Variable Weight Theory And The Multi-objective Decision-making

Professor WANG Pei-zhuang proposed most early variable weights thought, and gives one empirical formula of variable weights in order to solve the kind of problem that constant weight synthesis merely considers each kind of indicator in the decision-making relation importance, but has neglected to the condition balanced degree by personal preference, which "remain unchanged" (constant weight) should be changing (indicator value) practice has a certain one-sidedness. Professor LI Hong-xing gives axiomatic definition of variable weights, state variable weights vector and balance function through systematic research factor space theory and thorough analysis variable weights synthesis. Later Professor LIU Wen-qi proposed compromise of the variable weights pattern according to these two big decision theories relation between the utility analysis and variable weights synthesis, concluded balance function is extended form of synthesis utility and between the single factor utility type and balance function relation, obtained through the compromised utility function compromised variable weights, thus expended method of variable weights. Professor LI De-qing proposed a series of theories, for example hierarchy variable weight, the equivalent relations, state variable weights vector s determination when variable weights synthesis and so on.This article has completed the following several aspect work in the above achievement s foundation. First this article introduces some basic concepts of factor space theory. The main conclusion of the article is that discussed the algebraic properties of balance function, points out under certain premise, balance function collection can constitute semi-group by addition and multiple operation, then carries on the equal kind of division according to the equivalent relation, and verifies that between the new balance function which picks out different equal kinds by addition, multiply, power operation and original balance function are not the equivalent, and has given the concrete algorithm which the computer can do, at the same time, introduces the balance function with parameter in order to solve that variable weight s model can be effective choose, according to balance request of decision-making. Because the gradient vector of balance function is state variable weights vector, this article discusses the algebraic properties of state variable weights vector s with the similar method, obtains the state variable weights vector s collection can also constitute the semi-group by Hadamard multiple operator, then carries on the equal kind of division according to the equivalent relation, and verifies the between the new state variable weights vectors which picks out different equal kinds by Hadamard multiple operation and original state variable weight vectors are not equivalent, and has given the concrete algorithm which the computer can do and give two kind of new state variable weight vectors, based on geometry mean value and the information entropy. Variable weights synthesis and multi-objective decision-makings unify that may provide the more reasonable solution for many actual problems, therefore this article has given arrangement method of the feasible plan collection, then has confirmed this method scientific nature and the validity with the example. Finally, the article proposes the research tentative plan of variable weights theory.