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

Masoud Rabbani; Nastaran Oladzad-Abbasabady; Niloofar Akbarian-Saravi

doi:10.3934/jimo.2021007

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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

Masoud Rabbani ^, , Nastaran Oladzad-Abbasabady and Niloofar Akbarian-Saravi

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.

Keywords: Ambulance routing, MILP, NSGA-II, MOPSO, Patient groups.

Mathematics Subject Classification: Primary: 90C11.

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 Tirkolaee, A. Goli, M. 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. Başar, B. 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. Berkoune, J. Renaud, M. 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. Berman, Z. 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-Amiri, S. Tavakoli, H. 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-Vallejo, E. González-Rodríguez, F.-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 Coello, G. 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. Deb, A. Pratap, S. 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. Erdemir, R. Batta, P. A. Rogerson, A. 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. Gendreau, G. 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. Ghelichi, M. 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. Ghelichi, J. 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. Ghodratnama, H. 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-Veras, M. Jaller, L. N. Van Wassenhove, N. 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. Jotshi, Q. 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. Kalantari, A. Yousefli, M. 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. Knight, P. 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. Navazi, R. 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. Patwal, N. 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. Sasaki, T. 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. Talarico, F. 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. Tirkolaee, S. Hadian, G.-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. Tlili, S. 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-DíAz, M. E. Mayorga, S. 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. Zheng, S.-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. Zhou, B.-Y. Qu, H. Li, S.-Z. Zhao, P. 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 Tirkolaee, A. Goli, M. 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. Başar, B. 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. Berkoune, J. Renaud, M. 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. Berman, Z. 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-Amiri, S. Tavakoli, H. 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-Vallejo, E. González-Rodríguez, F.-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 Coello, G. 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. Deb, A. Pratap, S. 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. Erdemir, R. Batta, P. A. Rogerson, A. 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. Gendreau, G. 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. Ghelichi, M. 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. Ghelichi, J. 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. Ghodratnama, H. 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-Veras, M. Jaller, L. N. Van Wassenhove, N. 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. Jotshi, Q. 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. Kalantari, A. Yousefli, M. 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. Knight, P. 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. Navazi, R. 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. Patwal, N. 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. Sasaki, T. 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. Talarico, F. 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. Tirkolaee, S. Hadian, G.-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. Tlili, S. 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-DíAz, M. E. Mayorga, S. 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. Zheng, S.-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. Zhou, B.-Y. Qu, H. Li, S.-Z. Zhao, P. 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

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

Algorithm	Parameter
Algorithm	$ 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

Problem Num.	Dimension	NSGA-II					MOPSO					Optimal
Problem Num.	Dimension	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

Criteria	Optimal method	Average results		Two-sample t-test
Criteria	Optimal method	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

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%

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

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