Existence and construction of weight-set for satisfying preference orders of alternatives based on additive multi-attribute value model

Based on the additive multi-attribute value model for multiple attribute decision making (MADM) problems, this paper investigates how the set of attribute weights (or weight-set thereafter) is determined according to the preference orders of alternatives given by decision makers. The weight-set is a bounded convex polyhedron and can be written as a convex combination of the extreme points. We give the sufficient and necessary conditions for the weight-set to be not empty and present the structures of the weight-set for satisfying the preference orders of alternatives. A method is also proposed to determine the weight-set. The structure of the weight-set is used to determine the interval of weights for every attribute in the decision analysis and to judge whether there exists a positive weight in the weight-set. The research results are applied to several MADM problems such as the geometric additive multi-attribute value model and the MADM problem with cone structure.