Knowledge Measure-Based q-Rung Orthopair Fuzzy Inventory Model

Abstract

As a generalization of intuitionistic fuzzy set and Pythagorean fuzzy set, the q-Rung Orthopair Fuzzy Set (qROFS) and its application are implemented in some decision-making problems. So far the notion of qROFS is not yet applied in any inventory management problems. This chapter analyzed an inventory model under the q-rung orthopair fuzzy environment and utilized the knowledge measure of qROFS to the proposed model. This study explores an economic order quantity model with faulty products and screening errors. To satisfy the customers’ demand with perfect items, the proposed research utilized the product warranty claim strategy from the supplier. Since some of the faulty products are returned from the supplier without being replaced as good ones, this study examines under two cases say, the warranty unclaimed products are restored by mending option and the warranty unclaimed products are recovered by the emergency purchase option. For both cases, the q-Rung Orthopair Fuzzy (qROF) inventory model is framed by presuming the proportion of faulty products and the proportion of misclassification errors as q-Rung Orthopair Fuzzy Variables (qROFVs). The Knowledge Measure-based q-Rung Orthopair Fuzzy (KM-qROF) inventory model is proposed by computing the knowledge measure of qROFVs. The proposed model is illustrated with a numerical example along with the sensitivity analysis.

Appendix

Computation of storage charge.

The holding charge is obtained by adding the five parts (a), (b), (c), (d), and (e) in Fig. 6.6.

Fig. 6.6

Holding Charge

Using the substitutions, ${\mathsf{I}}_{1} = {\mathsf{I}} - d\mathsf{t}_{1}$, ${\mathsf{I}}_{2} = {\mathsf{I}}_{1} - {\mathscr{Q}}_{1}$, ${\mathsf{I}}_{3} = {\mathsf{I}} - d\left( {\mathsf{t}_{1} + \mathsf{t}_{2} } \right) - {\mathscr{Q}}_{1}$,

${\mathscr{Q}}^{\prime } = {\mathsf{I}}_{3} + \left( {{\mathscr{Q}}_{1} + {\mathscr{Q}}_{2} - \lambda \sigma {\mathsf{I}}} \right)$; where $\mathsf{t}_{1} = \frac{{\mathsf{I}}}{\beta }$, $\mathsf{t}_{2} = \frac{{{\mathscr{Q}}_{1} + {\mathscr{Q}}_{2} }}{{\mathscr{W}}} + \mathsf{t}_{{\text{w}}}$, $\mathsf{t}_{3} = \frac{{{\mathscr{Q}}^{\prime } }}{d}$, $\mathsf{t}_{4} = \frac{{\lambda \sigma {\mathsf{I}}}}{d}$ ${\mathscr{Q}}_{1} = \varepsilon_{2} {\mathsf{I}}$ and ${\mathscr{Q}}_{2} = \varepsilon_{3} {\mathsf{I}}$

• $\left({\text{a}}\right)=\frac{\mathsf{h}}{2}{\mathsf{I}}^{2}\left({\frac{2}{\beta}-\frac{\mathsf{d}}{{\beta^{2}}}}\right)$

• $\left( {\text{b}} \right) = \frac{\mathsf{h}}{2}{\mathsf{I}}^{2} \left( {\frac{{\left( {1 - \varepsilon_{2} - \frac{\mathsf{d}}{\beta }} \right)^{2} }}{\mathsf{d}}} \right)$

• $\begin{aligned} \left( {\text{c}} \right) & = \frac{{\mathsf{h}_{W} }}{2}\left( {\frac{{{\mathscr{Q}}^{\prime } \,^{2} - {\mathsf{I}}_{3}^{2} }}{\mathsf{d}}} \right) \\ & = \mathsf{h}_{W} {\mathsf{I}}^{2} \left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\left[ {\frac{1}{\mathsf{d}} - \frac{1}{\beta } - \frac{{\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)}}{{\mathscr{W}}} - \frac{{\left( {\varepsilon_{2} - \varepsilon_{3} + \lambda \sigma } \right)}}{{2\mathsf{d}}}} \right] \\ & \quad \quad \quad \quad - \mathsf{h}_{W} {\mathsf{I}}\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\mathsf{t}_{{\text{w}}} { } \\ \end{aligned}$

• $\left({\text{d}}\right)=\frac{{\mathsf{h}{\mathscr{Q}}_{2}{\mathscr{T}}^{\prime}}}{2}=\frac{{\mathsf{h}{\mathsf{I}}^{2}\varepsilon_{3}}}{{2\mathsf{d}}}$

• $\left( {\text{e}} \right) = \frac{{\mathsf{h}_{R} \lambda \sigma {\mathsf{I}}\mathsf{t}_{4} }}{2} = \frac{{\mathsf{h}_{R} \left( {\lambda \sigma } \right)^{2} {\mathsf{I}}^{2} }}{{2\mathsf{d}}}\;\;[{\text{For}}\;{\text{case}}({\text{i}})]$

• $\left( {\text{e}} \right) = \frac{{\mathsf{h}_{E} \lambda \sigma {\mathsf{I}}\mathsf{t}_{4} }}{2} = \frac{{\mathsf{h}_{E} \left( {\lambda \sigma } \right)^{2} {\mathsf{I}}^{2} }}{{2\mathsf{d}}}\;\;[{\text{For}}\;{\text{case}}\;({\text{ii}})]$

On adding (a), (b), (c), (d), and (e), the holding charge for case (i) and case (ii) are given as follows:

$$\begin{aligned} HC & = \frac{\mathsf{h}}{2}{\mathsf{I}}^{2} \left( {\frac{2}{\beta } - \frac{\mathsf{d}}{{\beta^{2} }} + \frac{{\left( {1 - \varepsilon_{2} - \frac{\mathsf{d}}{\beta }} \right)^{2} }}{\mathsf{d}}} \right) + \frac{{\mathsf{h}{\mathsf{I}}^{2} \varepsilon_{3} }}{{2\mathsf{d}}} \\ & \quad + \mathsf{h}_{W} {\mathsf{I}}^{2} \left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\left[ {\frac{1}{\mathsf{d}} - \frac{1}{\beta } - \frac{{\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)}}{{\mathscr{W}}} - \frac{{\left( {\varepsilon_{2} - \varepsilon_{3} + \lambda \sigma } \right)}}{{2\mathsf{d}}}} \right] \\ & \quad - \mathsf{h}_{W} {\mathsf{I}}\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\mathsf{t}_{{\text{w}}} + \frac{{\mathsf{h}_{R} \left( {\lambda \sigma } \right)^{2} {\mathsf{I}}^{2} }}{{2\mathsf{d}}}. \\ \end{aligned}$$

and

$$\begin{aligned} HC & = \frac{\mathsf{h}}{2}{\mathsf{I}}^{2} \left( {\frac{2}{\beta } - \frac{\mathsf{d}}{{\beta^{2} }} + \frac{{\left( {1 - \varepsilon_{2} - \frac{\mathsf{d}}{\beta }} \right)^{2} }}{\mathsf{d}}} \right) + \frac{{\mathsf{h}{\mathsf{I}}^{2} \varepsilon_{3} }}{{2\mathsf{d}}} \\ & \quad + \mathsf{h}_{W} {\mathsf{I}}^{2} \left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\left[ {\frac{1}{\mathsf{d}} - \frac{1}{\beta } - \frac{{\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)}}{{\mathscr{W}}} - \frac{{\left( {\varepsilon_{2} - \varepsilon_{3} + \lambda \sigma } \right)}}{{2\mathsf{d}}}} \right] \\ & \quad - \mathsf{h}_{W} {\mathsf{I}}\left( {\varepsilon_{2} + \varepsilon_{3} - \lambda \sigma } \right)\mathsf{t}_{{\text{w}}} + \frac{{\mathsf{h}_{E} \left( {\lambda \sigma } \right)^{2} {\mathsf{I}}^{2} }}{{2\mathsf{d}}}. \\ \end{aligned}$$