Some Properties Of Intuitionistic Judgment Matrix And Interval Complementary Judgment Matrix

In this process of comparing alternatives or attributes with each other, decision makers always give uncertain judgment matrices due to the uncertainty and the complexity of things as well as the vagueness of people’s understanding on things and their knowledge limitation. Uncertain judgment matrices usually include interval-valued ones, intuitionistic ones, etc. The researches about these two kinds of judgment matrices are focused on checking the consistencies of these judgment matrices and using these consistencies to rank alternatives, etc. On one hand, the consistency of intuitionistic judgment matrix and that of interval complementary matrix are considered, and the problem how to use these consistencies to determine the weights of alternatives is studied in this paper; On the other hand, because of individual differences decision makers usually provide preference information over alternatives or attributes in different types such as fuzzy complementary judgment matrix, interval complementary judgment matrix, and intuitionistic judgment matrix, so we study the method to transform interval complementary judgment matrix,intuitionistic judgment matrix into fuzzy complementary judgment matrix and the properties in transformation process. The major research contents in this paper are given in details as follows:(l)The ranking method for alternatives is studied based on the directed graph of intuitionistic judgment matrix (IGM). According to the peculiarity of intuitionistic fuzzy number, the conception of IGM directed graph is defined. The conditions which IGM need to meet are given when the directed graph of IGM has unique directed path. Based on directed graph of IGM, a ranking method is provided for IGM which has weak transitivity.(2)The new definitions of the additive consistency of IGM and the corresponding methods to determine the weights of alternatives are explored. The strong and weak relationship between the weak transitivity of IGM and the additive consistency of IGM does not exist,and it is illustrated by counter-examples. And the defects of the traditional definition of intuitionistic judgment matrix’s additive consistency are pointed out. Then a new definition about the additive consistency of IGM is given based on the character of intuitionistic fuzzy number. And this new definition of intuitionistic judgment matrix’s additive consistency is utilized to develop a middle translation method for obtaining the weights of alternatives. In addition, another new definition of additive consistency of IGM is given from the perspective of the relationship between IGM and its weights and the problem how to obtain the weights of IGM using this definition is discussed.(3)The quasi-consistency of interval complementary judgment matrix (ICJM) is discussed. The characters of the additive consistency and the multiplicative consistency of fuzzy complementary judgment matrix (FCJM) are analyzed and the feature of relative superiority degree is found in them. The conceptions called as the quasi-additive consistency and the quasi-multiplicative consistency of ICJM are given based on this feature. It is illustrated that they are in fact identical. So they are uniformly called as interval complementary judgment matrix’s quasi-consistency. Models are respectively established to get the interval-valued weights of ICJM with quasi-consistency and of general ICJM.(4)We study the properties in transformation process of interval complementary judgment matrix,intuitionistic judgment matrix are transformed into fuzzy complementary judgment matrix. Considering fuzzy complementary judgment matrix, interval complementary judgment matrix and intuitionistic judgment matrix, the methods how to make these preference relations unified and how to make them aggregated are given. First,The pecularity of fuzzy complementary judgment matrix is analysized. Then interval complementary judgment matrix and IGM are transformed into fuzzy complementary judgment matrix based on this peculiarity. So these three kinds of matrices are unified. Then, the rationalities in transformation process are discussed.