From numerical to heterogeneous linguistic best–worst method: Impacts of personalized individual semantics on consistency and consensus

Abstract

This paper investigates the impacts of personalized individual semantics (PIS) on consistency and consensus in Best–Worst method (BWM) with heterogeneous linguistic preference information. Firstly, a PIS driven consistency measurement and information transformation method is presented to analyze the consistency level of linguistic preference information (i.e., best-to-others and others-to-worst vectors) of each decision maker, and this is conducted by maximizing the consistency level of additive preference relation that converted from linguistic preference information via personalized numerical scales. Secondly, a PIS driven maximum consensus optimization model within the BWM framework is designed to yield the maximum consensus level among the additive preference relations generated from heterogeneous linguistic preference information by means of personalized numerical scales. Thirdly, a PIS-based heterogeneous linguistic consensus reaching process is put forward to promote the consensus establishment among decision makers. Finally, the validity of the proposed BWM framework is verified by a case study, a sensitive analysis, and a comparison analysis.

Introduction

The essence of group decision making (GDM) is the task of consolidating and fusing individual assessments so as to produce a collective ranking of a set of feasible alternatives (Chen et al., 2016, Dong et al., 2021a, Hochbaum and Levin, 2006, Tigran and Zvi, 2015, Wu et al., 2021a, Zhang et al., 2022). As an effective decision-making instrument, pairwise comparison method has been utilized in GDM to elicit the assessments of decision makers (Altuzarra et al., 2010). Several famous pairwise comparison methods have been reported over the last decades, including multiplicative preference relation (Saaty, 1980), additive preference relation (APR) (Orlovsky, 1978), and linguistic preference relation (Herrera et al., 1996). Among them, the multiplicative preference relation initialized by Saaty is one of the mostly used pairwise comparison methods (Saaty, 1980). Rezaei (2015) developed the Best–Worst method (BWM) on the basis of the multiplicative preference relation, in which the best and worst objects are first identified, and the best-to-others (BO, i.e., the best object over others) and others-to-worst (OW, i.e., the other objects over the worst object) preference vectors are then given by decision maker(s). Compared to the most popular pairwise comparison method, such as analytic hierarchy process (AHP), the BWM not only requires less preference information, but also yields more consistent preference information (Liang et al., 2020, Rezaei, 2015, Rezaei, 2016).

In view of the above excellent properties, the BWM has attracted a lot of attention in recent years. In particular, the existing BWM researches are mainly centered on the following aspects:

(1) Preference modeling in BWM. The original version of the BWM employs discrete values from 1–9 to perform the pairwise comparisons (Rezaei, 2015). To deal with uncertainty issues in practice, some improved preference modeling methods within the BWM framework have been raised with the examples of the simple linguistic terms (Guo and Zhao, 2017), the Z-number (Aboutorab et al., 2018), hesitant fuzzy set (Li et al., 2019), and 2-tuple linguistic model (Labella et al., 2021).

(2) Consistency issue in BWM. The consistency measurement is at the core of implementation of the pairwise comparison method, and it is devoted to analyzing whether preference information is logical or not (Aguarón and Moreno-Jiménez, 2003, Chiclana et al., 2009, Saaty, 1980). Rezaei (2015) presented a multiplicative transitivity-based consistency measurement for the BWM with 1–9 numerical points. Guo and Zhao (2017) reported a multiplicative transitivity-based consistency measurement method for BWM with linguistic information that described by triangular fuzzy numbers. Moreover, Li et al. (2019) introduced an additive transitivity-based approach for conducting consistency measurement in BWM. Liang et al. (2020) reported cardinal and ordinal consistency measurements to analyze coherence of the result in BWM and a method to determine consistency threshold. For more approaches for consistency measurement, see (Aboutorab et al., 2018, Labella et al., 2021).

(3) Preference fusion in BWM. As decision context becomes more and more complex, one individual may not consider all aspects of the decision problem. To this end, the BWM is extended into the group context, and several techniques have been established to reconcile and fuse the preferences of multiple decision makers. Hafezalkotob and Hafezalkotob (2017) proposed a hierarchical BWM framework to integrate the preferences of the senior decision makers and a group of decision makers. Mohammadi and Rezaei (2020) integrated a Bayesian hierarchical model to BWM to fuse the preferences of decision makers. Liang et al. (2021) investigated the preference fusion issue in belief-based BWM, in which the Dempster–Shafer​ theory is adopted to depict the preference degree of decision makers. Labella et al. (2021) proposed a sub-groups partition based preference fusion approach for large-scale linguistic BWM.

(4) Application of BWM. To date, BWM has been successfully applied in various areas, including supply chain management (Ahamadi et al., 2017), ecological risk assessment (Qin et al., 2021), quality assessment of airline systems (Rezaei et al., 2018), bioethanol facility location selection (Kheybari et al., 2019) and the smart catering (Van de Kaa et al., 2020).

Although great progress has been achieved in the study of BWM, they still need some improvements for better coping with practical decision-making problems.

(1) Numerical values vary from 1 to 9 were utilized by the original BWM to depict the preferences of decision maker(s). Subsequently, some improved preference modeling methods within the BWM framework were further reported to model fuzzy and uncertainty information. Among them, preference modeling methods with linguistic information have been widely applied. As far as we know, the existing BWM assumed that decision makers use the homogeneous linguistic information formats to express their preferences. Nevertheless, different decision makers may prefer to express preferences via heterogeneous linguistic information formats because they may have different knowledge structures, cultural backgrounds and practical experiences. Accordingly, it is necessary to develop a flexible BWM framework that decision makers are allowed to express preferences using heterogeneous linguistic information.

(2) It is common that same linguistic preference often means different things to different people, which is called personalized individual semantics (PIS) in linguistic GDM (Gao and Zhang, 2021, Herrera et al., 2000, Li et al., 2021, Mendel and Wu, 2010, Mendel et al., 2010, Wu et al., 2021b). For example, when evaluating a paper, two referees both think the quality of this paper is “good”. However, the linguistic term “good” may have different meanings for these two referees. Notably, one referee may think 85 points should be given to this paper should get, while the other referee may think the paper should get 98 points, when using a percentage point system. Although the PIS issue has recently attracted attention in the field of linguistic decision-making (Fan et al., 2021, Li et al., 2017, Li et al., 2021, Zhang and Li, 2022, Zhang et al., 2021), it has not been considered and addressed by existing linguistic BWM.

(3) In GDM context, there may be a large conflict between the preferences of decision makers because they may have different experiences and knowledge backgrounds (Altuzarra et al., 2010, Berekméri and Zafeiris, 2020, Chen et al., 2019, Labella et al., 2020, Meng et al., 2020, Sun et al., 2022, Wang et al., 2022, Wu et al., 2018, Wu et al., 2019, Wu et al., 2021c, Wan et al., 2022). However, to our knowledge, the preference divergence issue has not been considered in the existing BWM researches, which may hinder the smooth implementation of the final group decision result. Consequently, it is necessary to address the preference divergence issue by integrating a consensus reaching process into the BWM framework.

To overcome the above limitations of the extant BWM, this paper develops a PIS-based heterogeneous linguistic BWM framework with the following three main contributions.

(1) The preferences of decision makers within the BWM framework are modeling using heterogeneous linguistic preference information, which provides a flexible preference expression tool that conforms the preference expression habits of decision makers. In particular, three commonly used linguistic information formats, including simple linguistic terms, hesitant fuzzy linguistic term sets (HFLTSs) and linguistic distribution assessment (LDA), are used for the preference modeling.

(2) Considering the individualization phenomenon in linguistic preferences of decision makers, a PIS-based consistency measurement and information transformation method within the heterogeneous linguistic BWM framework is devised. Specifically, the proposed method achieves its objective by maximizing the consistency level of the numerical preference information generated from the linguistic preference information via setting personalized numerical scales (PNSs).

(3) The consensus issue is considered and addressed in the PIS-based heterogeneous linguistic BWM framework. First, a PIS-driven maximum consensus optimization model (PIS-MCOM) is devised to maximize the consensus level of the transformed numerical preference information from the heterogeneous linguistic preference information using PNSs, while ensuring that the consistency level of the individual numerical information is meeting the requirement. Then, a minimum adjustments consensus model is developed to promote the consensus among decision makers.

The rest of this paper is now described briefly. Section 2 reviews the basic concepts of the original BWM, the 2-tuple linguistic model, HFLTSs, LDA and APR. Section 3 defines the heterogeneous linguistic BWM based GDM problem and designs its resolution framework. Section 4 devises a PIS-driven consistency measurement and information transformation method. Section 5 put forwards a PIS-based consensus reaching process in heterogeneous linguistic BWM. After that, Section 6 offers a case study to show the application value of the PIS-based heterogeneous linguistic BWM. Section 7 presents sensitivity and comparative analysis to analyze the performance of the proposal. Finally, Section 8 concludes the main works and contributions of this paper.