Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information
Research highlights
► The 2-tuple linguistic multiple attribute group decision making problems with incomplete weight information are investigated. An optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. According to the traditional ideas of grey relational analysis (GRA), the optimal alternative(s) is determined by calculating the linguistic degree of grey relation of every alternative and 2-tuple linguistic positive ideal solution and 2-tuple linguistic negative ideal solution.
Introduction
The increasing complexity of the socio-economic environment makes it less and less possible for s a single expert or decision maker to consider all relevant aspects of a problem (Kim, Choi, & Kim, 1999). Therefore, complex decision problems usually have to be conducted by integrating a group of experts’ knowledge and experiences. Generally, the practice of multiple attribute group decision making (MAGDM) is to invite internal experts or external experts or their combination of related fields to evaluate each attribute of every alternative individually. Recently, many multiple attribute group decision making problems were studied in linguistic environment (Ben-Arieh & Chen, 2006; Herrera et al., 1996a, Herrera et al., 1996b, Herrera et al., 1997; Herrera and Herrera-Viedma, 2000a, Herrera and Herrera-Viedma, 2000b, Herrera-Viedma et al., 2005, Huynh and Nakamori, 2005, Jiang and Fan, 2003, Li and Fan, 2003, Liao et al., 2006, Tai and Chen, 2009, Tang and Zheng, 2006, Wang, 2009, Wang and Fan, 2003, Wei et al., 2006, Zhang and Chu, 2009). Herrera et al. (1996a) developed a consensus model for group decision making with linguistic assessments information. Herrera et al. (1996b) presented several group decision making processes which are designed using the linguistic ordered weighted averaging (LOWA) operator. Herrera and Herrera-Viedma (2000a) presented a new consensus model in a linguistic framework for heterogeneous group decision making problems. Herrera and Herrera-Viedma (2000b) defined four classical choice sets of alternatives for a linguistic preference relation and pointed out some relations between them and presented some particular linguistic choice functions together with a study of their rationality properties, and presented two types of linguistic choice mechanisms to derive solution sets of alternatives from linguistic choice functions. Herrera, Herrera-Viedma, and Verdegay (1997) established three steps to follow in the linguistic decision making analysis of a group decision making problems with linguistic information. Herrera-Viedma et al. (2005) presented a consensus support system model to assist the experts in all phases of the consensus reaching process of group decision making problems with multi-granular linguistic preference relations. Huynh and Nakamori (2005) proposed a satisfactory-oriented approach in which a random preference is defined for each alternative in the aggregation phase instead of using an aggregation operator to obtain a collective preference value. Ben-Arieh and Chen (2006) also proposed a method to increase the consensus level by updating the importance of the group members. Tang and Zheng (2006) developed a new linguistic modeling technique based on the semantic similarity relation among linguistic labels.
Grey system theory (Deng, 1988, Deng, 1989, Deng, 2002) is one of the methods used to study uncertainty, being superior in the mathematical analysis of systems with uncertain information. In grey system theory, according to the degree of information, if the system information is fully known, the system is called a white system; if the information is unknown, it is called a black system. A system with information known partially is called a grey system. The grey system theory includes five major parts: grey prediction, grey relational analysis (GRA), grey decision, grey programming and grey control. GRA is part of grey system theory, which is suitable for solving problems with complicated interrelationships between multiple factors and variables. So, GRA method has been widely used to solve the uncertainty problems under the discrete data and incomplete information (Deng, 1988, Deng, 1989, Deng, 2002, Liu et al., 1999, Liu and Lin, 1998, Olson and Wu, 2006, Wei, 2006, Wu, 2009, Zhang et al., 2005). In addition, GRA method is one of the very popular methods to analyze various relationships among the discrete data sets and make decisions in multiple attribute situations. The major advantages of the GRA method are that the results are based on the original data, the calculations are simple and straightforward, and, finally, it is one of the best methods to make decisions under business environment.
In this paper, a new method for multiple attribute group decision making problems with 2-tuple linguistic information based on the traditional ideas of grey relational analysis is developed. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts and operational laws of 2-tuple linguistic variables. In Section 3 we establish an optimization model based on the maximizing deviation method, by which the attribute weights can be determined and then develop a practical method based on the traditional ideas of grey relational analysis for group decision making problem with 2-tuple linguistic information, which is straightforward and has no loss of information. In Section 4, we give an illustrative example to verify the developed approach and to demonstrate its feasibility and practicality. In Section 5 we conclude the paper and give some remarks.
Section snippets
Preliminaries
Let S = {si∣i = 0, 1, … , t} be a linguistic term set with odd cardinality. Any label, si represents a possible value for a linguistic variable, and it should satisfy the following characteristics (Herrera and Martínez, 2000a, Herrera and Martínez, 2000b, Herrera et al., 2005):
(1) The set is ordered: si > sj, if i > j; (2) Max operator: max(si, sj) = si, if si ⩾ sj; (3) Min operator: min(si, sj) = si, if si ⩽ sj. For example, S can be defined as:
GRA method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information
For the group decision making problems, in which both the attribute weights and the expert weights take the form of real numbers, and the attribute preference values take the form of 2-tuple linguistic variables, we shall develop an approach based on the traditional ideas of grey relational analysis to group decision making based on 2-tuple linguistic information processing.
Let A = {A1, A2, … , Am} be a discrete set of alternatives, G = {G1, G2, …, Gn} be the set of attributes, D = {D1, D2, … , Dt} be the set
Numerical example
Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from Fan et al., 2009, Herrera et al., 2000). There is a panel with five possible alternatives to invest the money: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is a arms company; (5) A5 is a TV company. The investment company must take a decision according to the following four attributes: (1) G1 is the risk analysis; (2) G2 is the growth
Conclusion
With respect to 2-tuple linguistic multiple attribute group decision making problems with incomplete weight information, some basic concepts and operational laws of 2-tuple linguistic variables are introduced. An optimization model based on the maximizing deviation method, by which the attribute weights can be determined, is established. According to the traditional ideas of grey relational analysis (GRA), the optimal alternative(s) is determined by calculating the linguistic degree of grey
Acknowledgment
The work was supported by the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 09XJA630010).
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