Figure 1.
A sample vector with 4 routes
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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
| 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 |
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.
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 and 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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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). |
| [13] |
L. Fan,
Metaheuristic Methods for the Urban Transit Routing Problem, Ph. D thesis, Cardiff University (United Kingdom), 2009. |
| [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. |
| [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. |
| [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). |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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). |
| [32] |
M. Nikolić and D. Teodorović,
Transit network design by bee colony optimization, Expert Systems with Applications, 40 (2013), 5945-5955.
|
| [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. |
| [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). |
| [35] |
S. Schéele,
A supply model for public transit services, Transportation Research Part B: Methodological, 14 (1980), 133-146.
|
| [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. |
| [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. |
| [38] |
F. Zhao and A. Gan, Optimization of transit network to minimize transfer,
Final Report BD015-02 Research Office Florida Department of Transportation, 2003. |
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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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). |
| [13] |
L. Fan,
Metaheuristic Methods for the Urban Transit Routing Problem, Ph. D thesis, Cardiff University (United Kingdom), 2009. |
| [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. |
| [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. |
| [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). |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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). |
| [32] |
M. Nikolić and D. Teodorović,
Transit network design by bee colony optimization, Expert Systems with Applications, 40 (2013), 5945-5955.
|
| [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. |
| [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). |
| [35] |
S. Schéele,
A supply model for public transit services, Transportation Research Part B: Methodological, 14 (1980), 133-146.
|
| [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. |
| [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. |
| [38] |
F. Zhao and A. Gan, Optimization of transit network to minimize transfer,
Final Report BD015-02 Research Office Florida Department of Transportation, 2003. |
Figure 2.
Mandl's Swiss Network
Figure 3.
Mumford Network
Figure 8.
Approximate Pareto Fronts achieved by the proposed DE for Mandl's Swiss Network
Figure 9.
Pareto Fronts achieved by the proposed DE for Mumford Network
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 | 90.88 | 61.08 | 90.43 | 61.08 | 91.84 | 61.08 | 91.46 | 61.08 | |
| 8.35 | 36.61 | 9.57 | 36.61 | 8.16 | 36.61 | 8.54 | 36.61 | ||
| 0.77 | 2.31 | 0.00 | 2.31 | 0.00 | 2.31 | 0.00 | 2.31 | ||
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | ||
| 10.65 | 13.88 | 10.57 | 13.88 | 10.50 | 13.88 | 10.80 | 13.88 | ||
| 126 | 63 | 149 | 63 | 150 | 63 | 133 | 63 | ||
| 6 | 93.13 | 65.18 | 95.38 | 70.91 | 96.79 | 70.91 | 97.24 | 70.46 | |
| 6.29 | 30.38 | 4.36 | 25.50 | 3.21 | 25.50 | 2.76 | 24.34 | ||
| 0.58 | 3.53 | 0.06 | 2.95 | 0.00 | 2.95 | 0.00 | 5.20 | ||
| 0.00 | 0.90 | 0.00 | 0.64 | 0.00 | 0.64 | 0.00 | 0.00 | ||
| 10.49 | 13.82 | 10.27 | 13.48 | 10.21 | 13.48 | 10.16 | 12.60 | ||
| 148 | 63 | 221 | 63 | 224 | 63 | 223 | 63 | ||
| 7 | 92.55 | 64.42 | 96.47 | 65.13 | 98.01 | 70.65 | 97.37 | 68.96 | |
| 6.68 | 26.20 | 3.34 | 22.93 | 1.99 | 21.13 | 2.63 | 26.29 | ||
| 0.77 | 8.16 | 0.19 | 10.34 | 0.00 | 7.13 | 0.00 | 3.92 | ||
| 0.00 | 1.22 | 0.00 | 1.61 | 0.00 | 1.09 | 0.00 | 0.83 | ||
| 10.44 | 14.13 | 10.22 | 14.25 | 10.16 | 13.76 | 10.21 | 13.46 | ||
| 166 | 63 | 264 | 63 | 239 | 63 | 235 | 63 | ||
| 8 | 91.33 | 55.17 | 97.56 | 57.93 | 99.04 | 61.91 | 98.20 | 60.76 | |
| 8.67 | 21.97 | 2.31 | 31.92 | 0.96 | 29.67 | 1.80 | 25.63 | ||
| 0.00 | 18.11 | 0.13 | 9.70 | 0.00 | 6.87 | 0.00 | 10.34 | ||
| 0.00 | 4.75 | 0.00 | 0.45 | 0.00 | 1.54 | 0.00 | 3.26 | ||
| 10.45 | 15.45 | 10.17 | 14.45 | 10.11 | 14.22 | 10.31 | 14.78 | ||
| 245 | 63 | 291 | 63 | 256 | 63 | 244 | 63 | ||
| Note: 1. best results for passenger 2. best results for operator |
|||||||||
| 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 | 90.88 | 61.08 | 90.43 | 61.08 | 91.84 | 61.08 | 91.46 | 61.08 | |
| 8.35 | 36.61 | 9.57 | 36.61 | 8.16 | 36.61 | 8.54 | 36.61 | ||
| 0.77 | 2.31 | 0.00 | 2.31 | 0.00 | 2.31 | 0.00 | 2.31 | ||
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | ||
| 10.65 | 13.88 | 10.57 | 13.88 | 10.50 | 13.88 | 10.80 | 13.88 | ||
| 126 | 63 | 149 | 63 | 150 | 63 | 133 | 63 | ||
| 6 | 93.13 | 65.18 | 95.38 | 70.91 | 96.79 | 70.91 | 97.24 | 70.46 | |
| 6.29 | 30.38 | 4.36 | 25.50 | 3.21 | 25.50 | 2.76 | 24.34 | ||
| 0.58 | 3.53 | 0.06 | 2.95 | 0.00 | 2.95 | 0.00 | 5.20 | ||
| 0.00 | 0.90 | 0.00 | 0.64 | 0.00 | 0.64 | 0.00 | 0.00 | ||
| 10.49 | 13.82 | 10.27 | 13.48 | 10.21 | 13.48 | 10.16 | 12.60 | ||
| 148 | 63 | 221 | 63 | 224 | 63 | 223 | 63 | ||
| 7 | 92.55 | 64.42 | 96.47 | 65.13 | 98.01 | 70.65 | 97.37 | 68.96 | |
| 6.68 | 26.20 | 3.34 | 22.93 | 1.99 | 21.13 | 2.63 | 26.29 | ||
| 0.77 | 8.16 | 0.19 | 10.34 | 0.00 | 7.13 | 0.00 | 3.92 | ||
| 0.00 | 1.22 | 0.00 | 1.61 | 0.00 | 1.09 | 0.00 | 0.83 | ||
| 10.44 | 14.13 | 10.22 | 14.25 | 10.16 | 13.76 | 10.21 | 13.46 | ||
| 166 | 63 | 264 | 63 | 239 | 63 | 235 | 63 | ||
| 8 | 91.33 | 55.17 | 97.56 | 57.93 | 99.04 | 61.91 | 98.20 | 60.76 | |
| 8.67 | 21.97 | 2.31 | 31.92 | 0.96 | 29.67 | 1.80 | 25.63 | ||
| 0.00 | 18.11 | 0.13 | 9.70 | 0.00 | 6.87 | 0.00 | 10.34 | ||
| 0.00 | 4.75 | 0.00 | 0.45 | 0.00 | 1.54 | 0.00 | 3.26 | ||
| 10.45 | 15.45 | 10.17 | 14.45 | 10.11 | 14.22 | 10.31 | 14.78 | ||
| 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 | ||
| 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 |
| Network | Nodes | Links | Routes | Min-Max Nodes | Demand | ||
| 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 |
| 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 |
| 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 | 63.20 | 18.42 | 65.41 | 16.99 | |
| 35.82 | 23.40 | 34.24 | 30.72 | ||
| 0.98 | 20.78 | 0.35 | 28.92 | ||
| 0.00 | 37.40 | 0.00 | 23.38 | ||
| 16.05 | 32.40 | 15.27 | 33.41 | ||
| 759 | 111 | 673 | 107 | ||
| Mumford1 | 36.6 | 16.35 | 38.77 | 22.39 | |
| 52.42 | 29.06 | 54.23 | 40.57 | ||
| 10.71 | 29.92 | 5.12 | 34.33 | ||
| 0.26 | 24.66 | 1.88 | 2.71 | ||
| 24.79 | 34.69 | 23.16 | 31.15 | ||
| 2038 | 568 | 1955 | 567 | ||
| Mumford2 | 30.92 | 13.76 | 31.47 | 15.32 | |
| 51.29 | 27.69 | 58.23 | 29.31 | ||
| 16.36 | 29.53 | 9.60 | 31.08 | ||
| 1.44 | 29.02 | 0.70 | 24.29 | ||
| 28.65 | 36.54 | 27.28 | 34.52 | ||
| 5632 | 2244 | 5268 | 2305 | ||
| Mumford3 | 27.46 | 16.71 | 28.12 | 25.42 | |
| 50.97 | 33.69 | 54.35 | 41.26 | ||
| 18.76 | 29.18 | 16.84 | 24.91 | ||
| 2.81 | 20.42 | 0.69 | 8.41 | ||
| 31.44 | 36.92 | 30.16 | 34.81 | ||
| 6665 | 2830 | 6547 | 2732 | ||
| Note: 1. best results for passenger 2. best results for operator |
|||||
| Instances | Parameters | Mumford (2013) | Proposed DE | ||
| 1 | 2 | 1 | 2 | ||
| Mumford0 | 63.20 | 18.42 | 65.41 | 16.99 | |
| 35.82 | 23.40 | 34.24 | 30.72 | ||
| 0.98 | 20.78 | 0.35 | 28.92 | ||
| 0.00 | 37.40 | 0.00 | 23.38 | ||
| 16.05 | 32.40 | 15.27 | 33.41 | ||
| 759 | 111 | 673 | 107 | ||
| Mumford1 | 36.6 | 16.35 | 38.77 | 22.39 | |
| 52.42 | 29.06 | 54.23 | 40.57 | ||
| 10.71 | 29.92 | 5.12 | 34.33 | ||
| 0.26 | 24.66 | 1.88 | 2.71 | ||
| 24.79 | 34.69 | 23.16 | 31.15 | ||
| 2038 | 568 | 1955 | 567 | ||
| Mumford2 | 30.92 | 13.76 | 31.47 | 15.32 | |
| 51.29 | 27.69 | 58.23 | 29.31 | ||
| 16.36 | 29.53 | 9.60 | 31.08 | ||
| 1.44 | 29.02 | 0.70 | 24.29 | ||
| 28.65 | 36.54 | 27.28 | 34.52 | ||
| 5632 | 2244 | 5268 | 2305 | ||
| Mumford3 | 27.46 | 16.71 | 28.12 | 25.42 | |
| 50.97 | 33.69 | 54.35 | 41.26 | ||
| 18.76 | 29.18 | 16.84 | 24.91 | ||
| 2.81 | 20.42 | 0.69 | 8.41 | ||
| 31.44 | 36.92 | 30.16 | 34.81 | ||
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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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