Decision SupportMultiple criteria group decision making with belief distributions and distributed preference relations
Introduction
Emerging information technologies (ITs), such as cloud computing, big data, and Internet of things, have propelled the transformation and upgrading of economy and the rapid development of society. In the process, researchers have recognized many important issues. Among them, one issue is that the individual capabilities of perceiving and understanding real problems are limited, and thus group capabilities are preferred to analyze real problems and generate satisfactory or acceptable solutions. Another important issue is that increasing attention has been paid to sociological and ecological perspectives except for economic perspective when real problems are analyzed in the current era. Over the past several years, many real situations have been managed by depending on group capabilities. For example, concerns about the environment and sustainability are addressed by ways that include evaluating the management plans of a region to protect the ecosystem of the region from economic, ecological, and social perspectives (Zendehdel, Rademaker, De Baets, & Van Huylenbroeck, 2010), evaluating and selecting e-waste recycling programs to protect the environment (Wibowo & Deng, 2015), selecting the recycling and reusing technologies of waste materials (Soltani, Sadiq, & Hewage, 2016), and evaluating alternative marine fuels from the sustainability dimensions of ecology, sociology, and technology (Ren & Liang, 2017). To focus on emergencies, representative studies include selecting rescue plans after an earthquake (Xu et al., 2015) and selecting emergency technologies to manage water source pollution accidents (Qu et al., 2016). In addition, studies that focus on traditional problems integrated with emerging ITs have addressed the selection of IT outsourcing providers (Li & Wan, 2014) and the evaluation and selection of green suppliers in green supply chain management for an automobile enterprise (Qin et al., 2017).
With the aim of handling real situations such as those listed above, a number of group decision making (GDM) methods have been developed. These methods have focused on the aggregation of decision makers’ preference information (Blagojevic et al., 2016, Bottero et al., 2018, Chen et al., 2016, Chiclana et al., 2007, Liu et al., 2016a, Liu, 2017, Merigó et al., 2016), the guaranteeing of both the consistency of individual preference information and group consensus (Altuzaarra et al., Altuzarra et al., 2010, Dong et al., 2010, Meng and Chen, 2015, Wan et al., 2017, Wu and Xu, 2016, Zhang et al., 2015), group consensus convergence (Dong et al., 2015a, Fu and Yang, 2010, Fu and Yang, 2012, Zhang, 2017) and large-scale GDM with more than 20 decision makers (Liu et al., 2014, Wu and Liu, 2016, Liu et al., 2016b). There are generally two modes of characterizing the preference information of decision makers about alternatives in these methods. One mode is to compare alternatives in pairs, such as multiplicative preference relations (Dong et al., 2010), fuzzy linguistic preference relations (Wu & Xu, 2016) and fuzzy preference relations (Zhang et al., 2015). The other mode is to evaluate alternatives solely, such as linguistic assessments (Dong et al., 2015a, Merigó et al., 2016), type-2 fuzzy assessments (Wu & Liu, 2016) and belief distributions (Fu & Yang, 2010). Although the sole application of the two modes is feasible in many situations, that is not always the case. In some situations in which decision makers in GDM have different backgrounds, knowledge, experience, and cognitive habits, it is a flexible way for decision makers to provide preference information about alternatives in their preferred modes.
Many prior studies have been conducted to address situations in which decision makers prefer different preference modes. As summarized by Chen et al. (2015), there are three types of methods to address GDM with the two modes of preference information, including indirect methods, direct methods, and optimization-based methods. With indirect methods, heterogeneous preference information is unified through transformation functions and then aggregated into collective preference to generate a group solution (Chiclana et al., 1998, Chiclana et al., 2001, Dong et al., 2009, Herrera-Viedma et al., 2002). However, with direct methods, individual preference (or priority) vectors are derived from heterogeneous preference information and then aggregated into a collective preference vector to generate a group solution. Two important properties, including the internal consistency and the Pareto principle of social choice theory, are required to be satisfied (Dong and Zhang, 2014, Dong et al., 2015b). Finally, in optimization-based methods, heterogeneous preference information is aggregated in an optimization model to determine the collective preference vector and then generate a group solution (Fan et al., 2006, Ma et al., 2006, Zhang and Guo, 2014). The existing studies on direct and optimization-based methods indicate that heterogeneous preference information in these two types of methods is usually associated with individual and (or) collective preference vectors, in which each element represents the priority or the ranking value of an alternative (Chen et al., 2015). However, in some situations, such association may be difficult to construct. For example, when decision makers select belief distributions (Fu and Yang, 2010, Voola and Babu, 2017, Zhang et al., 2017) and distributed preference relations (Fu et al., 2016) to provide assessments, it is not easy to derive preference vectors from preference information. As a result, the indirect method seems to be a feasible way to handle GDM with belief distributions (BDs) and distributed preference relations (DPRs). Meanwhile, this presents a challenge because little attention has been paid to the indirect GDM method with BDs and DPRs and the transformation between BDs and DPRs. However, some existing indirect methods (e.g., Chiclana et al., 1998, Chiclana et al., 2001, Herrera-Viedma et al., 2002) may violate the internal consistency and the Pareto principle of social choice theory, as numerically demonstrated by Dong and Zhang (2014). This results in another challenge of generating a rational solution by using an indirect GDM method.
In this paper, we propose a new method to address these two challenges. After presenting the modeling of multiple criteria group decision making (MCGDM) problems with BDs and DPRs, we describe the process of analyzing the MCGDM problems and identify why BDs are transformed into DPRs. With the aid of the utilities of the grades associated with BDs and the scores of the grades associated with DPRs (see Section 2.1), the transformation from BDs into DPRs is discussed in detail. Two important properties of the transformation, including the internal consistency and the Pareto principle of social choice theory, are analyzed and theoretically proven. By using the transformation, the process of generating solutions to the MCGDM problems is presented.
The rest of this paper is organized as follows. Section 2 introduces the modeling and analysis of MCGDM problems with BDs and DPRs. Section 3 presents the proposed method. In Section 4, a problem of selecting a key field for an enterprise is investigated to demonstrate the applicability of the proposed method. The paper's conclusions are presented in Section 5.
Section snippets
Modeling and analysis of MCGDM problems with BDs and DPRs
In this section, we simply present the modeling of MCGDM problems with BDs and DPRs and the process of analyzing the problems to find group solutions.
Proposed method
In this section, we present a method of solving MCGDM problems with BDs and DPRs, which is an indirect method. The transformation from BDs into DPRs is introduced in the method. To guarantee the validity of the transformation in MCGDM, it is theoretically proven that the transformation satisfies the internal consistency and the Pareto principle of social choice theory, which are two important properties of GDM methods with different formats of preferences (Dong & Zhang, 2014). The whole process
Case study
In this section, the problem of selecting a key field for an enterprise located in Hefei, Anhui Province, China, is solved to demonstrate the applicability of the proposed method. To facilitate the handling of the problem, a solution system is developed in the MATLAB environment.
Conclusions
When a group of decision makers face a common decision problem, they usually seek to provide preference information with the different modes that are consistent with their knowledge, experience, and cognitive habits as they are allowed. To facilitate decision makers in flexibly providing preference information, two modes of preference information including the sole evaluation of alternatives and the pairwise comparison between alternatives are generally adopted in the existing studies. These
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grant Nos. 71622003, 71571060 71690235, 71690230, and 71521001).
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