The multiple attribute decision-making (MADM) probl ems are of the most interesting problems for many decision-making experts. This problem aris es in various fields of the real life, and constitutes very important content in scientific research such as management science, decision-making theory, system theory, operational research and economics. Now, many effective methods to determine the att ributive weights have been studied for MADM. Those are the subjective weight determining methods such as the feature vector method ( Saaty T.L. 1977), the least square sum method (Chu A Tw, Kala ba R E, Spingarn K, 1979), Delphi and AHP method (Hwang C.L., Lin M, 1987), and the objective weight determining methods such as the entropy method (Hwang C.L., Yoon K, 1981), the principal component analysis (Yan Jian-huo, 1989) and DEA (Data Envelopment Analysis) (Ye Chen, Kevin W. Li, Haiyan Xu and Sifeng Liu, 2009). The final ranking method affects greatly on the dec ision-making process. Hwang and Yoon (1981) proposed a new approach, TOPSIS (Technique for Orde r Preference by Similarity to Ideal Solution) for solving MADM problem. Recently, TOPSIS methods with interval weights (Gao Feng-ji, et al, 2005) and multiple attribute interval number TOPSIS (Chu A Tw, Kalaba R E, Spingarn K, 1979) have been studied. Guo Kai-hong and Mu You-jing (2012) studied the relation between several possibility degree formulas and proposed a possibil ity degree matrices-based method that aimed to objectively determine the weights of criteria in MA DM with intervals. A hybrid approach integrating OWA (Ordered Weighted Averaging) aggreg ation into TOPSIS is proposed to tackle * This work was supported in part by Nanjing Univer sity of Aeronautics and Astronautics, China. 2

This paper proposes a grey interval relation TOPSIS for the decision making in which all of the attribute weights and attribute values are given by the interval grey numbers. The feature of our method different from other grey relation decision-making is that all of the subjective and objective weights are obtained by interval grey number and that decision-making is performed based on the relative approach degree of grey TOPSIS, the relative approach degree of grey incidence and the relative membership degree of grey incidence using 2-dimensional Euclidean distance. The weighted Borda method is used for combining the results of three methods. An example shows the applicability of the proposed approach.

The multiple attribute mixed type decision making is performed by four methods, that is, the relative approach degree of grey TOPSIS method, the relative approach degree of grey incidence, the relative membership degree of grey incidence and the grey relation relative approach degree method using the maximum entropy estimation, respectively. In these decision making methods, the grey incidence degree in four-dimensional Euclidean space is used. The final arrangement result is obtained by weighted Borda method. An example illustrates the applicability of the proposed approach.

A multiple attribute decision making (MADM), in which attributes are real number, interval real number, linguistic and uncertain linguistic value, has been already applied in practice such as the evaluation of enterprise effect, the selection of investment project, the selection of person, the research of military equipment scheme, the evaluation of strategy effect, the reliability assessment and the maintainability assessment, etc (Yongqi Xia, 2004, Dang Luo, Sifeng Liu, 2005, Yongqing Wei, Peide Liu, 2009). Extended TOPSIS Method with Interval-Valued Intuitionistic Fuzzy Numbers for Virtual Enterprise Partner Selection has been researched by Fei Ye(2010). Chuanming Ding (2007,a) defined a new similarity degree for various types of attribute and normalized the calculation of similarity degree of the attribute value of each type in unified metric space. Also, by this similarity degree, the comparison of each plan with ideal plan was performed and decision making method was given. Chuanming (2007,b), based on the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), transformed the attribute value of plan into four-dimensional attribute value, unified various types of attribute value, defined a fourdimensional approach degree, and by this approach degree, solved the multiple attribute mixed-type decision-making problem associated with real number, interval real number, linguistic and uncertain linguistic value. Yongqi Xia (2004) studied a method considering insufficiency degree of information and preference to danger on the basis of the grey-fuzzy comprehensive evaluation method of interval value preference. In the method, they represent the weight and the attribute value by two interval number pair by considering membership and grey degree at the same time. Sifeng Liu, Yaoguo Dang, Jiangling Wang, Zhengpeng Wu (2009), based on the definitions of entropy, proposed a method of getting weight that considers the character of grey cluster decision-making and 2-tuple linguistic assessment, and proposed the method of 2-tuple linguistic assessment based on grey cluster. Zhen Zhang, Chonghui Guo (2012) transformed uncertain linguistic evaluation information of each decision maker to trapezoidal fuzzy numbers, and then denoted, by solving two optimization models, the collective evaluation of the alternatives by trapezoidal fuzzy numbers.