Optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand

Abstract

An inventory model with stock-dependent demand and non-instantaneous deterioration is developed in this paper. It is assumed that the item starts deteriorating at a constant rate after a certain period of time from the instant of receiving the delivery by the retailer. The retailer can reduce the rate of deterioration by investing in preservation technology. Depending on the fact that the on-hand stock may be finished before or after deterioration starts, two different inventory scenarios have been considered and analyzed. Optimal length of an inventory cycle as well as investment in preservation technology have been obtained in both the scenarios. Certain conditions have also been derived to identify situations where the retailer should or should not invest in preservation technology. The proposed model is illustrated with a numerical example.

Appendices

Appendix 1

During $t_d\le t\le T$, we have

$$\begin{aligned} &\frac{dI(t)}{dt}= {} -\alpha {I(t)}^\beta -(1-m)\theta I(t)\\ &\hbox {or, } \frac{dI(t)}{dt}+(1-m)\theta I(t)= {} -\alpha {I(t)}^\beta \end{aligned}$$

Substituting $y={I(t)}^{1-\beta }$ and using the boundary condition $y(T)=0$, we get

$$\begin{aligned} y= & {} \frac{\alpha }{(1-m)\theta }\left[ e^{(1-m)(1-\beta )\theta (T-t)}-1\right] \\ \hbox {or}, I(t)= & {} \left[ \frac{\alpha }{(1-m)\theta }\left\{ e^{(1-m)(1-\beta )\theta (T-t)}-1\right\} \right] ^{\frac{1}{1-\beta }} . \end{aligned}$$

At $t=t_d$, we have $I(t_d)=\left[ \frac{\alpha }{(1-m)\theta }\left\{ e^{(1-m)(1-\beta )\theta (T-t_d)}-1\right\} \right] ^{\frac{1}{1-\beta }}$

During $0\le t<t_d$, we have,

$$\begin{aligned} \frac{dI(t)}{dt}= & {} {-}\alpha \{I(t)\}^\beta \\ \hbox {so that }\, \{I(t)\}^{1-\beta }= \,& {} \{I(t_d)\}^{1-\beta }+\alpha (t_d-t)(1-\beta )\\= & {} \frac{\alpha }{(1-m)\theta }\left\{ e^{(1-m)(1-\beta )\theta (T-t_d)}-1\right\} +\alpha (t_d-t)(1-\beta ) . \end{aligned}$$

Hence Eq. (2) is obtained.

Appendix 2

Substituting $e^{(1-m)(1-\beta )\theta (T-t_d)}=1+(1-m)(1-\beta )\theta (T-t_d)$ and $e^{(1-m)(1-\beta )\theta (T-t)}=1+(1-m)(1-\beta )\theta (T-t)$, and neglecting higher order terms, we get

$$\begin{aligned} \Pi _1(T,\xi )= & {} \frac{1}{T}\left[ (p-c)\left[ \alpha (1-\beta )\left\{ \frac{1}{(1-\beta )(1-m)\theta }(1-\beta )(1-m)\theta (T-t_d)+t_d\right\} \right] ^{\frac{1}{1-\beta }}\right. \\&-(p+c_d)(1-m)\theta \int _{t_d}^{T} {\left\{ \frac{\alpha }{(1-m)\theta }(1-\beta )(1-m)\theta (T-t)\right\} ^{\frac{1}{1-\beta }}}dt-k-T\xi \\&-h\int _{0}^{t_d}\left[ \frac{\alpha }{(1-m)\theta }(1-\beta )(1-m)\theta (T-t_d)+\alpha (1-\beta )(t_d-t)\right] ^{\frac{1}{1-\beta }} dt\\&\left. -h\int _{t_d}^{T}\left\{ \frac{\alpha }{(1-m)\theta }(1-\beta )(1-m)\theta (T-t)\right\} ^{\frac{1}{1-\beta }} dt\right] \\=\, & {} \frac{1}{T}\left[ (p-c)\{\alpha (1-\beta )T\}^{\frac{1}{1-\beta }}-(p+c_d)(1-m)\theta \int _{t_d}^{T}\{\alpha (1-\beta )(T-t)\}^{\frac{1}{1-\beta }}dt\right. \\&-k-T\xi -h\int _{0}^{T}\{\alpha (1-\beta )(T-t)\}^{\frac{1}{1-\beta }}dt\\=\, & {} \frac{1}{T}\left[ (p-c)\left\{ \alpha (1-\beta )\right\} ^{\frac{1}{1-\beta }}T^{\frac{1}{1-\beta }}-\frac{(p+c_d)(1-m)\theta \{\alpha (1-\beta )\}^{\frac{2-\beta }{1-\beta }}}{\alpha (2-\beta )}(T-t_d) ^{\frac{2-\beta }{1-\beta }}\right. \\&\left. -k-T\xi -h\frac{\{\alpha (1-\beta )\}^{\frac{2-\beta }{1-\beta }}}{\alpha (2-\beta )}T^{\frac{2-\beta }{1-\beta }}\right] . \end{aligned}$$