An asymmetric trapezoidal cloud-based linguistic group decision-making method under unbalanced linguistic distribution assessments
Introduction
With the rapid development of society, decision-making issues are becoming increasingly complex and challenging, particularly when they regard multiple criteria. Ideally, decisions should be made by groups instead of individuals, especially when the consequences affect a group of people. As an important component of multicriteria group decision-making (MCGDM), linguistic MCGDM involves a more uncertain and complicated decision-making circumstance. Under this circumstance, it is difficult for decision-makers to provide certain values when examining options due to the essence of human thinking and complications with the options and criteria. Therefore, capturing the uncertainty in decision-making has recently received much interest. Some scholars believe that a great understanding of linguistic variables is critical to capturing uncertainty for linguistic decision-making issues; thus, they have conducted a series of studies that utilize fuzzy information representations to describe linguistic variables, involving interval fuzzy numbers, trapezoidal fuzzy numbers, cloud models and others (Tian et al., 2018, Yu et al., 2018).
As a kind of fuzzy information representation, cloud models are very popular for describing linguistic variables because they can describe the uncertainty and randomness of the decision information simultaneously. Therefore, these models have a small information loss and distortion during a mapping from linguistic variables to numerical values, and the original information can be retained as much as possible. Cloud models were proposed by Li, Liu, and Gan (2009). On this basis, some scholars have utilized cloud models as quantitative tools to complete a conversion of linguistic evaluation terms, involving risk assessment (Song and Zhu, 2019, Song et al., 2020). With the development of the research in this area, some scholars found that cloud models cannot represent the information of some concepts well. Consequently, these scholars created some extensions for cloud models to solve this issue, such as interval integrated clouds (Wang, Peng, Zhang, Liu, & Chen, 2015) and probabilistic linguistic integrated clouds (Peng, Zhang, & Wang, 2018).
However, existing cloud models and their extensions still cannot create a good information representation for some qualitative concepts. Taking age as an example, there are six groups: babies (0–3 years old), children (5–10 years old), teenagers (12–16 years old), youths (18–32 years old), middle-aged people (40–55 years old) and the older people (after 65 years old). Considering the irregular and unbalanced distribution of these six groups on the age axis, existing cloud models and their extensions have difficulty representing the information effectively for these six linguistic terms, because their expectation is only a specific value, or they are just balanced concepts. The current models cannot meet the following conditions simultaneously: 1) effectively providing a description of linguistic terms that involve interval concepts instead of specific concepts, and 2) effectively distinguishing the difference in spacing for each linguistic term.
To solve this issue, further research is conducted. According to existing studies, an extended cloud model, namely asymmetric trapezoidal clouds (ATCs), is developed, followed by the related contents with respect to ATCs. Our research has conducted the following work.
- (1)
To fill the research gap, the ATC model is proposed. Compared with existing cloud models, the expectation of ATCs is an interval value instead of a specific value, and the entropy and hyper entropy of ATCs are different on left and right sides. Based on this extension, ATCs are suitable for a description of linguistic terms with respect to the above condition. Although ATCs are an excellent quantitative information representation, they are not directly utilized in the decision-making process, because their complex form is scarcely able to exist in practical conditions. Unbalanced linguistic distribution assessments (ULDAs) are common information representations in the group decision-making process, because they describe the decision information and the related probability distribution for decision makers well, but the linguistic information cannot be directly used in the calculation process. As a result, our study makes a link between ATCs and ULDAs. As an effective quantitative information representation with respect to ULDAs, ATCs can be thoroughly exploited in the group decision-making process.
- (2)
Based on the concept of ATCs, our research develops a series of related definitions to meet the requirements of the calculation process in relation to ATCs, and provides a conversion method to illustrate how to map ULDAs. Because traditional linguistic scale functions cannot satisfy the conversion demands in our research, extended linguistic scale functions are developed to make a mapping from linguistic terms to interval values, and they effectively embody differences of spacing instead of a symmetrical spacing for linguistic terms. Based on extended linguistic scale functions, the expectation interval value of ATCs can be determined. Furthermore, a method is provided to determine the entropy and hyper entropy of ATCs based on ‘3’ principle. For a valid operational and computational process, some definitions of ATCs are proposed that involve a comparison method, distance measure and operational rule.
- (3)
To address practical issues, the above theoretical knowledge is incorporated into decision support models. In our research, two kinds of models are provided that involve ATCPA-based and TODIM-PROMETHEE Ⅱ-based decision support models. The previous model is based on the ATCPA aggregation operators, which is an extension to apply the traditional PA operator to the ATC situation. The latter model is based on a combined MCGDM method called the TODIM-PROMETHEE Ⅱ method that overcomes the inherent restrictions of the TODIM and PROMETHEE Ⅱ methods. Then, these decision support models are utilized to address a practical issue involving the evaluation of industrial wastewater discharge, followed by a further analysis to provide a novel perspective that determines the prior governance factors by changing criteria weight values.
The rest of this paper is organized as follows. The related literatures are reviewed in Section 2, and Section 3 introduces some basic concepts. In Section 4, the definition of ATCs and their related knowledge are given, involving comparison method, distance measure and operational rule. Section 5 includes two kinds of decision support models. In Section 6, an illustrative case study is conducted with respect to the evaluation of industrial sewage discharge, followed by an analysis and discussion. The theoretical contributions, practical implications, limitations and future work are shown in Section 7.
Section snippets
ULDAs
Since the advent of linguistic variables (Zadeh, 1975), various kinds of linguistic information have been developed to quantify the uncertainty in decision-making issues, involving a model based on a single linguistic term, such as a 2-tuple linguistic representation model (Herrera & Martínez, 2000), a model based on multiple linguistic terms, such as hesitant fuzzy linguistic term sets (Wang, Wang, Tian, & Zhao, 2018), and linguistic distribution assessments (LDAs) (Nie, Tian, Wang, Wang, &
Preliminaries
In this section, some basic concepts that involve linguistic representation models, NS models and cloud models are briefly reviewed.
ATC model
To overcome the limitations of normal cloud models, a novel concept model is proposed, namely, the ATC model: its definition is provided below. Definition 7. Let the ATC model be denoted by and defined as
The ATC model is constructed with three numerical sections, involving the expectation interval , left-half cloud and right-half cloud . , and are the expectation, entropy and hyper entropy in the left-half cloud, respectively.
Decision support model
To apply our work to practical circumstances well, two ATC-based decision support models are developed to address MCGDM issues. One model is based on aggregation operators, while other is based on the methodology.
Illustrative case study
In this section, a background case study is introduced, followed by the implementation details and the calculation outcomes. Subsequently, the feasibility and rationality of the constructed decision support models are demonstrated via a comparison analysis with existing studies from Liu et al., 2021, Nie et al., 2020.Then, a further analysis and discussion are provided.
Conclusion
Our work makes some theoretical contributions to the fuzzy information representation and the fuzzy group decision-making method. First, a comprehensive information representation, namely ATCs, is proposed. The ATC model can distinguish the difference in spacing among linguistic terms. The development of ATCs overcomes these limitations in that existing cloud models cannot adequately describe some continuous linguistic concepts and cannot conduct a mapping of linguistic values for an ULDA. As a
CRediT authorship contribution statement
Xiao-kang Wang: Conceptualization, Methodology, Data curation, Writing - original draft, Visualization. Yi-ting Wang: Investigation, Data curation, Validation, Visualization. Hong-yu Zhang: Investigation, Validation, Data curation. Jian-qiang Wang: Supervision, Writing - review & editing, Funding acquisition. Lin Li: Investigation, Validation, Data curation. Mark Goh: Investigation, Validation.
Acknowledgements
The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper. This work was supported by China Scholarship Council and the National Natural Science Foundation of China (No. 71871228).
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