# An approach for solving a fuzzy bilevel programming problem through nearest interval approximation approach and KKT optimality conditions

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## Abstract

In this paper, we consider a kind of bilevel linear programming problem where the coefficients of both objective functions are fuzzy numbers. In order to deal with such a problem, the original problem can be approximated by an interval bilevel programming problem in terms of the nearest interval approximation of a fuzzy number. Based on the Karush–Kuhn–Tucker (KKT) optimality conditions for the optimization problem with an interval-valued objective function, the interval bilevel programming problem can be converted into a single-level programming problem with an interval-value objective function. To minimize the interval objective function, the order relations of interval numbers are used to transform the uncertain single-objective optimization into a multi-objective optimization solved by global criteria method (GCM). Finally, illustrative numerical examples are provided to demonstrate the feasibility of the proposed approach.

## Keywords

Bilevel programming problem Fuzzy number Interval programming Nearest interval approximation KKT conditions## Notes

### Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61272119), Science Foundation of the Education Department of Shaanxi Province of China (No. 15JK1036) and Science Foundation of Baoji University of Arts and Sciences (No. ZK16049).

## Compliance with ethical standards

## Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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