库存问题与参数测度 $m_λ$
Inventory problems and the parametric measure $m_λ$
AI总结：
本文利用参数测度 $m_λ$ 构建风险中性多产品模糊库存模型，给出最优解的一般公式及梯形、三角形模糊需求下的计算式。
摘要：
可信性理论由 B. Liu 提出，是描述模糊不确定性的一种新方法。可信性测度是可信性理论的基本概念。最近，L. Yang 和 K. Iwamura 通过定义参数测度 $m_λ$ 扩展了可信性测度（其中 $λ$ 是区间 $[0,1]$ 内的实参数，并且当 $λ=1/2$ 时，作为特例得到可信性测度这一概念）。本文利用 $m_λ$ 测度研究了一个风险中性的多产品库存问题。我们的构造推广了 Y. Li 和 Y. Liu 于 2019 年建立的可信性库存模型。在我们的模型中，需求向量的各分量是模糊变量，并且最大化问题通过使用 $m_λ$ 期望值的概念来表述。我们将证明优化问题解的一个一般公式，并由此得到在需求为梯形模糊数和三角形模糊数的特殊情形下计算最优解的有效公式。当 $λ=1/2$ 时，作为特例可得到 Li 和 Liu 的可信性库存问题最优解的计算公式。这些计算公式被应用于由数值数据得到的若干 $m_λ$ 模型。
英文摘要：
The credibility theory was introduced by B. Liu as a new way to describe the fuzzy uncertainty. The credibility measure is the fundamental notion of the credibility theory. Recently, L.Yang and K. Iwamura extended the credibility measure by defining the parametric measure $m_λ$ ($λ$ is a real parameter in the interval $[0,1]$ and for $λ= 1/2$ we obtain as a particular case the notion of credibility measure). By using the $m_λ$-measure, we studied in this paper a risk neutral multi-item inventory problem. Our construction generalizes the credibilistic inventory model developed by Y. Li and Y. Liu in 2019. In our model, the components of demand vector are fuzzy variables and the maximization problem is formulated by using the notion of $m_λ$-expected value. We shall prove a general formula for the solution of optimization problem, from which we obtained effective formulas for computing the optimal solutions in the particular cases where the demands are trapezoidal and triangular fuzzy numbers. For $λ=1/2$ we obtain as a particular case the computation formulas of the optimal solutions of the credibilistic inventory problem of Li and Liu. These computation formulas are applied for some $m_λ$-models obtained from numerical data.
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