Optimal Pricing and Ordering Policies for Non Instantaneous Deteriorating Items with Price Dependent Demand and Maximum Lifetime

Abstract

Owing to damage, spoil or decay, deteriorating items deteriorate continuously within their maximum lifetime. And deterioration not only causes quantity loss but also worsens their quality. Hence, due to different quality, selling price of deteriorating items may be different to stimulate the market demand. The paper considers an inventory model for non-instantaneous deteriorating item with price dependent demand and maximum lifetime. The system aims to maximize the retailer’s profit by determining its optimal price at deteriorated stage and replenishment cycle. And two distinct cases as well as the corresponding theorems are formulated. Finally, several theoretical results and managerial insights are obtained via numerical examples.

Appendices

Appendix A. Proof of Theorem 1

For simplicity, Eq. (11) is used to define the following equation

$$G\left( T \right) = \frac{{\Delta_{1} }}{1 + m - T} - \frac{1}{2}D\left( {p_{0} } \right)\left( {hT - hm} \right) +\Delta_{2}.$$

(15)

Then we have

$$G\left( {t_{d} } \right) = \frac{{\Delta_{1} }}{{1 + m - t_{d} }} - \frac{1}{2}h(t_{d} - m)D\left( {p_{0} } \right) +\Delta_{2} ,$$

(16)

and

$$G\left( m \right) =\Delta_{1} +\Delta_{2} .$$

(17)

We can obtain the following equation by taking the first order derivation of Eq. (15)

$$\frac{dG\left( T \right)}{dT} = \frac{{\Delta_{1} }}{{\left( {1 + m - T} \right)^{2} }} - \frac{1}{2}hD\left( {p_{0} } \right) < 0.$$

(18)

That is, $G\left( T \right)$ is decreasing on $T \in [t_{d} ,m]$. Hence, we should discuss various situations based on the values of $G\left( {t_{d} } \right)$ and $G\left( m \right)$.

• 1. If $\Delta_{1} +\Delta_{2} \ge 0$ or $G\left( m \right) \ge 0$, we can find that $G\left( {t_{d} } \right) > 0$. Thus, $G\left( T \right) \ge 0$ for $T \in [t_{d} ,m]$. That is, $\Pi_{2} \left( {T,p_{0} } \right)$ is increasing on $T \in [t_{d} ,m]$ and it can obtain its maximum value as T approaches to m.

• 2. If $\Delta_{1} +\Delta_{2} < 0$ or ${\text{G}}\left( m \right) < 0$,

  • (a) If $G\left( {t_{d} } \right) \ge 0$, there exists a unique solution (say $T^{A}$) over $[t_{d} ,m]$ satisfying the equation of $G\left( T \right) = 0$ by applying Intermediate Value Theorem. Since $\frac{dG\left( T \right)}{dT}|_{{T = T^{A} }} \le 0$, $T^{A}$ is the maximum value point in the feasible region.

  • (b) If $G\left( {t_{d} } \right) < 0$, then $G\left( T \right) < 0$ for $T \in [t_{d} ,m]$. That is, $\Pi_{2} \left( {T,p_{0} } \right)$ is decreasing on $T \in [t_{d} ,m]$ and it can obtain its maximum value as T approaches to $t_{d}$.

This completes the proof of Theorem 1.

Appendix B Proof of Theorem 2

For simplicity, Eq. (12) is used to define the following equation

$$H(p_{0} ) = D(p_{0} )(T - t_{d} ) + D^{\prime}(p_{0} )[p_{0} (T - t_{d} ) -\Delta_{3} ].$$

(19)

Then we have

$$H(c) = D(c)(T - t_{d} ) - D^{\prime}\left( c \right)[c(T - t_{d} ) -\Delta_{3} ],$$

(20)

and

$$H(p_{1} ) = D(p_{1} )(T - t_{d} ) + D^{\prime}(p_{1} )[p_{1} (T - t_{d} ) -\Delta_{3} ].$$

(21)

By taking the first order derivation of Eq. (19), we can obtain that

$$\frac{{dH(p_{0} )}}{{dp_{0} }} = 2(T - t_{d} )D^{\prime}(p_{0} ) + D^{\prime\prime}(p_{0} )[p_{0} (T - t_{d} ) -\Delta_{3} ].$$

Hence, we assume that $\Delta_{4} = \frac{{\Delta_{3} }}{{T - t_{d} }}$ and two situations are discussed to find the optimal solution.

• 1. If $p_{0} \ge\Delta_{4}$, then we have $\frac{{dH\left( {p_{0} } \right)}}{{dp_{0} }} < 0$ for $p_{0} \in (\Delta_{4} ,p_{1} ]$ and $H\left( {\Delta_{4} } \right) > 0$.

  • (a) If $H\left( {p_{1} } \right) \ge 0$, then we can find $H\left( {p_{0} } \right) > 0$ for $p_{0} \in (\Delta_{4} ,p_{1} ]$. That is, $\Pi_{2} \left( {T,p_{0} } \right)$ is increasing on $p_{0} \epsilon (\Delta_{4} ,p_{1} ]$ and it can obtain its maximum value as p₀ approaches to $p_{1}$.

  • (b) If $H\left( {p_{1} } \right) < 0$, there exists a unique solution (say $p_{0}^{A}$) over $(\Delta_{4} ,p_{1} ]$ satisfying the equation of $H\left( {p_{0} } \right) = 0$ by applying Intermediate Value Theorem. Since $\frac{{dH\left( {p_{0} } \right)}}{dT}|_{{p_{0} = p_{0}^{A} }} = < 0,\,p_{0}^{A}$ is the maximum value point in the feasible region. Hence, $p_{0} = p_{0}^{A}$ is the maximum solution to $\Pi_{2} \left( {T,p_{0} } \right)$.

• 2. If $p_{0} \le\Delta_{4}$, then we have $H\left( {p_{0} } \right) > 0$ for $p_{0} \, \epsilon \, (c,\Delta_{4} ]$. Thus, $\Pi_{2} \left( {T,p} \right)$ is increasing on $(c,\Delta_{4} ]$ and it can obtain its maximum value as $p_{0}$ approaches to $\Delta_{4}$.

To conclude, the following theorem can be obtained.

• 1. If $H\left( {p_{1} } \right) \ge 0,\,\Pi_{2} \left( {T,p_{0} } \right)$ can obtain its maximum value as $p_{0}$ approaches to $p_{1}$.

• 2. If $H\left( {p_{1} } \right) < 0,\,\Pi_{2} \left( {T,p_{0} } \right)$ can obtain its maximum value as $p_{0} = p_{0}^{A} ,\,p_{0}^{A} \in \left({\Delta_{4},p_{1}} \right]$.

This completes the proof of Theorem 2.