| Decision making is closely related to all aspects of our lives and happens every day.With the rapid development of society and the economy,decision-making problems are becoming more and more complex.The analytic hierarchy process(AHP)provides a feasible theoretical framework for simulating and solving complex decision-making problems.When using the analytic hierarchy process,decision-makers usually need to pairwisely compare alternatives and give preference relations.There are two typical ones: multiplicative reciprocal matrices and additive reciprocal matrices.It is important to investigate the transitivity indexes of matrices and prioritization methods.This thesis mainly investigates the consistency index and obtaining the priority vector from multiplicative reciprocal matrices.The transitivity indexes of multiplicative and additive reciprocal matrices are obtained and applied to the voting theory.The main results and novelties are given as follows:(1)The consistency index of multiplicative reciprocal matrices and the method of obtaining the weights are studied.Based on the cosine similarity of tworow/column vectors in multiplicative reciprocal judgment matrices,a new consistency index is proposed.Its properties are studied,and the threshold of acceptable consistency is discussed in detail.Moreover,a new method is proposed to obtain the priority vector from multiplicative reciprocal matrices — double cosine similarity maximization method.It is found that the existing cosine similarity maximization method is its particular case.Some numerical examples are carried out to show that the proposed method is effective and flexible.(2)The indexes of the weak consistency and weak transitivity of multiplicative reciprocal matrices are determined.Under the assumption of rational economics,the opinions of decision-makers should be transitive.The interesting properties of transitivity are studied using the internal logic relations of the elements of a multiplicative reciprocal matrix.Different from the existing ideas of constructing transitivity indexes,the new indexes are proposed to measure the weak consistency and weak transitivity of multiplicative reciprocal matrices respectively by considering the matrix theory and the rank theory,An optimization model is established to adjust a multiplicative reciprocal matrix without transitivity to a new one with transitivity.(3)Three kinds of transitivity indexes of additive reciprocal judgment matrices.Based on the internal logic relations among the elements in an additive reciprocal matrix,the equivalence conditions of weak consistency,strict max-min transitivity,and weak transitivity are proposed,respectively,and their quantification indexes are obtained.Furthermore,the possibility indexes of quantifying the Condorcet paradox and reverse voting paradox are defined.When the preference relation is not transitive,an iterative algorithm for improving transitivity is proposed.The above results show that based on the logical relationship of the elements in preference relations,the consistency and transitivity indexes are constructed.The prioritization method and the transitivity improving algorithm are proposed.The possibility of the occurrence of the two kinds of paradoxes is quantified,The observations enrich and develop the decision theory and method under complex environment. |
