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

Kebing Chen; Haijie Zhou; Dong Lei

doi:10.3934/jimo.2020111

Article Contents

2021, Volume 17, Issue 6: 3131-3163. Doi: 10.3934/jimo.2020111

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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
*Corresponding author: Dong Lei

Received: October 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: November 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 97M40, 90B50; Secondary: 91A10.

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Figure 1. Graphical representation of the two-period inventory system

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Figure 2. The optimal solution regions on the disrupted demand, where M is sufficiently large number

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Figure 3. Optimal order quantities versus low disruption amount

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Figure 4. Optimal prices versus low disruption amount

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Figure 5. Order quantity, $ \tilde{x}_{1H} $ versus disruption amounts

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Figure 6. Order quantity, $ \tilde{x}_{1L} $ versus disruption amounts

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Figure 7. Optimal pricing, $ \tilde{P}_{1H} $ versus disruption amounts

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Figure 8. Optimal pricing, $ \tilde{P}_{1L} $ versus disruption amounts

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Figure 9. Optimal expected profit, $ \textbf{E}[\tilde{\Pi}_{12}^{B*}] $ versus disruption amounts

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Figure 10. Continuous disruption versus discrete disruption $ \Delta R_H = 8 $, $ \Delta R_L = -8 $ and $ \beta = 0.5 $

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Figure 11. Continuous disruption versus discrete disruption $ \Delta R_H = 8 $, $ \Delta R_L = -8 $ and $ \overline{\Delta R} = 0 $

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Table 1. 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 $

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Table 2. 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}<q<\frac{\overline R-c+h}{\overline R} $	$ \frac{(1-q)\overline R-c+h}{2(1-q)} $	$ \frac{cq-h}{2q(1-q)} $	$ \frac{\overline R+c}{2} $	$ \frac{q\overline R+h}{2} $	$ \frac{{\overline R}^2-2\overline Rc}{4}+\frac{2qch-qc^2-h^2}{4q(q-1)} $
$ q \geq \frac{\overline R-c+h}{\overline R} $	$ 0 $	$ \frac{q\overline R-h}{2q} $	NA}	$ \frac{q\overline R+h}{2} $	$ \frac{(q\overline R-h)^2}{4q} $

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Table 3. 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<\frac{R-c+h}{R} $	$ \frac{cq-h}{2q(1-q)} $	$ \frac{R-c+E^*}{2} $	$ \frac{R+c-E^*}{2} $	$ \frac{(R-c-E^)^2}{4}-cE^+\rho\Pi_2^(E^) $
$ q \geq \frac{R-c+h}{R} $	$ \frac{qR-h}{2q} $	$ \frac{R-c+E^*}{2} $	$ \frac{R+c-E^*}{2} $	$ \frac{(R-c-E^)^2}{4}-cE^+\rho\Pi_2^(E^) $

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