# American Institute of Mathematical Sciences

September  2020, 16(5): 2117-2139. doi: 10.3934/jimo.2019046

## Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees

 1 School of Mathematics and Statistics, Central South University, Changsha 410083, China 2 Experimental Teaching Center, Guangdong University of Foreign Studies, Guangzhou 510420, China

* Corresponding authors: Zhong Wan and Jing Zhang

Received  November 2018 Revised  December 2018 Published  May 2019

Fund Project: The first and second authors are supported by the National Science Foundation of China (Grant No. 71671190)

In the past one decade, an increasing number of motor vehicles necessarily results in huge amounts of end-of-life vehicles (ELVs) in the future. From the view point of environment protection and resource utilization, government subsidy and public awareness of environmental protection play a critical role in promoting the formal recycle enterprises to recycle the ELVs as many as possible. Different from the existing similar models, a mixed integer nonlinear optimization model is established in this paper to formulate the management problems of recycling ELVs as a centralized decision-making system, where damaged and aging degrees, correlation between the recycled quantity and take-back price of ELVs, and the public environmental protection awareness are considered. Unlike the results available in the literature, take-back prices of the ELVs are the endogenous variables of the model (decision variables), which affect the collected quantity of ELVs and the profit of recycling system. Additionally, due to distinct damaged and aging degrees of the ELVs, the refurbished or dismantled amounts of ELVs are also regarded as the decision variables so that the recycle system is more applicable. By case study and sensitivity analysis, validity of the model is verified and impacts of the governmental subsidy and environmental awareness are analyzed. By the proposed model, it is revealed that: (1) Distinct treatment of ELVs with different damaged and aging degrees can increase the profit of recycling ELVs; (2) Compared with the transportation cost, higher processing cost is a main obstacle to the profit growth. Advanced processing technology plays the most important role in improving the ELV recovery efficiency. (3) Both of government subsidy and environmental awareness seriously affect decision-making of recycle enterprises.

Citation: Zhong Wan, Jingjing Liu, Jing Zhang. Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2117-2139. doi: 10.3934/jimo.2019046
##### References:

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##### References:
Material flow of the ELV recovery network
The map and the existent ELV recycling network in Hunan
Effect of public environmental protection awareness
Impacts of subsidy
Impact of different types of costs on profit
Impacts of different types of costs on the recycled quantities
Number of different types of nodes in ELV recovery network
 $I$ $J$ $O$ $K$ $L$ $U$ $S$ $P$ $Q$ $R$ $V$ $W$ $M$ $N$ 5 5 2 6 5 2 2 2 2 2 2 2 2 2
 $I$ $J$ $O$ $K$ $L$ $U$ $S$ $P$ $Q$ $R$ $V$ $W$ $M$ $N$ 5 5 2 6 5 2 2 2 2 2 2 2 2 2
Distribution of nodes in ELV recovery network
 Changsha Zhuzhou Xiangtan Hengyang Shaoyang Resource 1 2 3 4 5 Collection center 1 2 3 4 5 Repair center 1, 2 - - - - Dismantler 1 2, 3 4 6 5 Shredder 2 1 3 5 4 Landfill 2 - 1 - - Steel mill 1, 2 - - - - Non-ferrous smeltery 1 - - 2 - Oil factory 1, 2 - - - - Battery factory 1 - - 2 - Rubber factory - 1 - - 2 Glass factory 1, 2 - - - - Plastics factory - - 1 - 2
 Changsha Zhuzhou Xiangtan Hengyang Shaoyang Resource 1 2 3 4 5 Collection center 1 2 3 4 5 Repair center 1, 2 - - - - Dismantler 1 2, 3 4 6 5 Shredder 2 1 3 5 4 Landfill 2 - 1 - - Steel mill 1, 2 - - - - Non-ferrous smeltery 1 - - 2 - Oil factory 1, 2 - - - - Battery factory 1 - - 2 - Rubber factory - 1 - - 2 Glass factory 1, 2 - - - - Plastics factory - - 1 - 2
Distance between the nodes of network (km)
 Collection center 1 2 3 4 5 Resources 1 10 67.4 46.9 152 148.3 2 49.8 20.2 26.9 117 133.6 3 43.7 27.2 13.4 111.4 122.1 4 147.6 100.7 104.8 6.1 75.5 5 170.2 170.5 146.8 109.9 43.5 Repair certer 1 29.4 40.9 28 131 137.6 2 155.9 133.2 120.6 51.8 26.9 Dismantler 1 21.8 59.8 46.4 150.9 153.1 2 60.2 13.8 36.3 116.4 138.7 3 49.8 22.9 31.9 122.2 139.2 4 124.9 64.8 81.8 48.8 105.1 5 148.8 143.3 121.6 84.3 15.5 6 144.2 107.3 103.6 16.6 53.3
 Collection center 1 2 3 4 5 Resources 1 10 67.4 46.9 152 148.3 2 49.8 20.2 26.9 117 133.6 3 43.7 27.2 13.4 111.4 122.1 4 147.6 100.7 104.8 6.1 75.5 5 170.2 170.5 146.8 109.9 43.5 Repair certer 1 29.4 40.9 28 131 137.6 2 155.9 133.2 120.6 51.8 26.9 Dismantler 1 21.8 59.8 46.4 150.9 153.1 2 60.2 13.8 36.3 116.4 138.7 3 49.8 22.9 31.9 122.2 139.2 4 124.9 64.8 81.8 48.8 105.1 5 148.8 143.3 121.6 84.3 15.5 6 144.2 107.3 103.6 16.6 53.3
Distance between the nodes of network (Continued Table 3)
 Shredder Secondary market 1 2 3 4 5 1 2 Dismantler 1 123.2 25.1 39.1 194.6 42.0 10.2 2.3 2 75.3 50.7 33.4 182.8 21.9 39.5 46.7 3 85.8 42.6 27.2 183.0 17.8 28.8 36.1 4 40.4 129.2 87.1 145.2 80.1 111.6 119.8 5 160.2 185.1 128.4 31.7 134.1 155.2 160.8 6 100.7 165.6 110.9 87.1 110.3 140.1 147.7 Landfill 1 148.7 39.4 59.5 204.0 65.3 - - 2 88.1 50.9 14.4 170.2 4.7 - - Steel mill 1 117.8 33.0 29.7 185.1 33.8 - - 2 114.7 33.3 27.2 184.0 30.8 - - Non-ferrous smeltery 1 149.4 33.0 65.3 214.1 69.3 - - 2 75.7 156.5 106.2 113.9 102.7 - -
 Shredder Secondary market 1 2 3 4 5 1 2 Dismantler 1 123.2 25.1 39.1 194.6 42.0 10.2 2.3 2 75.3 50.7 33.4 182.8 21.9 39.5 46.7 3 85.8 42.6 27.2 183.0 17.8 28.8 36.1 4 40.4 129.2 87.1 145.2 80.1 111.6 119.8 5 160.2 185.1 128.4 31.7 134.1 155.2 160.8 6 100.7 165.6 110.9 87.1 110.3 140.1 147.7 Landfill 1 148.7 39.4 59.5 204.0 65.3 - - 2 88.1 50.9 14.4 170.2 4.7 - - Steel mill 1 117.8 33.0 29.7 185.1 33.8 - - 2 114.7 33.3 27.2 184.0 30.8 - - Non-ferrous smeltery 1 149.4 33.0 65.3 214.1 69.3 - - 2 75.7 156.5 106.2 113.9 102.7 - -
Distance between the nodes of network (Continued Table 4)
 Oil Battery Rubber Glass Plastics 1 2 1 2 1 2 1 2 1 2 ND 1 12.8 8.7 6.6 177.7 36.1 200.6 7.6 5.5 43.3 164.6 2 53.4 53.8 42.7 143.54 17.3 191.5 40.6 51.3 29.0 144.2 3 42.8 43.6 32.1 149.3 9.8 191.2 30.1 41.0 24.5 146.3 4 121.9 129.2 115.2 72.9 85.6 157.5 115.3 126.2 81.0 98.2 5 153.6 171.4 157.5 86.9 142.9 39.9 161.9 168.5 128.1 31.5 6 145.1 158.6 143.4 30.7 118.5 100.2 145.7 155.4 107.1 39.8
 Oil Battery Rubber Glass Plastics 1 2 1 2 1 2 1 2 1 2 ND 1 12.8 8.7 6.6 177.7 36.1 200.6 7.6 5.5 43.3 164.6 2 53.4 53.8 42.7 143.54 17.3 191.5 40.6 51.3 29.0 144.2 3 42.8 43.6 32.1 149.3 9.8 191.2 30.1 41.0 24.5 146.3 4 121.9 129.2 115.2 72.9 85.6 157.5 115.3 126.2 81.0 98.2 5 153.6 171.4 157.5 86.9 142.9 39.9 161.9 168.5 128.1 31.5 6 145.1 158.6 143.4 30.7 118.5 100.2 145.7 155.4 107.1 39.8
Capacity (ton) and unit processing cost (yuan RMB/ton)
 ${ca}_{j}$ ${ca}_{k}$ ${ca}_{l}$ ${ca}_{u}$ $pc_{1o}$ $pc_{2o}$ $pc_k$ $pc_l$ $pc_u$ 2000 2000 1500 500 2000 3000 1960 270 500
 ${ca}_{j}$ ${ca}_{k}$ ${ca}_{l}$ ${ca}_{u}$ $pc_{1o}$ $pc_{2o}$ $pc_k$ $pc_l$ $pc_u$ 2000 2000 1500 500 2000 3000 1960 270 500
Unit transportation cost (yuan RMB/ton·km)
 $tc_{ij}$ $tc_{jo}$ $tc_{jk}$ $tc_{kl}$ $tc_{lu}$ $tc_{ks}$ $tc_{kp}$ $tc_{lm}$ $tc_{ln}$ $tc_{kq}$ $tc_{kr}$ $tc_{kv}$ $tc_{kw}$ 2 1 0.8 0.4 1 1.5 0.7 0.6 0.5 0.5 0.5 0.7 0.7
 $tc_{ij}$ $tc_{jo}$ $tc_{jk}$ $tc_{kl}$ $tc_{lu}$ $tc_{ks}$ $tc_{kp}$ $tc_{lm}$ $tc_{ln}$ $tc_{kq}$ $tc_{kr}$ $tc_{kv}$ $tc_{kw}$ 2 1 0.8 0.4 1 1.5 0.7 0.6 0.5 0.5 0.5 0.7 0.7
Unit selling prices of recyled components (×103 yuan RMB/ton)
 $s$ $s_{0}$ $s_{1}$ $s_{2}$ $s_{3}$ $s_{4}$ $s_{5}$ $s_{6}$ $s_{7}$ $z_{1}$ $z_{2}$ 3000 50000 2400 12000 4000 600 150 450 6000 500 1500 $s^{'}_{1}$ $s^{'}_{2}$ $s^{'}_{3}$ $s^{'}_{4}$ $s^{'}_{5}$ $s^{'}_{6}$ $s^{'}_{7}$ $z^{'}_{1}$ $z^{'}_{2}$ 27360 136800 45600 6840 17100 5130 68400 5700 17100
 $s$ $s_{0}$ $s_{1}$ $s_{2}$ $s_{3}$ $s_{4}$ $s_{5}$ $s_{6}$ $s_{7}$ $z_{1}$ $z_{2}$ 3000 50000 2400 12000 4000 600 150 450 6000 500 1500 $s^{'}_{1}$ $s^{'}_{2}$ $s^{'}_{3}$ $s^{'}_{4}$ $s^{'}_{5}$ $s^{'}_{6}$ $s^{'}_{7}$ $z^{'}_{1}$ $z^{'}_{2}$ 27360 136800 45600 6840 17100 5130 68400 5700 17100
Weight percentages in the recycled ELVs
 $alpha$ $beta_{1}$ $beta_{2}$ $beta_{3}$ $beta_{4}$ $beta_{5}$ $beta_{6}$ $beta_{7}$ $eta$ $eta_{1}$ $eta_{2}$ 0.81 0.06 0.04 0.017 0.013 0.03 0.015 0.015 15/81 62/81 4/81
 $alpha$ $beta_{1}$ $beta_{2}$ $beta_{3}$ $beta_{4}$ $beta_{5}$ $beta_{6}$ $beta_{7}$ $eta$ $eta_{1}$ $eta_{2}$ 0.81 0.06 0.04 0.017 0.013 0.03 0.015 0.015 15/81 62/81 4/81
Optimal solution in case study
 DV OS DV OS DV OS DV OS DV OS $\rho_{1,1}$ 15000 $A_{3,1,3}$ 119 $E_{3,2,5}$ 468.2 $Q3_{3,1}$ 92.8 $Q6_{2,1}$ 8.7 $\rho_{1,2}$ 15000 $A_{3,2,2}$ 578 $E_{3,3,5}$ 442.3 $Q3_{5,1}$ 106.4 $Q6_{3,1}$ 8.2 $\rho_{1,3}$ 15000 $A_{3,2,3}$ 9 $E_{3,5,3}$ 507.1 $Q3_{6,1}$ 111.5 $Q6_{5,1}$ 9.4 $\rho_{1,4}$ 15000 $A_{3,3,3}$ 418 $E_{3,6,3}$ 531.4 $Q3^{'}_{1,2}$ 34 $Q6_{6,1}$ 9.9 $\rho_{1,5}$ 14900 $A_{3,4,4}$ 656 $F_{3,2}$ 352.7 $Q3^{'}_{2,1}$ 34 $Q6^{'}_{1,2}$ 3 $\rho_{2,1}$ 10000 $A_{3,5,5}$ 626 $F_{5,2}$ 228.4 $Q3^{'}_{3,1}$ 34 $Q6^{'}_{2,1}$ 3 $\rho_{2,2}$ 10000 $B_{1,1,1}$ 150 $Q1_{1,2}$ 40.3 $Q3^{'}_{6,1}$ 34 $Q6^{'}_{3,1}$ 3 $\rho_{2,3}$ 10000 $B_{1,2,1}$ 150 $Q1_{2,1}$ 34.7 $Q4_{1,1}$ 8.7 $Q6^{'}_{6,1}$ 3 $\rho_{2,4}$ 10000 $B_{1,3,1}$ 150 $Q1_{3,1}$ 32.8 $Q4_{2,1}$ 7.5 $Q7_{1,1}$ 10.1 $\rho_{2,5}$ 9950 $B_{1,4,2}$ 150 $Q1_{5,1}$ 37.6 $Q4_{3,1}$ 7.1 $Q7_{2,1}$ 8.7 $\rho_{3,1}$ 501.4 $B_{1,5,2}$ 149 $Q1_{6,1}$ 39.4 $Q4_{5,2}$ 8.1 $Q7_{3,1}$ 8.19 $\rho_{3,2}$ 577.1 $B_{2,5,2}$ 199 $Q1^{'}_{1,2}$ 12 $Q4_{6,2}$ 8.5 $Q7_{5,2}$ 9.4 $\rho_{3,3}$ 600 $C_{2,1,1}$ 200 $Q1^{'}_{2,1}$ 12 $Q4^{'}_{1,1}$ 2.6 $Q7_{6,2}$ 9.84 $\rho_{3,4}$ 474.3 $C_{2,2,2}$ 200 $Q1^{'}_{3,1}$ 12 $Q4^{'}_{2,1}$ 2.6 $Q7^{'}_{1,1}$ 3 $\rho_{3,5}$ 435.7 $C_{2,3,3}$ 200 $Q1^{'}_{6,1}$ 12 $Q4^{'}_{3,1}$ 2.6 $Q7^{'}_{2,1}$ 3 $A_{1,1,1}$ 150 $C_{2,4,6}$ 200 $Q2_{1,2}$ 26.9 $Q4^{'}_{6,2}$ 2.6 $Q7^{'}_{3,1}$ 3 $A_{1,2,2}$ 150 $C_{3,1,1}$ 672 $Q2_{2,1}$ 23.1 $Q5_{1,1}$ 20.16 $Q7^{'}_{6,2}$ 3 $A_{1,3,3}$ 150 $C_{3,2,2}$ 578 $Q2_{3,1}$ 21.8 $Q5_{2,1}$ 17.3 $Q8_{3,2}$ 1210.8 $A_{1,4,4}$ 150 $C_{3,3,3}$ 546 $Q2_{5,1}$ 25.0 $Q5_{3,1}$ 16.4 $Q8_{5,2}$ 696.5 $A_{1,5,5}$ 149 $C_{3,4,6}$ 656 $Q2_{6,1}$ 26.2 $Q5_{5,2}$ 18.8 $Q8^{'}_{3,2}$ 247.9 $A_{2,1,1}$ 200 $C_{3,5,5}$ 626 $Q2^{'}_{1,2}$ 8 $Q5_{6,2}$ 19.7 $Q8^{'}_{5,2}$ 247.9 $A_{2,2,2}$ 200 $E_{2,1,3}$ 162 $Q2^{}_{2,1}$ 8 $Q5^{'}_{1,1}$ 6 $Q9_{3,1}$ 79.1 $A_{2,3,3}$ 200 $E_{2,2,5}$ 162 $Q2^{'}_{3,1}$ 8 $Q5^{'}_{2,1}$ 6 $Q9_{5,1}$ 45.5 $A_{2,4,4}$ 200 $E_{2,3,5}$ 162 $Q2^{'}_{6,1}$ 8 $Q5^{'}_{3,1}$ 6 $Q9^{'}_{3,1}$ 16.2 $A_{2,5,5}$ 199 $E_{2,6,3}$ 162 $Q3_{1,2}$ 114.2 $Q5^{'}_{6,2}$ 6 $Q9^{'}_{5,1}$ 16.2 $A_{3,1,1}$ 672 $E_{3,1,3}$ 544.3 $Q3_{2,1}$ 98.3 $Q6_{1,2}$ 10.1
 DV OS DV OS DV OS DV OS DV OS $\rho_{1,1}$ 15000 $A_{3,1,3}$ 119 $E_{3,2,5}$ 468.2 $Q3_{3,1}$ 92.8 $Q6_{2,1}$ 8.7 $\rho_{1,2}$ 15000 $A_{3,2,2}$ 578 $E_{3,3,5}$ 442.3 $Q3_{5,1}$ 106.4 $Q6_{3,1}$ 8.2 $\rho_{1,3}$ 15000 $A_{3,2,3}$ 9 $E_{3,5,3}$ 507.1 $Q3_{6,1}$ 111.5 $Q6_{5,1}$ 9.4 $\rho_{1,4}$ 15000 $A_{3,3,3}$ 418 $E_{3,6,3}$ 531.4 $Q3^{'}_{1,2}$ 34 $Q6_{6,1}$ 9.9 $\rho_{1,5}$ 14900 $A_{3,4,4}$ 656 $F_{3,2}$ 352.7 $Q3^{'}_{2,1}$ 34 $Q6^{'}_{1,2}$ 3 $\rho_{2,1}$ 10000 $A_{3,5,5}$ 626 $F_{5,2}$ 228.4 $Q3^{'}_{3,1}$ 34 $Q6^{'}_{2,1}$ 3 $\rho_{2,2}$ 10000 $B_{1,1,1}$ 150 $Q1_{1,2}$ 40.3 $Q3^{'}_{6,1}$ 34 $Q6^{'}_{3,1}$ 3 $\rho_{2,3}$ 10000 $B_{1,2,1}$ 150 $Q1_{2,1}$ 34.7 $Q4_{1,1}$ 8.7 $Q6^{'}_{6,1}$ 3 $\rho_{2,4}$ 10000 $B_{1,3,1}$ 150 $Q1_{3,1}$ 32.8 $Q4_{2,1}$ 7.5 $Q7_{1,1}$ 10.1 $\rho_{2,5}$ 9950 $B_{1,4,2}$ 150 $Q1_{5,1}$ 37.6 $Q4_{3,1}$ 7.1 $Q7_{2,1}$ 8.7 $\rho_{3,1}$ 501.4 $B_{1,5,2}$ 149 $Q1_{6,1}$ 39.4 $Q4_{5,2}$ 8.1 $Q7_{3,1}$ 8.19 $\rho_{3,2}$ 577.1 $B_{2,5,2}$ 199 $Q1^{'}_{1,2}$ 12 $Q4_{6,2}$ 8.5 $Q7_{5,2}$ 9.4 $\rho_{3,3}$ 600 $C_{2,1,1}$ 200 $Q1^{'}_{2,1}$ 12 $Q4^{'}_{1,1}$ 2.6 $Q7_{6,2}$ 9.84 $\rho_{3,4}$ 474.3 $C_{2,2,2}$ 200 $Q1^{'}_{3,1}$ 12 $Q4^{'}_{2,1}$ 2.6 $Q7^{'}_{1,1}$ 3 $\rho_{3,5}$ 435.7 $C_{2,3,3}$ 200 $Q1^{'}_{6,1}$ 12 $Q4^{'}_{3,1}$ 2.6 $Q7^{'}_{2,1}$ 3 $A_{1,1,1}$ 150 $C_{2,4,6}$ 200 $Q2_{1,2}$ 26.9 $Q4^{'}_{6,2}$ 2.6 $Q7^{'}_{3,1}$ 3 $A_{1,2,2}$ 150 $C_{3,1,1}$ 672 $Q2_{2,1}$ 23.1 $Q5_{1,1}$ 20.16 $Q7^{'}_{6,2}$ 3 $A_{1,3,3}$ 150 $C_{3,2,2}$ 578 $Q2_{3,1}$ 21.8 $Q5_{2,1}$ 17.3 $Q8_{3,2}$ 1210.8 $A_{1,4,4}$ 150 $C_{3,3,3}$ 546 $Q2_{5,1}$ 25.0 $Q5_{3,1}$ 16.4 $Q8_{5,2}$ 696.5 $A_{1,5,5}$ 149 $C_{3,4,6}$ 656 $Q2_{6,1}$ 26.2 $Q5_{5,2}$ 18.8 $Q8^{'}_{3,2}$ 247.9 $A_{2,1,1}$ 200 $C_{3,5,5}$ 626 $Q2^{'}_{1,2}$ 8 $Q5_{6,2}$ 19.7 $Q8^{'}_{5,2}$ 247.9 $A_{2,2,2}$ 200 $E_{2,1,3}$ 162 $Q2^{}_{2,1}$ 8 $Q5^{'}_{1,1}$ 6 $Q9_{3,1}$ 79.1 $A_{2,3,3}$ 200 $E_{2,2,5}$ 162 $Q2^{'}_{3,1}$ 8 $Q5^{'}_{2,1}$ 6 $Q9_{5,1}$ 45.5 $A_{2,4,4}$ 200 $E_{2,3,5}$ 162 $Q2^{'}_{6,1}$ 8 $Q5^{'}_{3,1}$ 6 $Q9^{'}_{3,1}$ 16.2 $A_{2,5,5}$ 199 $E_{2,6,3}$ 162 $Q3_{1,2}$ 114.2 $Q5^{'}_{6,2}$ 6 $Q9^{'}_{5,1}$ 16.2 $A_{3,1,1}$ 672 $E_{3,1,3}$ 544.3 $Q3_{2,1}$ 98.3 $Q6_{1,2}$ 10.1
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