# An approach based on reliability-based possibility degree of interval for solving general interval bilevel linear programming problem

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## Abstract

This paper presents a new method based on reliability-based possibility degree of interval to handle the general interval bilevel linear programming problem involving interval coefficients in both objective functions and constraints. Considering reliability of the uncertain constraints, the interval inequality constraints are first converted into their deterministic equivalent forms by virtue of the reliability-based possibility degree of interval, and then the original problem is transformed into a bilevel linear programming with interval coefficients in the upper and lower level objective functions only. Then, the notion of the optimal solution of the problem is given by means of a type of the interval order relation. Based on this concept, the transformed problem is reduced into a deterministic bilevel programming with the aid of linear combination method. Furthermore, the proposed method is extended to deal with the fuzzy bilevel linear programming problem through the nearest interval approximation. Finally, three numerical examples are given to illustrate the effectiveness of the proposed approach.

## Keywords

Bilevel programming Interval number Fuzzy number Reliability-based possibility degree of interval## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61602010) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2017JQ6046).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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