Abstract
Taking into account human subjective measures, fuzzy measures and fuzzy integrals have been intensively discussed. Fuzzy measures have the monotonicity property instead of additivity. Fuzzy integrals have been applied to the evaluation of complex and vague objects. Weber proposed ⊥-decomposable measures and integrals by them. The first subject in this paper is to define an integral by pseudo addition (+) and pseudo multiplication (×) that are distributive and commutative Bemiring operations. This integral takes the value between the minimum and the maximum value of the integrand. Thus a class of decision problems can be expressed by this integral in a unified form.
A decision in a “fuzzy environment” has been defined by Bellman and Zadeh as the intersection of fuzzy sets of goals and constraints. The decision as the intersection of fuzzy sets by applying the minimum operater implies that there is no compensation between membership values of alternatives. The simple additive weighting method is the most widely used method of multi-attribute decision problems, where weights are considered to be an additive measure.
This paper proposes to use fuzzy integrals for the aggregation operator that implies various degree of compensation. (+)-decomposable measures can be regarded as a measure expressing the grade of importance.
A numerical example is shown to illustrate how to identify pseudo additive weights in a multi-attribute decision problem by the fuzzy integrals.
