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January  2019, 15(1): 199-219. doi: 10.3934/jimo.2018039

## Closed-loop supply chain network equilibrium model with retailer-collection under legislation

 1 School of Management, China University of Mining and Technology, Xuzhou 221116, China 2 College of Engineering, Nanjing Agricultural University, Nanjing 210031, China 3 School of Business, Qingdao University, Qingdao 266071, China

* Corresponding author: lijianzh@njau.edu.cn(Jian Li); rivaldoking@126.com(Hao Sun).

Received  May 2017 Revised  August 2017 Published  January 2019 Early access  April 2018

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities of China (Project No. 2017XKQY034).

This paper examines the waste of electrical and electronic equipments (WEEE) and draws on variational inequalities to model the closed-loop supply chain network. The network consists of manufacturers, retailers and consumer markets engaging in a Cournot-Nash game. Retailers are responsible for collecting WEEE in the network. It is assumed that the price of the remanufactured goods is different from that of the newly manufactured ones. The network equilibrium occurs when all players agree on volumes and prices. Several properties of the model are examined and the modified projection method is utilized to obtain the optimal solutions. Numerical examples are provided to illustrate the impact of CLSC parameters on the profits of channel members and consumer benefits, and to provide policy support for governments. We find that it is necessary to regulate a medium collection rate and a certain minimum recovery rate. This is also advantageous to manufacturers in producing new manufactured products. The impact of collection rate and recovery rate on manufacturers are greater than that on retailers. Consumers can benefit from the increase of the recovery rate as well as the collection rate.

Citation: Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039
##### References:

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##### References:
Closed-loop supply chain network with WEEE collected by retailers
The volume changes of three kinds of products with the changes of collection rate while $\beta = 0.5$ are fixed
The volume changes of three kinds of products with the changes of recovery rate while $\alpha = 1$ are fixed
The prices under different collection rate ($\alpha$), given $\beta = 0.5$
 $\alpha=0$ $\alpha=0.2$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $\alpha=1$ $w_i^{f^*}$ 352.8712 352.8712 358.2704 383.6992 414.7223 435.5493 $w_j^{r^*}$ 59.8369 59.8369 71.6460 98.1422 108.6234 106.8103 $w_i^{r^*}$ 384.0070 384.0070 376.6145 362.1067 359.6196 375.6179 $P_j^{f^*}$ 402.8527 402.8527 406.0255 421.8723 441.8835 456.1049 $P_k^{r^*}$ 39.8914 39.8914 47.7639 65.4282 72.4157 71.2072 $P_j^{r^*}$ 393.9832 393.9832 388.5579 378.4611 377.7207 390.6693
 $\alpha=0$ $\alpha=0.2$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $\alpha=1$ $w_i^{f^*}$ 352.8712 352.8712 358.2704 383.6992 414.7223 435.5493 $w_j^{r^*}$ 59.8369 59.8369 71.6460 98.1422 108.6234 106.8103 $w_i^{r^*}$ 384.0070 384.0070 376.6145 362.1067 359.6196 375.6179 $P_j^{f^*}$ 402.8527 402.8527 406.0255 421.8723 441.8835 456.1049 $P_k^{r^*}$ 39.8914 39.8914 47.7639 65.4282 72.4157 71.2072 $P_j^{r^*}$ 393.9832 393.9832 388.5579 378.4611 377.7207 390.6693
The prices under different recovery rate ($\beta$), given $\alpha = 1$
 $\beta=0.1$ $\beta=0.2$ $\beta=0.3$ $\beta=0.4$ $\beta=0.5$ $w_i^{f^*}$ 422.8170 424.1308 427.8478 433.8971 435.5493 $w_j^{r^*}$ 91.0883 97.7935 103.2339 106.4136 106.8103 $w_i^{r^*}$ 424.1402 409.5847 394.2371 378.9959 375.6179 $P_j^{f^*}$ 450.0756 450.2139 451.9405 455.1846 456.1049 $P_k^{r^*}$ 60.7962 65.1957 68.8227 70.9424 71.2072 $P_j^{r^*}$ 427.1726 416.1012 404.5575 393.1814 390.6693
 $\beta=0.1$ $\beta=0.2$ $\beta=0.3$ $\beta=0.4$ $\beta=0.5$ $w_i^{f^*}$ 422.8170 424.1308 427.8478 433.8971 435.5493 $w_j^{r^*}$ 91.0883 97.7935 103.2339 106.4136 106.8103 $w_i^{r^*}$ 424.1402 409.5847 394.2371 378.9959 375.6179 $P_j^{f^*}$ 450.0756 450.2139 451.9405 455.1846 456.1049 $P_k^{r^*}$ 60.7962 65.1957 68.8227 70.9424 71.2072 $P_j^{r^*}$ 427.1726 416.1012 404.5575 393.1814 390.6693
The profits of manufacturers and retailers with different collection rate ($\alpha$), given $\beta = 0.5$
 Total profits $\alpha \le 0.2$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $\alpha=1$ Manufactures 13642.876 17247.346 15531.509 12759.803 10108.842 Retailers 2995.498 2993.549 2794.858 2376.378 1916.658
 Total profits $\alpha \le 0.2$ $\alpha=0.4$ $\alpha=0.6$ $\alpha=0.8$ $\alpha=1$ Manufactures 13642.876 17247.346 15531.509 12759.803 10108.842 Retailers 2995.498 2993.549 2794.858 2376.378 1916.658
The profits of manufacturers and retailers with different recovery rate ($\beta$), given $\alpha = 1$
 Total profits $\beta=0.1$ $\beta=0.2$ $\beta=0.3$ $\beta=0.4$ $\beta\ge0.5$ Manufactures 9082.426 9616.632 9981.219 10112.544 10108.842 Retailers 1669.838 1785.409 1944.052 2113.821 1916.658
 Total profits $\beta=0.1$ $\beta=0.2$ $\beta=0.3$ $\beta=0.4$ $\beta\ge0.5$ Manufactures 9082.426 9616.632 9981.219 10112.544 10108.842 Retailers 1669.838 1785.409 1944.052 2113.821 1916.658
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