Robustness in deterministic multi-objective linear programming with respect to the relative interior and angle deviation

Abstract

This paper deals with the robustness issue in deterministic multi-objective linear programming from two new standpoints. It is shown that a robustness notion recently reported in the literature is equivalent to strict efficiency. Corresponding to an efficient solution, a new quantity, robustness order (RO) is defined with respect to the interiority order of the cost matrix in the binding cone. A linear programming problem is provided to calculate the RO of a given efficient solution. The second part of the paper is devoted to investigating the robustness with respect to the eligible angle deviation of the cost matrix in the binding cone. Theoretical results are given to obtain the maximum eligible angle deviation. Finally, the relationship between two above-mentioned robustness standpoints is established. To have a better geometrical view, we prove the results for single-objective LP problems at first, and then we extend them to the multi-objective case. In addition to the theoretical results, some clarifying examples are given.

Keywords:

• Multi-objective linear programming
• robustness
• angle deviation
• relative interior

Acknowledgements

The authors would like to express their gratitude to anonymous referees and the handling editor for helpful comments on the first version of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was in part supported by a grant from IPM [number 94260124].