A technical note on: An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment—revisited

Abstract

This technical note deals with the modifications of the revised optimal policies for a problem to meet price and stock dependent demand of a controllable deteriorating item by investing on preservation technology. Originally, Mishra et al. (Ann Oper. Res 254(1–2):165–190, 2017) proposed two models and the optimal solution policies to them considering both the complete and the partial backordering of shortages of an item. Priyamvada et al. (OPSEARCH 58(1): 181–202, 2021) found some anomalies in their optimal solution policies and revised them in order to make them viable. However, we have found the revised version incomplete in providing the accurate optimal solution to the problem, and hence our endeavor here is to modify it further. The potential significance of this revision is highlighted by comparative studies on the results of the studied numerical example problems.

Appendices

Appendix 1

Derivation of the optimal p, ${p}^{*}$ from the profit function in case of complete backordering.

Using $D(p)=a-bp$ in (2), equate the first partial derivative of this resulting $TP\left(n,\alpha ,p\right)$ with respect to $p$ to zero and obtain

$$\begin{gathered} - b\left[ {\frac{{p\beta \gamma ^{2} T^{2} + 2npT - c\left( {\lambda \left( \alpha \right) + \beta } \right)\gamma ^{2} T^{2} - 2ncT - h\gamma ^{2} T^{2} - sT^{2} \left( {1 - \gamma } \right)^{2} }}{{2n}}} \right] \hfill \\ \, \; + \left( {a - bp} \right)\left[ {\frac{{2nT + \beta \gamma ^{2} T^{2} }}{{2n}}} \right] = 0 \hfill \\ \end{gathered}$$

$$\Rightarrow pb( \beta {\gamma }^{2}{T}^{2}+2nT+2nT+ \beta {\gamma }^{2}{T}^{2})=bc\left(\lambda \left(\alpha \right)+\beta \right){\gamma }^{2}{T}^{2}+2bcnT+bh{\gamma }^{2}{T}^{2}+bs{T}^{2}{\left(1-\gamma \right)}^{2}+2anT+a\beta {\gamma }^{2}{T}^{2}$$

$$\Rightarrow {p}^{*}=\frac{b\{c\left(\lambda \left(\alpha \right)+\beta \right){\gamma }^{2}{T}^{2}+2cnT+h{\gamma }^{2}{T}^{2}+s{T}^{2}{\left(1-\gamma \right)}^{2}\}+2anT+a\beta {\gamma }^{2}{T}^{2}}{2b(\beta {\gamma }^{2}{T}^{2}+2nT)}$$

Appendix 2

Derivation of the optimal p,${p}^{*}$ from the profit function in case of partial backordering.

Using $D(p)=a-bp$ in (6), equate the first partial derivative of this resulting $TP\left(n,\alpha ,p\right)$ with respect to $p$ to zero and obtain

$$\begin{gathered} - b\left[ {2np\gamma T + \beta p\gamma ^{2} T^{2} + pT^{2} \left( {1 - \gamma } \right)^{2} \left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\} - c\gamma T\left\{ {2n + \gamma T\left( {\lambda \left( \alpha \right) + \beta } \right)} \right\}} \right. \hfill \\ \left. { - cT\left( {1 - \gamma } \right)\left\{ {2n - \eta T\left( {1 - \gamma } \right)} \right\} - h\gamma ^{2} T^{2} - sT^{2} \left( {1 - \gamma } \right)^{2} \left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\} - c_{1} \eta \left( {1 - \gamma } \right)^{2} T^{2} } \right] \hfill \\ + (a - bp)\left[ {2n\gamma T + \beta \gamma ^{2} T^{2} + T^{2} \left( {1 - \gamma } \right)^{2} \left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\}} \right] = 0 \hfill \\ \end{gathered}$$

$$\begin{aligned} \Rightarrow p\left[ {4nb\gamma T + 2b\beta \gamma ^{2} T^{2} + 2bT^{2} \left( {1 - \gamma } \right)^{2} \left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\}} \right] = & bc\gamma T\left\{ {2n + \gamma T\left( {\lambda \left( \alpha \right) + \beta } \right)} \right\} \\ & + bcT\left( {1 - \gamma } \right)\left\{ {2n - \eta T\left( {1 - \gamma } \right)} \right\} + bh\gamma ^{2} T^{2} \\ & + bsT^{2} \left( {1 - \gamma } \right)^{2} \left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\} + bc_{1} \eta \left( {1 - \gamma } \right)^{2} T^{2} + 2an\gamma T+\alpha\beta\gamma^{2}T^{2}+\alpha T^{2}(1 - \gamma)^2\left\{ {1 - \frac{{\eta T}}{n}\left( {1 - \gamma } \right)} \right\} \\ \end{aligned}$$

$$\Rightarrow {p}^{*}=\frac{{bc\gamma }^{2}{T}^{2}\left(\lambda \left(\alpha \right)+\beta \right)+2nT(bc+a\gamma ){+\gamma }^{2}{T}^{2}(bh+a\beta ){+T}^{2}{\left(1-\gamma \right)}^{2}\left[\left\{1-\frac{\eta T}{n}\left(1-\gamma \right)\right\}\left(a+bs\right)+b\eta ({c}_{1}-c)\right]}{2b\gamma T(2n +\beta \gamma T)+2b{T}^{2}{\left(1-\gamma \right)}^{2}\left\{1-\frac{\eta T}{n}\left(1-\gamma \right)\right\}}$$