An Extensional Signed Lambda-fuzzy Measure Based on Signed Monotonicity

In this paper, first of all, for generalizing the notion of monotonicity, a most important key word, called signed monotonicity, is proposed, it is a generalization of not only the non-negative monotonicity but also the revised monotonicity, furthermore, the signed measure satisfying signed monotonicity, called a generalized signed fuzzy measure, is also proposed. It is proved that this new measure is a generalization of not only the non-negative fuzzy measure but also the signed fuzzy measure, and the well-known signed additive measure is not only a signed fuzzy measure but also a generalized signed fuzzy measure, moreover, the author points out that the signed measure satisfying Lambda-rule , called generalized Lambda-measure is not a signed fuzzy measure and generalized signed fuzzy measure, since it does not satisfy the revised monotonicity and signed monotonicity. At last, a new signed measure satisfying signed monotonicity based on Sugeno´s Lambda-measure, called extensional sign Lambda-fuzzy measure is also proposed, it is a generalized signed fuzzy measure and more useful than above mentioned signed measures based on Lambda-measure for using to analysis multi-criteria decision making problem, since the multi-criteria decision making problem needs to consider the signed monotoniccity. Some related properties are also discussed.