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

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Summary

Introduction

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.

A2 A3 popularity is that Choquet integral is a generalization of the weighted
Conclusions

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