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Inventory problems and the parametric measure $m_λ$

Irina Georgescu

TL;DR

This paper develops a risk-neutral multi-item inventory model in which demands are fuzzy variables and the objective is the $m_{\lambda}$-expected value of total profit. By exploiting the linearity of $E_{\lambda}$, the authors derive a closed-form solution $x_i^*=\frac{d_i}{h_i\,E_{\lambda}(1/D_i)}$, and provide computable expressions for $E_{\lambda}(1/D_i)$ when demands are trapezoidal or triangular fuzzy numbers. They present explicit formulas for the optimal quantities under trapezoidal demands, along with their triangular special cases and notable choices of $\lambda$ (e.g., $\lambda=\tfrac{1}{2}$ recovers credibilistic results). A data-driven numerical example illustrates the method, showing how the percentile-based fuzzification of data via Vercher yields tractable inventory decisions, and demonstrates the monotone influence of $\lambda$ on the solution. The work opens avenues for further extension to other fuzzy demand types and risk-averse or variance-oriented $m_{\lambda}$-theories in inventory management.

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 $m_λ$ ($λ$ 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.

Inventory problems and the parametric measure $m_λ$

TL;DR

This paper develops a risk-neutral multi-item inventory model in which demands are fuzzy variables and the objective is the -expected value of total profit. By exploiting the linearity of , the authors derive a closed-form solution , and provide computable expressions for when demands are trapezoidal or triangular fuzzy numbers. They present explicit formulas for the optimal quantities under trapezoidal demands, along with their triangular special cases and notable choices of (e.g., recovers credibilistic results). A data-driven numerical example illustrates the method, showing how the percentile-based fuzzification of data via Vercher yields tractable inventory decisions, and demonstrates the monotone influence of on the solution. The work opens avenues for further extension to other fuzzy demand types and risk-averse or variance-oriented -theories in inventory management.

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 ( is a real parameter in the interval and for we obtain as a particular case the notion of credibility measure). By using the -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 -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 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 -models obtained from numerical data.
Paper Structure (8 sections, 12 theorems, 1 equation, 1 figure)

This paper contains 8 sections, 12 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The expected value associated with the measure $m_{\lambda}$
  4. An inventory problem with fuzzy variables as demands
  5. Solution form when the demands are trapezoidal fuzzy variables
  6. A numerical example
  7. Conclusions
  8. Appendix

Key Result

Proposition 3.1

dzuche1 Let $\xi_1,\xi_2$ be two fuzzy variables such that $E_{\lambda}(\xi_1)< \infty$, $E_{\lambda}(\xi_2)< \infty$ and $\alpha,\beta$ are two non - negative real numbers. Then the following hold: (3.3) $E_{\lambda}(\xi_1 + \xi_2)$ = $E_{\lambda}(\xi_1) + E_{\lambda}(\xi_2)$; (3.4) $E_{\lambda}(\a

Figures (1)

  • Figure 1: Trapezoidal fuzzy number

Theorems & Definitions (19)

  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Corollary 3.5
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Proposition 4.4
  • Proposition 5.1
  • ...and 9 more