Impact of reorder option in supply chain coordination

Na Song; Ximin Huang; Yue Xie; Wai-Ki Ching; Tak-Kuen Siu

doi:10.3934/jimo.2016026

Article Contents

2017, Volume 13, Issue 1: 449-475. Doi: 10.3934/jimo.2016026

This issue Previous Article Dynamic pricing of network goods in duopoly markets with boundedly rational consumers Next Article Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings

Impact of reorder option in supply chain coordination

1.
School of Management and Economics, University of Electronic Science and Technology, Chengdu, China
2.
Scheller College of Business, Georgia Institute of Technology, Atlanta, USA
3.
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
4.
Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, Australia

Received: May 31, 2015

Published: August 2017

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Abstract

This paper studies the impacts of some reorder options on the performance as well as the coordination issues in a supply chain. A large category of products requires a long procurement lead time yet only has a relatively short selling season. Hence the purchase decisions usually have to be made well in advance of the opening of the sales. However, when uncertainty exists, the actual market demand may turn out to severely deviate from the initial order amount. To make up for the deficiency arising from this situation, a reorder option is introduced which renders a second manufacturing chance available shortly before he selling season. This reorder option facilitates an adjustment of the inventory level according to the realization of market demand. Since the market under investigation is facing a downward sloping demand curve, the effect of implementing this option is multi-fold. Moreover, the launch of the reorder option may also affect the decision makings at other levels of operations, such as altering the size of the initial order. Therefore, the overall impact of such option is not immediately clear. In this paper, it is shown that a properly designed reorder option is able to bring in profit growth and stabilize the fluctuations in the market retail price. Besides, quantity discount contracts are constructed to coordinate decisions on the initial inventory amount within the supply chain, so as to achieve higher economic efficiency. Finally, numerical examples are given to demonstrate the conclusions obtained in this paper.

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Mathematics Subject Classification: Primary: 91A80; Secondary: 68M20.

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Figure 1. Time line of the base case without the reorder option

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Figure 2. Time line of the base case with the reorder option

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Figure 3. Total profits with and without reorder option under $\sigma_1=10$.

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Figure 4. Optimal initial order $Q^*$ with a reorder option under $\sigma_1=10$.

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Figure 5. Incomes with exercising or dropping the reorder option at the states of $a_H$ and $a_L$ under $\sigma_1=10$.

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Figure 6. Total profits with and without a reorder option under $\sigma_2=25$.

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Figure 7. Optimal initial order $Q^*$ with a reorder option under $\sigma_2=25$.

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Figure 8. Incomes with exercising or dropping the reorder option at the states of $a_H$ and $a_L$ under $\sigma_2=25$.

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Figure 9. Supply chain's total profit and the retailer's profit with respect to $Q$ and $Q_R$, respectively.

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Figure 10. Distribution of the maximised supply chain total profit between the retailer and the supplier with respect to $\eta$.

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Figure 11. Incomes of the retailer with respect to $X$.

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Table 1. A List of Notations

$P$	market clearing price of the product
$Q$	amount of products at time $0$ with the reorder option
$Q^*$	optimal amount of products at time $0$ with the reorder option
$Q_0$	amount of products at time $0$ without the reorder option
$Q_0^*$	optimal amount of products at time $0$ without the reorder option
$\delta$	the slope of the demand curve
$a_H$ ($a_L$)	the indicator of the market condition in the high (low) state
$e_H$ ($e_L$)	Arrow-Debreu state price for the high (low) state
$\rho$	$\rho=e_H/e_L$
$B$	present value of the risk-free coupon that pays 1 dollar regardless of the state
$A$	present value of the security that pays $a_H$ ($a_L$) in the high (low) state
$p_H$ ($p_L$)	risk-neutral probability of the occurrence of the high (low) state
$\mu$	the mean of the uncertain factor $a$
${\sigma}^2$	the variance of the uncertain factor $a$
$c$	the unit production cost of a product at time $0$
$c_1$	the unit production cost of a product at time $T$
$R$	a pre-determined amount of products in the reorder option

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Table 2. A List of Notations

$w$	a fixed wholesale price which is higher than $c$
$X$	strike price which is pre-determined by the supplier
$Q_R$	amount of products ordered by the retailer at time $0$ with the reorder option
$Q_R^*$	optimal amount of products ordered by the retailer at time $0$ with the reorder option
$Q_{max}$	maximum size of products ordered by the retailer at time $0$ with the reorder option
$a, b$	parameters in the function of wholesale price $w$
$\eta$ ($\overline{\eta}$)	the portion of the maximized supply chain total profit earned by the retailer (supplier)

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Table 3. A summary of parameters

A	B	c	$c_1$	$\rho$	$\delta$
16 dollars	0.8 dollar	4 dollars	7 dollars	4	1
$e_L$	$e_H$	$\mu$
0.64 dollar	0.16 dollar	20 dollars

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