# American Institute of Mathematical Sciences

November  2021, 17(6): 3131-3163. doi: 10.3934/jimo.2020111

## Two-period pricing and ordering decisions of perishable products with a learning period for demand disruption

 1 College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China 2 School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

*Corresponding author: Dong Lei

Received  November 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

In this paper, we develop a two-period inventory model of perishable products with considering the random demand disruption. Faced with the random demand disruption, the firm has two order opportunities: the initial order at the beginning of selling season (i.e., Period 1) is intended to learn the real information of the disrupted demand. When the information of disruption is realized, the firm places the second order, and also decides how many unsold units should be carried into the rest of selling season (i.e., Period 2). The firm may offer two products of different perceived quality in Period 2, and therefore it must trade-off between the quantity of carry-over units and the quantity of young units when the carry-over units cannibalize the sales of young units. Meanwhile, there is both price competition and substitutability between young and old units. We find that the quantity of young units ordered in Period 2 decreases with the quality of units ordered in Period 1, while the pricing of young units is independent of the quality level of old units. However, both the surplus inventory level and the pricing of old units monotonically increase with their quality. We also investigate the influence of two demand disruption scenarios on the optimal order quantity and the optimal pricing when considering different quality situations. We find that in the continuous random disruption scenario, the information value of disruption to the firm is only related to the disruption mean, while in the discrete random disruption scenario, it is related to both unit purchase cost of young units and the disruption levels.

Citation: Kebing Chen, Haijie Zhou, Dong Lei. Two-period pricing and ordering decisions of perishable products with a learning period for demand disruption. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3131-3163. doi: 10.3934/jimo.2020111
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##### References:
Graphical representation of the two-period inventory system
The optimal solution regions on the disrupted demand, where M is sufficiently large number
Optimal order quantities versus low disruption amount
Optimal prices versus low disruption amount
Order quantity, $\tilde{x}_{1H}$ versus disruption amounts
Order quantity, $\tilde{x}_{1L}$ versus disruption amounts
Optimal pricing, $\tilde{P}_{1H}$ versus disruption amounts
Optimal pricing, $\tilde{P}_{1L}$ versus disruption amounts
Optimal expected profit, $\textbf{E}[\tilde{\Pi}_{12}^{B*}]$ versus disruption amounts
Continuous disruption versus discrete disruption $\Delta R_H = 8$, $\Delta R_L = -8$ and $\beta = 0.5$
Continuous disruption versus discrete disruption $\Delta R_H = 8$, $\Delta R_L = -8$ and $\overline{\Delta R} = 0$
Notations used in the stochastic model
 Parameters $R$ Market potential in Period 1, $R>c$ $\Delta R$ Random disrupted amount in Period 1, i.e., continuous or discrete random variable $\overline{\Delta R}$ Mean of random disrupted amount $\Delta R$ $\overline R$ Determined market potential in Period 2, $\overline R>c$ $h$ Unit cost to carry old products $q$ Perceived quality of old units when the young units go on sale, $0\leq q \leq1$ $c$ Unit cost to purchase young products $c_u$ Unit penalty cost for a unit increased quantity, $c_u \geq 0$ $c_s$ Unit penalty cost for a unit decreased quantity, $c_s \geq 0$ $P_1$ Retail price of young products in Period 1 $P_2^n$ Retail price of young products in Period 2 $P_2^o$ Retail price of old products in Period 2 Decision Variables $x_i$ Order quantity of young products in Period $i$, $i=1, 2$ $E$ Reorder point, or ending-stock level at the end of Period 1, where $E \geq 0$ Functions $\Pi_2(x_2, E)$ Profit of Period 2 given the surplus inventory $E$ and the reorder quantity $x_2$ $\Pi_{12}(x_1)$ Expected profit of both periods given the initial order quantity $x_1$
 Parameters $R$ Market potential in Period 1, $R>c$ $\Delta R$ Random disrupted amount in Period 1, i.e., continuous or discrete random variable $\overline{\Delta R}$ Mean of random disrupted amount $\Delta R$ $\overline R$ Determined market potential in Period 2, $\overline R>c$ $h$ Unit cost to carry old products $q$ Perceived quality of old units when the young units go on sale, $0\leq q \leq1$ $c$ Unit cost to purchase young products $c_u$ Unit penalty cost for a unit increased quantity, $c_u \geq 0$ $c_s$ Unit penalty cost for a unit decreased quantity, $c_s \geq 0$ $P_1$ Retail price of young products in Period 1 $P_2^n$ Retail price of young products in Period 2 $P_2^o$ Retail price of old products in Period 2 Decision Variables $x_i$ Order quantity of young products in Period $i$, $i=1, 2$ $E$ Reorder point, or ending-stock level at the end of Period 1, where $E \geq 0$ Functions $\Pi_2(x_2, E)$ Profit of Period 2 given the surplus inventory $E$ and the reorder quantity $x_2$ $\Pi_{12}(x_1)$ Expected profit of both periods given the initial order quantity $x_1$
Optimal decisions and profits of Period 2 based on the perceived quality of old inventory
 Cases $x_2^*$ $E^*$ $P_2^{n*}$ $P_2^{o*}$ $\Pi_2(x_2^*, E^*)$ $q \leq \frac{h}{c}$ $\frac{\overline R-c}{2}$ $0$ $\frac{\overline R+c}{2}$ NA} $\frac{(\overline R-c)^2}{4}$ $\frac{h}{c}  Cases$ x_2^*  E^*  P_2^{n*}  P_2^{o*}  \Pi_2(x_2^*, E^*)  q \leq \frac{h}{c}  \frac{\overline R-c}{2}  0  \frac{\overline R+c}{2} $NA}$ \frac{(\overline R-c)^2}{4}  \frac{h}{c}
Optimal decisions and profits of Period 1 based on the perceived quality of old inventory
 $q$ $E^*$ $x_1^*$ $P_1^*$ $\Pi_{12}(x_1^*)$ $q \leq \frac{h}{c}$ 0 $\frac{R-c}{2}$ $\frac{R+c}{2}$ $\frac{(1+\rho)(R-c)^2}{4}$ $\frac{h}{c} $ q  E^*  x_1^*  P_1^*  \Pi_{12}(x_1^*)  q \leq \frac{h}{c} $0$ \frac{R-c}{2}  \frac{R+c}{2}  \frac{(1+\rho)(R-c)^2}{4}  \frac{h}{c}
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