March  2022, 18(2): 1035-1062. doi: 10.3934/jimo.2021007

Ambulance routing in disaster response considering variable patient condition: NSGA-II and MOPSO algorithms

1. 

School of Industrial Engineering, University of Tehran, Tehran, Iran

* Corresponding author: Masoud Rabbani, Tel: +98 21 88335605, Email: mrabani@ut.ac.ir

Received  May 2020 Revised  September 2020 Published  March 2022 Early access  January 2021

The shortage of relief vehicles capacity is a common issue throughout disastrous situations due to the abundance of injured people who need urgent medical aid. Hence, ambulances fleet management is highly important to save as many injured individuals as possible. In this regard, the present paper defines different patient groups based on their needs and characteristics. In order to provide the affected people with proper and timely medical aid, changes in their health status are also considered. A Mixed-integer Linear Programming (MILP) model is proposed to find the best sequence of routes for each ambulance and minimize the latest service completion time (SCT) as well as the number of patients whose condition gets worse because of receiving untimely medical services. Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO) are used to find high-quality solutions over a short time. In the end, Lorestan province, Iran, is considered as a case study to assess the model's performance and analyze the sensitivity of solutions with respect to the major parameters, which results in insightful managerial suggestions.

Citation: Masoud Rabbani, Nastaran Oladzad-Abbasabady, Niloofar Akbarian-Saravi. Ambulance routing in disaster response considering variable patient condition: NSGA-II and MOPSO algorithms. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1035-1062. doi: 10.3934/jimo.2021007
References:
[1]

T. Andersson and P. Värbrand, Decision support tools for ambulance dispatch and relocation, in Operational Research for Emergency Planning in Healthcare: Volume 1, Palgrave Macmillan, London, 195-201. doi: 10.1057/9781137535696_3.

[2]

E. Babaee TirkolaeeA. GoliM. Pahlevan and R. Malekalipour Kordestanizadeh, A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management & Research, 37 (2019), 1089-1101.  doi: 10.1177/0734242X19865340.

[3]

A. BaşarB. Çatay and T. Ünlüyurt, A taxonomy for emergency service station location problem, Optimization Letters, 6 (2012), 1147-1160.  doi: 10.1007/s11590-011-0376-1.

[4]

D. BerkouneJ. RenaudM. Rekik and A. Ruiz, Transportation in disaster response operations, Socio-Economic Planning Sciences, 46 (2012), 23-32.  doi: 10.1016/j.seps.2011.05.002.

[5]

O. BermanZ. Drezner and G. O. Wesolowsky, The facility and transfer points location problem, International Transactions in Operational Research, 12 (2005), 387-402.  doi: 10.1111/j.1475-3995.2005.00514.x.

[6]

A. Bozorgi-AmiriS. TavakoliH. Mirzaeipour and M. Rabbani, Integrated locating of helicopter stations and helipads for wounded transfer under demand location uncertainty, The American Journal of Emergency Medicine, 35 (2017), 410-417.  doi: 10.1016/j.ajem.2016.11.024.

[7]

C. C. Branas, E. J. MacKenzie and C. S. ReVelle, A trauma resource allocation model for ambulances and hospitals, Health Services Research, 35 (2000), 489.

[8]

J.-F. Camacho-VallejoE. González-RodríguezF.-J. Almaguer and R. G. González-Ramírez, A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production, 105 (2015), 134-145. 

[9]

C. A. Coello CoelloG. T. Pulido and Mk Salazar Lechuga, Handling multiple objectives with particle swarm optimization, IEEE Transactions on Evolutionary Computation, 8 (2004), 256-279.  doi: 10.1109/TEVC.2004.826067.

[10]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.

[11]

A. E. Eiben, Z. Michalewicz, M. Schoenauer and J. E. Smith, Parameter control in evolutionary algorithms, In Parameter Setting in Evolutionary Algorithms, Springer, 2007, 19-46.

[12]

E. T. ErdemirR. BattaP. A. RogersonA. Blatt and Ma rie Flanigan, Joint ground and air emergency medical services coverage models: A greedy heuristic solution approach, European Journal of Operational Research, 207 (2010), 736-749.  doi: 10.1016/j.ejor.2010.05.047.

[13]

J. A. Fitzsimmons and B. N. Srikar, Emergency ambulance location using the contiguous zone search routine, Journal of Operations Management, 2 (1982), 225-237.  doi: 10.1016/0272-6963(82)90011-0.

[14]

T. Furuta and K.-i. Tanaka, Minisum and minimax location models for helicopter emergency medical service systems, Journal of the Operations Research Society of Japan, 56 (2013), 221-242.  doi: 10.15807/jorsj.56.221.

[15]

H. Garg, An efficient biogeography based optimization algorithm for solving reliability optimization problems, Swarm and Evolutionary Computation, 24 (2015), 1-10.  doi: 10.1016/j.swevo.2015.05.001.

[16]

H. Garg, A hybrid PSO-GA algorithm for constrained optimization problems, Applied Mathematics and Computation, 274 (2016), 292-305.  doi: 10.1016/j.amc.2015.11.001.

[17]

H. Garg, A hybrid GSA-GA algorithm for constrained optimization problems, Information Sciences, 478 (2019), 499-523.  doi: 10.1016/j.ins.2018.11.041.

[18]

H. Garg and S. P. Sharma, Multi-objective reliability-redundancy allocation problem using particle swarm optimization, Computers & Industrial Engineering, 64 (2013), 247-255.  doi: 10.1016/j.cie.2012.09.015.

[19]

M. GendreauG. Laporte and F. Semet, A dynamic model and parallel tabu search heuristic for real-time ambulance relocation, Parallel Computing, 27 (2001), 1641-1653.  doi: 10.1016/S0167-8191(01)00103-X.

[20]

Z. GhelichiM. Saidi-Mehrabad and M. S. Pishvaee, A stochastic programming approach toward optimal design and planning of an integrated green biodiesel supply chain network under uncertainty: A case study, Energy, 156 (2018), 661-687.  doi: 10.1016/j.energy.2018.05.103.

[21]

Z. GhelichiJ. Tajik and M. S. Pishvaee, A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: A case study of Mashhad, Computers & Chemical Engineering, 110 (2018), 13-34.  doi: 10.1016/j.compchemeng.2017.11.017.

[22]

A. GhodratnamaH. R. Arbabi and A. Azaron, Production planning in industrial townships modeled as hub location-allocation problems considering congestion in manufacturing plants, Computers & Industrial Engineering, 129 (2019), 479-501.  doi: 10.1016/j.cie.2019.01.049.

[23]

R. Goldberg and P. Listowsky, Critical factors for emergency vehicle routing expert systems, Expert Systems with Applications, 7 (1994), 589-602.  doi: 10.1016/0957-4174(94)90082-5.

[24]

J. Holguín-VerasM. JallerL. N. Van WassenhoveN. Pérez and T. Wachtendorf, On the unique features of post-disaster humanitarian logistics, Journal of Operations Management, 30 (2012), 494-506. 

[25]

S. A. Hosseinijou and M. Bashiri, Stochastic models for transfer point location problem, The International Journal of Advanced Manufacturing Technology, 58 (2012), 211-225.  doi: 10.1007/s00170-011-3360-0.

[26]

A. JotshiQ. Gong and R. Batta, Dispatching and routing of emergency vehicles in disaster mitigation using data fusion, Socio-Economic Planning Sciences, 43 (2009), 1-24.  doi: 10.1016/j.seps.2008.02.005.

[27]

H. KalantariA. YousefliM. Ghazanfari and K. Shahanaghi, Fuzzy transfer point location problem: A possibilistic unconstrained nonlinear programming approach, The International Journal of Advanced Manufacturing Technology, 70 (2014), 1043-1051.  doi: 10.1007/s00170-013-5338-6.

[28]

V. A. KnightP. R. Harper and L. Smith, Ambulance allocation for maximal survival with heterogeneous outcome measures, Omega, 40 (2012), 918-926.  doi: 10.1016/j.omega.2012.02.003.

[29]

G. Mavrotas and K. Florios, An improved version of the augmented $\varepsilon$-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems, Applied Mathematics and Computation, 219 (2013), 9652-9669.  doi: 10.1016/j.amc.2013.03.002.

[30]

F. NavaziR. Tavakkoli-Moghaddam and Z. Sazvar, A multi-period location-allocation-inventory problem for ambulance and helicopter ambulance stations: Robust possibilistic approach, IFAC-PapersOnLine, 51 (2018), 322-327.  doi: 10.1016/j.ifacol.2018.08.303.

[31]

L. Özdamar and M. A. Ertem, Models, solutions and enabling technologies in humanitarian logistics, European Journal of Operational Research, 244 (2015), 55-65.  doi: 10.1016/j.ejor.2014.11.030.

[32]

R. S. PatwalN. Narang and H. Garg, A novel TVAC-PSO based mutation strategies algorithm for generation scheduling of pumped storage hydrothermal system incorporating solar units, Energy, 142 (2018), 822-837.  doi: 10.1016/j.energy.2017.10.052.

[33]

A. J. Pedraza-Martinez and L. N. Van Wassenhove, Transportation and vehicle fleet management in humanitarian logistics: Challenges for future research, EURO Journal on Transportation and Logistics, 1 (2012), 185-196.  doi: 10.1007/s13676-012-0001-1.

[34]

M. Rabbani, S. Momen, N. Akbarian-Saravi, H. Farrokhi-Asl and Z. Ghelichi, Optimal design for sustainable bioethanol supply chain considering the bioethanol production strategies: A case study, Computers & Chemical Engineering, 134 (2020), 106720. doi: 10.1016/j.compchemeng.2019.106720.

[35]

M. SasakiT. Furuta and A. Suzuki, Exact optimal solutions of the minisum facility and transfer points location problems on a network, International Transactions in Operational Research, 15 (2008), 295-306.  doi: 10.1111/j.1475-3995.2008.00602.x.

[36]

V. Schmid, Solving the dynamic ambulance relocation and dispatching problem using approximate dynamic programming, European Journal of Operational Research, 219 (2012), 611-621.  doi: 10.1016/j.ejor.2011.10.043.

[37]

N. Schuurman, N. J. Bell, R. L'Heureux and S. M Hameed, Modelling optimal location for pre-hospital helicopter emergency medical services, BMC Emergency Medicine, 9 (2009), Art. No. 6. doi: 10.1186/1471-227X-9-6.

[38]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.  doi: 10.1162/evco.1994.2.3.221.

[39]

G. Taguchi, Introduction to quality engineering: designing quality into products and processes, Technical report, 1986.

[40]

L. TalaricoF. Meisel and K. Sörensen, Ambulance routing for disaster response with patient groups, Computers & Operations Research, 56 (2015), 120-133.  doi: 10.1016/j.cor.2014.11.006.

[41]

H. Tikani and M. Setak, Ambulance routing in disaster response scenario considering different types of ambulances and semi soft time windows, Journal of Industrial and Systems Engineering, 12 (2019), 95-128. 

[42]

E. B. Tirkolaee, A. Goli and G.-W. Weber, Fuzzy mathematical programming and self-adaptive artificial fish swarm algorithm for just-in-time energy-aware flow shop scheduling problem with outsourcing option, IEEE Transactions on Fuzzy Systems, 2020.

[43]

E. B. TirkolaeeS. HadianG.-W. Weber and I. Mahdavi, A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.  doi: 10.1111/coin.12240.

[44]

T. TliliS. Abidi and S. Krichen, A mathematical model for efficient emergency transportation in a disaster situation, The American Journal of Emergency Medicine, 36 (2018), 1585-1590.  doi: 10.1016/j.ajem.2018.01.039.

[45]

H. Toro-DíAzM. E. MayorgaS. Chanta and and L. A. Mclay, Joint location and dispatching decisions for emergency medical services, Computers & Industrial Engineering, 64 (2013), 917-928. 

[46]

Y.-J. ZhengS.-Y. Chen and and H.-F. Ling, Evolutionary optimization for disaster relief operations: A survey, Applied Soft Computing, 27 (2015), 553-566.  doi: 10.1016/j.asoc.2014.09.041.

[47]

A. ZhouB.-Y. QuH. LiS.-Z. ZhaoP. N. Suganthan and Q. Zhang, Multiobjective evolutionary algorithms: A survey of the state of the art, Swarm and Evolutionary Computation, 1 (2011), 32-49.  doi: 10.1016/j.swevo.2011.03.001.

show all references

References:
[1]

T. Andersson and P. Värbrand, Decision support tools for ambulance dispatch and relocation, in Operational Research for Emergency Planning in Healthcare: Volume 1, Palgrave Macmillan, London, 195-201. doi: 10.1057/9781137535696_3.

[2]

E. Babaee TirkolaeeA. GoliM. Pahlevan and R. Malekalipour Kordestanizadeh, A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management & Research, 37 (2019), 1089-1101.  doi: 10.1177/0734242X19865340.

[3]

A. BaşarB. Çatay and T. Ünlüyurt, A taxonomy for emergency service station location problem, Optimization Letters, 6 (2012), 1147-1160.  doi: 10.1007/s11590-011-0376-1.

[4]

D. BerkouneJ. RenaudM. Rekik and A. Ruiz, Transportation in disaster response operations, Socio-Economic Planning Sciences, 46 (2012), 23-32.  doi: 10.1016/j.seps.2011.05.002.

[5]

O. BermanZ. Drezner and G. O. Wesolowsky, The facility and transfer points location problem, International Transactions in Operational Research, 12 (2005), 387-402.  doi: 10.1111/j.1475-3995.2005.00514.x.

[6]

A. Bozorgi-AmiriS. TavakoliH. Mirzaeipour and M. Rabbani, Integrated locating of helicopter stations and helipads for wounded transfer under demand location uncertainty, The American Journal of Emergency Medicine, 35 (2017), 410-417.  doi: 10.1016/j.ajem.2016.11.024.

[7]

C. C. Branas, E. J. MacKenzie and C. S. ReVelle, A trauma resource allocation model for ambulances and hospitals, Health Services Research, 35 (2000), 489.

[8]

J.-F. Camacho-VallejoE. González-RodríguezF.-J. Almaguer and R. G. González-Ramírez, A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production, 105 (2015), 134-145. 

[9]

C. A. Coello CoelloG. T. Pulido and Mk Salazar Lechuga, Handling multiple objectives with particle swarm optimization, IEEE Transactions on Evolutionary Computation, 8 (2004), 256-279.  doi: 10.1109/TEVC.2004.826067.

[10]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.  doi: 10.1109/4235.996017.

[11]

A. E. Eiben, Z. Michalewicz, M. Schoenauer and J. E. Smith, Parameter control in evolutionary algorithms, In Parameter Setting in Evolutionary Algorithms, Springer, 2007, 19-46.

[12]

E. T. ErdemirR. BattaP. A. RogersonA. Blatt and Ma rie Flanigan, Joint ground and air emergency medical services coverage models: A greedy heuristic solution approach, European Journal of Operational Research, 207 (2010), 736-749.  doi: 10.1016/j.ejor.2010.05.047.

[13]

J. A. Fitzsimmons and B. N. Srikar, Emergency ambulance location using the contiguous zone search routine, Journal of Operations Management, 2 (1982), 225-237.  doi: 10.1016/0272-6963(82)90011-0.

[14]

T. Furuta and K.-i. Tanaka, Minisum and minimax location models for helicopter emergency medical service systems, Journal of the Operations Research Society of Japan, 56 (2013), 221-242.  doi: 10.15807/jorsj.56.221.

[15]

H. Garg, An efficient biogeography based optimization algorithm for solving reliability optimization problems, Swarm and Evolutionary Computation, 24 (2015), 1-10.  doi: 10.1016/j.swevo.2015.05.001.

[16]

H. Garg, A hybrid PSO-GA algorithm for constrained optimization problems, Applied Mathematics and Computation, 274 (2016), 292-305.  doi: 10.1016/j.amc.2015.11.001.

[17]

H. Garg, A hybrid GSA-GA algorithm for constrained optimization problems, Information Sciences, 478 (2019), 499-523.  doi: 10.1016/j.ins.2018.11.041.

[18]

H. Garg and S. P. Sharma, Multi-objective reliability-redundancy allocation problem using particle swarm optimization, Computers & Industrial Engineering, 64 (2013), 247-255.  doi: 10.1016/j.cie.2012.09.015.

[19]

M. GendreauG. Laporte and F. Semet, A dynamic model and parallel tabu search heuristic for real-time ambulance relocation, Parallel Computing, 27 (2001), 1641-1653.  doi: 10.1016/S0167-8191(01)00103-X.

[20]

Z. GhelichiM. Saidi-Mehrabad and M. S. Pishvaee, A stochastic programming approach toward optimal design and planning of an integrated green biodiesel supply chain network under uncertainty: A case study, Energy, 156 (2018), 661-687.  doi: 10.1016/j.energy.2018.05.103.

[21]

Z. GhelichiJ. Tajik and M. S. Pishvaee, A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: A case study of Mashhad, Computers & Chemical Engineering, 110 (2018), 13-34.  doi: 10.1016/j.compchemeng.2017.11.017.

[22]

A. GhodratnamaH. R. Arbabi and A. Azaron, Production planning in industrial townships modeled as hub location-allocation problems considering congestion in manufacturing plants, Computers & Industrial Engineering, 129 (2019), 479-501.  doi: 10.1016/j.cie.2019.01.049.

[23]

R. Goldberg and P. Listowsky, Critical factors for emergency vehicle routing expert systems, Expert Systems with Applications, 7 (1994), 589-602.  doi: 10.1016/0957-4174(94)90082-5.

[24]

J. Holguín-VerasM. JallerL. N. Van WassenhoveN. Pérez and T. Wachtendorf, On the unique features of post-disaster humanitarian logistics, Journal of Operations Management, 30 (2012), 494-506. 

[25]

S. A. Hosseinijou and M. Bashiri, Stochastic models for transfer point location problem, The International Journal of Advanced Manufacturing Technology, 58 (2012), 211-225.  doi: 10.1007/s00170-011-3360-0.

[26]

A. JotshiQ. Gong and R. Batta, Dispatching and routing of emergency vehicles in disaster mitigation using data fusion, Socio-Economic Planning Sciences, 43 (2009), 1-24.  doi: 10.1016/j.seps.2008.02.005.

[27]

H. KalantariA. YousefliM. Ghazanfari and K. Shahanaghi, Fuzzy transfer point location problem: A possibilistic unconstrained nonlinear programming approach, The International Journal of Advanced Manufacturing Technology, 70 (2014), 1043-1051.  doi: 10.1007/s00170-013-5338-6.

[28]

V. A. KnightP. R. Harper and L. Smith, Ambulance allocation for maximal survival with heterogeneous outcome measures, Omega, 40 (2012), 918-926.  doi: 10.1016/j.omega.2012.02.003.

[29]

G. Mavrotas and K. Florios, An improved version of the augmented $\varepsilon$-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems, Applied Mathematics and Computation, 219 (2013), 9652-9669.  doi: 10.1016/j.amc.2013.03.002.

[30]

F. NavaziR. Tavakkoli-Moghaddam and Z. Sazvar, A multi-period location-allocation-inventory problem for ambulance and helicopter ambulance stations: Robust possibilistic approach, IFAC-PapersOnLine, 51 (2018), 322-327.  doi: 10.1016/j.ifacol.2018.08.303.

[31]

L. Özdamar and M. A. Ertem, Models, solutions and enabling technologies in humanitarian logistics, European Journal of Operational Research, 244 (2015), 55-65.  doi: 10.1016/j.ejor.2014.11.030.

[32]

R. S. PatwalN. Narang and H. Garg, A novel TVAC-PSO based mutation strategies algorithm for generation scheduling of pumped storage hydrothermal system incorporating solar units, Energy, 142 (2018), 822-837.  doi: 10.1016/j.energy.2017.10.052.

[33]

A. J. Pedraza-Martinez and L. N. Van Wassenhove, Transportation and vehicle fleet management in humanitarian logistics: Challenges for future research, EURO Journal on Transportation and Logistics, 1 (2012), 185-196.  doi: 10.1007/s13676-012-0001-1.

[34]

M. Rabbani, S. Momen, N. Akbarian-Saravi, H. Farrokhi-Asl and Z. Ghelichi, Optimal design for sustainable bioethanol supply chain considering the bioethanol production strategies: A case study, Computers & Chemical Engineering, 134 (2020), 106720. doi: 10.1016/j.compchemeng.2019.106720.

[35]

M. SasakiT. Furuta and A. Suzuki, Exact optimal solutions of the minisum facility and transfer points location problems on a network, International Transactions in Operational Research, 15 (2008), 295-306.  doi: 10.1111/j.1475-3995.2008.00602.x.

[36]

V. Schmid, Solving the dynamic ambulance relocation and dispatching problem using approximate dynamic programming, European Journal of Operational Research, 219 (2012), 611-621.  doi: 10.1016/j.ejor.2011.10.043.

[37]

N. Schuurman, N. J. Bell, R. L'Heureux and S. M Hameed, Modelling optimal location for pre-hospital helicopter emergency medical services, BMC Emergency Medicine, 9 (2009), Art. No. 6. doi: 10.1186/1471-227X-9-6.

[38]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.  doi: 10.1162/evco.1994.2.3.221.

[39]

G. Taguchi, Introduction to quality engineering: designing quality into products and processes, Technical report, 1986.

[40]

L. TalaricoF. Meisel and K. Sörensen, Ambulance routing for disaster response with patient groups, Computers & Operations Research, 56 (2015), 120-133.  doi: 10.1016/j.cor.2014.11.006.

[41]

H. Tikani and M. Setak, Ambulance routing in disaster response scenario considering different types of ambulances and semi soft time windows, Journal of Industrial and Systems Engineering, 12 (2019), 95-128. 

[42]

E. B. Tirkolaee, A. Goli and G.-W. Weber, Fuzzy mathematical programming and self-adaptive artificial fish swarm algorithm for just-in-time energy-aware flow shop scheduling problem with outsourcing option, IEEE Transactions on Fuzzy Systems, 2020.

[43]

E. B. TirkolaeeS. HadianG.-W. Weber and I. Mahdavi, A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.  doi: 10.1111/coin.12240.

[44]

T. TliliS. Abidi and S. Krichen, A mathematical model for efficient emergency transportation in a disaster situation, The American Journal of Emergency Medicine, 36 (2018), 1585-1590.  doi: 10.1016/j.ajem.2018.01.039.

[45]

H. Toro-DíAzM. E. MayorgaS. Chanta and and L. A. Mclay, Joint location and dispatching decisions for emergency medical services, Computers & Industrial Engineering, 64 (2013), 917-928. 

[46]

Y.-J. ZhengS.-Y. Chen and and H.-F. Ling, Evolutionary optimization for disaster relief operations: A survey, Applied Soft Computing, 27 (2015), 553-566.  doi: 10.1016/j.asoc.2014.09.041.

[47]

A. ZhouB.-Y. QuH. LiS.-Z. ZhaoP. N. Suganthan and Q. Zhang, Multiobjective evolutionary algorithms: A survey of the state of the art, Swarm and Evolutionary Computation, 1 (2011), 32-49.  doi: 10.1016/j.swevo.2011.03.001.

Figure 1.  Requests classification and changes in patient condition
Figure 2.  The flowchart of NSGA-II
Figure 3.  An illustrative example of the decoding process
Figure 4.  An illustrative example of the classical order crossover
Figure 5.  An illustrative example of the two-point swap mutation
Figure 6.  The flowchart of MOPSO
Figure 7.  Results of NSGA-II parameter tuning based on the Taguchi method
Figure 8.  Results of MOPSO parameter tuning based on the Taguchi method
Figure 9.  Optimal Pareto front for test instance 1
Figure 10.  Pareto front approximation for test instance 10
Figure 11.  Comparison of NSGA-II and MOPSO based on CPU time
Figure 12.  Comparison of NSGA-II and MOPSO based on RNI
Figure 13.  Comparison of NSGA-II and MOPSO based on DM
Figure 14.  Comparison of NSGA-II and MOPSO based on SM
Figure 15.  Comparison of NSGA-II and MOPSO based on GD
Figure 16.  Lorestan province hospital locations and post-disaster potential locations for medical demands
Figure 17.  Alternation of OFVs according to the variation in the number of hospitals
Figure 20.  Alternation of OFVs according to the variation of travel time
Figure 18.  Alternation of OFVs according to the variation in the number of ambulances
Figure 19.  Alternation of OFVs according to the variation in the number of patients
Table 1.  Assumptions considered in the related works on emergency medical service
Authors Objective function Problem Modes Variable condition Patient groups Solution approach
Talarico et al. [40] Min max response time Routing _ _ X Heuristic
Jotshi et al. [26] Max demand coverage/Min coverage cost Routing _ _ X Heuristic
Schuurman et al. [37] Max demand coverage Locating _ _ _ Exact
Erdemir et al. [12] Min establishment cost/Max demand coverage Locating X _ _ Heuristic
Branas et al. [7] Min average response time Locating _ _ _ Heuristic
Furuta and Tanaka [14] Min max transfer time/Min response time Locating X _ _ Exact
Sasaki et al. [35] Min transfer time Locating _ _ _ Exact
Berman et al. [5] Min transfer time Locating _ _ _ Heuristic
Hosseinijou and Bashiri [25] Min max distance Locating _ _ _ Exact: analytical/numerical
Kalantari et al. [27] Min transfer time Locating _ _ _ Heuristic
Camacho et al. [8] Min response time Distribution X _ _ Exact
Tikani and Setak [41] Min max response time Routing _ _ X Heuristic
Fitzsimmons and Srikar [13] Min max response time Locating _ _ _ Heuristic
Gendreau et al. [19] Max demand coverage Relocating _ _ _ Heuristic
Knight et al. [28] Max survival probability Locating _ _ X Heuristic
Toro-Díaz et al. [45] Min response time/Max demand coverage Locating/dispatching _ _ _ Heuristic
Andersson and Värbrand [1] Min max transfer time Locating/dispatching _ _ _ Heuristic
Schmid [36] Min average response time Relocating/dispatching _ _ _ Heuristic
Goldberg and Listowsky [23] _ Routing _ _ _ Survey
Tlili et al. [44] Min travel cost Routing _ _ _ Heuristic
Bozorgi-Amiri et al. [6] Min transfer time Locating X _ _ Exact
Navazi et al. [30] Min establishment cost/Min response time Locating X _ _ Exact
This paper Min max response time/Min number of patients Routing _ X X Heuristic
Authors Objective function Problem Modes Variable condition Patient groups Solution approach
Talarico et al. [40] Min max response time Routing _ _ X Heuristic
Jotshi et al. [26] Max demand coverage/Min coverage cost Routing _ _ X Heuristic
Schuurman et al. [37] Max demand coverage Locating _ _ _ Exact
Erdemir et al. [12] Min establishment cost/Max demand coverage Locating X _ _ Heuristic
Branas et al. [7] Min average response time Locating _ _ _ Heuristic
Furuta and Tanaka [14] Min max transfer time/Min response time Locating X _ _ Exact
Sasaki et al. [35] Min transfer time Locating _ _ _ Exact
Berman et al. [5] Min transfer time Locating _ _ _ Heuristic
Hosseinijou and Bashiri [25] Min max distance Locating _ _ _ Exact: analytical/numerical
Kalantari et al. [27] Min transfer time Locating _ _ _ Heuristic
Camacho et al. [8] Min response time Distribution X _ _ Exact
Tikani and Setak [41] Min max response time Routing _ _ X Heuristic
Fitzsimmons and Srikar [13] Min max response time Locating _ _ _ Heuristic
Gendreau et al. [19] Max demand coverage Relocating _ _ _ Heuristic
Knight et al. [28] Max survival probability Locating _ _ X Heuristic
Toro-Díaz et al. [45] Min response time/Max demand coverage Locating/dispatching _ _ _ Heuristic
Andersson and Värbrand [1] Min max transfer time Locating/dispatching _ _ _ Heuristic
Schmid [36] Min average response time Relocating/dispatching _ _ _ Heuristic
Goldberg and Listowsky [23] _ Routing _ _ _ Survey
Tlili et al. [44] Min travel cost Routing _ _ _ Heuristic
Bozorgi-Amiri et al. [6] Min transfer time Locating X _ _ Exact
Navazi et al. [30] Min establishment cost/Min response time Locating X _ _ Exact
This paper Min max response time/Min number of patients Routing _ X X Heuristic
Table 2.  Employed notation for modeling the ARP
Table 3.  Adjusted parameters for NSGA-II and MOPSO
Algorithm Parameter
$ N_{pop} $ $ max-it $ $ p_c $ $ p_m $ $ p_m $ $ w $ $ c_1 $ $ c_2 $
NSGA-II 75 100 0.6 0.2 _ _ _ _
MOPSO 100 125 _ _ 75 0.5 1.25 1.5
Algorithm Parameter
$ N_{pop} $ $ max-it $ $ p_c $ $ p_m $ $ p_m $ $ w $ $ c_1 $ $ c_2 $
NSGA-II 75 100 0.6 0.2 _ _ _ _
MOPSO 100 125 _ _ 75 0.5 1.25 1.5
Table 4.  Results of the 15 problem instances using NSGA-II and MOPSO
Problem Num. Dimension NSGA-II MOPSO Optimal
CPU time (s) RNI DM SM GD CPU time (s) RNI DM SM GD CPU time (s)
1 (3, 4, 4, 3) 124.23 0.31 2.1253 0.0079 0.005 136.61 0.16 1.9872 0.0098 0.005 15
2 (5, 6, 8, 6) 132.4 0.37 1.5231 0.0091 0.003 148.25 0.11 1.9865 0.0111 0.005 89
3 (8, 10, 10, 7) 139.87 0.29 3.2964 0.0082 0.003 161.2 0.2 5.2964 0.0109 0.006 236
4 (12, 15, 13, 11) 149.12 0.4 0.4374 0.011 0.004 169.59 0.33 0.1381 0.0147 0.004 894
5 (15, 17, 15, 13) 156.76 0.64 1.0964 0.0054 0.006 175.32 0.42 2.1784 0.0021 0.006 2601
6 (20, 23, 20, 18) 161.2 0.57 1.4384 0.0066 0.003 183.82 0.62 5.9824 0.0069 0.002 8732
7 (20, 24, 22, 20) 168.94 0.59 1.9842 0.0069 0.001 187.48 0.48 3.9855 0.0132 0.003 15000
8 (25, 28, 21, 20) 175.54 0.21 4.7337 0.0074 0.005 193.56 0.15 2.9156 0.0097 0.007 15000
9 (25, 27, 25, 21) 179.36 0.28 3.341 0.0043 0.002 195.7 0.27 3.4415 0.004 0.008 15000
10 (30, 32, 24, 22) 188.87 0.88 3.7609 0.0121 0.003 210.1 0.42 4.5699 0.0158 0.005 15000
11 (40, 43, 29, 25) 194.3 0.91 1.2745 0.0108 0.001 223.24 0.81 6.1801 0.0124 0.001 15000
12 (50, 55, 41, 32) 212.25 0.73 2.8954 0.0039 0.002 240.51 0.83 4.0643 0.0061 0.001 15000
13 (75, 83, 66, 45) 237.86 0.77 5.762 0.0089 0.006 271.24 0.5 5.957 0.009 0.008 15000
14 (90, 95, 70, 51) 264.22 0.86 3.8713 0.0077 0.005 284.5 0.46 3.869 0.0111 0.005 15000
15 (100, 112, 84, 62) 283.67 0.94 1.8751 0.0106 0.005 303.2 0.73 5.5147 0.0131 0.006 15000
Problem Num. Dimension NSGA-II MOPSO Optimal
CPU time (s) RNI DM SM GD CPU time (s) RNI DM SM GD CPU time (s)
1 (3, 4, 4, 3) 124.23 0.31 2.1253 0.0079 0.005 136.61 0.16 1.9872 0.0098 0.005 15
2 (5, 6, 8, 6) 132.4 0.37 1.5231 0.0091 0.003 148.25 0.11 1.9865 0.0111 0.005 89
3 (8, 10, 10, 7) 139.87 0.29 3.2964 0.0082 0.003 161.2 0.2 5.2964 0.0109 0.006 236
4 (12, 15, 13, 11) 149.12 0.4 0.4374 0.011 0.004 169.59 0.33 0.1381 0.0147 0.004 894
5 (15, 17, 15, 13) 156.76 0.64 1.0964 0.0054 0.006 175.32 0.42 2.1784 0.0021 0.006 2601
6 (20, 23, 20, 18) 161.2 0.57 1.4384 0.0066 0.003 183.82 0.62 5.9824 0.0069 0.002 8732
7 (20, 24, 22, 20) 168.94 0.59 1.9842 0.0069 0.001 187.48 0.48 3.9855 0.0132 0.003 15000
8 (25, 28, 21, 20) 175.54 0.21 4.7337 0.0074 0.005 193.56 0.15 2.9156 0.0097 0.007 15000
9 (25, 27, 25, 21) 179.36 0.28 3.341 0.0043 0.002 195.7 0.27 3.4415 0.004 0.008 15000
10 (30, 32, 24, 22) 188.87 0.88 3.7609 0.0121 0.003 210.1 0.42 4.5699 0.0158 0.005 15000
11 (40, 43, 29, 25) 194.3 0.91 1.2745 0.0108 0.001 223.24 0.81 6.1801 0.0124 0.001 15000
12 (50, 55, 41, 32) 212.25 0.73 2.8954 0.0039 0.002 240.51 0.83 4.0643 0.0061 0.001 15000
13 (75, 83, 66, 45) 237.86 0.77 5.762 0.0089 0.006 271.24 0.5 5.957 0.009 0.008 15000
14 (90, 95, 70, 51) 264.22 0.86 3.8713 0.0077 0.005 284.5 0.46 3.869 0.0111 0.005 15000
15 (100, 112, 84, 62) 283.67 0.94 1.8751 0.0106 0.005 303.2 0.73 5.5147 0.0131 0.006 15000
Table 5.  The analytical results of two-sample t-test
Criteria Optimal method Average results Two-sample t-test
NSGA-II MOPSO Comparison of means P-value
CPU time (s) Both methods 194.5727 205.6213 $ \mu_{NSGA-II}=\mu_{MOPSO} $ 0
RNI NSGA-II 0.5833 0.4326 $ \mu_{NSGA-II}> \mu_{MOPSO} $ 0.32
DM MOPSO 2.6276 3.8711 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.212
SM NSGA-II 0.0079 0.0098 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.901
GD NSGA-II 0.0036 0.0048 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.625
Criteria Optimal method Average results Two-sample t-test
NSGA-II MOPSO Comparison of means P-value
CPU time (s) Both methods 194.5727 205.6213 $ \mu_{NSGA-II}=\mu_{MOPSO} $ 0
RNI NSGA-II 0.5833 0.4326 $ \mu_{NSGA-II}> \mu_{MOPSO} $ 0.32
DM MOPSO 2.6276 3.8711 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.212
SM NSGA-II 0.0079 0.0098 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.901
GD NSGA-II 0.0036 0.0048 $ \mu_{NSGA-II}< \mu_{MOPSO} $ 0.625
Table 6.  Data related to the Lorestan province cities
City name Population % of injured people % of GC demands % of RC demands
Aleshtar 33, 132 30% 88% 12%
Aligudarz 89, 521 21% 49% 51%
Azna 41, 703 27% 83% 17%
Borujerd 245, 730 19% 77% 23%
Dorud 100, 979 32% 66% 34%
Khorramabad 354, 854 15% 41% 59%
Kuhdasht 111, 737 28% 58% 42%
Nur Abad 62, 195 34% 71% 29%
Pol Dokhtar 32, 590 26% 92% 8%
City name Population % of injured people % of GC demands % of RC demands
Aleshtar 33, 132 30% 88% 12%
Aligudarz 89, 521 21% 49% 51%
Azna 41, 703 27% 83% 17%
Borujerd 245, 730 19% 77% 23%
Dorud 100, 979 32% 66% 34%
Khorramabad 354, 854 15% 41% 59%
Kuhdasht 111, 737 28% 58% 42%
Nur Abad 62, 195 34% 71% 29%
Pol Dokhtar 32, 590 26% 92% 8%
Table 7.  Data related to the Lorestan province hospitals
Hospital name RC patient capacity Number of ambulances
Ibn Sina 120 25
Imam Ali 215 30
Imam Jafar Sadegh 148 28
Imam Khomeini 270 41
Kuhdasht 132 27
Shohada-ye Ashayer 347 40
Shahid Chamran 221 36
Shohada-ye Haftom Tir 150 20
Hospital name RC patient capacity Number of ambulances
Ibn Sina 120 25
Imam Ali 215 30
Imam Jafar Sadegh 148 28
Imam Khomeini 270 41
Kuhdasht 132 27
Shohada-ye Ashayer 347 40
Shahid Chamran 221 36
Shohada-ye Haftom Tir 150 20
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