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April  2017, 13(2): 1065-1084. doi: 10.3934/jimo.2016062

Solving routing and wavelength assignment problem with maximum edge-disjoint paths

1. 

Department of Transportation and Logistics Management, National Chiao Tung University, Hsinchu 300, Taiwan

2. 

Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, USA

* Corresponding author

Received  October 2014 Revised  August 2016 Published  October 2016

The routing and wavelength assignment (RWA) problem in wave-length-division multiplexing optical networks is critically important for increasing the efficiency of wavelength-routed all-optical networks. Given the physical network structure and a set of connection requests, the problem aims to establishing routes for the requests and assigning a wavelength to each of them, subject to the wavelength continuous and wavelength clash constraints. The objective is to minimize he number of required wavelengths to satisfy all requests. The RWA problem was proven to be NP-hard and has been investigated for decades. In this paper, an algorithm based on the maximum edge-disjoint paths is proposed to tackle the RWA problem. Its performance is verified by experiments on some realistic network topologies. Compared with the state-of-the-art bin-packing based methods and the particle swarm optimization algorithm, the proposed method can find the best solution among all testing instances in reasonable computing time.

Citation: Chia-Chun Hsu, Hsun-Jung Cho, Shu-Cherng Fang. Solving routing and wavelength assignment problem with maximum edge-disjoint paths. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1065-1084. doi: 10.3934/jimo.2016062
References:
[1]

D. Banerjee and B. Mukherjee, A practical approach for routing and wavelength assignment in large wavelength-routed optical networks, IEEE Journal on Selected Areas in Communications, 14 (1996), 903-908. doi: 10.1109/49.510913. Google Scholar

[2]

M. J. Blesa and C. Blum, Findinge edge-disjoint paths in networks: An ant colony optimization algorithm, Journal of Mathematical Modeling and Algorithms, 6 (2007), 367-391. doi: 10.1007/s10852-007-9060-y. Google Scholar

[3]

I. ChlamtacA. Ganz and G. Karmi, Lightpath communications: an approach to high bandwidth optical WAN's, IEEE Transactions on Communications, 40 (1992), 1171-1182. doi: 10.1109/26.153361. Google Scholar

[4]

J. S. Choi, N. Golmie, F. Lapeyrere, F. Mouveaux and D. Su, A functional classification of routing and wavelength assignment schemes in DWDM networks: Static Case, In Proceedings of the 7th International Conference on Optical Communications and Networks, (2000), 1109-1115.Google Scholar

[5]

H. Choo and V. V. Shakhov, Routing and wavelength assignment in optical WDM networks with maximum quantity of edge disjoint paths, Computational Science -ICCS 2004, 3038 (2004), 1138-1145. doi: 10.1007/978-3-540-24688-6_147. Google Scholar

[6]

T. FischerK. BauerP. Merz and K. Bauer, Solving the routing and wavelength assignment problem with a multilevel distributed memetic algorithm, Memetic Computing, 1 (2009), 101-123. doi: 10.1007/s12293-008-0006-3. Google Scholar

[7]

X. GuanS. GuoW. Gong and C. Qiao, A new method for solving routing and wavelength assignment problems in optical networks, Journal of Lightwave Technology, 25 (2007), 1895-1909. doi: 10.1109/JLT.2007.901344. Google Scholar

[8]

A. Hassan, Particle Swarm Optimization for Routing and Wavelength Assignment in Next Generation WDM Networks, Ph. D thesis, Queen Mary University of London, 2010.Google Scholar

[9]

C.-C. Hsu and H.-J. Cho, A genetic algorithm for the maximum edge-disjoint paths problem, Neurocomputing, 148 (2015), 17-22. doi: 10.1016/j.neucom.2012.10.046. Google Scholar

[10]

C.-C. HsuH.-J. Cho and S.-C. Fang, Routing and wavelength assignment in optical networks from maximum edge-disjoint paths, Genetic and Evolutionary Computing: Advances in Intelligent Systems and Computing, 238 (2014), 95-103. doi: 10.1007/978-3-319-01796-9_10. Google Scholar

[11]

Y. S. KavianA. RashediA. Mahani and Z. Ghassemlooy, Routing and wavelength assignment in optical networks using artificial bee colony algorithm, European Journal of Operational Research, 124 (2013), 1243-1249. doi: 10.1016/j.ijleo.2012.03.022. Google Scholar

[12]

J. Kleinberg, Approximation Algorithms for Disjoint Paths Problems, Ph. D thesis, Massachusetts Institute of Technology, 1996.Google Scholar

[13]

P. LeesutthipornchaiC. Charnsripinyo and N. Wattanapongsakorn, Solving multi-objective routing and wavelength assignment in WDM network using hybrid evolutionary computation approach, Computer Communications, 33 (2010), 2246-2259. doi: 10.1016/j.comcom.2010.07.029. Google Scholar

[14]

G. Li and R. Shmha, The partition coloring problem and its application to wavelength routing and assignment, In Proceedings of the First Workshop on Optical Networks, 2000.Google Scholar

[15]

P. ManoharD. Manjunath and R. K. Shevgaonkar, Routing and wavelength assignment in optical networks from edge disjoint path algorithms, IEEE Communications Letters, 6 (2002), 211-213. doi: 10.1109/4234.1001667. Google Scholar

[16]

A. X. MartinsC. DuhamelP. MaheyR. R. Saldanha and M. C. de Souza, Variable neighborhood descent with iterated local search for routing and wavelength assignment, Computers and Operations Research, 39 (2012), 2133-2141. doi: 10.1016/j.cor.2011.10.022. Google Scholar

[17]

T. F. NoronhaM. G. C. Resende and C. C. Ribeiro, A biased random-key genetic algorithm for routing and wavelength assignment, Journal of Global Optimization, 50 (2011), 503-518. doi: 10.1007/s10898-010-9608-7. Google Scholar

[18]

T. F. Noronha and C. C. Ribeiro, Routing and wavelength assignment by partition colouring, European Journal of Operational Research, 171 (2006), 797-810. doi: 10.1016/j.ejor.2004.09.007. Google Scholar

[19]

A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks, IEEE/ACM Transactions on Networking, 11 (2003), 259-272. doi: 10.1109/TNET.2003.810321. Google Scholar

[20]

R. PoliJ. Kennedy and T. Blackwell, Particle swarm optimization -An overview, Swarm intelligence, 1 (2007), 33-57. Google Scholar

[21]

N. Skorin-Kapov, Routing and wavelength assignment in optical networks using bin packing based algorithms, European Journal of Operational Research, 177 (2007), 1167-1179. doi: 10.1016/j.ejor.2006.01.003. Google Scholar

[22]

Y. WangT. H. Cheng and M. H. Lim, A Tabu search algorithm for static routing and wavelength assignment problem, IEEE Communications Letters, 9 (2005), 841-843. doi: 10.1109/LCOMM.2005.1506721. Google Scholar

[23]

W. J. YoonD. H. KimM. Y. ChungT. J. Lee and H. Choo, Routing with maximum EDPs and wavelength assignment with path conflict graphs, In Proceedings of the 2006 international conference on Computational Science and Its Applications, 3981 (2006), 856-865. doi: 10.1007/11751588_89. Google Scholar

[24]

H. ZangJ. P. Jue and B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, 1 (2000), 47-60. Google Scholar

show all references

References:
[1]

D. Banerjee and B. Mukherjee, A practical approach for routing and wavelength assignment in large wavelength-routed optical networks, IEEE Journal on Selected Areas in Communications, 14 (1996), 903-908. doi: 10.1109/49.510913. Google Scholar

[2]

M. J. Blesa and C. Blum, Findinge edge-disjoint paths in networks: An ant colony optimization algorithm, Journal of Mathematical Modeling and Algorithms, 6 (2007), 367-391. doi: 10.1007/s10852-007-9060-y. Google Scholar

[3]

I. ChlamtacA. Ganz and G. Karmi, Lightpath communications: an approach to high bandwidth optical WAN's, IEEE Transactions on Communications, 40 (1992), 1171-1182. doi: 10.1109/26.153361. Google Scholar

[4]

J. S. Choi, N. Golmie, F. Lapeyrere, F. Mouveaux and D. Su, A functional classification of routing and wavelength assignment schemes in DWDM networks: Static Case, In Proceedings of the 7th International Conference on Optical Communications and Networks, (2000), 1109-1115.Google Scholar

[5]

H. Choo and V. V. Shakhov, Routing and wavelength assignment in optical WDM networks with maximum quantity of edge disjoint paths, Computational Science -ICCS 2004, 3038 (2004), 1138-1145. doi: 10.1007/978-3-540-24688-6_147. Google Scholar

[6]

T. FischerK. BauerP. Merz and K. Bauer, Solving the routing and wavelength assignment problem with a multilevel distributed memetic algorithm, Memetic Computing, 1 (2009), 101-123. doi: 10.1007/s12293-008-0006-3. Google Scholar

[7]

X. GuanS. GuoW. Gong and C. Qiao, A new method for solving routing and wavelength assignment problems in optical networks, Journal of Lightwave Technology, 25 (2007), 1895-1909. doi: 10.1109/JLT.2007.901344. Google Scholar

[8]

A. Hassan, Particle Swarm Optimization for Routing and Wavelength Assignment in Next Generation WDM Networks, Ph. D thesis, Queen Mary University of London, 2010.Google Scholar

[9]

C.-C. Hsu and H.-J. Cho, A genetic algorithm for the maximum edge-disjoint paths problem, Neurocomputing, 148 (2015), 17-22. doi: 10.1016/j.neucom.2012.10.046. Google Scholar

[10]

C.-C. HsuH.-J. Cho and S.-C. Fang, Routing and wavelength assignment in optical networks from maximum edge-disjoint paths, Genetic and Evolutionary Computing: Advances in Intelligent Systems and Computing, 238 (2014), 95-103. doi: 10.1007/978-3-319-01796-9_10. Google Scholar

[11]

Y. S. KavianA. RashediA. Mahani and Z. Ghassemlooy, Routing and wavelength assignment in optical networks using artificial bee colony algorithm, European Journal of Operational Research, 124 (2013), 1243-1249. doi: 10.1016/j.ijleo.2012.03.022. Google Scholar

[12]

J. Kleinberg, Approximation Algorithms for Disjoint Paths Problems, Ph. D thesis, Massachusetts Institute of Technology, 1996.Google Scholar

[13]

P. LeesutthipornchaiC. Charnsripinyo and N. Wattanapongsakorn, Solving multi-objective routing and wavelength assignment in WDM network using hybrid evolutionary computation approach, Computer Communications, 33 (2010), 2246-2259. doi: 10.1016/j.comcom.2010.07.029. Google Scholar

[14]

G. Li and R. Shmha, The partition coloring problem and its application to wavelength routing and assignment, In Proceedings of the First Workshop on Optical Networks, 2000.Google Scholar

[15]

P. ManoharD. Manjunath and R. K. Shevgaonkar, Routing and wavelength assignment in optical networks from edge disjoint path algorithms, IEEE Communications Letters, 6 (2002), 211-213. doi: 10.1109/4234.1001667. Google Scholar

[16]

A. X. MartinsC. DuhamelP. MaheyR. R. Saldanha and M. C. de Souza, Variable neighborhood descent with iterated local search for routing and wavelength assignment, Computers and Operations Research, 39 (2012), 2133-2141. doi: 10.1016/j.cor.2011.10.022. Google Scholar

[17]

T. F. NoronhaM. G. C. Resende and C. C. Ribeiro, A biased random-key genetic algorithm for routing and wavelength assignment, Journal of Global Optimization, 50 (2011), 503-518. doi: 10.1007/s10898-010-9608-7. Google Scholar

[18]

T. F. Noronha and C. C. Ribeiro, Routing and wavelength assignment by partition colouring, European Journal of Operational Research, 171 (2006), 797-810. doi: 10.1016/j.ejor.2004.09.007. Google Scholar

[19]

A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks, IEEE/ACM Transactions on Networking, 11 (2003), 259-272. doi: 10.1109/TNET.2003.810321. Google Scholar

[20]

R. PoliJ. Kennedy and T. Blackwell, Particle swarm optimization -An overview, Swarm intelligence, 1 (2007), 33-57. Google Scholar

[21]

N. Skorin-Kapov, Routing and wavelength assignment in optical networks using bin packing based algorithms, European Journal of Operational Research, 177 (2007), 1167-1179. doi: 10.1016/j.ejor.2006.01.003. Google Scholar

[22]

Y. WangT. H. Cheng and M. H. Lim, A Tabu search algorithm for static routing and wavelength assignment problem, IEEE Communications Letters, 9 (2005), 841-843. doi: 10.1109/LCOMM.2005.1506721. Google Scholar

[23]

W. J. YoonD. H. KimM. Y. ChungT. J. Lee and H. Choo, Routing with maximum EDPs and wavelength assignment with path conflict graphs, In Proceedings of the 2006 international conference on Computational Science and Its Applications, 3981 (2006), 856-865. doi: 10.1007/11751588_89. Google Scholar

[24]

H. ZangJ. P. Jue and B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, 1 (2000), 47-60. Google Scholar

Figure 1.  Pseudocode of FD(FFD) Algorithm
Figure 2.  Pseudocode of FD(FFD) Algorithm
Figure 3.  Pseudocode of PSO Algorithm
Figure 4.  Pseudocode of a basic MEDP-RWA algorithm
Figure 5.  Pseudocode of GMR algorithm
Figure 6.  Number of times that the best value is achieved by different methods
Figure 7.  Frequency graphs of testing results on zib54
Figure 8.  Frequency graphs of testing results on ta2
Figure 9.  Frequency graphs of testing results on Germany50
Figure 10.  Relative difference of computational time on Norway
Figure 11.  Relative difference of computational time on giul
Figure 12.  Relative difference of computational time on graph Germany
Figure 13.  Relative difference of computational time on graph ta2
Table 1.  Main quantitative characteristics of testing networks
Graph$\left | V \right |$$\left | E \right |$Min.Avg.Max.Diameter
CHNNET152733.655
NSF162523.144
New York164926.1113
APPANET203233.246
EON203923.975
France254523.6105
Norway275123.867
cost266375723.158
janos-us-ca396723.1510
giul398634.486
piro40408944.557
USAnet467523.3511
Germany50508823.559
zib54548013.0108
ta26510813.3108
(Min., Avg. and Max. denote the minimum, average and maximum degree, respectively)
Graph$\left | V \right |$$\left | E \right |$Min.Avg.Max.Diameter
CHNNET152733.655
NSF162523.144
New York164926.1113
APPANET203233.246
EON203923.975
France254523.6105
Norway275123.867
cost266375723.158
janos-us-ca396723.1510
giul398634.486
piro40408944.557
USAnet467523.3511
Germany50508823.559
zib54548013.0108
ta26510813.3108
(Min., Avg. and Max. denote the minimum, average and maximum degree, respectively)
Table 2.  Results of GMR and bin-packing based methods (time unit: sec)
GA_MEDP_RWA (GMR)FFFFDBFBFD
instancemaxavgminavg time# wl$t_{FF}$# wl$t_{FFD}$# wl$t_{BF}$# wl$t_{BFD}$
CHNNET_0244.042.0740.2440.1440.1150.14
CHNNET_0454.443.0250.1840.1550.2340.26
CHNNET_061111.01110.66110.70110.62110.81110.96
CHNNET_081414.01413.50151.02141.17141.21141.51
CHNNET_11615.01516.31151.48151.42161.75152.20
NSF_0255.052.4160.3450.1760.1960.18
NSF_0487.375.8090.3790.3580.5280.50
NSF_0698.887.97100.46100.4990.5990.64
NSF_081413.21315.11171.06151.10141.50141.47
NSF_11716.61622.89191.70171.45171.96172.08
NewYork_0222.021.3720.1420.0820.0720.08
NewYork_0433.032.6530.2330.2030.2330.23
NewYork_0654.143.9340.3340.3440.3440.33
NewYork_0876.066.7570.6460.5870.6260.81
NewYork_188.087.0380.7380.8081.0282.30
ARPANET_02109.1910.5890.5590.5390.6490.86
ARPANET_041312.11216.96121.05121.29121.31121.76
ARPANET_062121.02125.98212.70212.68213.39214.58
ARPANET_082929.02932.31305.46296.14307.822910.94
ARPANET_13333.03362.85346.85337.773410.193312.68
EON_0243.133.1930.2240.1740.3440.36
EON_0498.3814.74101.1091.1481.5381.54
EON_061111.01117.52131.21111.18111.94111.90
EON_081413.71326.63162.20132.99133.61133.72
EON_11918.11838.17223.64183.58185.77185.73
France_0288.088.8781.3081.2781.5781.72
France _041312.81216.05144.29134.90135.14136.03
France _062222.02239.07228.732210.552211.692215.27
France _082726.32647.092813.252714.002718.532624.20
France _13434.03460.093422.543423.993429.933435.84
GA_MEDP_RWA (GMR)FFFFDBFBFD
instancemaxavgminavg time# wl$t_{FF}$# wl$t_{FFD}$# wl$t_{BF}$# wl$t_{BFD}$
CHNNET_0244.042.0740.2440.1440.1150.14
CHNNET_0454.443.0250.1840.1550.2340.26
CHNNET_061111.01110.66110.70110.62110.81110.96
CHNNET_081414.01413.50151.02141.17141.21141.51
CHNNET_11615.01516.31151.48151.42161.75152.20
NSF_0255.052.4160.3450.1760.1960.18
NSF_0487.375.8090.3790.3580.5280.50
NSF_0698.887.97100.46100.4990.5990.64
NSF_081413.21315.11171.06151.10141.50141.47
NSF_11716.61622.89191.70171.45171.96172.08
NewYork_0222.021.3720.1420.0820.0720.08
NewYork_0433.032.6530.2330.2030.2330.23
NewYork_0654.143.9340.3340.3440.3440.33
NewYork_0876.066.7570.6460.5870.6260.81
NewYork_188.087.0380.7380.8081.0282.30
ARPANET_02109.1910.5890.5590.5390.6490.86
ARPANET_041312.11216.96121.05121.29121.31121.76
ARPANET_062121.02125.98212.70212.68213.39214.58
ARPANET_082929.02932.31305.46296.14307.822910.94
ARPANET_13333.03362.85346.85337.773410.193312.68
EON_0243.133.1930.2240.1740.3440.36
EON_0498.3814.74101.1091.1481.5381.54
EON_061111.01117.52131.21111.18111.94111.90
EON_081413.71326.63162.20132.99133.61133.72
EON_11918.11838.17223.64183.58185.77185.73
France_0288.088.8781.3081.2781.5781.72
France _041312.81216.05144.29134.90135.14136.03
France _062222.02239.07228.732210.552211.692215.27
France _082726.32647.092813.252714.002718.532624.20
France _13434.03460.093422.543423.993429.933435.84
Table 3.  Results of GMR and bin-packing based methods (cont'd)
GA_MEDP_RWA (GMR)FFFFDBFBFD
instancemaxavgminavg time# wl$t_{FF}$# wl$t_{FFD}$# wl$t_{BF}$# wl$t_{BFD}$
Norway_0298.689.26101.6781.7891.9082.17
Norway _041514.61418.47153.75154.12154.56155.43
Norway _062221.42132.84227.87228.20229.732212.32
Norway _083029.52946.413115.053016.593119.793024.59
Norway _13736.63663.773721.993623.753828.583636.13
cost266_021918.21846.68197.34187.441910.381912.40
cost266_043534.133159.693527.283328.583636.353547.28
cost266_065453.153237.555467.185370.905598.0053117.30
cost266_086867.267275.2967120.7368144.2269190.5568273.40
janos-us-ca_022626.02654.752716.612616.532727.402727.58
janos-us-ca_044039.63997.424249.523959.494370.754088.45
janos-us-ca_066867.867210.3671110.9969124.1072157.5068200.32
janos-us-ca_088988.288326.3992199.0992212.1693257.3991345.67
giul_021110.31026.93107.40108.11108.891011.24
giul_042120.21982.661929.101931.701937.421946.16
giul_062524.624150.732548.912555.962559.102473.12
giul_083634.934226.8535113.8534125.2036134.2534181.70
piro40_0217171755.511710.671712.581715.201718.50
piro40_04282828118.502834.982838.002843.052854.44
piro40_06464646208.814661.984668.004684.3346109.15
piro40_08606060338.6160105.9760119.3560140.2060181.52
USAnet_022524.02386.132535.622637.952645.152556.09
USAnet_044745.945355.6447130.4345147.7648176.4345228.50
USAnet_066766.165422.6368244.0065287.5868346.6666432.89
USAnet_088886.585490.2088475.6985544.9889587.1186791.38
Germany50_022120.720109.282135.862136.102146.102258.58
Germany50_044544.243350.2445152.9044163.0048217.6048276.50
Germany50_066160.560584.9263347.7061410.6163430.8965544.74
Germany50_087674.773921.5977552.5075725.9079843.70761098.40
zib54_023231.130149.443352.283155.013577.703391.67
zib54_046765.765406.9067209.8065220.5073304.6471379.55
zib54_069290.088762.5691442.9589522.8599696.7898889.98
zib54_08122119.21171446.41117994.60117939.601301229.601271457.22
ta2_023534.634411.5935148.9334163.0035188.7535250.99
ta2_047270.169945.7972507.3069542.5073635.5770813.04
ta2_069997.4961865.20981311.70981484.001001881.30972047.40
ta2_08131129.71282835.571301935.301292577.901352644.431283713.46
GA_MEDP_RWA (GMR)FFFFDBFBFD
instancemaxavgminavg time# wl$t_{FF}$# wl$t_{FFD}$# wl$t_{BF}$# wl$t_{BFD}$
Norway_0298.689.26101.6781.7891.9082.17
Norway _041514.61418.47153.75154.12154.56155.43
Norway _062221.42132.84227.87228.20229.732212.32
Norway _083029.52946.413115.053016.593119.793024.59
Norway _13736.63663.773721.993623.753828.583636.13
cost266_021918.21846.68197.34187.441910.381912.40
cost266_043534.133159.693527.283328.583636.353547.28
cost266_065453.153237.555467.185370.905598.0053117.30
cost266_086867.267275.2967120.7368144.2269190.5568273.40
janos-us-ca_022626.02654.752716.612616.532727.402727.58
janos-us-ca_044039.63997.424249.523959.494370.754088.45
janos-us-ca_066867.867210.3671110.9969124.1072157.5068200.32
janos-us-ca_088988.288326.3992199.0992212.1693257.3991345.67
giul_021110.31026.93107.40108.11108.891011.24
giul_042120.21982.661929.101931.701937.421946.16
giul_062524.624150.732548.912555.962559.102473.12
giul_083634.934226.8535113.8534125.2036134.2534181.70
piro40_0217171755.511710.671712.581715.201718.50
piro40_04282828118.502834.982838.002843.052854.44
piro40_06464646208.814661.984668.004684.3346109.15
piro40_08606060338.6160105.9760119.3560140.2060181.52
USAnet_022524.02386.132535.622637.952645.152556.09
USAnet_044745.945355.6447130.4345147.7648176.4345228.50
USAnet_066766.165422.6368244.0065287.5868346.6666432.89
USAnet_088886.585490.2088475.6985544.9889587.1186791.38
Germany50_022120.720109.282135.862136.102146.102258.58
Germany50_044544.243350.2445152.9044163.0048217.6048276.50
Germany50_066160.560584.9263347.7061410.6163430.8965544.74
Germany50_087674.773921.5977552.5075725.9079843.70761098.40
zib54_023231.130149.443352.283155.013577.703391.67
zib54_046765.765406.9067209.8065220.5073304.6471379.55
zib54_069290.088762.5691442.9589522.8599696.7898889.98
zib54_08122119.21171446.41117994.60117939.601301229.601271457.22
ta2_023534.634411.5935148.9334163.0035188.7535250.99
ta2_047270.169945.7972507.3069542.5073635.5770813.04
ta2_069997.4961865.20981311.70981484.001001881.30972047.40
ta2_08131129.71282835.571301935.301292577.901352644.431283713.46
Table 4.  Results of GMR and PSO (time unit: sec)
GA_MEDP_RWAPSO
instancemaxavgminavg timemaxavgminavg time
CHNNET_0244.042.0775.9512.97
CHNNET_0454.443.0297.5718.60
CHNNET_061111.01110.661815.51330.39
CHNNET_081414.01413.502420.41848.95
CHNNET_11615.01516.312521.51878.96
NSF_0255.052.4176.056.75
NSF_0487.375.80109.0813.75
NSF_0698.887.971110.0916.19
NSF_081413.21315.111715.51429.02
NSF_11716.61622.892018.91836.03
NewYork_0222.021.3732.927.37
NewYork _0433.032.6565.2418.78
NewYork _0654.143.9386.8625.51
NewYork _0876.066.75109.1838.35
NewYork _188.087.031210.71050.17
ARPANET_02109.1910.581311.31021.30
ARPANET_041312.11216.961916.31462.15
ARPANET_062121.02125.982824.92293.86
ARPANET_082929.02932.314036.533133.70
ARPANET_13333.03362.854641.236172.60
EON_0243.133.1954.3410.40
EON_0498.3814.741210.61035.34
EON_061111.01117.521513.01246.61
EON_081413.71326.632018.21748.92
EON_11918.11838.172522.92273.32
France_0288.088.871411.51050.60
France _041312.81216.052219.31786.48
France _062222.02239.073430.427129.16
France _082726.32647.094540.338290.72
France _13434.03460.095448.945236.64
GA_MEDP_RWAPSO
instancemaxavgminavg timemaxavgminavg time
CHNNET_0244.042.0775.9512.97
CHNNET_0454.443.0297.5718.60
CHNNET_061111.01110.661815.51330.39
CHNNET_081414.01413.502420.41848.95
CHNNET_11615.01516.312521.51878.96
NSF_0255.052.4176.056.75
NSF_0487.375.80109.0813.75
NSF_0698.887.971110.0916.19
NSF_081413.21315.111715.51429.02
NSF_11716.61622.892018.91836.03
NewYork_0222.021.3732.927.37
NewYork _0433.032.6565.2418.78
NewYork _0654.143.9386.8625.51
NewYork _0876.066.75109.1838.35
NewYork _188.087.031210.71050.17
ARPANET_02109.1910.581311.31021.30
ARPANET_041312.11216.961916.31462.15
ARPANET_062121.02125.982824.92293.86
ARPANET_082929.02932.314036.533133.70
ARPANET_13333.03362.854641.236172.60
EON_0243.133.1954.3410.40
EON_0498.3814.741210.61035.34
EON_061111.01117.521513.01246.61
EON_081413.71326.632018.21748.92
EON_11918.11838.172522.92273.32
France_0288.088.871411.51050.60
France _041312.81216.052219.31786.48
France _062222.02239.073430.427129.16
France _082726.32647.094540.338290.72
France _13434.03460.095448.945236.64
Table 5.  Results of GMR and PSO (con't)
GA_MEDP_RWAPSO
instancemaxavgminavg timemaxavgminavg time
Norway_0298.689.261513.11161.56
Norway _041514.61418.472522.520101.40
Norway _062221.42132.843732.629202.11
Norway _083029.52946.414843.640254.75
Norway _13736.63663.775954.149323.64
cost266_021918.21846.682623.821105.22
cost266_043534.133159.695450.947239.95
cost266_065453.153237.557973.270436.38
cost266_086867.267275.299993.488703.53
janos-us-ca_022626.02654.754036.131288.18
janos-us-ca_044039.63997.426660.556529.74
janos-us-ca_066867.867210.36111102.794959.93
janos-us-ca_088988.288326.39144134.11221239.46
giul_021110.31026.931614.513294.44
giul_042120.21982.663228.826451.61
giul_062524.624150.734238.334738.65
giul_083634.934226.855953.549990.63
piro40_0217171755.512220.019159.88
piro40_04282828118.503633.732300.88
piro40_06464646208.816658.752474.73
piro40_08606060338.618679.473684.40
USAnet_022524.02386.134036.432446.56
USAnet_044745.945355.647669.565983.93
USAnet_066766.165422.6310698.1921418.73
USAnet_088886.585490.20139126.51201715.20
Germany50_022120.720109.283733.531379.56
Germany50_044544.243350.247469.466683.95
Germany50_066160.560584.9210295.791969.87
Germany50_087674.773921.59126120.51151271.90
zib54_023231.130149.447564.158619.63
zib54_046765.765406.90159141.61331462.19
zib54_069290.088762.56208195.21872955.29
zib54_08122119.21171446.41289266.32504147.09
ta2_023534.634411.597570.3661453.68
ta2_047270.169945.79152141.81332852.21
ta2_069997.4961865.20208198.71813867.39
ta2_08131129.71282835.57284271.02585953.52
GA_MEDP_RWAPSO
instancemaxavgminavg timemaxavgminavg time
Norway_0298.689.261513.11161.56
Norway _041514.61418.472522.520101.40
Norway _062221.42132.843732.629202.11
Norway _083029.52946.414843.640254.75
Norway _13736.63663.775954.149323.64
cost266_021918.21846.682623.821105.22
cost266_043534.133159.695450.947239.95
cost266_065453.153237.557973.270436.38
cost266_086867.267275.299993.488703.53
janos-us-ca_022626.02654.754036.131288.18
janos-us-ca_044039.63997.426660.556529.74
janos-us-ca_066867.867210.36111102.794959.93
janos-us-ca_088988.288326.39144134.11221239.46
giul_021110.31026.931614.513294.44
giul_042120.21982.663228.826451.61
giul_062524.624150.734238.334738.65
giul_083634.934226.855953.549990.63
piro40_0217171755.512220.019159.88
piro40_04282828118.503633.732300.88
piro40_06464646208.816658.752474.73
piro40_08606060338.618679.473684.40
USAnet_022524.02386.134036.432446.56
USAnet_044745.945355.647669.565983.93
USAnet_066766.165422.6310698.1921418.73
USAnet_088886.585490.20139126.51201715.20
Germany50_022120.720109.283733.531379.56
Germany50_044544.243350.247469.466683.95
Germany50_066160.560584.9210295.791969.87
Germany50_087674.773921.59126120.51151271.90
zib54_023231.130149.447564.158619.63
zib54_046765.765406.90159141.61331462.19
zib54_069290.088762.56208195.21872955.29
zib54_08122119.21171446.41289266.32504147.09
ta2_023534.634411.597570.3661453.68
ta2_047270.169945.79152141.81332852.21
ta2_069997.4961865.20208198.71813867.39
ta2_08131129.71282835.57284271.02585953.52
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