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doi: 10.3934/jimo.2020142

The joint location-transportation model based on grey bi-level programming for early post-earthquake relief

Yufeng Zhou 1, , Bin Zheng 2,, , Jiafu Su 1, and  Yufeng Li 3,

1. 

Chongqing Engineering Technology Research Center for Information Management in, Development, Chongqing Technology and Business University, Chongqing, 400067, China
2. 

School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China
3. 

Research center for economy of upper researches of the Yangtze River, Chongqing Technology, and Business University, Chongqing, 400067, China

* Corresponding author: Bin Zheng

Received  May 2020 Revised  July 2020 Published  September 2020

The initial period after the earthquake is the prime time for disaster relief. During this period, it is of great value to rationally locate the transfer facilities of relief materials and effectively arrange the transportation of relief materials. Considering the characteristics of the two-level emergency logistics system including uncertain demand, uncertain transportation time, multiple varieties of relief materials, shortage of supply, multi-transportation modes and different urgencies of relief material demand, the integrated issue with the concern of transfer facility location and relief material transportation is studied. Then, this problem is formulated as a grey mixed integer bi-level nonlinear programming in which the upper-level aims at the shortest relief material transportation time and the lower-level aims at the maximum fairness of relief material distribution. According to the characteristics of the model, a hybrid genetic algorithm is designed to solve the proposed model. Finally, a numerical simulation is carried out on the background of 5.12 Wenchuan Earthquake. In addition, the validation of the proposed model and algorithm is verified.

Keywords: Earthquake disaster, joint location-transportation problem, grey bi-level programming, facility location, hybrid genetic algorithm.
Mathematics Subject Classification: Primary: 90B50; Secondary: 90B06.
Citation: Yufeng Zhou, Bin Zheng, Jiafu Su, Yufeng Li. The joint location-transportation model based on grey bi-level programming for early post-earthquake relief. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020142
References:
[1]

A. Afshar and A. Haghani, Modeling integrated supply chain logistics in real-time large-scale disaster relief operations, Socio-Economic Plan. Sci., 46 (2012), 327-338.  doi: 10.1016/j.seps.2011.12.003.  Google Scholar

[2]

J. F. Bard, Some properties of the bilevel programming problem, J. Optim. Theory Appl., 68 (1991), 371-378.  doi: 10.1007/BF00941574.  Google Scholar

[3]

V. Bélanger, A. Ruiz and P. Soriano, Recent optimization models and trends in location, relocation, and dispatching of emergency medical vehicles, European J. Oper. Res., 272 (2019), 1-23.  doi: 10.1016/j.ejor.2018.02.055.  Google Scholar

[4]

C. Blair, The computational complexity of multi-level linear programs, Ann. Oper. Res., 34 (1992), 13-19.  doi: 10.1007/BF02098170.  Google Scholar

[5]

A. Bozorgi-Amiri and M. Khorsi, A dynamic multi-objective location-routing model for relief logistic planning under uncertainty on demand, travel time, and cost parameters, The International Journal of Advanced Manufacturing Technology, 85 (2016), 1633-1648.   Google Scholar

[6]

C. J. Cao, C. D. Li, Q. Yang, Y. Liu and T. Qu, A novel multi-objective programming model of relief distribution for sustainable disaster supply chain in large-scale natural disasters, J. Cleaner Production, 174 (2018), 1422-1435.  doi: 10.1016/j.jclepro.2017.11.037.  Google Scholar

[7]

A. Y. Chen and T. Y. Yu, Network based temporary facility location for the Emergency Medical Services considering the disaster induced demand and the transportation infrastructure in disaster response, Transport Res. B. Meth., 91 (2016), 408-423.  doi: 10.1016/j.trb.2016.06.004.  Google Scholar

[8]

Y. Dai, Z. J. Ma, D. L. Zhu and T. Fang, Fuzzy dynamic location-routing problem in post-earthquake delivery of relief materials, Chinese Journal of Management Science, 15 (2012), 60-70.   Google Scholar

[9]

X. T. Deng, Complexity issues in bi-level linear programming, in Multilevel Optimization: Algorithms and Applications, Vol. 20, Kluwer Acad. Publ., Dordrecht, 1998,149–164. doi: 10.1007/978-1-4613-0307-7_6.  Google Scholar

[10]

M. Haghi, S. M. T. F. Ghomi and F. Jolai, Developing a robust multi-objective model for pre/post disaster times under uncertainty in demand and resource, J. Cleaner Production, 154 (2017), 188-202.  doi: 10.1016/j.jclepro.2017.03.102.  Google Scholar

[11]

H. Hu, X. Li, Y. Y. Zhang, C. J. Shang and S. C. Zhang, Multi-objective location-routing model for hazardous material logistics with traffic restriction constraint in inter-city roads, Comput. Ind. Eng., 128 (2019), 861-876.  doi: 10.1016/j.cie.2018.10.044.  Google Scholar

[12]

J. Jian, Y. Zhang, L. Jiang and J. Su, Coordination of supply chains with competing manufacturers considering fairness concerns, Complexity, 2020 (2020), 4372603. doi: 10.1155/2020/4372603.  Google Scholar

[13]

S. Li and K. L. Teo, Post-disaster multi-period road network repair: Work scheduling and relief logistics optimization, Ann. Oper. Res., 283 (2019), 1345-1385.  doi: 10.1007/s10479-018-3037-2.  Google Scholar

[14]

S. Li, Z. J. Ma and K. L. Teo, A new model for road network repair after natural disaster: Integrating logistics support scheduling with repair crew scheduling and routing activities, Comput. Ind. Eng., 145 (2020), 106506. Google Scholar

[15]

S. L. Li, Z. J. Ma, B. Zheng and Y. Dai, Fuzzy multi-objective location-multimodal transportation problem for relief delivery during the initial post-earthquake period, Chinese Journal of Management Science, 21 (2013), 144-151.   Google Scholar

[16]

C. S. Liu, L. Luo, X. C. Zhou and F. H. Huang, Collaborative decision-making of relief allocation-transportation in early post-earthquake: Considering both fairness and efficiency, Control and Decision, 33 (2018), 2057-2063.   Google Scholar

[17]

C. S. Liu, G. Kou, Y. Peng and F. E. Alsaadi, Location-routing problem for relief distribution in the early post-earthquake stage from the perspective of fairness, Sustainability, 11 (2019), 3420. doi: 10.3390/su11123420.  Google Scholar

[18]

R. Lotfian and M. Najafi, Optimal location of emergency stations in underground mine networks using a multiobjective mathematical model, Injury Prevention, 25 (2019), 264-272.  doi: 10.1136/injuryprev-2017-042657.  Google Scholar

[19]

H. M. Rizeei, B. Pradhan and M. A. Saharkhiz, Allocation of emergency response centres in response to pluvial flooding-prone demand points using integrated multiple layer perceptron and maximum coverage location problem models, Int. J. Disast. Risk. Red., 38 (2019), 101205. doi: 10.1016/j.ijdrr.2019.101205.  Google Scholar

[20]

A. S. Safaei, S. Farsad and M. M. Paydar, Robust bi-level optimization of relief logistics operations, Appl. Math. Model., 56 (2018), 359-380.  doi: 10.1016/j.apm.2017.12.003.  Google Scholar

[21]

A. S. Safaei, S. Farsad and M. M. Paydar, Emergency logistics planning under supply risk and demand uncertainty, Oper. Res., 20 (2020), 1437-1460.  doi: 10.1007/s12351-018-0376-3.  Google Scholar

[22]

F. S. Salman and E. Yücel, Emergency facility location under random network damage: Insights from the Istanbul case, Comput. Oper. Res., 62 (2015), 266-281.  doi: 10.1016/j.cor.2014.07.015.  Google Scholar

[23]

J. Su, Q. Bai, S. Sindakis, X Zhang and T Yang, Vulnerability of multinational corporation knowledge network facing resource loss, Management Decision, 2020. doi: 10.1108/MD-02-2019-0227.  Google Scholar

[24]

B. Vahdani, D. Veysmoradi, N. Shekari and S. M. Mousavi, Multi-objective, multi-period location-routing model to distribute relief after earthquake by considering emergency roadway repair, Neural Comput. Appl., 30 (2018), 835-854.  doi: 10.1007/s00521-016-2696-7.  Google Scholar

[25]

B. Vahdani, D. Veysmoradi, F. Noori and F. Mansour, Two-stage multi-objective location-routing-inventory model for humanitarian logistics network design under uncertainty, Int. J. Disast. Risk. Re., 27 (2018), 290-306.   Google Scholar

[26]

H. J. Wang, L. J. Du and S. H. Ma, Multi-objective open location-routing model with split delivery for optimized relief distribution in post-earthquake, Transport Research Part E: Logistics and Transportation Review, 69 (2014), 160-179.  doi: 10.1016/j.tre.2014.06.006.  Google Scholar

[27]

J. Xu, Z. Wang, M. Zhang and Y. Tu, A new model for a 72-h post-earthquake emergency logistics location-routing problem under a random fuzzy environment, Transportation Letters, 8 (2016), 270-285.  doi: 10.1080/19427867.2015.1126064.  Google Scholar

[28]

M. Yahyaei and A. Bozorgi-Amiri, Robust reliable humanitarian relief network design: An integration of shelter and supply facility location, Ann. Oper. Res., 283 (2019), 897-916.  doi: 10.1007/s10479-018-2758-6.  Google Scholar

[29]

W. Yi and L. Özdamar, A dynamic logistics coordination model for evacuation and support in disaster response activities, European J. Oper. Res., 179 (2007), 1177-1193.  doi: 10.1016/j.ejor.2005.03.077.  Google Scholar

[30]

B. Zeng, M. Tong and X. Ma, A new-structure grey Verhulst model: Development and performance comparison, Appl. Math. Model., 81 (2020), 522-537.  doi: 10.1016/j.apm.2020.01.014.  Google Scholar

[31]

B. Zeng, M. Zhou, X. Liu and and Z. Zhang, Application of a new grey prediction model and grey average weakening buffer operator to forecast China's shale gas output, Energy Reports, 6 (2020), 1608-1618.   Google Scholar

[32]

S. Zhang and H. Yi, Emergency material transportation considering the impact of secondary disasters, in Proceedings of the 2019 10th International Conference on E-business, Management and Economics, 2019,279–285. doi: 10.1145/3345035.3345041.  Google Scholar

[33]

S. W. Zhang, H. X. Guo, K. J. Zhu, S. W. Yu and J. L. Li, Multistage assignment optimization for emergency rescue teams in the disaster chain, Knowledge-Based Systems, 137 (2017), 123-137.  doi: 10.1016/j.knosys.2017.09.024.  Google Scholar

[34]

B. Zheng, Z. J. Ma and Y. F. Zhou, Bi-level model for dynamic location-transportation problem for post earthquake relief distribution, J. Systems & Management, 26 (2017), 326-337.   Google Scholar

[35]

Y. Zhou and N. Chen, The LAP under facility disruptions during early post-earthquake rescue using PSO-GA hybrid algorithm, Fresen. Environ. Bull., 28 (2019), 9906-9914.   Google Scholar

[36]

Y. F. Zhou, H. X. Yu, Z. Li and J. F. Su, Robust optimization of a distribution network location-routing problem under carbon trading policies, IEEE Access, 8 (2020), 46288-46306.  doi: 10.1109/ACCESS.2020.2979259.  Google Scholar

[37]

Y. Zhou, L. Yufeng and L. Zhi, A grey target group decision method with dual hesitant fuzzy information considering decision-maker's loss aversion, Sci. Programming, 2020 (2020), 8930387. doi: 10.1155/2020/8930387.  Google Scholar

show all references

References:
[1]

A. Afshar and A. Haghani, Modeling integrated supply chain logistics in real-time large-scale disaster relief operations, Socio-Economic Plan. Sci., 46 (2012), 327-338.  doi: 10.1016/j.seps.2011.12.003.  Google Scholar

[2]

J. F. Bard, Some properties of the bilevel programming problem, J. Optim. Theory Appl., 68 (1991), 371-378.  doi: 10.1007/BF00941574.  Google Scholar

[3]

V. Bélanger, A. Ruiz and P. Soriano, Recent optimization models and trends in location, relocation, and dispatching of emergency medical vehicles, European J. Oper. Res., 272 (2019), 1-23.  doi: 10.1016/j.ejor.2018.02.055.  Google Scholar

[4]

C. Blair, The computational complexity of multi-level linear programs, Ann. Oper. Res., 34 (1992), 13-19.  doi: 10.1007/BF02098170.  Google Scholar

[5]

A. Bozorgi-Amiri and M. Khorsi, A dynamic multi-objective location-routing model for relief logistic planning under uncertainty on demand, travel time, and cost parameters, The International Journal of Advanced Manufacturing Technology, 85 (2016), 1633-1648.   Google Scholar

[6]

C. J. Cao, C. D. Li, Q. Yang, Y. Liu and T. Qu, A novel multi-objective programming model of relief distribution for sustainable disaster supply chain in large-scale natural disasters, J. Cleaner Production, 174 (2018), 1422-1435.  doi: 10.1016/j.jclepro.2017.11.037.  Google Scholar

[7]

A. Y. Chen and T. Y. Yu, Network based temporary facility location for the Emergency Medical Services considering the disaster induced demand and the transportation infrastructure in disaster response, Transport Res. B. Meth., 91 (2016), 408-423.  doi: 10.1016/j.trb.2016.06.004.  Google Scholar

[8]

Y. Dai, Z. J. Ma, D. L. Zhu and T. Fang, Fuzzy dynamic location-routing problem in post-earthquake delivery of relief materials, Chinese Journal of Management Science, 15 (2012), 60-70.   Google Scholar

[9]

X. T. Deng, Complexity issues in bi-level linear programming, in Multilevel Optimization: Algorithms and Applications, Vol. 20, Kluwer Acad. Publ., Dordrecht, 1998,149–164. doi: 10.1007/978-1-4613-0307-7_6.  Google Scholar

[10]

M. Haghi, S. M. T. F. Ghomi and F. Jolai, Developing a robust multi-objective model for pre/post disaster times under uncertainty in demand and resource, J. Cleaner Production, 154 (2017), 188-202.  doi: 10.1016/j.jclepro.2017.03.102.  Google Scholar

[11]

H. Hu, X. Li, Y. Y. Zhang, C. J. Shang and S. C. Zhang, Multi-objective location-routing model for hazardous material logistics with traffic restriction constraint in inter-city roads, Comput. Ind. Eng., 128 (2019), 861-876.  doi: 10.1016/j.cie.2018.10.044.  Google Scholar

[12]

J. Jian, Y. Zhang, L. Jiang and J. Su, Coordination of supply chains with competing manufacturers considering fairness concerns, Complexity, 2020 (2020), 4372603. doi: 10.1155/2020/4372603.  Google Scholar

[13]

S. Li and K. L. Teo, Post-disaster multi-period road network repair: Work scheduling and relief logistics optimization, Ann. Oper. Res., 283 (2019), 1345-1385.  doi: 10.1007/s10479-018-3037-2.  Google Scholar

[14]

S. Li, Z. J. Ma and K. L. Teo, A new model for road network repair after natural disaster: Integrating logistics support scheduling with repair crew scheduling and routing activities, Comput. Ind. Eng., 145 (2020), 106506. Google Scholar

[15]

S. L. Li, Z. J. Ma, B. Zheng and Y. Dai, Fuzzy multi-objective location-multimodal transportation problem for relief delivery during the initial post-earthquake period, Chinese Journal of Management Science, 21 (2013), 144-151.   Google Scholar

[16]

C. S. Liu, L. Luo, X. C. Zhou and F. H. Huang, Collaborative decision-making of relief allocation-transportation in early post-earthquake: Considering both fairness and efficiency, Control and Decision, 33 (2018), 2057-2063.   Google Scholar

[17]

C. S. Liu, G. Kou, Y. Peng and F. E. Alsaadi, Location-routing problem for relief distribution in the early post-earthquake stage from the perspective of fairness, Sustainability, 11 (2019), 3420. doi: 10.3390/su11123420.  Google Scholar

[18]

R. Lotfian and M. Najafi, Optimal location of emergency stations in underground mine networks using a multiobjective mathematical model, Injury Prevention, 25 (2019), 264-272.  doi: 10.1136/injuryprev-2017-042657.  Google Scholar

[19]

H. M. Rizeei, B. Pradhan and M. A. Saharkhiz, Allocation of emergency response centres in response to pluvial flooding-prone demand points using integrated multiple layer perceptron and maximum coverage location problem models, Int. J. Disast. Risk. Red., 38 (2019), 101205. doi: 10.1016/j.ijdrr.2019.101205.  Google Scholar

[20]

A. S. Safaei, S. Farsad and M. M. Paydar, Robust bi-level optimization of relief logistics operations, Appl. Math. Model., 56 (2018), 359-380.  doi: 10.1016/j.apm.2017.12.003.  Google Scholar

[21]

A. S. Safaei, S. Farsad and M. M. Paydar, Emergency logistics planning under supply risk and demand uncertainty, Oper. Res., 20 (2020), 1437-1460.  doi: 10.1007/s12351-018-0376-3.  Google Scholar

[22]

F. S. Salman and E. Yücel, Emergency facility location under random network damage: Insights from the Istanbul case, Comput. Oper. Res., 62 (2015), 266-281.  doi: 10.1016/j.cor.2014.07.015.  Google Scholar

[23]

J. Su, Q. Bai, S. Sindakis, X Zhang and T Yang, Vulnerability of multinational corporation knowledge network facing resource loss, Management Decision, 2020. doi: 10.1108/MD-02-2019-0227.  Google Scholar

[24]

B. Vahdani, D. Veysmoradi, N. Shekari and S. M. Mousavi, Multi-objective, multi-period location-routing model to distribute relief after earthquake by considering emergency roadway repair, Neural Comput. Appl., 30 (2018), 835-854.  doi: 10.1007/s00521-016-2696-7.  Google Scholar

[25]

B. Vahdani, D. Veysmoradi, F. Noori and F. Mansour, Two-stage multi-objective location-routing-inventory model for humanitarian logistics network design under uncertainty, Int. J. Disast. Risk. Re., 27 (2018), 290-306.   Google Scholar

[26]

H. J. Wang, L. J. Du and S. H. Ma, Multi-objective open location-routing model with split delivery for optimized relief distribution in post-earthquake, Transport Research Part E: Logistics and Transportation Review, 69 (2014), 160-179.  doi: 10.1016/j.tre.2014.06.006.  Google Scholar

[27]

J. Xu, Z. Wang, M. Zhang and Y. Tu, A new model for a 72-h post-earthquake emergency logistics location-routing problem under a random fuzzy environment, Transportation Letters, 8 (2016), 270-285.  doi: 10.1080/19427867.2015.1126064.  Google Scholar

[28]

M. Yahyaei and A. Bozorgi-Amiri, Robust reliable humanitarian relief network design: An integration of shelter and supply facility location, Ann. Oper. Res., 283 (2019), 897-916.  doi: 10.1007/s10479-018-2758-6.  Google Scholar

[29]

W. Yi and L. Özdamar, A dynamic logistics coordination model for evacuation and support in disaster response activities, European J. Oper. Res., 179 (2007), 1177-1193.  doi: 10.1016/j.ejor.2005.03.077.  Google Scholar

[30]

B. Zeng, M. Tong and X. Ma, A new-structure grey Verhulst model: Development and performance comparison, Appl. Math. Model., 81 (2020), 522-537.  doi: 10.1016/j.apm.2020.01.014.  Google Scholar

[31]

B. Zeng, M. Zhou, X. Liu and and Z. Zhang, Application of a new grey prediction model and grey average weakening buffer operator to forecast China's shale gas output, Energy Reports, 6 (2020), 1608-1618.   Google Scholar

[32]

S. Zhang and H. Yi, Emergency material transportation considering the impact of secondary disasters, in Proceedings of the 2019 10th International Conference on E-business, Management and Economics, 2019,279–285. doi: 10.1145/3345035.3345041.  Google Scholar

[33]

S. W. Zhang, H. X. Guo, K. J. Zhu, S. W. Yu and J. L. Li, Multistage assignment optimization for emergency rescue teams in the disaster chain, Knowledge-Based Systems, 137 (2017), 123-137.  doi: 10.1016/j.knosys.2017.09.024.  Google Scholar

[34]

B. Zheng, Z. J. Ma and Y. F. Zhou, Bi-level model for dynamic location-transportation problem for post earthquake relief distribution, J. Systems & Management, 26 (2017), 326-337.   Google Scholar

[35]

Y. Zhou and N. Chen, The LAP under facility disruptions during early post-earthquake rescue using PSO-GA hybrid algorithm, Fresen. Environ. Bull., 28 (2019), 9906-9914.   Google Scholar

[36]

Y. F. Zhou, H. X. Yu, Z. Li and J. F. Su, Robust optimization of a distribution network location-routing problem under carbon trading policies, IEEE Access, 8 (2020), 46288-46306.  doi: 10.1109/ACCESS.2020.2979259.  Google Scholar

[37]

Y. Zhou, L. Yufeng and L. Zhi, A grey target group decision method with dual hesitant fuzzy information considering decision-maker's loss aversion, Sci. Programming, 2020 (2020), 8930387. doi: 10.1155/2020/8930387.  Google Scholar

Figure 1.  Schematic Diagram of the Post-Earthquake Emergency Logistics System
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Figure 2.  Relief materials from different temporary transfer facilities
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Figure 3.  Relief materials from the same temporary transfer facility and do not need waiting
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Figure 4.  Relief materials from the same temporary transfer facility and need waiting
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Figure 5.  Flow Chart of Genetic Algorithm
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Figure 6.  Schematic diagram of chromosome coding
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Figure 7.  Operational processes of decision-making in upper-level and lower-level
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Figure 8.  Schematic of arithmetic crossover operator
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Figure 9.  Schematic diagram of partial matching arithmetic crossover
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Figure 10.  Convergence diagram of hybrid genetic algorithm
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Table 1.  The amount of relief materials supplied by collection centers
Num. Collection centers Food(Units) Daily necessities(Units)
Ⅰ Chengdu Military Airport 200000 90000
Ⅱ Shuangliu Airport 800000 100000
Ⅲ Chengdu North Railway Station 1200000 130000
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Num. Collection centers Food(Units) Daily necessities(Units)
Ⅰ Chengdu Military Airport 200000 90000
Ⅱ Shuangliu Airport 800000 100000
Ⅲ Chengdu North Railway Station 1200000 130000
Table 2.  The amount of relief materials supplied by collection centers
Candidate temporary transfer facilities Num Affected area $ T{h_j} $(tons) Candidate temporary transfer facilities Num. Affected area $ T{h_j} $(tons)
Dujiangyan (1) ① 800 Pingwu (10) ② 700
Maoxian (2) ① 800 Jiangyou (11) ② 800
Pengzhou (3) ① 800 Deyang (12) ② 1500
Wenchuan (4) ① 500 Mianyang (13) ② 1500
Jiuzhaigou (5) ① 1000 Guanghan (14) ② 1300
Chongzhou (6) ① 800 Guangyuan (15) ③ 1500
Dayi (7) ① 1200 Qingchuan (16) ③ 1000
Shifang (8) ② 1000 Wangcang (17) ③ 1000
Beichuan (9) ② 800 Jiange (18) ③ 1000
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Candidate temporary transfer facilities Num Affected area $ T{h_j} $(tons) Candidate temporary transfer facilities Num. Affected area $ T{h_j} $(tons)
Dujiangyan (1) ① 800 Pingwu (10) ② 700
Maoxian (2) ① 800 Jiangyou (11) ② 800
Pengzhou (3) ① 800 Deyang (12) ② 1500
Wenchuan (4) ① 500 Mianyang (13) ② 1500
Jiuzhaigou (5) ① 1000 Guanghan (14) ② 1300
Chongzhou (6) ① 800 Guangyuan (15) ③ 1500
Dayi (7) ① 1200 Qingchuan (16) ③ 1000
Shifang (8) ② 1000 Wangcang (17) ③ 1000
Beichuan (9) ② 800 Jiange (18) ③ 1000
Table 3.  The demand for relief materials at affected points
Affected points Num. Affected area Food(units) Daily necessities(Units)
Dujiangyan 1 ① [224000, 336000] [22400, 33600]
Guankou 2 ① [42400, 63600] [4240, 6360]
Qingchengshan 3 ① [38880, 58320] [3888, 5832]
Zipingpu 4 ① [35200, 52800] [3520, 5280]
Hongkou 5 ① [38400, 57600] [3840, 5760]
Maoxian 6 ① [173200, 259800] [17320, 25980]
Fushun 7 ① [28800, 43200] [2880, 4320]
Feihong 8 ① [30400, 45600] [3040, 4560]
Heihu 9 ① [32000, 48000] [3200, 4800]
Taiping 10 ① [25600, 38400] [2560, 3840]
Lixian 11 ① [171200, 256800] [17120, 25680]
Putouxiang 12 ① [25600, 38400] [2560, 38400]
Mukaxiang 13 ① [16000, 24000] [1600, 2400]
Tonghuaxiang 14 ① [38400, 57600] [3840, 5760]
Wenchuan 15 ① [246800, 370200] [24680, 37020]
Yingxiu 16 ① [32000, 48000] [3200, 4800]
Shuimo 17 ① [28800, 43200] [2880, 4320]
Wolong 18 ① [27200, 40800] [2720, 4080]
Yanmeng 19 ① [27840, 41760] [2784, 4176]
Sanjiang 20 ① [24640, 36960] [2464, 3696]
Xiaojin 21 ① [20000, 30000] [2000, 3000]
Heishui 22 ① [49600, 74400] [4960, 7440]
Songpan 23 ① [19200, 28800] [1920, 2880]
Anxian 24 ② [191600, 287400] [19160, 28740]
Xiushui 25 ② [28480, 42720] [2848, 4272]
Baolin 26 ② [33600, 50400] [3360, 5040]
Gaochuan 27 ② [18240, 27360] [1824, 2736]
Shifang 28 ② [192000, 288000] [19200, 28800]
Luoshui 29 ② [30400, 45600] [3040, 4560]
Shuangsheng 30 ② [41600, 62400] [4160, 6240]
Yinghua 31 ② [38400, 57600] [3840, 5760]
Mianzhu 32 ② [232400, 348600] [23240, 34860]
Jiannan 33 ② [28160, 42240] [2816, 4224]
Mianyuan 34 ② [24960, 37440] [2496, 3744]
Hanwang 35 ② [21760, 32640] [2176, 3264]
Beichuan 36 ② [242800, 364200] [24280, 36420]
Yong'an 37 ② [31040, 46560] [3104, 4656]
Yongchang 38 ② [33600, 50400] [3360, 5040]
Kaiping 39 ② [30400, 45600] [3040, 4560]
Luojiang 40 ② [69600, 104400] [6960, 10440]
Zhongjiang 41 ② [84800, 127200] [8480, 12720]
Santai 42 ② [109600, 164400] [10960, 16440]
Yanting 43 ② [110000, 165000] [11000, 16500]
Zitong 44 ② [125200, 187800] [12520, 18780]
Deyang 45 ② [173760, 260640] [17376, 26064]
Mianyang 46 ② [184000, 276000] [18400, 27600]
Guangyuan 47 ③ [75040, 112560] [7504, 11256]
Qingchuan 48 ③ [216400, 324600] [21640, 32460]
Walixiang 49 ③ [32960, 49440] [3296, 4944]
Banqiaoxiang 50 ③ [39040, 58560] [3904, 5856]
Qimaxiang 51 ③ [36000, 54000] [3600, 5400]
Yingpanxiang 52 ③ [32800, 49200] [3280, 4920]
Lizhou 53 ③ [71200, 106800] [7120, 10680]
Chaotian 54 ③ [66400, 99600] [6640, 9960]
Cangxi 55 ③ [125200, 187800] [12520, 18780]
Jiange 56 ③ [62000, 93000] [6200, 9300]
Yuanba 57 ③ [71200, 106800] [7120, 10680]
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Affected points Num. Affected area Food(units) Daily necessities(Units)
Dujiangyan 1 ① [224000, 336000] [22400, 33600]
Guankou 2 ① [42400, 63600] [4240, 6360]
Qingchengshan 3 ① [38880, 58320] [3888, 5832]
Zipingpu 4 ① [35200, 52800] [3520, 5280]
Hongkou 5 ① [38400, 57600] [3840, 5760]
Maoxian 6 ① [173200, 259800] [17320, 25980]
Fushun 7 ① [28800, 43200] [2880, 4320]
Feihong 8 ① [30400, 45600] [3040, 4560]
Heihu 9 ① [32000, 48000] [3200, 4800]
Taiping 10 ① [25600, 38400] [2560, 3840]
Lixian 11 ① [171200, 256800] [17120, 25680]
Putouxiang 12 ① [25600, 38400] [2560, 38400]
Mukaxiang 13 ① [16000, 24000] [1600, 2400]
Tonghuaxiang 14 ① [38400, 57600] [3840, 5760]
Wenchuan 15 ① [246800, 370200] [24680, 37020]
Yingxiu 16 ① [32000, 48000] [3200, 4800]
Shuimo 17 ① [28800, 43200] [2880, 4320]
Wolong 18 ① [27200, 40800] [2720, 4080]
Yanmeng 19 ① [27840, 41760] [2784, 4176]
Sanjiang 20 ① [24640, 36960] [2464, 3696]
Xiaojin 21 ① [20000, 30000] [2000, 3000]
Heishui 22 ① [49600, 74400] [4960, 7440]
Songpan 23 ① [19200, 28800] [1920, 2880]
Anxian 24 ② [191600, 287400] [19160, 28740]
Xiushui 25 ② [28480, 42720] [2848, 4272]
Baolin 26 ② [33600, 50400] [3360, 5040]
Gaochuan 27 ② [18240, 27360] [1824, 2736]
Shifang 28 ② [192000, 288000] [19200, 28800]
Luoshui 29 ② [30400, 45600] [3040, 4560]
Shuangsheng 30 ② [41600, 62400] [4160, 6240]
Yinghua 31 ② [38400, 57600] [3840, 5760]
Mianzhu 32 ② [232400, 348600] [23240, 34860]
Jiannan 33 ② [28160, 42240] [2816, 4224]
Mianyuan 34 ② [24960, 37440] [2496, 3744]
Hanwang 35 ② [21760, 32640] [2176, 3264]
Beichuan 36 ② [242800, 364200] [24280, 36420]
Yong'an 37 ② [31040, 46560] [3104, 4656]
Yongchang 38 ② [33600, 50400] [3360, 5040]
Kaiping 39 ② [30400, 45600] [3040, 4560]
Luojiang 40 ② [69600, 104400] [6960, 10440]
Zhongjiang 41 ② [84800, 127200] [8480, 12720]
Santai 42 ② [109600, 164400] [10960, 16440]
Yanting 43 ② [110000, 165000] [11000, 16500]
Zitong 44 ② [125200, 187800] [12520, 18780]
Deyang 45 ② [173760, 260640] [17376, 26064]
Mianyang 46 ② [184000, 276000] [18400, 27600]
Guangyuan 47 ③ [75040, 112560] [7504, 11256]
Qingchuan 48 ③ [216400, 324600] [21640, 32460]
Walixiang 49 ③ [32960, 49440] [3296, 4944]
Banqiaoxiang 50 ③ [39040, 58560] [3904, 5856]
Qimaxiang 51 ③ [36000, 54000] [3600, 5400]
Yingpanxiang 52 ③ [32800, 49200] [3280, 4920]
Lizhou 53 ③ [71200, 106800] [7120, 10680]
Chaotian 54 ③ [66400, 99600] [6640, 9960]
Cangxi 55 ③ [125200, 187800] [12520, 18780]
Jiange 56 ③ [62000, 93000] [6200, 9300]
Yuanba 57 ③ [71200, 106800] [7120, 10680]
Table 4.  Relevant parameters in upper-level transportation
Num. Transportation modes $ T_{ijm}^0 $ Departure point($ i \in SN $) Destination point($ j \in EN $) $ {t_{ijm}}( \otimes ) $ $ ca{p_{ijm}} $ $ I{T_{ijm}} $
1 HC 8 2 2 0.15 100 0
2 HC 8 2 4 0.18 100 0
3 HC 8 2 9 0.43 100 0
4 HC 8 2 16 0.47 100 0
5 R 8.2 1 5 0.32 150 6
6 R 9.3 1 13 0.32 150 6
7 R 9 1 14 0.23 170 6
8 R 10.2 1 15 0.42 170 6
9 A 8.2 3 11 3 800 4
10 A 8.8 3 12 0.9 800 4
11 A 9 3 13 1.8 800 4
12 A 8.5 3 14 0.6 800 4
13 A 8.7 3 15 6.6 800 4
14 HC 8 1 1 [1.60, 2.40] 80 0
15 HC 8 1 3 [1.50, 2.26] 80 0
16 HC 8 1 12 [1.95, 2.93] 80 0
17 HC 8 1 13 [3.01, 4.51] 80 0
18 HC 8 1 11 [5.09, 7.63] 80 0
19 HC 8 1 8 [1.95, 2.93] 80 0
20 HC 8 1 14 [2.45, 3.67] 90 0
21 HC 8 1 15 [6.77, 10.15] 90 0
22 HC 8 1 16 [9.92, 14.88] 90 0
23 HC 8 1 18 [6.53, 9.79] 90 0
24 HC 8 1 5 [18.88, 28.32] 90 0
25 HC 8 1 6 [2.08, 3.12] 90 0
26 HC 8 1 7 [2.08, 3.12] 90 0
27 HC 8 1 10 [13.44, 20.16] 90 0
28 HC 8 1 17 [9.68, 14.52] 85 0
29 HC 8 2 1 [1.60, 2.40] 85 0
30 HC 8 2 3 [1.52, 2.28] 85 0
31 HC 8 2 12 [1.92, 2.88] 85 0
32 HC 8 2 13 [3.04, 4.56] 85 0
33 HC 8 2 11 [5.28, 7.92] 85 0
34 HC 8 2 8 [2.27, 3.41] 85 0
35 HC 8 2 14 [2.77, 4.15] 85 0
36 HC 8 2 15 [7.09, 10.63] 85 0
37 HC 8 2 16 [10.24, 15.36] 85 0
38 HC 8 2 18 [6.85, 10.27] 85 0
39 HC 8 2 5 [19.20, 28.80] 80 0
40 HC 8 2 6 [2.40, 3.60] 80 0
41 HC 8 2 7 [2.40, 3.60] 80 0
42 HC 8 2 10 [13.76, 20.64] 80 0
43 HC 8 2 17 [10.00, 15.00] 80 0
44 HC 8 3 1 [1.36, 2.04] 90 0
45 HC 8 3 3 [1.26, 1.90] 90 0
46 HC 8 3 12 [1.71, 2.57] 90 0
47 HC 8 3 13 [2.77, 4.15] 90 0
48 HC 8 3 11 [4.85, 7.27] 90 0
49 HC 8 3 8 [1.71, 2.57] 90 0
50 HC 8 3 14 [2.21, 3.31] 90 0
51 HC 8 3 15 [6.53, 9.79] 90 0
52 HC 8 3 16 [9.68, 14.52] 90 0
53 HC 8 3 18 [6.29, 9.43] 85 0
54 HC 8 3 5 [18.64, 27.96] 85 0
55 HC 8 3 6 [1.84, 2.76] 85 0
56 HC 8 3 7 [1.84, 2.76] 85 0
57 HC 8 3 10 [13.20, 19.80] 85 0
58 HC 8 3 17 [9.44, 14.16] 85 0
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Num. Transportation modes $ T_{ijm}^0 $ Departure point($ i \in SN $) Destination point($ j \in EN $) $ {t_{ijm}}( \otimes ) $ $ ca{p_{ijm}} $ $ I{T_{ijm}} $
1 HC 8 2 2 0.15 100 0
2 HC 8 2 4 0.18 100 0
3 HC 8 2 9 0.43 100 0
4 HC 8 2 16 0.47 100 0
5 R 8.2 1 5 0.32 150 6
6 R 9.3 1 13 0.32 150 6
7 R 9 1 14 0.23 170 6
8 R 10.2 1 15 0.42 170 6
9 A 8.2 3 11 3 800 4
10 A 8.8 3 12 0.9 800 4
11 A 9 3 13 1.8 800 4
12 A 8.5 3 14 0.6 800 4
13 A 8.7 3 15 6.6 800 4
14 HC 8 1 1 [1.60, 2.40] 80 0
15 HC 8 1 3 [1.50, 2.26] 80 0
16 HC 8 1 12 [1.95, 2.93] 80 0
17 HC 8 1 13 [3.01, 4.51] 80 0
18 HC 8 1 11 [5.09, 7.63] 80 0
19 HC 8 1 8 [1.95, 2.93] 80 0
20 HC 8 1 14 [2.45, 3.67] 90 0
21 HC 8 1 15 [6.77, 10.15] 90 0
22 HC 8 1 16 [9.92, 14.88] 90 0
23 HC 8 1 18 [6.53, 9.79] 90 0
24 HC 8 1 5 [18.88, 28.32] 90 0
25 HC 8 1 6 [2.08, 3.12] 90 0
26 HC 8 1 7 [2.08, 3.12] 90 0
27 HC 8 1 10 [13.44, 20.16] 90 0
28 HC 8 1 17 [9.68, 14.52] 85 0
29 HC 8 2 1 [1.60, 2.40] 85 0
30 HC 8 2 3 [1.52, 2.28] 85 0
31 HC 8 2 12 [1.92, 2.88] 85 0
32 HC 8 2 13 [3.04, 4.56] 85 0
33 HC 8 2 11 [5.28, 7.92] 85 0
34 HC 8 2 8 [2.27, 3.41] 85 0
35 HC 8 2 14 [2.77, 4.15] 85 0
36 HC 8 2 15 [7.09, 10.63] 85 0
37 HC 8 2 16 [10.24, 15.36] 85 0
38 HC 8 2 18 [6.85, 10.27] 85 0
39 HC 8 2 5 [19.20, 28.80] 80 0
40 HC 8 2 6 [2.40, 3.60] 80 0
41 HC 8 2 7 [2.40, 3.60] 80 0
42 HC 8 2 10 [13.76, 20.64] 80 0
43 HC 8 2 17 [10.00, 15.00] 80 0
44 HC 8 3 1 [1.36, 2.04] 90 0
45 HC 8 3 3 [1.26, 1.90] 90 0
46 HC 8 3 12 [1.71, 2.57] 90 0
47 HC 8 3 13 [2.77, 4.15] 90 0
48 HC 8 3 11 [4.85, 7.27] 90 0
49 HC 8 3 8 [1.71, 2.57] 90 0
50 HC 8 3 14 [2.21, 3.31] 90 0
51 HC 8 3 15 [6.53, 9.79] 90 0
52 HC 8 3 16 [9.68, 14.52] 90 0
53 HC 8 3 18 [6.29, 9.43] 85 0
54 HC 8 3 5 [18.64, 27.96] 85 0
55 HC 8 3 6 [1.84, 2.76] 85 0
56 HC 8 3 7 [1.84, 2.76] 85 0
57 HC 8 3 10 [13.20, 19.80] 85 0
58 HC 8 3 17 [9.44, 14.16] 85 0
Table 5.  Relevant parameters in lower-level transportation
Depart Destin Depart Destin
ure ation ure ation
Num. point point $ {t'_{jk}}( \otimes ) $ $ ca{p'_{jk}} $ Num. point point $ {t'_{jk}}( \otimes ) $ $ ca{p'_{jk}} $
($ j \in $ ($ k \in $ ($ j \in $ ($ k \in $
$ EN $) $ DN $) $ EN $) $ DN $)
1 1 1 0 40 124 10 45 [9.36, 14.04] 38
2 1 2 [0.32, 0.48] 40 125 10 46 [8.69, 13.03] 38
3 1 3 [0.70, 1.06] 40 126 11 24 [1.89, 2.83] 38
4 1 4 [0.96, 1.44] 40 127 11 25 [2.45, 3.67] 38
5 1 5 [1.41, 2.11] 40 128 11 26 [1.89, 2.83] 39
6 1 21 [10.40, 15.60] 38 129 11 27 [3.81, 5.71] 39
7 2 6 [0.00, 0.00] 38 130 11 28 [2.77, 4.15] 39
8 2 7 [0.22, 0.34] 38 131 11 29 [3.01, 4.51] 39
9 2 8 [0.29, 0.43] 38 132 11 30 [2.64, 3.96] 39
10 2 9 [0.42, 0.62] 38 133 11 31 [3.36, 5.04] 39
11 2 10 [0.18, 0.26] 38 134 11 32 [2.48, 3.72] 39
12 2 11 [6.56, 9.84] 35 135 11 33 [2.56, 3.84] 39
13 2 12 [7.01, 10.51] 35 136 11 34 [2.75, 4.13] 39
14 2 13 [4.96, 7.44] 35 137 11 35 [2.88, 4.32] 39
15 2 14 [4.16, 6.24] 35 138 11 40 [2.40, 3.60] 39
16 2 15 [11.06, 16.58] 35 139 11 41 [4.00, 6.00] 40
17 2 16 [7.86, 11.78] 35 140 11 42 [2.48, 3.72] 40
18 2 17 [9.78, 14.66] 35 141 11 43 [4.19, 6.29] 40
19 2 18 [9.58, 14.38] 43 142 11 44 [3.01, 4.51] 40
20 2 19 [11.38, 17.06] 43 143 11 45 [2.40, 3.60] 40
21 2 20 [10.69, 16.03] 43 144 11 46 [1.65, 2.47] 38
22 2 22 [7.52, 11.28] 43 145 12 24 [1.63, 2.45] 38
23 2 23 [7.22, 10.82] 43 146 12 25 [2.24, 3.36] 38
24 3 1 [1.33, 1.99] 43 147 12 26 [1.63, 2.45] 38
25 3 2 [1.65, 2.47] 43 148 12 27 [3.60, 5.40] 38
26 3 3 [2.13, 3.19] 43 149 12 28 [1.09, 1.63] 38
27 3 4 [2.13, 3.19] 43 150 12 29 [1.60, 2.40] 35
28 3 5 [2.67, 4.01] 43 151 12 30 [1.12, 1.68] 35
29 3 21 [11.36, 17.04] 38 152 12 31 [2.00, 3.00] 35
30 4 6 [2.61, 3.91] 38 153 12 32 [1.44, 2.16] 35
31 4 7 [10.61, 15.91] 38 154 12 33 [1.60, 2.40] 35
32 4 8 [4.16, 6.24] 38 155 12 34 [2.08, 3.12] 35
33 4 9 [4.53, 6.79] 38 156 12 35 [1.84, 2.76] 35
34 4 10 [4.35, 6.53] 38 157 12 40 [0.93, 1.39] 43
35 4 11 [5.04, 7.56] 38 158 12 41 [2.03, 3.05] 43
36 4 12 [5.20, 7.80] 39 159 12 42 [2.85, 4.27] 43
37 4 13 [3.60, 5.40] 39 160 12 43 [4.67, 7.01] 43
38 4 14 [3.76, 5.64] 39 161 12 44 [3.52, 5.28] 43
39 4 15 0 39 162 12 45 0 43
40 4 16 [4.24, 6.36] 39 163 12 46 [1.55, 2.33] 43
41 4 17 [5.12, 7.68] 39 164 13 24 [0.67, 1.01] 43
42 4 18 [5.28, 7.92] 39 165 13 25 [1.23, 1.85] 43
43 4 19 [0.32, 0.48] 39 166 13 26 [1.01, 1.51] 43
44 4 20 [5.76, 8.64] 39 167 13 27 [2.59, 3.89] 38
45 4 22 [9.84, 14.76] 39 168 13 28 [1.81, 2.71] 38
46 4 23 [9.60, 14.40] 39 169 13 29 [2.08, 3.12] 38
47 5 1 [18.61, 27.91] 40 170 13 30 [1.68, 2.52] 38
48 5 2 [18.85, 28.27] 40 171 13 31 [2.40, 3.60] 38
49 5 3 [18.99, 28.49] 40 172 13 32 [1.52, 2.28] 38
50 5 4 [19.73, 29.59] 40 173 13 33 [1.68, 2.52] 38
51 5 5 [19.78, 29.66] 40 174 13 34 [1.52, 2.28] 39
52 5 6 [15.20, 22.80] 38 175 13 35 [1.95, 2.93] 39
53 5 7 [22.21, 33.31] 38 176 13 40 [0.99, 1.49] 39
54 5 8 [13.38, 20.06] 38 177 13 41 [2.88, 4.32] 39
55 5 9 [13.20, 19.80] 38 178 13 42 [2.08, 3.12] 39
56 5 10 [13.14, 19.70] 38 179 13 43 [3.81, 5.71] 39
57 5 11 [21.44, 32.16] 38 180 13 44 [2.56, 3.84] 39
58 5 12 [22.72, 34.08] 35 181 13 45 [1.55, 2.33] 39
59 5 13 [22.56, 33.84] 35 182 13 46 0 39
60 5 14 [22.72, 34.08] 35 183 14 24 [2.08, 3.12] 39
61 5 15 [17.60, 26.40] 35 184 14 25 [2.64, 3.96] 39
62 5 16 [21.12, 31.68] 35 185 14 26 [2.00, 3.00] 40
63 5 17 [22.72, 34.08] 35 186 14 27 [4.00, 6.00] 40
64 5 18 [22.56, 33.84] 35 187 14 28 [1.04, 1.56] 40
65 5 19 [19.04, 28.56] 43 188 14 29 [1.49, 2.23] 40
66 5 20 [24.32, 36.48] 43 189 14 30 [1.15, 1.73] 40
67 5 21 [16.00, 24.00] 43 190 14 31 [1.87, 2.81] 38
68 5 22 [17.92, 26.88] 43 191 14 32 [1.68, 2.52] 38
69 5 23 [8.00, 12.00] 43 192 14 33 [1.84, 2.76] 38
70 6 1 [1.81, 2.71] 43 193 14 34 [2.53, 3.79] 38
71 6 2 [1.92, 2.88] 43 194 14 35 [2.11, 3.17] 38
72 6 3 [1.39, 2.09] 43 195 14 40 [1.55, 2.33] 38
73 6 4 [2.40, 3.60] 43 196 14 41 [2.61, 3.91] 35
74 6 5 [2.99, 4.49] 43 197 14 42 [3.20, 4.80] 35
75 6 21 [11.44, 17.16] 38 198 14 43 [5.04, 7.56] 35
76 6 22 [16.00, 24.00] 38 199 14 44 [3.89, 5.83] 35
77 6 23 [16.00, 24.00] 38 200 14 45 [0.91, 1.37] 35
78 7 1 [2.03, 3.05] 38 201 14 46 [1.92, 2.88] 35
79 7 2 [2.16, 3.24] 38 202 15 47 0 35
80 7 3 [1.60, 2.40] 38 203 15 48 [5.12, 7.68] 43
81 7 4 [2.67, 4.01] 38 204 15 49 [5.39, 8.09] 43
82 7 5 [3.20, 4.80] 39 205 15 50 [4.40, 6.60] 43
83 7 21 [11.68, 17.52] 39 206 15 51 [4.59, 6.89] 43
84 8 24 [1.92, 2.88] 39 207 15 52 [3.63, 5.45] 43
85 8 25 [2.26, 3.38] 39 208 15 53 [0.08, 0.12] 43
86 8 26 [1.33, 1.99] 39 209 15 54 [1.84, 2.76] 43
87 8 27 [3.30, 4.94] 39 210 15 55 [3.57, 5.35] 43
88 8 28 0 39 211 15 56 [1.28, 1.92] 43
89 8 29 [0.83, 1.25] 39 212 15 57 [1.17, 1.75] 43
90 8 30 [0.48, 0.72] 39 213 16 47 [0.37, 0.55] 38
91 8 31 [1.20, 1.80] 39 214 16 48 0 38
92 8 32 [1.01, 1.51] 39 215 16 49 [0.88, 1.32] 38
93 8 33 [1.17, 1.75] 40 216 16 50 [0.72, 1.08] 38
94 8 34 [1.79, 2.69] 40 217 16 51 [1.01, 1.51] 38
95 8 35 [1.41, 2.11] 40 218 16 52 [1.60, 2.40] 38
96 8 40 [1.94, 2.90] 40 219 16 53 [5.09, 7.63] 38
97 8 41 [3.31, 4.97] 40 220 16 54 [6.66, 9.98] 39
98 8 42 [3.07, 4.61] 38 221 16 55 [6.75, 10.13] 39
99 8 43 [4.88, 7.32] 38 222 16 56 [4.53, 6.79] 39
100 8 44 [3.73, 5.59] 38 223 16 57 [5.60, 8.40] 39
101 8 45 [1.36, 2.04] 38 224 17 47 [2.16, 3.24] 39
102 8 46 [1.76, 2.64] 38 225 17 48 [6.48, 9.72] 39
103 9 36 [1.60, 2.40] 38 226 17 49 [6.56, 9.84] 39
104 9 37 [0.93, 1.39] 35 227 17 50 [5.79, 8.69] 39
105 9 38 [1.55, 2.33] 35 228 17 51 [5.97, 8.95] 39
106 9 39 [2.03, 3.05] 35 229 17 52 [5.01, 7.51] 39
107 10 24 [8.67, 13.01] 35 230 17 53 [2.29, 3.43] 39
108 10 25 [9.44, 14.16] 35 231 17 54 [4.00, 6.00] 40
109 10 26 [8.85, 13.27] 35 232 17 55 [3.55, 5.33] 40
110 10 27 [10.80, 16.20] 35 233 17 56 [2.83, 4.25] 40
111 10 28 [9.68, 14.52] 43 234 17 57 [1.20, 1.80] 40
112 10 28 [9.92, 14.88] 43 235 18 47 [1.49, 2.23] 40
113 10 30 [9.55, 14.33] 43 236 18 48 [4.53, 6.79] 38
114 10 31 [10.27, 15.41] 43 237 18 49 [4.40, 6.60] 38
115 10 32 [9.39, 14.09] 43 238 18 50 [4.13, 6.19] 38
116 10 33 [9.52, 14.28] 43 239 18 51 [4.32, 6.48] 38
117 10 34 [9.71, 14.57] 43 240 18 52 [3.36, 5.04] 38
118 10 35 [9.79, 14.69] 43 241 18 53 [1.47, 2.21] 38
119 10 40 [8.80, 13.20] 43 242 18 54 [2.82, 4.22] 35
120 10 41 [11.01, 16.51] 43 243 18 55 [2.69, 4.03] 35
121 10 42 [9.33, 13.99] 38 244 18 56 0 35
122 10 43 [11.15, 16.73] 38 245 18 57 [1.55, 2.33] 35
123 10 44 [9.89, 14.83] 38 - - - - -
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Depart Destin Depart Destin
ure ation ure ation
Num. point point $ {t'_{jk}}( \otimes ) $ $ ca{p'_{jk}} $ Num. point point $ {t'_{jk}}( \otimes ) $ $ ca{p'_{jk}} $
($ j \in $ ($ k \in $ ($ j \in $ ($ k \in $
$ EN $) $ DN $) $ EN $) $ DN $)
1 1 1 0 40 124 10 45 [9.36, 14.04] 38
2 1 2 [0.32, 0.48] 40 125 10 46 [8.69, 13.03] 38
3 1 3 [0.70, 1.06] 40 126 11 24 [1.89, 2.83] 38
4 1 4 [0.96, 1.44] 40 127 11 25 [2.45, 3.67] 38
5 1 5 [1.41, 2.11] 40 128 11 26 [1.89, 2.83] 39
6 1 21 [10.40, 15.60] 38 129 11 27 [3.81, 5.71] 39
7 2 6 [0.00, 0.00] 38 130 11 28 [2.77, 4.15] 39
8 2 7 [0.22, 0.34] 38 131 11 29 [3.01, 4.51] 39
9 2 8 [0.29, 0.43] 38 132 11 30 [2.64, 3.96] 39
10 2 9 [0.42, 0.62] 38 133 11 31 [3.36, 5.04] 39
11 2 10 [0.18, 0.26] 38 134 11 32 [2.48, 3.72] 39
12 2 11 [6.56, 9.84] 35 135 11 33 [2.56, 3.84] 39
13 2 12 [7.01, 10.51] 35 136 11 34 [2.75, 4.13] 39
14 2 13 [4.96, 7.44] 35 137 11 35 [2.88, 4.32] 39
15 2 14 [4.16, 6.24] 35 138 11 40 [2.40, 3.60] 39
16 2 15 [11.06, 16.58] 35 139 11 41 [4.00, 6.00] 40
17 2 16 [7.86, 11.78] 35 140 11 42 [2.48, 3.72] 40
18 2 17 [9.78, 14.66] 35 141 11 43 [4.19, 6.29] 40
19 2 18 [9.58, 14.38] 43 142 11 44 [3.01, 4.51] 40
20 2 19 [11.38, 17.06] 43 143 11 45 [2.40, 3.60] 40
21 2 20 [10.69, 16.03] 43 144 11 46 [1.65, 2.47] 38
22 2 22 [7.52, 11.28] 43 145 12 24 [1.63, 2.45] 38
23 2 23 [7.22, 10.82] 43 146 12 25 [2.24, 3.36] 38
24 3 1 [1.33, 1.99] 43 147 12 26 [1.63, 2.45] 38
25 3 2 [1.65, 2.47] 43 148 12 27 [3.60, 5.40] 38
26 3 3 [2.13, 3.19] 43 149 12 28 [1.09, 1.63] 38
27 3 4 [2.13, 3.19] 43 150 12 29 [1.60, 2.40] 35
28 3 5 [2.67, 4.01] 43 151 12 30 [1.12, 1.68] 35
29 3 21 [11.36, 17.04] 38 152 12 31 [2.00, 3.00] 35
30 4 6 [2.61, 3.91] 38 153 12 32 [1.44, 2.16] 35
31 4 7 [10.61, 15.91] 38 154 12 33 [1.60, 2.40] 35
32 4 8 [4.16, 6.24] 38 155 12 34 [2.08, 3.12] 35
33 4 9 [4.53, 6.79] 38 156 12 35 [1.84, 2.76] 35
34 4 10 [4.35, 6.53] 38 157 12 40 [0.93, 1.39] 43
35 4 11 [5.04, 7.56] 38 158 12 41 [2.03, 3.05] 43
36 4 12 [5.20, 7.80] 39 159 12 42 [2.85, 4.27] 43
37 4 13 [3.60, 5.40] 39 160 12 43 [4.67, 7.01] 43
38 4 14 [3.76, 5.64] 39 161 12 44 [3.52, 5.28] 43
39 4 15 0 39 162 12 45 0 43
40 4 16 [4.24, 6.36] 39 163 12 46 [1.55, 2.33] 43
41 4 17 [5.12, 7.68] 39 164 13 24 [0.67, 1.01] 43
42 4 18 [5.28, 7.92] 39 165 13 25 [1.23, 1.85] 43
43 4 19 [0.32, 0.48] 39 166 13 26 [1.01, 1.51] 43
44 4 20 [5.76, 8.64] 39 167 13 27 [2.59, 3.89] 38
45 4 22 [9.84, 14.76] 39 168 13 28 [1.81, 2.71] 38
46 4 23 [9.60, 14.40] 39 169 13 29 [2.08, 3.12] 38
47 5 1 [18.61, 27.91] 40 170 13 30 [1.68, 2.52] 38
48 5 2 [18.85, 28.27] 40 171 13 31 [2.40, 3.60] 38
49 5 3 [18.99, 28.49] 40 172 13 32 [1.52, 2.28] 38
50 5 4 [19.73, 29.59] 40 173 13 33 [1.68, 2.52] 38
51 5 5 [19.78, 29.66] 40 174 13 34 [1.52, 2.28] 39
52 5 6 [15.20, 22.80] 38 175 13 35 [1.95, 2.93] 39
53 5 7 [22.21, 33.31] 38 176 13 40 [0.99, 1.49] 39
54 5 8 [13.38, 20.06] 38 177 13 41 [2.88, 4.32] 39
55 5 9 [13.20, 19.80] 38 178 13 42 [2.08, 3.12] 39
56 5 10 [13.14, 19.70] 38 179 13 43 [3.81, 5.71] 39
57 5 11 [21.44, 32.16] 38 180 13 44 [2.56, 3.84] 39
58 5 12 [22.72, 34.08] 35 181 13 45 [1.55, 2.33] 39
59 5 13 [22.56, 33.84] 35 182 13 46 0 39
60 5 14 [22.72, 34.08] 35 183 14 24 [2.08, 3.12] 39
61 5 15 [17.60, 26.40] 35 184 14 25 [2.64, 3.96] 39
62 5 16 [21.12, 31.68] 35 185 14 26 [2.00, 3.00] 40
63 5 17 [22.72, 34.08] 35 186 14 27 [4.00, 6.00] 40
64 5 18 [22.56, 33.84] 35 187 14 28 [1.04, 1.56] 40
65 5 19 [19.04, 28.56] 43 188 14 29 [1.49, 2.23] 40
66 5 20 [24.32, 36.48] 43 189 14 30 [1.15, 1.73] 40
67 5 21 [16.00, 24.00] 43 190 14 31 [1.87, 2.81] 38
68 5 22 [17.92, 26.88] 43 191 14 32 [1.68, 2.52] 38
69 5 23 [8.00, 12.00] 43 192 14 33 [1.84, 2.76] 38
70 6 1 [1.81, 2.71] 43 193 14 34 [2.53, 3.79] 38
71 6 2 [1.92, 2.88] 43 194 14 35 [2.11, 3.17] 38
72 6 3 [1.39, 2.09] 43 195 14 40 [1.55, 2.33] 38
73 6 4 [2.40, 3.60] 43 196 14 41 [2.61, 3.91] 35
74 6 5 [2.99, 4.49] 43 197 14 42 [3.20, 4.80] 35
75 6 21 [11.44, 17.16] 38 198 14 43 [5.04, 7.56] 35
76 6 22 [16.00, 24.00] 38 199 14 44 [3.89, 5.83] 35
77 6 23 [16.00, 24.00] 38 200 14 45 [0.91, 1.37] 35
78 7 1 [2.03, 3.05] 38 201 14 46 [1.92, 2.88] 35
79 7 2 [2.16, 3.24] 38 202 15 47 0 35
80 7 3 [1.60, 2.40] 38 203 15 48 [5.12, 7.68] 43
81 7 4 [2.67, 4.01] 38 204 15 49 [5.39, 8.09] 43
82 7 5 [3.20, 4.80] 39 205 15 50 [4.40, 6.60] 43
83 7 21 [11.68, 17.52] 39 206 15 51 [4.59, 6.89] 43
84 8 24 [1.92, 2.88] 39 207 15 52 [3.63, 5.45] 43
85 8 25 [2.26, 3.38] 39 208 15 53 [0.08, 0.12] 43
86 8 26 [1.33, 1.99] 39 209 15 54 [1.84, 2.76] 43
87 8 27 [3.30, 4.94] 39 210 15 55 [3.57, 5.35] 43
88 8 28 0 39 211 15 56 [1.28, 1.92] 43
89 8 29 [0.83, 1.25] 39 212 15 57 [1.17, 1.75] 43
90 8 30 [0.48, 0.72] 39 213 16 47 [0.37, 0.55] 38
91 8 31 [1.20, 1.80] 39 214 16 48 0 38
92 8 32 [1.01, 1.51] 39 215 16 49 [0.88, 1.32] 38
93 8 33 [1.17, 1.75] 40 216 16 50 [0.72, 1.08] 38
94 8 34 [1.79, 2.69] 40 217 16 51 [1.01, 1.51] 38
95 8 35 [1.41, 2.11] 40 218 16 52 [1.60, 2.40] 38
96 8 40 [1.94, 2.90] 40 219 16 53 [5.09, 7.63] 38
97 8 41 [3.31, 4.97] 40 220 16 54 [6.66, 9.98] 39
98 8 42 [3.07, 4.61] 38 221 16 55 [6.75, 10.13] 39
99 8 43 [4.88, 7.32] 38 222 16 56 [4.53, 6.79] 39
100 8 44 [3.73, 5.59] 38 223 16 57 [5.60, 8.40] 39
101 8 45 [1.36, 2.04] 38 224 17 47 [2.16, 3.24] 39
102 8 46 [1.76, 2.64] 38 225 17 48 [6.48, 9.72] 39
103 9 36 [1.60, 2.40] 38 226 17 49 [6.56, 9.84] 39
104 9 37 [0.93, 1.39] 35 227 17 50 [5.79, 8.69] 39
105 9 38 [1.55, 2.33] 35 228 17 51 [5.97, 8.95] 39
106 9 39 [2.03, 3.05] 35 229 17 52 [5.01, 7.51] 39
107 10 24 [8.67, 13.01] 35 230 17 53 [2.29, 3.43] 39
108 10 25 [9.44, 14.16] 35 231 17 54 [4.00, 6.00] 40
109 10 26 [8.85, 13.27] 35 232 17 55 [3.55, 5.33] 40
110 10 27 [10.80, 16.20] 35 233 17 56 [2.83, 4.25] 40
111 10 28 [9.68, 14.52] 43 234 17 57 [1.20, 1.80] 40
112 10 28 [9.92, 14.88] 43 235 18 47 [1.49, 2.23] 40
113 10 30 [9.55, 14.33] 43 236 18 48 [4.53, 6.79] 38
114 10 31 [10.27, 15.41] 43 237 18 49 [4.40, 6.60] 38
115 10 32 [9.39, 14.09] 43 238 18 50 [4.13, 6.19] 38
116 10 33 [9.52, 14.28] 43 239 18 51 [4.32, 6.48] 38
117 10 34 [9.71, 14.57] 43 240 18 52 [3.36, 5.04] 38
118 10 35 [9.79, 14.69] 43 241 18 53 [1.47, 2.21] 38
119 10 40 [8.80, 13.20] 43 242 18 54 [2.82, 4.22] 35
120 10 41 [11.01, 16.51] 43 243 18 55 [2.69, 4.03] 35
121 10 42 [9.33, 13.99] 38 244 18 56 0 35
122 10 43 [11.15, 16.73] 38 245 18 57 [1.55, 2.33] 35
123 10 44 [9.89, 14.83] 38 - - - - -
Table 6.  Transportation of relief materials from collection centers to opened temporary transfer facilities
Collection centers Opened temporary transfer facility Transportation mode Food (units) Collection centers Opened temporary transfer facility Transportation mode Daily necessities (units)
facility facility
Ⅱ (1) H 54849 Ⅰ (1) H 15869
Ⅲ (1) H 71647 Ⅱ (12) H 13904
Ⅲ (17) H 80471 Ⅰ (16) H 13037
Ⅲ (12) A 384669 Ⅰ (17) H 9487
Ⅲ (13) A 304150 Ⅲ (12) A 49057
Ⅲ (14) A 153849 Ⅲ (13) A 80943
Ⅲ (15) A 205214 Ⅰ (14) R 25568
Ⅰ (13) R 182926 Ⅰ (15) R 26039
Ⅰ (15) R 17074 Ⅱ (2) HC 12652
Ⅱ (2) HC 113786 Ⅱ (4) HC 18159
Ⅱ (4) HC 80529 Ⅱ (9) HC 5805
Ⅱ (9) HC 77345 Ⅱ (16) HC 11686
Ⅱ (6) HC 107638 - - - -
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Collection centers Opened temporary transfer facility Transportation mode Food (units) Collection centers Opened temporary transfer facility Transportation mode Daily necessities (units)
facility facility
Ⅱ (1) H 54849 Ⅰ (1) H 15869
Ⅲ (1) H 71647 Ⅱ (12) H 13904
Ⅲ (17) H 80471 Ⅰ (16) H 13037
Ⅲ (12) A 384669 Ⅰ (17) H 9487
Ⅲ (13) A 304150 Ⅲ (12) A 49057
Ⅲ (14) A 153849 Ⅲ (13) A 80943
Ⅲ (15) A 205214 Ⅰ (14) R 25568
Ⅰ (13) R 182926 Ⅰ (15) R 26039
Ⅰ (15) R 17074 Ⅱ (2) HC 12652
Ⅱ (2) HC 113786 Ⅱ (4) HC 18159
Ⅱ (4) HC 80529 Ⅱ (9) HC 5805
Ⅱ (9) HC 77345 Ⅱ (16) HC 11686
Ⅱ (6) HC 107638 - - - -
Table 7.  Transportation of relief materials from opened temporary transfer facilities to affected points
Opened temporary transfer facility Affected point Food (units) Opened temporary transfer facility Affected point Daily necessities (units)
(1) 1 71036 (1) 1 8753
(1) 2 13446 (1) 2 1657
(1) 3 12330 (1) 3 1519
(1) 4 11163 (1) 4 1376
(1) 5 12178 (1) 5 1501
(2) 6 54926 (2) 6 6768
(2) 7 9133 (2) 7 1126
(2) 8 9641 (2) 8 1188
(2) 9 10148 (2) 9 1251
(2) 10 8119 (2) 10 1001
(4) 11 54292 (4) 11 6690
(4) 12 8119 (4) 12 1001
(4) 13 5074 (4) 13 626
(4) 14 12178 (4) 14 1501
(4) 15 78267 (4) 15 9644
(4) 16 10148 (4) 16 1251
(4) 17 9133 (4) 17 1125
(4) 18 8626 (4) 18 1063
(4) 19 8829 (4) 19 1088
(4) 20 7814 (4) 20 963
(1) 21 6343 (1) 21 1063
(2) 22 15730 (2) 22 1938
(2) 23 6089 (2) 23 719
(13) 24 113026 (13) 24 18783
(13) 25 16801 (13) 25 2792
(13) 26 19821 (13) 26 3294
(13) 27 10760 (13) 27 1789
(14) 28 113262 (14) 28 18822
(14) 29 17934 (14) 29 2981
(12) 30 24541 (12) 30 4078
(14) 31 22653 (14) 31 3765
(12) 32 137095 (12) 32 22782
(12) 33 16612 (12) 33 2761
(13) 34 14725 (13) 34 2447
(12) 35 12837 (12) 35 2134
(9) 36 143230 (9) 36 23802
(9) 37 18311 (9) 37 3043
(9) 38 19821 (9) 38 3294
(9) 39 17934 (9) 39 2981
(12) 40 41058 (12) 40 6643
(12) 41 50024 (12) 41 7529
(13) 42 64654 (13) 42 10744
(13) 43 64890 (13) 43 10783
(13) 44 73856 (13) 44 12273
(12) 45 102502 (12) 45 17034
(13) 46 108543 (13) 46 18038
(15) 47 48232 (15) 47 5687
(16) 48 139090 (16) 48 16399
(16) 49 21185 (16) 49 2498
(16) 50 25093 (16) 50 2959
(16) 51 23139 (16) 51 2728
(16) 52 21082 (16) 52 2486
(15) 53 45764 (15) 53 5396
(15) 54 42678 (15) 54 4862
(17) 55 80471 (17) 55 9487
(15) 56 39850 (15) 56 4698
(15) 57 45764 (15) 57 5396
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Opened temporary transfer facility Affected point Food (units) Opened temporary transfer facility Affected point Daily necessities (units)
(1) 1 71036 (1) 1 8753
(1) 2 13446 (1) 2 1657
(1) 3 12330 (1) 3 1519
(1) 4 11163 (1) 4 1376
(1) 5 12178 (1) 5 1501
(2) 6 54926 (2) 6 6768
(2) 7 9133 (2) 7 1126
(2) 8 9641 (2) 8 1188
(2) 9 10148 (2) 9 1251
(2) 10 8119 (2) 10 1001
(4) 11 54292 (4) 11 6690
(4) 12 8119 (4) 12 1001
(4) 13 5074 (4) 13 626
(4) 14 12178 (4) 14 1501
(4) 15 78267 (4) 15 9644
(4) 16 10148 (4) 16 1251
(4) 17 9133 (4) 17 1125
(4) 18 8626 (4) 18 1063
(4) 19 8829 (4) 19 1088
(4) 20 7814 (4) 20 963
(1) 21 6343 (1) 21 1063
(2) 22 15730 (2) 22 1938
(2) 23 6089 (2) 23 719
(13) 24 113026 (13) 24 18783
(13) 25 16801 (13) 25 2792
(13) 26 19821 (13) 26 3294
(13) 27 10760 (13) 27 1789
(14) 28 113262 (14) 28 18822
(14) 29 17934 (14) 29 2981
(12) 30 24541 (12) 30 4078
(14) 31 22653 (14) 31 3765
(12) 32 137095 (12) 32 22782
(12) 33 16612 (12) 33 2761
(13) 34 14725 (13) 34 2447
(12) 35 12837 (12) 35 2134
(9) 36 143230 (9) 36 23802
(9) 37 18311 (9) 37 3043
(9) 38 19821 (9) 38 3294
(9) 39 17934 (9) 39 2981
(12) 40 41058 (12) 40 6643
(12) 41 50024 (12) 41 7529
(13) 42 64654 (13) 42 10744
(13) 43 64890 (13) 43 10783
(13) 44 73856 (13) 44 12273
(12) 45 102502 (12) 45 17034
(13) 46 108543 (13) 46 18038
(15) 47 48232 (15) 47 5687
(16) 48 139090 (16) 48 16399
(16) 49 21185 (16) 49 2498
(16) 50 25093 (16) 50 2959
(16) 51 23139 (16) 51 2728
(16) 52 21082 (16) 52 2486
(15) 53 45764 (15) 53 5396
(15) 54 42678 (15) 54 4862
(17) 55 80471 (17) 55 9487
(15) 56 39850 (15) 56 4698
(15) 57 45764 (15) 57 5396
Table 8.  Transportation of relief materials from opened temporary transfer facilities to affected points
Affected point $ {\tilde T_{kc}}( \otimes ) $(hours) Affected point $ {\tilde T_{kc}}( \otimes ) $(hours)
food Daily necessities food Daily necessities
1 10 10 30 11.1 11.8
2 10.4 10.4 31 11.44 11.57
3 10.88 10.88 32 22.3 23
4 11.2 11.2 33 11.7 12.4
5 11.76 11.76 34 12.7 12.7
6 8.15 16.3 35 12 12.7
7 8.43 16.58 36 30.86 39.29
8 8.51 16.66 37 18.02 26.45
9 8.67 16.82 38 18.8 27.23
10 8.37 16.52 39 19.4 27.83
11 35.26 22.66 40 10.86 11.56
12 22.86 22.86 41 12.24 12.94
13 20.86 20.86 42 18.6 18.6
14 21.06 21.06 43 25.08 25.08
15 16.36 16.36 44 20.4 20.4
16 21.66 21.66 45 9.7 10.4
17 22.76 22.76 46 10.8 10.8
18 22.96 22.96 47 15.3 10.62
19 16.76 16.76 48 16.94 20.4
20 23.56 23.56 49 18.04 21.5
21 23 23 50 17.84 21.3
22 17.55 25.7 51 18.2 21.66
23 17.17 25.32 52 18.94 22.4
24 15 15 53 15.4 10.72
25 12.34 12.34 54 17.6 12.92
26 12.06 12.06 55 33.12 33.42
27 14.04 14.04 56 16.9 12.22
28 15.6 15.73 57 16.76 12.08
29 10.96 11.09 - - -
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Affected point $ {\tilde T_{kc}}( \otimes ) $(hours) Affected point $ {\tilde T_{kc}}( \otimes ) $(hours)
food Daily necessities food Daily necessities
1 10 10 30 11.1 11.8
2 10.4 10.4 31 11.44 11.57
3 10.88 10.88 32 22.3 23
4 11.2 11.2 33 11.7 12.4
5 11.76 11.76 34 12.7 12.7
6 8.15 16.3 35 12 12.7
7 8.43 16.58 36 30.86 39.29
8 8.51 16.66 37 18.02 26.45
9 8.67 16.82 38 18.8 27.23
10 8.37 16.52 39 19.4 27.83
11 35.26 22.66 40 10.86 11.56
12 22.86 22.86 41 12.24 12.94
13 20.86 20.86 42 18.6 18.6
14 21.06 21.06 43 25.08 25.08
15 16.36 16.36 44 20.4 20.4
16 21.66 21.66 45 9.7 10.4
17 22.76 22.76 46 10.8 10.8
18 22.96 22.96 47 15.3 10.62
19 16.76 16.76 48 16.94 20.4
20 23.56 23.56 49 18.04 21.5
21 23 23 50 17.84 21.3
22 17.55 25.7 51 18.2 21.66
23 17.17 25.32 52 18.94 22.4
24 15 15 53 15.4 10.72
25 12.34 12.34 54 17.6 12.92
26 12.06 12.06 55 33.12 33.42
27 14.04 14.04 56 16.9 12.22
28 15.6 15.73 57 16.76 12.08
29 10.96 11.09 - - -
Table 9.  Sensitivity analysis of $ P $
P Amount of opened facilities Opened temporary transfer facilities (the optimal solution) Objective function value Running time(s)
MIN MAX AVG Optimal AVG SD AVG minimum SD
4 4 4 4 (5), (9), (12), (15) 13273776.83 14868939.06 1451416.62 53.48 16.16 42.31
5 5 5 5 (2), (3), (9), (13), (15) 8848462.92 9220754.33 302415.93 207.17 165.37 25.97
6 6 6 6 (1), (4), (9), (12), (13), (15) 6853198.43 7271162.90 290586.68 244.28 183.08 36.24
7 7 7 7 (1), (2), (4), (9), (13), (14), (15) 5781612.49 6227580.83 273625.21 257.40 234.93 12.80
8 8 8 8 (1), (2), (4), (9), (12), (13), (15), (16) 5072974.53 5295173.25 224424.74 244.43 178.19 49.48
9 9 9 9 (1), (2), (4), (9), (13), (14), (15), (16), (17) 4858475.72 5124041.30 150278.08 244.43 178.19 49.48
10 9 10 9.8 (1), (2), (4), (9), (12), (13), (14), (15), (16), (17) 4782950.76 4924828.55 113448.04 233.81 193.92 26.61
11 11 11 11 (1), (2), (3), (4), (9), (12), (13), (14), (15), (16), (17) 4778328.99 4925983.68 107442.10 228.31 213.40 16.51
12 11 12 11.2 (1), (2), (4), (6), (9), (12), (13), (14), (15), (16), (18) 4776395.25 4878544.65 110360.54 232.67 198.17 27.35
13 12 13 12.4 (1), (2), (3), (4), (5), (9), (10), (12), (13), (14), (15), (16), (18) 4783358.20 4892302.40 102522.06 233.92 221.46 7.82
14 11 14 12.5 (1), (2), (4), (6), (9), (12), (13), (14), (15), (16), (17) 4757147.13 4856777.97 106004.73 237.36 224.33 12.72
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P Amount of opened facilities Opened temporary transfer facilities (the optimal solution) Objective function value Running time(s)
MIN MAX AVG Optimal AVG SD AVG minimum SD
4 4 4 4 (5), (9), (12), (15) 13273776.83 14868939.06 1451416.62 53.48 16.16 42.31
5 5 5 5 (2), (3), (9), (13), (15) 8848462.92 9220754.33 302415.93 207.17 165.37 25.97
6 6 6 6 (1), (4), (9), (12), (13), (15) 6853198.43 7271162.90 290586.68 244.28 183.08 36.24
7 7 7 7 (1), (2), (4), (9), (13), (14), (15) 5781612.49 6227580.83 273625.21 257.40 234.93 12.80
8 8 8 8 (1), (2), (4), (9), (12), (13), (15), (16) 5072974.53 5295173.25 224424.74 244.43 178.19 49.48
9 9 9 9 (1), (2), (4), (9), (13), (14), (15), (16), (17) 4858475.72 5124041.30 150278.08 244.43 178.19 49.48
10 9 10 9.8 (1), (2), (4), (9), (12), (13), (14), (15), (16), (17) 4782950.76 4924828.55 113448.04 233.81 193.92 26.61
11 11 11 11 (1), (2), (3), (4), (9), (12), (13), (14), (15), (16), (17) 4778328.99 4925983.68 107442.10 228.31 213.40 16.51
12 11 12 11.2 (1), (2), (4), (6), (9), (12), (13), (14), (15), (16), (18) 4776395.25 4878544.65 110360.54 232.67 198.17 27.35
13 12 13 12.4 (1), (2), (3), (4), (5), (9), (10), (12), (13), (14), (15), (16), (18) 4783358.20 4892302.40 102522.06 233.92 221.46 7.82
14 11 14 12.5 (1), (2), (4), (6), (9), (12), (13), (14), (15), (16), (17) 4757147.13 4856777.97 106004.73 237.36 224.33 12.72
Table 10.  Performance comparison between genetic algorithm (GA) and immune optimization algorithm (IOA)
$ P $ Algorithm $ Z_{_1}^* $ $ {\bar Z_1} $ Time(AVG)
9 GA 4858475.72 5124041.30 244.43
IOA 4936050.86 5244041.02 250.67
10 GA 4782950.76 4924828.55 233.81
IOA 4821963.21 4959828.55 230.14
11 GA 4778328.99 4925983.68 228.31
IOA 4790375.49 4965983.68 225.63
12 GA 4776395.25 4878544.65 232.67
IOA 4795203.36 4879121.35 229.43
13 GA 4783358.20 4892302.40 233.92
IOA 4790522.90 4898580.12 228.45
16 GA 4757147.13 4856777.97 237.36
IOA 4767658.14 4848676.16 233.97
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$ P $ Algorithm $ Z_{_1}^* $ $ {\bar Z_1} $ Time(AVG)
9 GA 4858475.72 5124041.30 244.43
IOA 4936050.86 5244041.02 250.67
10 GA 4782950.76 4924828.55 233.81
IOA 4821963.21 4959828.55 230.14
11 GA 4778328.99 4925983.68 228.31
IOA 4790375.49 4965983.68 225.63
12 GA 4776395.25 4878544.65 232.67
IOA 4795203.36 4879121.35 229.43
13 GA 4783358.20 4892302.40 233.92
IOA 4790522.90 4898580.12 228.45
16 GA 4757147.13 4856777.97 237.36
IOA 4767658.14 4848676.16 233.97
[1]

Gaidi Li, Zhen Wang, Dachuan Xu. An approximation algorithm for the $k$-level facility location problem with submodular penalties. Journal of Industrial & Management Optimization, 2012, 8 (3) : 521-529. doi: 10.3934/jimo.2012.8.521
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