Dynamic pricing and promotion expenditures in an EOQ model of perishable products

Abstract

This study considers dynamic decisions of a retailer who seeks to determine the selling price and promotion expenditures associated with a perishable product, as well as to set the order quantity and the inter-replenishment time (cycle length). We propose an EOQ model in which the retailer faces a general demand function that is separable into multiplicative components of selling price, products’ age and promotion expenditure. We find analytical expressions for the optimal price and promotion trajectories; and we show that the former increases and the latter decreases in the products’ age, and that both are independent of the cycle length. Moreover, we show that the selling price is independent of the promotion expenditure, but not vice versa. It is also proved that under these optimal trajectories, the profit rate is strictly pseudo-concave in the cycle length. A comparison between dynamic and stationary strategies is given.

Appendices

Appendix 1

Proof of Lemma 4

Since $\pi \left( {p(t),a(t),T} \right) =\Pi \left( {p(t),a(t),T} \right) /T$, then

$$\begin{aligned} \frac{\partial \pi (p(t),a(t),T)}{\partial T}=\frac{1}{T}\left( {\frac{\partial \Pi (p(t),a(t),T)}{\partial T}-\pi (p(t),a(t),T)} \right) . \end{aligned}$$

By applying Leibniz’s rule to the profit function

$$\begin{aligned} \Pi \left( {p(t),a(t),T} \right) =-K+\int \limits _0^T {\left[ {-a(t)+\left( {p(t)-C(t)} \right) \lambda \left( {p(t),a(t),t} \right) } \right] dt} , \end{aligned}$$

we obtain

$$\begin{aligned} \frac{\partial \Pi (p(t),a(t),T)}{\partial T}=-a(T)+\left( {p(T)-C(T)} \right) \lambda \left( {p(T),a(T),T} \right) , \end{aligned}$$

so the necessary condition for maximizing $\pi (p(t),a(t),T)$ is

$$\begin{aligned} \left( {p(T)-C(T)} \right) \lambda \left( {p(T),a(T),T} \right) =\pi (p(t),a(t),T)+a(T). \end{aligned}$$

Hence, if $\pi \left( {p^{*}(t),a^{*}(t),T^{*}} \right) >0$ then $C(T^{*})<p(T^{*})\le p_{\max }$. $\square$

Appendix 2

Proof of Lemma 5

Differentiating Z(t) results in

$$\begin{aligned} Z^{\prime }(t)=\lambda _0 \left\{ {\begin{array}{l} \lambda _2 (t)\left[ {\left( {p^{*{\prime }}(t)-C^{\prime }(t)} \right) \lambda _{ 1} \left( {p^{*}(t)} \right) +\left( {p^{*}(t)-C(t)} \right) \lambda _{ 1} ^{\prime }\left( {p^{*}(t)} \right) p^{*{\prime }}(t)} \right] \\ +\lambda _2 ^{\prime }(t)\left( {p^{*}(t)-C(t)} \right) \lambda _{ 1} \left( {p^{*}(t)} \right) \\ \end{array}} \right\} . \end{aligned}$$

Using algebraic manipulations and substituting Eq. (7), we obtain

$$\begin{aligned} Z^{\prime }(t)=\lambda _0 \left\{ {\begin{array}{l} \lambda _2 (t)\left[ {-\lambda _{ 1} \left( {p^{*}(t)} \right) C^{\prime }(t)+\left( {r\left( {p^{*}(t)} \right) -C(t)} \right) \lambda _{ 1} ^{\prime }\left( {p^{*}(t)} \right) p^{*{\prime }}(t)} \right] \\ +\lambda _2 ^{\prime }(t)\left( {p^{*}(t)-C(t)} \right) \lambda _{ 1} \left( {p^{*}(t)} \right) \\ \end{array}} \right\} . \end{aligned}$$

By Eq. (6), $r\left( {p^{*}(t)} \right) -C(t)=0$, so

$$\begin{aligned} Z^{\prime }(t)=\lambda _0 \lambda _{ 1} \left( {p^{*}(t)} \right) \left\{ {-\lambda _2 (t)C^{\prime }(t)+\lambda _2 ^{\prime }(t)\left( {p^{*}(t)-C(t)} \right) } \right\} . \end{aligned}$$

By using the result of Corollary 1(ii), we obtain $Z^{\prime }(t)<0$. $\square$