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September  2020, 16(5): 2369-2387. doi: 10.3934/jimo.2019058

Emergency logistics for disaster management under spatio-temporal demand correlation: The earthquakes case

Rodrigo A. Garrido 1,, and  Ivan Aguirre 2,

1. 

Faculty of Engineering and Sciences, Universidad Diego Portales, Av. Ejercito 441, Santiago Centro, Santiago, Chile
2. 

Diagonal Las Torres 2300, Santiago, Chile

* Corresponding author: rodrigo.garrido@udp.cl

Received  September 2018 Revised  February 2019 Published  May 2019

Emergency logistics is crucial to ameliorate the impact of large earthquakes on society. We present a modeling framework to assist decision makers in strategic and tactical planning for effective relief operations after an earthquake's occurrence. The objective is to perform these operations quickly while keeping its total expenses under a budget. The modeling framework locates/allocates resources in potentially affected zones, and transportation capacity is dynamically deployed in those zones. Demand uncertainty is directly incorporated through an impulse stochastic process. The novelty of this approach is threefold. It incorporates temporo-spatial dependence and demands heterogeneity. It incorporates the availability of transportation capacity at different zones. It incorporates tight budget constraints that precludes the total satisfaction of demands. The resulting model is a large size stochastic mixed-integer programming model, which can be approximately solved through Sample Average Approximation. An example is provided and a thorough sensitivity analysis is performed. The numerical results suggest that that the response times are highly sensitive to the availability of inventory at each period. In addition, all logistics parameters (except for inventory capacity) have approximately the same impact on the total response time. The elasticity for all these parameters indicate constant returns to scale.

Keywords: OR in disaster relief, OR in developing countries, humanitarian logistics, stochastic programming, distribution.
Mathematics Subject Classification: Primary: 90C15, 90C11; Secondary: 90B15.
Citation: Rodrigo A. Garrido, Ivan Aguirre. Emergency logistics for disaster management under spatio-temporal demand correlation: The earthquakes case. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2369-2387. doi: 10.3934/jimo.2019058
References:
[1]

M. Ahmadi, A. Seifi and B. Tootooni, A humanitarian logistics model for disaster relief operation considering network failure and standard relief time: A case study on san francisco district, Transportation Research Part E: Logistics and Transportation Review, 75 (2015), 145-163.  doi: 10.1016/j.tre.2015.01.008.  Google Scholar

[2]

A. R. Akkihal, Inventory Pre-positioning for Humanitarianoperations, Master's thesis, Massachusetts Institute of Technology, USA, 2006. Google Scholar

[3]

D. Alem, A. Clark and A. Moreno, Stochastic network models for logistics planning in disaster relief, European Journal of Operational Research, 255 (2016), 187-206.  doi: 10.1016/j.ejor.2016.04.041.  Google Scholar

[4]

N. Altay and W. G. Green Ⅲ, Or/ms research in disaster operations management, European Journal of Operational Research, 175 (2006), 475-493.  doi: 10.1016/j.ejor.2005.05.016.  Google Scholar

[5]

T. Anagnos and A. Kiremidjian, A review of earthquake occurrence models for seismic hazard analysis, Probabilistic Engineering Mechanics, 3 (1988), 3-11.  doi: 10.1016/0266-8920(88)90002-1.  Google Scholar

[6]

J. G. Anderson and M. D. Trifunac, On uniform risk functionals which describe strong earthquake ground motion: Definition, numerical estimation, and an application to the fourier amplitude spectrum of acceleration, Report CE 77-02 University of Southern California, Los Angeles, U.S.A., 1977. Google Scholar

[7]

J. G. Anderson and M. D. Trifunac, Uniform risk functionals for characterization of strong earthquake ground motion, Bulletin of the Seismological Society of America, 68 (1978), 205-218.   Google Scholar

[8]

A. Ben-Tala, B. Do Chung, S. R. Mandala and T. Yao, Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains, Transportation Research Part B: Methodological, 45 (2011), 1177-1189.  doi: 10.1016/j.trb.2010.09.002.  Google Scholar

[9]

C. Boonmee, M. Arimura and T. Asada, Facility location optimization model for emergency humanitarian logistics, International Journal of Disaster Risk Reduction, 24 (2017), 485-498.  doi: 10.1016/j.ijdrr.2017.01.017.  Google Scholar

[10]

A. M. Caunhyea, X. Niea and S. Pokharel, Optimization models in emergency logistics: A literature review, Socio-Economic Planning Sciences, Special Issue: Disaster Planning and Logistics: Part 1, 46 (2012), 4-13.  doi: 10.1016/j.seps.2011.04.004.  Google Scholar

[11]

C. A. Cornell, Engineering seismic risk analysis, Bulletin of the Seismological Society of America, 58 (1968), 1583-1606.   Google Scholar

[12]

A. Coskun, W. Elmaghraby, M. Karaman and F. S. Salman, Relief aid stocking decisions under bilateral agency cooperation, Socio-Economic Planning Sciences, 2018. doi: 10.1016/j.seps.2018.10.009.  Google Scholar

[13]

V. Del Gaudio and J. Wasowski, Time probabilistic evaluation of seismically induced landslide hazard in irpinia (southern italy), Soil Dynamics and Earthquake Engineering, 24 (2004), 915-928.   Google Scholar

[14]

F. Fiedrich, F. Gehbauer and U. Rickers, Optimized resource allocation for emergency response after earthquake disasters, Safety Science, 35 (2000), 41-57.  doi: 10.1016/S0925-7535(00)00021-7.  Google Scholar

[15]

G. Galindo and R. Batta, Review of recent developments in or/ms research in disaster operations management, European Journal of Operational Research, 230 (2013), 201-211.  doi: 10.1016/j.ejor.2013.01.039.  Google Scholar

[16]

R. A. Garrido, Optimal emergency resources deployment under a terrorist threat: The hazmat case and beyond, In Handbook of OR/MS Models in Hazardous Materials Transportation, International Series in Operations Research and Management Science, chapter 8, pages 245–267. Springer, New York, 2013. doi: 10.1007/978-1-4614-6794-6_8.  Google Scholar

[17]

R. A. Garrido and P. Lamas, Optimal logistics and transportation decisions for emergency response to natural disasters, In World Academy of Science, Engineering and Technology 76, chapter 8, pages 201–213. WASET, Johannesburg, South Africa, 2013. Google Scholar

[18]

R. A. Garrido, P. Lamas and F. J. Pino, A stochastic programming approach for floods emergency logistics, Transportation Research Part E: Logistics and Transportation Review, 75 (2015), 18-31.  doi: 10.1016/j.tre.2014.12.002.  Google Scholar

[19]

I. D. Gupta, Probabilistic seismic hazard analysis method for mapping of spectral amplitudes and other design-specific quantities to estimate the earthquake effects on man-made structures, ISET Journal of Earthquake Technology, 44 (2007), 127-167.   Google Scholar

[20]

J. Holguin-Veras, N. Pérez, S. Ukkusuri, T. Wachtendorf and B. Brown, Emergency Logistics Issues Affecting the Response to Katrina: A Synthesis and Preliminary Suggestions for Improvement, Transportation Research Record: Journal of the Transportation Research Board, 2022 (2007), 76-82.  doi: 10.3141/2022-09.  Google Scholar

[21]

E. L. Krinitzsky, Earthquake probability in engineering – part 1: The use and misuse of expert opinion: The third richard h. jahns distinguished lecture in engineering geology, Engineering Geology, 33 (1993), 257-288.  doi: 10.1016/0013-7952(93)90030-G.  Google Scholar

[22]

J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM Journal on Optimization, 19 (2008), 674-699.  doi: 10.1137/070702928.  Google Scholar

[23]

E. Mas, A. Suppasri, Sh. Koshimura and F. Imamura, Agent based simulation of the 2011 great east japan earthquake tsunami evacuation procedure. introduction to an integrated model of tsunami inundation and evacuation, Journal of Natural Disaster Science, 34 (2012), 41-57.   Google Scholar

[24]

T. Matisziw, A. Murray and T. Grubesic, Strategic network restoration, Networks and Spatial Economics, 10 (2010), 345-361.  doi: 10.1007/s11067-009-9123-x.  Google Scholar

[25]

R. K. McGuire, Seismic design spectra and mapping procedures using hazard analysis based directly on oscillator response, Earthquake Engineering and Structural Dynamics, 5 (1977), 211-234.  doi: 10.1002/eqe.4290050302.  Google Scholar

[26]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2007), 969-996.  doi: 10.1137/050622328.  Google Scholar

[27]

L. Ozdamar, D. Tuzun-Aksu, E. Yasa and B. Ergunes, Disaster relief routing in limited capacity road networks with heterogeneous flows, Journal of Industrial and Management Optimization, 14 (2018), 1367-1380.  doi: 10.3934/jimo.2018011.  Google Scholar

[28]

B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Sample average approximation method for chance constrained programming: Theory and applications, Journal of Optimization Theory and Applications, 142 (2009), 399-416.  doi: 10.1007/s10957-009-9523-6.  Google Scholar

[29]

B. Papazachos, E. E. Papadimitriou, A. A. Kiratzi, Ch. A. Papaioannou and G. F. Karakaisis, Probabilities of occurrence of large earthquakes in the aegean and surrounding area during the period 1986–2006, Pure and Applied Geophysics, 125 (1987), 597-612.  doi: 10.1007/BF00879574.  Google Scholar

[30]

B. C. Papazachos, Ch. A. Papaioannou, V. N. Margaris and N. P. Theodulidis, Seismic hazard assessment in greece based on strong motion duration, Proceedings of the Tenth World Conference on Earthquake Engineeing, 2 (1992), 425-430.   Google Scholar

[31]

T. Parsons, Recalculated probability of m greater than 7 earthquakes beneath the sea of marmara, turkey, Journal of Geophysical Research, 109 (2004), 1-21.   Google Scholar

[32]

T. Parsons, Significance of stress transfer in time-dependent earthquake probability calculations, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978-2012.  doi: 10.1029/2004JB003190.  Google Scholar

[33]

M. Peng, Y. Peng and H. Chen, Post-seismic supply chain risk management: A system dynamics disruption analysis approach for inventory and logistics planning, Computers & Operations Research, Special issue Multiple Criteria Decision Making in Emergency Management, 42 (2014), 14-24.  doi: 10.1016/j.cor.2013.03.003.  Google Scholar

[34]

D. Richardson, S. de Leeuw and I. F. A. Vis, Conceptualising inventory prepositioning in the humanitarian sector, In Luis M. Camarinha-Matos, Xavier Boucher, and Hamideh Afsarmanesh, editors, Collaborative Networks for a Sustainable World, pages 149–156, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. doi: 10.1007/978-3-642-15961-9_17.  Google Scholar

[35]

D. A. Richardson, S. Leeuw and W. Dullaert, Factors affecting global inventory prepositioning locations in humanitarian operations–a delphi study, Journal of Business Logistics, 37 (2016), 59-74.  doi: 10.1111/jbl.12112.  Google Scholar

[36]

F. S. Salman and E. Yücel, Emergency facility location under random network damage: Insights from the istanbul case, Computers & Operations Research, 62 (2015), 266-281.  doi: 10.1016/j.cor.2014.07.015.  Google Scholar

[37]

J.-B. Sheu, Dynamic relief-demand management for emergency logistics operations under large-scale disasters, Transportation Research Part E: Logistics and Transportation Review, 46 (2010), 1-17.  doi: 10.1016/j.tre.2009.07.005.  Google Scholar

[38]

J.-B. Sheu, An emergency logistics distribution approach for quick response to urgent relief demand in disasters, Transportation Research Part E: Logistics and Transportation Review, 43 (2010), 687-709.  doi: 10.1016/j.tre.2006.04.004.  Google Scholar

[39]

J. Tobita, N. Fukuwa and M. Mori, Integrated disaster simulator using webgis and its application to community disaster mitigation activities, Journal of Natural Disaster Science, 30 (2008), 71-82.  doi: 10.2328/jnds.30.71.  Google Scholar

[40]

M. I. Todorovska and M. D. Trifunac, Liquefaction opportunity mapping via seismic wave energy, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 125 (1999), 1032-1042.  doi: 10.1061/(ASCE)1090-0241(1999)125:12(1032).  Google Scholar

[41]

S. Tofighi, S. A. Torabi and S. A. Mansouri, Humanitarian logistics network design under mixed uncertainty, European Journal of Operational Research, 250 (2016), 239-250.  doi: 10.1016/j.ejor.2015.08.059.  Google Scholar

[42]

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

[43]

T. Yao, S. R. Mandala and B. D. Chung, Evacuation transportation planning under uncertainty: A robust optimization approach, Networks and Spatial Economics, 9 (2009), 171-189.  doi: 10.1007/s11067-009-9103-1.  Google Scholar

[44]

E. Yücel, F. S. Salman and I. Arsik, Improving post-disaster road network accessibility by strengthening links against failures, European Journal of Operational Research, 269 (2018), 406-422.  doi: 10.1016/j.ejor.2018.02.015.  Google Scholar

[45]

W. Yushimito, M. Jaller and S. Ukkusuri, A voronoi-based heuristic algorithm for locating distribution centers in disasters, Networks and Spatial Economics, 12 (2012), 21-39.  doi: 10.1007/s11067-010-9140-9.  Google Scholar

show all references

References:
[1]

M. Ahmadi, A. Seifi and B. Tootooni, A humanitarian logistics model for disaster relief operation considering network failure and standard relief time: A case study on san francisco district, Transportation Research Part E: Logistics and Transportation Review, 75 (2015), 145-163.  doi: 10.1016/j.tre.2015.01.008.  Google Scholar

[2]

A. R. Akkihal, Inventory Pre-positioning for Humanitarianoperations, Master's thesis, Massachusetts Institute of Technology, USA, 2006. Google Scholar

[3]

D. Alem, A. Clark and A. Moreno, Stochastic network models for logistics planning in disaster relief, European Journal of Operational Research, 255 (2016), 187-206.  doi: 10.1016/j.ejor.2016.04.041.  Google Scholar

[4]

N. Altay and W. G. Green Ⅲ, Or/ms research in disaster operations management, European Journal of Operational Research, 175 (2006), 475-493.  doi: 10.1016/j.ejor.2005.05.016.  Google Scholar

[5]

T. Anagnos and A. Kiremidjian, A review of earthquake occurrence models for seismic hazard analysis, Probabilistic Engineering Mechanics, 3 (1988), 3-11.  doi: 10.1016/0266-8920(88)90002-1.  Google Scholar

[6]

J. G. Anderson and M. D. Trifunac, On uniform risk functionals which describe strong earthquake ground motion: Definition, numerical estimation, and an application to the fourier amplitude spectrum of acceleration, Report CE 77-02 University of Southern California, Los Angeles, U.S.A., 1977. Google Scholar

[7]

J. G. Anderson and M. D. Trifunac, Uniform risk functionals for characterization of strong earthquake ground motion, Bulletin of the Seismological Society of America, 68 (1978), 205-218.   Google Scholar

[8]

A. Ben-Tala, B. Do Chung, S. R. Mandala and T. Yao, Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains, Transportation Research Part B: Methodological, 45 (2011), 1177-1189.  doi: 10.1016/j.trb.2010.09.002.  Google Scholar

[9]

C. Boonmee, M. Arimura and T. Asada, Facility location optimization model for emergency humanitarian logistics, International Journal of Disaster Risk Reduction, 24 (2017), 485-498.  doi: 10.1016/j.ijdrr.2017.01.017.  Google Scholar

[10]

A. M. Caunhyea, X. Niea and S. Pokharel, Optimization models in emergency logistics: A literature review, Socio-Economic Planning Sciences, Special Issue: Disaster Planning and Logistics: Part 1, 46 (2012), 4-13.  doi: 10.1016/j.seps.2011.04.004.  Google Scholar

[11]

C. A. Cornell, Engineering seismic risk analysis, Bulletin of the Seismological Society of America, 58 (1968), 1583-1606.   Google Scholar

[12]

A. Coskun, W. Elmaghraby, M. Karaman and F. S. Salman, Relief aid stocking decisions under bilateral agency cooperation, Socio-Economic Planning Sciences, 2018. doi: 10.1016/j.seps.2018.10.009.  Google Scholar

[13]

V. Del Gaudio and J. Wasowski, Time probabilistic evaluation of seismically induced landslide hazard in irpinia (southern italy), Soil Dynamics and Earthquake Engineering, 24 (2004), 915-928.   Google Scholar

[14]

F. Fiedrich, F. Gehbauer and U. Rickers, Optimized resource allocation for emergency response after earthquake disasters, Safety Science, 35 (2000), 41-57.  doi: 10.1016/S0925-7535(00)00021-7.  Google Scholar

[15]

G. Galindo and R. Batta, Review of recent developments in or/ms research in disaster operations management, European Journal of Operational Research, 230 (2013), 201-211.  doi: 10.1016/j.ejor.2013.01.039.  Google Scholar

[16]

R. A. Garrido, Optimal emergency resources deployment under a terrorist threat: The hazmat case and beyond, In Handbook of OR/MS Models in Hazardous Materials Transportation, International Series in Operations Research and Management Science, chapter 8, pages 245–267. Springer, New York, 2013. doi: 10.1007/978-1-4614-6794-6_8.  Google Scholar

[17]

R. A. Garrido and P. Lamas, Optimal logistics and transportation decisions for emergency response to natural disasters, In World Academy of Science, Engineering and Technology 76, chapter 8, pages 201–213. WASET, Johannesburg, South Africa, 2013. Google Scholar

[18]

R. A. Garrido, P. Lamas and F. J. Pino, A stochastic programming approach for floods emergency logistics, Transportation Research Part E: Logistics and Transportation Review, 75 (2015), 18-31.  doi: 10.1016/j.tre.2014.12.002.  Google Scholar

[19]

I. D. Gupta, Probabilistic seismic hazard analysis method for mapping of spectral amplitudes and other design-specific quantities to estimate the earthquake effects on man-made structures, ISET Journal of Earthquake Technology, 44 (2007), 127-167.   Google Scholar

[20]

J. Holguin-Veras, N. Pérez, S. Ukkusuri, T. Wachtendorf and B. Brown, Emergency Logistics Issues Affecting the Response to Katrina: A Synthesis and Preliminary Suggestions for Improvement, Transportation Research Record: Journal of the Transportation Research Board, 2022 (2007), 76-82.  doi: 10.3141/2022-09.  Google Scholar

[21]

E. L. Krinitzsky, Earthquake probability in engineering – part 1: The use and misuse of expert opinion: The third richard h. jahns distinguished lecture in engineering geology, Engineering Geology, 33 (1993), 257-288.  doi: 10.1016/0013-7952(93)90030-G.  Google Scholar

[22]

J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM Journal on Optimization, 19 (2008), 674-699.  doi: 10.1137/070702928.  Google Scholar

[23]

E. Mas, A. Suppasri, Sh. Koshimura and F. Imamura, Agent based simulation of the 2011 great east japan earthquake tsunami evacuation procedure. introduction to an integrated model of tsunami inundation and evacuation, Journal of Natural Disaster Science, 34 (2012), 41-57.   Google Scholar

[24]

T. Matisziw, A. Murray and T. Grubesic, Strategic network restoration, Networks and Spatial Economics, 10 (2010), 345-361.  doi: 10.1007/s11067-009-9123-x.  Google Scholar

[25]

R. K. McGuire, Seismic design spectra and mapping procedures using hazard analysis based directly on oscillator response, Earthquake Engineering and Structural Dynamics, 5 (1977), 211-234.  doi: 10.1002/eqe.4290050302.  Google Scholar

[26]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2007), 969-996.  doi: 10.1137/050622328.  Google Scholar

[27]

L. Ozdamar, D. Tuzun-Aksu, E. Yasa and B. Ergunes, Disaster relief routing in limited capacity road networks with heterogeneous flows, Journal of Industrial and Management Optimization, 14 (2018), 1367-1380.  doi: 10.3934/jimo.2018011.  Google Scholar

[28]

B. K. Pagnoncelli, S. Ahmed and A. Shapiro, Sample average approximation method for chance constrained programming: Theory and applications, Journal of Optimization Theory and Applications, 142 (2009), 399-416.  doi: 10.1007/s10957-009-9523-6.  Google Scholar

[29]

B. Papazachos, E. E. Papadimitriou, A. A. Kiratzi, Ch. A. Papaioannou and G. F. Karakaisis, Probabilities of occurrence of large earthquakes in the aegean and surrounding area during the period 1986–2006, Pure and Applied Geophysics, 125 (1987), 597-612.  doi: 10.1007/BF00879574.  Google Scholar

[30]

B. C. Papazachos, Ch. A. Papaioannou, V. N. Margaris and N. P. Theodulidis, Seismic hazard assessment in greece based on strong motion duration, Proceedings of the Tenth World Conference on Earthquake Engineeing, 2 (1992), 425-430.   Google Scholar

[31]

T. Parsons, Recalculated probability of m greater than 7 earthquakes beneath the sea of marmara, turkey, Journal of Geophysical Research, 109 (2004), 1-21.   Google Scholar

[32]

T. Parsons, Significance of stress transfer in time-dependent earthquake probability calculations, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978-2012.  doi: 10.1029/2004JB003190.  Google Scholar

[33]

M. Peng, Y. Peng and H. Chen, Post-seismic supply chain risk management: A system dynamics disruption analysis approach for inventory and logistics planning, Computers & Operations Research, Special issue Multiple Criteria Decision Making in Emergency Management, 42 (2014), 14-24.  doi: 10.1016/j.cor.2013.03.003.  Google Scholar

[34]

D. Richardson, S. de Leeuw and I. F. A. Vis, Conceptualising inventory prepositioning in the humanitarian sector, In Luis M. Camarinha-Matos, Xavier Boucher, and Hamideh Afsarmanesh, editors, Collaborative Networks for a Sustainable World, pages 149–156, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. doi: 10.1007/978-3-642-15961-9_17.  Google Scholar

[35]

D. A. Richardson, S. Leeuw and W. Dullaert, Factors affecting global inventory prepositioning locations in humanitarian operations–a delphi study, Journal of Business Logistics, 37 (2016), 59-74.  doi: 10.1111/jbl.12112.  Google Scholar

[36]

F. S. Salman and E. Yücel, Emergency facility location under random network damage: Insights from the istanbul case, Computers & Operations Research, 62 (2015), 266-281.  doi: 10.1016/j.cor.2014.07.015.  Google Scholar

[37]

J.-B. Sheu, Dynamic relief-demand management for emergency logistics operations under large-scale disasters, Transportation Research Part E: Logistics and Transportation Review, 46 (2010), 1-17.  doi: 10.1016/j.tre.2009.07.005.  Google Scholar

[38]

J.-B. Sheu, An emergency logistics distribution approach for quick response to urgent relief demand in disasters, Transportation Research Part E: Logistics and Transportation Review, 43 (2010), 687-709.  doi: 10.1016/j.tre.2006.04.004.  Google Scholar

[39]

J. Tobita, N. Fukuwa and M. Mori, Integrated disaster simulator using webgis and its application to community disaster mitigation activities, Journal of Natural Disaster Science, 30 (2008), 71-82.  doi: 10.2328/jnds.30.71.  Google Scholar

[40]

M. I. Todorovska and M. D. Trifunac, Liquefaction opportunity mapping via seismic wave energy, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 125 (1999), 1032-1042.  doi: 10.1061/(ASCE)1090-0241(1999)125:12(1032).  Google Scholar

[41]

S. Tofighi, S. A. Torabi and S. A. Mansouri, Humanitarian logistics network design under mixed uncertainty, European Journal of Operational Research, 250 (2016), 239-250.  doi: 10.1016/j.ejor.2015.08.059.  Google Scholar

[42]

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

[43]

T. Yao, S. R. Mandala and B. D. Chung, Evacuation transportation planning under uncertainty: A robust optimization approach, Networks and Spatial Economics, 9 (2009), 171-189.  doi: 10.1007/s11067-009-9103-1.  Google Scholar

[44]

E. Yücel, F. S. Salman and I. Arsik, Improving post-disaster road network accessibility by strengthening links against failures, European Journal of Operational Research, 269 (2018), 406-422.  doi: 10.1016/j.ejor.2018.02.015.  Google Scholar

[45]

W. Yushimito, M. Jaller and S. Ukkusuri, A voronoi-based heuristic algorithm for locating distribution centers in disasters, Networks and Spatial Economics, 12 (2012), 21-39.  doi: 10.1007/s11067-010-9140-9.  Google Scholar

Figure 1.  Level of Service vs. Budget
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Table 1.  Sensitivity scenarios and parameters' variations
Parameters Base Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 4 2 3 5 6
Number of Products 2 1 3 4 5
Number of Zones 6 2 4 8 10
Inventory capacity per period (units) 100 90 95 105 110
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Parameters Base Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 4 2 3 5 6
Number of Products 2 1 3 4 5
Number of Zones 6 2 4 8 10
Inventory capacity per period (units) 100 90 95 105 110
Table 2.  Sensitivity of the average objective function with respect to each parameter's variation
Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods -52% -27% 26% 52%
Number of Products -52% 52% 105% 158%
Number of Zones -69% -35% 34% 69%
Inventory Capacity per Period 9% 6% -8% -19%
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Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods -52% -27% 26% 52%
Number of Products -52% 52% 105% 158%
Number of Zones -69% -35% 34% 69%
Inventory Capacity per Period 9% 6% -8% -19%
Table 3.  Objective function's Coefficient of Variation for each sensitivity scenario
Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 0.08 0.06 0.05 0.04
Number of Products 0.08 0.04 0.04 0.06
Number of Zones 0.11 0.08 0.04 0.04
Inventory Capacity per Period 0.03 0.04 0.07 0.10
Table Options

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Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 0.08 0.06 0.05 0.04
Number of Products 0.08 0.04 0.04 0.06
Number of Zones 0.11 0.08 0.04 0.04
Inventory Capacity per Period 0.03 0.04 0.07 0.10
Table 4.  Elasticity of the average objective function with respect to each parameter's variation
Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 1.04 1.08 1.04 1.04
Number of Products 1.04 1.04 1.05 1.05
Number of Zones 1.03 1.06 1.03 1.03
Inventory Capacity per Period -0.87 -1.20 -1.60 -1.85
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Parameters Scen. 1 Scen. 2 Scen. 3 Scen. 4
Number of Periods 1.04 1.08 1.04 1.04
Number of Products 1.04 1.04 1.05 1.05
Number of Zones 1.03 1.06 1.03 1.03
Inventory Capacity per Period -0.87 -1.20 -1.60 -1.85
Table 5.  Variation of the solution under different probabilities of earthquake's occurrence
Total Access Time Earthquake Pr.
= 20%		 Earthquake Pr.
= 10%		 PEarthquake
Pr. = 2%		 Earthquake Pr.
= 1%
Lower Bound 500 700 2,900 4,000
Upper Bound 1,000 1,200 3,900 4,800
Relative Gap 100% 71% 34% 20%
Absolute Gap 500 700 1000 800
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Total Access Time Earthquake Pr.
= 20%		 Earthquake Pr.
= 10%		 PEarthquake
Pr. = 2%		 Earthquake Pr.
= 1%
Lower Bound 500 700 2,900 4,000
Upper Bound 1,000 1,200 3,900 4,800
Relative Gap 100% 71% 34% 20%
Absolute Gap 500 700 1000 800
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