American Institute of Mathematical Sciences

Advanced
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

Zhong Wan 1,, , Jingjing Liu 1, and  Jing Zhang 2,,

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  September 2020 Early access  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.

Keywords: Optimization model, end-of-life vehicles, government subsidy, branch-and-bound algorithm.
Mathematics Subject Classification: Primary: 90B06, 62K05; Secondary: 68T37.
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:
[1]

J. P. Arnaout, G. Arnaout and J. Khoury, Simulation and optimization of ant colony optimization algorithm for the stochastic uncapacitated location-allocation problem, Journal of Industrial and Management Optimization, 12 (2016), 1215-1225.  doi: 10.3934/jimo.2016.12.1215.  Google Scholar

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[4]

X. R. Chen, Y. M. Liu and Z. Wan, Optimal decision-making for the online and offline retailers under BOPS model, The ANZIAM Journal, 58 (2016), 187-208.  doi: 10.1017/S1446181116000201.  Google Scholar

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J. Y. Li, X. Hu and Z. Wan, An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints, Journal of Industrial and Management Optimization, online first (2018). doi: 10.3934/jimo.2018200.  Google Scholar

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Y. X. Li, Z. Wan and J. J. Liu, Bi-level programming approach to optimal strategy for VMI problems under random demand, The ANZIAM Journal, 59 (2017), 247-270.  doi: 10.1017/S1446181117000384.  Google Scholar

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V. Simic, Interval-parameter chance-constraint programming model for end-of-life vehicles management under rigorous environmental regulations, Waste Management, 52 (2016), 180-192.  doi: 10.1016/j.wasman.2016.03.044.  Google Scholar

[17]

K. Subulan, A. S. Taşan and A. Baykasoğlu, A fuzzy goal programming model to strategic planning problem of a lead/acid battery closed-loop supply chain, Journal of Manufacturing Systems, 37 (2015), 243-264.  doi: 10.1016/j.jmsy.2014.09.001.  Google Scholar

[18]

Q. Sun, C. Wang, L. S. Zuo and F. H. Lu, Digital empowerment in a WEEE collection business ecosystem: A comparative study of two typical cases in China, Journal of Cleaner Production, 184 (2018), 414-422.  doi: 10.1016/j.jclepro.2018.02.114.  Google Scholar

[19]

Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028.  Google Scholar

[20]

Z. Wan, S. J. Zhang and K. L. Teo, Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving, Optimization and Engineering, 15 (2014), 267-292.  doi: 10.1007/s11081-012-9205-3.  Google Scholar

[21]

Y. X. Wang, X. Y. Chang, Z. G. Chen, Y. G. Zhong and T. J. Fan, Impact of subsidy policies on recycling and remanufacturing using system dynamics methodology: a case of auto parts in China, Journal of Cleaner Production, 74 (2014), 161-171.  doi: 10.1016/j.jclepro.2014.03.023.  Google Scholar

[22]

X. Wang and M. Chen, Implementing extended producer responsibility: Vehicle remanufacturing in China, Journal of Cleaner Production, 19 (2011), 680-686.   Google Scholar

[23]

L. Wang and M. Chen, Policies and perspective on end-of-life vehicles in China, Journal of Cleaner Production, 44 (2013), 168-176.  doi: 10.1016/j.jclepro.2012.11.036.  Google Scholar

[24]

H. Wu and Z. Wan, A multi-objective optimization model and orthogonal design based hybrid heuristic algorithm for regional urban mining management problems., Journal of the Air and Waste Management Association, 68 (2018), 146-169.   Google Scholar

[25]

T. Z. Zhang, J. W. Chu, X. P. Wang, X. H. Liu and P. F. Cui, Development pattern and enhancing system of automotive components remanufacturing industry in China, Resources, Conservation and Recycling, 55 (2011), 613-622.  doi: 10.1016/j.resconrec.2010.09.015.  Google Scholar

[26]

X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

[27]

Q. H. Zhao and M. Chen, A comparison of ELV recycling system in China and Japan and China's strategies, Resources, Conservation and Recycling, 57 (2011), 15-21.  doi: 10.1016/j.resconrec.2011.09.010.  Google Scholar

show all references

References:
[1]

J. P. Arnaout, G. Arnaout and J. Khoury, Simulation and optimization of ant colony optimization algorithm for the stochastic uncapacitated location-allocation problem, Journal of Industrial and Management Optimization, 12 (2016), 1215-1225.  doi: 10.3934/jimo.2016.12.1215.  Google Scholar

[2]

P. Ávila-Torres, F. López-Irarragorri, R. Caballero and et al., The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty Journal of Industrial and Management Optimization, 14 (2018), 447-472. doi: 10.3934/jimo.2017055.  Google Scholar

[3]

J. B. Bahersa and J. Kim, Regional approach of waste electrical and electronic equipment (WEEE) management in France, Resources, Conservation and Recycling, 129 (2018), 45-55.  doi: 10.1016/j.resconrec.2017.10.016.  Google Scholar

[4]

X. R. Chen, Y. M. Liu and Z. Wan, Optimal decision-making for the online and offline retailers under BOPS model, The ANZIAM Journal, 58 (2016), 187-208.  doi: 10.1017/S1446181116000201.  Google Scholar

[5]

E. Demirel, N. Demirel and H. Gökçen, A mixed integer linear programming model to optimize reverse logistics activities of end-of-life vehicles in Turkey, Journal of Cleaner Production, 112 (2016), 2101-2113.  doi: 10.1016/j.jclepro.2014.10.079.  Google Scholar

[6]

S. H. Hu and Z. G. Wen, Monetary evaluation of end-of-life vehicle treatment from a social perspective for different scenarios in China, Journal of Cleaner Production, 159 (2017), 257-270.  doi: 10.1016/j.jclepro.2017.05.042.  Google Scholar

[7]

S. T. John, R. Sridharan, P. N. Ram Kumar and M. Krishnamoorthy, Multi-period reverse logistics network design for used refrigerators., Applied Mathematical Modelling, 54 (2018), 311-331.  doi: 10.1016/j.apm.2017.09.053.  Google Scholar

[8]

O. Kaya and B. Urek, A mixed integer nonlinear programming model and heuristic solutions for location, inventory and pricing decisions in a closed loop supply chain, Computers and Operations Research, 65 (2016), 93-103.  doi: 10.1016/j.cor.2015.07.005.  Google Scholar

[9]

W. C. Li, H. T. Bai, J. F. Yin and H. Xu, Life cycle assessment of end-of-life vehicle recycling processes in China-take Corolla taxis for example, Journal of Cleaner Production, 117 (2016), 176-187.  doi: 10.1016/j.jclepro.2016.01.025.  Google Scholar

[10]

J. Y. Li, X. Hu and Z. Wan, An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints, Journal of Industrial and Management Optimization, online first (2018). doi: 10.3934/jimo.2018200.  Google Scholar

[11]

Y. X. Li, Z. Wan and J. J. Liu, Bi-level programming approach to optimal strategy for VMI problems under random demand, The ANZIAM Journal, 59 (2017), 247-270.  doi: 10.1017/S1446181117000384.  Google Scholar

[12]

H. H. Liu, M. Lei, H. H. Deng, G. K. Leong and T. Huang, A dual channel, quality-based price competition model for the WEEE recycling market with government subsidy, Omega, 59 (2016), 290-302.  doi: 10.1016/j.omega.2015.07.002.  Google Scholar

[13]

Y. X. Pan and H. T. Li, Sustainability evaluation of end-of-life vehicle recycling based on emergy analysis: a case study of an end-of-life vehicle recycling enterprise in China, Journal of Cleaner Production, 131 (2016), 219-227.  doi: 10.1016/j.jclepro.2016.05.045.  Google Scholar

[14]

P. N. K. Phuc, V. F. Yu and Y. C. Tsao, Optimizing fuzzy reverse supply chain for end-of-life vehicles, Computers and Industrial Engineering, 113 (2017), 757-765.  doi: 10.1016/j.cie.2016.11.007.  Google Scholar

[15]

I. Polak and N. Privault, A stochastic newsvendor game with dynamic retail prices, Journal of Industrial and Management Optimization, 14 (2018), 731-742.  doi: 10.3934/jimo.2017072.  Google Scholar

[16]

V. Simic, Interval-parameter chance-constraint programming model for end-of-life vehicles management under rigorous environmental regulations, Waste Management, 52 (2016), 180-192.  doi: 10.1016/j.wasman.2016.03.044.  Google Scholar

[17]

K. Subulan, A. S. Taşan and A. Baykasoğlu, A fuzzy goal programming model to strategic planning problem of a lead/acid battery closed-loop supply chain, Journal of Manufacturing Systems, 37 (2015), 243-264.  doi: 10.1016/j.jmsy.2014.09.001.  Google Scholar

[18]

Q. Sun, C. Wang, L. S. Zuo and F. H. Lu, Digital empowerment in a WEEE collection business ecosystem: A comparative study of two typical cases in China, Journal of Cleaner Production, 184 (2018), 414-422.  doi: 10.1016/j.jclepro.2018.02.114.  Google Scholar

[19]

Z. Wan, H. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Applied Mathematical Modelling, 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028.  Google Scholar

[20]

Z. Wan, S. J. Zhang and K. L. Teo, Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving, Optimization and Engineering, 15 (2014), 267-292.  doi: 10.1007/s11081-012-9205-3.  Google Scholar

[21]

Y. X. Wang, X. Y. Chang, Z. G. Chen, Y. G. Zhong and T. J. Fan, Impact of subsidy policies on recycling and remanufacturing using system dynamics methodology: a case of auto parts in China, Journal of Cleaner Production, 74 (2014), 161-171.  doi: 10.1016/j.jclepro.2014.03.023.  Google Scholar

[22]

X. Wang and M. Chen, Implementing extended producer responsibility: Vehicle remanufacturing in China, Journal of Cleaner Production, 19 (2011), 680-686.   Google Scholar

[23]

L. Wang and M. Chen, Policies and perspective on end-of-life vehicles in China, Journal of Cleaner Production, 44 (2013), 168-176.  doi: 10.1016/j.jclepro.2012.11.036.  Google Scholar

[24]

H. Wu and Z. Wan, A multi-objective optimization model and orthogonal design based hybrid heuristic algorithm for regional urban mining management problems., Journal of the Air and Waste Management Association, 68 (2018), 146-169.   Google Scholar

[25]

T. Z. Zhang, J. W. Chu, X. P. Wang, X. H. Liu and P. F. Cui, Development pattern and enhancing system of automotive components remanufacturing industry in China, Resources, Conservation and Recycling, 55 (2011), 613-622.  doi: 10.1016/j.resconrec.2010.09.015.  Google Scholar

[26]

X. B. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

[27]

Q. H. Zhao and M. Chen, A comparison of ELV recycling system in China and Japan and China's strategies, Resources, Conservation and Recycling, 57 (2011), 15-21.  doi: 10.1016/j.resconrec.2011.09.010.  Google Scholar

Figure 1.  Material flow of the ELV recovery network
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Figure 2.  The map and the existent ELV recycling network in Hunan
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Figure 3.  Effect of public environmental protection awareness
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Figure 4.  Impacts of subsidy
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Figure 5.  Impact of different types of costs on profit
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Figure 6.  Impacts of different types of costs on the recycled quantities
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Table 1.  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
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$ 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
Table 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
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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
Table 3.  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
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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
Table 4.  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 - -
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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 - -
Table 5.  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
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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
Table 6.  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
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$ {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
Table 7.  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
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$tc_{ij}$ $ 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
Table 8.  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
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$ 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
Table 9.  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
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$alpha$ $beta_{1}$ $beta_{2}$ $beta_{3}$ $beta_{4}$ $beta_{5}$ $beta_{6}$ $beta_{7}$ $eta$ $eta_{1}$ $eta_{2}$
0.81 0.06 0.04 0.017 0.013 0.03 0.015 0.015 15/81 62/81 4/81
Table 10.  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
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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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