# Strategy Analysis of Recycling and Remanufacturing by Remanufacturers in Closed-Loop Supply Chain

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Description

#### 3.1. Joint Strategies by Manufacturer and Remanufacturer—Model I

**Proposition 1.**

#### 3.2. Sensitivity Analysis

## 4. Strategies Analysis of the Decentralized Model D

**Proposition 2.**

**Proposition 3.**

- In Case I, with $0\le v<{v}_{1}^{D}$,$${c}_{3}^{D}-{c}_{1}^{D}=v-{c}_{n}\rho +\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)<-\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)\le 0$$$${c}_{3}^{D}-{c}_{2}^{D}=v-{c}_{n}\rho -\frac{\delta {\rho}^{2}\left(1-{c}_{n}\right)}{2}\le 0$$
- In Case II, with ${v}_{1}^{D}\le v\le {v}_{2}^{D}$,$${c}_{3}^{D}-{c}_{1}^{D}=v-{c}_{n}\rho +\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)\le 0$$
- In Case III, with ${v}_{2}^{D}<v\le {v}_{3}^{D}$,$${c}_{8}^{D}-{c}_{1}^{D}=\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}-\frac{\left(2-\rho \right)\left({c}_{n}\rho -v\right)}{2\left(1-\rho \right)}>0$$$${c}_{8}^{D}-{c}_{2}^{D}=-\frac{\left(2-\rho \right)\left({c}_{n}\rho -v\right)}{2\left(1-\rho \right)}\le 0$$
- In Case IV, with ${v}_{3}^{D}<v$,when ${c}_{r}<{c}_{1}^{D}$, the remanufacturer opts for the strategy of ${q}_{2i}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2}$, whereas, when ${c}_{1}^{D}\le {c}_{r}\le {c}_{2}^{D}$, the remanufacturer opts for the strategy of ${q}_{2i}^{D}=\frac{\rho \left(1+{c}_{n}\right)-2\left({c}_{r}+v\right)}{2\rho \left(2-\rho \right)}$. When ${c}_{2}^{D}<{c}_{r}$, the remanufacturer opts for the strategy of ${q}_{2i}^{I}=0$.

**Proposition 4.**

**Strategy D-a**

**Strategy D-b**

**Strategy D-c**

**Strategy D-d**

**Strategy D-e**

**Strategy D-f**

**Strategy D-g**

## 5. Sensitivity Analysis

## 6. Contrastive Analyses of Models I and D

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

- Case I, ${\lambda}_{1}=0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}>0$${q}_{2n}^{I}=\frac{1-{c}_{n}}{2}$, ${q}_{2r}^{I}=0$, ${\lambda}_{3}^{I}=v-\rho {c}_{n}$when ${q}_{2n}^{I}\ge 0$, ${c}_{n}\le 1$ is derived.when ${\lambda}_{3}^{I}>0$, $v>\rho {c}_{n}={v}_{1}^{I}$ is derived.
- Case II, ${\lambda}_{1}=0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}=0$${q}_{2n}^{I}=\frac{1+v-\rho -{c}_{n}}{2\left(1-\rho \right)}$, ${q}_{2r}^{I}=\frac{\rho {c}_{n}-v}{2\rho \left(1-\rho \right)}$when ${q}_{2n}^{I}\ge 0$, $v\ge {c}_{n}-1+\rho ={v}_{2}^{I}$ is derived.when ${q}_{2r}^{I}\ge 0$, $v\le \rho {c}_{n}={v}_{1}^{I}$ is derived.when $\delta {q}_{1}\ge {q}_{2r}^{I}$, $v\ge \rho {c}_{n}-\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)={v}_{3}^{I}$$${v}_{1}^{I}-{v}_{2}^{I}=\left(1-\rho \right)\left(1-{c}_{n}\right)\ge 0$$
- Case III, ${\lambda}_{1}>0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}=0$${q}_{2n}^{I}=\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}$, ${q}_{2r}^{I}=\frac{\delta \left(1-{c}_{n}\right)}{2}$, ${\lambda}_{1}^{I}=\rho {c}_{n}-v-\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$when ${q}_{2n}^{II}\ge 0$, ${c}_{n}\le 1$ is derived.when ${\lambda}_{1}^{II}>0$, $v<\rho {c}_{n}-\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)={v}_{3}^{I}$ is derived.
- Case IV, ${\lambda}_{1}=0$, ${\lambda}_{2}>0$ and ${\lambda}_{3}=0$${q}_{2n}^{I}=0$, ${q}_{2r}^{I}=\frac{\rho -v}{2\rho}$, ${\lambda}_{2}^{I}=\rho +{c}_{n}-v-1$when ${q}_{2r}^{I}\ge 0$, $v\le \rho ={v}_{4}^{I}$ is derived.when ${\lambda}_{1}^{I}>0$, $v<{c}_{n}-1+\rho ={v}_{2}^{I}$ is derived.when $\delta {q}_{1}\ge {q}_{2r}^{I}$, $v\ge \rho -\delta \rho \left(1-{c}_{n}\right)={v}_{5}^{I}$ and ${v}_{2}^{I}-{v}_{5}^{I}=-\left(1-\delta \rho \right)\left(1-{c}_{n}\right)\le 0$ are derived.Therefore, there is no $v$ satisfying the condition to enable the case IV to occur.
- Case V, ${\lambda}_{1}>0$, ${\lambda}_{2}>0$ and ${\lambda}_{3}=0$${q}_{2n}^{I}=0$, ${q}_{2r}^{I}=\frac{\delta \left(1-{c}_{n}\right)}{2}$, ${\lambda}_{1}^{I}=\rho -v-\delta \rho \left(1-{c}_{n}\right)$, ${\lambda}_{2}^{I}=-\left(1-\delta \rho \right)\left(1-{c}_{n}\right)$when ${\lambda}_{1}^{I}>0$, $v<\rho -\delta \rho \left(1-{c}_{n}\right)={v}_{5}^{I}$ is derived.when ${\lambda}_{2}^{I}>0$, it is false.

## Appendix B. Proof of Proposition 2

- Case I, ${\lambda}_{1}=0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}>0$${q}_{2n}^{D}=\frac{1-{c}_{n}-\rho {q}_{2i}}{2}$, ${q}_{2r}^{D}=0$, ${\lambda}_{3}^{D}=\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho $when ${q}_{2n}^{D}\ge 0$, ${q}_{2i}\le \frac{1-{c}_{n}}{\rho}$ is true.when ${\lambda}_{3}^{D}>0$, ${w}_{2r}>{c}_{n}\rho -\rho {q}_{2i}\left(1-\rho \right)={w}_{1}^{D}$ is derived.
- Case II ${\lambda}_{1}=0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}=0$${q}_{2n}^{D}=\frac{1+{w}_{2r}-{c}_{n}-\rho}{2\left(1-\rho \right)}$, ${q}_{2r}^{D}=-\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}$when ${q}_{2n}^{D}\ge 0$, ${w}_{2r}\ge {c}_{n}+\rho -1={w}_{2}^{D}$ is derived.when ${q}_{2r}^{D}\ge 0$, ${w}_{2r}\le {c}_{n}\rho -\rho {q}_{2i}\left(1-\rho \right)={w}_{1}^{D}$ is derived.when $\delta {q}_{1}\ge {q}_{2r}^{D}+{q}_{2i}$, ${w}_{2r}\ge \rho \left[{c}_{n}\left(1+\delta -\delta \rho \right)+\left(1-\rho \right)\left({q}_{2i}-\delta \right)\right]={w}_{3}^{D}$$${w}_{1}^{D}-{w}_{2}^{D}=\left(1-\rho \right)\left(1-{c}_{n}-\rho {q}_{2i}\right)\ge 0$$$${w}_{1}^{D}-{w}_{3}^{D}=\rho \left(1-\rho \right)\left(\delta \left(1-{c}_{n}\right)-2{q}_{2i}\right)\ge 0$$
- Case III, ${\lambda}_{1}>0$, ${\lambda}_{2}=0$ and ${\lambda}_{3}=0$${q}_{2n}^{D}=\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}}{2}$, ${q}_{2r}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}$, ${\lambda}_{1}^{D}=\rho \left[{c}_{n}\left(1+\delta -\delta \rho \right)+\left(1-\rho \right)\left({q}_{2i}-\delta \right)-{w}_{2r}\right]$when ${q}_{2n}^{D}\ge 0$, ${q}_{2i}\ge -\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{\rho}$ is true.when ${q}_{2r}^{D}\ge 0$, ${q}_{2i}\le \frac{\delta \left(1-{c}_{n}\right)}{2}$ is true.when ${\lambda}_{1}^{D}>0$, ${w}_{2r}<\rho \left[{c}_{n}\left(1+\delta -\delta \rho \right)+\left(1-\rho \right)\left({q}_{2i}-\delta \right)\right]={w}_{3}^{D}$ is derived.
- Case IV ${\mathsf{\lambda}}_{1}=0$, ${\mathsf{\lambda}}_{2}>0$ and ${\mathsf{\lambda}}_{3}=0$${q}_{2n}^{D}=0$, ${q}_{2r}^{D}=\frac{\rho -\rho {q}_{2i}-{w}_{2r}}{2\rho}$, ${\lambda}_{2}^{D}=\rho -{w}_{2r}+{c}_{n}-1$when ${q}_{2r}^{D}\ge 0$, ${w}_{2r}\le \rho -\rho {q}_{2i}={w}_{4}^{D}$ is derived.when ${\lambda}_{1}^{D}>0$, ${w}_{2r}<\rho +{c}_{n}-1={w}_{2}^{D}$ is derived.when $\delta {q}_{1}\ge {q}_{2r}^{D}+{q}_{2i}$, ${w}_{2r}\ge \rho \left(1+{q}_{2i}+\delta {c}_{n}-\delta \right)={w}_{5}^{D}$$${w}_{4}^{D}-{w}_{5}^{D}=\rho \left(\delta \left(1-{c}_{n}\right)-2{q}_{2i}\right)\ge 0$$$${w}_{5}^{D}-{w}_{2}^{D}=\left(1-\delta \rho \right)\left(1-{c}_{n}\right)+\rho {q}_{2i}\ge 0$$There is no appropriate ${w}_{2r}$ to enable Case IV to exist.
- Case V ${\lambda}_{1}>0$, ${\lambda}_{2}>0$ and ${\lambda}_{3}=0$${q}_{2n}^{D}=0$, ${q}_{2r}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}$, ${\lambda}_{1}^{D}=\rho {q}_{2i}-{w}_{2r}-\delta \rho \left(1-{c}_{n}\right)+\mathsf{\rho}$, ${\lambda}_{2}^{D}=-\left(1-\delta \rho \right)\left(1-{c}_{n}\right)-\rho {q}_{2i}$when ${q}_{2r}^{D}\ge 0$, ${q}_{2i}\le \frac{\delta \left(1-{c}_{n}\right)}{2}$ is derived.when ${\lambda}_{1}^{D}>0$, ${w}_{2r}<\rho \left(1+{q}_{2i}+\delta {c}_{n}-\delta \right)={w}_{5}^{D}$ is derived.when ${\lambda}_{2}^{D}>0$, ${q}_{2i}<-\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{\rho}$ is not true.

## Appendix C. Proof of Proposition 3

- (1)
- when ${w}_{1}^{D}<{w}_{2r}$, ${q}_{2n}^{D}=\frac{1-{c}_{n}-\rho {q}_{2i}}{2}$, ${q}_{2r}^{D}=0$ is derived.$$\mathrm{L}\left({q}_{2i},{w}_{2r}\right)=\left({w}_{2r}-v\right){q}_{2r}^{D}+\left({p}_{2i}-{c}_{r}-v\right){q}_{2i}+{\lambda}_{1}\left(\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}\right)+{\lambda}_{2}{q}_{2i}$$$$s.t.\{\begin{array}{c}{\lambda}_{1}\left(\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}\right)=0\\ {\lambda}_{2}{q}_{2i}=0\\ {w}_{1}<{w}_{2r}\\ {\lambda}_{1},{\lambda}_{2}\ge 0\\ \frac{\delta \left(1-{c}_{n}\right)}{2}\ge {q}_{2i}\ge 0\end{array}$$
- ●
- Case I ${\lambda}_{1}>0$ and ${\lambda}_{2}=0$${q}_{2i}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2}$, ${\lambda}_{1}^{D}=-\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}+\frac{\rho \left(1+{c}_{n}\right)}{2}-{c}_{r}-v$when ${\lambda}_{1}^{D}>0$, ${c}_{r}<-\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}+\frac{\rho \left(1+{c}_{n}\right)}{2}-v={c}_{1}^{D}$ is derived.
- ●
- Case II ${\lambda}_{1}=0$ and ${\lambda}_{2}=0$$${q}_{2i}^{D}=\frac{\rho \left(1+{c}_{n}\right)-2\left({c}_{r}+v\right)}{2\rho \left(2-\rho \right)}$$when $\frac{\delta \left(1-{c}_{n}\right)}{2}\ge {q}_{2i}^{D}$, ${c}_{r}\ge -\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}+\frac{\rho \left(1+{c}_{n}\right)}{2}-v={c}_{1}^{D}$${c}_{2}^{D}-{c}_{1}^{D}=\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}\ge 0$ are derived.
- ●
- Case III ${\lambda}_{1}=0$ and ${\lambda}_{2}>0$${q}_{2i}^{D}=0$, ${\lambda}_{2}^{D}={c}_{r}+v-\frac{\rho \left(1+{c}_{n}\right)}{2}$when ${\lambda}_{2}^{D}>0$, ${c}_{r}>\frac{\rho \left(1+{c}_{n}\right)}{2}-v={c}_{2}^{D}$ is derived.
- (2)
- when ${w}_{3}^{D}\le {w}_{2r}\le {w}_{1}^{D}$, ${q}_{2n}^{D}=\frac{1+{w}_{2r}-{c}_{n}-\rho}{2\left(1-\rho \right)}$, ${q}_{2r}^{D}=-\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}$ are derived.$$\begin{array}{ll}\mathrm{L}\left({q}_{2i},{w}_{2r}\right)=& \left({w}_{2r}-v\right){q}_{2r}^{D}+\left({p}_{2i}-{c}_{r}-v\right){q}_{2i}\\ & \hspace{1em}+{\lambda}_{1}\left(\frac{\delta \left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}-{q}_{2i}\right)+{\lambda}_{2}{q}_{2i}\\ & \hspace{1em}+{\lambda}_{3}({c}_{n}\rho -\rho {q}_{2i}\left(1-\rho \right)-{w}_{2r})+{\lambda}_{4}({w}_{2r}-\rho [{c}_{n}(1+\delta -\delta \rho )\\ & \hspace{1em}+(1-\rho )({q}_{2i}-\delta )\left]\right)\end{array}$$$$s.t.\{\begin{array}{c}{\lambda}_{1}\left(\frac{\delta \left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}-{q}_{2i}\right)=0\\ {\lambda}_{2}{q}_{2i}=0\\ {\lambda}_{3}({c}_{n}\rho -\rho {q}_{2i}\left(1-\rho \right)-{w}_{2r})=0\\ {\lambda}_{4}({w}_{2r}-\rho [{c}_{n}(1+\delta -\delta \rho )+(1-\rho )({q}_{2i}-\delta )\left]\right)=0\\ {w}_{3}\le {w}_{2r}\le {w}_{1}\\ v\le {w}_{2r}\\ {\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}\ge 0\\ \frac{\delta \left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}\ge {q}_{2i}\ge 0\end{array}$$
- ●
- Case I ${\lambda}_{1}>0$, ${\lambda}_{2}=0$, ${\lambda}_{3}>0$ and ${\lambda}_{4}=0$$${q}_{2i}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2},{w}_{2r}^{D}={c}_{n}\rho -\frac{\delta \rho \left(1-{c}_{n}\right)\left(1-\rho \right)}{2},{\lambda}_{1}^{D}=-\frac{\delta \rho \left(1-{c}_{n}\right)\left(2-\rho \right)}{2}+\frac{\rho \left(1+{c}_{n}\right)}{2}-{c}_{r}-v,\phantom{\rule{0ex}{0ex}}{\lambda}_{3}^{D}=\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{4\left(1-\rho \right)}-\frac{{c}_{r}}{2\rho \left(1-\rho \right)}$$when ${\lambda}_{3}^{D}>0$, ${c}_{r}<\frac{\rho \left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}={c}_{3}^{D}$${c}_{3}^{D}-{c}_{1}^{D}=v-{c}_{n}\rho +\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ are derived.
- ●
- Case II ${\lambda}_{1}>0$, ${\lambda}_{2}=0$, ${\lambda}_{3}=0$ and ${\lambda}_{4}=0$${q}_{2i}^{D}=-\frac{{c}_{r}}{\rho \left(2-\rho \right)}+\frac{\left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2\left(2-\rho \right)}$, ${\lambda}_{1}^{D}=\frac{{c}_{n}\rho \left(1+2\delta -2\delta \rho \right)}{2-\rho}-\mathrm{v}-\frac{2{c}_{r}\left(1-\rho \right)}{\left(2-\rho \right)}+\frac{\rho \left(1-\rho \right)\left(1-2\delta \right)}{2-\rho}$, ${w}_{2r}^{D}={c}_{n}\rho -\frac{{c}_{r}\left(1-\rho \right)}{\left(2-\rho \right)}+\frac{\rho \left(1-\rho \right)\left(1-2\delta \right)\left(1-{c}_{n}\right)}{2\left(2-\rho \right)}$when ${q}_{2i}^{D}\ge 0$, ${c}_{r}\le \frac{\rho \left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2}={c}_{4}^{D}$ is derived.when ${\lambda}_{1}^{D}>0$, ${c}_{r}<\frac{{c}_{n}\rho \left(1+2\delta -2\delta \rho \right)}{2\left(1-\rho \right)}-\frac{v\left(2-\rho \right)}{2\left(1-\rho \right)}+\frac{\rho \left(1-2\delta \right)}{2}={c}_{5}^{D}$ is derived.when ${w}_{2r}^{D}\ge v$, ${c}_{r}\le \frac{{c}_{n}\rho \left(3+2\delta -2\delta \rho -\rho \right)}{2\left(1-\rho \right)}-\frac{v\left(2-\rho \right)}{\left(1-\rho \right)}+\frac{\rho \left(1-2\delta \right)}{2}={c}_{6}^{D}$ is derived.when ${w}_{2r}^{D}\le {w}_{1}$, ${c}_{r}\ge \frac{\rho \left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}={c}_{3}^{D}$$${c}_{5}^{D}-{c}_{6}^{D}=\frac{v\left(2-\rho \right)}{2\left(1-\rho \right)}-\frac{{c}_{n}\rho \left(2-\rho \right)}{2\left(1-\rho \right)}$$$${c}_{5}^{D}-{c}_{4}^{D}=\frac{\left({c}_{n}\rho -v\right)\left(2-\rho \right)}{2\left(1-\rho \right)}-\rho \delta \left(2-\rho \right)\left(1-{c}_{n}\right)$$$${c}_{4}^{D}-{c}_{3}^{D}=\frac{\rho \delta \left(2-\rho \right)\left(1-{c}_{n}\right)}{2}\ge 0$$$${c}_{5}^{D}-{c}_{3}^{D}=\frac{\left({c}_{n}\rho -v\right)\left(2-\rho \right)}{2\left(1-\rho \right)}-\frac{\rho \delta \left(2-\rho \right)\left(1-{c}_{n}\right)}{2}$$$${c}_{6}^{D}-{c}_{3}^{D}=\frac{\left({c}_{n}\rho -v\right)\left(2-\rho \right)}{\left(1-\rho \right)}-\frac{\rho \delta \left(2-\rho \right)\left(1-{c}_{n}\right)}{2}$$when $0\le v<{c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$, ${c}_{3}^{D}<{c}_{4}^{D}<{c}_{5}^{D}<{c}_{6}^{D}$ is derived.when ${c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)\le \mathrm{v}\le {c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$, ${c}_{3}\le {c}_{5}^{D}\le {c}_{4}^{D}\le {c}_{6}^{D}$ is derived.when ${c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)<v\le {c}_{n}\rho $, ${c}_{5}^{D}<{c}_{3}^{D}$ is derived and ${c}_{r}$ does not satisfy the constraint.
- ●
- Case III ${\lambda}_{1}=0$, ${\lambda}_{2}=0$, ${\lambda}_{3}=0$ and ${\lambda}_{4}=0$${q}_{2i}^{D}=\frac{\rho -2{c}_{r}-v}{2\rho}$, ${w}_{2r}^{D}=\frac{v+{c}_{n}\rho}{2}$when ${q}_{2i}^{D}\ge 0$, ${c}_{r}\le \frac{\rho -v}{2}={c}_{7}^{D}$ is derived.when ${w}_{2r}^{D}\ge v$, $\mathrm{v}\le {c}_{n}\rho $ is derived.when ${w}_{2r}^{D}\ge {w}_{3}$, ${c}_{r}\ge \frac{{c}_{n}\rho \left(1+2\delta -2\delta \rho \right)}{2\left(1-\rho \right)}-\frac{v\left(2-\rho \right)}{2\left(1-\rho \right)}+\frac{\rho \left(1-2\delta \right)}{2}={c}_{5}^{D}$ is derived.when ${w}_{2r}^{D}\le {w}_{1}$, ${c}_{r}\ge \frac{\rho \left(v-{c}_{n}+1-\rho \right)}{2\left(1-\rho \right)}={c}_{8}^{D}$ is derived.when $\frac{\delta \left(1-{c}_{n}\right)}{2}+{q}_{2r}^{D}\ge {q}_{2i}^{D}$, ${c}_{r}\ge \frac{{c}_{n}\rho \left(1+2\delta -2\delta \rho \right)}{2\left(1-\rho \right)}-\frac{v\left(2-\rho \right)}{2\left(1-\rho \right)}+\frac{\rho \left(1-2\delta \right)}{2}={c}_{5}^{D}$$${c}_{7}^{D}-{c}_{8}^{D}=\frac{\rho {c}_{n}-v}{2\left(1-\rho \right)}\ge 0$$$${c}_{5}^{D}-{c}_{8}^{D}=\frac{\rho {c}_{n}-v}{\left(1-\rho \right)}-\mathsf{\delta}\mathsf{\rho}\left(1-{c}_{n}\right)$$when $0\le v<{c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$, ${c}_{8}^{D}<{c}_{7}^{D}<{c}_{5}^{D}$ is derived.when ${c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)\le \mathrm{v}\le {c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$, ${c}_{8}^{D}\le {c}_{5}^{D}\le {c}_{7}^{D}$ is derived.when ${c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)<v\le {c}_{n}\rho $, ${c}_{5}^{D}<{c}_{8}^{D}<{c}_{7}^{D}$ is derived.
- ●
- Case IV ${\lambda}_{1}=0$, ${\mathsf{\lambda}}_{2}>0$, ${\mathsf{\lambda}}_{3}>0$ and ${\lambda}_{4}=0$${q}_{2i}^{D}=0$, ${w}_{2r}^{D}={c}_{n}\rho $, ${\lambda}_{1}^{D}={c}_{r}+v-\frac{\rho \left(1+{c}_{n}\right)}{2}$, ${\lambda}_{3}^{D}=\frac{v-{c}_{n}\rho}{2\rho \left(1-\rho \right)}$when ${w}_{2r}^{D}\ge v$, $v\le {c}_{n}\rho $ is derived.when ${\lambda}_{1}^{D}>0$, ${c}_{r}>\frac{\rho \left(1+{c}_{n}\right)}{2}-v={c}_{2}^{D}$ is derived.when ${\lambda}_{3}^{D}>0$, $v>{c}_{n}\rho $ is derived.when ${w}_{2r}^{D}\ge {w}_{3}$, ${c}_{n}\le 1$ is derived.
- ●
- Case V ${\lambda}_{1}=0$, ${\lambda}_{2}>0$, ${\lambda}_{3}=0$ and ${\lambda}_{4}=0$${q}_{2i}^{D}=0$, ${w}_{2r}^{D}=\frac{v+{c}_{n}\rho}{2}$, ${\lambda}_{1}^{I}={c}_{r}-\frac{\rho}{2}+\frac{v}{2}$when ${w}_{2r}^{D}\ge v$, $v\le {c}_{n}\rho $ is derived.when ${\lambda}_{1}^{D}>0$, ${c}_{r}>\frac{\rho -v}{2}={c}_{7}^{D}$ is derived.when ${w}_{2r}^{D}\ge {w}_{3}$, $v\ge {c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ is derived.when ${w}_{2r}^{D}\le {w}_{1}$, $v\le {c}_{n}\rho $ is derived.
- ●
- Case VI ${\lambda}_{1}=0$, ${\lambda}_{2}>0$, ${\lambda}_{3}=0$ and ${\lambda}_{4}>0$${q}_{2i}^{D}=0$, ${w}_{2r}^{D}={c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$, ${\lambda}_{1}^{D}={c}_{r}-\frac{\rho \left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2}$, ${\lambda}_{4}^{D}=\frac{{c}_{n}\left(1+2\delta -2\delta \rho \right)}{2\left(1-\rho \right)}-\mathsf{\delta}-\frac{v}{2\rho \left(1-\rho \right)}$when ${w}_{2r}^{D}\ge v$, $v\le {c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ is derived.when ${\lambda}_{1}^{D}>0$, ${c}_{r}>\frac{\rho \left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2}={c}_{4}^{D}$ is derived.when ${\lambda}_{4}^{D}>0$, $v<{c}_{n}\rho -2\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ is derived.when ${w}_{2r}<{w}_{3}^{D}$, ${q}_{2n}^{D}=\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}}{2}$, ${q}_{2r}^{D}=\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}$ is derived.$$\mathrm{L}\left({q}_{2i},{w}_{2r}\right)=\left({w}_{2r}-v\right){q}_{2r}^{D}+\left({p}_{2i}-{c}_{r}-v\right){q}_{2i}$$$$s.t.\{\begin{array}{c}{w}_{2r}<{w}_{3}\\ v\le {w}_{2r}\\ {\lambda}_{1},{\lambda}_{2}\ge 0\\ \frac{\delta \left(1-{c}_{n}\right)}{2}\ge {q}_{2i}\ge 0\end{array}$$

**Table A1.**The remanufacturer’s optimal sales and wholesale prices of remanufactured products in Model D.

${\mathit{q}}_{2\mathit{i}}^{\mathit{D}}$ | ${\mathit{w}}_{2\mathit{r}}^{\mathit{D}}$ | |
---|---|---|

${c}_{2}^{D}<{c}_{r}$ | $0$ | none |

${c}_{1}^{D}\le {c}_{r}\le {c}_{2}^{D}$ | $\frac{\rho \left(1+{c}_{n}\right)-2\left({c}_{r}+v\right)}{2\rho \left(2-\rho \right)}$ | none |

${c}_{r}<{c}_{1}^{D}$ | $\frac{\delta \left(1-{c}_{n}\right)}{2}$ | none |

${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{8}^{D}\le {c}_{r}\le {c}_{7}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{5}^{D}\le {c}_{r}\le {c}_{7}^{D}$ | $\frac{\rho -2{c}_{r}-v}{2\rho}$ | $\frac{\rho {c}_{n}+v}{2}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{4}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{5}^{D}$ | $-\frac{{c}_{r}}{\rho \left(2-\rho \right)}+\frac{\left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2\left(2-\rho \right)}$ | ${c}_{n}\rho -\frac{{c}_{r}\left(1-\rho \right)}{\left(2-\rho \right)}+\frac{\rho \left(1-\rho \right)\left(1-2\delta \right)\left(1-{c}_{n}\right)}{2\left(2-\rho \right)}$ |

${v}_{1}^{D}\le \mathrm{v}\le {v}_{3}^{D}$ and ${c}_{r}>{c}_{7}^{D}$ | $0$ | $\frac{v+{c}_{n}\rho}{2}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{4}^{D}<{c}_{r}$ | $0$ | ${c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ |

**Table A2.**The remanufacturer’s optimal sales and wholesale prices of remanufactured products in Model D.

${\mathit{q}}_{2\mathit{i}}^{\mathit{D}}$ | ${\mathit{w}}_{2\mathit{r}}^{\mathit{D}}$ | |
---|---|---|

${v}_{3}^{D}<v$ and ${c}_{2}^{D}<{c}_{r}$ | $0$ | none |

${v}_{3}^{D}<v$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{2}^{D}$ or ${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{8}^{D}$ | $\frac{\rho \left(1+{c}_{n}\right)-2\left({c}_{r}+v\right)}{2\rho \left(2-\rho \right)}$ | none |

$0\le \mathrm{v}<{v}_{2}^{D}$ and ${c}_{r}<{c}_{3}^{D}$ or ${v}_{2}^{D}<v$ and ${c}_{r}<{c}_{1}^{D}$ | $\frac{\delta \left(1-{c}_{n}\right)}{2}$ | none |

${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{8}^{D}\le {c}_{r}\le {c}_{7}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{5}^{D}\le {c}_{r}\le {c}_{7}^{D}$ | $\frac{\rho -2{c}_{r}-v}{2\rho}$ | $\frac{\rho {c}_{n}+v}{2}$ |

$0\le \mathrm{v}\le {v}_{1}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{4}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{5}^{D}$ | $-\frac{{c}_{r}}{\rho \left(2-\rho \right)}+\frac{\left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)}{2\left(2-\rho \right)}$ | ${c}_{n}\rho -\frac{{c}_{r}\left(1-\rho \right)}{\left(2-\rho \right)}+\frac{\rho \left(1-\rho \right)\left(1-2\delta \right)\left(1-{c}_{n}\right)}{2\left(2-\rho \right)}$ |

${v}_{1}^{D}\le \mathrm{v}\le {v}_{3}^{D}$ and ${c}_{r}>{c}_{7}^{D}$ | $0$ | $\frac{v+{c}_{n}\rho}{2}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{4}^{D}<{c}_{r}$ | $0$ | ${c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)$ |

${\mathit{q}}_{2\mathit{n}}^{\mathit{D}}$ | ${\mathit{q}}_{2\mathit{r}}^{\mathit{D}}$ | |
---|---|---|

${v}_{3}^{D}<v$ and ${c}_{2}^{D}<{c}_{r}$ | $\frac{\left(1-{c}_{n}\right)}{2}$ | $0$ |

${v}_{3}^{D}<v$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{2}^{D}$ or ${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{8}^{D}$ | $\frac{2\left({c}_{r}+v\right)+4\left(1-{c}_{n}\right)+\rho \left({c}_{n}-3\right)}{4\left(2-\rho \right)}$ | $0$ |

$0\le \mathrm{v}<{v}_{2}^{D}$ and ${c}_{r}<{c}_{3}^{D}$ or ${v}_{2}^{D}<v$ and ${c}_{r}<{c}_{1}^{D}$ | $\frac{\left(2-\rho \delta \right)\left(1-{c}_{n}\right)}{4}$ | $0$ |

${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{8}^{D}\le {c}_{r}\le {c}_{7}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{5}^{D}\le {c}_{r}\le {c}_{7}^{D}$ | $\frac{1}{2}+\frac{v}{4\left(1-\rho \right)}-\frac{{c}_{n}\left(2-\rho \right)}{4\left(1-\rho \right)}$ | $\frac{{c}_{r}}{2\rho}-\frac{1}{4}+\frac{{c}_{n}-v}{4\left(1-\rho \right)}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{4}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{5}^{D}$ | $-\frac{{c}_{r}}{2\left(2-\rho \right)}+\frac{\left(1-{c}_{n}\right)\left(4-\rho -2\delta \rho \right)}{4\left(2-\rho \right)}$ | $\frac{{c}_{r}}{\rho \left(2-\rho \right)}-\frac{\left(1-\rho \delta \right)\left(1-{c}_{n}\right)}{2\left(2-\rho \right)}$ |

${v}_{1}^{D}\le \mathrm{v}\le {v}_{3}^{D}$ and ${c}_{r}>{c}_{7}^{D}$ | $\frac{1}{2}+\frac{v}{4\left(1-\rho \right)}-\frac{{c}_{n}\left(2-\rho \right)}{4\left(1-\rho \right)}$ | $\frac{{c}_{n}\rho -v}{4\rho \left(1-\rho \right)}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{4}^{D}<{c}_{r}$ | $\frac{\left(1-\rho \delta \right)\left(1-{c}_{n}\right)}{2}$ | $\frac{\delta \left(1-{c}_{n}\right)}{2}$ |

${\mathit{\pi}}_{2\mathit{M}}^{\mathit{D}}$ | ${\mathit{\pi}}_{2\mathit{R}}^{\mathit{D}}$ | |
---|---|---|

${v}_{3}^{D}<v$ and ${c}_{2}^{D}<{c}_{r}$ | $\frac{{\left(1-{c}_{n}\right)}^{2}}{4}$ | $0$ |

${v}_{3}^{D}<v$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{2}^{D}$ or ${v}_{2}^{D}<v\le {v}_{3}^{D}$ and ${c}_{1}^{D}\le {c}_{r}\le {c}_{8}^{D}$ | $\frac{{\left(4-4{c}_{n}-3\rho +2v+2{c}_{r}+{c}_{n}\rho \right)}^{2}}{16{\left(2-\rho \right)}^{2}}$ | $\frac{{\left({c}_{n}\rho +\rho -2v-2{c}_{r}\right)}^{2}}{8\rho \left(2-\rho \right)}$ |

$0\le \mathrm{v}<{v}_{2}^{D}$ and ${c}_{r}<{c}_{3}^{D}$ or ${v}_{2}^{D}<v$ and ${c}_{r}<{c}_{1}^{D}$. | $\frac{{[\left(2-\rho \delta \right)\left(1-{c}_{n}\right)]}^{2}}{16}$ | $\frac{\delta \rho \left(1-{c}_{n}^{2}\right)}{4}-\frac{\rho \left(2-\rho \right){\left[\delta \left(1-{c}_{n}\right)\right]}^{2}}{8}-\frac{\delta \left(1-{c}_{n}\right)\left({c}_{r}+v\right)}{2}$ |

$\begin{array}{l}\left[\frac{2{c}_{r}+\rho -\rho {c}_{n}}{4}\right]\left[\frac{{c}_{r}}{2\rho}-\frac{1}{4}+\frac{{c}_{n}-v}{4\left(1-\rho \right)}\right]\\ +\left[\frac{2+v+2{c}_{r}-2{c}_{n}-\rho}{4}\right][\frac{2+v+\rho {c}_{n}-2{c}_{n}-2\rho}{4\left(1-\rho \right)}]\end{array}$ | $\begin{array}{l}\left[\frac{{c}_{r}}{2\rho}-\frac{1}{4}+\frac{{c}_{n}-v}{4\left(1-\rho \right)}\right]\left[\frac{\rho {c}_{n}-v}{2}\right]\\ +\left[\frac{\rho -2{c}_{r}-v}{2\rho}\right][\frac{{c}_{n}\rho +\rho -2v-2{c}_{r}}{4}]\end{array}$ | |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{4}^{D}$ or ${v}_{1}^{D}\le \mathrm{v}\le {v}_{2}^{D}$ and ${c}_{3}^{D}\le {c}_{r}\le {c}_{5}^{D}$ | $\begin{array}{l}\frac{\left[\left(1-{c}_{n}\right)\left(4-\rho -2\delta \rho \right)-2{c}_{r}\right]\left[2{c}_{r}+\left(1-{c}_{n}\right)\left(4+2\delta {\rho}^{2}-3\rho -2\delta \rho \right)\right]}{16{\left(2-\rho \right)}^{2}}\\ +\frac{\left[2{c}_{r}-\rho \left(1-\rho \delta \right)\left(1-{c}_{n}\right)\right]\left[\rho +2{c}_{r}-{c}_{n}\rho \right]}{8\rho \left(2-\rho \right)}\end{array}$ | $\begin{array}{l}\frac{2{c}_{r}-\rho \left(1-\rho \delta \right)\left(1-{c}_{n}\right)}{2\rho \left(2-\rho \right)}[{c}_{n}\rho -\frac{{c}_{r}\left(1-\rho \right)}{\left(2-\rho \right)}\\ +\frac{\rho \left(1-\rho \right)\left(1-2\delta \right)\left(1-{c}_{n}\right)}{2\left(2-\rho \right)}-v]\\ +\frac{\rho \left(1-{c}_{n}\right)\left(1+2\delta -2\delta \rho \right)-2{c}_{r}}{2\rho \left(2-\rho \right)}[{c}_{n}\rho \\ +\frac{\rho \left(1-{c}_{n}\right)\left(4-4\delta -3\rho +4\delta \rho \right)}{4\left(2-\rho \right)}\\ -\frac{{c}_{r}\left(4-3\rho \right)}{2\left(2-\rho \right)}-v]\end{array}$ |

${v}_{1}^{D}\le \mathrm{v}\le {v}_{3}^{D}$ and ${c}_{r}>{c}_{7}^{D}$ | $\frac{\left({c}_{n}\rho -v\right)\left(2\rho -v-{c}_{n}\rho \right)}{16\rho \left(1-\rho \right)}+\frac{\left(1-{c}_{n}\right)}{2}[\frac{1}{2}+\frac{v}{4\left(1-\rho \right)}-\frac{{c}_{n}\left(2-\rho \right)}{4\left(1-\rho \right)}]$ | $\frac{{\left({c}_{n}\rho -v\right)}^{2}}{8\rho \left(1-\rho \right)}$ |

$0\le \mathrm{v}<{v}_{1}^{D}$ and ${c}_{4}^{D}<{c}_{r}$ | $\frac{\left(1+\rho {\delta}^{2}-{\rho}^{2}{\delta}^{2}\right){\left(1-{c}_{n}\right)}^{2}}{4}$ | $[\frac{\delta \left(1-{c}_{n}\right)}{2}][{c}_{n}\rho -\delta \rho \left(1-\rho \right)\left(1-{c}_{n}\right)-v]$ |

## References

- Guide, V.D.R., Jr.; Li, J. The Potential for Cannibalization of New Products Sales by Remanufactured Products. Decis. Sci.
**2010**, 41, 547–572. [Google Scholar] [CrossRef] - Tripathi, V.; Weilerstein, K.; Mclella, L. Marketing Essentials: What Printer OEMs Must Do to Compete against Low-Cost Remanufactured Supplies; Gartner Inc.: Stamford, CT, USA, 2009; Available online: http://www.gartner.com/id=1035914 (accessed on 23 June 2009).
- Wang, K.; Xiong, Y. Remanufacturer–manufacturer Collaborative Model in the Same Channel under Three Channel Power Structures. Key Eng. Mater.
**2014**, 572, 699–702. [Google Scholar] [CrossRef] - Wang, K.; Xiong, Z.K.; Xiong, Y.; Yan, W. Remanufacturer-Manufacturer Collaboration in a Supply Chain: The Manufacturer Plays the Leader Role. Asia-Pac. J. Oper. Res.
**2015**, 32, 1550040-1–1550040-17. [Google Scholar] [CrossRef] - Savaskan, R.C.; Bhattacharya, S.; van Wassenhove, L.N. Closed-loop Supply Chain Models with Product Remanufacturing. Manag. Sci.
**2004**, 50, 239–252. [Google Scholar] [CrossRef] - Bulmus, S.C.; Zhu, S.X.; Teunter, R. Capacity and Production Decisions under a Remanufacturing Strategy. Int. J. Prod. Econ.
**2013**, 145, 359–370. [Google Scholar] [CrossRef] - Kuhn, H.W.; Tucker, A.W. Nonlinear Programming. In Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 1951; pp. 481–492. [Google Scholar]
- Debo, L.G.; Toktay, L.B.; van Wassenhove, L.N. Market Segmentation and Production Technology Selection for Remanufacturable Products. Manag. Sci.
**2005**, 51, 1193–1205. [Google Scholar] [CrossRef] - Atasu, A.; Guide, V.D., Jr.; van Wassenhove, L.N. Product Reuse Economics in Closed-loop Supply Chain Research. Prod. Oper. Manag.
**2008**, 17, 483–496. [Google Scholar] [CrossRef] - Wu, C.H. OEM Product Design in a Price Competition with Remanufactured Product. Omega
**2013**, 41, 287–298. [Google Scholar] [CrossRef] - Ma, Z.J.; Zhang, N.; Dai, Y.; Hu, S. Managing Channel Profits of Different Cooperative Models in Closed-loop Supply Chains. Omega
**2016**, 59, 251–262. [Google Scholar] - Cerchione, R.; Esposito, E. A Systematic Review of Supply Chain Knowledge Management Research: State of the Art and Research Opportunities. Int. J. Prod. Econ.
**2016**, 182, 276–292. [Google Scholar] [CrossRef] - Centobelli, P.; Cerchione, R.; Esposito, E. Knowledge Management in Startups: Systematic Literature Review and Future Research Agenda. Sustainability
**2017**, 9, 361. [Google Scholar] [CrossRef] - Centobelli, P.; Cerchione, R.; Esposito, E. Environmental Sustainability in the Service Industry of Transportation and Logistics Service Providers: Systematic Literature Review and Research Directions. Transp. Res. Part D
**2017**, 53, 454–470. [Google Scholar] [CrossRef] - Rajeev, A.; Pati, R.K.; Padhi, S.S.; Govindan, K. Evolution of Sustainability in Supply Chain Management: A Literature Review. J. Clean. Prod.
**2017**, 162, 299–314. [Google Scholar] [CrossRef] - Souza, G.C. Closed-loop Supply Chains: A Critical Review, and Future Research. Decis. Sci.
**2013**, 44, 7–38. [Google Scholar] [CrossRef] - Govindan, K.; Soleimani, H.; Kannan, D. Reverse Logistics and Closed-loop Supply Chain: A Comprehensive Review to Explore the Future. Eur. J. Oper. Res.
**2015**, 240, 603–626. [Google Scholar] [CrossRef][Green Version] - Battini, D.; Bogataj, M.; Choudhary, A. Closed Loop Supply Chain (CLSC): Economics, Modeling Management and Control. Int. J. Prod. Econ.
**2017**, 183, 319–321. [Google Scholar] [CrossRef] - Ferrer, G.; Swaminathan, J.M. Managing New and Remanufactured Products. Manag. Sci.
**2006**, 52, 15–26. [Google Scholar] [CrossRef][Green Version] - Ferrer, G.; Swaminathan, J.M. Managing New and Differentiated Remanufactured Products. Eur. J. Oper. Res.
**2010**, 203, 370–379. [Google Scholar] [CrossRef][Green Version] - Ferguson, M.; Toktay, B. The Effect of Competition on Recovery Strategies. Prod. Oper. Manag.
**2006**, 15, 351–368. [Google Scholar] [CrossRef] - Atasu, A.; van Wassenhove, L.N. Environmental Legislation on Product Take-back and Recovery. In Closed-loop Supply Chains: New Developments to Improve the Sustainability of Business Practices; Ferguson, M., Souza, G., Eds.; CRC Press: Boca Raton, FL, USA, 2010; pp. 23–38. [Google Scholar]
- Giovanni, P.D.; Zaccour, G. A Two-period Game of a Closed-loop Supply Chain. Eur. J. Oper. Res.
**2014**, 232, 22–40. [Google Scholar] [CrossRef] - Wu, C.H. Strategic and Operational Decisions under Sales Competition and Collection Competition for End-of-use Products in Remanufacturing. Int. J. Prod. Econ.
**2015**, 169, 11–20. [Google Scholar] [CrossRef] - Gan, S.S.; Pujawan, I.N.; Suparno; Widodo, B. Pricing Decision for New and Remanufactured Product in a Closed-loop Supply Chain with Separate Sales-Channel. Int. J. Prod. Econ.
**2016**, 190, 120–132. [Google Scholar] [CrossRef] - Karakayali, I.; Emir-Farinas, H.; Akcali, E. An Analysis of Decentralized Collection and Processing of End-of-life Products. J. Oper. Manag.
**2007**, 25, 1161–1183. [Google Scholar] [CrossRef] - Jung, K.S.; Hwang, H. Competition and Cooperation in a Remanufacturing System with Take-back Requirement. J. Intell. Manuf.
**2011**, 22, 427–433. [Google Scholar] [CrossRef] - Chen, J.M.; Chang, C.I. The Co-operative Strategy of a Closed-loop Supply Chain with Remanufacturing. Transp. Res. Part E
**2012**, 48, 387–400. [Google Scholar] [CrossRef] - The Scottish Government. Remanufacture, Refurbishment, Reuse and Recycling of Vehicles: Trends and Opportunities; The Scottish Government: Edinburgh, UK, 2013.
- Wu, C.H. Price and Service Competition between New and Remanufactured Products in a Two-echelon Supply Chain. Int. J. Prod. Econ.
**2012**, 140, 496–507. [Google Scholar] [CrossRef] - Wu, C.H. Product-design and Pricing Strategies with Remanufacturing. Eur. J. Oper. Res.
**2012**, 222, 204–215. [Google Scholar] [CrossRef] - Bulmus, S.C.; Zhu, S.X.; Teunter, R. Competition for Cores in Remanufacturing. Eur. J. Oper. Res.
**2014**, 233, 105–113. [Google Scholar] [CrossRef] - Xiong, Y.; Zhao, Q.; Zhou, Y. Manufacturer-remanufacturing vs Supplier-remanufacturing in a Closed-loop Supply Chain. Int. J. Prod. Econ.
**2016**, 176, 21–28. [Google Scholar] [CrossRef] - Miao, Z.; Fu, K.; Xia, Z.; Wang, Y. Models for Closed-loop Supply Chain with Trade-ins. Omega
**2015**, 66, 308–326. [Google Scholar] [CrossRef] - Xiong, Y.; Zhou, Y.; Li, G.; Chan, H.K.; Xiong, Z.K. Don’t Forget Your Supplier When Remanufacturing. Eur. J. Oper. Res.
**2013**, 230, 15–25. [Google Scholar] [CrossRef]

**Figure 3.**The impact of parameter variations on the optimal profits of the overall supply chains in Model I.

**Figure 7.**The impact of parameter variations on manufacturer’s and remanufacturer’s profits in Model D.

Parameters | Meanings |
---|---|

${p}_{1}$ | The price of new products at which the manufacturer sells in the first period |

${q}_{1}$ | The quantity of new products sold by the manufacturer in the first period |

${p}_{2n}$ | The price of new products at which the manufacturer sells in the second period |

${p}_{2r}$ | The price of remanufactured products at which the manufacturer sells in the second period |

${p}_{2i}$ | The price of remanufactured products at which the remanufacturer sells in the second period |

${q}_{2n}$ | The quantity of new products sold by the manufacturer in the second period |

${q}_{2r}$ | The quantity of remanufactured products sold by the manufacturer in the second period |

${q}_{2i}$ | The quantity of remanufactured products sold by the remanufacturer in the second period |

$v$ | The cost of recycling used products made in the first period and the production cost of remanufactured products in the second period |

${c}_{n}$ | The production cost of new products |

${c}_{r}$ | The cost of remanufactured products sold by the remanufacturer |

$\delta $ | Recyclable rate of used products |

$\rho $ | Consumers’ preference for remanufactured products in relation to new products |

${\pi}_{ij}$ | Participants’ profits in each period ( $i=1,2$；$j=M,R$), with $M$ as the manufacturer and $R$ as the remanufacturer |

${\mathit{q}}_{2\mathit{n}}^{\mathit{I}\mathit{a}}$ | ${\mathit{q}}_{2\mathit{n}}^{\mathit{I}\mathit{b}}$ | ${\mathit{q}}_{2\mathit{n}}^{\mathit{I}\mathit{c}}$ | ${\mathit{q}}_{2\mathit{r}}^{\mathit{I}\mathit{a}}$ | ${\mathit{q}}_{2\mathit{r}}^{\mathit{I}\mathit{b}}$ | ${\mathit{q}}_{2\mathit{r}}^{\mathit{I}\mathit{c}}$ | |
---|---|---|---|---|---|---|

${c}_{n}$ | $-$ | $-$ | $-$ | $-$ | $+$ | $0$ |

$v$ | $0$ | $+$ | $0$ | $0$ | $-$ | $0$ |

$\delta $ | $-$ | $0$ | $0$ | $+$ | $0$ | $0$ |

**Table 3.**The impact of parameter variation on the optimal profits of the overall supply chains in Model I.

${\mathit{\pi}}_{2\mathit{M}}^{\mathit{I}\mathit{a}}$ | ${\mathit{\pi}}_{2\mathit{M}}^{\mathit{I}\mathit{b}}$ | ${\mathit{\pi}}_{2\mathit{M}}^{\mathit{I}\mathit{c}}$ | |
---|---|---|---|

${c}_{n}$ | $-$ | $-$ | $-$ |

$v$ | $-$ | $-$ | $0$ |

$\delta $ | $+$ | $0$ | $0$ |

${\mathit{q}}_{2\mathit{n}}^{\mathit{D}}$ | ${\mathit{q}}_{2\mathit{r}}^{\mathit{D}}$ | |
---|---|---|

${w}_{1}^{D}<{w}_{2r}$ | $\frac{1-{c}_{n}-\rho {q}_{2i}}{2}$ | $0$ |

${w}_{3}^{D}\le {w}_{2r}\le {w}_{1}^{D}$ | $\frac{1+{w}_{2r}-{c}_{n}-\rho}{2\left(1-\rho \right)}$ | $-\frac{\rho {q}_{2i}\left(1-\rho \right)+{w}_{2r}-{c}_{n}\rho}{2\rho \left(1-\rho \right)}$ |

${w}_{2r}<{w}_{3}^{D}$ | $\frac{\left(1-\delta \rho \right)\left(1-{c}_{n}\right)}{2}+\frac{\rho {q}_{2i}}{2}$ | $\frac{\delta \left(1-{c}_{n}\right)}{2}-{q}_{2i}$ |

${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{a}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{b}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{c}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{d}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{e}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{f}}$ | ${\mathit{w}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $0$ | $0$ | $0$ | $+$ | $+$ | $+$ | $+$ |

$\mathrm{v}$ | $0$ | $0$ | $0$ | $+$ | $0$ | $+$ | $0$ |

${c}_{r}$ | $0$ | $0$ | $0$ | $0$ | $-$ | $0$ | $0$ |

$\mathsf{\delta}$ | $0$ | $0$ | $0$ | $0$ | $-$ | $0$ | $-$ |

${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{a}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{b}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{c}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{d}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{e}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{f}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{i}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $0$ | $+$ | $-$ | $0$ | $-$ | $0$ | $0$ |

$\mathrm{v}$ | $0$ | $-$ | $0$ | $-$ | $0$ | $0$ | $0$ |

${c}_{r}$ | $0$ | $-$ | $0$ | $-$ | $-$ | $0$ | $0$ |

$\mathsf{\delta}$ | $0$ | $0$ | $+$ | $0$ | $+$ | $0$ | $0$ |

${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{a}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{b}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{c}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{d}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{e}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{f}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{n}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ |

$v$ | $0$ | $+$ | $0$ | $+$ | $0$ | $+$ | $0$ |

${c}_{r}$ | $0$ | $+$ | $0$ | $0$ | $-$ | $0$ | $0$ |

$\delta $ | $0$ | $0$ | $-$ | $0$ | $-$ | $0$ | $-$ |

${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{a}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{b}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{c}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{d}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{e}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{f}}$ | ${\mathit{q}}_{\mathbf{2}\mathit{r}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $0$ | $0$ | $0$ | $+$ | $+$ | $+$ | $-$ |

$v$ | $0$ | $0$ | $0$ | $-$ | $0$ | $-$ | $0$ |

${c}_{r}$ | $0$ | $0$ | $0$ | $+$ | $+$ | $0$ | $0$ |

$\delta $ | $0$ | $0$ | $0$ | $0$ | $+$ | $0$ | $+$ |

**Table 7.**The impact of parameter variations on the manufacturer’s and remanufacturer’s optimal profits in Model D.

${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{a}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{b}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{c}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{d}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{e}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{f}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{M}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ |

$\mathrm{v}$ | $0$ | $+$ | $0$ | $+$ | $0$ | $-$ | $0$ |

${c}_{r}$ | $0$ | $+$ | $0$ | $+$ | $+$ | $0$ | $0$ |

$\mathsf{\delta}$ | $0$ | $0$ | $-$ | $0$ | $-$ | $0$ | $+$ |

${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{a}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{b}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{c}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{d}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{e}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{f}}$ | ${\mathit{\pi}}_{\mathbf{2}\mathit{R}}^{\mathit{D}\mathit{g}}$ | |

${c}_{n}$ | $0$ | $+$ | $-$ | $+$ | $-$ | $+$ | $-$ |

$\mathrm{v}$ | $0$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ |

${c}_{r}$ | $0$ | $-$ | $-$ | $-$ | $-$ | $0$ | $0$ |

$\mathsf{\delta}$ | $0$ | $0$ | $+$ | $0$ | $+$ | $0$ | $+$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Long, X.; Shu, T.; Chen, S.; Wang, S.; Lai, K.K.; Yang, Y. Strategy Analysis of Recycling and Remanufacturing by Remanufacturers in Closed-Loop Supply Chain. *Sustainability* **2017**, *9*, 1818.
https://doi.org/10.3390/su9101818

**AMA Style**

Long X, Shu T, Chen S, Wang S, Lai KK, Yang Y. Strategy Analysis of Recycling and Remanufacturing by Remanufacturers in Closed-Loop Supply Chain. *Sustainability*. 2017; 9(10):1818.
https://doi.org/10.3390/su9101818

**Chicago/Turabian Style**

Long, Xiaofeng, Tong Shu, Shou Chen, Shouyang Wang, Kin Keung Lai, and Yan Yang. 2017. "Strategy Analysis of Recycling and Remanufacturing by Remanufacturers in Closed-Loop Supply Chain" *Sustainability* 9, no. 10: 1818.
https://doi.org/10.3390/su9101818