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September  2018, 8(3): 351-376. doi: 10.3934/naco.2018023

Differential evolution with improved sub-route reversal repair mechanism for multiobjective urban transit routing problem

Ahmed Tarajo Buba 1, and  Lai Soon Lee 1,2,,

1. 

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
2. 

Laboratory of Computational Statistics and Operations Research, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

* Corresponding author: Lai Soon LEE

Received  May 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author is supported by FRGS2016 (01-01-16-1867FR).

The urban transit routing problem (UTRP) deals with public transport systems in determining a set of efficient transit routes on existing road networks to meet transit demands. The UTRP is a complex combinatorial optimization problem characterized with a large search space, multi-constraint, and multiobjective nature where the likelihood of generating infeasible route sets is high. In this paper, an improved sub-route reversal repair mechanism is proposed to deal with the infeasibility. A population-based metaheuristic, namely, Differential Evolution (DE) algorithm is then proposed to handle the multiobjective UTRP with the aim of devising an efficient transit route network that optimizes both passengers' and operators' costs. Computational experiments are performed on well-known benchmark instances to evaluate the effectiveness of the proposed repair mechanism and the DE algorithm. The computational results are reported to have better parameter values in most cases when compared to other approaches in the literature.

Keywords: Urban transit, routing, multiobjective, repair mechanism, differential evolution.
Mathematics Subject Classification: Primary: 90B06, 90B10; Secondary: 65K05, 68W99.
Citation: Ahmed Tarajo Buba, Lai Soon Lee. Differential evolution with improved sub-route reversal repair mechanism for multiobjective urban transit routing problem. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 351-376. doi: 10.3934/naco.2018023
References:
[1]

R. O. Arbex and C. B. da Cunha, Efficient transit network design and frequencies setting multi-objective optimization by alternating objective genetic algorithm, Transportation Research Part B: Methodological, 81 (2015), 355-376.  doi: 10.1016/j.trb.2015.06.014.  Google Scholar

[2]

M. H. Baaj and H. S. Mahmassani, An AI-based approach for transit route system planning and design, Journal of Advanced Transportation, 25 (1991), 187-209.  doi: 10.1002/atr.5670250205.  Google Scholar

[3]

M. H. Baaj and H. S. Mahmassani, Hybrid route generation heuristic algorithm for the design of transit networks, Transportation Research Part C: Emerging Technologies, 3 (1995), 31-50.  doi: 10.1016/0968-090X(94)00011-S.  Google Scholar

[4]

M. Bagherian, S. Massah and S. Kermanshahi, A swarm based method for solving transit network design problem, Proceedings of the 36th Australasian Transport Research Forum, (2013), 2-4.   Google Scholar

[5]

D. Beasley, D.R. Bull and R.R. Martin, An overview of genetic algorithms: part 2, research topics, University Computing, 15 (1993), 170-181.   Google Scholar

[6]

M. Bielli, M. Caramia and P. Carotenuto, Genetic algorithms in bus network optimization, Transportation Research Part C: Emerging Technologies, 10 (2002), 19-34.  doi: 10.1016/S0968-090X(00)00048-6.  Google Scholar

[7]

A. T. Buba and L. S. Lee, Differential evolution for urban transit routing problem, Journal of Computer and Communications, 4 (2016), 11-25.  doi: 10.4236/jcc.2016.414002.  Google Scholar

[8]

A. Ceder and N. H. M. Wilson, Bus network design, Transportation Research Part B: Methodological, 20 (1986), 331-344.  doi: 10.1016/0191-2615(86)90047-0.  Google Scholar

[9]

P. Chakroborty, Genetic algorithms for optimal urban transit network design, Computer-Aided Civil and Infrastructure Engineering, 18 (2003), 184-200.  doi: 10.1111/1467-8667.00309.  Google Scholar

[10]

P. Chakroborty and T. Wivedi, Optimal route network design for transit systems using genetic algorithms, Engineering Optimization, 34 (2002), 83-100.  doi: 10.1080/03052150210909.  Google Scholar

[11]

J. S. C. Chew, L. S. Lee and H. V. Seow, Genetic algorithm for biobjective urban transit routing problem, Journal of Applied Mathematics, 2013 (2013), Article ID 698645, 15 pages.  Google Scholar

[12]

S. Chien, Z. Yang and E. Hou, Genetic algorithm approach for transit route planning and design, Journal of Transportation Engineering, 127 (2001), 200-207.  doi: 10.1061/(ASCE)0733-947X(2001)127:3(200).  Google Scholar

[13]

L. Fan, Metaheuristic Methods for the Urban Transit Routing Problem, Ph. D thesis, Cardiff University (United Kingdom), 2009. Google Scholar

[14]

L. Fan and C. L. Mumford, A metaheuristic approach to the urban transit routing problem, Journal of Heuristics, 16 (2010), 353-372.  doi: 10.1007/s10732-008-9089-8.  Google Scholar

[15]

L. Fan, C. L. Mumford and D. Evans, A simple multi-objective optimization algorithm for the urban transit routing problem, IEEE Congress on Evolutionary Computation, (2009), 1-7.  doi: 10.1109/CEC.2009.4982923.  Google Scholar

[16]

W. Fan and R. B. Machemehl, Optimal transit route network design problem with variable transit demand: genetic algorithm approach, Journal of Transportation Engineering, 132 (2006), 40-51.  doi: 10.1061/(ASCE)0733-947X(2006)132:1(40).  Google Scholar

[17]

W. Fan and R. B. Machemehl, Tabu search strategies for the public transportation network optimizations with variable transit demand, Computer-Aided Civil and Infrastructure Engineering, 23 (2008), 502-520.  doi: 10.1111/j.1467-8667.2008.00556.x.  Google Scholar

[18]

J. Guan, H. Yang and S. Wirasinghe, Simultaneous optimization of transit line configuration and passenger line assignment, Transportation Research Part B: Methodological, 40 (2006), 885-902.  doi: 10.1016/j.trb.2005.12.003.  Google Scholar

[19]

A. Jaszkiewicz, On the computational efficiency of multiple objective metaheuristics. The knapsack problem case study, European Journal of Operational Research, 158 (2004), 418-433.  doi: 10.1016/j.ejor.2003.06.015.  Google Scholar

[20]

M. P. John, C. L. Mumford and R. Lewis, An improved multi-objective algorithm for the urban transit routing problem, in Evolutionary Computation in Combinatorial Optimisation, EVOCOP Lecture Series in Computer Science (eds. C. Blum and G. Ochoa), Springer, 2014. doi: 10.1007/978-3-662-44320-0_5.  Google Scholar

[21]

P. N. Kechagiopoulos and G. N. Beligiannis, Solving the urban transit routing problem using a particle swarm optimization based algorithm, Applied Soft Computing, 21 (2014), 654-676.  doi: 10.1016/j.asoc.2014.04.005.  Google Scholar

[22]

F. Kiliç and M. Gök, A demand based route generation algorithm for public transit network design, Computers & Operations Research, 51 (2014), 21-29.  doi: 10.1016/j.cor.2014.05.001.  Google Scholar

[23]

H. N. Koutsopoulos and N. H. M. Wilson, Determination of headways as a function of time varying characteristics on a transit network, in Computer Scheduling of Public Transport 2, (1985), 391–414. Google Scholar

[24]

T. I. Magnanti and R. T. Wong, Network design and transportation planning: models and algorithms, Transportation Science, 18 (1984), 1-55.  doi: 10.1287/trsc.18.1.1.  Google Scholar

[25]

C. E. Mandl, Evaluation and optimization of urban public transportation networks, European Journal of Operational Research, 5 (1980), 396-404.  doi: 10.1016/0377-2217(80)90126-5.  Google Scholar

[26]

E. Mazloumi, M. Mesbah, A. Ceder, S. Moridpour and G. Currie, Efficient transit schedule design of timing points: a comparison of ant colony and genetic algorithms, Transportation Research Part B: Methodological, 46 (2012), 217-234.  doi: 10.1016/j.trb.2011.09.010.  Google Scholar

[27]

C. L. Mumford, New heuristic and evolutionary operators for the multi-objective urban transit routing problem, IEEE Congress on Evolutionary Computation, (2013), 939-946.  doi: 10.1109/CEC.2013.6557668.  Google Scholar

[28]

A. T. Murray, A coverage model for improving public transit system accessibility and expanding access, Annals of Operations Research, 123 (2003), 143-156.  doi: 10.1023/A:1026123329433.  Google Scholar

[29]

M. A. Nayeem, M. K. Rahman and M.S. Rahman, Transit network design by genetic algorithm with elitism, Transportation Research Part C: Emerging Technologies, 46 (2014), 30-45.  doi: 10.1016/j.trc.2014.05.002.  Google Scholar

[30]

G. Newell, Some issues relating to the optimal design of bus routes, Transportation Science, 13 (1979), 20-35.  doi: 10.1287/trsc.13.1.20.  Google Scholar

[31]

S. Ngamchai and D. J. Lovell, Optimal time transfer in bus transit route network design using a genetic algorithm, Journal of Transportation Engineering, 129 (2003), 510-521.  doi: 10.1061/(ASCE)0733-947X(2003)129:5(510).  Google Scholar

[32]

M. Nikolić and D. Teodorović, Transit network design by bee colony optimization, Expert Systems with Applications, 40 (2013), 5945-5955.   Google Scholar

[33]

J. Pacheco, A. Alvarez, S. Casado and J. L. González-Velarde, A tabu search approach to an urban transport problem in northern Spain, Computers & Operations Research, 36 (2009), 967-979.  doi: 10.1016/j.cor.2007.12.002.  Google Scholar

[34]

S. Pattnaik, S. Mohan and V. Tom, Urban bus transit route network design using genetic algorithm, Journal of Transportation Engineering, 124 (1998), 368-375.  doi: 10.1061/(ASCE)0733-947X(1998)124:4(368).  Google Scholar

[35]

S. Schéele, A supply model for public transit services, Transportation Research Part B: Methodological, 14 (1980), 133-146.   Google Scholar

[36]

R. Storn and K. Price, Differrential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report, International Computer Science Institute Berkeley, CA, 1995. Google Scholar

[37]

Q. K. Wan and H. K. Lo, A mixed integer formulation for multiple-route transit network design, Journal of Mathematical Modelling and Algorithms, 2 (2003), 299-308.  doi: 10.1023/B:JMMA.0000020425.99217.cd.  Google Scholar

[38]

F. Zhao and A. Gan, Optimization of transit network to minimize transfer, Final Report BD015-02 Research Office Florida Department of Transportation, 2003. Google Scholar

show all references

References:
[1]

R. O. Arbex and C. B. da Cunha, Efficient transit network design and frequencies setting multi-objective optimization by alternating objective genetic algorithm, Transportation Research Part B: Methodological, 81 (2015), 355-376.  doi: 10.1016/j.trb.2015.06.014.  Google Scholar

[2]

M. H. Baaj and H. S. Mahmassani, An AI-based approach for transit route system planning and design, Journal of Advanced Transportation, 25 (1991), 187-209.  doi: 10.1002/atr.5670250205.  Google Scholar

[3]

M. H. Baaj and H. S. Mahmassani, Hybrid route generation heuristic algorithm for the design of transit networks, Transportation Research Part C: Emerging Technologies, 3 (1995), 31-50.  doi: 10.1016/0968-090X(94)00011-S.  Google Scholar

[4]

M. Bagherian, S. Massah and S. Kermanshahi, A swarm based method for solving transit network design problem, Proceedings of the 36th Australasian Transport Research Forum, (2013), 2-4.   Google Scholar

[5]

D. Beasley, D.R. Bull and R.R. Martin, An overview of genetic algorithms: part 2, research topics, University Computing, 15 (1993), 170-181.   Google Scholar

[6]

M. Bielli, M. Caramia and P. Carotenuto, Genetic algorithms in bus network optimization, Transportation Research Part C: Emerging Technologies, 10 (2002), 19-34.  doi: 10.1016/S0968-090X(00)00048-6.  Google Scholar

[7]

A. T. Buba and L. S. Lee, Differential evolution for urban transit routing problem, Journal of Computer and Communications, 4 (2016), 11-25.  doi: 10.4236/jcc.2016.414002.  Google Scholar

[8]

A. Ceder and N. H. M. Wilson, Bus network design, Transportation Research Part B: Methodological, 20 (1986), 331-344.  doi: 10.1016/0191-2615(86)90047-0.  Google Scholar

[9]

P. Chakroborty, Genetic algorithms for optimal urban transit network design, Computer-Aided Civil and Infrastructure Engineering, 18 (2003), 184-200.  doi: 10.1111/1467-8667.00309.  Google Scholar

[10]

P. Chakroborty and T. Wivedi, Optimal route network design for transit systems using genetic algorithms, Engineering Optimization, 34 (2002), 83-100.  doi: 10.1080/03052150210909.  Google Scholar

[11]

J. S. C. Chew, L. S. Lee and H. V. Seow, Genetic algorithm for biobjective urban transit routing problem, Journal of Applied Mathematics, 2013 (2013), Article ID 698645, 15 pages.  Google Scholar

[12]

S. Chien, Z. Yang and E. Hou, Genetic algorithm approach for transit route planning and design, Journal of Transportation Engineering, 127 (2001), 200-207.  doi: 10.1061/(ASCE)0733-947X(2001)127:3(200).  Google Scholar

[13]

L. Fan, Metaheuristic Methods for the Urban Transit Routing Problem, Ph. D thesis, Cardiff University (United Kingdom), 2009. Google Scholar

[14]

L. Fan and C. L. Mumford, A metaheuristic approach to the urban transit routing problem, Journal of Heuristics, 16 (2010), 353-372.  doi: 10.1007/s10732-008-9089-8.  Google Scholar

[15]

L. Fan, C. L. Mumford and D. Evans, A simple multi-objective optimization algorithm for the urban transit routing problem, IEEE Congress on Evolutionary Computation, (2009), 1-7.  doi: 10.1109/CEC.2009.4982923.  Google Scholar

[16]

W. Fan and R. B. Machemehl, Optimal transit route network design problem with variable transit demand: genetic algorithm approach, Journal of Transportation Engineering, 132 (2006), 40-51.  doi: 10.1061/(ASCE)0733-947X(2006)132:1(40).  Google Scholar

[17]

W. Fan and R. B. Machemehl, Tabu search strategies for the public transportation network optimizations with variable transit demand, Computer-Aided Civil and Infrastructure Engineering, 23 (2008), 502-520.  doi: 10.1111/j.1467-8667.2008.00556.x.  Google Scholar

[18]

J. Guan, H. Yang and S. Wirasinghe, Simultaneous optimization of transit line configuration and passenger line assignment, Transportation Research Part B: Methodological, 40 (2006), 885-902.  doi: 10.1016/j.trb.2005.12.003.  Google Scholar

[19]

A. Jaszkiewicz, On the computational efficiency of multiple objective metaheuristics. The knapsack problem case study, European Journal of Operational Research, 158 (2004), 418-433.  doi: 10.1016/j.ejor.2003.06.015.  Google Scholar

[20]

M. P. John, C. L. Mumford and R. Lewis, An improved multi-objective algorithm for the urban transit routing problem, in Evolutionary Computation in Combinatorial Optimisation, EVOCOP Lecture Series in Computer Science (eds. C. Blum and G. Ochoa), Springer, 2014. doi: 10.1007/978-3-662-44320-0_5.  Google Scholar

[21]

P. N. Kechagiopoulos and G. N. Beligiannis, Solving the urban transit routing problem using a particle swarm optimization based algorithm, Applied Soft Computing, 21 (2014), 654-676.  doi: 10.1016/j.asoc.2014.04.005.  Google Scholar

[22]

F. Kiliç and M. Gök, A demand based route generation algorithm for public transit network design, Computers & Operations Research, 51 (2014), 21-29.  doi: 10.1016/j.cor.2014.05.001.  Google Scholar

[23]

H. N. Koutsopoulos and N. H. M. Wilson, Determination of headways as a function of time varying characteristics on a transit network, in Computer Scheduling of Public Transport 2, (1985), 391–414. Google Scholar

[24]

T. I. Magnanti and R. T. Wong, Network design and transportation planning: models and algorithms, Transportation Science, 18 (1984), 1-55.  doi: 10.1287/trsc.18.1.1.  Google Scholar

[25]

C. E. Mandl, Evaluation and optimization of urban public transportation networks, European Journal of Operational Research, 5 (1980), 396-404.  doi: 10.1016/0377-2217(80)90126-5.  Google Scholar

[26]

E. Mazloumi, M. Mesbah, A. Ceder, S. Moridpour and G. Currie, Efficient transit schedule design of timing points: a comparison of ant colony and genetic algorithms, Transportation Research Part B: Methodological, 46 (2012), 217-234.  doi: 10.1016/j.trb.2011.09.010.  Google Scholar

[27]

C. L. Mumford, New heuristic and evolutionary operators for the multi-objective urban transit routing problem, IEEE Congress on Evolutionary Computation, (2013), 939-946.  doi: 10.1109/CEC.2013.6557668.  Google Scholar

[28]

A. T. Murray, A coverage model for improving public transit system accessibility and expanding access, Annals of Operations Research, 123 (2003), 143-156.  doi: 10.1023/A:1026123329433.  Google Scholar

[29]

M. A. Nayeem, M. K. Rahman and M.S. Rahman, Transit network design by genetic algorithm with elitism, Transportation Research Part C: Emerging Technologies, 46 (2014), 30-45.  doi: 10.1016/j.trc.2014.05.002.  Google Scholar

[30]

G. Newell, Some issues relating to the optimal design of bus routes, Transportation Science, 13 (1979), 20-35.  doi: 10.1287/trsc.13.1.20.  Google Scholar

[31]

S. Ngamchai and D. J. Lovell, Optimal time transfer in bus transit route network design using a genetic algorithm, Journal of Transportation Engineering, 129 (2003), 510-521.  doi: 10.1061/(ASCE)0733-947X(2003)129:5(510).  Google Scholar

[32]

M. Nikolić and D. Teodorović, Transit network design by bee colony optimization, Expert Systems with Applications, 40 (2013), 5945-5955.   Google Scholar

[33]

J. Pacheco, A. Alvarez, S. Casado and J. L. González-Velarde, A tabu search approach to an urban transport problem in northern Spain, Computers & Operations Research, 36 (2009), 967-979.  doi: 10.1016/j.cor.2007.12.002.  Google Scholar

[34]

S. Pattnaik, S. Mohan and V. Tom, Urban bus transit route network design using genetic algorithm, Journal of Transportation Engineering, 124 (1998), 368-375.  doi: 10.1061/(ASCE)0733-947X(1998)124:4(368).  Google Scholar

[35]

S. Schéele, A supply model for public transit services, Transportation Research Part B: Methodological, 14 (1980), 133-146.   Google Scholar

[36]

R. Storn and K. Price, Differrential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report, International Computer Science Institute Berkeley, CA, 1995. Google Scholar

[37]

Q. K. Wan and H. K. Lo, A mixed integer formulation for multiple-route transit network design, Journal of Mathematical Modelling and Algorithms, 2 (2003), 299-308.  doi: 10.1023/B:JMMA.0000020425.99217.cd.  Google Scholar

[38]

F. Zhao and A. Gan, Optimization of transit network to minimize transfer, Final Report BD015-02 Research Office Florida Department of Transportation, 2003. Google Scholar

Figure 1.  A sample vector with 4 routes
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Figure 2.  Mandl's Swiss Network
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Figure 3.  Mumford Network
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Table 4)">Figure 4.  Best Route Network for Operator and Passenger with 4 Routes (see, Table 4)
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Table 4)">Figure 5.  Best Route Network for Operator and Passenger with 6 Routes (see, Table 4)
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Table 4)">Figure 6.  Best Route Network for Operator and Passenger with 7 Routes (see, Table 4)
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Table 4)">Figure 7.  Best Route Network for Operator and Passenger with 8 Routes (see, Table 4)
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Figure 8.  Approximate Pareto Fronts achieved by the proposed DE for Mandl's Swiss Network
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Figure 9.  Pareto Fronts achieved by the proposed DE for Mumford Network
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Table 3.  Comparison results (passenger and operator) of Mandl's Swiss network
Number of Routes Para-meters Fan et al. (2009) Mumford (2013) Chew et al. (2013) Proposed DE
1 2 1 2 1 2 1 2
4 $d_{0}$ 90.88 61.08 90.43 61.08 91.84 61.08 91.46 61.08
$d_{1}$ 8.35 36.61 9.57 36.61 8.16 36.61 8.54 36.61
$d_{2}$ 0.77 2.31 0.00 2.31 0.00 2.31 0.00 2.31
$d_{un}$ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
$C_{p}$ 10.65 13.88 10.57 13.88 10.50 13.88 10.80 13.88
$C_{o}$ 126 63 149 63 150 63 133 63
6 $d_{0}$ 93.13 65.18 95.38 70.91 96.79 70.91 97.24 70.46
$d_{1}$ 6.29 30.38 4.36 25.50 3.21 25.50 2.76 24.34
$d_{2}$ 0.58 3.53 0.06 2.95 0.00 2.95 0.00 5.20
$d_{un}$ 0.00 0.90 0.00 0.64 0.00 0.64 0.00 0.00
$C_{p}$ 10.49 13.82 10.27 13.48 10.21 13.48 10.16 12.60
$C_{o}$ 148 63 221 63 224 63 223 63
7 $d_{0}$ 92.55 64.42 96.47 65.13 98.01 70.65 97.37 68.96
$d_{1}$ 6.68 26.20 3.34 22.93 1.99 21.13 2.63 26.29
$d_{2}$ 0.77 8.16 0.19 10.34 0.00 7.13 0.00 3.92
$d_{un}$ 0.00 1.22 0.00 1.61 0.00 1.09 0.00 0.83
$C_{p}$ 10.44 14.13 10.22 14.25 10.16 13.76 10.21 13.46
$C_{o}$ 166 63 264 63 239 63 235 63
8 $d_{0}$ 91.33 55.17 97.56 57.93 99.04 61.91 98.20 60.76
$d_{1}$ 8.67 21.97 2.31 31.92 0.96 29.67 1.80 25.63
$d_{2}$ 0.00 18.11 0.13 9.70 0.00 6.87 0.00 10.34
$d_{un}$ 0.00 4.75 0.00 0.45 0.00 1.54 0.00 3.26
$C_{p}$ 10.45 15.45 10.17 14.45 10.11 14.22 10.31 14.78
$C_{o}$ 245 63 291 63 256 63 244 63
Note:     1. best results for passenger
          2. best results for operator
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Number of Routes Para-meters Fan et al. (2009) Mumford (2013) Chew et al. (2013) Proposed DE
1 2 1 2 1 2 1 2
4 $d_{0}$ 90.88 61.08 90.43 61.08 91.84 61.08 91.46 61.08
$d_{1}$ 8.35 36.61 9.57 36.61 8.16 36.61 8.54 36.61
$d_{2}$ 0.77 2.31 0.00 2.31 0.00 2.31 0.00 2.31
$d_{un}$ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
$C_{p}$ 10.65 13.88 10.57 13.88 10.50 13.88 10.80 13.88
$C_{o}$ 126 63 149 63 150 63 133 63
6 $d_{0}$ 93.13 65.18 95.38 70.91 96.79 70.91 97.24 70.46
$d_{1}$ 6.29 30.38 4.36 25.50 3.21 25.50 2.76 24.34
$d_{2}$ 0.58 3.53 0.06 2.95 0.00 2.95 0.00 5.20
$d_{un}$ 0.00 0.90 0.00 0.64 0.00 0.64 0.00 0.00
$C_{p}$ 10.49 13.82 10.27 13.48 10.21 13.48 10.16 12.60
$C_{o}$ 148 63 221 63 224 63 223 63
7 $d_{0}$ 92.55 64.42 96.47 65.13 98.01 70.65 97.37 68.96
$d_{1}$ 6.68 26.20 3.34 22.93 1.99 21.13 2.63 26.29
$d_{2}$ 0.77 8.16 0.19 10.34 0.00 7.13 0.00 3.92
$d_{un}$ 0.00 1.22 0.00 1.61 0.00 1.09 0.00 0.83
$C_{p}$ 10.44 14.13 10.22 14.25 10.16 13.76 10.21 13.46
$C_{o}$ 166 63 264 63 239 63 235 63
8 $d_{0}$ 91.33 55.17 97.56 57.93 99.04 61.91 98.20 60.76
$d_{1}$ 8.67 21.97 2.31 31.92 0.96 29.67 1.80 25.63
$d_{2}$ 0.00 18.11 0.13 9.70 0.00 6.87 0.00 10.34
$d_{un}$ 0.00 4.75 0.00 0.45 0.00 1.54 0.00 3.26
$C_{p}$ 10.45 15.45 10.17 14.45 10.11 14.22 10.31 14.78
$C_{o}$ 245 63 291 63 256 63 244 63
Note:     1. best results for passenger
          2. best results for operator
Table 1.  The parameters of benchmark instances
Network Nodes Links Routes Min-Max Nodes Demand $LB_{Cp}$ $LB_{Co}$
Mandl 15 21 4, 6, 7, 8 2 - 8 15,570 10.0058 63
Mumford0 30 90 12 2 - 15 342,160 13.0121 94
Mumford1 70 210 15 10 - 30 1,926,170 19.2695 345
Mumford2 110 385 56 10 - 22 4,847,900 22.1689 864
Mumford3 127 425 60 12 - 25 6,394,950 24.7453 982
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Network Nodes Links Routes Min-Max Nodes Demand $LB_{Cp}$ $LB_{Co}$
Mandl 15 21 4, 6, 7, 8 2 - 8 15,570 10.0058 63
Mumford0 30 90 12 2 - 15 342,160 13.0121 94
Mumford1 70 210 15 10 - 30 1,926,170 19.2695 345
Mumford2 110 385 56 10 - 22 4,847,900 22.1689 864
Mumford3 127 425 60 12 - 25 6,394,950 24.7453 982
Table 2.  The comparison of feasible route sets repaired by the four repair mechanisms
Case Number of Routes Repair Mechanism Average Minimum Maximum CPU Time (sec)
Ⅰ 4 TR 47 34 68 0.013
MSC 53 39 69 0.052
SRR 56 47 68 0.040
iSRR 79 44 86 0.031
Ⅱ 6 TR 149 133 170 0.029
MSC 206 193 220 0.490
SRR 169 157 181 0.032
iSRR 232 198 254 0.023
Ⅲ 7 TR 201 184 219 0.029
MSC 293 267 317 0.066
SRR 223 210 234 0.025
iSRR 312 296 335 0.029
Ⅳ 8 TR 231 207 250 0.02
MSC 356 319 376 0.042
SRR 265 242 284 0.025
iSRR 358 324 382 0.004
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Case Number of Routes Repair Mechanism Average Minimum Maximum CPU Time (sec)
Ⅰ 4 TR 47 34 68 0.013
MSC 53 39 69 0.052
SRR 56 47 68 0.040
iSRR 79 44 86 0.031
Ⅱ 6 TR 149 133 170 0.029
MSC 206 193 220 0.490
SRR 169 157 181 0.032
iSRR 232 198 254 0.023
Ⅲ 7 TR 201 184 219 0.029
MSC 293 267 317 0.066
SRR 223 210 234 0.025
iSRR 312 296 335 0.029
Ⅳ 8 TR 231 207 250 0.02
MSC 356 319 376 0.042
SRR 265 242 284 0.025
iSRR 358 324 382 0.004
Table 4.  Best route sets constructed by the proposed DE algorithm for Mandl's Swiss network
Case Routes Best route sets for Passenger Best route sets for Operator
Ⅰ 4 0 - 1 - 2 - 5 - 7 - 14 - 8 14 - 8
4 - 3 - 5 - 7 - 14 - 6 - 9 - 10 4 - 3 – 1 – 0
0 – 1 - 4 - 3 - 5 - 7 - 9 - 6 10 - 9 - 6 - 14 - 7 - 5 - 2 - 1
9 - 7 - 5 - 3 - 11 - 10 - 12 - 13 11 - 10 - 12 - 13
Ⅱ 6 8 - 14 - 7 - 9 - 10 - 11 - 3 - 4 11 - 3
4 - 3 - 1 - 2 - 5 - 7 - 14 - 8 13 - 12 - 10 - 9
14 - 6 - 9 - 7 - 5 - 3 - 1 - 0 9 - 6 - 14 - 7 - 5 - 2 - 1 - 3
8 - 14 - 6 - 9 - 7 - 5 - 2 - 1 1 - 0
13 - 12 - 10 - 9 - 7 - 5 - 14 - 8 8 - 14
12 - 10 - 11 - 3 - 5 - 2 - 1 - 0 3 - 4
Ⅲ 7 1 - 4 - 3 - 5 – 7 - 9 - 13 - 12 0 - 1 - 2 - 5 - 7 - 14 - 6 - 9
0 - 1 - 2 - 5 - 3 - 11 - 10 - 9 3 - 5
1 - 2 - 5 - 7 - 9 - 10 - 12 - 13 4 – 3 - 1
9 - 6 - 14 - 7 - 5 - 2 - 1 - 0 10 - 11
0 - 1 - 4 - 3 - 5 - 7 - 14 - 8 14 – 8
8 – 14 - 6 - 9 - 13 - 12 - 10 - 11 10 - 9
8 - 14 - 7 – 5 - 2 - 1 - 3 - 11 10 - 12 - 13
Ⅳ 8 0 - 1 - 2 - 5 - 14 - 7 1 - 3
0 – 1 - 4 - 3 - 5 - 7 - 14 - 8 10 - 11
0 – 1 - 2 - 5 - 7 - 9 - 6 - 14 3 - 5 - 7 - 14 - 6 - 9 - 10 - 12
11 - 3 - 5 - 7 - 14 - 6 - 9 14 - 8
0 – 1 - 4 - 3 - 5 - 7 - 9 - 10 0 - 1
13 – 12 - 10 - 9 - 6 - 14 - 8 3 - 4
12 – 10 - 11 - 3 - 5 - 2 - 1 - 0 5 - 2
13 – 12 - 10 - 9 - 6 - 14 - 7 - 5 12 - 13
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Case Routes Best route sets for Passenger Best route sets for Operator
Ⅰ 4 0 - 1 - 2 - 5 - 7 - 14 - 8 14 - 8
4 - 3 - 5 - 7 - 14 - 6 - 9 - 10 4 - 3 – 1 – 0
0 – 1 - 4 - 3 - 5 - 7 - 9 - 6 10 - 9 - 6 - 14 - 7 - 5 - 2 - 1
9 - 7 - 5 - 3 - 11 - 10 - 12 - 13 11 - 10 - 12 - 13
Ⅱ 6 8 - 14 - 7 - 9 - 10 - 11 - 3 - 4 11 - 3
4 - 3 - 1 - 2 - 5 - 7 - 14 - 8 13 - 12 - 10 - 9
14 - 6 - 9 - 7 - 5 - 3 - 1 - 0 9 - 6 - 14 - 7 - 5 - 2 - 1 - 3
8 - 14 - 6 - 9 - 7 - 5 - 2 - 1 1 - 0
13 - 12 - 10 - 9 - 7 - 5 - 14 - 8 8 - 14
12 - 10 - 11 - 3 - 5 - 2 - 1 - 0 3 - 4
Ⅲ 7 1 - 4 - 3 - 5 – 7 - 9 - 13 - 12 0 - 1 - 2 - 5 - 7 - 14 - 6 - 9
0 - 1 - 2 - 5 - 3 - 11 - 10 - 9 3 - 5
1 - 2 - 5 - 7 - 9 - 10 - 12 - 13 4 – 3 - 1
9 - 6 - 14 - 7 - 5 - 2 - 1 - 0 10 - 11
0 - 1 - 4 - 3 - 5 - 7 - 14 - 8 14 – 8
8 – 14 - 6 - 9 - 13 - 12 - 10 - 11 10 - 9
8 - 14 - 7 – 5 - 2 - 1 - 3 - 11 10 - 12 - 13
Ⅳ 8 0 - 1 - 2 - 5 - 14 - 7 1 - 3
0 – 1 - 4 - 3 - 5 - 7 - 14 - 8 10 - 11
0 – 1 - 2 - 5 - 7 - 9 - 6 - 14 3 - 5 - 7 - 14 - 6 - 9 - 10 - 12
11 - 3 - 5 - 7 - 14 - 6 - 9 14 - 8
0 – 1 - 4 - 3 - 5 - 7 - 9 - 10 0 - 1
13 – 12 - 10 - 9 - 6 - 14 - 8 3 - 4
12 – 10 - 11 - 3 - 5 - 2 - 1 - 0 5 - 2
13 – 12 - 10 - 9 - 6 - 14 - 7 - 5 12 - 13
Table 5.  Comparison results (passenger and operator) of Large Mumford networks
Instances Parameters Mumford (2013) Proposed DE
1 2 1 2
Mumford0 $d_{0}$ 63.20 18.42 65.41 16.99
$d_{1}$ 35.82 23.40 34.24 30.72
$d_{2}$ 0.98 20.78 0.35 28.92
$d_{un}$ 0.00 37.40 0.00 23.38
$C_{p}$ 16.05 32.40 15.27 33.41
$C_{o}$ 759 111 673 107
Mumford1 $d_{0}$ 36.6 16.35 38.77 22.39
$d_{1}$ 52.42 29.06 54.23 40.57
$d_{2}$ 10.71 29.92 5.12 34.33
$d_{un}$ 0.26 24.66 1.88 2.71
$C_{p}$ 24.79 34.69 23.16 31.15
$C_{o}$ 2038 568 1955 567
Mumford2 $d_{0}$ 30.92 13.76 31.47 15.32
$d_{1}$ 51.29 27.69 58.23 29.31
$d_{2}$ 16.36 29.53 9.60 31.08
$d_{un}$ 1.44 29.02 0.70 24.29
$C_{p}$ 28.65 36.54 27.28 34.52
$C_{o}$ 5632 2244 5268 2305
Mumford3 $d_{0}$ 27.46 16.71 28.12 25.42
$d_{1}$ 50.97 33.69 54.35 41.26
$d_{2}$ 18.76 29.18 16.84 24.91
$d_{un}$ 2.81 20.42 0.69 8.41
$C_{p}$ 31.44 36.92 30.16 34.81
$C_{o}$ 6665 2830 6547 2732
Note:     1. best results for passenger
          2. best results for operator
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Instances Parameters Mumford (2013) Proposed DE
1 2 1 2
Mumford0 $d_{0}$ 63.20 18.42 65.41 16.99
$d_{1}$ 35.82 23.40 34.24 30.72
$d_{2}$ 0.98 20.78 0.35 28.92
$d_{un}$ 0.00 37.40 0.00 23.38
$C_{p}$ 16.05 32.40 15.27 33.41
$C_{o}$ 759 111 673 107
Mumford1 $d_{0}$ 36.6 16.35 38.77 22.39
$d_{1}$ 52.42 29.06 54.23 40.57
$d_{2}$ 10.71 29.92 5.12 34.33
$d_{un}$ 0.26 24.66 1.88 2.71
$C_{p}$ 24.79 34.69 23.16 31.15
$C_{o}$ 2038 568 1955 567
Mumford2 $d_{0}$ 30.92 13.76 31.47 15.32
$d_{1}$ 51.29 27.69 58.23 29.31
$d_{2}$ 16.36 29.53 9.60 31.08
$d_{un}$ 1.44 29.02 0.70 24.29
$C_{p}$ 28.65 36.54 27.28 34.52
$C_{o}$ 5632 2244 5268 2305
Mumford3 $d_{0}$ 27.46 16.71 28.12 25.42
$d_{1}$ 50.97 33.69 54.35 41.26
$d_{2}$ 18.76 29.18 16.84 24.91
$d_{un}$ 2.81 20.42 0.69 8.41
$C_{p}$ 31.44 36.92 30.16 34.81
$C_{o}$ 6665 2830 6547 2732
Note:     1. best results for passenger
          2. best results for operator
Table A1.  Best route sets for the operator constructed by the proposed DE algorithm for Mumford0 network
Routes Sequence of Routes
1 1-23-14-11-25- 0-17-22
2 9 - 1 - 3
3 0-12-8-26
4 22-18-13-6
5 19-18
6 25-28-16-27-2-29-15-10
7 20-14
8 1 - 3
9 3-24-4
10 21 - 10
11 20 - 7
12 5 - 21
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Routes Sequence of Routes
1 1-23-14-11-25- 0-17-22
2 9 - 1 - 3
3 0-12-8-26
4 22-18-13-6
5 19-18
6 25-28-16-27-2-29-15-10
7 20-14
8 1 - 3
9 3-24-4
10 21 - 10
11 20 - 7
12 5 - 21
Table A2.  Best route sets for the passenger constructed by the proposed DE algorithm for Mumford0 network
Routes Sequence of Routes
1 20-24-7-28-17-11-25-22-18-19-12-8-26-0-13
2 12-19-22-0-25-28-7-4-24-14-9-1-3-23-20
3 12-19-8-26-0-18-17-11-14-9-3-1-23-20-24
4 29-2-27-16-7-14-24-3-11-25-22-18-19-12-17
5 27-10-15-21-5-6-13-18-12-19-22-17-0-26-8
6 3-1-4-23-20-14-11-25-0-19-17-18-13-6-15
7 19-8-26-0 -8-17-11-14-24-3-1-9-23-20-7
8 26-0-22-19-12-18-17-28-16-10-29-2-15-21-5
9 19-8-26-0-22-18-12-17-28-25-11-14-24-20-23
10 0-13-6-2-15-16-7-14-24-3-23-2-4-1-9
11 24-4-20-23-9-3-1-14-7-25-22-0-13-18-17
12 13-18-12-17-19-0-25-11-14-23-9-1-24-4-7
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Routes Sequence of Routes
1 20-24-7-28-17-11-25-22-18-19-12-8-26-0-13
2 12-19-22-0-25-28-7-4-24-14-9-1-3-23-20
3 12-19-8-26-0-18-17-11-14-9-3-1-23-20-24
4 29-2-27-16-7-14-24-3-11-25-22-18-19-12-17
5 27-10-15-21-5-6-13-18-12-19-22-17-0-26-8
6 3-1-4-23-20-14-11-25-0-19-17-18-13-6-15
7 19-8-26-0 -8-17-11-14-24-3-1-9-23-20-7
8 26-0-22-19-12-18-17-28-16-10-29-2-15-21-5
9 19-8-26-0-22-18-12-17-28-25-11-14-24-20-23
10 0-13-6-2-15-16-7-14-24-3-23-2-4-1-9
11 24-4-20-23-9-3-1-14-7-25-22-0-13-18-17
12 13-18-12-17-19-0-25-11-14-23-9-1-24-4-7
Table A3.  Best route sets for the operator constructed by the proposed DE algorithm for Mumford1 network
Routes Sequence of Routes
1 22-33- 43-13-20-10-59-68-11-31
2 18-32-37-67-27-25-4-40-46-7
3 30-42-62-15-49-11-31-9-16-10
4 21-63-30-24-42-15-49-26-16-10
5 67-65-37-5-54-19-36-45-34-35
6 28-56-8-26-49-1-16-55-20-10
7 47-40-46-7-53-4-12-25-34-35
8 12-41-34-35-6-21-63-24-30-26
9 12-41-34-51-56-6-21-63-24-30
10 57-25-34-58-64-52-20-17-50-23
11 22-33-13-43-61-44- 60-14-2-50
12 50-0-2-23-3-66-38-36-54-18
13 15- 49-1-31- 9-16-10-52-64-58
14 6-55-52- 64-35-51-8-10-17-50
15 39-48-66-29-69-19-54-5-37-65
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Routes Sequence of Routes
1 22-33- 43-13-20-10-59-68-11-31
2 18-32-37-67-27-25-4-40-46-7
3 30-42-62-15-49-11-31-9-16-10
4 21-63-30-24-42-15-49-26-16-10
5 67-65-37-5-54-19-36-45-34-35
6 28-56-8-26-49-1-16-55-20-10
7 47-40-46-7-53-4-12-25-34-35
8 12-41-34-35-6-21-63-24-30-26
9 12-41-34-51-56-6-21-63-24-30
10 57-25-34-58-64-52-20-17-50-23
11 22-33-13-43-61-44- 60-14-2-50
12 50-0-2-23-3-66-38-36-54-18
13 15- 49-1-31- 9-16-10-52-64-58
14 6-55-52- 64-35-51-8-10-17-50
15 39-48-66-29-69-19-54-5-37-65
Table A4.  Best route sets for the passenger constructed by the proposed DE algorithm for Mumford1 network
Routes Sequence of Routes
1 48-69-38- 66-19- 65-54-5-37- 67-27-25-57-34-35-6-47-53-40-46-7-4-12- 41-58-33-50-2-0-23
2 59-1-31-11- 68- 9- 49- 62- 42-30-26-16-10-17-50-2-0-69-48-29-3-66-18-65-67-27-12- 40-53-7
3 27-25-34-51-35-58-64-52-20-55-8-26- 49-15-30-42-62-24-21- 6- 47-7- 46-40- 4-12-41-36-38-69
4 63-24-21-30-15-49-9-11-1-16-55-20-10-28-56-51-57-58-33-43-61-60-44-14-2-50-23-3-48-29
5 27-25-34-51-35-57-41-58-64-52-20-43-13-50-0-2-14-61-44-60-17-10-8-26-49-15-30-42-62-21
6 5-8-34-35-57-41-51-8-56-28-8-26-62-15-49-1- 59-17-50-23-3-48-29-19-54-65-67-27-12-40-53
7 45-36-38-69-29-19-18-54-5-65-37-32-39-48-3-23- 0-50-2-14-44-60-61-43-13-22-33-20-10-16
8 57-58-64-52-55-10-28-8-56-6-47-7-46-40-4-25-34-51-41-36-54-18-39-48-66-19-69-0-50-33
9 18-65-54-5-67-27-12-41-34-45-36-38-66-29-69-23-0-50-13-43-61-60-44-17-55-8-52-64-51-56
10 26-8-51-64-52-55-16-10-28-56-6-47-53-7-4-25-34-58-33-50-2-14-61-43-13-22-19-3-66-69
11 52-20-17-43-13-33-58-64-35-41-27-25-34-45-22-36-54-19-69-0-50-23-3-48-39-65-32-18-66
12 49-1-16-10-59-68-9-31-11-15-62-42-24-63-6-35-41-36-19-66-48-3-38-69-23-50-17-55-8-52
13 10-52-64-58-57-25-12-27-67-37-18-66-3-29-48-39-65-32-54-38-69-23-50-17-61-60-44-14-2-0
14 25-4-7-46-40-47-6-56-51-58-33-43-61-60-44-14-2-23-29-66-19-65-37-5-54-36-45-34-41-27
15 49-15-30-24-62-26-1-9-68-11-59-17-43-13-33-58-64-35-41-34-57-51-8-55-20-10-28-56-6-21
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Routes Sequence of Routes
1 48-69-38- 66-19- 65-54-5-37- 67-27-25-57-34-35-6-47-53-40-46-7-4-12- 41-58-33-50-2-0-23
2 59-1-31-11- 68- 9- 49- 62- 42-30-26-16-10-17-50-2-0-69-48-29-3-66-18-65-67-27-12- 40-53-7
3 27-25-34-51-35-58-64-52-20-55-8-26- 49-15-30-42-62-24-21- 6- 47-7- 46-40- 4-12-41-36-38-69
4 63-24-21-30-15-49-9-11-1-16-55-20-10-28-56-51-57-58-33-43-61-60-44-14-2-50-23-3-48-29
5 27-25-34-51-35-57-41-58-64-52-20-43-13-50-0-2-14-61-44-60-17-10-8-26-49-15-30-42-62-21
6 5-8-34-35-57-41-51-8-56-28-8-26-62-15-49-1- 59-17-50-23-3-48-29-19-54-65-67-27-12-40-53
7 45-36-38-69-29-19-18-54-5-65-37-32-39-48-3-23- 0-50-2-14-44-60-61-43-13-22-33-20-10-16
8 57-58-64-52-55-10-28-8-56-6-47-7-46-40-4-25-34-51-41-36-54-18-39-48-66-19-69-0-50-33
9 18-65-54-5-67-27-12-41-34-45-36-38-66-29-69-23-0-50-13-43-61-60-44-17-55-8-52-64-51-56
10 26-8-51-64-52-55-16-10-28-56-6-47-53-7-4-25-34-58-33-50-2-14-61-43-13-22-19-3-66-69
11 52-20-17-43-13-33-58-64-35-41-27-25-34-45-22-36-54-19-69-0-50-23-3-48-39-65-32-18-66
12 49-1-16-10-59-68-9-31-11-15-62-42-24-63-6-35-41-36-19-66-48-3-38-69-23-50-17-55-8-52
13 10-52-64-58-57-25-12-27-67-37-18-66-3-29-48-39-65-32-54-38-69-23-50-17-61-60-44-14-2-0
14 25-4-7-46-40-47-6-56-51-58-33-43-61-60-44-14-2-23-29-66-19-65-37-5-54-36-45-34-41-27
15 49-15-30-24-62-26-1-9-68-11-59-17-43-13-33-58-64-35-41-34-57-51-8-55-20-10-28-56-6-21
Table A5.  Best route sets for the operator constructed by the proposed DE algorithm for Mumford2 network
Routes Sequence of Routes
1 85-54-34-35-91-52-100-18-9-19
2 17-50-32-95-20-68-21-13-32-59
3 47-41-16-103-7-52-89-11-79-72
4 31-24-85-94-101-55-87-56-39-65
5 102-71-51-46-86-28-21-32-17-20
6 77-4-107-5-11-89-96-42-106-57
7 36-72-108-4-107-77-5-42-89-11-106-73
8 77-4-108-79-72-44-82-99-76-31
9 27-17-39-55-80-63-12-74-47-103
10 20-32-50-68-59-27-94-83-35-49
11 24-6-94-105-27-32-13-21-86-28
12 34-54-22-3-93-100-52-10-40-7
13 15-35-34-66-18-92-72-108-36-64
14 64-36-4-107-77-5-11-109-89-42
15 109-64-96-64-92-9-3-81-94-105
16 77-5-107-36-48-79-73-106-57-37
17 101-94-83-15-2-63-45-53-90-49
18 30-44-43-72-79-73-106-57-40-10
19 3-22-54-85-66-15-83-35-8-90
20 15-35-49-52-89-106-42-57-10-37
21 56-0-78-17-65-61-1-97-58-47
22 0-97-80-55-39-65-61-1-0-87
23 39-56-61-65-78-95-50-20-68-75
24 17-65-39-87-45-53-12-38-80-63
25 58-80-87-56-1-97-0-65-78-17
26 25-84-37-40-57-10-70-104-7-49
27 105-81-6-66-34-35-8-62-12-8
28 47-41-16-60-12-8-63-55-39-78
29 84-70-41-104-40-37-7-57-42-96
30 34-54-3-9-93-98-67-18-66-15
31 91-62-90-53-8-12-41-60-47-16
32 74-58-80-63-2-45-38-53-8-49
33 20-32-50-95-78-65-61-1-87-45
34 77-107-108-79-64-92-9-19-85-3
35 102-14-69-99-82-23-31-71-88-51
36 109-64-89-106-11-42-5-36-107-77
37 17-32-21-86-51-71-31-76-33-99
38 58-80-63-53-90-91-52-10-57-7
39 60-41-70-104-37-7-49-90-8-12
40 67-98-18-93-9-92-44-43-29-26
41 85-22-3-81-6-24-31-23-82-19
42 51-71-102-76-33-31-24-6-54-22
43 24-31-76-14-69-99-82-26-43-30
44 90-8-63-53-62-12-16-41-74-38
45 34-54-3-81-6-24-31-71-102-88
46 101-27-32-20-75-50-13-28-86-24
47 108-36-48-44-72-26-29-43-30-69
48 17-50-75-68-13-32-59-27-95-20
49 62-90-91-35-34-3-85-24-22-66
50 24-6-54-85-66-15-83-2-35-90
51 48-36-108-4-77-5-42-73-109-89
52 65-78-17-20-50-68-59-28-46-21
53 81-6-94-2-63-8-90-35-49-91
54 36-64-73-89-52-49-35-34-18-93
55 17-39-55-45-2-94-83-35-90-91
56 36-72-108-4-107-77-5-11-73-96
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Routes Sequence of Routes
1 85-54-34-35-91-52-100-18-9-19
2 17-50-32-95-20-68-21-13-32-59
3 47-41-16-103-7-52-89-11-79-72
4 31-24-85-94-101-55-87-56-39-65
5 102-71-51-46-86-28-21-32-17-20
6 77-4-107-5-11-89-96-42-106-57
7 36-72-108-4-107-77-5-42-89-11-106-73
8 77-4-108-79-72-44-82-99-76-31
9 27-17-39-55-80-63-12-74-47-103
10 20-32-50-68-59-27-94-83-35-49
11 24-6-94-105-27-32-13-21-86-28
12 34-54-22-3-93-100-52-10-40-7
13 15-35-34-66-18-92-72-108-36-64
14 64-36-4-107-77-5-11-109-89-42
15 109-64-96-64-92-9-3-81-94-105
16 77-5-107-36-48-79-73-106-57-37
17 101-94-83-15-2-63-45-53-90-49
18 30-44-43-72-79-73-106-57-40-10
19 3-22-54-85-66-15-83-35-8-90
20 15-35-49-52-89-106-42-57-10-37
21 56-0-78-17-65-61-1-97-58-47
22 0-97-80-55-39-65-61-1-0-87
23 39-56-61-65-78-95-50-20-68-75
24 17-65-39-87-45-53-12-38-80-63
25 58-80-87-56-1-97-0-65-78-17
26 25-84-37-40-57-10-70-104-7-49
27 105-81-6-66-34-35-8-62-12-8
28 47-41-16-60-12-8-63-55-39-78
29 84-70-41-104-40-37-7-57-42-96
30 34-54-3-9-93-98-67-18-66-15
31 91-62-90-53-8-12-41-60-47-16
32 74-58-80-63-2-45-38-53-8-49
33 20-32-50-95-78-65-61-1-87-45
34 77-107-108-79-64-92-9-19-85-3
35 102-14-69-99-82-23-31-71-88-51
36 109-64-89-106-11-42-5-36-107-77
37 17-32-21-86-51-71-31-76-33-99
38 58-80-63-53-90-91-52-10-57-7
39 60-41-70-104-37-7-49-90-8-12
40 67-98-18-93-9-92-44-43-29-26
41 85-22-3-81-6-24-31-23-82-19
42 51-71-102-76-33-31-24-6-54-22
43 24-31-76-14-69-99-82-26-43-30
44 90-8-63-53-62-12-16-41-74-38
45 34-54-3-81-6-24-31-71-102-88
46 101-27-32-20-75-50-13-28-86-24
47 108-36-48-44-72-26-29-43-30-69
48 17-50-75-68-13-32-59-27-95-20
49 62-90-91-35-34-3-85-24-22-66
50 24-6-54-85-66-15-83-2-35-90
51 48-36-108-4-77-5-42-73-109-89
52 65-78-17-20-50-68-59-28-46-21
53 81-6-94-2-63-8-90-35-49-91
54 36-64-73-89-52-49-35-34-18-93
55 17-39-55-45-2-94-83-35-90-91
56 36-72-108-4-107-77-5-11-73-96
Table A6.  Best route sets for the passenger constructed by the proposed DE algorithm for Mumford2 network
Routes Sequence of Routes
1 96-11-73-89-42-5-36-107-48-79-64-92-9-66-18-3-98-67-100-93-34
2 90-53-38-12-62-91-52-100-18-3-98-93-67-92-72-44-82-19-33-14-31
3 77-5-36-108-4-107-48-92-64-109-11-73-96-42-89-52-100-18-9-19-102-88
4 26-72-36-79-108-4-107-5-11-106-57-37-104-7-49-35-15-54-6-86-46-21
5 26-30-44-72-36-48-108-4-77-5-11-73-64-89-52-91-62-12-60-58-80
6 21-59-27-17-50-32-95-68-13-28-86-81-3-22-85-19-82-23-9-92-44-72
7 88-71-51-31-24-6-22-3-34-18-67-100-93-35-8-90-62-103-70-10
8 77-5-11-73-109-89-96-64-36-79-108-4-107-48-92-44-82-19-24-86-28-46
9 67-100-93-98-18-3-54-24-19-85-22-66-9-92-72-79-73-96-42-11-89-52
10 84-25-70-103-12-41-74-58-38-80-55-87-39-0-61-65-56-17-50-32-95-78
11 21-32-13-68-50-17-39-55-63-12-103-70-7-52-100-18-9-19-82-44-48-107
12 29-26-44-92-72-36-79-73-89-52-49-90-91-8-53-38-87-55-45-2-35-15
13 69-99-33-19-82-44-72-92-18-66-34-35-91-100-52-10-40-57-106-73-79
14 50-13-21-86-81-105-83-3-9-92-64-36-4-107-77-5-11-109-89-42-96-73
15 35-15-83-34-54-6-81-86-28-13-68-17-27-94-101-55-87-56-39-65-61
16 74-60-47-58-97-80-38-87-56-17-78-39-65-50-20-75-68-95-59-27-105-94
17 79-64-73-109-11-106-89-96-42-57-10-70-84-104-40-25-7-49-35-83-94-27
18 24-85-22-3-67-18-93-9-19-23-82-33-31-51-46-21-32-17-39-55-63-12
19 74-60-47-58-97-80-38-12-63-53-62-90-49-52-89-64-36-4-107-77
20 84-7-10-70-104-25-40-37-57-42-106-73-89-109-11-79-36-108-72-92-18-66
21 90-53-8-91-35-2-15-34-93-67-3-66-18-9-92-44-82-33-31-51-46-21
22 83-105-94-6-81-85-24-86-28-21-27-101-55-87-56-65-50-75-17-78-0-1
23 28-13-50-65-56-0-87-97-80-45-38-58-47-103-12-63-55-39-17-50-68
24 96-64-109-11-5-42-57-106-73-89-52-91-90-35-93-98-18-9-19-33-76
25 89-42-106-57-37-104-7-49-35-15-54-6-86-24-19-82-44-72-79
26 22-85-54-6-66-18-9-19-3-81-86-51-71-88-102-76-69-30-26-82-33-31
27 96-89-109-73-42-106-11-64-79-48-92-67-3-93-100-18-9-19-33-14-102
28 23-82-26-72-92-48-108-5-77-4-36-64-89-52-49-90-8-53-38-63-80-58
29 101-55-39-87-80-0-56-65-78-17-95-59-21-86-51-102-76-69-30-43-29-26
30 22-85-54-6-66-18-100-93-98-67-3-34-35-90-8-62-12-38-80-55-101-27
31 62-53-38-58-80-55-39-78-95-17-27-94-101-55-87-56-39-65-61
32 36-48-92-44-82-23-9-3-83-2-45-87-38-53-90-49-52-89-64-109-73-89
33 106-11-73-96-42-5-77-4-107-108-48-92-64-79-72-43-44-82-19-24-86-21
34 62-91-52-100-18-98-93-35-34-66-6-24-31-23-9-19-102-88-71-51-46
35 47-62-103-12-38-58-97-1-61-56-65-78-95-27-94-85-19-82-26-30-44
36 50-95-78-17-68-59-27-94-83-35-49-52-89-106-73-79-72-44-82-99-76
37 9-3-66-85-22-3-93-100-52-10-70-16-74-58-97-0-61-1-87-38-53
38 36-48-92-67-3-15-83-34-66-18-93-9-19-33-76-31-24-6-94-101-55-39
39 66-85-6-81-86-46-28-21-13-32-20-95-17-27-94-105-83-35-8-62-91
40 107-77-4-36-5-42-11-96-73-106-57-40-7-103-12-63-55-39-17-50-68
41 100-18-67-93-9-92-72-36-4-77-5-107-108-79-48-44-30-43-26-82-19-24
42 62-91-52-100-18-98-93-35-34-66-6-24-54-3-19-33-99-69-30-29-43
43 83-15-35-90-49-91-62-12-103-70-10-52-100-93-3-22-85-66-34-35
44 2-83-3-34-54-66-6-94-105-81-86-46-21-68-20-32-13-28-59-95-27-101
45 86-6-54-22-3-9-19-23-33-99-69-14-76-31-24-85-94-101-55-39-87-1-97
46 69-43-29-26-44-92-18-9-19-102-88-71-51-86-21-59-95-78-0-1
47 52-100-91-8-35-15-2-34-66-3-67-98-93-9-18-92-64-109-73-79-72
48 24-6-54-22-66-85-3-98-67-92-44-43-69-30-26-82-33-31-51-46-21-32
49 57-42-96-11-79-36-5-107-77-4-108-48-44-26-82-19-24-86-21-13-32-59
50 75-68-21-32-50-65-56-61-0-80-63-45-2-15-18-92-48-107-77-5-11-89
51 17-68-59-32-95-27-94-83-35-15-3-9-18-100-52-10-70-41-12-63-55-39
52 38-80-45-87-56-61-0-1-97-58-60-12-62-91-52-89-64-36-107-48-72-26
53 71-88-51-102-33-99-14-69-30-29-43-72-79-73-106-57-37-104-41-12-63-55
54 15-34-54-85-3-93-67-98-18-66-22-31-23-9-19-102-88-51-86-21-27-101
55 33-19-3-34-66-93-100-52-89-109-73-11-42-96-64-92-9-3-81-94-105-27
56 41-16-47-74-58-80-55-87-0-1-61-65-39-55-101-94-81-85-24-19-33-76
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Routes Sequence of Routes
1 96-11-73-89-42-5-36-107-48-79-64-92-9-66-18-3-98-67-100-93-34
2 90-53-38-12-62-91-52-100-18-3-98-93-67-92-72-44-82-19-33-14-31
3 77-5-36-108-4-107-48-92-64-109-11-73-96-42-89-52-100-18-9-19-102-88
4 26-72-36-79-108-4-107-5-11-106-57-37-104-7-49-35-15-54-6-86-46-21
5 26-30-44-72-36-48-108-4-77-5-11-73-64-89-52-91-62-12-60-58-80
6 21-59-27-17-50-32-95-68-13-28-86-81-3-22-85-19-82-23-9-92-44-72
7 88-71-51-31-24-6-22-3-34-18-67-100-93-35-8-90-62-103-70-10
8 77-5-11-73-109-89-96-64-36-79-108-4-107-48-92-44-82-19-24-86-28-46
9 67-100-93-98-18-3-54-24-19-85-22-66-9-92-72-79-73-96-42-11-89-52
10 84-25-70-103-12-41-74-58-38-80-55-87-39-0-61-65-56-17-50-32-95-78
11 21-32-13-68-50-17-39-55-63-12-103-70-7-52-100-18-9-19-82-44-48-107
12 29-26-44-92-72-36-79-73-89-52-49-90-91-8-53-38-87-55-45-2-35-15
13 69-99-33-19-82-44-72-92-18-66-34-35-91-100-52-10-40-57-106-73-79
14 50-13-21-86-81-105-83-3-9-92-64-36-4-107-77-5-11-109-89-42-96-73
15 35-15-83-34-54-6-81-86-28-13-68-17-27-94-101-55-87-56-39-65-61
16 74-60-47-58-97-80-38-87-56-17-78-39-65-50-20-75-68-95-59-27-105-94
17 79-64-73-109-11-106-89-96-42-57-10-70-84-104-40-25-7-49-35-83-94-27
18 24-85-22-3-67-18-93-9-19-23-82-33-31-51-46-21-32-17-39-55-63-12
19 74-60-47-58-97-80-38-12-63-53-62-90-49-52-89-64-36-4-107-77
20 84-7-10-70-104-25-40-37-57-42-106-73-89-109-11-79-36-108-72-92-18-66
21 90-53-8-91-35-2-15-34-93-67-3-66-18-9-92-44-82-33-31-51-46-21
22 83-105-94-6-81-85-24-86-28-21-27-101-55-87-56-65-50-75-17-78-0-1
23 28-13-50-65-56-0-87-97-80-45-38-58-47-103-12-63-55-39-17-50-68
24 96-64-109-11-5-42-57-106-73-89-52-91-90-35-93-98-18-9-19-33-76
25 89-42-106-57-37-104-7-49-35-15-54-6-86-24-19-82-44-72-79
26 22-85-54-6-66-18-9-19-3-81-86-51-71-88-102-76-69-30-26-82-33-31
27 96-89-109-73-42-106-11-64-79-48-92-67-3-93-100-18-9-19-33-14-102
28 23-82-26-72-92-48-108-5-77-4-36-64-89-52-49-90-8-53-38-63-80-58
29 101-55-39-87-80-0-56-65-78-17-95-59-21-86-51-102-76-69-30-43-29-26
30 22-85-54-6-66-18-100-93-98-67-3-34-35-90-8-62-12-38-80-55-101-27
31 62-53-38-58-80-55-39-78-95-17-27-94-101-55-87-56-39-65-61
32 36-48-92-44-82-23-9-3-83-2-45-87-38-53-90-49-52-89-64-109-73-89
33 106-11-73-96-42-5-77-4-107-108-48-92-64-79-72-43-44-82-19-24-86-21
34 62-91-52-100-18-98-93-35-34-66-6-24-31-23-9-19-102-88-71-51-46
35 47-62-103-12-38-58-97-1-61-56-65-78-95-27-94-85-19-82-26-30-44
36 50-95-78-17-68-59-27-94-83-35-49-52-89-106-73-79-72-44-82-99-76
37 9-3-66-85-22-3-93-100-52-10-70-16-74-58-97-0-61-1-87-38-53
38 36-48-92-67-3-15-83-34-66-18-93-9-19-33-76-31-24-6-94-101-55-39
39 66-85-6-81-86-46-28-21-13-32-20-95-17-27-94-105-83-35-8-62-91
40 107-77-4-36-5-42-11-96-73-106-57-40-7-103-12-63-55-39-17-50-68
41 100-18-67-93-9-92-72-36-4-77-5-107-108-79-48-44-30-43-26-82-19-24
42 62-91-52-100-18-98-93-35-34-66-6-24-54-3-19-33-99-69-30-29-43
43 83-15-35-90-49-91-62-12-103-70-10-52-100-93-3-22-85-66-34-35
44 2-83-3-34-54-66-6-94-105-81-86-46-21-68-20-32-13-28-59-95-27-101
45 86-6-54-22-3-9-19-23-33-99-69-14-76-31-24-85-94-101-55-39-87-1-97
46 69-43-29-26-44-92-18-9-19-102-88-71-51-86-21-59-95-78-0-1
47 52-100-91-8-35-15-2-34-66-3-67-98-93-9-18-92-64-109-73-79-72
48 24-6-54-22-66-85-3-98-67-92-44-43-69-30-26-82-33-31-51-46-21-32
49 57-42-96-11-79-36-5-107-77-4-108-48-44-26-82-19-24-86-21-13-32-59
50 75-68-21-32-50-65-56-61-0-80-63-45-2-15-18-92-48-107-77-5-11-89
51 17-68-59-32-95-27-94-83-35-15-3-9-18-100-52-10-70-41-12-63-55-39
52 38-80-45-87-56-61-0-1-97-58-60-12-62-91-52-89-64-36-107-48-72-26
53 71-88-51-102-33-99-14-69-30-29-43-72-79-73-106-57-37-104-41-12-63-55
54 15-34-54-85-3-93-67-98-18-66-22-31-23-9-19-102-88-51-86-21-27-101
55 33-19-3-34-66-93-100-52-89-109-73-11-42-96-64-92-9-3-81-94-105-27
56 41-16-47-74-58-80-55-87-0-1-61-65-39-55-101-94-81-85-24-19-33-76
Table A7.  Best route sets for the operator constructed by the proposed DE algorithm for Mumford3 network
Routes Sequence of Routes
1 122-94-27-118-46-11-0-101-26-126-92-85-20-50-76-19-13-83
2 29-36-24-87-61-102-108-115-3-96-78-114-23-2-37
3 99-48-79-0-7-40-44-82-88-39
4 35-91-82-95-13-83-50-92-76-20
5 105-42-89-31-39-88-82-40-50-76
6 66-53-23-2-48-126-92-83-13-19
7 38-25-88-121-125-91-82-40-50-20
8 81-9-66-80-29-122-52-51-54-30
9 81-27-94-29-5-36-58-104-43-28
10 107-12-3-96-71-100-14-106-108-102
11 82-116-101-0-48-2-49-72-9-5
12 105-42-43-104-59-113-90-103-98-17
13 27-94-122-29-36-80-5-24-87-110-68-34-56
14 105-42-89-31-39-88-82-40-50-13
15 116-120-7-11-0-77-67-1-27-29
16 105-58-36-5-24-117-86-109-69-10-4-18
17 81-94-29-9-66-53-23-2-79-0
18 44-65-50-83-13-95-123-35-91-25-38-54-51-47
19 85-20-92-126-48-79-0-101-88-121
20 60-55-99-73-6-112-62-16-114-78-71-100-14-108-61-106
21 81-29-80-63-105-42-28-103-43-104-18-10
22 15-3-96-100-12-78-114-74-23-72-124
23 31-38-52-27-1-77-67-46-39-25
24 74-49-124-70-66-9-29-5-24-117
25 48-26-93-44-40-65-116-82-88-39
26 122-94-29-27-67-11-79-48-97-2
27 102-108-115-106-61-53-23-2-48-73
28 92-76-20-85-75-60-73-6-32-45
29 83-13-95-123-35-91-25-38-54-51
30 98-42-104-105-63-36-5-24-87-61-115-3-12-56
31 38-31-54-51-52-118-27-122-29-80-66-124-9
32 23-2-11-1-118-52-51-47-31-54
33 57-21-8-119-22-30-41-33-31-38-52-122
34 99-48-79-2-74-66-80-5-24-86
35 84-68-3-71-96-78-100-14-61-102
36 102-87-84-68-34-56-15-12-3-115-61-14-72-66
37 81-27-29-94-122-52-51-54-30-38
38 72-124-70-66-9-80-63-105-104-18-69-103-109-10-86
39 43-105-104-59-89-31-38-25-91-35
40 110-84-68-56-15-12-107-3-100-14
41 94-29-36-5-63-105-42-98-90-17
42 81-1-67-11-0-120-44-40-65-95
43 9-29-80-66-74-114-78-3-56-34
44 2-79-11-46-39-25-41-8-35-21
45 103-28-64-111-69-10-109-4-43-90-113-42
46 97-79-48-2-23-53-61-87-84-110
47 66-70-72-49-2-79-0-101-88-91
48 81-27-94-122-52-51-54-30-31-39
49 43-90-59-42-104-18-109-10-86-117-24-5-58-105
50 9-29-80-66-74-114-2-11-46-39
51 84-87-110-68-12-107-15-3-71-96-78-114-74-37-79-11-7-101-26
52 0-48-79-2-49-66-80-29-122-52
53 111-64-4-18-103-109-43-17-89-52
54 122-94-29-36-5-58-104-59-113-90
55 27-81-67-46-39-31-54-51-47-31
56 0-48-79-2-49-72-9-63-105-43
57 31-39-88-82-40-92-76-20-19-85
58 98-113-59-90-103-69-10-18-117-86
59 55-60-112-6-45-73-126-48-0-77-1-118
60 109-86-117-18-10-4-43-42-89-31
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Routes Sequence of Routes
1 122-94-27-118-46-11-0-101-26-126-92-85-20-50-76-19-13-83
2 29-36-24-87-61-102-108-115-3-96-78-114-23-2-37
3 99-48-79-0-7-40-44-82-88-39
4 35-91-82-95-13-83-50-92-76-20
5 105-42-89-31-39-88-82-40-50-76
6 66-53-23-2-48-126-92-83-13-19
7 38-25-88-121-125-91-82-40-50-20
8 81-9-66-80-29-122-52-51-54-30
9 81-27-94-29-5-36-58-104-43-28
10 107-12-3-96-71-100-14-106-108-102
11 82-116-101-0-48-2-49-72-9-5
12 105-42-43-104-59-113-90-103-98-17
13 27-94-122-29-36-80-5-24-87-110-68-34-56
14 105-42-89-31-39-88-82-40-50-13
15 116-120-7-11-0-77-67-1-27-29
16 105-58-36-5-24-117-86-109-69-10-4-18
17 81-94-29-9-66-53-23-2-79-0
18 44-65-50-83-13-95-123-35-91-25-38-54-51-47
19 85-20-92-126-48-79-0-101-88-121
20 60-55-99-73-6-112-62-16-114-78-71-100-14-108-61-106
21 81-29-80-63-105-42-28-103-43-104-18-10
22 15-3-96-100-12-78-114-74-23-72-124
23 31-38-52-27-1-77-67-46-39-25
24 74-49-124-70-66-9-29-5-24-117
25 48-26-93-44-40-65-116-82-88-39
26 122-94-29-27-67-11-79-48-97-2
27 102-108-115-106-61-53-23-2-48-73
28 92-76-20-85-75-60-73-6-32-45
29 83-13-95-123-35-91-25-38-54-51
30 98-42-104-105-63-36-5-24-87-61-115-3-12-56
31 38-31-54-51-52-118-27-122-29-80-66-124-9
32 23-2-11-1-118-52-51-47-31-54
33 57-21-8-119-22-30-41-33-31-38-52-122
34 99-48-79-2-74-66-80-5-24-86
35 84-68-3-71-96-78-100-14-61-102
36 102-87-84-68-34-56-15-12-3-115-61-14-72-66
37 81-27-29-94-122-52-51-54-30-38
38 72-124-70-66-9-80-63-105-104-18-69-103-109-10-86
39 43-105-104-59-89-31-38-25-91-35
40 110-84-68-56-15-12-107-3-100-14
41 94-29-36-5-63-105-42-98-90-17
42 81-1-67-11-0-120-44-40-65-95
43 9-29-80-66-74-114-78-3-56-34
44 2-79-11-46-39-25-41-8-35-21
45 103-28-64-111-69-10-109-4-43-90-113-42
46 97-79-48-2-23-53-61-87-84-110
47 66-70-72-49-2-79-0-101-88-91
48 81-27-94-122-52-51-54-30-31-39
49 43-90-59-42-104-18-109-10-86-117-24-5-58-105
50 9-29-80-66-74-114-2-11-46-39
51 84-87-110-68-12-107-15-3-71-96-78-114-74-37-79-11-7-101-26
52 0-48-79-2-49-66-80-29-122-52
53 111-64-4-18-103-109-43-17-89-52
54 122-94-29-36-5-58-104-59-113-90
55 27-81-67-46-39-31-54-51-47-31
56 0-48-79-2-49-72-9-63-105-43
57 31-39-88-82-40-92-76-20-19-85
58 98-113-59-90-103-69-10-18-117-86
59 55-60-112-6-45-73-126-48-0-77-1-118
60 109-86-117-18-10-4-43-42-89-31
Table A8.  Best route sets for the passenger constructed by the proposed DE algorithm for Mumford3 network
Routes Sequence of Routes
1 23-114-16-62-112-6-73-48-26-7-40-82-44-93-88-39-46-1-118-27-29-5-24-87-110
2 27-118-67-81-80-70-72-74-2-23-37-97-11-7-40-50-13-19-85-75-20-32-60-73-48
3 12-3-78-114-74-23-37-2-11-46-39-30-22-21-35-91-25-38-31-89-59-104-58-63-5
4 46-1-11-2-74-49-66-14-100-12-56-15-3-78-114-16-55-60-32-20-85-19-13-83-92
5 57-8-21-125-22-41-30-39-46-11-2-74-23-53-61-108-102-24-5-63-36-9-80-81-67
6 17-98-113-89-31-39-88-93-44-40-95-65-50-20-19-32-45-60-6-73-126-92-76-85-75
7 41-33-54-31-89-59-104-43-105-63-80-66-74-114-16-112-60-73-48-2-23-53-61-102-87
8 13-92-85-75-60-55-45-73-48-2-49-72-9-63-105-43-69-10-86-24-102-61-14-100-78
9 102-106-115-61-14-100-71-3-96-78-114-23-37-74-2-79-0-120-65-101-88-82-40-92-85
10 29-81-94-122-52-27-67-46-118-1-11-0-48-73-60-75-20-50-40-82-91-125-22-8-21
11 63-36-80-66-49-124-23-72-9-5-24-102-61-87-84-68-56-12-107-3-96-100-71-78-114
12 86-10-69-111-28-64-98-90-103-43-42-89-31-39-88-82-40-50-76-20-75-85-19-13-83
13 27-94-122-29-36-80-5-24-87-110-68-34-56-3-78-114-2-79-0-101-88-121-125-123-91
14 111-64-103-69-109-10-18-104-58-63-9-72-49-2-97-48-79-11-67-27-29-5-24-87-110
15 70-9-81-27-1-11-2-74-23-53-61-106-108-102-24-5-29-122-52-51-54-30-31-39-88
16 102-24-5-29-27-1-46-39-25-41-57-22-125-21-8-35-91-121-88-101-7-120-116-93
17 63-36-9-72-23-37-74-114-2-97-0-48-26-101-88-125-21-119-22-30-39-46-11-7-65
18 35-8-22-119-21-125-121-25-91-123-95-65-7-11-2-114-78-12-100-14-66-80-29-94-81
19 39-46-11-2-23-53-14-106-108-61-102-87-110-84-56-15-107-3-78-71-96-100-12-68-34
20 70-72-124-49-2-74-66-9-63-80-36-5-29-122-27-1-67-11-77-0-120-26-7-40-50
21 9-72-66-53-23-114-2-11-26-7-101-120-116-95 40 44 82 65 50 76 85 75 20 32 126
22 62-16-55-6-73-48-97-2-23-74-72-66-14-108-115-3-78-71-96-100-12-56-15-107
23 74-49-23-72-66-70-9-5-24-36-29-27-67-46-118-122-52-38-30-22-119-8-41-25-88
24 105-58-63-36-24-102-61-115-106-14-108-87-84-110-68-56-15-107-3-96-100-12-78-114-2
25 47-51-89-113-59-42-104-43-105-63-80-66-74-114-16-112-60-73-48-2-23-53-61-102-87
26 96-71-78-12-68-84-56-3-100-14-61-87-106-115-108-102-24-5-58-104-43-98-42-89-31
27 102-87-84-68-34-56-15-12-3-115- 61-14-72-66-53-23-114-16-62-112-6-73-48-26-101
28 23-72-124-66-9-80-63-105-58-104-18-117-24-87-110-68-3-78-114-16-55-60-32-20-85
29 107-15-3-96-71-78-114-2-37-97-0-7-11-1-81-80-66-14-100-12-56-34-68-84-87
30 1-77-11-2-37-79-48-26-120-65-116-50-83-13-92-20-32-60-55-16-114-78-3-68-110
31 56-34-68-84-110-87-106-108-61-102-24-117-86-18-69-43-98-113-90-17-42-104-58-36-5
32 79-37-74-114-78-71-100-12-68-34-56-15-3-115-61-102-24-5-58-104-43-98-42-89-31
33 82-91-25-39-46-118-1-11-0-77-67-27-52-51-89-113-17-59-90-42-43-98-64-103-28
34 56-34-68-12-107-15-3-100-14-72-70-9-36-80-81-122-27-1-11-0-120-26-7-40-50
35 33-38-25-91-121-88-101-93-7-26-40-82-116-44-95-123-35-8-119-22-30-39-46-11-2
36 43-90-59-113-98-103-4-10-117-24-5-58-63-80-66-124-72-23-37-74-49-2-48-73-60
37 7-93-120-116-40-44-65-95-13-76-85-19-32-75-60-73-48-0-11-1-118-52-89-17-64
38 107-3-56-15-12-78-71-100-14-108-87-106-115-61-102-24-36-9-72-66-53-23-2-79-0
39 7-0-48-79-2-23-53-61-87-110-68-3-78-114-16-55-60-32-20-85-92-40-82-88-39
40 9-72-66-53-23-114-2-11-26-7-101-120-116-50-83-92-76-85-19-32-60-6-73-48-126
41 117-10-86-24-5-9-72-66-53-23-114-78-96-71-100-14-106-87-108-115-3-107-15-56-34
42 32-75-60-73-48-0-11-1-118-52-89-17-98-90-42-104-58-63-80-70-72-49-66-9-124
43 107-3-71-78-100-14-72-23-53-61-108-115-106-102-24-5-9-63-58-104-59-17-89-31-39
44 41-57-22-125-121-25-88-101-116-82-65-40-26-120-7-0-97-37-79-48-2-11-46-39-30
45 29-122-27-118-52-38-39-88-101-93-116-82-40-92-85-19-13-50-20-75-60-55-99-48-0
46 5-80-63-36-29-81-9-124-49-72-70-66-14-100-3-107-12-56-68-110-84-87-61-115-108
47 5-63-9-29-81-1-77-11-2-97-37-23-74-49-124-70-80-36-24-117-10-69-103-111-28
48 29-122-27-118-52-38-39-88-101-93-116-82-65-7-0-11-1-81-80-63-58-104-59-89-31
49 95-116-93-101-88-121-25-38-52-122-27-67-1-46-11-79-48-0-97-37-23-53-66-14-100
50 124-23-53-14-66-70-80-9-63-58-36-29-81-27-1-11-2-74-114-78-3-56-34-68-12
51 40-50-76-19-20-75-60-112-55-45-6-73-99-48-0-77-1-81-80-63-58-104-59-89-31
52 81-122-29-27-1-67-46-11-77-0-97-37-23-114-74-72-9-36-24-86-10-69-103-4-43
53 62-16-55-60-45-112-32-19-85-75-20-92-126-48-0-101-116-82-95-65-7-11-2-114-78
54 27-29-5-58-105-63-9-124-66-72-74-2-114-78-12-107-3-100-14-53-61-87-84-110-68
55 66-53-23-114-16-55-112-45-6-73-99-48-2-49-72-9-5-24-117-18-10-86-109-4-64
56 8-22-125-123-35-91-88-101-0-79-2-114-78-3-15-56-34-68-110-84-87-61-14-108-106
57 107-15-12-78-96-71-3-56-68-84-87-106-115-108-102-61-14-53-23-2-11-46-39-30-22
58 84-110-68-34-56-3-71-78-114-74-49-23-2-79-0-101-88-91-123-125-22-8-41-25-38
59 78-96-3-56-15-12-100-14-108-61-115-106-87-102-24-5-80-70-9-72-49-2-48-73-60
60 112-60-6-73-99-55-16-114-78-3-71-96-100-12-68-56-84-87-24-5-29-122-52-38-30
Table Options

Download as excel

Routes Sequence of Routes
1 23-114-16-62-112-6-73-48-26-7-40-82-44-93-88-39-46-1-118-27-29-5-24-87-110
2 27-118-67-81-80-70-72-74-2-23-37-97-11-7-40-50-13-19-85-75-20-32-60-73-48
3 12-3-78-114-74-23-37-2-11-46-39-30-22-21-35-91-25-38-31-89-59-104-58-63-5
4 46-1-11-2-74-49-66-14-100-12-56-15-3-78-114-16-55-60-32-20-85-19-13-83-92
5 57-8-21-125-22-41-30-39-46-11-2-74-23-53-61-108-102-24-5-63-36-9-80-81-67
6 17-98-113-89-31-39-88-93-44-40-95-65-50-20-19-32-45-60-6-73-126-92-76-85-75
7 41-33-54-31-89-59-104-43-105-63-80-66-74-114-16-112-60-73-48-2-23-53-61-102-87
8 13-92-85-75-60-55-45-73-48-2-49-72-9-63-105-43-69-10-86-24-102-61-14-100-78
9 102-106-115-61-14-100-71-3-96-78-114-23-37-74-2-79-0-120-65-101-88-82-40-92-85
10 29-81-94-122-52-27-67-46-118-1-11-0-48-73-60-75-20-50-40-82-91-125-22-8-21
11 63-36-80-66-49-124-23-72-9-5-24-102-61-87-84-68-56-12-107-3-96-100-71-78-114
12 86-10-69-111-28-64-98-90-103-43-42-89-31-39-88-82-40-50-76-20-75-85-19-13-83
13 27-94-122-29-36-80-5-24-87-110-68-34-56-3-78-114-2-79-0-101-88-121-125-123-91
14 111-64-103-69-109-10-18-104-58-63-9-72-49-2-97-48-79-11-67-27-29-5-24-87-110
15 70-9-81-27-1-11-2-74-23-53-61-106-108-102-24-5-29-122-52-51-54-30-31-39-88
16 102-24-5-29-27-1-46-39-25-41-57-22-125-21-8-35-91-121-88-101-7-120-116-93
17 63-36-9-72-23-37-74-114-2-97-0-48-26-101-88-125-21-119-22-30-39-46-11-7-65
18 35-8-22-119-21-125-121-25-91-123-95-65-7-11-2-114-78-12-100-14-66-80-29-94-81
19 39-46-11-2-23-53-14-106-108-61-102-87-110-84-56-15-107-3-78-71-96-100-12-68-34
20 70-72-124-49-2-74-66-9-63-80-36-5-29-122-27-1-67-11-77-0-120-26-7-40-50
21 9-72-66-53-23-114-2-11-26-7-101-120-116-95 40 44 82 65 50 76 85 75 20 32 126
22 62-16-55-6-73-48-97-2-23-74-72-66-14-108-115-3-78-71-96-100-12-56-15-107
23 74-49-23-72-66-70-9-5-24-36-29-27-67-46-118-122-52-38-30-22-119-8-41-25-88
24 105-58-63-36-24-102-61-115-106-14-108-87-84-110-68-56-15-107-3-96-100-12-78-114-2
25 47-51-89-113-59-42-104-43-105-63-80-66-74-114-16-112-60-73-48-2-23-53-61-102-87
26 96-71-78-12-68-84-56-3-100-14-61-87-106-115-108-102-24-5-58-104-43-98-42-89-31
27 102-87-84-68-34-56-15-12-3-115- 61-14-72-66-53-23-114-16-62-112-6-73-48-26-101
28 23-72-124-66-9-80-63-105-58-104-18-117-24-87-110-68-3-78-114-16-55-60-32-20-85
29 107-15-3-96-71-78-114-2-37-97-0-7-11-1-81-80-66-14-100-12-56-34-68-84-87
30 1-77-11-2-37-79-48-26-120-65-116-50-83-13-92-20-32-60-55-16-114-78-3-68-110
31 56-34-68-84-110-87-106-108-61-102-24-117-86-18-69-43-98-113-90-17-42-104-58-36-5
32 79-37-74-114-78-71-100-12-68-34-56-15-3-115-61-102-24-5-58-104-43-98-42-89-31
33 82-91-25-39-46-118-1-11-0-77-67-27-52-51-89-113-17-59-90-42-43-98-64-103-28
34 56-34-68-12-107-15-3-100-14-72-70-9-36-80-81-122-27-1-11-0-120-26-7-40-50
35 33-38-25-91-121-88-101-93-7-26-40-82-116-44-95-123-35-8-119-22-30-39-46-11-2
36 43-90-59-113-98-103-4-10-117-24-5-58-63-80-66-124-72-23-37-74-49-2-48-73-60
37 7-93-120-116-40-44-65-95-13-76-85-19-32-75-60-73-48-0-11-1-118-52-89-17-64
38 107-3-56-15-12-78-71-100-14-108-87-106-115-61-102-24-36-9-72-66-53-23-2-79-0
39 7-0-48-79-2-23-53-61-87-110-68-3-78-114-16-55-60-32-20-85-92-40-82-88-39
40 9-72-66-53-23-114-2-11-26-7-101-120-116-50-83-92-76-85-19-32-60-6-73-48-126
41 117-10-86-24-5-9-72-66-53-23-114-78-96-71-100-14-106-87-108-115-3-107-15-56-34
42 32-75-60-73-48-0-11-1-118-52-89-17-98-90-42-104-58-63-80-70-72-49-66-9-124
43 107-3-71-78-100-14-72-23-53-61-108-115-106-102-24-5-9-63-58-104-59-17-89-31-39
44 41-57-22-125-121-25-88-101-116-82-65-40-26-120-7-0-97-37-79-48-2-11-46-39-30
45 29-122-27-118-52-38-39-88-101-93-116-82-40-92-85-19-13-50-20-75-60-55-99-48-0
46 5-80-63-36-29-81-9-124-49-72-70-66-14-100-3-107-12-56-68-110-84-87-61-115-108
47 5-63-9-29-81-1-77-11-2-97-37-23-74-49-124-70-80-36-24-117-10-69-103-111-28
48 29-122-27-118-52-38-39-88-101-93-116-82-65-7-0-11-1-81-80-63-58-104-59-89-31
49 95-116-93-101-88-121-25-38-52-122-27-67-1-46-11-79-48-0-97-37-23-53-66-14-100
50 124-23-53-14-66-70-80-9-63-58-36-29-81-27-1-11-2-74-114-78-3-56-34-68-12
51 40-50-76-19-20-75-60-112-55-45-6-73-99-48-0-77-1-81-80-63-58-104-59-89-31
52 81-122-29-27-1-67-46-11-77-0-97-37-23-114-74-72-9-36-24-86-10-69-103-4-43
53 62-16-55-60-45-112-32-19-85-75-20-92-126-48-0-101-116-82-95-65-7-11-2-114-78
54 27-29-5-58-105-63-9-124-66-72-74-2-114-78-12-107-3-100-14-53-61-87-84-110-68
55 66-53-23-114-16-55-112-45-6-73-99-48-2-49-72-9-5-24-117-18-10-86-109-4-64
56 8-22-125-123-35-91-88-101-0-79-2-114-78-3-15-56-34-68-110-84-87-61-14-108-106
57 107-15-12-78-96-71-3-56-68-84-87-106-115-108-102-61-14-53-23-2-11-46-39-30-22
58 84-110-68-34-56-3-71-78-114-74-49-23-2-79-0-101-88-91-123-125-22-8-41-25-38
59 78-96-3-56-15-12-100-14-108-61-115-106-87-102-24-5-80-70-9-72-49-2-48-73-60
60 112-60-6-73-99-55-16-114-78-3-71-96-100-12-68-56-84-87-24-5-29-122-52-38-30
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Mauro Garavello, Benedetto Piccoli. On fluido-dynamic models for urban traffic. Networks & Heterogeneous Media, 2009, 4 (1) : 107-126. doi: 10.3934/nhm.2009.4.107

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