Retailer’s ordering policies for time-varying deteriorating items with partial backlogging and permissible delay in payments in a two-warehouse environment

Abstract

With the convergence of innovation, technology, and supply chain, the world has been shrinking, and the retail industry is one of the largest spread across the globe in the past few decades. Consumer expectations are on priority for the retailers. Most of the retail sector deals with the items whose usefulness declines with time and reaches the expiration date, resulting in a decrease in sales and eventually diminishing revenues for the retailers. In such cases, effective replenishment decisions and ordering policies may yield a significant increase in revenues. Further, with emerging retail trends, providing trade credit is considered a price reduction tool and an alternative to price discounts. Motivated by this, an inventory model developed and analyzed for items exhibiting time-varying deterioration with partially backlogged shortages and permissible delay in payment in the two-warehouse environment. The primary objective is to obtain the optimal ordering and backlogging policies for the retailer by minimizing the relevant cost. The optimal solution is obtained, solved analytically, and the inventory model validated with the help of numerical illustrations. The sensitivity analysis of the optimal solution with respect to key parameters and the managerial implications are also provided. The model is applicable to perishable items such as baked products, fruits, vegetables, groceries, meat, and seafood, where the deterioration is time-dependent and is perceived by its expiration date.

Appendices

Appendix 1

Proof of Theorem 1 Let $g_{1} (T) = A + HC_{rw} + HC_{ow} + BC + LC + PC - IE_{1} + IP_{1}$ and $h\left( T \right) = T > 0$.

Therefore,

$$ q_{1} (T) = \frac{{g_{1} (T)}}{h(T)} = TC_{1} (t_{r} ,\;T) $$

For the given value of $t_{r}$, the first order derivative of $g_{1} (T)$ is obtained as

$$ g_{1}^{^{\prime}} (T) = \left\{ {c_{b} \delta (T - t_{w} ) - c_{o} - c - pI_{e} M} \right\}De^{{ - \delta (T - t_{w} )}} + c_{o} D $$

(36)

A derivative of (36) is further obtained as,

$$ g_{1}^{^{\prime\prime}} (T) = \left\{ {c_{o} + c + pI_{e} M + c_{b} - c_{b} (T - t_{w} )} \right\}D\delta e^{{ - \delta (T - t_{w} )}} $$

This implies, if $J > 0$, then $g_{1}^{^{\prime\prime}} (T) > 0$ and hence $g_{1} (T)$ is non-negative, differentiable and strictly convex.

It has been calculated and verified that $g_{2}^{^{\prime}} (T)$ and $g_{3}^{^{\prime}} (T)$ is the same as $g_{1}^{^{\prime}} (T)$, where

$$ g_{2} (T) = A + HC_{rw} + HC_{ow} + BC + LC + PC - IE_{2} + IP_{2} $$

$$ g_{3} (T) = A + HC_{rw} + HC_{ow} + BC + LC + PC - IE_{3} $$

Thus, if $J > 0$ then the total cost $TC(t_{r} ,\;T)$ in Eq. (28) is a strictly pseudo-convex function in T, and there exists a unique optimal solution.

Appendix 2

It is observed that $\frac{{g_{i} (T)}}{h(T)} = TC_{i} (t_{r} ,\;T),\;\;i = 1,2,3$. Hence given $t_{r}$, taking the first order derivative of $TC_{1} (t_{r} ,\;T)$ with respect to T, and setting the result to zero, the necessary and sufficient condition to find $T^{*}$, is obtained as follows:

$$ \begin{aligned} & \frac{{dTC_{1} (t_{r} ,\;T)}}{dT} = \frac{{g_{1}^{^{\prime}} (T)}}{T} - \frac{{g_{1} (T)}}{{T^{2} }} = 0 \\ & \quad \Rightarrow g_{1}^{^{\prime}} (T)T - g_{1} (T) = 0 \\ \end{aligned} $$

Thus from Eqs. (29) and (36), if $J > 0$ then the necessary and sufficient condition for $T^{*}$ is

$$ \begin{aligned} & \left[ {\left\{ {c_{b} \delta \left( {T - t_{w} } \right) - c_{o} - c - pI_{e} M} \right\}De^{{ - \delta \left( {T - t_{w} } \right)}} + c_{o} D} \right]T \\ & \quad = A + c\left\{ {W + D\int\limits_{0}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} + \frac{D}{\delta }\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} + FD\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } \int\limits_{t}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} dt \\ & \quad \quad + H\left\{ {W\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{0} \left( t \right)}} dt} + D\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} \int\limits_{t}^{{t_{w} }} {e^{{Z_{0} \left( u \right)}} du} dt} } \right\} + \frac{{c_{b} D}}{{\delta^{2} }}\left\{ {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} \left\{ {1 + \delta \left( {T - t_{w} } \right)} \right\}} \right\}\quad \quad 0 < M \le t_{r} \\ & \quad \quad + \frac{{c_{o} D}}{\delta }\left\{ {\delta \left( {T - t_{w} } \right) - \left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} - \left\{ {\frac{1}{2}pI_{e} DM^{2} + pI_{e} \frac{D}{\delta }M\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} \\ & \quad \quad + cI_{p} \left\{ {\int\limits_{M}^{{t_{r} }} {De^{{ - Z_{r} \left( t \right)}} \int\limits_{t}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} dt} + \int\limits_{M}^{{t_{r} }} {We^{{ - Z_{0} \left( t \right)}} dt} + \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} D\int\limits_{t}^{{t_{w} }} {e^{{Z_{0} \left( u \right)}} du} dt} } \right\} \\ \end{aligned} $$

Similarly, the necessary and sufficient conditions for $TC_{2} \left( {t_{r} ,T} \right)$, $TC_{3} \left( {t_{r} ,T} \right)$ with respect to T are obtained as follows:

$$ \begin{aligned} & \left[ {\left\{ {c_{b} \delta \left( {T - t_{w} } \right) - c_{o} - c - pI_{e} M} \right\}De^{{ - \delta \left( {T - t_{w} } \right)}} + c_{o} D} \right]T \\ & \quad = A + c\left\{ {W + D\int\limits_{0}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} + \frac{D}{\delta }\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} + FD\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } \int\limits_{t}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} dt \\ & \quad \quad + H\left\{ {W\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{0} \left( t \right)}} dt} + D\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} \int\limits_{t}^{{t_{w} }} {e^{{Z_{0} \left( u \right)}} du} dt} } \right\} \\ & \quad \quad + \frac{{c_{b} D}}{{\delta^{2} }}\left\{ {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} \left\{ {1 + \delta \left( {T - t_{w} } \right)} \right\}} \right\} + \frac{{c_{o} D}}{\delta }\left\{ {\delta \left( {T - t_{w} } \right) - \left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} \\ & \quad \quad - \left\{ {\frac{1}{2}pI_{e} DM^{2} + pI_{e} \frac{D}{\delta }M\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} + cI_{p} D\int\limits_{M}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} \int\limits_{t}^{{t_{w} }} {e^{{Z_{0} \left( u \right)}} du} dt} \\ & \quad \quad t_{r} < M \le t_{w} \\ \end{aligned} $$

$$ \begin{aligned} & \left[ {\left\{ {c_{b} \delta \left( {T - t_{w} } \right) - c_{o} - c - pI_{e} M} \right\}De^{{ - \delta \left( {T - t_{w} } \right)}} + c_{o} D} \right]T \\ & \quad = A + c\left\{ {W + D\int\limits_{0}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} + \frac{D}{\delta }\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} + FD\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } \int\limits_{t}^{{t_{r} }} {e^{{Z_{r} \left( u \right)}} du} dt \\ & \quad \quad + H\left\{ {W\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{0} \left( t \right)}} dt} + D\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} \int\limits_{t}^{{t_{w} }} {e^{{Z_{0} \left( u \right)}} du} dt} } \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \begin{array}{*{20}c} {t_{w} < M \le T} \\ {{\text{or }}T < M} \\ \end{array} \\ & \quad \quad + \frac{{c_{b} D}}{{\delta^{2} }}\left\{ {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} \left\{ {1 + \delta \left( {T - t_{w} } \right)} \right\}} \right\} + \frac{{c_{o} D}}{\delta }\left\{ {\delta \left( {T - t_{w} } \right) - \left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} \\ & \quad \quad - \left\{ {\frac{1}{2}pI_{e} Dt_{w}^{2} + pI_{e} D\left( {M - t_{w} } \right)t_{w} + pI_{e} \frac{D}{\delta }M\left( {1 - e^{{ - \delta \left( {T - t_{w} } \right)}} } \right)} \right\} \\ \end{aligned} $$

Appendix 3

Proof of Theorem 2 For any given T, the first and second order derivatives of $TC_{1} \left( {t_{r} ,T} \right)$ with respect to $t_{r}$ are obtained as:

$$ \frac{{dTC_{1} \left( {t_{r} ,T} \right)}}{dt} = \frac{1}{T}\left[ \begin{gathered} FDe^{{Z_{r} \left( {t_{r} } \right)}} \int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + cDe^{{Z_{r} \left( {t_{r} } \right)}} + cI_{p} De^{{Z_{r} \left( {t_{r} } \right)}} \int\limits_{M}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt \hfill \\ + \left\{ \begin{gathered} \left( {H + cI_{p} } \right)De^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} - c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} \left( {T - t_{w} } \right) \hfill \\ + c_{o} D\left( {e^{{ - \delta \left( {T - t_{w} } \right)}} - 1} \right) - cDe^{{ - \delta \left( {T - t_{w} } \right)}} + pI_{e} DMe^{{ - \delta \left( {T - t_{w} } \right)}} \hfill \\ \end{gathered} \right\}\frac{{dt_{w} }}{{dt_{r} }} \hfill \\ \end{gathered} \right];\quad 0 < M \le t_{r} $$

$$ \frac{{d^{2} TC_{1} \left( {t_{r} ,T} \right)}}{{dt^{2} }} = \frac{1}{T}\left[ \begin{gathered} FD\left\{ {e^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right)\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + 1} \right\} + cDe^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right) + cI_{p} De^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right)\int\limits_{M}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right)De^{{ - \delta \left( {T - t_{w} } \right)}} - c_{o} D} \right\}\frac{{d^{2} t_{w} }}{{dt_{r}^{2} }} \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \theta_{o} \left( {t_{w} } \right)\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right)D\delta e^{{ - \delta \left( {T - t_{w} } \right)}} + c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} } \right\}\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} \hfill \\ cI_{p} D\left\{ {\left( {e^{{Z_{0} \left( {t_{w} } \right)}} \theta_{o} \left( {t_{w} } \right)\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + 1} \right)\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} - 1 + e^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} \frac{{d^{2} t_{w} }}{{dt_{r}^{2} }}} \right\} \hfill \\ + pI_{e} DMe^{{ - \delta \left( {T - t_{w} } \right)}} \left\{ {\delta \left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} + \frac{{d^{2} t_{w} }}{{dt_{r}^{2} }}} \right\}^{2} \hfill \\ \end{gathered} \right];\quad 0 < M \le t_{r} $$

$$ \frac{{dTC_{2} \left( {t_{r} ,T} \right)}}{dt} = \frac{1}{T}\left[ \begin{gathered} FDe^{{Z_{r} \left( {t_{r} } \right)}} \int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + cDe^{{Z_{r} \left( {t_{r} } \right)}} + \hfill \\ + \left\{ \begin{gathered} HDe^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} - c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} \left( {T - t_{w} } \right) \hfill \\ + c_{o} D\left( {e^{{ - \delta \left( {T - t_{w} } \right)}} - 1} \right) - cDe^{{ - \delta \left( {T - t_{w} } \right)}} + pI_{e} DMe^{{ - \delta \left( {T - t_{w} } \right)}} \hfill \\ + cI_{p} De^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{M}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} \hfill \\ \end{gathered} \right\}\frac{{dt_{w} }}{{dt_{r} }} \hfill \\ \end{gathered} \right];\quad t_{r} < M \le t_{w} $$

$$ \frac{{d^{2} TC_{2} \left( {t_{r} ,T} \right)}}{{dt^{2} }} = \frac{1}{T}\left[ \begin{gathered} FD\left\{ {e^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right)\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + 1} \right\} + cDe^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right) \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right)De^{{ - \delta \left( {T - t_{w} } \right)}} - c_{o} D} \right\}\frac{{d^{2} t_{w} }}{{dt_{r}^{2} }} \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \theta_{o} \left( {t_{w} } \right)\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + D\delta e^{{ - \delta \left( {T - t_{w} } \right)}} \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right) + c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} } \right\}\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} \hfill \\ cI_{p} D\left\{ {\left( {e^{{Z_{0} \left( {t_{w} } \right)}} \theta_{o} \left( {t_{w} } \right)\int\limits_{M}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + 1} \right)\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} + e^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{M}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} \frac{{d^{2} t_{w} }}{{dt_{r}^{2} }}} \right\} \hfill \\ + pI_{e} DMe^{{ - \delta \left( {T - t_{w} } \right)}} \left\{ {\delta \left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} + \frac{{d^{2} t_{w} }}{{dt_{r}^{2} }}} \right\}^{2} \hfill \\ \end{gathered} \right]; \, \quad \, t_{r} < M \le t_{w} $$

$$ \frac{{dTC_{3} \left( {t_{r} ,T} \right)}}{dt} = \frac{1}{T}\left[ \begin{gathered} FDe^{{Z_{r} \left( {t_{r} } \right)}} \int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + cDe^{{Z_{r} \left( {t_{r} } \right)}} + \hfill \\ + \left\{ \begin{gathered} HDe^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} - c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} \left( {T - t_{w} } \right) \hfill \\ + c_{o} D\left( {e^{{ - \delta \left( {T - t_{w} } \right)}} - 1} \right) - cDe^{{ - \delta \left( {T - t_{w} } \right)}} \hfill \\ + pI_{e} D\left( {M - t_{w} - Me^{{ - \delta \left( {T - t_{w} } \right)}} } \right) \hfill \\ \end{gathered} \right\}\frac{{dt_{w} }}{{dt_{r} }} \hfill \\ \end{gathered} \right]; \, \quad \, t_{w} < M \le T\;{\text{ or}}\; \, T < M $$

$$ \frac{{d^{2} TC_{3} \left( {t_{r} ,T} \right)}}{{dt^{2} }} = \frac{1}{T}\left[ \begin{gathered} FD\left\{ {e^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right)\int\limits_{0}^{{t_{r} }} {e^{{ - Z_{r} \left( t \right)}} } dt + 1} \right\} + cDe^{{Z_{r} \left( {t_{r} } \right)}} \theta_{r} \left( {t_{r} } \right) \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right)De^{{ - \delta \left( {T - t_{w} } \right)}} - c_{o} D} \right\}\frac{{d^{2} t_{w} }}{{dt_{r}^{2} }} \hfill \\ + \left\{ {HDe^{{Z_{0} \left( {t_{w} } \right)}} \theta_{o} \left( {t_{w} } \right)\int\limits_{{t_{r} }}^{{t_{w} }} {e^{{ - Z_{0} \left( t \right)}} dt} + D\delta e^{{ - \delta \left( {T - t_{w} } \right)}} \left( {c_{o} - c_{b} \left( {T - t_{w} } \right) - c} \right) + c_{b} De^{{ - \delta \left( {T - t_{w} } \right)}} } \right\}\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} \hfill \\ + pI_{e} D\left\{ {\left( {M\delta e^{{ - \delta \left( {T - t_{w} } \right)}} + 1} \right)\left( {\frac{{dt_{w} }}{{dt_{r} }}} \right)^{2} + \left( {M\left( {e^{{ - \delta \left( {T - t_{w} } \right)}} - 1} \right) + t_{w} } \right)\frac{{d^{2} t_{w} }}{{dt_{r}^{2} }}} \right\} \hfill \\ \end{gathered} \right];\quad t_{w} < M \le T \, \;{\text{or }}\;T < M $$

Let $(c_{o} - c_{b} (T - t_{w} ) - c) = L$, then for any given T, if $L > 0$, then $TC_{1} (t_{r} ,\;T)$, $TC_{2} (t_{r} ,\;T)$ and $TC_{3} (t_{r} ,\;T)$ given in Eqs. (29), (30) and (31) respectively are a strictly convex function of $t_{r}$. Consequently, there exists a unique optimal solution for a total cost $TC(t_{r} ,\;T)$ in Eq. (28).