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The necessarily optimal solution is known as the most reasonable solution to linear programming problems with interval/fuzzy objective function coefficients. As it remains optimal against the certain fluctuations of objective function coefficients, the necessarily optimal solution can be seen as a robust optimal solution. In this paper, we demonstrate that the necessary optimality degree of a non-degenerate basic feasible solution can be obtained easily by utilizing the tolerance approach. The necessary optimality degree evaluates to what extent the solution remains optimal against the fluctuations of objective function coefficients. Several types of fuzzy subsets showing the possible range of the objective function coefficient vector are considered. For each type of fuzzy subset, an efficient calculation method of necessary optimality degree is proposed. Numerical examples are given to illustrate the proposed methods.

Robust optimality analysis of non-degenerate basic feasible solutions in linear programming problems with fuzzy objective coefficients