ]0, 1[ ∈ (i.e., the unique one which satisfies Note that by taking ϵ = 0 we would obtain the m α family of EVAs.

| Comparing sin-based EVAs, polynomial EVAs, and piecewise polynomial EVAs
The main difference between these families of EVAs is the value of their derivative in 1 2 . Note that in all cases the derivative function reaches its minimum value at this point.
For sin-based EVAs, the derivative at x = for the variable α > 1. The piecewise polynomial EVA allows both to choose how much the intermediate values will become closer, by adjusting λ, and how much the extreme values will become distant, by choosing α. However, this family of EVAs is more complex and loses the regularity of the other two families.

| A THEORETIC DISCUSSION ABOUT THE PERFORMANCE OF EVAS AND EVRS IN CRPS FOR LSGDM
In Section 4 the EVA approach and the EVR approach have been introduced as models for nonlinear preferences. However, the strategy to deal with polarized opinions is totally different in EVA and EVR. According to References [24,25], the less extreme values have a more cohesive effect and greater success to reach an agreement. Therefore, this section is devoted to provide a sustained proof about why EVRs are not a good strategy to remap FPRs in CRPs meanwhile EVAs tend to improve the performance of the consensus models.

| EVAs and order of alternatives in CRPs
First a study of how much EVAs deform the original preferences is provided.
be an EVA whose first derivative is strictly increasing in , 1 has an unique solution x , 1 Proof. The existence and the uniqueness of x 0 are given by the bijectivity of D′. Now consider the function g: , 1 the candidates to be relative extremes for the Note that we do not need that D′ is strictly increasing in , 1 1 2 ⎡ ⎣ ⎤ ⎦ . It suffices to consider an EVA D whose first derivative is strictly increasing in a neighborhood of 1 which contains a value r > 1 2 such that D r ′( ) < 1. The general version, whose proof is analogous, is stated: be an EVA whose first derivative is strictly increasing in r has an unique solution x r ] , 1[ 0 ∈ which satisfies Remark 3. Note that the symmetry of D around 1 2 would provide another x 0, from a certain expert. We want to analyze how different the order of the alternatives chosen by that expert will be after applying an EVA.
To compute the order of the alternatives we just assign each alternative a score depending on the value of the preferences: Then we order the alternatives according to the score they have received.
We cannot prove that the order of the alternatives will not change after applying any EVA, but we can show that there exists a threshold which enables us to control the distance between the score obtained for the deformed preferences and the original score.
is not only a threshold for the amplification of the distances, but also provides a bound to study how similar is the order alternatives after applying the EVA with respect to the original order. Therefore, the value x 0 will receive the name of amplification threshold, and the value R x D x − ( ) 0 0 0 ≔ | | will be called maximum deformation.
Let us study these quantities for different families of EVAs.

| Sin-based EVAs
Let α ]0, ] For α = 2 we obtain R = Let us consider all of these together: when the EVA approach is used on a consensus model which aggregates the preferences by prioritizing intermediate values, the model will ignore the extreme values and the intermediate ones will become closer because of the properties of the EVA function. In this case, the model will need a lower amount of rounds to reach the consensus, since the EVA has done part of the work by making the intermediate values closer. Since the order of the alternatives has not been changed too much, a consensus model which uses the EVA approach will choose a similar alternative faster than the original one.
This also explains why we are not using EVRs to model the nonlinear approach in CRPs. Although EVRs also modify the preferences in a nonlinear way, in this case the distances between extreme preferences are reduced and the distance between intermediate preferences is amplified. When EVRs are implemented in a consensus model, which usually prioritizes intermediate preferences to facilitate the consensus, the model will probably need more rounds to reach the consensus, since intermediate elements are less similar.
Both EVAs and EVRs are valid approaches to model nonlinear scales in CRPs (RQ1), but to improve classic models the EVA approach outperforms both the linear approach and the EVR approach (RQ2).

| EXTREME VALUES AMPLIFICATIONS IN GDM
This section aims at verifying, validating, and showing the better performance of EVA functions in LSGDM problems with respect to lineal preference modeling. Therefore, an illustrative LSGDM problem is solved by using the specialized CRP software AFRYCA 10 and comparing the performance of two widely used consensus models (see Remark 7) when they use linear and nonlinear preference scales. Therefore, we have implemented the EVA families s α (Equation 1) and m α (Equation 2) into the CRPs introduced in References [16,26] (both included in AFRYCA) and then carried out several simulations to compare their performance by using EVAs and linear preferences.
Remark 6. In Section 6 it was pointed out that extreme values make more difficult the achievement of agreements. 24,25 Therefore, due to the fact that EVRs amplify the distances between less extreme values, we will only consider the study in further detail of EVAs for CRPs in LSGDM because EVRs are not suitable for smoothing the achievement of agreements.
Remark 7. The consensus model proposed by Herrera-Viedma et al. 16 has been selected since it has been widely used in the literature and several consensus models are based on its performance. 18  This section is divided into two subsections. In both, the performance of the classical models Herrera-Viedma et al. 16 and Quesada et al. 26 is compared with the EVA-modified models. To do so, for each consensus model five different scenarios are considered: the classical model (no EVA is used), the EVAs s 0.08 and s 0.09 (defined by Equation 1), and the EVAs m 2 and m 3.39 (defined by Equation 2). In Section 7.1, 500 simulations are developed for each one of these scenarios. In all of them, 100 randomly defined FPRs are used to model experts' preferences. For these 500 simulations, both the average number of rounds required to obtain the consensus and the average degree of consensus are computed. In Section 7.2, the simulations are developed by using concrete values for the experts' FPRs 27 to be able to compare graphically the evolution of the experts' opinions through the different rounds of the CRPs.

| Average performance of EVAs
To validate the EVA approach, the average performance of EVA-modified models has been compared with the average performance of the classic models. To do so, 500 simulations with 100 randomly defined FPRs have been developed for each EVA in both Herrera-Viedma et al. 16 and Quesada et al. 26 models.
The obtained results, which are summarized in Tables 3 and 4, show that the EVA approach always outperforms the classic approach in terms of convergence, by keeping a similar average T A B L E 3 Average results on Herrera-Viedma et al. 16  consensus. This fact provides a clear answer to the second research question: the classic consensus models improve on average when the nonlinear approach is modeled by an EVA.

| Performance of EVAs in a concrete example
To clarify the performance of EVAs in CRPs for LSGDM, we have chosen an individual simulation with the FPRs values provided in Reference [27] and then show the results obtained in Tables 4 and 5 together a graphical evolution of the consensus progress. The results obtained are summarized in Tables 5 and 6.
To facilitate the understanding of the simulation results, AFRYCA provides a visualization of the different CRP simulations based on the multidimensional scaling technique 28 (see Figures 7 and 8). This representation shows the collective opinion of the experts' group in the center of the plot. Around the collective opinion, the experts' preferences are represented. The closer the experts' preferences to the collective opinion, the greater the consensus reached. In this way, we can appreciate the state of the experts' preferences for each round and the evolution of the CRPs in the simulations. Furthermore, we have also shown the results obtained from AFRYCA in Tables 5 and 6.
The classical model 16 reached a consensus level of 0.87 in six rounds. For this model the EVAs s 0.08 and s 0.09 have not reduced the number of rounds required to reach the consensus, but have improved the consensus level reached. The latter can be appreciated in Figure 7, since in the last round (round 6) the experts are closer each other than with linear preferences. In addition, the polynomial-based EVA m 2 has reduced the amount of rounds required to reach a similar consensus level, whereas the EVA m 3.39 has improved both aspects by needing just five rounds to reach a consensus level greater than the obtained in the original model. Again, the latter can be visualized in Figure 7.
On the other hand, the classical model 26 obtained a consensus level of 0.85 in 10 rounds (see Table 6). In this case, both families of EVAs have obtained significantly better performance than the original model (see Figure 8). The EVAs s s , 0.08 0.09 , and m 2 have slightly improved the consensus level in just seven rounds. In this case, the EVA m 3.39 has performed surprisingly well by increasing the level of consensus reached in only five rounds.
The simulation has shown that the implemented EVAs improve the performance of both models. By keeping the same order for the alternatives, after using the EVAs either the number of rounds has been reduced or the consensus level is increased. These simulations clarify and reinforce the positive answer to the second research question when the nonlinear approach used is an EVA.

| CONCLUSIONS
Nowadays, CRPs are a prominent line of research in GDM. Several models have been proposed in the literature, but usually these models assume linear scales for experts' preferences. 16,26 This contribution has studied and proposed the use of nonlinear scales to obtain more realistic preference modeling from the original experts' preferences, even in large-scale contexts.
We have exhaustively studied the analytical properties of these nonlinear scales, obtaining the main mathematical characteristics of those functions which are good candidates to become a proper nonlinear deformation for the original preferences. These particular deformations of the preferences have received the name of EVAs. These EVA functions remap linear-scaled FPRs into nonlinearscaled FPRs and deform the preferences in the way that the distances between extreme values are increased and the distances between intermediate values are decreased. In addition, we have stated the dual definition of EVRs, that is, those functions that reduce the distance between extreme values by amplifying the distance between the intermediate ones.
After introducing a general method to construct EVAs and EVRs, we have proposed several families of EVAs: s α (Equation 1), m α (Equation 2), and b α r s , (Equation 3). The first one is based on the sin function, the second one is constructed from a polynomial and the last one is obtained from a piecewise polynomial function. Finally, we have simulated the performance of some of these EVAs in two classical consensus models by using the software AFRYCA. 10 The use of the nonlinear scales provided by the EVAs improves the performance of both classical models used in this study. The simulations with random FPRs showed that the EVA approach reduces the average number of rounds required to reach the consensus in both models. In addition, when using the same FPRs for the comparisons, the novel EVA-modified models either reach the consensus in a faster way or increase the level of consensus when we use the proper EVA.
Further studies should focus on either suggesting new EVAs or optimizing the parameters of the existing EVAs for concrete CRPs. In addition, a deeper study of the performance of EVAs in different consensus models should be developed. Additionally, since every EVA (resp., EVR) induces a similarity measure, it is also interesting to study the effects of using these proximity measures when comparing FPRs by moving closer (resp., bringing near) extreme values and bringing near (resp., moving closer) the intermediate ones. Another possible research work would be finding a concrete GDM problem adequate to the properties of EVRs. Furthermore, future works could be related to the application of the proposed framework to real-world problems, such as the high-speed rail passenger satisfaction and bid evaluation with LSGDM. 29,30