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Production-quality policy for a make-from-stock remanufacturing system

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Abstract

Remanufacturing capacities are constrained by the collected amount of end-of-life (EOL) products resulting in the make-from-stock model for remanufacturing industries. Remanufacturers may select quality choices in a strategic manner to improve their remanufacturing capacities. Thus, we examine a Cournot duopoly in which an original equipment manufacturer (OEM) sells new products and an independent remanufacturer (IR) remanufactures EOL products, and we determine the production quantities of low- and high-quality remanufactured products. The IR can select pure low-quality, pure high-quality, or mixed-quality policies. We formulate the dynamics between the OEM and IR in a two-period game, solve for the firms’ equilibrium quantities through dynamic programming, and derive the conditions of the remanufacturing-quality policies. Furthermore, we characterize the firms’ equilibrium decisions and analyze the parametric effects on profits, consumer surplus, and social welfare. When the availability of EOL products is sufficient, a high-quality policy is the dominant strategy for the IR and is also beneficial to both the OEM and consumers in most cases. However, when the amount of EOL products is not sufficient, the IR may choose to reduce the average quality level to obtain ample capacity; this situation is beneficial to consumers but harmful to the OEM.

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Notes

  1. As stated by Melumad and Ziv (2004), an alternative approach is that a manufacturer selects its average quality level from a continuous bounded set, inevitably leading to model complexity. The two-level representation of quality selection enables us to simplify the explanation and derive the key insights analytically.

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Acknowledgements

This research was partially supported by the Ministry of Science and Technology, Taiwan, ROC, under #MOST 108-2221-E-006-222-MY3.

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Appendix: Derivation of the condition for duopoly in the second period

Appendix: Derivation of the condition for duopoly in the second period

For simplicity, we let \(\mathbb {D}_i\equiv p_i-\alpha \,\chi _i\), where \(i\in \{o,r\}\); thus, the consumer utilities can be rewritten as \(U_o=\theta -\mathbb {D}_o\) and \(U_r=\rho \,\theta -\mathbb {D}_r\). Each consumer purchases product i, \(i\in \{o,r\}\), when \(U_i>0\) and \(U_i>U_{i'}\), where \(i'\in \{o,r\}/i\). Solving \(U_o=0\) and \(U_r=0\) for \(\theta\), we obtain two points, \(\theta _o\equiv \mathbb {D}_o\) and \(\theta _r\equiv \mathbb {D}_r/\rho\), respectively. There are two possible cases: \(\theta _o\ge \theta _r\) and \(\theta _o<\theta _r\). However, under the case \(\theta _o<\theta _r\Leftrightarrow \mathbb {D}_o<\mathbb {D}_r/\rho\), \(U_o>U_r\) always holds because \(U_o=\theta -\mathbb {D}_o>\theta -\mathbb {D}_r/\rho >\rho (\theta -\mathbb {D}_r/\rho )=U_r\). Hence, we know that the duopoly does not exist under the case \(\theta _o<\theta _r\). The condition of \(\theta _o\ge \theta _r\Leftrightarrow \mathbb {D}_o\ge \mathbb {D}_r/\rho\), called “Condition 1”, must be held for duopoly, corresponding to the works of (Chiang et al. 2003) and (Atasu et al. 2008).

Next, we derive the indifferent point by solving \(U_o=U_r\) for \(\theta\) and obtain \(\theta _{ro}\equiv (\mathbb {D}_o-\mathbb {D}_r)/(1-\rho )\). Because \(\partial \,U_o/\partial \,\theta >\partial \,U_r/\partial \,\theta\), \(U_r\ge U_o\) if \(\theta \le \theta _{ro}\), and \(U_r< U_o\) otherwise. Hence, consumers located between \(\theta _r\) and \(\theta _{ro}\) choose remanufactured products, and consumers located between \(\theta _{ro}\) and 1 choose new products. Notably, we can observe that the total demand is dependent of \(\theta _r\) but independent of \(\theta _o\). For the existence of duopolistic competition, we have to verify that \(\theta _{ro}\) is located between \(\theta _r\) and 1. (1) It is easily to verify that \(\theta _{ro}\ge \theta _r\) because \(\text{ Condition } \text{1: }~\mathbb {D}_o\ge \mathbb {D}_r/\rho \Rightarrow \theta _{ro}-\theta _r=(\rho \mathbb {D}_o-\mathbb {D}_r)/((1-\rho ) \rho )\ge 0\). (2) For \(\theta _{ro}\le 1\), we can obtain the condition \(\rho -(1 -\mathbb {D}_o+\mathbb {D}_r)\le 0\), called “Condition 2”.

Condition 1 and Condition 2 must hold throughout the paper. Then, we incorporate the forms of prices, i.e., \(p_o=1+\alpha -q_o-\rho \left( q_{rh}+q_{rl}\right)\) and \(p_r=\rho \left( 1-\left( q_o+q_{rh}\right) \right) +\alpha \,q_{rh}/(q_{rh}+q_{rl})-\rho q_{rl}\), into Condition 1 and Condition 2, and the conditions can be rewritten as follows:

$$\begin{aligned} \text{ Condition } \text{1: }&\mathbb {D}_o\ge \mathbb {D}_r/\rho \Leftrightarrow (1-\rho ) \rho \left( q_{rh}+q_{rl}\right) \ge 0,\\ \text{ Condition } \text{2: }&\rho -(1 -\mathbb {D}_o+\mathbb {D}_r)\le 0 \Leftrightarrow (1-\rho ) q_o\le 0. \end{aligned}$$

Clearly, Condition 1 innately holds. For Condition 2, we derive the best-response decision of \(p_o\), i.e., \(p_o^{\dag }\equiv \frac{1}{2} \left( \alpha -c-\rho \left( q_{rh}+q_{rl}\right) +1\right)\), by maximizing Eq. (2). Incorporating \(p_o^{\dag }\) into Condition 2 implies that \(\rho - (\alpha -c+1)/(q_{rh}+q_{rl})\le 0\) must be true. As a result, we assume that the condition \(\rho - (\alpha -c+1)/(q_{rh}+q_{rl})\le 0\), is always true throughout the paper.

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Wu, CH. Production-quality policy for a make-from-stock remanufacturing system. Flex Serv Manuf J 33, 425–456 (2021). https://doi.org/10.1007/s10696-020-09379-3

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