Abstract
The Choquet integral is a popular tool for dealing with multiple criteria decision making. In practice, if we have a fuzzy measure on the set of criteria, we can use Choquet integral to rank alternatives. In this paper, we consider the inverse problem of Choquet integral: Given a ranking of alternatives, does there exist a fuzzy measure by which we can get the ranking through Choquet integral.
Highlights
The Choquet integral [1] is a generalization of the Lebesgue integral, and like it, defined with respect to a measure
Alternative Ai : In this paper, we discuss the inverse problem of Choquet integral: Given a ranking of alternatives, does there exist a fuzzy measure by which we can get the ranking through Choquet integral? The inverse problem of Choquet integral, first of all, is theoretically important
In some multiple criteria decision making situations, it’s difficult to define a fuzzy measure on the set of criteria and rank alternatives with Choquet integral
Summary
The Choquet integral [1] is a generalization of the Lebesgue integral, and like it, defined with respect to a measure. Choquet measure is defined with respect to fuzzy measures. Fuzzy measures have been a very useful tool in multiple criteria decision making, since introduced in 1974 by Sugeno [2]. If we have a fuzzy measure on the set of criteria, we can use Choquet integral to rank alternatives. At the beginning of nineties, Sugeno integral [2] was the main tool to example: Let’s consider two criteria B1 and B2 , and three alternatives A1, A3 , and A3. Murofushi and Sugeno [6,7] proposed to use the Choquet integral. It quickly became popular and has been widely used in many decision making situations.
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