The concept of A-equitable efficiency in solving the multiobjective optimisation problems have recently been introduced by Mut and Wiecek [Generalised equitable preference in multiobjective programming. Euro J Oper Res. 2011;212:535–551], where A is an arbitrary matrix with non-negative entries. The preference relation of this concept solution, ${⪯}_{e\left(A\right)}$, does not satisfy the strict monotonicity and strict principle of transfers axioms in general, therefore the set of A-equitably efficient solutions is not contained within the set of equitably efficient solutions and the set of Pareto-optimal solutions for the same problem. In this paper, we extend the work done by Mut and Wiecek and state the new conditions on the matrix A that guarantee to satisfy these axioms by the preference relation ${⪯}_{e\left(A\right)}$. The other contribution of this paper is the use of the preference relation ${⪯}_{e\left(\mathrm{\Delta }A\right)}$ to solve the multio objective optimisation problems instead of ${⪯}_{e\left(A\right)}$. This has the advantage that the decision-maker has more freedom to choose the preferences matrix A. The relation ${⪯}_{e\left(\mathrm{\Delta }A\right)}$ has become an equitable rational preference relation by imposing the weaker conditions on entries of A, in comparison to the relation ${⪯}_{e\left(A\right)}$.

概念一-equitable效率在解决多目标优化问题最近被MUT和Wiecek [多目标规划的广义公平优先引进。欧洲 J 歌剧院 Res。2011;212:535–551]，其中A是具有非负项的任意矩阵。这个概念解的偏好关系，${⪯}_{\mathrm{电子}\left(\mathrm{一种}\right)}$，一般不满足传递公理的严格单调性和严格原则，因此对于同一问题，A -公平有效解集不包含在公平有效解集和帕累托最优解集内。在本文中，我们扩展了 Mut 和 Wiecek 所做的工作，并说明了矩阵A上的新条件，这些条件保证通过偏好关系满足这些公理${⪯}_{\mathrm{电子}\left(\mathrm{一种}\right)}$. 本文的另一个贡献是使用了偏好关系${⪯}_{\mathrm{电子}\left(\mathrm{\Delta }\mathrm{一种}\right)}$ 解决多目标优化问题而不是 ${⪯}_{\mathrm{电子}\left(\mathrm{一种}\right)}$. 这样做的好处是决策者可以更自由地选择偏好矩阵A。关系${⪯}_{\mathrm{电子}\left(\mathrm{\Delta }\mathrm{一种}\right)}$通过对A 的条目施加较弱的条件，与关系相比，已经成为公平理性的偏好关系${⪯}_{\mathrm{电子}\left(\mathrm{一种}\right)}$.