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doi: 10.3934/jimo.2021051

Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences

Shoufeng Ji 1, , Jinhuan Tang 2, , Minghe Sun 3, and  Rongjuan Luo 1,,

1. 

School of Business Administration, Northeastern University, Shenyang 110167, China
2. 

School of Economics and Management, Shenyang Aerospace University, Shenyang 110136, China
3. 

Department of Management Science and Statistics, University of Texas at San Antonio, San Antonio, TX 78249-0632, USA

* Corresponding author: Dr Rongjuan Luo

Received  July 2020 Revised  January 2021 Published  March 2021

Fund Project: This research was supported by National Science Foundation of China [grant numbers 71572031, 71971049 and 71702112]

A combined location-routing-inventory system (CLRIS) in a three-echelon supply chain network is studied with environmental considerations. Specifically, a bi-objective mixed integer programming model is formulated for the CLRIS to deal with the trade-offs between the total cost and the carbon-capped difference (CCD). A multi-objective particle swarm optimization (MOPSO) heuristic solution procedure is developed and implemented to solve the bi-objective mixed integer programming problem. The bi-objective mixed integer programming model and the MOPSO heuristic procedure are applied to a real-life problem as an illustrative example. The approximate nondominated frontier formed by solutions not dominated by others can be used for the decision makers to make trade-offs between the total cost and the CCD. Sensitivity analyses are conducted, and the relationship among the carbon cap, CCD, the total cost and the carbon prices are examined, and relevant managerial insights are provided. Comparisons with other existing algorithms show that the MSPSO heuristic procedure has very good performance.

Keywords: Location-routing-inventory, Carbon emissions, Carbon-capped difference, MOPSO heuristic procedure, Nondominated frontier.
Mathematics Subject Classification: Primary: 65K05; Secondary: 90C08.
Citation: Shoufeng Ji, Jinhuan Tang, Minghe Sun, Rongjuan Luo. Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021051
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show all references

References:
[1]

M. A. Alhaj, D. Svetinovic and A. Diabat, A carbon-sensitive two-echelon-inventory supply chain model with stochastic demand, Resources Conservation & Recycling, 108 (2016), 82-87.  doi: 10.1016/j.resconrec.2015.11.011.  Google Scholar

[2]

M. J. Amoshahy, M. Shamsi and M. H. Sedaaghi, A novel flexible inertia weight particle swarm optimization algorithm, PLoS One, 11 (2016), e0161558. doi: 10.1371/journal.pone.0161558.  Google Scholar

[3]

Z. N. Ansari and R. Kant, A state-of-art literature review reflecting 15 years of focus on sustainable supply chain management, Journal of Cleaner Production, 142 (2017), 2524-2543.  doi: 10.1016/j.jclepro.2016.11.023.  Google Scholar

[4]

V. Artale, C. L. Milazzo, C. Orlando and A. Ricciardello, Comparison of GA and PSO approaches for the direct and LQR tuning of a multirotor PD controller, Journal of Industrial & Management Optimization, 13 (2017), 2067-2091. doi: 10.3934/jimo.2017032.  Google Scholar

[5]

Q. Bai, J. Xu, F. Meng and N. Yu, Impact of cap-and-trade regulation on coordinating perishable products supply chain with cost learning, Journal of Industrial & Management Optimization, 2020. doi: 10.3934/jimo.2020126.  Google Scholar

[6]

J. C. Bansal, P. Singh, M. Saraswat, A. Verma, S. S. Jadon and A. Abraham, Inertia weight strategies in particle swarm optimization, in Third World Congress on Nature and Biologically Inspired Computing, Salamanca, 2011,633-640. doi: 10.1109/NaBIC.2011.6089659.  Google Scholar

[7]

E. Bazan, M. Y. Jaber and S. Zanoni, Carbon emissions and energy effects on a two-level manufacturer-retailer closed-loop supply chain model with remanufacturing subject to different coordination mechanisms, International Journal of Production Economics, 183 (2017), 394-408.  doi: 10.1016/j.ijpe.2016.07.009.  Google Scholar

[8]

S. Benjaafar, Y. Li and M. Daskin, Carbon footprint and the management of supply chains: Insights from simple models, IEEE Transactions on Automation Science and Engineering, 10 (2012), 99-116. doi: 10.1109/TASE.2012.2203304.  Google Scholar

[9]

J. Cong, T. Pang and H. Peng, Optimal strategies for capital constrained low-carbon supply chains under yield uncertainty, Journal of Cleaner Production, 256 (2020), 120339. doi: 10.1016/j.jclepro.2020.120339.  Google Scholar

[10]

Y. Daryanto, H. M. Wee and R. D. Astanti, Three-echelon supply chain model considering carbon emission and item deterioration, Transportation Research Part E: Logistics and Transportation Review, 122 (2019), 368-383.  doi: 10.1016/j.tre.2018.12.014.  Google Scholar

[11]

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[12]

M. Desrochers and G. Laporte, Improvements and extensions to the miller-tucker-zemlin subtour elimination constraints, Operations Research Letters, 10 (1991), 27-36. doi: 10.1016/0167-6377(91)90083-2.  Google Scholar

[13]

S. Du, J. Zhu, H. Jiao and W. Ye, Game-theoretical analysis for supply chain with consumer preference to low carbon, International Journal of Production Research, 53 (2015), 3753-3768. doi: 10.1080/00207543.2014.988888.  Google Scholar

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[16]

A. Ghorbani and M. R. A. Jokar, A hybrid imperialist competitive-simulated annealing algorithm for a multisource multi-product location-routing-inventory problem, Computers & Industrial Engineering, 101 (2016), 116-127.  doi: 10.1016/j.cie.2016.08.027.  Google Scholar

[17]

K. Hoen, T. Tan, J. Fransoo and G. Van Houtum, Effect of carbon emission regulations on transport mode selection under stochastic demand, Flexible Services and Manufacturing Journal, 26 (2014), 170-195. doi: 10.1007/s10696-012-9151-6.  Google Scholar

[18]

Y.-S. Huang, C. C. Fang and Y. A. Lin, Inventory management in supply chains with consideration of logistics, green investment and different carbon emissions policies, Computers & Industrial Engineering, 139 (2020), 106207. doi: 10.1016/j.cie.2019.106207.  Google Scholar

[19]

M. Y. Jaber, C. H. Glock and A. M. El. Saadany, Supply chain coordination with emissions reduction incentives, International Journal of Production Research, 51 (2013), 69-82. doi: 10.1080/00207543.2011.651656.  Google Scholar

[20]

A. A. Javid and N. Azad, Incorporating location, routing and inventory decisions in supply chain network design, Transportation Research Part E: Logistics and Transportation Review, 46 (2010), 582-597. Google Scholar

[21]

S. F. Ji, R. J. Luo and X. S. Peng, A probability guided evolutionary algorithm for multi-objective green express cabinet assignment in urban last-mile logistics, International Journal of Production Research, 57 (2019), 3382-3404. doi: 10.1080/00207543.2018.1533653.  Google Scholar

[22]

J. Kennedy and R. Eberhart, Particle swarm optimization, in, IEEE International Conference on Neural Networks, Perth, Australia, 1995, 1942-1948. doi: 10.1109/ICNN.1995.488968.  Google Scholar

[23]

J. S. L. Lam and Y. Gu, A market-oriented approach for intermodal network optimisation meeting cost, time and environmental requirements, International Journal of Production Economics, 171 (2016), 266-274.  doi: 10.1016/j.ijpe.2015.09.024.  Google Scholar

[24]

H. F. Ling, X. Z. Zhou, X. L. Jiang and Y. H. Xiao, Improved constrained multi-objective particle swarm optimization algorithm, Journal of Computer Applications, 32 (2012), 1320-1324. doi: 10.3724/SP.J.1087.2012.01320.  Google Scholar

[25]

R. J. Luo, S. F. Ji and B. L. Zhu, A Pareto evolutionary algorithm based on incremental learning for a kind of multi-objective multidimensional knapsack problem, Computers & Industrial Engineering, 135 (2019), 537-559.  doi: 10.1016/j.cie.2019.06.027.  Google Scholar

[26]

J. M. C. Martí, J. S. Tancrez and R. W. Seifert, Carbon footprint and responsiveness trade-offs in supply chain network design, International Journal of Production Economics, 166 (2015), 129-142.   Google Scholar

[27]

H. Min, V. Jayaraman and R. Srivastava, Combined location-routing problems: A synthesis and future research directions, European Journal of Operational Research, 108 (1998), 1-15.  doi: 10.1016/S0377-2217(97)00172-0.  Google Scholar

[28]

S. M. Mousavi, A. Bahreininejad, S. N. Musa and F. Yusof, A modified particle swarm optimization for solving the integrated location and inventory control problems in a two-echelon supply chain network, Journal of Intelligent Manufacturing, 28 (2017), 191-206. doi: 10.1007/s10845-014-0970-z.  Google Scholar

[29]

M. Musavi and A. Bozorgi-Amiri, A multi-objective sustainable hub location-scheduling problem for perishable food supply chain, Computers & Industrial Engineering, 113 (2017), 766-778.  doi: 10.1016/j.cie.2017.07.039.  Google Scholar

[30]

T. Paksoy, E. }Ozceylan and G. W. Weber, A multi objective model for optimization of a green supply chain network, AIP Conference Proceedings, 311 (2010), 1239. doi: 10.1063/1.3459765.  Google Scholar

[31]

A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D thesis, Cranfield University, London, 2007. Google Scholar

[32]

B. Qian, L. Wang, D.-X. Huang and X. Wang, Scheduling multi-objective job shops using a memetic algorithm based on differential evolution, The International Journal of Advanced Manufacturing Technology, 35 (2008), 1014-1027. doi: 10.1007/s00170-006-0787-9.  Google Scholar

[33]

C. N. Samuel, U. Venkatadri, C. Diallo and A. Khatab, Robust closed-loop supply chain design with presorting, return quality and carbon emission considerations, Journal of Cleaner Production, 247 (2020), 119086. doi: 10.1016/j.jclepro.2019.119086.  Google Scholar

[34]

L. K. Saxena, P. K. Jain and A. K. Sharma, Tactical supply chain planning for tyre remanufacturing considering carbon tax policy, The International Journal of Advanced Manufacturing Technology, 97 (2018), 1505-1528. doi: 10.1007/s00170-018-1972-3.  Google Scholar

[35]

B. L. Shankar, S. Basavarajappa, J. C. Chen and R. S. Kadadevaramath, Location and allocation decisions for multi-echelon supply chain network-a multi-objective evolutionary approach, Expert Systems with Applications, 40 (2013), 551-562. doi: 10.1016/j.eswa.2012.07.065.  Google Scholar

[36]

Y. Shi and R. C. Eberhart, Empirical study of particle swarm optimization, in Proceedings of Twelfth IEEE International Conference on Artificial Intelligence (IJCA), Washington, D. C., USA, 1999, 1945-1950. doi: 10.1109/CEC.1999.785511.  Google Scholar

[37]

M. R. Sierra and C. A. C. Coello, Improving Pso-Based Multi-Objective Optimization Using Crowding, Mutation and $\varepsilon$-dominance, in, Third International Conference on Evolutionary Multi-Criterion Optimization, Evolutionary Multi-Criterion Optimization, Guanajuato, Mexico, 2005. Google Scholar

[38]

R. E. Steuer, Multiple Criteria Optimization: Theory, Computation and Applications, John Wiley and Sons, New York, 1986.  Google Scholar

[39]

M. Sun, Interactive Multiple Objective Programming Procedures Via Adaptive Random Search and Feed-Forward Artificial Neural Networks, Ph.D. dissertation, the University of Georgia, Athens, GA, 1992. Google Scholar

[40]

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Figure 1.  A three-echelon supply chain network
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Figure 2.  Locations of the Plants, the potential DCs and the Retailers
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Figure 3.  The nondominated frontier of the CLRIS considering CCD
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Figure 4.  The nondominated frontier of the CLRIS considering carbon emissions
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Figure 5.  Nondominated frontiers of the model minimizing CCD by varying the carbon cap
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Figure 6.  Box plots for the three evaluation indicators
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Figure 7.  The obtained nondominated frontiers of the all three procedures
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Table 1.  Summary of relevant literature for low carbon supply chains
Literature category Issues considered Related publications
Translating carbon emissions into cost Carbon cost; carbon price; pollution cost; cap-and-trade; emissions trading scheme Tseng and Hung [44]; Treitl et al. [43]; Alhaj et al. [1]; Bai et al. [5]; Wang, Tao and Shi [50].
Treating carbon emissions as a constraint in supply chain models Carbon footprints; carbon-constrained Benjaafar et al. [8]; Xu et al. [53]; Lam and Gu [23]; Martí et al. [26].
Policies for carbon emission reduction Carbon trading; carbon tax Bazan et al. [7]; Wang et al. [47]; Wang, Zhao and Herty [49]; Saxena et al. [34]; Xin et al. [51]; Huang et al. [18]; Samuel et al. [33].
Multiple objectives with carbon emissions Trade-offs between cost and environment Paksoy et al. [30]; Musavi and Bozorgi-Amiri [29]; Daryanto et al. [10].
Carbon emission reduction with coordination mechanisms Green production; green purchasing; coordination mechanism; life cycle assessment Palmer [31]; Jaber et al. [19]; Du et al. [13]; Xu et al. [52]; Zhang et al. [56].
Supply chain network design Location; routing; inventory Min et al. [27]; Wang, Tao and Shi [50]; Javid and Azad [20]; Farahani et al. [15]; Tang et al. [42]; Zhalechian et al. [55].
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Literature category Issues considered Related publications
Translating carbon emissions into cost Carbon cost; carbon price; pollution cost; cap-and-trade; emissions trading scheme Tseng and Hung [44]; Treitl et al. [43]; Alhaj et al. [1]; Bai et al. [5]; Wang, Tao and Shi [50].
Treating carbon emissions as a constraint in supply chain models Carbon footprints; carbon-constrained Benjaafar et al. [8]; Xu et al. [53]; Lam and Gu [23]; Martí et al. [26].
Policies for carbon emission reduction Carbon trading; carbon tax Bazan et al. [7]; Wang et al. [47]; Wang, Zhao and Herty [49]; Saxena et al. [34]; Xin et al. [51]; Huang et al. [18]; Samuel et al. [33].
Multiple objectives with carbon emissions Trade-offs between cost and environment Paksoy et al. [30]; Musavi and Bozorgi-Amiri [29]; Daryanto et al. [10].
Carbon emission reduction with coordination mechanisms Green production; green purchasing; coordination mechanism; life cycle assessment Palmer [31]; Jaber et al. [19]; Du et al. [13]; Xu et al. [52]; Zhang et al. [56].
Supply chain network design Location; routing; inventory Min et al. [27]; Wang, Tao and Shi [50]; Javid and Azad [20]; Farahani et al. [15]; Tang et al. [42]; Zhalechian et al. [55].
Table 2.  Distances from the plants to the potential DCs (km)
Shenbei Yuhong Sujiatun Dongling
Fushun 77.5 71.1 64.6 36.5
Liaoyang 112.8 65.5 44.7 35.8
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Shenbei Yuhong Sujiatun Dongling
Fushun 77.5 71.1 64.6 36.5
Liaoyang 112.8 65.5 44.7 35.8
Table 3.  Parameters of the potential DCs
Shenbei Yuhong Sujiatun Dongling
Lead time (days) 3 2 3 4
Service level 95% 95% 95% 95%
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Shenbei Yuhong Sujiatun Dongling
Lead time (days) 3 2 3 4
Service level 95% 95% 95% 95%
Table 4.  Areas and fixed costs of the potential DCs
Capacity (ton) Fixed cost per period () Holding cost (/ton/day) Areas (m2)
Shenbei 340 20000 0.20 1100
Yuhong 500 18000 0.25 1600
Sujiatun 310 11200 0.20 1000
Dongling 380 15600 0.25 1200
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Capacity (ton) Fixed cost per period () Holding cost (/ton/day) Areas (m2)
Shenbei 340 20000 0.20 1100
Yuhong 500 18000 0.25 1600
Sujiatun 310 11200 0.20 1000
Dongling 380 15600 0.25 1200
Table 5.  Details of some typical solutions on the nondominated frontier
Solutions Plants DC Status Total cost CCD
Shenbei Yuhong Sujiatun Dongling
1 Fushun 1 0 1 1 80153.5 4451.7
Liaoyang 1 0 0 0
2 Fushun 1 1 0 1 81875.5 3677.2
Liaoyang 1 1 0 0
3 Fushun 0 1 0 1 88125.9 3025.7
Liaoyang 1 1 0 0
4 Fushun 1 1 0 1 93750.3 2418.4
Liaoyang 1 1 0 1
5 Fushun 1 1 0 1 100104.9 1975.1
Liaoyang 1 1 1 0
6 Fushun 1 1 1 0 110139.3 1575.7
Liaoyang 1 0 1 0
7 Fushun 1 1 1 0 117697.4 1273.5
Liaoyang 1 1 1 0
8 Fushun 0 1 1 1 122385.8 968.1
Liaoyang 0 1 1 1
9 Fushun 1 1 1 1 134982.2 613.9
Liaoyang 0 1 1 1
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Solutions Plants DC Status Total cost CCD
Shenbei Yuhong Sujiatun Dongling
1 Fushun 1 0 1 1 80153.5 4451.7
Liaoyang 1 0 0 0
2 Fushun 1 1 0 1 81875.5 3677.2
Liaoyang 1 1 0 0
3 Fushun 0 1 0 1 88125.9 3025.7
Liaoyang 1 1 0 0
4 Fushun 1 1 0 1 93750.3 2418.4
Liaoyang 1 1 0 1
5 Fushun 1 1 0 1 100104.9 1975.1
Liaoyang 1 1 1 0
6 Fushun 1 1 1 0 110139.3 1575.7
Liaoyang 1 0 1 0
7 Fushun 1 1 1 0 117697.4 1273.5
Liaoyang 1 1 1 0
8 Fushun 0 1 1 1 122385.8 968.1
Liaoyang 0 1 1 1
9 Fushun 1 1 1 1 134982.2 613.9
Liaoyang 0 1 1 1
Table 6.  Values of the evaluation indicators for MOPSO, NSGAII and MOEA/D
Convergence Diversity Dominance
MOPSO NSGAII MOEA/D MOPSO NSGAII MOEA/D MOPSO NSGAII MOEA/D
1 0.258 0.179 2.320 0.507 0.518 0.691 0.909 0.476 0.000
2 0.126 0.251 2.261 0.520 0.494 0.775 0.878 0.466 0.000
3 0.193 0.142 1.208 0.508 0.540 0.694 0.814 0.485 0.000
4 0.170 0.254 1.273 0.516 0.482 0.841 0.828 0.441 0.000
5 0.160 0.244 1.679 0.468 0.471 0.738 0.913 0.428 0.000
6 0.158 0.260 1.741 0.559 0.479 0.808 0.820 0.463 0.000
7 0.137 0.241 1.375 0.493 0.497 0.698 0.826 0.535 0.000
8 0.177 0.106 2.207 0.480 0.627 0.817 0.775 0.517 0.000
9 0.233 0.092 1.471 0.516 0.545 0.728 0.789 0.371 0.000
10 0.250 0.103 1.050 0.455 0.493 0.713 0.834 0.526 0.000
11 0.239 0.171 1.607 0.521 0.598 0.789 0.817 0.330 0.000
12 0.106 0.179 1.800 0.485 0.557 0.804 0.892 0.326 0.000
13 0.146 0.188 1.952 0.406 0.533 0.724 0.884 0.478 0.000
14 0.131 0.181 1.667 0.490 0.493 0.704 0.871 0.448 0.000
15 0.115 0.158 2.385 0.433 0.449 0.779 0.905 0.476 0.000
16 0.158 0.195 2.158 0.530 0.482 0.662 0.868 0.557 0.000
17 0.122 0.268 1.276 0.480 0.485 0.710 0.841 0.486 0.000
18 0.193 0.169 1.727 0.489 0.517 0.663 0.892 0.546 0.000
19 0.168 0.220 1.700 0.550 0.576 0.637 0.926 0.403 0.000
20 0.172 1.295 1.333 0.478 0.567 0.787 0.833 0.353 0.000
21 0.143 1.724 1.696 0.502 0.517 0.708 0.866 0.303 0.000
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Convergence Diversity Dominance
MOPSO NSGAII MOEA/D MOPSO NSGAII MOEA/D MOPSO NSGAII MOEA/D
1 0.258 0.179 2.320 0.507 0.518 0.691 0.909 0.476 0.000
2 0.126 0.251 2.261 0.520 0.494 0.775 0.878 0.466 0.000
3 0.193 0.142 1.208 0.508 0.540 0.694 0.814 0.485 0.000
4 0.170 0.254 1.273 0.516 0.482 0.841 0.828 0.441 0.000
5 0.160 0.244 1.679 0.468 0.471 0.738 0.913 0.428 0.000
6 0.158 0.260 1.741 0.559 0.479 0.808 0.820 0.463 0.000
7 0.137 0.241 1.375 0.493 0.497 0.698 0.826 0.535 0.000
8 0.177 0.106 2.207 0.480 0.627 0.817 0.775 0.517 0.000
9 0.233 0.092 1.471 0.516 0.545 0.728 0.789 0.371 0.000
10 0.250 0.103 1.050 0.455 0.493 0.713 0.834 0.526 0.000
11 0.239 0.171 1.607 0.521 0.598 0.789 0.817 0.330 0.000
12 0.106 0.179 1.800 0.485 0.557 0.804 0.892 0.326 0.000
13 0.146 0.188 1.952 0.406 0.533 0.724 0.884 0.478 0.000
14 0.131 0.181 1.667 0.490 0.493 0.704 0.871 0.448 0.000
15 0.115 0.158 2.385 0.433 0.449 0.779 0.905 0.476 0.000
16 0.158 0.195 2.158 0.530 0.482 0.662 0.868 0.557 0.000
17 0.122 0.268 1.276 0.480 0.485 0.710 0.841 0.486 0.000
18 0.193 0.169 1.727 0.489 0.517 0.663 0.892 0.546 0.000
19 0.168 0.220 1.700 0.550 0.576 0.637 0.926 0.403 0.000
20 0.172 1.295 1.333 0.478 0.567 0.787 0.833 0.353 0.000
21 0.143 1.724 1.696 0.502 0.517 0.708 0.866 0.303 0.000
Table A1.  Distances from the potential DCs to the retailers (km)
Shenbei Yuhong Dongling
1 82.6 47.7 55.6
2 80.1 47.0 37.3
3 80.5 27.9 54.2
4 64.5 35.7 42.3
5 62.3 40.0 47.3
6 64.6 49.8 15.7
7 61.2 55.6 7.0
8 58.2 45.0 27.3
9 64.2 36.1 49.1
10 66.1 27.2 49.5
11 66.0 12.0 50.0
12 66.2 48.2 60.1
13 63.8 60.6 12.0
14 56.6 40.2 58.9
15 55.2 35.2 67.1
16 58.1 7.0 55.3
17 62.1 54.2 42.0
18 56.9 42.1 41.1
19 50.3 38.4 47.9
20 45.7 39.0 52.1
21 50.1 32.2 50.0
22 50.1 12.0 57.1
23 40.1 36.1 37.0
24 50.1 48.9 12.2
25 30.2 45.9 52.1
26 25.8 40.1 67.1
27 32.1 40.2 55.3
28 23.2 38.4 55.3
29 25.8 42.6 65.1
30 22.4 48.9 54.2
31 20.5 52.6 62.1
32 23.7 46.0 70.1
33 7.0 55.6 66.2
34 15.0 61.3 72.1
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Shenbei Yuhong Dongling
1 82.6 47.7 55.6
2 80.1 47.0 37.3
3 80.5 27.9 54.2
4 64.5 35.7 42.3
5 62.3 40.0 47.3
6 64.6 49.8 15.7
7 61.2 55.6 7.0
8 58.2 45.0 27.3
9 64.2 36.1 49.1
10 66.1 27.2 49.5
11 66.0 12.0 50.0
12 66.2 48.2 60.1
13 63.8 60.6 12.0
14 56.6 40.2 58.9
15 55.2 35.2 67.1
16 58.1 7.0 55.3
17 62.1 54.2 42.0
18 56.9 42.1 41.1
19 50.3 38.4 47.9
20 45.7 39.0 52.1
21 50.1 32.2 50.0
22 50.1 12.0 57.1
23 40.1 36.1 37.0
24 50.1 48.9 12.2
25 30.2 45.9 52.1
26 25.8 40.1 67.1
27 32.1 40.2 55.3
28 23.2 38.4 55.3
29 25.8 42.6 65.1
30 22.4 48.9 54.2
31 20.5 52.6 62.1
32 23.7 46.0 70.1
33 7.0 55.6 66.2
34 15.0 61.3 72.1
Table A2.  Distances between the retailers (km)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
1 0.0 17.1 15.9 18.7 50.2 52.5 55.6 52.5 48.9 50.2 49.5 48.9 58.8 52.5 47.9 51.1 60.0 50.1 52.5 49.1 50.2 50.2 55.3 55.3 56.3 52.1 56.3 62.3 67.3 69.2 70.1 75.2 77.1
2 17.1 0.0 18.7 7.0 10.1 18.8 50.2 20.1 15.9 15.9 24.1 52.5 55.1 18.8 19.8 45.1 53.2 47.1 20.1 50.2 50.2 51.2 53.2 54.7 55.7 53.2 54.9 66.1 66.1 67.1 67.1 69.2 70.1
3 15.9 18.7 0.0 17.3 47.5 49.9 52.1 48.5 45.2 17.3 16.6 16.8 60.0 45.2 47.5 45.7 60.0 49.9 45.1 46.2 44.1 50.3 60.3 60.0 57.3 49.8 55.1 57.3 57.3 60.1 61.3 67.1 68.9
4 18.7 7.0 17.3 0.0 7.0 55.1 40.1 30.2 25.1 7.0 15.2 32.1 41.9 30.2 27.1 25.7 41.9 56.2 30.2 29.5 35.1 26.7 36.9 36.3 41.9 39.1 36.9 50.1 53.4 57.3 59.9 64.7 66.3
5 50.2 10.1 47.5 7.0 0.0 7.0 49.6 7.2 5.9 28.1 50.5 25.8 32.6 27.1 13.8 45.1 32.6 44.3 40.1 46.2 49.9 43.2 32.6 36.6 39.8 39.8 37.6 55.3 55.3 59.9 62.0 65.0 66.0
6 52.5 18.8 49.9 55.1 7.0 0.0 18.8 8.7 20.4 24.1 29.4 48.9 55.1 55.1 57.8 35.9 54.1 55.1 54.1 32.4 48.9 58.8 57.8 48.9 53.9 54.8 55.5 58.4 58.7 62.4 67.1 68.6 68.1
7 55.6 50.2 52.1 40.1 49.6 18.8 0.0 18.8 49.6 50.2 47.1 52.1 5.1 49.6 50.2 54.1 10.0 49.8 47.1 50.1 52.1 48.6 19.8 19.6 23.8 50.1 50.2 49.2 46.3 54.1 59.2 59.2 60.1
8 52.5 20.1 48.5 30.2 7.2 8.7 18.8 0.0 6.4 15.6 20.6 28.7 19.6 14.2 20.6 35.9 19.6 24.2 22.1 20.6 28.7 19.6 20.6 28.7 30.7 32.5 34.6 50.3 45.3 50.3 54.2 55.2 66.4
9 48.9 15.9 45.2 25.1 5.9 20.4 49.6 6.4 0.0 15.2 18.9 25.8 26.8 6.4 15.2 35.6 26.8 27.8 10.1 35.6 38.9 30.2 38.9 39.7 40.1 37.8 41.1 45.6 45.8 47.6 50.2 50.6 50.9
10 50.2 15.9 17.3 7.0 28.1 24.1 50.2 15.6 15.2 0.0 5.0 17.3 55.1 15.2 5.0 15.9 45.2 36.8 15.2 36.8 38.8 39.6 45.2 43.2 43.2 40.1 43.2 45.4 47.9 47.4 45.2 52.1 52.0
11 50.2 24.1 16.6 15.2 50.5 29.4 47.1 20.6 18.9 5.0 0.0 10.0 45.9 27.1 9.0 7.1 27.1 20.1 16.9 16.9 17.5 18.9 27.1 26.7 27.1 25.9 32.2 36.5 45.9 47.2 46.2 53.1 53.0
12 48.9 52.5 16.8 32.1 25.8 48.9 52.1 28.7 25.8 17.3 10.0 0.0 60.0 25.8 9.2 10.0 50.0 40.3 25.8 11.2 12.3 40.3 49.0 45.1 44.3 40.3 44.5 40.3 50.1 53.7 56.5 58.2 59.9
13 58.8 55.1 60.0 41.9 32.6 55.1 5.1 19.6 26.8 55.1 45.9 60.0 0.0 35.6 41.9 45.9 12.1 20.3 32.6 50.1 45.9 35.6 17.0 43.1 45.1 55.1 58.1 59.8 45.1 62.8 69.1 59.8 60.2
14 52.5 18.8 45.2 30.2 27.1 55.1 49.6 14.2 6.4 15.2 27.1 25.8 35.6 0.0 6.2 15.2 45.1 16.1 5.9 23.3 29.1 20.1 40.1 38.7 38.5 29.1 32.1 37.7 39.8 42.3 45.2 46.6 47.9
15 47.9 19.8 47.5 27.1 13.8 57.8 50.2 20.6 15.2 5.0 9.0 9.2 41.9 6.2 0.0 15.2 45.7 48.9 10.2 15.6 20.1 21.2 40.7 41.2 40.2 38.0 40.2 43.5 45.1 47.9 47.1 48.2 48.9
16 51.1 45.1 45.7 25.7 45.1 35.9 54.1 35.9 35.6 15.9 7.1 10.0 45.9 15.2 15.2 0.0 37.1 29.1 25.7 10.1 11.1 23.4 37.1 32.7 33.3 23.4 30.0 31.2 36.1 46.1 45.2 47.7 49.2
17 60.0 53.2 60.0 41.9 32.6 54.1 10.0 19.6 26.8 45.2 27.1 50.0 12.1 45.1 45.7 37.1 0.0 18.7 21.1 45.2 44.5 25.6 8.8 18.7 25.5 45.2 46.6 55.8 45.1 50.1 68.1 63.1 60.0
18 52.1 50.2 45.2 39.1 49.8 18.8 10.0 22.1 25.9 40.1 23.1 48.9 15.0 25.9 57.8 31.2 11.9 7.7 18.8 40.1 44.4 21.8 7.5 44.4 24.5 43.1 46.8 48.7 40.1 45.1 55.1 58.1 58.7
19 50.1 47.1 49.9 56.2 44.3 55.1 49.8 24.2 27.8 36.8 20.1 40.3 20.3 16.1 48.9 29.1 18.7 0.0 7.4 35.9 40.6 10.0 22.7 20.0 24.7 27.8 35.7 40.7 49.7 50.7 52.3 57.9 58.7
20 52.5 20.1 45.1 30.2 40.1 54.1 47.1 22.1 10.1 15.2 16.9 25.8 32.6 5.9 10.2 25.7 21.1 7.4 0.0 20.1 25.5 5.6 27.6 28.7 32.5 30.7 35.7 45.1 45.0 43.1 48.7 49.2 50.1
21 49.1 50.2 46.2 29.5 46.2 32.4 50.1 20.6 35.6 36.8 16.9 11.2 50.1 23.3 15.6 10.1 45.2 35.9 20.1 0.0 8.5 22.2 40.7 34.5 32.1 18.7 24.5 38.0 42.7 43.0 43.7 45.6 46.5
22 50.2 50.2 44.1 35.1 49.9 48.9 52.1 28.7 38.9 38.8 17.5 12.3 45.9 29.1 20.1 11.1 44.5 40.6 25.5 8.5 0.0 27.8 45.2 40.1 38.1 18.9 26.0 30.1 44.1 44.6 45.0 46.7 48.0
23 50.2 51.2 50.3 26.7 43.2 58.8 48.6 19.6 30.2 39.6 18.9 40.3 35.6 20.1 21.2 23.4 25.6 10.0 5.6 22.2 27.8 0.0 27.1 14.2 15.8 20.7 24.2 33.3 38.9 40.1 43.3 45.1 45.2
24 55.3 53.2 60.3 36.9 32.6 57.8 19.8 20.6 38.9 45.2 27.1 49.0 17.0 40.1 40.7 37.1 8.8 22.7 27.6 40.7 45.2 27.1 0.0 13.6 19.5 33.2 34.8 45.2 33.8 38.5 46.1 55.1 50.1
25 55.3 54.7 60.0 36.3 36.6 48.9 19.6 28.7 39.7 43.2 26.7 45.1 43.1 38.7 41.2 32.7 18.7 20.0 28.7 34.5 40.1 14.2 13.6 0.0 6.1 18.9 17.0 41.8 20.0 25.0 40.0 45.2 46.1
26 56.3 55.7 57.3 41.9 39.8 53.9 23.8 30.7 40.1 43.2 27.1 44.3 45.1 38.5 40.2 33.3 25.5 24.7 32.5 32.1 38.1 15.8 19.5 6.1 0.0 20.0 17.9 36.1 30.1 36.1 36.2 37.1 38.2
27 52.1 53.2 49.8 39.1 39.8 54.8 50.1 32.5 37.8 40.1 25.9 40.3 55.1 29.1 38.0 23.4 45.2 27.8 30.7 18.7 18.9 20.7 33.2 18.9 20.0 0.0 7.3 18.2 36.2 38.1 38.1 40.2 44.2
28 56.3 54.9 55.1 36.9 37.6 55.5 50.2 34.6 41.1 43.2 32.2 44.5 58.1 32.1 40.2 30.0 46.6 35.7 35.7 24.5 26.0 24.2 34.8 17.0 17.9 7.3 0.0 13.5 30.1 36.9 40.2 42.1 43.5
29 62.3 66.1 57.3 50.1 55.3 58.4 49.2 50.3 45.6 45.4 36.5 40.3 59.8 37.7 43.5 31.2 55.8 40.7 45.1 38.0 30.1 33.3 45.2 41.8 36.1 18.2 13.5 0.0 35.6 37.3 20.0 45.2 46.2
30 67.3 66.1 57.3 53.4 55.3 58.7 46.3 45.3 45.8 47.9 45.9 50.1 45.1 39.8 45.1 36.1 45.1 49.7 45.0 42.7 44.1 38.9 33.8 20.0 30.1 36.2 30.1 35.6 0.0 7.2 37.3 30.2 32.1
31 69.2 67.1 60.1 57.3 59.9 62.4 54.1 50.3 47.6 47.4 47.2 53.7 62.8 42.3 47.9 46.1 50.1 50.7 43.1 43.0 44.6 40.1 38.5 25.0 36.1 38.1 36.9 37.3 7.2 0.0 25.6 20.1 26.1
32 70.1 67.1 61.3 59.9 62.0 67.1 59.2 54.2 50.2 45.2 46.2 56.5 69.1 45.2 47.1 45.2 68.1 52.3 48.7 43.7 45.0 43.3 46.1 40.0 36.2 38.1 40.2 20.0 37.3 25.6 0.0 38.2 45.6
33 75.2 69.2 67.1 64.7 65.0 68.6 59.2 55.2 50.6 52.1 53.1 58.2 59.8 46.6 48.2 47.7 63.1 57.9 49.2 45.6 46.7 45.1 55.1 45.2 37.1 40.2 42.1 45.2 30.2 20.1 38.2 0.0 8.2
34 77.1 70.1 68.9 66.3 66.0 68.1 60.1 66.4 50.9 52.0 53.0 59.9 60.2 47.9 48.9 49.2 60.0 58.7 50.1 46.5 48.0 45.2 50.1 46.1 38.2 44.2 43.5 46.2 32.1 26.1 45.6 8.2 0.0
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
1 0.0 17.1 15.9 18.7 50.2 52.5 55.6 52.5 48.9 50.2 49.5 48.9 58.8 52.5 47.9 51.1 60.0 50.1 52.5 49.1 50.2 50.2 55.3 55.3 56.3 52.1 56.3 62.3 67.3 69.2 70.1 75.2 77.1
2 17.1 0.0 18.7 7.0 10.1 18.8 50.2 20.1 15.9 15.9 24.1 52.5 55.1 18.8 19.8 45.1 53.2 47.1 20.1 50.2 50.2 51.2 53.2 54.7 55.7 53.2 54.9 66.1 66.1 67.1 67.1 69.2 70.1
3 15.9 18.7 0.0 17.3 47.5 49.9 52.1 48.5 45.2 17.3 16.6 16.8 60.0 45.2 47.5 45.7 60.0 49.9 45.1 46.2 44.1 50.3 60.3 60.0 57.3 49.8 55.1 57.3 57.3 60.1 61.3 67.1 68.9
4 18.7 7.0 17.3 0.0 7.0 55.1 40.1 30.2 25.1 7.0 15.2 32.1 41.9 30.2 27.1 25.7 41.9 56.2 30.2 29.5 35.1 26.7 36.9 36.3 41.9 39.1 36.9 50.1 53.4 57.3 59.9 64.7 66.3
5 50.2 10.1 47.5 7.0 0.0 7.0 49.6 7.2 5.9 28.1 50.5 25.8 32.6 27.1 13.8 45.1 32.6 44.3 40.1 46.2 49.9 43.2 32.6 36.6 39.8 39.8 37.6 55.3 55.3 59.9 62.0 65.0 66.0
6 52.5 18.8 49.9 55.1 7.0 0.0 18.8 8.7 20.4 24.1 29.4 48.9 55.1 55.1 57.8 35.9 54.1 55.1 54.1 32.4 48.9 58.8 57.8 48.9 53.9 54.8 55.5 58.4 58.7 62.4 67.1 68.6 68.1
7 55.6 50.2 52.1 40.1 49.6 18.8 0.0 18.8 49.6 50.2 47.1 52.1 5.1 49.6 50.2 54.1 10.0 49.8 47.1 50.1 52.1 48.6 19.8 19.6 23.8 50.1 50.2 49.2 46.3 54.1 59.2 59.2 60.1
8 52.5 20.1 48.5 30.2 7.2 8.7 18.8 0.0 6.4 15.6 20.6 28.7 19.6 14.2 20.6 35.9 19.6 24.2 22.1 20.6 28.7 19.6 20.6 28.7 30.7 32.5 34.6 50.3 45.3 50.3 54.2 55.2 66.4
9 48.9 15.9 45.2 25.1 5.9 20.4 49.6 6.4 0.0 15.2 18.9 25.8 26.8 6.4 15.2 35.6 26.8 27.8 10.1 35.6 38.9 30.2 38.9 39.7 40.1 37.8 41.1 45.6 45.8 47.6 50.2 50.6 50.9
10 50.2 15.9 17.3 7.0 28.1 24.1 50.2 15.6 15.2 0.0 5.0 17.3 55.1 15.2 5.0 15.9 45.2 36.8 15.2 36.8 38.8 39.6 45.2 43.2 43.2 40.1 43.2 45.4 47.9 47.4 45.2 52.1 52.0
11 50.2 24.1 16.6 15.2 50.5 29.4 47.1 20.6 18.9 5.0 0.0 10.0 45.9 27.1 9.0 7.1 27.1 20.1 16.9 16.9 17.5 18.9 27.1 26.7 27.1 25.9 32.2 36.5 45.9 47.2 46.2 53.1 53.0
12 48.9 52.5 16.8 32.1 25.8 48.9 52.1 28.7 25.8 17.3 10.0 0.0 60.0 25.8 9.2 10.0 50.0 40.3 25.8 11.2 12.3 40.3 49.0 45.1 44.3 40.3 44.5 40.3 50.1 53.7 56.5 58.2 59.9
13 58.8 55.1 60.0 41.9 32.6 55.1 5.1 19.6 26.8 55.1 45.9 60.0 0.0 35.6 41.9 45.9 12.1 20.3 32.6 50.1 45.9 35.6 17.0 43.1 45.1 55.1 58.1 59.8 45.1 62.8 69.1 59.8 60.2
14 52.5 18.8 45.2 30.2 27.1 55.1 49.6 14.2 6.4 15.2 27.1 25.8 35.6 0.0 6.2 15.2 45.1 16.1 5.9 23.3 29.1 20.1 40.1 38.7 38.5 29.1 32.1 37.7 39.8 42.3 45.2 46.6 47.9
15 47.9 19.8 47.5 27.1 13.8 57.8 50.2 20.6 15.2 5.0 9.0 9.2 41.9 6.2 0.0 15.2 45.7 48.9 10.2 15.6 20.1 21.2 40.7 41.2 40.2 38.0 40.2 43.5 45.1 47.9 47.1 48.2 48.9
16 51.1 45.1 45.7 25.7 45.1 35.9 54.1 35.9 35.6 15.9 7.1 10.0 45.9 15.2 15.2 0.0 37.1 29.1 25.7 10.1 11.1 23.4 37.1 32.7 33.3 23.4 30.0 31.2 36.1 46.1 45.2 47.7 49.2
17 60.0 53.2 60.0 41.9 32.6 54.1 10.0 19.6 26.8 45.2 27.1 50.0 12.1 45.1 45.7 37.1 0.0 18.7 21.1 45.2 44.5 25.6 8.8 18.7 25.5 45.2 46.6 55.8 45.1 50.1 68.1 63.1 60.0
18 52.1 50.2 45.2 39.1 49.8 18.8 10.0 22.1 25.9 40.1 23.1 48.9 15.0 25.9 57.8 31.2 11.9 7.7 18.8 40.1 44.4 21.8 7.5 44.4 24.5 43.1 46.8 48.7 40.1 45.1 55.1 58.1 58.7
19 50.1 47.1 49.9 56.2 44.3 55.1 49.8 24.2 27.8 36.8 20.1 40.3 20.3 16.1 48.9 29.1 18.7 0.0 7.4 35.9 40.6 10.0 22.7 20.0 24.7 27.8 35.7 40.7 49.7 50.7 52.3 57.9 58.7
20 52.5 20.1 45.1 30.2 40.1 54.1 47.1 22.1 10.1 15.2 16.9 25.8 32.6 5.9 10.2 25.7 21.1 7.4 0.0 20.1 25.5 5.6 27.6 28.7 32.5 30.7 35.7 45.1 45.0 43.1 48.7 49.2 50.1
21 49.1 50.2 46.2 29.5 46.2 32.4 50.1 20.6 35.6 36.8 16.9 11.2 50.1 23.3 15.6 10.1 45.2 35.9 20.1 0.0 8.5 22.2 40.7 34.5 32.1 18.7 24.5 38.0 42.7 43.0 43.7 45.6 46.5
22 50.2 50.2 44.1 35.1 49.9 48.9 52.1 28.7 38.9 38.8 17.5 12.3 45.9 29.1 20.1 11.1 44.5 40.6 25.5 8.5 0.0 27.8 45.2 40.1 38.1 18.9 26.0 30.1 44.1 44.6 45.0 46.7 48.0
23 50.2 51.2 50.3 26.7 43.2 58.8 48.6 19.6 30.2 39.6 18.9 40.3 35.6 20.1 21.2 23.4 25.6 10.0 5.6 22.2 27.8 0.0 27.1 14.2 15.8 20.7 24.2 33.3 38.9 40.1 43.3 45.1 45.2
24 55.3 53.2 60.3 36.9 32.6 57.8 19.8 20.6 38.9 45.2 27.1 49.0 17.0 40.1 40.7 37.1 8.8 22.7 27.6 40.7 45.2 27.1 0.0 13.6 19.5 33.2 34.8 45.2 33.8 38.5 46.1 55.1 50.1
25 55.3 54.7 60.0 36.3 36.6 48.9 19.6 28.7 39.7 43.2 26.7 45.1 43.1 38.7 41.2 32.7 18.7 20.0 28.7 34.5 40.1 14.2 13.6 0.0 6.1 18.9 17.0 41.8 20.0 25.0 40.0 45.2 46.1
26 56.3 55.7 57.3 41.9 39.8 53.9 23.8 30.7 40.1 43.2 27.1 44.3 45.1 38.5 40.2 33.3 25.5 24.7 32.5 32.1 38.1 15.8 19.5 6.1 0.0 20.0 17.9 36.1 30.1 36.1 36.2 37.1 38.2
27 52.1 53.2 49.8 39.1 39.8 54.8 50.1 32.5 37.8 40.1 25.9 40.3 55.1 29.1 38.0 23.4 45.2 27.8 30.7 18.7 18.9 20.7 33.2 18.9 20.0 0.0 7.3 18.2 36.2 38.1 38.1 40.2 44.2
28 56.3 54.9 55.1 36.9 37.6 55.5 50.2 34.6 41.1 43.2 32.2 44.5 58.1 32.1 40.2 30.0 46.6 35.7 35.7 24.5 26.0 24.2 34.8 17.0 17.9 7.3 0.0 13.5 30.1 36.9 40.2 42.1 43.5
29 62.3 66.1 57.3 50.1 55.3 58.4 49.2 50.3 45.6 45.4 36.5 40.3 59.8 37.7 43.5 31.2 55.8 40.7 45.1 38.0 30.1 33.3 45.2 41.8 36.1 18.2 13.5 0.0 35.6 37.3 20.0 45.2 46.2
30 67.3 66.1 57.3 53.4 55.3 58.7 46.3 45.3 45.8 47.9 45.9 50.1 45.1 39.8 45.1 36.1 45.1 49.7 45.0 42.7 44.1 38.9 33.8 20.0 30.1 36.2 30.1 35.6 0.0 7.2 37.3 30.2 32.1
31 69.2 67.1 60.1 57.3 59.9 62.4 54.1 50.3 47.6 47.4 47.2 53.7 62.8 42.3 47.9 46.1 50.1 50.7 43.1 43.0 44.6 40.1 38.5 25.0 36.1 38.1 36.9 37.3 7.2 0.0 25.6 20.1 26.1
32 70.1 67.1 61.3 59.9 62.0 67.1 59.2 54.2 50.2 45.2 46.2 56.5 69.1 45.2 47.1 45.2 68.1 52.3 48.7 43.7 45.0 43.3 46.1 40.0 36.2 38.1 40.2 20.0 37.3 25.6 0.0 38.2 45.6
33 75.2 69.2 67.1 64.7 65.0 68.6 59.2 55.2 50.6 52.1 53.1 58.2 59.8 46.6 48.2 47.7 63.1 57.9 49.2 45.6 46.7 45.1 55.1 45.2 37.1 40.2 42.1 45.2 30.2 20.1 38.2 0.0 8.2
34 77.1 70.1 68.9 66.3 66.0 68.1 60.1 66.4 50.9 52.0 53.0 59.9 60.2 47.9 48.9 49.2 60.0 58.7 50.1 46.5 48.0 45.2 50.1 46.1 38.2 44.2 43.5 46.2 32.1 26.1 45.6 8.2 0.0
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