Inventory models for non-instantaneous deteriorating items with expiration dates under the joined effect of preservation technology and linearly time-dependent holding cost when order-size linked to advance payment

Abstract

The aim of this study is to investigate a joint replenishment and preservation technology investment problem for non-instantaneous deteriorating items in which preservation technology investment affects not only the deterioration rate but also the length of the non-deterioration period. Regarding efforts towards delaying the deterioration process, the longer these items are kept in storage, the higher is the holding cost. Therefore, we consider the holding cost as a linearly increasing function of storage period. Along these lines, the retailer needs to balance the cost of investment in the preservation technology to reduce the deterioration rate and the increased total profit due to the decreased deterioration rate. This study discusses the scenario where the supplier offers the retailer the option to prepay a predefined percentage of the purchase cost if their order quantity exceeds or equals the minimum order quantity. Otherwise, the retailer takes a loan from a bank to prepay the entire purchasing cost. The main purpose of this study is to provide the best combined replenishment strategies and preservation technology investment under the abovementioned scenario by ensuring the maximum profit. A solution procedure is developed to determine the optimal solutions, and an algorithm is created by combining all theoretical results derived from the analytical study. Numerical examples are used to verify the theoretical results. Sensitivity analysis revealed some useful managerial insights. Results reveal that the retailer places orders under the conditions of a higher frequency with smaller order quantities to avoid excessive shortage costs when the holding cost is a linearly increasing function of the storage period.

Appendices

Appendix A

Proof of Lemma 1

• (A) For a fixed $\xi$, the first order derivative of $G_{11} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)$ with respect to $T$ is given by:

  $\frac{{\partial G_{11} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)}}{\partial T} = \delta_{12} (\xi )m(\xi )T^{2} \left( {1 - m(\xi )} \right)\left( {2 - m(\xi )} \right)\left( {1 + \lambda - T + t_{d} (\xi )} \right)^{m(\xi ) - 3} > 0$ if $T > 0$.

  So, $G_{11} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)$ is increasing on $T > 0$. Additionally,

  $G_{11} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > G_{11} \left( {\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t_{d} (\xi )} \right) = t_{d}^{2} (\xi )m(\xi )\left( {1 - m(\xi )} \right)(1 + \lambda )^{ - 1} > 0$ if $T > t_{d} (\xi )$.

  Furthermore, for any given $\xi$, $G_{11} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > 0$ if $T \ge t_{d} (\xi )$.

• (B) The proof of Lemma 1(B) is similar to the proof of Lemma 1(A).

• (C) The first order derivative of $G_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)$ for a fixed $\xi$ with respect to $T$ is given by:

  $$ \frac{{\partial G_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)}}{\partial T} = T^{2} \cdot F(\xi ,T) $$

  where

  $$ \begin{aligned} F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) &= \frac{{\delta_{12} (\xi )}}{2}t_{d}^{2} (\xi )m(\xi )\left[ {1 - m(\xi )} \right]\left[ {2 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 3} \\ &\quad + \frac{2}{2 - m(\xi )}\left[ {1 + m(\xi )} \right] + \left( {1 + \lambda } \right)^{2 - m(\xi )} t_{d} (\xi )m(\xi )\left[ {1 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 3} \\ &\quad + \frac{1}{3 - m(\xi )}\left( {1 + \lambda } \right)^{3 - m(\xi )} m(\xi )\left[ {1 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 3}\\ &\quad - \frac{6}{[2 - m(\xi )][3 - m(\xi )]} \\ \end{aligned} $$

Then, for a fixed $\xi$, taking the first order derivative of $F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)$ with respect to $T$ is given by:

$$ \begin{aligned} \frac{{\partial F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)}}{\partial T} &= \frac{{\delta_{12} (\xi )}}{2}t_{d}^{2} (\xi )m(\xi )\left[ {1 - m(\xi )} \right]\left[ {2 - m(\xi )} \right]\left[ {3 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 4} \\ &\quad + \left( {1 + \lambda } \right)^{2 - m(\xi )} t_{d} (\xi )m(\xi )\left[ {1 - m(\xi )} \right]\left[ {(3 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 4} \\ &\quad + \left( {1 + \lambda } \right)^{3 - m(\xi )} m(\xi )\left[ {1 - m(\xi )} \right]\left[ {1 + \lambda - T + t_{d} (\xi )} \right]^{m(\xi ) - 4} \\ & > 0 \\ \end{aligned} $$

So, we have $F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T)$ is increasing on $T > 0$. Additionally,

$$ \begin{gathered} F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > F(\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t_{d} (\xi )) = \frac{m(\xi )}{{2\left( {1 + \lambda } \right)^{2} }}\left[ {1 - m(\xi )} \right]\left[ {2 - m(\xi )} \right]t_{d}^{2} (\xi ) + m(\xi ) + \frac{m(\xi )}{{1 + \lambda }}\left[ {1 - m(\xi )} \right]t_{d} (\xi ) \\ > 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} if{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T \ge t_{d} (\xi ) \\ \end{gathered} $$

The above arguments imply that for any given $\xi$, $G^{\prime}_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > 0$ if $T \ge t_{d} (\xi )$, so we have $G_{31} (T)$ is increasing on $T \ge t_{d} (\xi )$. Moreover,

$G_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > G_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t_{d} (\xi )) = \frac{m(\xi )}{{2(1 + \lambda )}}\left( {1 - m(\xi )} \right)t_{d}^{4} (\xi ) + \frac{m(\xi )}{3}t_{d}^{3} (\xi ) > 0$ if $T > t_{d} (\xi )$.

Furthermore, for any given $\xi$, $G_{31} (\xi {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} T) > 0$ if $T \ge t_{d} (\xi )$.

Incorporating the above argument, we have completed the proof of Lemma 1.□

Appendix B

Proof of Lemma 2

• (A) For any given $\xi$, by the concavity of $Z_{1} (\xi ,T)$, if $\Omega_{1} \le 0$ and $\Omega_{0} \le 0$ which imply $Z^{\prime}_{1} (T|\xi ) \le 0$ on $T \in {[}T_{u} (\xi ),\lambda ]$, we have $Z_{1} (\xi ,T)$ is decreasing on ${[}T_{u} (\xi ),\lambda ]$.

• (B) For any given $\xi$, by the concavity of $Z_{1} (\xi ,T)$, if $\Omega_{1} { > }0$ and $\Omega_{0} \le 0$ which imply there exists a unique value of $T$ such that $\frac{{\partial Z_{1} (T|\xi )}}{\partial T}\left| {_{{T = T_{1}^{*} }} } \right. = 0$. Furthermore, we have ${\text{Z}}_{1} (\xi ,T)$ is increasing on ${[}T_{u} (\xi ),T_{1}^{*} ]$ and decreasing on ${[}T_{1}^{*} ,\lambda ]$.

• (C) For any given $\xi$, by the concavity of $Z_{1} (\xi ,T)$, if $\Omega_{1} { > }0$ and $\Omega_{0} > 0$ which imply $Z^{\prime}_{1} (T|\xi ) > 0$ on $T \in {[}T_{u} (\xi ),\lambda ]$, we have $Z_{1} (\xi ,T)$ is increasing on ${[}T_{u} (\xi ),\lambda ]$.

• (D) For any given $\xi$, by the concavity of $Z_{2} (\xi ,T)$, if $\Omega_{2} \le 0$ and $\Omega_{3} \le 0$ which imply $Z^{\prime}_{2} (T|\xi ) \le 0$ on $T \in {[}t_{d} (\xi ),T_{u} (\xi )]$, we have $Z_{2} (\xi ,T)$ is decreasing on ${[}t_{d} {(}\xi {), }T_{u} (\xi )]$.

• (E) For any given $\xi$, by the concavity of $Z_{2} (\xi ,T)$, if $\Omega_{2} \le 0$ and $\Omega_{3} { > }0$ which imply there exists a unique value of $T$ such that $\frac{{\partial Z_{2} (T|\xi )}}{\partial T}\left| {_{{T = T_{2}^{*} }} } \right. = 0$. Furthermore, $Z_{2} (\xi ,T)$ is increasing on $\left[ {t_{d} (\xi ),T_{2}^{*} } \right]$ and decreasing on $\left[ {T_{2}^{*} ,T_{u} (\xi )} \right]$.

• (F) For any given $\xi$, by the concavity of $Z_{2} (\xi ,T)$, if $\Omega_{2} { > }0$ and $\Omega_{3} { > }0$ which imply $Z^{\prime}_{2} (T|\xi ) > 0$ on $T \in \left[ {t_{d} (\xi ),T_{u} (\xi )} \right]$, we have $Z_{2} (\xi ,T)$ is increasing on ${[}t_{d} {(}\xi {), }T_{u} (\xi )]$.

Incorporating the above argument, we have completed the proof of Lemma 2.□

Appendix C

Proof of Lemma 3

• (A) For any given $T$, if $\Omega_{11} \le 0$ which implies $\frac{{\partial Z_{1} (\xi |T)}}{\partial \xi } \le 0$ on $\xi_{1} \in \left[ {\xi_{u} ,t_{d}^{ - 1} (T)} \right]$ because of $\frac{{\partial Z_{1}^{2} (\xi {|}T)}}{{\partial \xi^{2} }} \le 0$, we have $Z_{1} (\xi |T)$ is decreasing on $\left[ {\xi_{u} ,t_{d}^{ - 1} (T)} \right]$ and $\xi_{1}^{*} { = }\xi_{u}$.

• (B) For any given $T$, if $\Omega_{12} > 0$ which implies $\frac{{\partial Z_{1} (\xi |T)}}{\partial \xi } > 0$ on $\xi_{1} \in \left[ {\xi_{u} ,t_{d}^{ - 1} (T)} \right]$ because of $\frac{{\partial Z_{1}^{2} (\xi {|}T)}}{{\partial \xi^{2} }} \le 0$, we have $Z_{1} (\xi |T)$ is increasing on $\left[ {\xi_{u} ,t_{d}^{ - 1} (T)} \right]$ and $\xi_{1}^{*} {\text{ = t}}_{d}^{ - 1} (T)$.

• (G) For any given $T$, if $\Omega_{11} > 0$ and $\Omega_{12} \le 0$ which implies there exists a unique value of $\xi$ such that $\frac{{\partial Z_{1} (\xi |T)}}{\partial \xi }\left| {_{{\xi = \xi_{1}^{*} }} } \right. = 0$. Furthermore, $Z_{1} (\xi |T)$ is increasing on ${[}\xi_{u} ,\xi^{*} ]$ and decreasing on ${[}\xi^{*} ,t_{d}^{ - 1} (T)]$, then we have $\xi^{*} { = }\xi_{1}^{*}$.

Incorporating the above argument, we have completed the proof of Lemma 3.□