An MCGDM problem consists of several elements (Ishizaka and Nemery, 2013):

The group of evaluators evaluates the alternatives based on some criteria to select the best alternative/s as the solution to the problem (Tüysüz and Kahraman, 2023). In this sense, the importance of the criteria plays a key role in the resolution of an MCGDM problem (Rezaei, 2015; Saaty, 1977). The classical MCGDM resolution scheme carries out an aggregation process of the evaluators’ assessments before selecting the best alternative, which should properly rate the relevance of the criteria to reach reliable results (see Fig. 1). For instance, it is logical to assume that when buying a car, safety has a greater weight than colour in the decision-making process.

Inevitably, society evolves and demands even more complex decision problems to be solved. Consequently, evaluators have to deal with problems in which there is a lack of information, or that are defined in vague and imprecise contexts. Under these uncertain conditions, evaluators may not feel comfortable providing numerically precise assessments, and they may prefer to use expressions closer to their natural way of thinking. The fuzzy linguistic approach addresses the uncertainty modelling in human beings’ opinions through the use of linguistic variables defined by Zadeh as “a variable whose values are not numbers, but words or sentences in a natural or artificial language” (Zadeh, 1975). When evaluators face MCGDM problems by using linguistic information, we talk about LiMCGDM. Classically, in the resolution of a LiMCGDM problem, evaluators use a linguistic term set, i.e. an ordered set of linguistic descriptors, S = { s 0 , s 1 , … , s G }, to provide their assessments (see Fig. 2).

The use of linguistic information for uncertainty modelling involves performing computations based on it. The aim of the CW approach (Zadeh, 1996) is to emulate the reasoning process of humans by generating linguistic results from linguistic inputs, which are easily comprehensible and accurately depicted. Various CW proposals have been put forth in the literature (González and Lantigua, 2024; Li et al., 2023; Martínez et al., 2015; Wang and Hao, 2006). In particular, this paper adopts the 2-tuple linguistic representation model (Martínez et al., 2015) for modelling linguistic information and performing CW processes. This approach allows the modelling of uncertainty using the fuzzy linguistic approach while retaining the integrity, accuracy, and interpretability of the results.

The 2-tuple linguistic model (Martínez et al., 2015) uses the fuzzy linguistic approach to enable linguistic computations in a precise and understandable manner. This model represents linguistic information using a pair of values ( s l , α ) ∈ S ‾, where s l corresponds to a linguistic term from a pre-defined set of terms S = { s 0 , s 1 , … , s G }, and α ∈ [ − 0.5 , 0.5 ) represents the displacement of the membership function of the linguistic term s l to the right or left. Specifically, the symbolic translation is defined as follows: (1) α ∈ [ − 0.5 , 0.5 ) , if s l ∈ { s 1 , s 2 , … , s G − 1 } , [ 0 , 0.5 ) , if s l = s 0 , [ − 0.5 , 0 ] , if s l = s G .

One significant feature of 2-tuple linguistic expressions is their ability to be converted into a numerical value x ∈ [ 0 , G ], which streamlines the computational process.

Consequently, to make computations with linguistic terms using the 2-tuple linguistic model, it suffices to use Proposition 1 to transform the linguistic inputs into numerical values, carry out the corresponding computational processes, and use Proposition 1 to re-translate the numeric results into linguistic expressions. Note that the resulting 2-tuple linguistic value will be of the form ( s l , α ), and usually, it will not correspond to one of the original linguistic terms in S, i.e. α ≠ 0. In that case, the numeric value Δ S − 1 ( s l , α ) will be located between two consecutive linguistic terms, namely s l − 1 and s l , or s l and s l + 1 , and therefore it will have a positive membership degree for the triangular fuzzy numbers associated to these two labels, which depend on the value of α.

Many problems that involve modelling information by using linguistic assessments typically rely on linguistic variables with uniformly and symmetrically distributed values (see Fig. 2). However, some problems require the use of unbalanced linguistic term sets, which are asymmetrically distributed (Herrera et al., 2008; Licerán-Gutiérrez et al., 2022). Unbalanced linguistic scales are necessary when the linguistic variables involved in the problem require different degrees of specificity on each side of a central label. Figure 3 illustrates an example of an unbalanced linguistic scale asymmetrically distributed, with the central label being represented by the linguistic term “Fair”.

When multiple evaluators with diverse viewpoints and expertise are involved in an MCGDM process, conflicts can arise among them. However, the resolution scheme depicted in Fig. 1 overlooks such potential conflicts and performs an aggregation process that yields a solution based on the majority opinion, which may not be agreeable to all evaluators (Butler and Rothstein, 2006). To address this issue, CRPs were introduced as an additional step in resolving group decision-making problems (García-Zamora et al., 2023a). In a CRP, evaluators (or decision-makers) express and justify their opinions and strive to converge on a common ground before reaching a decision that is acceptable to everyone.

In the specialized literature, CRPs have been classified into two main categories: CRPs with and without a feedback process (García-Zamora et al., 2023a). The CRPs with the feedback process are usually guided by a moderator who identifies the changes to make over the evaluators’ preferences and asks evaluators if they want to accept or reject such changes. The CRPs without a feedback process replace the role of the moderator with an automatic process that detects the changes to the evaluators’ preferences and applies such changes automatically without checking with them. Among the latter proposals, MCC models (Ben-Arieh et al., 2008; García-Zamora et al., 2023b) are especially convenient when used as a consensus support system to guide (decision) analysts/moderators in interactive CRPs to quickly detect disagreements and suggest changes in a more precise way (García-Zamora et al., 2025).

CRPs can help to enhance the level of consensus among groups of evaluators. However, the notion of consensus is not straightforward. Traditionally, it has been defined as the unanimous agreement of all evaluators involved in a problem (Butler and Rothstein, 2006), which is practically unattainable in real-world decision-making scenarios. Therefore, an alternative concept of consensus has been proposed in the literature. This paper focuses on the soft consensus introduced by Kacprzyk and Fedrizzi (1988), which posits that consensus exists when “most of the important individuals agree as to (their testimonies concerning) almost all the relevant options”.

Therefore, instead of a unanimous consensus, we consider here a more flexible consensus definition based on consensus measures (Palomares et al., 2014). Let us denote by P the set consisting of all the possible ratings that a group of evaluators may provide. Then, a consensus measure is a mathematical function κ : P → R 0 + that, intuitively, allows computing the distance/dissimilarity between these opinions (García-Zamora et al., 2023a). Therefore, given the opinions of a certain group of evaluators P ∈ P, the value ε = κ ( P ) is known as the consensus degree of such preferences, and the closer to one, the higher the similarity (consensus) between the opinions P. In the CRPs that follow the consensus measure-based approach, a certain consensus threshold ε 0 is fixed a priori, and the goal of the CRP is to modify the initial evaluators’ preferences P 0 into new values P that satisfy κ ( P ) ⩽ ε 0 . This approach enables the assessment of how far the opinions of evaluators are from unanimity. Note that ε 0 is a unitless magnitude that represents the desired level of consensus to achieve in the group. The range of values is the positive real numbers, in which 0 means total agreement. Consequently, this value should not be predefined by the evaluators involved in the negotiation but by a selected person who could be an analyst, the moderator of the process, or even the decision-makers. The value assigned depends on the context of the problem. Some aspects could be considered to set a proper value of ε 0 , such as the number of evaluators, which plays an important role here; a decision involving hundreds of evaluators cannot be expected to reach a very high consensus within a few modifications. Another aspect to consider may be the initial consensus degree κ ( P 0 ) within the group before the CRP.

The BWM (Brunelli and Rezaei, 2019; Rezaei, 2015; Ortega-Rodríguez and Labella, 2026) was proposed as a prioritization methodology for deriving the weights of the criteria by reducing inconsistencies in the elicitation process and obtaining more reliable solutions. To do so, the evaluator chooses the best and the worst criteria, which are denoted by C B and C W , respectively. Afterward, the evaluator must point out the comparisons of C B regarding the remaining criteria and all the criteria regarding C W . To model such comparisons, BWM makes use of Saaty’s multiplicative scale (Saaty, 1977; Adali and Tuş, 2026), which is decomposed into two parts: numerical scale and linguistic scale. Whereas the linguistic scale consists of 9 gradations, i.e. 1: Equally important, 2: Weakly more important, 3: Moderately more important, 4: Moderately plus more important, 5: Strongly more important, 6: Strongly plus more important, 7: Demonstrated more important, 8: Very, very strongly more important, and 9: Extremely more important, the numerical scale is composed of 17 numerical values described as: { 1 f t , f 1 , f t } , t = 2 , 3 , … , 9 , where f 1 = 1 and f t + 1 > f t > 1 and the value of f t correspond to the t grade of the linguistic scale.

The pairwise comparisons among C B and the remaining criteria are collected in the Best-to-Others (BO) vector, B O = ( b B 1 , b B 2 , … , b B n ) , where b B i ∈ [ 1 , 9 ] ∩ N denotes the degree of preference of C B over the criterion C i .

In the same way, the evaluator’s pairwise comparisons regarding all the criteria with C W are collected in the Others-to-Worst (OW) vector, O W = ( b 1 W , b 2 W , … , b n W ) , where b i W ∈ [ 1 , 9 ] ∩ N denotes the preference degree of the criterion C i over C W . These values are then used as the input of an optimization model to obtain the weights for the criteria { w 1 ∗ , w 2 ∗ , … , w n ∗ }: (5) min w max i = 1 , 2 , … , n { | b B i − w B w i | , | b i W − w i w W | } , s.t. ∑ i = 1 n w i = 1 , w i ⩾ 0 , ∀ i = 1 , 2 , … , n .

The BWM produces weights that allow constructing a fully consistent multiplicative pairwise comparison matrix ( b ˆ i j : = w i w j ) close to the original preferences B O and O W specified by the evaluator, thanks to the full consistency property of the multiplicative pairwise comparison matrix in 1-9 Saaty’s scale (Saaty, 1977; Majumder et al., 2026), where b i j = b i k b k j for i , j , k ∈ 1 , 2 , … , n. Additionally, the BWM optimization model proposes a consistency index to measure the reliability of evaluators’ preferences (see Rezaei, 2015 for further details).

Lately, the classical BWM has been extended to versions with additional features. Particularly, this proposal takes advantage of one of them introduced in Licerán-Gutiérrez et al. (2022), which is the use of an unbalanced linguistic scale (Herrera et al., 2008) to represent the BWM weights linguistically. This proposal argued that to represent the importance of the BWM weights in this format, it is necessary to define a suitable unbalanced linguistic scale (Licerán-Gutiérrez et al., 2022), here noted as S n U , whose semantics depends on the n number of the elements to be weighted (see Table 1).

The 2-tuple linguistic approach is usually based on a symmetrically distributed additive scale, which suggests that the pairwise comparisons necessary to apply the linguistic BWM should also be modelled as an additive scale (Orlovski, 1978), rather than the classic multiplicative Saaty’s scale (Saaty, 1977). The use of the 2-tuple linguistic model within the BWM method allows not only the evaluators to provide their preferences using a linguistic scale but also offers an output in terms of linguistic values. This is extremely important to develop a feedback mechanism, since the analyst/moderator may provide the recommendations using expressions such as “the importance of the criterion C1 is average” or “criterion C2 is unimportant”.

This section aims to clarify how to remap linguistic preferences into the interval [ 0 , 1 ] to compute weights using BWM.

Let us consider a linguistic term set S = { s 0 , … , s G } of granularity G + 1 ∈ N. To remap the linguistic labels into a numeric additive scale, we can define the mapping Δ ˆ S : [ 0 , 1 ] → S ‾ as (7) Δ ˆ S ( β ) = ( s round ( G β ) , G β − round ( G β ) ) , ∀ β ∈ [ 0 , 1 ] , whose inverse Δ ˆ S − 1 : S ‾ → [ 0 , 1 ] is (8) Δ ˆ S − 1 ( s l , α ) = ( l + α ) / G , ∀ ( s l , α ) ∈ S ‾ . Note that the mapping Δ ˆ S is slightly different from the mapping Δ S . Whereas the input of the first one is the centroid of the triangular fuzzy number associated with the linguistic label, the input of the second one is the index of the linguistic label.

The linguistic scale S represents the element importance in both directions, so that the preference degree between the criterion C i and C j is s l , then the inverse preference is s G − l . Nevertheless, the linguistic scale S consists of a set of linguistic descriptors with an uncertain and imprecise nature that should be represented in some way. In this sense, the fuzzy linguistic approach uses the fuzzy sets theory to handle vague and imprecise information (Parsons, 1996) (see Section 2.1). To do so, a fuzzy membership function is associated with each linguistic descriptor, now a fuzzy linguistic term, resulting in a fuzzy linguistic term set. Therefore, we set a fuzzy membership function for each linguistic descriptor of S to define a fuzzy linguistic term set S. For instance, using these mappings for G = 16 may lead to the linguistic scale given in Table 2, whose semantics are represented in Fig. 4.

Consequently, the evaluators may use the linguistic scale S to provide their BWM preferences. However, because of the nature of the BWM performance in which the evaluators compare how important the best criterion is with the remainder and how important the rest of the criteria are concerning the worst criterion, the evaluators will only use the subset {EI, WMI, MMI, MPMI, SMI, SPMI, DMI, VSMI, EMI} of the scale.

In addition, after applying BWM to derive numerical weights in [ 0 , 1 ], it will be necessary to retrieve the linguistic values and represent such weights in a format closer to how human beings express their opinions. Therefore, such numerical weights must be remapped into 2-tuple linguistic values in the unbalanced linguistic scale S n U (see Table 1) according to the following process (Licerán-Gutiérrez et al., 2022):

Let x ‾ h represent the centroid (Cheng, 1998) of the linguistic term s h , which is assumed to be an element of the unbalanced scale S n U with granularity G. Consider the mapping Δ ˆ U : [ 0 , 1 ] → S n U defined as Δ ˆ U ( w ) = ( s l , α ), ∀ w ∈ [ 0 , 1 ] where: (9) l = arg min h ∈ { 0 , … , G } | w − x ‾ h | , (10) α = 0 , if w = x ‾ l , w − x ‾ l x ‾ l + 1 − x ‾ l , if w > x ‾ l , w − x ‾ l x ‾ l − x ‾ l − 1 , if w < x ‾ l .

As an illustrative example, Fig. 5 shows the graphic representation of an unbalanced linguistic scale with 5 fuzzy linguistic terms, in which the centroid of the central label ( A I) is the closest one to a numerical weight w, i.e. Δ ˆ U ( w ) = ( A I , α ).

MCC models have been proven to be extremely useful in supporting interactive CRPs by facilitating the achievement of consensus by the group (García-Zamora et al., 2023a). Although they are automatic consensus models without feedback (and thus they cannot ensure the evaluators’ acceptance of the generated collective opinion), they can help the analyst/moderator in a CRP to identify the stronger disagreement points and provide precise recommendations to smooth them. In this view, here we introduce a BWM consensus model inspired by MCC. This model could act as a consensus support tool for CRPs.

Let us suppose an MCGDM in which a group of m ∈ N evaluators E = { E 1 , E 2 , … , E m } wants to reach a collectively agreed solution about the importance of n ∈ N criteria. First, the evaluators provide their opinions by using BW preferences, which implies the evaluator, E k , chooses the best ( B k ) and the worst ( W k ) criteria according to his/her opinion and also provides two pairwise comparison vectors:

for the best criteria C B k B O k = ( a B k 1 , a B k 2 , … , a B k n ) and for the worst criteria C W k O W k = ( a 1 W k , a 2 W k , … , a n W k ) , where a B k i , a i W k ∈ S, i = 1 , 2 … , n

Classically, these preferences were given by using a 1–9 Saaty scale (Saaty, 1977). Nevertheless, in this proposal, the evaluators are using fuzzy linguistic terms that belong to a fuzzy linguistic term set S. To facilitate the linearization of the BWM optimization model, these linguistic preferences are remapped into a linear scale in [ 0 , 1 ] by using the function Δ ˆ S − 1 . To simplify the notation, the information is stored as follows: •

The vectors B O k = ( a B k 1 , a B k 2 , … , a B k n ), k = 1 , … , m, containing the linguistic comparisons among the best criterion and the remainder for each evaluator,

Whereas the classical BWM obtains the weights for only one evaluator, our proposal derives the individual weight for each evaluator involved in the decision process and stores them in a matrix U = ( u k i ) ∈ M m × n , where u k i represents the individual weight corresponding to E k for the criterion C i . For the sake of simplicity, let us consider the preference structure consisting of the weighting vectors of dimension n P U = { u ∈ [ 0 , 1 ] n : ∑ i = 1 n u i = 1 }. In such a case, our goal is to associate a weighting vector u k = ( u k 1 , … , u k n ) ∈ P U to each evaluator. In addition, our proposal also computes the collective weights from the aggregation of each evaluator’s weights: g = ( g 1 , g 2 , … , g n ) ∈ P U , where g i = F i ( u 1 , … , u m ) for a certain fusion mapping F : P U m → P U . In such a case, after running the model, each evaluator will be assigned a weighting vector u k ∈ P U , and we will also obtain the collective weighting vector g ∈ P U . For the sake of clarity, in this paper, we consider the arithmetic mean to compute the collective weight vector, i.e. F i ( u 1 , … , u m ) = 1 m ∑ k = 1 m u k i ∀ i ∈ { 1 , 2 , … , n }, which also allows linearizing the resulting optimization model.

To guarantee agreed weights, let us consider the consensus measure κ : P U m → R 0 + defined by κ ( u 1 , … , u m ) = 1 m m ∑ i = 1 n ∑ k = 1 m | u k i − F i ( u 1 , … , u m ) | ∀ ( u 1 , … , u m ) ∈ P U m .

Therefore, in our model, individual and collective weights must satisfy the following consensus constraint: (11) κ ( u 1 , … , u m ) ⩽ ε 0 .

This constraint computes the level of agreement in the group by using a consensus measure based on the distance between the individual weights and the collective ones and ensures the achievement of a desired level of consensus ε 0 (García-Zamora et al., 2023a). Note that this consensus measure could be replaced by any other with a similar interpretation. For instance, to avoid compensatory behaviours, we might define κ ˆ : P U m → R 0 + by κ ˆ ( u 1 , … , u m ) = max i = 1 , … , m k = 1 , … , n { | u k i − F i ( u 1 , … , u m ) | } ∀ ( u 1 , … , u m ) ∈ P U m .

On the other hand, due to the transformation to the 0 − 1 linear scale, the distance between the weights that the method aims to obtain, and the original preferences elicited from the evaluators can be defined by ξ : M m , n × M m , n × R m × R m × P U m ) → R 0 + (12) ξ ( B O , O W , B , W , u ) = 1 2 m n ∑ k = 1 m ∑ i = 1 n ( | Δ ˆ S − 1 ( a B k i ) − 0.5 − u k B k + u k i | + | Δ ˆ S − 1 ( a i W k ) − 0.5 − u k i + u k W k | ) .

Therefore, for fixed values B O, O W, B, W, the proposed consensual BWM (LC-BWM) is defined as

To quantify the consistency of the preferences provided by the evaluators, here we combine some statistical insight with the value of the function ξ ( B O , O W , B , W , u ), which measures the distance between the original preferences given by the evaluators ( B O , O W , B , W ) and the output of the L-BWM u.

We will conduct several Montecarlo simulations to determine the percentiles of the random variable ξ ( B O , O W , B , W , u ). Since the smaller the value ξ ( B O , O W , B , W , u ), the more consistent the results of the L-BWM, we can analyse the consistency of the preferences ( B O , O W , B , W ) by checking the percentile corresponding to the value ξ ( B O , O W , B , W , u ).

First, note that for n ∈ N, BWM preferences may be randomly generated as follows:

Under these conditions, for the most common values of n ∈ N, i.e. natural numbers between 4 and 9, we have generated a sample of 50000 linguistic BWM preferences and computed the corresponding weights. Then, for each BWM preference ( B O , O W , B , W ) and resulting output u, the objective value has been computed X = ξ ( B O , O W , B , W , u ). Subsequently, the cumulative density function for X has been approximated using the function ecdf in the package StatsBase for Julia (Bezanson et al., 2017). The obtained values for the percentiles are shown in Table 3, and allow classifying a BW preference according to the probability of obtaining a more consistent preference.

For instance, let us consider 5 criteria whose importance is evaluated from some BW preferences provided by evaluators. After solving the LC-BWM optimization model, the value of the objective function is ξ ( B O , O W , B , W , u ) = 0.016. Then, to determine the consistency index, we need to check the row that corresponds to 5 criteria ( n = 5) in Table 3. According to the table, the first value in that row that is greater than 0.016 is 0.025, which corresponds to the percentile 40 %. Therefore, the given preferences are allocated in the top 40 % of the most consistent preferences for 5 criteria. Note that the acceptable values of the consistency index must be chosen according to the needs of the MCDM problem, taking into account that the lower the value ξ ( B O , O W , B , W , u ), the higher the reliability of the preferences ( B O , O W , B , W ). On the contrary, preferences whose consistency index is closer to the percentiles 80 % − 100 %, will show a poor level of consistency and, consequently, are not reliable. As a general rule, those preferences whose consistency index is allocated in the percentiles 20 − 40 %, are reliable enough to generate reasonable results.

Therefore, we define an individual consistency index that measures how far the individual preferences are from the ideal consistency scenario, i.e. ξ ( B O , O W , B , W , u ) = 0. Since our proposal aims to deal with LiMCGDM problems, the group consistency index is computed from the average of the individuals’ consistency indices. The LC-BWM derives the additive weights minimizing the group consistency, and thus keeping as much as possible the initial evaluators’ view, and satisfying the consensus condition to obtain an agreed solution. In any case, instead of the arithmetic mean, the maximum operator could be used for computing the group’s consistency index and avoiding compensatory effects without impacting the structure of the model (García-Zamora et al., 2023b; García-Zamora et al., 2025).

The case study presented in the previous section was intentionally limited to five evaluators and four criteria in order to facilitate the explanation of the proposed LC-BWM model and allow the reader to clearly follow each stage of the decision-making process. Although such a configuration is representative of many real-world group decision-making problems, it does not fully illustrate the computational capabilities of the proposed approach when dealing with larger problem instances.

Therefore, an additional computational scalability analysis was conducted to evaluate the performance of the model under increasingly demanding conditions. Specifically, the number of evaluators was progressively increased from 5 to 100 evaluators. This allows us to assess whether the proposed formulation remains computationally efficient when applied to large-scale group decision-making scenarios involving a substantial number of participants.

Regarding the number of criteria, the experiments were restricted to scenarios involving 3, 5, 7, and 9 criteria. This decision is motivated by the cognitive limitations associated with pairwise comparison-based preference elicitation methods. According to Miller’s well-known “seven plus or minus two” principle (Miller, 1956), individuals have a limited capacity to simultaneously process information, and several studies have highlighted that excessively large sets of elements to compare may impose a significant cognitive burden on evaluators (Goodwin and Wright, 2014; Rezaei, 2015; Saaty, 1990). Consequently, considering more than nine criteria would be less representative of realistic applications of the BWM framework and similar pairwise comparison approaches.

To perform this analysis, consistent synthetic BWM preference relations were automatically generated for each combination of evaluators and criteria. The execution time required to solve each instance was then recorded. Table 6 summarizes the obtained results. As expected, the execution time increases with both the number of evaluators and the number of criteria. Nevertheless, even for the largest tested scenario involving 100 evaluators and 9 criteria, the proposed LC-BWM model required less than one second to obtain the optimal solution. These results suggest that the computational burden introduced by the consensus mechanism remains moderate and that the proposed formulation is suitable for large-scale group decision-making problems.

This section compares the LC-BWM approach with other existing group BWM approaches. The comparison focuses on G-BWM and MCC-BWM because these approaches share the same objective of deriving criteria weights from BWM preference structures in group decision-making environments. Moreover, all three methods operate on the same type of preference information, which allows a direct and fair comparison of their weighting and consensus mechanisms. In contrast, many well-known MCDM methods either address different decision-making tasks or require alternative preference representations. Consequently, comparing the proposed LC-BWM with such methods would require transforming the original BWM preference information into different formats. This transformation process may introduce information loss or distort the original evaluators’ assessments, potentially affecting the validity and fairness of the comparison. Among criteria-weighting approaches, AHP is one of the most widely adopted methods. However, AHP relies on a different preference elicitation process based on complete pairwise comparison matrices. In fact, one of the main motivations behind the development of the BWM was to reduce the number of required comparisons and the associated cognitive burden while maintaining satisfactory consistency levels. These advantages have already been extensively discussed and validated in the BWM literature. Therefore, the objective of this study is not to reassess BWM against AHP, but rather to analyse how consensus mechanisms can be incorporated into the BWM framework. For this reason, G-BWM and MCC-BWM constitute the most appropriate benchmark methods for evaluating the contribution of the proposed consensus-based BWM approach.

To do this comparison, we first define default BW preferences that are then used to run the different proposals based on the BWM. Note that the proposals sometimes present different characteristics, so it will be necessary to perform some operations on the preferences and optimization models to make a fair comparison.

In this comparative analysis, for the sake of simplicity, we consider 3 evaluators that provide the following BW preferences B = ( C 1 , C 5 , C 3 ), W = ( C 5 , C 1 , C 2 )

This section provides a sensitivity analysis to provide the robustness and validity of the proposal. To do so, we carry out two different analyses. The former is regarding the LC-BWM behaviour when the initial evaluators’ preferences are changed to a greater or lesser extent. The latter analyses the results obtained from the LC-BWM, considering different values of ε 0 .

The authors have no relevant financial or non-financial interests to disclose.

This study does not involve human participants and/or animals.

Álvaro Labella: Writing – review & editing, Writing original draft, Validation, Software, Formal analysis, Conceptualization. Diego García-Zamora: Writing – review & editing, Writing original draft, Validation, Software, Formal analysis, Conceptualization. Bapi Dutta: Writing – review & editing, Writing original draft, Validation, Software, Formal analysis, Conceptualization. Luis Martínez: Writing – review & editing, Writing original draft, Supervision, Formal analysis, Conceptualization.