On The Consistency And Priority Methods Of Comparison Matrices

In decision making, decision makers' preferences on decision alternatives or criteria are often characterized by different kinds of pairwise comparison matrices. The study of consistency and priority methods for comparison matrices becomes a very important aspect in designing good decision making models. In this dissertation, to provide the theoretical base for the application of comparison matrices in decision making problems, we mainly discuss the consistency and priority methods of the crisp number comparison matrices, interval multiplicative comparison matrices, the linguistic comparison matrix and the intuitionistic fuzzy comparison matrix.We study the consistency and priorty method of the crisp number comparison matrix. Prom the viewpoint of graph theory, we firstly discuss the properties of a multiplicative comparison matrix with ordinal consistency and propose an algorithm to measure the ordinal consistency of the multiplicative comparison matrix; using the algorithm, we can find out all the illegitimate elements in the comparison matrix and provide a method for measuring the ordinal consistency of a multiplicative comparison matrix with non-strict pairwise comparison information. Then we complete the theorem that is about the relation between the elements and the priority weights of a fuzzy comparison matrix (also called a complementary comparison matrix or a fuzzy preference relation) having additive consistency. On the basis of the theorem, we analyze and improve some existing priority methods. For the acceptable incomplete fuzzy comparison matrix, we introduce the notion of strict additive consistency, and generalize the sufficient and necessary condition on the additive consistent fuzzy comparison matrix, which would provide the theoretical base for deriving rational priority methods of the incomplete fuzzy comparison matrix.We study the rationality of the priority weights and propose two new priority methods for the interval multiplicative comaprison matrices. Firstly, we discuss the properties of an interval multiplicative comaprison matrix with transitivity and then propose a sufficient and necessary condition that a consistent interval comparison matrix is of transitivity. On the base of these theory, we prove that any vector in the feasible region of the priority weights on the consistent interval judgment matrix, exhibits the same rank order in the sense of the ith alternative not being inferior to the jth alternative. Thus, it is a simple and reasonable priority method using the arithmetic mean or geometric mean of all the vertices of the feasible region to generate the priority weights of a consistent interval judgment matrix. Then, on the base of Geometric Consistecy Index, we give the notion of cardinal satisfactory consistency of the interval comparison matrix, and develop two mathmatical progrminming models to derive the weights for the inteval comparison matrix with cardinal satisfactory consistency. At last, some examples are provided to illustrate the validity and practicality of the proposed methods.We give two method for measuring the satisfying consistency of a linguistic comparison matrix with non-strict pairwise comparison information. We first analyze the rationality of the existing definitions on the satisfactory consistency of a linguistic judgement matrix, then give a notion of the satisfactory consistency index and present a method of computing the index. Using the method, we can measure the satisfactory consistency of the linguistic judgement matrix, find out all the teams of the judgment elements that are illegitimate in the linguistic judgement matrix. Then we give another measuring method by discussing the properties of the linguistic judgement matrix with satisfactory consistency and solve the problem that there is no judgement method for the satisfactory consistency of a linguistic judgement matrix with non-strict pairwise comparison information. Finally, we illustrate the validity and potential practicality of the two methods with three examples.We study the methods of ranking intuitionistic fuzzy numbers and the methods of measuring the similarity between intuitionistic fuzzy numbers (sets). We firstly define a possibility degree formula for the comparison between two intuitionistic fuzzy numbers. Then we prove that the ranking order of two intuitionistic fuzzy numbers which uses the possibility degree formula is the same as the one which uses the score function defined by Chen and Tan, and point out that the possibility degree formula can give more information for the comparison between two intuitionistic fuzzy numbers. On the basis of the possibility degree formula, we give a possibility degree method for ranking n intuitionistic fuzzy numbers and utilize the method to rank the alternatives in multi-attribute decision making. At last, we present a new similarity measure formula between intutionistic fuzzy number (sets), then prove the properties and analyze the advantages and the faults of the similarity measure formula. These methods offer rational tools for the application of intuitionistic fuzzy numbers (sets) in decision making, pattern recognition and medical diagnostic reasoning.