doi: 10.3934/jimo.2022145
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A hybrid metaheuristic algorithm for the multi-objective location-routing problem in the early post-disaster stage

1. 

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110819, China

2. 

School of Economics and Management, Yanshan University, Qinhuangdao, Hebei 066004, China

*Corresponding author: Fuqiang Lu

Received  December 2021 Revised  June 2022 Early access August 2022

Disasters such as earthquakes, typhoons, floods and COVID-19 continue to threaten the lives of people in all countries. In order to cover the basic needs of the victims, emergency logistics should be implemented in time. Location-routing problem (LRP) tackles facility location problem and vehicle routing problem simultaneously to obtain the overall optimization. In response to the shortage of relief materials in the early post-disaster stage, a multi-objective model for the LRP considering fairness is constructed by evaluating the urgency coefficients of all demand points. The objectives are the lowest cost, delivery time and degree of dissatisfaction. Since LRP is a NP-hard problem, a hybrid metaheuristic algorithm of Discrete Particle Swarm Optimization (DPSO) and Harris Hawks Optimization (HHO) is designed to solve the model. In addition, three improvement strategies, namely elite-opposition learning, nonlinear escaping energy, multi-probability random walk, are introduced to enhance its execution efficiency. Finally, the effectiveness and performance of the LRP model and the hybrid metaheuristic algorithm are verified by a case study of COVID-19 in Wuhan. It demonstrates that the hybrid metaheuristic algorithm is more competitive with higher accuracy and the ability to jump out of the local optimum than other metaheuristic algorithms.

Citation: Tongren Yan, Fuqiang Lu, Suxin Wang, Leizhen Wang, Hualing Bi. A hybrid metaheuristic algorithm for the multi-objective location-routing problem in the early post-disaster stage. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022145
References:
[1]

K. AlamatsazA. Ahmadi and S. M. J. M. Al-e-hashem, A multiobjective model for the green capacitated location-routing problem considering drivers' satisfaction and time window with uncertain demand, Environmental Science and Pollution Research, 29 (2022), 5052-5071.  doi: 10.1007/s11356-021-15907-x.

[2]

R. BandyopadhyayR. Kundu and D. Oliva, Segmentation of brain MRI using an altruistic Harris Hawks' Optimization algorithm, Knowledge-Based Systems, 232 (2021), 107468. 

[3]

K. L. Chang, H. Zhou, G. J. Chen and H. Q. Chen, Multiobjective location routing problem considering uncertain data after disasters, Discrete Dyn. Nat. Soc., (2017), Art. ID 1703608, 7 pp. doi: 10.1155/2017/1703608.

[4]

L. Cooper, The transportation-location problem, Operations Research, 20 (1972), 94-108.  doi: 10.1287/opre.20.1.94.

[5]

G. Dhiman and V. Kumar, Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems, Knowledge-Based Systems, 165 (2019), 169-196.  doi: 10.1016/j.knosys.2018.11.024.

[6]

R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, (1995), 39–43. doi: 10.1109/MHS.1995.494215.

[7]

A. M. Fathollahi-FardM. Hajiaghaei-KeshteliR. Tavakkoli-Moghaddam and N. R. Smith, Bi-level programming for home health care supply chain considering outsourcing, Journal of Industrial Information Integration, 25 (2021), 100246.  doi: 10.1016/j.jii.2021.100246.

[8]

X. H. Gao, A bi-level stochastic optimization model for multi-commodity rebalancing under uncertainty in disaster response, Annals of Operations Research, (2019), 1–34. doi: 10.1007/s10479-019-03506-6.

[9]

A. Gogna and A. Tayal, Metaheuristics: Review and application, Journal of Experimental & Theoretical Artificial Intelligence, 25 (2013), 503-526.  doi: 10.1080/0952813X.2013.782347.

[10]

A. A. HeidariS. MirjaliliH. FarisI. AljarahM. Mafarja and H. L. Chen, Harris hawks optimization: Algorithm and applications, Future Generation Computer Systems, 97 (2019), 849-872.  doi: 10.1016/j.future.2019.02.028.

[11]

J. H. Holland, Genetic algorithms, Scientific American, 267 (1992), 66-73. 

[12]

M. IrfanA. WadoodT. KhurshaidB. M. KhanK.-C. KimS.-R. Oh and S.-B. Rhee, An optimized adaptive protection scheme for numerical and directional overcurrent relay coordination using Harris Hawk Optimization, Energies, 14 (2021), 5603.  doi: 10.3390/en14185603.

[13]

M. Issa and A. Samn, Passive vehicle suspension system optimization using Harris Hawk Optimization algorithm, Math. Comput. Simulation, 191 (2022), 328-345.  doi: 10.1016/j.matcom.2021.08.016.

[14]

Y. P. Jiang and Y. F. Yuan, Emergency logistics in a large-scale disaster context: Achievements and challenges, International Journal of Environmental Research and Public Health, 16 (2019), 779.  doi: 10.3390/ijerph16050779.

[15]

J. Kennedy and R. Eberhart, A discrete binary version of the particle swarm algorithm, International Conference on Systems, Man, and Cybernetics, 5 (1997), 4104-4108.  doi: 10.1109/ICSMC.1997.637339.

[16]

S. KhanchehzarrinM. P. PanahN. Mahdavi-Amiri and S. Shiripour, A bi-level multi-objective location-routing optimization model for disaster relief operations considering public donations, Socio-Economic Planning Sciences, 80 (2021), 101165.  doi: 10.1016/j.seps.2021.101165.

[17]

G. Kou, H. Xiao, M. H. Cao and L. H. Lee, Optimal computing budget allocation for the vector evaluated genetic algorithm in multi-objective simulation optimization, Automatica, 129 (2021), 109599, 14 pp. doi: 10.1016/j.automatica.2021.109599.

[18]

Y. H. Li and W. D. Chen, Entropy method of constructing a combined model for improving loan default prediction: A case study in China, Journal of the Operational Research Society, 72 (2021), 1099-1109.  doi: 10.1080/01605682.2019.1702905.

[19]

B. LiangD. P. YangX. H. Qin and T. Tinta, A risk-averse shelter location and evacuation routing assignment problem in an uncertain environment, International Journal of Environmental Research and Public Health, 16 (2019), 4007.  doi: 10.3390/ijerph16204007.

[20]

C. S. LiuG. KouY. 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.

[21]

R. B. LopesC. Ferreira and B. S. Santos, A simple and effective evolutionary algorithm for the capacitated location-routing problem, Comput. Oper. Res., 70 (2016), 155-162.  doi: 10.1016/j.cor.2016.01.006.

[22]

Z. Mahtab, A. Azeem, S. M. Ali, S. K. Paul and A. M. Fathollahi-Fard, Multi-objective robust-stochastic optimisation of relief goods distribution under uncertainty: A real-life case study, International Journal of Systems Science: Operations & Logistics, (2021), 241–262. doi: 10.1080/23302674.2021.1879305.

[23]

Y. Marinakis, An improved particle swarm optimization algorithm for the capacitated location routing problem and for the location routing problem with stochastic demands, Applied Soft Computing, 37 (2015), 680-701.  doi: 10.1016/j.asoc.2015.09.005.

[24]

S. Mirjalili and A. Lewis, The whale optimization algorithm, Advances in Engineering Software, 95 (2016), 51-67.  doi: 10.1016/j.advengsoft.2016.01.008.

[25]

M. MojtahediR. Y. SunindijoF. LestariSu parni and O. Wijaya, Developing hospital emergency and disaster management index using TOPSIS method, Sustainability, 13 (2021), 5213.  doi: 10.3390/su13095213.

[26]

A. M. NezhadroshanA. M. Fathollahi-Fard and M. Hajiaghaei-Keshteli, A scenario-based possibilistic-stochastic programming approach to address resilient humanitarian logistics considering travel time and resilience levels of facilities, International Journal of Systems Science: Operations & Logistics, 8 (2021), 321-347.  doi: 10.1080/23302674.2020.1769766.

[27]

A. F. Pena-Delgado, H. Peraza-Vazquez and J. H. Almazan-Covarrubias, A novel bio-inspired algorithm applied to selective harmonic elimination in a three-phase eleven-level inverter, Mathematical Problems in Engineering, (2020).

[28]

J. Piri and P. Mohapatra, An analytical study of modified multi-objective Harris Hawk Optimizer towards medical data feature selection, Computers in Biology and Medicine, 135 (2021), 104558.  doi: 10.1016/j.compbiomed.2021.104558.

[29]

B. RabtaC. Wankmuller and G. Reiner, A drone fleet model for last-mile distribution in disaster relief operations, International Journal of Disaster Risk Reduction, 28 (2018), 107-112.  doi: 10.1016/j.ijdrr.2018.02.020.

[30]

B. SaeidianM. S. Mesgari and M. Ghodousi, Evaluation and comparison of Genetic Algorithm and Bees Algorithm for location-allocation of earthquake relief centers, International Journal of Disaster Risk Reduction, 15 (2016), 94-107.  doi: 10.1016/j.ijdrr.2016.01.002.

[31]

S. M. Shavarani, Multi-level facility location-allocation problem for post-disaster humanitarian relief distribution: A case study, Journal of Humanitarian Logistics and Supply Chain Management, 9 (2019), 70-81.  doi: 10.1108/JHLSCM-05-2018-0036.

[32]

L. ShenF. M. TaoY. H. Shi and R. R. Qin, Optimization of location-routing problem in emergency logistics considering carbon emissions, International Journal of Environmental Research and Public Health, 16 (2019), 2982.  doi: 10.3390/ijerph16162982.

[33]

H. R. Tizhoosh, Opposition-based learning: A new scheme for machine intelligence, International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, 1 (2015), 695-701.  doi: 10.1109/CIMCA.2005.1631345.

[34]

B. VahdaniD. VeysmoradiN. Shekari and S. M. Mousavi, Multi-objective, multi-period location-routing model to distribute relief after earthquake by considering emergency roadway repair, Neural Computing and Applications, 30 (2018), 835-854.  doi: 10.1007/s00521-016-2696-7.

[35]

D. VeysmoradiB. VahdaniM. F. Sartangi and S. M. Mousavi, Multi-objective open location-routing model for relief distribution networks with split delivery and multi-mode transportation under uncertainty, Scientia Iranica, 25 (2018), 3635-3653.  doi: 10.24200/sci.2017.4572.

[36]

S. X. WangH. Wen and F. Q. Lu, Modeling of 3D cargo loading problem and optimization of crow search algorithm, Journal of Hunan University Natural Sciences, 47 (2020), 21-30. 

[37]

X. W. Xiong, F. Zhao, Y. D. Wang and Y. P. Wang, Research on the model and algorithm for multimodal distribution of emergency supplies after earthquake in the perspective of fairness, Mathematical Problems in Engineering, (2019), 1629321. doi: 10.1155/2019/1629321.

[38]

R. Y. Yin and P. X. Lu, A cluster-first route-second constructive heuristic method for emergency logistics scheduling in urban transport networks, Sustainability, 14 (2022), 2301.  doi: 10.3390/su14042301.

[39]

S. Y. Yin, L. J. Du, Y. F. Xia and Z. Ye, Review on model and algorithms of the post-disaster relief distribution problem, Journal of Physics: Conference Series, 1302 (2019), 022003. doi: 10.1088/1742-6596/1302/2/022003.

[40]

S. ZhangQ. F. Luo and Y. Q. Zhou, Hybrid grey wolf optimizer using elite opposition-based learning strategy and simplex method, International Journal of Computational Intelligence and Applications, 16 (2017), 1750012.  doi: 10.1142/S1469026817500122.

show all references

References:
[1]

K. AlamatsazA. Ahmadi and S. M. J. M. Al-e-hashem, A multiobjective model for the green capacitated location-routing problem considering drivers' satisfaction and time window with uncertain demand, Environmental Science and Pollution Research, 29 (2022), 5052-5071.  doi: 10.1007/s11356-021-15907-x.

[2]

R. BandyopadhyayR. Kundu and D. Oliva, Segmentation of brain MRI using an altruistic Harris Hawks' Optimization algorithm, Knowledge-Based Systems, 232 (2021), 107468. 

[3]

K. L. Chang, H. Zhou, G. J. Chen and H. Q. Chen, Multiobjective location routing problem considering uncertain data after disasters, Discrete Dyn. Nat. Soc., (2017), Art. ID 1703608, 7 pp. doi: 10.1155/2017/1703608.

[4]

L. Cooper, The transportation-location problem, Operations Research, 20 (1972), 94-108.  doi: 10.1287/opre.20.1.94.

[5]

G. Dhiman and V. Kumar, Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems, Knowledge-Based Systems, 165 (2019), 169-196.  doi: 10.1016/j.knosys.2018.11.024.

[6]

R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, (1995), 39–43. doi: 10.1109/MHS.1995.494215.

[7]

A. M. Fathollahi-FardM. Hajiaghaei-KeshteliR. Tavakkoli-Moghaddam and N. R. Smith, Bi-level programming for home health care supply chain considering outsourcing, Journal of Industrial Information Integration, 25 (2021), 100246.  doi: 10.1016/j.jii.2021.100246.

[8]

X. H. Gao, A bi-level stochastic optimization model for multi-commodity rebalancing under uncertainty in disaster response, Annals of Operations Research, (2019), 1–34. doi: 10.1007/s10479-019-03506-6.

[9]

A. Gogna and A. Tayal, Metaheuristics: Review and application, Journal of Experimental & Theoretical Artificial Intelligence, 25 (2013), 503-526.  doi: 10.1080/0952813X.2013.782347.

[10]

A. A. HeidariS. MirjaliliH. FarisI. AljarahM. Mafarja and H. L. Chen, Harris hawks optimization: Algorithm and applications, Future Generation Computer Systems, 97 (2019), 849-872.  doi: 10.1016/j.future.2019.02.028.

[11]

J. H. Holland, Genetic algorithms, Scientific American, 267 (1992), 66-73. 

[12]

M. IrfanA. WadoodT. KhurshaidB. M. KhanK.-C. KimS.-R. Oh and S.-B. Rhee, An optimized adaptive protection scheme for numerical and directional overcurrent relay coordination using Harris Hawk Optimization, Energies, 14 (2021), 5603.  doi: 10.3390/en14185603.

[13]

M. Issa and A. Samn, Passive vehicle suspension system optimization using Harris Hawk Optimization algorithm, Math. Comput. Simulation, 191 (2022), 328-345.  doi: 10.1016/j.matcom.2021.08.016.

[14]

Y. P. Jiang and Y. F. Yuan, Emergency logistics in a large-scale disaster context: Achievements and challenges, International Journal of Environmental Research and Public Health, 16 (2019), 779.  doi: 10.3390/ijerph16050779.

[15]

J. Kennedy and R. Eberhart, A discrete binary version of the particle swarm algorithm, International Conference on Systems, Man, and Cybernetics, 5 (1997), 4104-4108.  doi: 10.1109/ICSMC.1997.637339.

[16]

S. KhanchehzarrinM. P. PanahN. Mahdavi-Amiri and S. Shiripour, A bi-level multi-objective location-routing optimization model for disaster relief operations considering public donations, Socio-Economic Planning Sciences, 80 (2021), 101165.  doi: 10.1016/j.seps.2021.101165.

[17]

G. Kou, H. Xiao, M. H. Cao and L. H. Lee, Optimal computing budget allocation for the vector evaluated genetic algorithm in multi-objective simulation optimization, Automatica, 129 (2021), 109599, 14 pp. doi: 10.1016/j.automatica.2021.109599.

[18]

Y. H. Li and W. D. Chen, Entropy method of constructing a combined model for improving loan default prediction: A case study in China, Journal of the Operational Research Society, 72 (2021), 1099-1109.  doi: 10.1080/01605682.2019.1702905.

[19]

B. LiangD. P. YangX. H. Qin and T. Tinta, A risk-averse shelter location and evacuation routing assignment problem in an uncertain environment, International Journal of Environmental Research and Public Health, 16 (2019), 4007.  doi: 10.3390/ijerph16204007.

[20]

C. S. LiuG. KouY. 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.

[21]

R. B. LopesC. Ferreira and B. S. Santos, A simple and effective evolutionary algorithm for the capacitated location-routing problem, Comput. Oper. Res., 70 (2016), 155-162.  doi: 10.1016/j.cor.2016.01.006.

[22]

Z. Mahtab, A. Azeem, S. M. Ali, S. K. Paul and A. M. Fathollahi-Fard, Multi-objective robust-stochastic optimisation of relief goods distribution under uncertainty: A real-life case study, International Journal of Systems Science: Operations & Logistics, (2021), 241–262. doi: 10.1080/23302674.2021.1879305.

[23]

Y. Marinakis, An improved particle swarm optimization algorithm for the capacitated location routing problem and for the location routing problem with stochastic demands, Applied Soft Computing, 37 (2015), 680-701.  doi: 10.1016/j.asoc.2015.09.005.

[24]

S. Mirjalili and A. Lewis, The whale optimization algorithm, Advances in Engineering Software, 95 (2016), 51-67.  doi: 10.1016/j.advengsoft.2016.01.008.

[25]

M. MojtahediR. Y. SunindijoF. LestariSu parni and O. Wijaya, Developing hospital emergency and disaster management index using TOPSIS method, Sustainability, 13 (2021), 5213.  doi: 10.3390/su13095213.

[26]

A. M. NezhadroshanA. M. Fathollahi-Fard and M. Hajiaghaei-Keshteli, A scenario-based possibilistic-stochastic programming approach to address resilient humanitarian logistics considering travel time and resilience levels of facilities, International Journal of Systems Science: Operations & Logistics, 8 (2021), 321-347.  doi: 10.1080/23302674.2020.1769766.

[27]

A. F. Pena-Delgado, H. Peraza-Vazquez and J. H. Almazan-Covarrubias, A novel bio-inspired algorithm applied to selective harmonic elimination in a three-phase eleven-level inverter, Mathematical Problems in Engineering, (2020).

[28]

J. Piri and P. Mohapatra, An analytical study of modified multi-objective Harris Hawk Optimizer towards medical data feature selection, Computers in Biology and Medicine, 135 (2021), 104558.  doi: 10.1016/j.compbiomed.2021.104558.

[29]

B. RabtaC. Wankmuller and G. Reiner, A drone fleet model for last-mile distribution in disaster relief operations, International Journal of Disaster Risk Reduction, 28 (2018), 107-112.  doi: 10.1016/j.ijdrr.2018.02.020.

[30]

B. SaeidianM. S. Mesgari and M. Ghodousi, Evaluation and comparison of Genetic Algorithm and Bees Algorithm for location-allocation of earthquake relief centers, International Journal of Disaster Risk Reduction, 15 (2016), 94-107.  doi: 10.1016/j.ijdrr.2016.01.002.

[31]

S. M. Shavarani, Multi-level facility location-allocation problem for post-disaster humanitarian relief distribution: A case study, Journal of Humanitarian Logistics and Supply Chain Management, 9 (2019), 70-81.  doi: 10.1108/JHLSCM-05-2018-0036.

[32]

L. ShenF. M. TaoY. H. Shi and R. R. Qin, Optimization of location-routing problem in emergency logistics considering carbon emissions, International Journal of Environmental Research and Public Health, 16 (2019), 2982.  doi: 10.3390/ijerph16162982.

[33]

H. R. Tizhoosh, Opposition-based learning: A new scheme for machine intelligence, International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, 1 (2015), 695-701.  doi: 10.1109/CIMCA.2005.1631345.

[34]

B. VahdaniD. VeysmoradiN. Shekari and S. M. Mousavi, Multi-objective, multi-period location-routing model to distribute relief after earthquake by considering emergency roadway repair, Neural Computing and Applications, 30 (2018), 835-854.  doi: 10.1007/s00521-016-2696-7.

[35]

D. VeysmoradiB. VahdaniM. F. Sartangi and S. M. Mousavi, Multi-objective open location-routing model for relief distribution networks with split delivery and multi-mode transportation under uncertainty, Scientia Iranica, 25 (2018), 3635-3653.  doi: 10.24200/sci.2017.4572.

[36]

S. X. WangH. Wen and F. Q. Lu, Modeling of 3D cargo loading problem and optimization of crow search algorithm, Journal of Hunan University Natural Sciences, 47 (2020), 21-30. 

[37]

X. W. Xiong, F. Zhao, Y. D. Wang and Y. P. Wang, Research on the model and algorithm for multimodal distribution of emergency supplies after earthquake in the perspective of fairness, Mathematical Problems in Engineering, (2019), 1629321. doi: 10.1155/2019/1629321.

[38]

R. Y. Yin and P. X. Lu, A cluster-first route-second constructive heuristic method for emergency logistics scheduling in urban transport networks, Sustainability, 14 (2022), 2301.  doi: 10.3390/su14042301.

[39]

S. Y. Yin, L. J. Du, Y. F. Xia and Z. Ye, Review on model and algorithms of the post-disaster relief distribution problem, Journal of Physics: Conference Series, 1302 (2019), 022003. doi: 10.1088/1742-6596/1302/2/022003.

[40]

S. ZhangQ. F. Luo and Y. Q. Zhou, Hybrid grey wolf optimizer using elite opposition-based learning strategy and simplex method, International Journal of Computational Intelligence and Applications, 16 (2017), 1750012.  doi: 10.1142/S1469026817500122.

Figure 1.  A simplified diagram of location-routing problem
Figure 2.  Encoding scheme
Figure 3.  An example of the encoding scheme
Figure 4.  The change curve of sigmoid function
Figure 5.  The change curve of unimproved $ E $
Figure 6.  The change curve of improved $ E $
Figure 7.  The flow chart of IHDH
Figure 8.  The geographical location information of the 7 candidate delivery centers and the 13 demand points
Figure 9.  The iterative convergence curve of IHDH
Figure 10.  The convergence curves of the average of 30 running results
Figure 11.  The box-plot for the seven metaheuristic algorithms
Table 1.  Explanations of corresponding notations
Notation Explanation
Sets
$ I $ Set of candidate distribution centers.
$ J $ Set of demand points.
$ K $ Set of vehicles.
Parameters
$ Q $ Maximum supply of relief materials.
$ Q_{1i} $ Maximum capacity of distribution center $ i $.
$ Q_{2k} $ Maximum capacity of vehicle $ k $.
$ C_{1i} $ Construction and operation costs of distribution center $ i $.
$ C_{2k} $ Unit transportation cost of vehicle $ k $.
$ q_j $ Demand of demand point $ j $.
$ \lambda_j $ Urgency coefficient of demand point $ j $.
$ d_{ij} $ Distance between node $ i $ and node $ j $.
$ t_{ijk} $ Time for vehicle $ k $ to transport from node $ i $ to node $ j $.
$ u_j $ Quantity of relief materials received at demand point $ j $.
Decisions variables
$ z_i $ Whether candidate distribution center $ i $ is employed.
$ y_{ijk} $ Whether vehicle $ k $ delivers relief materials from node $ i $ to $ j $.
$ x_{ij} $ Whether demand point $ j $ is served by distribution center $ i $.
$ X_{ij} $ Quantity of relief materials provided by distribution center
$ i $ to demand point $ j $.
Notation Explanation
Sets
$ I $ Set of candidate distribution centers.
$ J $ Set of demand points.
$ K $ Set of vehicles.
Parameters
$ Q $ Maximum supply of relief materials.
$ Q_{1i} $ Maximum capacity of distribution center $ i $.
$ Q_{2k} $ Maximum capacity of vehicle $ k $.
$ C_{1i} $ Construction and operation costs of distribution center $ i $.
$ C_{2k} $ Unit transportation cost of vehicle $ k $.
$ q_j $ Demand of demand point $ j $.
$ \lambda_j $ Urgency coefficient of demand point $ j $.
$ d_{ij} $ Distance between node $ i $ and node $ j $.
$ t_{ijk} $ Time for vehicle $ k $ to transport from node $ i $ to node $ j $.
$ u_j $ Quantity of relief materials received at demand point $ j $.
Decisions variables
$ z_i $ Whether candidate distribution center $ i $ is employed.
$ y_{ijk} $ Whether vehicle $ k $ delivers relief materials from node $ i $ to $ j $.
$ x_{ij} $ Whether demand point $ j $ is served by distribution center $ i $.
$ X_{ij} $ Quantity of relief materials provided by distribution center
$ i $ to demand point $ j $.
Table 2.  The overall vehicle delivery routes of the example
Vehicle number Delivery route of the vehicle
1 $ 3 \longrightarrow 6 \longrightarrow 8 \longrightarrow 3 $
2 $ 1 \longrightarrow 4 \longrightarrow 1 $
3 $ 3 \longrightarrow 3 $
4 $ 1 \longrightarrow 5 \longrightarrow 7 \longrightarrow 1 $
Vehicle number Delivery route of the vehicle
1 $ 3 \longrightarrow 6 \longrightarrow 8 \longrightarrow 3 $
2 $ 1 \longrightarrow 4 \longrightarrow 1 $
3 $ 3 \longrightarrow 3 $
4 $ 1 \longrightarrow 5 \longrightarrow 7 \longrightarrow 1 $
Table 3.  The comparison results between the precise algorithm and the hybrid metaheuristic algorithm in the small scale experiment
Solving algorithm Fitness CPU time (s)
Precise algorithm 1.2763 224.169
Metaheuristic algorithm 1.2814 3.465
Solving algorithm Fitness CPU time (s)
Precise algorithm 1.2763 224.169
Metaheuristic algorithm 1.2814 3.465
Table 4.  The parameters of candidate delivery centers
Candidate delivery center Number Cost (CNY) Capacity (kg)
Wuhan Railway Station 1 40000 20000
Wuchang Railway Station 2 36000 18000
Hankou Railway Station 3 34000 17000
Huashannan Railway Station 4 32000 16000
Tangxunhu Railway Station 5 30000 15000
Tianhe International Airport 6 40000 20000
Hannan General Airport 7 34000 17000
Candidate delivery center Number Cost (CNY) Capacity (kg)
Wuhan Railway Station 1 40000 20000
Wuchang Railway Station 2 36000 18000
Hankou Railway Station 3 34000 17000
Huashannan Railway Station 4 32000 16000
Tangxunhu Railway Station 5 30000 15000
Tianhe International Airport 6 40000 20000
Hannan General Airport 7 34000 17000
Table 5.  The parameters of demand vehicles
Number of vehicles Cost (CNY) Velocity (km/h) Capacity (kg)
1–3 8000 45 320
4–6 9000 45 360
7–10 10000 45 400
Number of vehicles Cost (CNY) Velocity (km/h) Capacity (kg)
1–3 8000 45 320
4–6 9000 45 360
7–10 10000 45 400
Table 6.  The parameters of demand points
Demand point Number Demand (kg)
Jiangan District 8 5100
Jianghan District 9 9400
Qiaokou District 10 9700
Hanyang District 11 4600
Wuchang District 12 8900
Qingshan District 13 4200
Hongshan District 14 4500
Hannan District 15 3900
Caidian District 16 2400
Jiangxia District 17 1700
Huangpi District 18 1400
Xinzhou District 19 1200
Dongxihu District 20 1900
Demand point Number Demand (kg)
Jiangan District 8 5100
Jianghan District 9 9400
Qiaokou District 10 9700
Hanyang District 11 4600
Wuchang District 12 8900
Qingshan District 13 4200
Hongshan District 14 4500
Hannan District 15 3900
Caidian District 16 2400
Jiangxia District 17 1700
Huangpi District 18 1400
Xinzhou District 19 1200
Dongxihu District 20 1900
Table 7.  The relevant information of the 13 districts in Wuhan on February 29
Number Cumulative cases Additional cases Percentage Density (/km$ ^{2} $) GDP (CNY)
8 4300 183 0.4466% 11993 $ 1335.9 \times 10^8 $
9 7290 191 0.9989% 25797 $ 1428.5 \times 10^8 $
10 7239 739 0.8331% 21689 $ 874.18 \times 10^8 $
11 3579 407 0.5342% 6007 $ 761.39 \times 10^8 $
12 8224 351 0.6398% 19904 $ 1522.04 \times 10^8 $
13 2958 143 0.5592% 9261 $ 863.14 \times 10^8 $
14 4999 351 0.2918% 2988 $ 1058.58 \times 10^8 $
15 1517 47 1.1154% 474 $ 1643.53 \times 10^8 $
16 1974 120 0.2515% 718 $ 400.53 \times 10^8 $
17 1661 196 0.1683% 489 $ 948.8 \times 10^8 $
18 1775 106 0.1727% 456 $ 1071.1 \times 10^8 $
19 1018 22 0.1111% 626 $ 951.61 \times 10^8 $
20 2588 65 0.4304% 1214 $ 1351.63 \times 10^8 $
Number Cumulative cases Additional cases Percentage Density (/km$ ^{2} $) GDP (CNY)
8 4300 183 0.4466% 11993 $ 1335.9 \times 10^8 $
9 7290 191 0.9989% 25797 $ 1428.5 \times 10^8 $
10 7239 739 0.8331% 21689 $ 874.18 \times 10^8 $
11 3579 407 0.5342% 6007 $ 761.39 \times 10^8 $
12 8224 351 0.6398% 19904 $ 1522.04 \times 10^8 $
13 2958 143 0.5592% 9261 $ 863.14 \times 10^8 $
14 4999 351 0.2918% 2988 $ 1058.58 \times 10^8 $
15 1517 47 1.1154% 474 $ 1643.53 \times 10^8 $
16 1974 120 0.2515% 718 $ 400.53 \times 10^8 $
17 1661 196 0.1683% 489 $ 948.8 \times 10^8 $
18 1775 106 0.1727% 456 $ 1071.1 \times 10^8 $
19 1018 22 0.1111% 626 $ 951.61 \times 10^8 $
20 2588 65 0.4304% 1214 $ 1351.63 \times 10^8 $
Table 8.  The weights of these evaluation indicators
Evaluation indicator Indicator weight
Cumulative cases 0.181597
Additional cases 0.191141
Percentage 0.161722
Density 0.35917
GDP 0.10637
Evaluation indicator Indicator weight
Cumulative cases 0.181597
Additional cases 0.191141
Percentage 0.161722
Density 0.35917
GDP 0.10637
Table 9.  The urgent coefficients of the 13 districts in Wuhan
Number of demand points Urgent coefficient
8 0.0879481
9 0.155844
10 0.18134
11 0.0765009
12 0.145336
13 0.0749613
14 0.0624538
15 0.0583092
16 0.0450429
17 0.0323935
18 0.0247054
19 0.0244273
20 0.0307375
Number of demand points Urgent coefficient
8 0.0879481
9 0.155844
10 0.18134
11 0.0765009
12 0.145336
13 0.0749613
14 0.0624538
15 0.0583092
16 0.0450429
17 0.0323935
18 0.0247054
19 0.0244273
20 0.0307375
Table 10.  The specific information of the selected distribution centers in the optimal LRP scheme
Number of selected delivery centers Capacity (kg) Actual supply (kg)
1 20000 17000
2 18000 8400
3 17000 17000
5 30000 12600
Number of selected delivery centers Capacity (kg) Actual supply (kg)
1 20000 17000
2 18000 8400
3 17000 17000
5 30000 12600
Table 11.  The vehicle delivery routes in the optimal LRP scheme
Number of vehicles Delivery route of the vehicle
1 $ 1 \longrightarrow 20 \longrightarrow 19 \longrightarrow 18 \longrightarrow 13 \longrightarrow 1 $
2 $ 5 \longrightarrow 16 \longrightarrow 11 \longrightarrow 5 $
3 $ 5 \longrightarrow 17 \longrightarrow 15 \longrightarrow 5 $
4 $ 2 \longrightarrow 12 \longrightarrow 2 $
5 $ 1 \longrightarrow 8 \longrightarrow 14 \longrightarrow 1 $
6 $ 1 \longrightarrow 1 $
7 $ 3 \longrightarrow 10 \longrightarrow 3 $
8 $ 3 \longrightarrow 3 $
9 $ 5 \longrightarrow 5 $
10 $ 3 \longrightarrow 9 \longrightarrow 3 $
Number of vehicles Delivery route of the vehicle
1 $ 1 \longrightarrow 20 \longrightarrow 19 \longrightarrow 18 \longrightarrow 13 \longrightarrow 1 $
2 $ 5 \longrightarrow 16 \longrightarrow 11 \longrightarrow 5 $
3 $ 5 \longrightarrow 17 \longrightarrow 15 \longrightarrow 5 $
4 $ 2 \longrightarrow 12 \longrightarrow 2 $
5 $ 1 \longrightarrow 8 \longrightarrow 14 \longrightarrow 1 $
6 $ 1 \longrightarrow 1 $
7 $ 3 \longrightarrow 10 \longrightarrow 3 $
8 $ 3 \longrightarrow 3 $
9 $ 5 \longrightarrow 5 $
10 $ 3 \longrightarrow 9 \longrightarrow 3 $
Table 12.  The specific information of the demand points in the optimal LRP scheme
Number of demand points Urgency coefficient Demand (kg) (kg) Quantities received (kg)
8 0.0879481 5100 5100
9 0.155844 9400 7300
10 0.18134 9700 9700
11 0.0765009 4600 4600
12 0.145336 8900 8400
13 0.0749613 4200 4200
14 0.0624538 4500 3900
15 0.0583092 3900 3900
16 0.0450429 3400 2400
17 0.0323935 1700 1700
18 0.0247054 1400 1400
19 0.0244273 1200 500
20 0.0307375 1900 1900
Number of demand points Urgency coefficient Demand (kg) (kg) Quantities received (kg)
8 0.0879481 5100 5100
9 0.155844 9400 7300
10 0.18134 9700 9700
11 0.0765009 4600 4600
12 0.145336 8900 8400
13 0.0749613 4200 4200
14 0.0624538 4500 3900
15 0.0583092 3900 3900
16 0.0450429 3400 2400
17 0.0323935 1700 1700
18 0.0247054 1400 1400
19 0.0244273 1200 500
20 0.0307375 1900 1900
Table 13.  The experiment results of these metaheuristic algorithms
Algorithm Best fitness Worst fitness Average fitness Standard deviation Average iterations to its optimum Time (s)
SA 1.5038 1.6284 1.5557 0.0342 599 3.762
DPSO 1.5821 1.6807 1.6201 0.0337 503 4.753
WOA 1.5515 1.6512 1.5952 0.0328 459 5.668
SOA 1.5065 1.6351 1.5859 0.0373 481 5.343
BWO 1.5006 1.6649 1.6005 0.0472 514 2.805
HDH 1.5131 1.6058 1.5701 0.0274 372 6.164
IHDH 1.4029 1.5056 1.4595 0.0365 251 13.567
Algorithm Best fitness Worst fitness Average fitness Standard deviation Average iterations to its optimum Time (s)
SA 1.5038 1.6284 1.5557 0.0342 599 3.762
DPSO 1.5821 1.6807 1.6201 0.0337 503 4.753
WOA 1.5515 1.6512 1.5952 0.0328 459 5.668
SOA 1.5065 1.6351 1.5859 0.0373 481 5.343
BWO 1.5006 1.6649 1.6005 0.0472 514 2.805
HDH 1.5131 1.6058 1.5701 0.0274 372 6.164
IHDH 1.4029 1.5056 1.4595 0.0365 251 13.567
[1]

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 and Management Optimization, 2022, 18 (3) : 1949-1977. doi: 10.3934/jimo.2021051

[2]

Lixin Wei, Bin Feng, Qingsong Liu. Picker routing optimization of storage stacker based on improved multi-objective iterative local search algorithm. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022187

[3]

Anh Son Ta, Le Thi Hoai An, Djamel Khadraoui, Pham Dinh Tao. Solving Partitioning-Hub Location-Routing Problem using DCA. Journal of Industrial and Management Optimization, 2012, 8 (1) : 87-102. doi: 10.3934/jimo.2012.8.87

[4]

Xiliang Sun, Wanjie Hu, Xiaolong Xue, Jianjun Dong. Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study. Journal of Industrial and Management Optimization, 2023, 19 (1) : 170-196. doi: 10.3934/jimo.2021179

[5]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1651-1663. doi: 10.3934/jimo.2021037

[6]

Min Zhang, Gang Li. Multi-objective optimization algorithm based on improved particle swarm in cloud computing environment. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1413-1426. doi: 10.3934/dcdss.2019097

[7]

Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 121-134. doi: 10.3934/naco.2021055

[8]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial and Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130

[9]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001

[10]

Jian Xiong, Zhongbao Zhou, Ke Tian, Tianjun Liao, Jianmai Shi. A multi-objective approach for weapon selection and planning problems in dynamic environments. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1189-1211. doi: 10.3934/jimo.2016068

[11]

Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487

[12]

Hamed Fazlollahtabar, Mohammad Saidi-Mehrabad. Optimizing multi-objective decision making having qualitative evaluation. Journal of Industrial and Management Optimization, 2015, 11 (3) : 747-762. doi: 10.3934/jimo.2015.11.747

[13]

Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2669-2683. doi: 10.3934/jimo.2020088

[14]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[15]

Fan Zhang, Guifa Teng, Mengmeng Gao, Shuai Zhang, Jingjing Zhang. Multi-machine and multi-task emergency allocation algorithm based on precedence rules. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1501-1513. doi: 10.3934/dcdss.2019103

[16]

Shungen Luo, Xiuping Guo. Multi-objective optimization of multi-microgrid power dispatch under uncertainties using interval optimization. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021208

[17]

Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial and Management Optimization, 2022, 18 (2) : 897-932. doi: 10.3934/jimo.2021002

[18]

Jiao-Yan Li, Xiao Hu, Zhong 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, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200

[19]

Giuseppe Buttazzo, Serena Guarino Lo Bianco, Fabrizio Oliviero. Optimal location problems with routing cost. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1301-1317. doi: 10.3934/dcds.2014.34.1301

[20]

Adriel Cheng, Cheng-Chew Lim. Optimizing system-on-chip verifications with multi-objective genetic evolutionary algorithms. Journal of Industrial and Management Optimization, 2014, 10 (2) : 383-396. doi: 10.3934/jimo.2014.10.383

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (131)
  • HTML views (247)
  • Cited by (0)

[Back to Top]