Inventory problems and the parametric measure $m_{\lambda}$

TL;DR

This paper extends credibilistic inventory models by introducing the parametric measure $m_{}$, providing a generalized framework for fuzzy demand inventory problems and deriving explicit solutions for specific fuzzy demand types.

Contribution

It generalizes the credibilistic inventory model using the $m_{}$-measure, offering new formulas for optimal solutions with fuzzy demands and extending previous models.

Findings

Derived a general formula for the $m_{}$-expected value optimization.

Provided explicit formulas for trapezoidal and triangular fuzzy demands.

Validated formulas with numerical data applications.

Abstract

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 $m_{\lambda}$ ($\lambda$ is a real parameter in the interval $[0,1]$ and for $\lambda= 1/2$ we obtain as a particular case the notion of credibility measure). By using the $m_{\lambda}$-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 $m_{\lambda}$-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 $\lambda=1/2$ 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 $m_{\lambda}$-models obtained from numerical data.