Decision SupportA limited cost consensus approach with fairness concern and its application
Introduction
Group decision-making (GDM) is a decision-making activity carried out by multiple individual decision makers (DMs), which solves unstructured decision-making problems. GDM theory was proposed with the development of welfare economics. On the basis of various social choice methods, Arrow (1951) proposed the famous Impossibility Theorem. Around Arrow's theory, a variety of GDM methods have been produced. There are two key processes to solve the GDM problem: consensus process and selection process (Herrera-Viedma, Herrera & Chiclana, 2002, 2007). In the consensus process, multiple DMs express, discuss, and modify their preferences or opinions in multiple rounds to reach the desired consensus level (Ben-Arieh & Easton, 2007; Cabrerizo, Moreno, Pérez & Herrera-Viedma, 2010; Dong & Cooper, 2016; Perez, Cabrerizo, Alonso & Herrera-Viedma, 2014). Generally, there is a moderator in consensus process, who does not participate in specific discussion and expression of opinions. The moderator is responsible for coordinating multiple DMs to reach a maximum consensus level efficiently and supervising the behavior of DMs (Cabrerizo et al., 2010; Dong & Cooper, 2016; Herrera, Herrera-Viedma & Verdegay, 1996; Xu, Cabrerizo & Herrera-Viedma, 2017).
In GDM, the traditional concept of consensus is defined as full and unanimous agreement that all DMs reach on alternatives (Bezdek, Spillman & Spillman, 1978; Kacprzyk & Fedrizzi, 1989; Spillman, Spillman & Bezdek, 1980). However, due to the harsh conditions, consensus is difficult to achieve in reality. Even if such a consensus is obtained, a great price (time, capital, and other resources) is required. There is more than one way to measure consensus level. Bordogna, Fedrizzi and Pasi (1997) divided these measurements into “hard” consensus and “soft” consensus based on numerical or linguistic opinions. The “hard” consensus requires a consensus level: non-consensus “0″ or complete consensus “1” (Bezdek et al., 1978, 1979). The “soft” consensus requires a consensus level in the interval [0,1] (Ben-Arieh & Chen, 2006). For instance, “Most of DMs agree with most of the other DMs on scheme A”. Another soft consensus is measured by the similarity between preference relations of DMs (Chao, Kou, Li & Peng, 2018). Obviously, hard consensus is a special case of soft consensus. The scholars mainly use opinion distance and preference relation to measure consensus level. The distance measurement between individual opinion and consensus opinion is widely used in GDM (González-Arteaga, Alcantud & de Andrés Calle, 2016; Roselló, Sánchez, Agell, Prats & Mazaira, 2014). The research on measurement of preference relations mainly involves DM's fuzzy preference relation with self-confidence in large-scale decision-making groups (Liu, Xu & Herrera, 2019), multi-granular linguistic preference relation (Morente-Molinera, Wu, Morfeq, Al-Hmouz & Herrera-Viedma, 2020), distributed preference relations with dynamic expert reliability (Xue, Fu & Yang, 2021), average similarity between intuitionistic fuzzy preference relations (Ureña, Chiclana, Melançon & Herrera-Viedma, 2019). Gong, Guo, Herrera-Viedma, Gong and Wei (2020) proposed the concept of uncertain preference relations, they proved that interval preference relation and their additive consistency are special cases of uncertain preference relation. In addition, some scholars have studied the consensus measurement with double hierarchy linguistic preference relations (Gou, Xu, Wang & Liao, 2021), heterogeneous preference structures (Zhang, Dong & Herrera-Viedma, 2019), interval additive preference relations (Wu, Yang, Tu & Chen, 2020), and so on.
When the decision-making problem is complex and resource consumption is serious, cost becomes a key factor in GDM. Some scholars proposed a minimum cost consensus model with linear cost function and quadratic cost function (Ben-Arieh & Easton, 2007; Ben-Arieh, Easton & Evans, 2009). The moderator adopts effective measures such as financial compensation to prompt DMs to adjust their initial opinions, so as to reach a consensus level. From the perspective of moderator, he/she always hopes to pay less consensus cost to reach a consensus. From the perspective of DMs, they always seek the greatest compensation. Therefore, the minimum cost consensus and the maximum compensation consensus form a dual relationship (Gong et al., 2015; Gong, Zhang, Forrest, Li & Xu, 2015). Since satisfaction level of DMs and moderator is crucial in GDM, we can use utility function to describe satisfaction level. Taking maximum utility as the optimization goal, the maximum utility (linear utility and non-linear utility) consensus models with limited cost are constructed (Gong, Xu, Chiclana & Xu, 2017; Gong, Xu, Li & Xu, 2015, 2015). Furthermore, the maximum utility consensus model with cost chance constraint is more applicable to real-world decision making (Tan, Gong, Chiclana & Zhang, 2018). Cost consensus shows an economic behavior, so there must be a game behavior among DMs. Therefore, there have been some studies on cost consensus with Stackelberg game (Zhang, Dong, Zhang & Pedrycz, 2020) and cost consensus with non-cooperative behavior (Xu, Chen, Dong & Chiclana, 2020).
Due to differences in knowledge, abilities, and background, DMs’ perception in consensus and interest is also different. If the consensus satisfies DM's perception and interest, then he/she will feel fair. Otherwise, he/she will feel unfair and destroy consensus. The fairness theory (Adams, 1963, 1965) believes that whether people are motivated is not only affected by the results they obtained, but also by the fairness of their comparison with others. In other words, one cares not only about own gains and losses, but also the others' gains and losses. If the ratio of one's gains and losses is roughly the same as that of others, he/she will think that the consensus is fair. If the ratio of one's gains is higher than others, he/she will receive additional incentives and may have a certain pride. If the ratio of one's gains is lower than others, he/she will feel unfair. Therefore, consensus fairness is crucial to consensus process and selection process in GDM (Kacprzyk & Zadrozny, 2016; Swiechowski, Kacprzyk & Zadrozny, 2016; Zhang et al., 2020). Justice and fairness concerns extend from the negotiation process to the outcome and into the implementation stage. Outcomes built on justice and fairness principles will enhance the efficiency, stability, and implementation of the negotiated agreements (Druckman & Wagner, 2017; Jacobs, Christensen & Prislin, 2009). To fairly weigh the DM's preferences, Fedrizzi and Brunelli (2009) designed a new consistency evaluation method. Fu, Zhou and Xue (2018) proposed a fairness framework for MCDM, it considered the fairness among criteria and the fairness among alternatives. In order to guarantee fairness for disadvantaged candidates, Kuhlman and Rundensteiner (2020) proposed a family of exact fairness algorithms for rank aggregation. Ducassé and Cellier (2014) proposed a Logical Multicriteria Sort (LMS) process to support multi-criteria sorting within islands of agreement. Alternatives were systematically analyzed along the selected criteria, which can guarantee some fairness. Boiney (1995) pointed out that moderator's intervening raises additional concerns for the DM in GDM, one of which is fairness. DMs’ horizontal recognition of compensation will affect their adjusted opinions. Therefore, we believe that there is necessary to study the influence of DMs' fairness concerns on GDM.
Existing studies on GDM rarely focus on the role of fairness, or only discuss the fairness of evaluation attributes, and do not systematically study fairness preference of DMs in consensus modeling. In the cost consensus, DMs may pursue the fair consensus results and maximum compensation. In view of this, this paper will propose a limited cost consensus approach with fairness concern of DMs. It has the following contributions:
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Based on fairness preference theory, consensus fairness is measured by defining the fairness utility function and fairness utility level. We explore some properties of the model and propose a generalized form of limited cost consensus model with fairness concern. In consensus reaching process, the DM will constantly compare the consensus compensation with other DMs to judge whether it is fair. Therefore, studying the fairness behavior of DMs is fully in line with the actual economic and management activities.
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The model is applied to the emission reduction among Chinese manufacturing enterprises. Through empirical analysis, we can obtain the influence of some parameters on consensus, especially the parameters related to fairness concern. These research results have crucial reference value for the government's environmental protection governance.
The rest of this paper is arranged as follows. Section 2 introduces the basic fairness preference theory and the minimum cost consensus. Section 3 defines the fair utility function and fair utility level in GDM. Based on this, we propose a limited cost consensus model with group fairness concern under a certain consensus level. In Section 4, we apply the model to the consensus issue of emission reduction among manufacturing enterprises. Through comparative analysis and sensitivity analysis, we verify the rationality of the model and put forward some management suggestions. The conclusions and directions for future research are provided in Section 5.
Section snippets
Preliminaries
In this section, we briefly introduce the basic notations, fairness preference theory and minimum cost consensus, which are necessary to understand the model proposed in this paper.
Model construction
The existing research on GDM consensus does not consider the fairness concern behavior of DMs. Consensus fairness is crucial to the consensus process and selection process in GDM problems (Kacprzyk & Zadrozny, 2016; Swiechowski et al., 2016; Zhang et al., 2020). In addition, the moderator needs to adopt some means to prompt DMs to adjust their opinions in consensus process. In what follows, we will define fairness utility level, consensus level, aggregation operator, and limited consensus cost
Empirical analysis
China's development has attracted worldwide attention, but rapid development has brought many social problems, especially manufacturing pollution. During the period of development and transformation of manufacturing, environmental pollution caused by high consumption makes China face severe environmental pressure. From the perspective of decision analysis, the negotiation between the government and manufacturing companies on pollutant emission reduction is a typical GDM problem. In summary, it
Conclusions
Consensus fairness is crucial to the consensus process and selection process in GDM problems. Based on fairness preference theory, this study developed a limited cost group consensus method with fairness concern behaviors of DMs. The proposed model more truly describes the decision-making behavior and activities of DMs in consensus process, and can obtain a stable and balanced consensus result. The main conclusions are shown as follows:
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Based on the fairness preference theory, this paper defines
Acknowledgments
The authors would like to thank the Editor-in-Chief and the anonymous reviewers for their insightful and constructive comments that have led to an improved version of this paper. This work was supported by the projects of the National Natural Science Foundation of China (72071111, 71801127, 71671091), the Fundamental Research Funds for the Central Universities of China (NC2019003), and a project of Intelligence Introduction Base of the Ministry of Science and Technology (G20190010178). At the
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