Decision Support
An interval-valued intuitionistic fuzzy principal component analysis model-based method for complex multi-attribute large-group decision-making
Introduction
Multi-attribute group decision-making (MAGDM) problems have been the focus of substantial research since the sorting model of group alternatives was developed by the French mathematician Borda in 1781 (Ahn, 2015, Altuzarra, Moreno-Jimenez and Salvador, 2007, Brunelli and Fedrizzi, 2015, Chen, Chang and Lu, 2013, Dede, Kamalakis and Sphicopoulos, 2015, Huang, Chang and Li, 2013, Lahdelma and Salminen, 2001, Merigó, Casanovas and Yang, 2014, Ondemir and Gupta, 2014, Peng, Kou, Wang and Shi, 2011, Wang and Chin, 2009, Xia and Chen, 2015, Xia, Xu and Chen, 2013, Zhu and Xu, 2014). Due to the development of information techniques, the general public can participate in public decision-making problems, which has led to the evolution of the electronic democracy. The traditional MAGDM problem is a type of small-group decision-making process in which the scale of the decision maker (DM) is small (e.g., 3–5 persons), and the complexity of the decision-making problem is not high (e.g., the investment decision-making of the enterprise). However, decision-making in an electronic democracy is a type of public affairs decision-making process that exhibits a greater influence. Chen (2006) referred to these problems as complex multi-attribute large-group decision-making (CMALGDM) problems and characterized them by the following four features: (a) members of the group can render decisions at different times and in different places within a network environment, (b) the group usually consists of more than 20 DMs, (c) connections may exist between various decision attributes, and (d) the preference information of the DMs is uncertain.
In an uncertain environment, a fuzzy set (Zadeh, 1965) is limited by its failure to provide a broad description of all the information that is involved in a specific problem. To address this issue, Atanassov (1986) introduced the concept of the intuitionistic fuzzy (IF) set, which simultaneously considers membership and non-membership degree information. Atanassov and Gargov (1989) introduced the concept of the interval-valued intuitionistic fuzzy (IVIF) set, which denotes the degrees of membership and non-membership by closed subintervals of the interval [0, 1]. This approach not only expands the ability of the IF set to handle uncertain information but also improves its ability to solve practical decision-making problems. In this paper, we choose IVIF numbers as the preference information that is provided by the DMs.
Similar to MAGDM problems (Tan, 2011a, Tan, 2011b), CMALGDM problems involve the following two stages: (a) the aggregation stage, which combines individual preferences to obtain an overall preference value for each alternative, and (b) the exploitation stage, which orders the overall preference values to obtain the best alternative(s). Xu and Chen (2007), Wang, Guan, and Wu (2012), Lin and Zhang (2012), and Yu, Wu, and Lu (2012) developed several aggregation operators for IVIF information. These operators are based on the assumption that the attributes or DM preferences are independent; they are linear operators that are based on additive measures, which are characterized by an independence axiom (Keeney and Raiffa, 1976, Wakker, 1999). However, this assumption is not appropriate in CMALGDM problems. Due to the complexity of the CMALGDM problem, alternatives are usually described by multiple attributes that exhibit a high degree of interdependence or interactive characteristics. Because DMs are normally from a number of interest groups, the assumption of independence between the DM preferences in the same interest group is violated. As a result, we cannot adopt these operators to directly aggregate the alternative evaluation information in the CMALGDM problem.
To overcome the limitation of these operators, Tan (2011a, 2011b), Xu (2010), Xu and Xia (2011), and Wei (2012) considered the correlation between attributes or that between DMs from the perspective of fuzzy measures and proposed several Choquet integral (Choquet, 1953) based aggregation operators to aggregate the IVIF information. However, two drawbacks exist: (a) the method to confirm the fuzzy measures of the attributes or DMs is not advanced and (b) the number of fuzzy measures increases exponentially with an increase in the number of attributes or DMs. If there are n attributes or DMs, 2n fuzzy measures are required. In the CMALGDM problem, n is typically a large number, and acquiring 2n fuzzy measures is challenging. Thus, these operators cannot be conveniently applied to solve the CMALGDM problem.
Because the traditional aggregation operators are based on the independence axiom, should we treat the attributes and DMs as variables and then transform them into several independent variables? Principal component analysis (PCA) (Hotelling, 1933, Jolliffe, 1986, Pearson, 1901) is a well-known multivariate statistical technique that involves the orthogonal transformation of a specific number of correlated variables to an equivalent number of uncorrelated variables named principal components (PCs). Cazes, Chouakria, Diday, and Schektman (1997), Lauro and Palumbo (2000), Palumbo and Lauro (2003), Gioia and Lauro (2006), Guo and Li (2007), Hladik, Daney, and Tsigaridas (2010), D'Urso and Giordani (2004), Douzal-Chouakria, Billard, and Diday (2011), and Wang, Guan, and Wu (2012) generalized the PCA model for interval-valued variables. Based on this previous research, we propose a new PCA model for IVIF variables, called the IVIF-PCA model. Using this model, we compress the attributes and the DMs into several independent variables. This model not only reduces the complexity of the aggregation procedure but also improves the reliability of the aggregation results.
The remainder of this paper is organized as follows: In Section 2, we review several basic concepts about the IF set and the IVIF set and briefly introduce the traditional PCA model. In Section 3, we propose new concepts that form the foundation of the IVIF-PCA model. We describe the IVIF-PCA model in Section 4. In Section 5, we adopt the proposed model for the CMALGDM problem and provide a decision-making method. An example is provided to investigate the feasibility and validity of the proposed decision-making method in Section 6. In Section 7, we compare our approach to an interval PCA approach and give the decision-making results for the case in which we neglect the dependence between the attributes and that between the DMs. A practical optimal alternative selection method is also provided in this section. Finally, the conclusions are discussed in Section 8.
Section snippets
Preliminaries
The basic concepts of the IF set and IVIF set are introduced in Section 2.1. Section 2.2 presents a literature review of the traditional PCA model, which is beneficial for the proposal of the IVIF-PCA model.
New concepts
By reviewing the traditional PCA model, we discovered that when numerous correlated variables are obtained, we can construct linear combinations of these variables to form an equivalent number of new variables. These new variables are independent and contain all of the information about the original variables. The coefficients of the combinations (i.e., PC coefficients) are derived by maximizing the variances of these new variables. To solve the correlation problem between the IVIF variables,
The IVIF-PCA model
Based on the new concepts presented in Section 3, we derive the IVIF-PCA algorithm in Section 4.1, which is a new PCA algorithm for IVIF information. Several properties of IVIF-PCA are discussed in Section 4.2. To ensure that the final assessment information appears in the form of IVIF numbers, a transformation method is proposed in Section 4.3.
The CMALGDM method using the IVIF-PCA model
In this section, we provide a brief description of the CMALGDM problem. We then substitute the original variables (attributes or DMs) with their PCs in the IVIF-PCA method and use the operators to correctly aggregate the information. We consider the weights of the PCs according to their contributions. The decision-making steps are provided at the end of this section.
An illustrative example
An actual CMALGDM problem is provided as follows. A department that is responsible for a certain river basin in China plans to build a large hydropower station. To ensure scientific decision-making, the department invites 20 DMs dj(j = 1, 2, …, 20), including government departments, environmental experts, engineering experts, regional economy experts and public representatives, to evaluate 5 preliminary design alternatives xi(i = 1, 2, 3, 4, 5) during the preliminary stage of the project. In
Comparison
This section comprises three main parts. We first compare the IVIF-PCA method with the interval PCA based method, and we then give the decision results for the case in which we do not consider the independence of the attributes as well as that of the DMs. Finally, an optimal alternative determination method is given.
Conclusions
Due to the complexity of the CMALGDM problem, alternatives are usually described by multiple attributes that exhibit a high degree of interdependent or interactive characteristics. Because DMs may come from different interest groups, the independence between these DM preferences in the same interest group may be violated. Conventional information aggregation operators are proposed based on the independence axiom; as a result, these operators are not suitable for directly aggregating alternative
Acknowledgments
We would like to thank the editors Robert Dyson and Roman Slowinski, and three anonymous reviewers for their constructive comments that have helped to improve the presentation and quality of the paper. This work is partly supported by the National Natural Science Foundation of China (No. 71102072).
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