Informatica

Making decisions is a routine activity for human beings. Typically, a decision-making problem involves selecting the best solution among several alternatives. However, decision-making problems become more complex when multiple criteria must be considered. These types of problems are referred to as multi-criteria decision-making (MCDM) problems and arise in a variety of contexts, such as supplier selection (Kabadayi and Dehghanimohammadabadi, 2022; Sahoo et al., 2024), renewable energy (Asakereh et al., 2022; Jameel et al., 2026), healthcare (Bouraima et al., 2024; Puška et al., 2022), and education (Srivastava et al., 2024). Multi-criteria group decision-making (MCGDM) problems occur when the decision-making process for MCDM problems is carried out by taking into account the preferences of several evaluators.

The participation of multiple evaluators in an MCGDM problem allows the decision problem to be analysed from different perspectives. In the specialized literature, there are multiple rules for solving MCGDM problems. Among others, Authority Rule (Pingle, 1997), in which a leader has the authority to make the final decision, Majority Rule (Straffin Jr, 1977) based on a voting process, Minority Rule (Butler and Rothstein, 2006), in which the decision is assigned to a subgroup of people, Negative Minority Rule (Butler and Rothstein, 2006), where the group votes for the least popular alternatives to be removed and several rounds are carried out, or Consensus Rules, in which all group members agreed that the solution chosen is a satisfactory solution (García-Zamora et al., 2023a). This paper follows the latter type of rules and the concept of consensus-reaching process (CRP) to address MCGDM problems. In a CRP, usually supervised by a moderator, evaluators discuss among themselves and try to increase the level of agreement within the group to reach a solution that satisfies its members. However, CRPs are usually time-consuming and become increasingly complex as the number of evaluators grows. Under these circumstances, automatic CRP models have proven to facilitate the task of the moderator, supporting the consensus process in an effective and accurate way, because these models are able to detect disagreements and provide agreed solutions without evaluators’ feedback (García-Zamora et al., 2023b; García-Zamora et al., 2025). Obviously, these solutions are valid from a theoretical point of view, but they can serve as a guide for moderators to guide a CRP through the feedback process. In this regard, our contribution addresses the handling of disagreements between evaluators in MCGDM problems from an automatic CRP perspective and subsequently helps the moderators to conduct CRPs with feedback in a faster and more efficient way if it is necessary.

Solving MCGDM problems is becoming increasingly complex due to the lack of information and uncertainty often present in decision-making contexts. This can lead to doubts and vague judgments that cannot be easily modelled by using numerical values. To address this challenge, fuzzy linguistic approaches have been widely adopted (Zadeh, 1975), allowing evaluators to express their preferences by using linguistic expressions that align better with human communication patterns. MCGDM problems in which evaluators provide their opinions by means of linguistic expressions are known as linguistic MCGDM (LiMCGDM) problems. Several linguistic models have been proposed in the literature (Du et al., 2023; Martínez et al., 2015; Wang and Hao, 2006), but the 2-tuple linguistic model (Martínez et al., 2015) is particularly noteworthy for its ability to perform precise computing with words (CW) processes.

One of the most common tasks in MCDM problems is evaluating the importance of the criteria involved in the problem. Various decision methods have been developed to derive the importance of criteria from evaluators’ opinions, such as the expert judgment method proposed by Kendall based on expert questioning (Kendall, 1970), the Analytic Hierarchy Process (AHP) introduced by Saaty (1977, 1990) that obtains criteria weights from pairwise comparisons among the criteria, or the step-wise weight assessment ratio analysis (SWARA) by Keršuliene et al. (2010) to estimate the opinions of experts or interest groups on the significance ratio of criteria.

The BWM has gained popularity due to its simplicity and reliability in deriving criteria weights (Rezaei, 2015). The BWM relies on pairwise comparisons between the best and worst criteria, which are chosen by the evaluator, and the remaining criteria. It addresses some behavioural errors related to the consistency of the preferences that arise in similar MCDM methods. Despite its effectiveness, the first version of this method has limitations:

Extant works have already presented several attempts to address the previous limitations, either by adapting the BWM to MCGDM (Moreno-Albarracín et al., 2020; Safarzadeh et al., 2018) or even by using linguistic information (Licerán-Gutiérrez et al., 2022). Safarzadeh et al. (2018) proposed a Group BWM (G-BWM) to manage MCGDM problems. However, it still has some drawbacks, such as the imposition that all evaluators select the same best and worst criterion, the omission of the LiMCGDM problems, or its inability to deal with the disagreements that may arise among evaluators. On the other hand, the Minimum Cost Consensus BWM (MCC-BWM) versions introduced in Licerán-Gutiérrez et al. (2022); Moreno-Albarracín et al. (2020) follow a similar approach. First, they applied an optimization BWM model for each evaluator to obtain individual weights. Afterward, such individual weights are used as inputs of an MCC optimization model (Ben-Arieh et al., 2008) to obtain collectively agreed weights. The MCC model allowed for deriving collective weights by changing as little as possible the individual weights and obtaining an agreed solution. However, this way of dealing with disagreements can lead to inconsistent results.

Therefore, this paper aims to support CRPs with feedback and deal properly with the conflicts that may appear among the participants in a LiMCGDM problem to fix the importance of the criteria. A novel MCC-BWM method is then proposed to adapt the traditional BWM to a 2-tuple linguistic setting, and generates optimal agreed collective weights based on pairwise linguistic comparisons of their views on the best to others, and others to the worst criteria. These collective weights are directly obtained as numeric values from a single mathematical programming model that integrates consensus constraints within classic BWM to obtain the theoretical optimum for the CRP. To retrieve a linguistic output, an unbalanced 2-tuple linguistic scale (Licerán-Gutiérrez et al., 2022; Martínez et al., 2015) is then used to guarantee the interpretability of the results. In comparison to a balanced linguistic scale, unbalanced scales allow describing situations in which the linguistic labels are not symmetrically distributed in the universe of discourse in which they are defined (Herrera et al., 2008). These linguistic outputs can be used by the moderators involved in a CRP to (i) identify disagreements among the evaluators, (ii) identify the changes in the preferences to be suggested, and, ultimately, (iii) to reach the desired consensus, keeping the evaluators’ initial opinions as much as possible. In addition, the proposal includes a novel consistency measure to quantify the consistency of the preferences provided by the evaluators.

To sum up, the main novelties of the proposal are listed below:

This paper is structured as follows: Section 2 provides a review of fundamental concepts related to the proposed method. Section 3 presents an extension of the BWM for handling disagreements in LiMCGDM problems. In Section 4, the feasibility and applicability of the proposed method are illustrated through a case study related to sustainable packaging. Section 5 further validates the proposal by means of a comprehensive numerical analysis, including computational scalability experiments, comparative analyses with related BWM approaches, sensitivity analyses, and a discussion of the main findings. Finally, Section 6 summarizes the main conclusions and identifies avenues for future research.

In this section, some preliminary concepts related to the proposal are revised. Firstly, the notion of LiMCGDM is reviewed. Afterward, the 2-tuple linguistic model is described. Then, CRPs are introduced as a tool to soften the disagreements in MCGDM processes. Finally, the classical BWM and some of its extensions are discussed.

This section first introduces the selected linguistic additive scale that replaces Saaty’s multiplicative scale used in the classical BWM to provide the evaluators’ preferences. Afterward, a novel consensus BWM optimization model is introduced, which allows facing LiMCGDM problems regarding evaluating criteria importance and obtaining consensual solutions represented both numerically and linguistically. Finally, a novel consistency index is provided to evaluate the reliability of evaluators’ preferences.

The food industry is a major contributor to global environmental challenges such as climate change, deforestation, and plastic pollution (D’Adamo, 2023; Prasanna et al., 2025). One area where food companies can make a significant impact on sustainability is in their choice of packaging materials.

Packaging plays an important role in protecting food and extending its shelf life, but it also generates a significant amount of waste, particularly in the form of single-use plastics. In recent years, there has been a growing interest in sustainable packaging options, such as those made from biodegradable materials or those that are easily recyclable.

In this case study, a food company aims at reducing its environmental impact and is exploring new packaging options. The company’s sustainability team has studied some potential packaging options and is seeking input from a diverse group of evaluators to help make a decision.

This case study includes 5 representatives from the packaging industry, environmental organizations, and the company’s internal departments $E=\{{E_{1}},{E_{2}},{E_{3}},{E_{4}},{E_{5}}\}$. Each evaluator brings their own perspectives, values, and priorities to the decision-making process, highlighting the importance of a structured and transparent decision-making approach.

Before considering the new packaging options to reduce the environmental impact, the evaluators also have to identify the criteria to consider for evaluating such options. They have identified 4 main criteria $C=\{{C_{1}}:\textit{recyclability},\hspace{2.5pt}{C_{2}}:\textit{cost},\hspace{2.5pt}{C_{3}}:\textit{energy usage},\hspace{2.5pt}{C_{4}}:\textit{carbon emissions}\}$. However, the evaluators have different views on the importance of each criterion, which led to differences in their evaluations. To ensure a unified judgment, the panel of members has agreed to participate in a CRP supervised by a moderator to achieve an agreement on the importance of each criterion. In this regard, the LC-BWM method will be used to support the moderator during the consensus process.

Following the BWM approach, the members of the panel first have to choose the best and worst criteria according to their expertise. In the LC-BWM, the vectors B and W, which contain the best and worst criteria selected by the evaluators, respectively, are described below: Once the members have chosen the best and worst criteria, they provide linguistic pairwise comparisons. The evaluators use the fuzzy linguistic term set S with $G=16$ described in Section 3 to assess their pairwise comparisons, which are collected in the LC-BWM through the matrices $BO$, which contains the comparisons between the best criteria and the remainder, and $OW$, which contains the comparisons between the remainder criteria and the worst ones as follows:

The moderator can also compute the consistency index of these preferences by using the LC-BWM to check if they are reasonable. This is key because if the preferences are not consistent (contradictory/random), the results are not reliable, and the whole process may be compromised. In this regard, all of them are allocated in the top 40% of the most consistent preferences. In addition, note that we have transformed the single linguistic terms provided by the evaluators into 2-tuple linguistic values by adding a symbolic translation equal to 0 (see Remark 1). This is necessary to apply the LC-BWM later on.

Before carrying out a CRP, let us use the L-BWM model to compute the importance of the criteria according to each evaluator’s opinion (see Table 4). The computation of the consensus degree of such preferences (0.93) evidences that there are some disagreements in the evaluators’ opinions, a CRP is initiated to increase the level of agreement in the group. To easily identify such disagreements and provide precise suggestions to reach a consensus faster, the CRP is first theoretically computed by the LC-BWM.

To apply the LC-BWM, we transform the linguistic preferences into a linear scale in $[0,1]$ by using the mapping ${\hat{\Delta }_{S}^{-1}}$:

From the previous numerical matrices and the vectors B and W, we apply the LC-BWM optimization model considering ${\varepsilon _{0}}=0.05$ to obtain the agreed modified evaluators’ opinions, and the agreed individual and group weights, which are expressed in a multiplicative scale by using Eq. (13) (see Table 5).

Note that the values of the modified preferences may be obtained from the modified individual weights by:

Thus, we can obtain the feedback for the evaluators whose opinions are represented in $\overline{BO}$ and $\overline{OW}$, in which changes with respect to their initial opinions have been highlighted in bold.

Note that most of the evaluators’ preferences remain the same. LC-BWM suggests that ${E_{3}}$ and ${E_{4}}$ should change their view regarding the importance of ${C_{1}}$ and ${C_{2}}$, respectively. The highest number of changes is recommended for ${E_{5}}$. It is suggested that all preferences of this evaluator’s regarding the best-to-other comparison change slightly (no change involves more than one linguistic label away) since this evaluator’s opinion is the furthest away from the rest. These changes are reflected in the resulting weights shown in Table 5.

Table 5 shows the optimal theoretical weights for the problem that allow achieving the desired level of consensus. In this table, we can see even more clearly that the changes suggested by the LC-BWM imply minimal changes in the weights obtained from the evaluators’ initial preferences. The most significant change occurs in the weight given by ${E_{4}}$ to ${C_{2}}$ (-40%).

In addition, the agreed group solution identifies ${C_{4}}$ (carbon emissions) as the most important criterion and ${C_{3}}$ (energy usage) as the least important one (see the row Group in Table 5). To facilitate the understanding of the results from the evaluators, the numerically agreed weights can be retranslated into a linguistic representation (see the row Linguistic in Table 5). Notice that the most important criterion ${C_{4}}$ is labelled with “important”, whereas the least important criterion ${C_{3}}$ is labelled as “unimportant”. This linguistic representation can be useful for both moderators and evaluators, as it allows them to immediately identify the importance of a criterion by using linguistic terms that people often use to express degrees of importance.

Therefore, this information can be used by the moderator to provide suggestions in the following round of the CRP and achieve the desired level of consensus efficiently. Notice that the changes suggested by the LC-BWM are quite slight since the model finds a solution that changes the initial preferences of the evaluators as little as possible. This is key for the use of the LC-BWM in CRPs with feedback, since it is expected that the evaluators will be willing to modify their opinions if these changes are minimal.

For all experiments reported in this manuscript, we have used JuMP (Julia for Mathematical Programming), a domain-specific modelling language for mathematical optimization embedded in Julia (Dunning et al., 2017). Specifically, optimization experiments are conducted in Julia 1.12.6 on a laptop with Windows 11 Professional OS, 1.4 GHz Intel Core Ultra i7-155H CPU, and 32 GB RAM by invoking the Gurobi 9.0.3 optimizer.

This study proposes a novel extension of the BWM based on the fuzzy linguistic approach that uses the 2-tuple linguistic model to derive agreed criteria weights in LiMCGDM problems. Evaluators express pairwise comparisons through a linguistic scale, enabling the modelling of uncertainty in their assessments. A novel BWM optimization model has been presented to derive consensual individual and group weights by preserving as much as possible the initial evaluators’ views. The resulting weights are represented both numerically and linguistically to enhance their interpretability. In addition, we have provided a novel consistency index based on a cumulative density function that evaluates the consistency in evaluators’ preferences according to the probability of finding more consistent values. Finally, we have proved the feasibility of the model by solving a real-world case study and comparing the performance of the proposal with other versions of BWM for groups.

Beyond extending the BWM to a linguistic group decision-making environment, the main novelty of the proposed LC-BWM lies in the integration of consensus requirements directly into the weighting process. Unlike existing approaches that first derive individual weights and subsequently apply a consensus model, LC-BWM simultaneously obtains individual and collective consensual weights within a single optimization framework. This allows the method to preserve the evaluators’ original opinions as much as possible while ensuring a predefined level of agreement among the participants.

The key advantages of the proposal are listed below:

From a practical perspective, the proposed LC-BWM provides moderators and decision analysts with a support tool for CRPs. By simultaneously deriving individual and collective consensual weights while preserving the evaluators’ original views as much as possible, the approach facilitates the identification of disagreement sources and supports the generation of targeted recommendations. Furthermore, the linguistic representation of preferences and results enhances the interpretability of the obtained solutions.

Despite these advantages, several limitations should be acknowledged. The proposed methodology cannot guarantee that evaluators will accept the suggested preference modifications during a real CRP. In addition, the computational representation of linguistic assessments relies on predefined semantics that may not perfectly match the interpretation of all evaluators. Finally, further empirical validation in additional decision-making domains would strengthen the practical evidence supporting the proposal.

Regarding future research, several directions deserve further investigation. First, although the proposed LC-BWM can theoretically handle large groups of evaluators, its performance in large-scale group decision-making scenarios should be analysed in greater detail, particularly from a computational and consensus-management perspective. Second, future studies could investigate alternative consensus measures and aggregation operators to analyse their impact on the resulting collective weights. Third, it would be interesting to extend the proposal to more sophisticated uncertainty modelling frameworks, such as hesitant, probabilistic, or multi-granular linguistic information.

Furthermore, future work will explore the simulation of different evaluators’ behaviours during CRPs, including situations in which some evaluators partially accept or reject the suggested modifications. The integration of intelligent support mechanisms to automatically generate personalized feedback recommendations for evaluators also represents a promising research direction.

Finally, we will consider the use of co-constructive preference modelling approaches to validate and refine the conversion of evaluators’ linguistic assessments into computational representations (Corrente et al., 2021). Such approaches could provide a more faithful representation of the semantics associated with linguistic expressions and further improve the interpretability and acceptance of the generated recommendations.