Pricing and ordering strategies of supply chain with selling gift cards

Jingming Pan; Wenqing Shi; Xiaowo Tang

doi:10.3934/jimo.2017050

Article Contents

2018, Volume 14, Issue 1: 349-369. Doi: 10.3934/jimo.2017050

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Pricing and ordering strategies of supply chain with selling gift cards

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 611731, China

^* Corresponding author: Jingming Pan
^* Corresponding author: Jingming Pan

Received: March 31, 2016

Revised: April 30, 2016

Early access: May 2017

Published: January 2018

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Abstract

Gift cards is frequently used to replace traditional gift cash and gift products, especially when gift givers do not know gift receivers' performances. Basing on this phenomenon, we analyze the supplier's and the retailer's strategies with selling gift cards. First, we develop a Stackelberg model without selling gift cards. Next, we develop two models with selling gift cards when unredeemed gift cards balances become the retailer's property and the state's property, respectively. We present the optimal solutions and exam the impacts of parameters on the optimal decisions and the supply chain performance. When the retailer sells gift cards, the optimal order quantity is smaller than that without selling gift cards. The optimal wholesale price with selling gift cards is related to the treatment of unredeemed gift card balances. When unredeemed gift cards balances become the retailer's property, the wholesale price is lower than that without selling gift cards. However, when unredeemed gift cards balances become the state's property, the wholesale price is lower than that without selling gift cards in some conditions. And with selling gift cards, the optimal expected profits of retailer and supply chain are better off, but the optimal expected profit of supplier is worse off.

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Mathematics Subject Classification: Primary: 90B05, 90B60; Secondary: 91B42.

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Figure 1. Decision behaviors in supply chain without gift cards

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Figure 2. Decision behaviors in supply chain with gift cards

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Figure 3. $w^*$, $q^*$ and $\pi^*$ vs. $CV$ (Note: $c=5, \theta=0.15, \alpha=0.8, \beta=0.5$ and $m=0.3$)

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Figure 4. $w^*$, $q^*$ and $\pi^*$ vs. $c/p$ (Note: $\sigma=40, \theta=0.15, \alpha=0.8, \beta=0.5$ and $m=0.3$)

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Figure 5. $w^*$, $q^*$ and $\pi^*$ vs. $\alpha$ (Note: $\sigma=40, c=5, \theta=0.15, \beta=0.5$ and $m=0.3$)

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Figure 6. $w^*$, $q^*$ and $\pi^*$ vs. $\beta$ (Note: $\sigma=40, c=5, \theta=0.15, \alpha=0.5$ and $m=0.3$)

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Figure 7. $w^*$, $q^*$ and $\pi^*$ vs. $\beta$ (Note: $\sigma=40, c=5, \theta=0.15, \alpha=0.8$ and $m=0.3$)

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Figure 8. $w^*$, $q^*$ and $\pi^*$ vs. $m$ (Note: $\sigma=40, c=5, \theta=0.15, \alpha=0.8$ and $\beta=0.5$)

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Table 1. Notation

Decision Variables:
q	order quantity of gift product of the retailer
w	wholesale price of gift product of the supplier
Parameters:
x	demand for gift product/gift cards from gift givers before the holiday
f(x), F(x)	the pdf and cdf of demand from gift givers before the holiday
p	sale price of unit gift product
c	cost of unit gift product
v	salvage value of unit gift product
θ	return rate of gift product/probability of a gift giving consumer buying gift card before the holiday
β	probability of gift product buyers buying gift cards when gift product is stock-out before the holiday
α	average redemption rate of gift cards after the holiday
m	profit margin of non-gift products

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Table 2. Optimal solutions for uniformly distributed demand

Optimal wholesale price	Optimal order quantity	Conditions
$w_{NG}^* = \frac{1}{2}\left[ {c + \left( {1 - \theta } \right)p + \theta v} \right]$	$q_{NG}^* = \frac{{b\left[ {\left( {1 - \theta } \right)p + \theta v - c} \right]}}{{2\left( {1 - \theta } \right)\left( {p - v} \right)}}$	--
$w_{RG}^* = \frac{1}{2}\left[ {c + \left( {1 - \beta - m\alpha \beta + \alpha \beta } \right)p} \right]$	$q_{RG}^* = \frac{{b\left( {1 - \theta } \right)\left[ {\left( {1 - \beta - m\alpha \beta + \alpha \beta } \right)p - c} \right]}}{{2\left( {1 -\beta- m\alpha \beta + \alpha \beta } \right)p - v}}$	$\left( {1 - \beta - m\alpha \beta + \alpha \beta } \right)p - v \ge 0$
$w_{SG}^* = \frac{1}{2}\left[ {c + \left( {1 - m\alpha \beta } \right)p} \right]$	$q_{SG}^* = \frac{{b\left( {1 - \theta } \right)\left[ {\left( {1 - m\alpha \beta } \right)p - c} \right]}}{{2\left( {1 - m\alpha \beta } \right)p - v}}$	$\left( {1 - m\alpha \beta } \right)p - v \ge 0$

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