October  2018, 14(4): 1367-1380. doi: 10.3934/jimo.2018011

Disaster relief routing in limited capacity road networks with heterogeneous flows

Department of Industrial and Systems Engineering, Yeditepe University, 26 Agustos Yerlesimi, Kayisdagi Cad., 34755 Atasehir, Istanbul, Turkey

* Corresponding author: Dilek Tuzun Aksu

Received  September 2016 Revised  October 2017 Published  January 2018

In the aftermath of a major earthquake, delivery of essential services to survivors is of utmost importance and in urban areas it is conducted using road networks that are already stressed by road damages, other urban traffic and evacuation. Relief distribution efforts should be planned carefully in order to create minimal additional traffic congestion. We propose a dynamic relief distribution model where relief trucks share limited capacity road networks with counterflows resulting from car traffic. We develop a MIP model for this problem and solve it by decomposing the road network geographically and solving each subnetwork iteratively using the Relax and Fix method.

Citation: Linet Ozdamar, Dilek Tuzun Aksu, Elifcan Yasa, Biket Ergunes. Disaster relief routing in limited capacity road networks with heterogeneous flows. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1367-1380. doi: 10.3934/jimo.2018011
References:
[1]

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

[2]

V. Akbari and F. S. Salman, Multi-vehicle synchronized arc routing problem to restore post-disaster network connectivity, European Journal of Operational Research, 257 (2017), 625-640.  doi: 10.1016/j.ejor.2016.07.043.  Google Scholar

[3]

M. Albareda-SambolaA. Alonso-AyusoL. F. EscuderoE. Fernandez and C. Pizarro, Fix-and-relax-coordination for a multi-period location-allocation problem under uncertainty, Computers & Operations Research, 40 (2013), 2878-2892.  doi: 10.1016/j.cor.2013.07.004.  Google Scholar

[4]

A. M. Anaya-ArenasJ. Renaud and A. Ruiz, Relief distribution networks: A systematic review, Annals of Operations Research, 223 (2014), 53-79.  doi: 10.1007/s10479-014-1581-y.  Google Scholar

[5]

J. AraozE. Fernandez and C. Zoltan, Privatized rural postman problems, Computers & Operations Research, 33 (2006), 3432-3449.  doi: 10.1016/j.cor.2005.02.013.  Google Scholar

[6]

J. AraozE. Fernandez and O. Meza, Solving the prize-collecting rural postman problem, European Journal of Operational Research, 196 (2009), 886-896.  doi: 10.1016/j.ejor.2008.04.037.  Google Scholar

[7]

C. ArchettiD. FeilletA. Hertz and M. G. Speranza, The undirected capacitated arc routing problem with profits, Computers & Operations Research, 37 (2010), 1860-1869.  doi: 10.1016/j.cor.2009.05.005.  Google Scholar

[8]

B. BalcikB. M. Beamon and K. Smilowitz, Last mile distribution in humanitarian relief, Journal of Intelligent Transportation Systems, 12 (2008), 51-63.  doi: 10.1080/15472450802023329.  Google Scholar

[9]

T. A. BaldoM. O. SantosB. Almada-Lobo and R. Morabito, An optimization approach for the lot sizing and scheduling problem in the brewery industry, Computers & Industrial Engineering, 72 (2014), 58-71.  doi: 10.1016/j.cie.2014.02.008.  Google Scholar

[10]

P. BeraldiG. GhianiE. Guerriero and A. Grieco, Scenario-based planning for lot-sizing and scheduling with uncertain processing times, International Journal of Production Economics, 101 (2006), 140-149.  doi: 10.1016/j.ijpe.2005.05.018.  Google Scholar

[11]

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.  Google Scholar

[12]

V. CamposR. Bandeira and A. Bandeira, A method for evacuation route planning in disaster situations, Procedia-Social and Behavioral Sciences, 54 (2012), 503-512.  doi: 10.1016/j.sbspro.2012.09.768.  Google Scholar

[13]

A. M. CaunhyeX. Nie and S. Pokharel, Optimization models in emergency logistics: A literature review, Socio-Economic Planning Sciences, 46 (2012), 4-13.  doi: 10.1016/j.seps.2011.04.004.  Google Scholar

[14]

A. Y. ChenF. Pena-Mora and Y. Ouyang, A collaborative GIS framework to support equipment distribution for civil engineering disaster response operations, Automation in Construction, 20 (2011), 637-648.  doi: 10.1016/j.autcon.2010.12.007.  Google Scholar

[15]

T. Cova and J. Johnson, A network flow model for lane-based evacuation routing, Transportation Research Part A: Policy and Practice, 37 (2003), 579-604.  doi: 10.1016/S0965-8564(03)00007-7.  Google Scholar

[16]

J. Cui, S. An and M. Zhao, A generalized minimum cost flow model for multiple emergency flow routing Mathematical Problems in Engineering 2014 (2014), Article ID 832053, 12 pages. doi: 10.1155/2014/832053.  Google Scholar

[17]

C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research Part B: Methodological, 28 (1994), 269-287.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[18]

L. E. de la TorreI. S. Dolinskaya and K. R. Smilowitz, Disaster relief routing: Integrating research and practice, Socio-Economic Planning Sciences, 46 (2012), 88-97.   Google Scholar

[19]

L. EscuderoA. GarinM. Merino and G. Perez, BFC-MSMIP: An exact branch-and-fix coordination approach for solving multistage stochastic mixed 0--1 problems, TOP, 17 (2009), 96-122.  doi: 10.1007/s11750-009-0083-6.  Google Scholar

[20]

D. FeilletP. Dejax and M. Gendreau, The profitable arc tour problem: Solution with a branch-and-price algorithm, Transportation Science, 39 (2005), 539-552.  doi: 10.1287/trsc.1040.0106.  Google Scholar

[21]

C. M. Feng and C. C. Wen, A fuzzy bi-level and multi-objective model to control traffic flow into disaster area post earthquake, Journal of the Eastern Asia Society for Transportation Studies, 6 (2005), 4253-4268.   Google Scholar

[22]

D. FerreiraR. Morabito and S. Rangel, Relax and fix heuristics to solve one-stage one machine lot-scheduling models for small-scale soft drink plants, Computers & Operations Research, 37 (2010), 684-691.  doi: 10.1016/j.cor.2009.06.007.  Google Scholar

[23]

S. GuptaM. StarrR. Zanjirani Farahani and N. Matinrad, Disaster management from a POM perspective: Mapping a new domain, Production and Operations Management, 25 (2016), 1611-1637.  doi: 10.1111/poms.12591.  Google Scholar

[24]

A. Haghani and S. C. Oh, Formulation and solution of a multi-commodity, multi-modal network flow model for disaster relief operations, Transportation Research Part A: Policy and Practice, 30 (1996), 231-250.  doi: 10.1016/0965-8564(95)00020-8.  Google Scholar

[25]

F. LiberatoreM. T. OrtunoG. TiradoB. Vitoriano and M. P. Scaparra, A hierarchical compromise model for the joint optimization of recovery operations and distribution of emergency goods in Humanitarian Logistics, Computers & Operations Research, 42 (2014), 3-13.  doi: 10.1016/j.cor.2012.03.019.  Google Scholar

[26]

G. J. LimS. ZangenehM. R. Baharnemati and T. Assavapokee, A capacitated network flow optimization approach for short notice evacuation planning, European Journal of Operational Research, 223 (2012), 234-245.  doi: 10.1016/j.ejor.2012.06.004.  Google Scholar

[27]

Y. H. LinR. BattaP. A. RogersonA. Blatt and M. Flanigan, A logistics model for emergency supply of critical items in the aftermath of a disaster, Socio-Economic Planning Sciences, 45 (2011), 132-145.  doi: 10.1016/j.seps.2011.04.003.  Google Scholar

[28]

Y. LiuX. R. Lai and G. L. Chang, Two-level integrated optimization system for planning emergency evacuation, J. Of Transportation Engineering, 132 (2006), 800-807.  doi: 10.1061/(ASCE)0733-947X(2006)132:10(800).  Google Scholar

[29]

E. Miller-Hooks and S. S. Patterson, On solving quickest time problems in time-dependent, dynamic networks, Journal of Mathematical Modelling and Algorithms, 3 (2004), 39-71.  doi: 10.1023/B:JMMA.0000026708.57419.6d.  Google Scholar

[30]

M. NajafiK. Eshghi and W. Dullaert, A multi-objective robust optimization model for logistics planning in the earthquake response phase, Transportation Research Part E: Logistics and Transportation Review, 49 (2013), 217-249.  doi: 10.1016/j.tre.2012.09.001.  Google Scholar

[31]

L. Ozdamar and O. Demir, A hierarchical clustering and routing procedure for large scale disaster relief logistics planning, Transportation Research Part E: Logistics and Transportation Review, 48 (2012), 591-602.  doi: 10.1016/j.tre.2011.11.003.  Google Scholar

[32]

L. OzdamarE. Ekinci and B. Kucukyazici, Emergency logistics planning in natural disasters, Annals of Operations Research, 129 (2004), 217-245.  doi: 10.1023/B:ANOR.0000030690.27939.39.  Google Scholar

[33]

L. Ozdamar 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.  Google Scholar

[34]

B. VitorianoM. T. OrtunoG. Tirado and J. Montero, A multi-criteria optimization model for humanitarian aid distribution, Journal of Global Optimization, 51 (2011), 189-208.  doi: 10.1007/s10898-010-9603-z.  Google Scholar

[35]

B. VitorianoT. Ortuno and G. Tirado, HADS, a goal programming-based humanitarian aid distribution system, Journal of Multi-Criteria Decision Analysis, 16 (2009), 55-64.  doi: 10.1002/mcda.439.  Google Scholar

[36]

S. Yan and Y. L. Shih, Optimal scheduling of emergency roadway repair and subsequent relief distribution, Computers & Operations Research, 36 (2009), 2049-2065.  doi: 10.1016/j.cor.2008.07.002.  Google Scholar

show all references

References:
[1]

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

[2]

V. Akbari and F. S. Salman, Multi-vehicle synchronized arc routing problem to restore post-disaster network connectivity, European Journal of Operational Research, 257 (2017), 625-640.  doi: 10.1016/j.ejor.2016.07.043.  Google Scholar

[3]

M. Albareda-SambolaA. Alonso-AyusoL. F. EscuderoE. Fernandez and C. Pizarro, Fix-and-relax-coordination for a multi-period location-allocation problem under uncertainty, Computers & Operations Research, 40 (2013), 2878-2892.  doi: 10.1016/j.cor.2013.07.004.  Google Scholar

[4]

A. M. Anaya-ArenasJ. Renaud and A. Ruiz, Relief distribution networks: A systematic review, Annals of Operations Research, 223 (2014), 53-79.  doi: 10.1007/s10479-014-1581-y.  Google Scholar

[5]

J. AraozE. Fernandez and C. Zoltan, Privatized rural postman problems, Computers & Operations Research, 33 (2006), 3432-3449.  doi: 10.1016/j.cor.2005.02.013.  Google Scholar

[6]

J. AraozE. Fernandez and O. Meza, Solving the prize-collecting rural postman problem, European Journal of Operational Research, 196 (2009), 886-896.  doi: 10.1016/j.ejor.2008.04.037.  Google Scholar

[7]

C. ArchettiD. FeilletA. Hertz and M. G. Speranza, The undirected capacitated arc routing problem with profits, Computers & Operations Research, 37 (2010), 1860-1869.  doi: 10.1016/j.cor.2009.05.005.  Google Scholar

[8]

B. BalcikB. M. Beamon and K. Smilowitz, Last mile distribution in humanitarian relief, Journal of Intelligent Transportation Systems, 12 (2008), 51-63.  doi: 10.1080/15472450802023329.  Google Scholar

[9]

T. A. BaldoM. O. SantosB. Almada-Lobo and R. Morabito, An optimization approach for the lot sizing and scheduling problem in the brewery industry, Computers & Industrial Engineering, 72 (2014), 58-71.  doi: 10.1016/j.cie.2014.02.008.  Google Scholar

[10]

P. BeraldiG. GhianiE. Guerriero and A. Grieco, Scenario-based planning for lot-sizing and scheduling with uncertain processing times, International Journal of Production Economics, 101 (2006), 140-149.  doi: 10.1016/j.ijpe.2005.05.018.  Google Scholar

[11]

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.  Google Scholar

[12]

V. CamposR. Bandeira and A. Bandeira, A method for evacuation route planning in disaster situations, Procedia-Social and Behavioral Sciences, 54 (2012), 503-512.  doi: 10.1016/j.sbspro.2012.09.768.  Google Scholar

[13]

A. M. CaunhyeX. Nie and S. Pokharel, Optimization models in emergency logistics: A literature review, Socio-Economic Planning Sciences, 46 (2012), 4-13.  doi: 10.1016/j.seps.2011.04.004.  Google Scholar

[14]

A. Y. ChenF. Pena-Mora and Y. Ouyang, A collaborative GIS framework to support equipment distribution for civil engineering disaster response operations, Automation in Construction, 20 (2011), 637-648.  doi: 10.1016/j.autcon.2010.12.007.  Google Scholar

[15]

T. Cova and J. Johnson, A network flow model for lane-based evacuation routing, Transportation Research Part A: Policy and Practice, 37 (2003), 579-604.  doi: 10.1016/S0965-8564(03)00007-7.  Google Scholar

[16]

J. Cui, S. An and M. Zhao, A generalized minimum cost flow model for multiple emergency flow routing Mathematical Problems in Engineering 2014 (2014), Article ID 832053, 12 pages. doi: 10.1155/2014/832053.  Google Scholar

[17]

C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research Part B: Methodological, 28 (1994), 269-287.  doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[18]

L. E. de la TorreI. S. Dolinskaya and K. R. Smilowitz, Disaster relief routing: Integrating research and practice, Socio-Economic Planning Sciences, 46 (2012), 88-97.   Google Scholar

[19]

L. EscuderoA. GarinM. Merino and G. Perez, BFC-MSMIP: An exact branch-and-fix coordination approach for solving multistage stochastic mixed 0--1 problems, TOP, 17 (2009), 96-122.  doi: 10.1007/s11750-009-0083-6.  Google Scholar

[20]

D. FeilletP. Dejax and M. Gendreau, The profitable arc tour problem: Solution with a branch-and-price algorithm, Transportation Science, 39 (2005), 539-552.  doi: 10.1287/trsc.1040.0106.  Google Scholar

[21]

C. M. Feng and C. C. Wen, A fuzzy bi-level and multi-objective model to control traffic flow into disaster area post earthquake, Journal of the Eastern Asia Society for Transportation Studies, 6 (2005), 4253-4268.   Google Scholar

[22]

D. FerreiraR. Morabito and S. Rangel, Relax and fix heuristics to solve one-stage one machine lot-scheduling models for small-scale soft drink plants, Computers & Operations Research, 37 (2010), 684-691.  doi: 10.1016/j.cor.2009.06.007.  Google Scholar

[23]

S. GuptaM. StarrR. Zanjirani Farahani and N. Matinrad, Disaster management from a POM perspective: Mapping a new domain, Production and Operations Management, 25 (2016), 1611-1637.  doi: 10.1111/poms.12591.  Google Scholar

[24]

A. Haghani and S. C. Oh, Formulation and solution of a multi-commodity, multi-modal network flow model for disaster relief operations, Transportation Research Part A: Policy and Practice, 30 (1996), 231-250.  doi: 10.1016/0965-8564(95)00020-8.  Google Scholar

[25]

F. LiberatoreM. T. OrtunoG. TiradoB. Vitoriano and M. P. Scaparra, A hierarchical compromise model for the joint optimization of recovery operations and distribution of emergency goods in Humanitarian Logistics, Computers & Operations Research, 42 (2014), 3-13.  doi: 10.1016/j.cor.2012.03.019.  Google Scholar

[26]

G. J. LimS. ZangenehM. R. Baharnemati and T. Assavapokee, A capacitated network flow optimization approach for short notice evacuation planning, European Journal of Operational Research, 223 (2012), 234-245.  doi: 10.1016/j.ejor.2012.06.004.  Google Scholar

[27]

Y. H. LinR. BattaP. A. RogersonA. Blatt and M. Flanigan, A logistics model for emergency supply of critical items in the aftermath of a disaster, Socio-Economic Planning Sciences, 45 (2011), 132-145.  doi: 10.1016/j.seps.2011.04.003.  Google Scholar

[28]

Y. LiuX. R. Lai and G. L. Chang, Two-level integrated optimization system for planning emergency evacuation, J. Of Transportation Engineering, 132 (2006), 800-807.  doi: 10.1061/(ASCE)0733-947X(2006)132:10(800).  Google Scholar

[29]

E. Miller-Hooks and S. S. Patterson, On solving quickest time problems in time-dependent, dynamic networks, Journal of Mathematical Modelling and Algorithms, 3 (2004), 39-71.  doi: 10.1023/B:JMMA.0000026708.57419.6d.  Google Scholar

[30]

M. NajafiK. Eshghi and W. Dullaert, A multi-objective robust optimization model for logistics planning in the earthquake response phase, Transportation Research Part E: Logistics and Transportation Review, 49 (2013), 217-249.  doi: 10.1016/j.tre.2012.09.001.  Google Scholar

[31]

L. Ozdamar and O. Demir, A hierarchical clustering and routing procedure for large scale disaster relief logistics planning, Transportation Research Part E: Logistics and Transportation Review, 48 (2012), 591-602.  doi: 10.1016/j.tre.2011.11.003.  Google Scholar

[32]

L. OzdamarE. Ekinci and B. Kucukyazici, Emergency logistics planning in natural disasters, Annals of Operations Research, 129 (2004), 217-245.  doi: 10.1023/B:ANOR.0000030690.27939.39.  Google Scholar

[33]

L. Ozdamar 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.  Google Scholar

[34]

B. VitorianoM. T. OrtunoG. Tirado and J. Montero, A multi-criteria optimization model for humanitarian aid distribution, Journal of Global Optimization, 51 (2011), 189-208.  doi: 10.1007/s10898-010-9603-z.  Google Scholar

[35]

B. VitorianoT. Ortuno and G. Tirado, HADS, a goal programming-based humanitarian aid distribution system, Journal of Multi-Criteria Decision Analysis, 16 (2009), 55-64.  doi: 10.1002/mcda.439.  Google Scholar

[36]

S. Yan and Y. L. Shih, Optimal scheduling of emergency roadway repair and subsequent relief distribution, Computers & Operations Research, 36 (2009), 2049-2065.  doi: 10.1016/j.cor.2008.07.002.  Google Scholar

Figure 1.  Road network of Fatih County in Istanbul
Table .   
Algorithm 1
1: Initialize $k=0$
2:While $(k < \frac{T}{\Delta t})$ do
3:     $k++$
4:     Define $y_{ijcq}$ and $v_{ijcq}$ for $q=(k-1)\Delta t+1, ..., k\Delta t$ as integers, let $y_{ijcq}$ and $v_{ijcq}$ be positive float variables for $q= k\Delta t+1, ..., T$
5:     Solve Model R
6:     Fix $y_{ijcq}$ and $v_{ijcq}$ for $q=(k-1)\Delta t+1, ..., k\Delta t$ at their optimal integer values
7: end while
8: Report final solution
9: End (Algorithm1)
Algorithm 1
1: Initialize $k=0$
2:While $(k < \frac{T}{\Delta t})$ do
3:     $k++$
4:     Define $y_{ijcq}$ and $v_{ijcq}$ for $q=(k-1)\Delta t+1, ..., k\Delta t$ as integers, let $y_{ijcq}$ and $v_{ijcq}$ be positive float variables for $q= k\Delta t+1, ..., T$
5:     Solve Model R
6:     Fix $y_{ijcq}$ and $v_{ijcq}$ for $q=(k-1)\Delta t+1, ..., k\Delta t$ at their optimal integer values
7: end while
8: Report final solution
9: End (Algorithm1)
Table 1.  Performance of the RF Algorithm as a function of $\Delta t$
Region Method Obj. CPU(secs.) Rel.Gap % Dev
1
(56 links)
No RF 1948.89 590.48 0.001 -
RF ($\Delta t$=2) 1928.38 680.01 -1.052
RF ($\Delta t$=4) 1931.90 415.80 -0.872
RF ($\Delta t$=8) 1932.05 374.31 -0.864
2
(54 links)
No RF 2055.46 3609.47 0.017 -
RF ($\Delta t$=2) 2037.86 734.53 -0.856
RF ($\Delta t$=4) 2052.20 489.48 -0.159
RF ($\Delta t$=8) 2053.23 744.67 -0.108
3
(65 links)
No RF 1659.32 3611.89 0.021 -
RF ($\Delta t$=2) 1667.39 999.43 0.486
RF ($\Delta t$=4) 1655.30 636.56 -0.242
RF ($\Delta t$=8) 1656.58 645.14 -0.165
4
(52 links)
No RF 3967.29 3609.02 0.003 -
RF ($\Delta t$=2) 3964.52 713.17 -0.070
RF ($\Delta t$=4) 3961.60 424.16 -0.143
RF ($\Delta t$=8) 3967.04 278.05 -0.006
5
(46 links)
No RF 2969.91 3610.08 0.009 -
RF ($\Delta t$=2) 2969.68 304.56 -0.008
RF ($\Delta t$=4) 2973.97 164.88 0.137
RF ($\Delta t$=8) 2972.40 140.04 0.084
6
(63 links)
No RF 1347.78 3612.22 0.018 -
RF ($\Delta t$=2) 1346.13 948.06 -0.122
RF ($\Delta t$=4) 1353.34 533.05 0.413
RF ($\Delta t$=8) 1353.42 472.09 0.418
7
(64 links)
No RF 1834.74 3608.97 0.011 -
RF ($\Delta t$=2) 1822.31 999.92 -0.677
RF ($\Delta t$=4) 1824.13 539.54 -0.578
RF ($\Delta t$=8) 1830.28 475.05 -0.243
Region Method Obj. CPU(secs.) Rel.Gap % Dev
1
(56 links)
No RF 1948.89 590.48 0.001 -
RF ($\Delta t$=2) 1928.38 680.01 -1.052
RF ($\Delta t$=4) 1931.90 415.80 -0.872
RF ($\Delta t$=8) 1932.05 374.31 -0.864
2
(54 links)
No RF 2055.46 3609.47 0.017 -
RF ($\Delta t$=2) 2037.86 734.53 -0.856
RF ($\Delta t$=4) 2052.20 489.48 -0.159
RF ($\Delta t$=8) 2053.23 744.67 -0.108
3
(65 links)
No RF 1659.32 3611.89 0.021 -
RF ($\Delta t$=2) 1667.39 999.43 0.486
RF ($\Delta t$=4) 1655.30 636.56 -0.242
RF ($\Delta t$=8) 1656.58 645.14 -0.165
4
(52 links)
No RF 3967.29 3609.02 0.003 -
RF ($\Delta t$=2) 3964.52 713.17 -0.070
RF ($\Delta t$=4) 3961.60 424.16 -0.143
RF ($\Delta t$=8) 3967.04 278.05 -0.006
5
(46 links)
No RF 2969.91 3610.08 0.009 -
RF ($\Delta t$=2) 2969.68 304.56 -0.008
RF ($\Delta t$=4) 2973.97 164.88 0.137
RF ($\Delta t$=8) 2972.40 140.04 0.084
6
(63 links)
No RF 1347.78 3612.22 0.018 -
RF ($\Delta t$=2) 1346.13 948.06 -0.122
RF ($\Delta t$=4) 1353.34 533.05 0.413
RF ($\Delta t$=8) 1353.42 472.09 0.418
7
(64 links)
No RF 1834.74 3608.97 0.011 -
RF ($\Delta t$=2) 1822.31 999.92 -0.677
RF ($\Delta t$=4) 1824.13 539.54 -0.578
RF ($\Delta t$=8) 1830.28 475.05 -0.243
Table 2.  Comparison of solution quality for various decomposition strategies
Subnetworks Obj. CPU Rel.Gap No. of Links
67 3527.10 7251.00 0.03 127
6+7 3182.52 7221.19
6+7 (RF) 3183.70 947.14
12 3998.60 7241.77 0.02 110
1+2 4004.35 4199.95
1+2 (RF) 3985.28 1118.97
45 8499.28 7232.73 0.01 98
4+5 6937.20 7219.10
4+5 (RF) 6939.44 418.10
37 3744.10 7243.97 0.10 129
3+7 3494.06 7220.86
3+7 (RF) 3486.86 1120.19
34 5621.70 7246.52 0.01 117
3+4 5626.61 7220.91
3+4 (RF) 5623.62 923.19
345 9050.53 10897.60 0.12 163
34+5 8591.61 10856.60
3+4+5 8596.52 10830.99
3+4+5 (RF) 8596.02 1063.23
456 9713.23 10905.90 0.08 161
45+6 9847.06 10844.95
4+5+6 8284.98 10831.32
4+5+6 (RF) 8292.86 890.18
567 5520.59 10914.90 0.24 173
5+67 6497.01 10861.08
5+6+7 6152.43 10831.27
5+6+7 (RF) 6156.10 1087.18
123 650.07 10939.80 1.00 175
12+3 5657.92 10853.66
1+2+3 5663.67 7811.84
1+2+3 (RF) 5641.86 1764.11
12345 1145.08 18450.00 1.00 273
12+345 13049.13 18139.37
123+45 9149.35 18172.53
1+2+3+4+5 12600.87 15030.94
1+2+3+4+5 (RF) 12581.30 2182.21
34567 1170.16 18560.90 1.00 290
34+567 11142.29 18161.42
37+456 13457.33 18149.87
3+4+5+6+7 11779.04 18052.18
3+4+5+6+7 (RF) 11779.72 2010.37
Subnetworks Obj. CPU Rel.Gap No. of Links
67 3527.10 7251.00 0.03 127
6+7 3182.52 7221.19
6+7 (RF) 3183.70 947.14
12 3998.60 7241.77 0.02 110
1+2 4004.35 4199.95
1+2 (RF) 3985.28 1118.97
45 8499.28 7232.73 0.01 98
4+5 6937.20 7219.10
4+5 (RF) 6939.44 418.10
37 3744.10 7243.97 0.10 129
3+7 3494.06 7220.86
3+7 (RF) 3486.86 1120.19
34 5621.70 7246.52 0.01 117
3+4 5626.61 7220.91
3+4 (RF) 5623.62 923.19
345 9050.53 10897.60 0.12 163
34+5 8591.61 10856.60
3+4+5 8596.52 10830.99
3+4+5 (RF) 8596.02 1063.23
456 9713.23 10905.90 0.08 161
45+6 9847.06 10844.95
4+5+6 8284.98 10831.32
4+5+6 (RF) 8292.86 890.18
567 5520.59 10914.90 0.24 173
5+67 6497.01 10861.08
5+6+7 6152.43 10831.27
5+6+7 (RF) 6156.10 1087.18
123 650.07 10939.80 1.00 175
12+3 5657.92 10853.66
1+2+3 5663.67 7811.84
1+2+3 (RF) 5641.86 1764.11
12345 1145.08 18450.00 1.00 273
12+345 13049.13 18139.37
123+45 9149.35 18172.53
1+2+3+4+5 12600.87 15030.94
1+2+3+4+5 (RF) 12581.30 2182.21
34567 1170.16 18560.90 1.00 290
34+567 11142.29 18161.42
37+456 13457.33 18149.87
3+4+5+6+7 11779.04 18052.18
3+4+5+6+7 (RF) 11779.72 2010.37
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