When some coefficients of the constraints are uncertain with only their possible ranges being given, a conventional linear programming (LP) problem can be generalised to the one with set-inclusive constraints. We consider the case where the possible ranges are given by fuzzy sets in this paper. The set-inclusive constraints with fuzzy coefficients have been treated by a necessity measure. However, the usual necessity measure cannot express well the decision-maker’s requirement about the trade-off between the robustness level and the satisfaction level of the constraints. We extend the necessity measure to incorporate the trade-off between the robustness level and the satisfaction level of the constraints. We apply the extended necessity measure to LP problems with fuzzy coefficients. After the problem is formulated and reduced to the conventional programming problem, we propose a solution algorithm. Numerical examples are given to illustrate the proposed approach.

当约束的某些系数不确定且仅给出它们的可能范围时，传统的线性规划 (LP) 问题可以推广到具有集合包含约束的问题。在本文中，我们考虑了可能范围由模糊集给出的情况。具有模糊系数的集合包含约束已经通过必要性度量来处理。然而，通常的必要性度量不能很好地表达决策者对约束的稳健性水平和满足性水平之间权衡的要求。我们扩展了必要性度量，以纳入约束的稳健性水平和满意度水平之间的权衡。我们将扩展必要性度量应用于具有模糊系数的 LP 问题。在问题被公式化并简化为常规编程问题后，我们提出了一种解决算法。给出了数值示例来说明所提出的方法。