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May  2020, 16(3): 1203-1220. doi: 10.3934/jimo.2018200

An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints

Jiao-Yan Li 1,2, , Xiao Hu 1, and  Zhong Wan 1,,

1. 

School of Mathematics and Statistics, Central South University, Changsha, 410083, China
2. 

Hunan University of Commerce, Shool of Computer and Information Engineering, Changsha, 410205, China

* Corresponding author: Zhong Wan

Received  June 2017 Revised  September 2017 Published  May 2020 Early access  December 2018

Fund Project: All the three authors are supported by the National Science Foundation of China (Grant No. 71671190)

Vehicle routing problem (VRP) is a typical and important combinatorial optimization problem, and is often involved with complicated temporal and spatial constraints in practice. In this paper, the VRP is formulated as an optimization model for minimizing the number of vehicles and the total transportation cost subject to constraints on loading plan, service time and weight capacity. The transportation cost consists of the rent charge of vehicles, fuel cost, and carbon tax. Owing to complexity of the built model, it is divided into two subproblems by a two-stage optimization approach: at the first stage, the number of vehicles is minimized, then the routing plan is optimized at the second stage. For solving the sequential subproblems, two correlated genetic algorithms are developed, which share the same initial population to reduce their computational costs. Numerical results indicate that the developed algorithms are efficient, and a number of important managerial insights are revealed from the model.

Keywords: Vehicle routing problems, optimization model, algorithms, carbon emissions, bi-objective optimization.
Mathematics Subject Classification: Primary: 90B06, 62K05; Secondary: 68T37.
Citation: Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200
References:
[1]

R. Akcelik and M. Besley, Operating cost, fuel consumption, and emission models in aaSIDRA and aaMOTION, 25th Conference of Australian Institutes of Transport Research (CAITR 2003), 2003 (2003), 1-15.   Google Scholar

[2]

J. P. Arnaout, G. Arnaout and J. EIKhoury, Simulation and optimization of ant colony optimization algorithm for the stochastic uncapacitated location-allocation problem, Journal of Industrial and Management Optimization, 12 (2016), 1215-1225.  doi: 10.3934/jimo.2016.12.1215.  Google Scholar

[3]

R. Baldacci, N. Christofides and A. Mingozzi, An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts, Mathematical Programming, 115 (2008), 351-385.  doi: 10.1007/s10107-007-0178-5.  Google Scholar

[4]

M. Barth and K. Boriboonsomsin, Real-world carbon dioxide impacts of traffic congestion, Transportation Research Record: Journal of the Transportation Research Board, 2058 (2008), 163-171.   Google Scholar

[5]

T. Bektas and G. Laporte, The pollution-routing problem, Transportation Research Part B: Methodological, 45 (2011), 1232-1250.   Google Scholar

[6]

J. F. Cordeau, G. Laporte, M. W. P. Savelsbergh, et al. Vehicle routing, Transportation, Handbooks in Operations Research and Management Science, 14 (2006), 367-428. Google Scholar

[7]

G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 80-91.  doi: 10.1287/mnsc.6.1.80.  Google Scholar

[8]

J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich., 1975  Google Scholar

[9]

J. Homberger and H. Gehring, Two evolutionary metaheuristics for the vehicle routing problem with time windows, INFOR: Information Systems and Operational Research, 37 (1999), 297-318.   Google Scholar

[10]

J. Homberger and H. Gehring, A two-phase hybrid metaheuristic for the vehicle routing problem with time windows, European Journal of Operational Research, 162 (2005), 220-238.   Google Scholar

[11]

S. Huang and Z. Wan, A new nonmonotone spectral residual method for nonsmooth nonlinear equations, Journal of Computational and Applied Mathematics, 313 (2017), 82-101.  doi: 10.1016/j.cam.2016.09.014.  Google Scholar

[12]

S. Huang, Z. Wan and X. H. Chen, A new nonmonotone line search technique for unconstrained optimization, Numerical Algorithms, 68 (2015), 671-689.  doi: 10.1007/s11075-014-9866-4.  Google Scholar

[13]

S. Huang, Z. Wan and J. Zhang, An extended nonmonotone line search technique for large-scale unconstrained optimization, Journal of Computational and Applied Mathematics, 330 (2018), 586-604.  doi: 10.1016/j.cam.2017.09.026.  Google Scholar

[14]

O. Jabali, T. Woensel and A. G. de Kok, Analysis of travel times and CO$_2$ emissions in time-dependent vehicle routing, Production and Operations Management, 21 (2012), 1060-1074.   Google Scholar

[15]

I. Kara, B. Y. Kara and M. K. Yetis, Energy minimizing vehicle routing problem, Combinatorial Optimization and Applications, Springer Berlin Heidelberg, 2007, 62-71. doi: 10.1007/978-3-540-73556-4_9.  Google Scholar

[16]

M. Lai, H. Yang, S. Yang, et al., Cyber-physical logistics system-based vehicle routing optimization, Journal of Industrial and Management Optimization, 10 (2014), 701-715. doi: 10.3934/jimo.2014.10.701.  Google Scholar

[17]

Y. X. Li and Z. Wan, Bi-level programming approach to optimal strategy for VMI problems under random demand, ANZIAM Journal, 59 (2017), 247-270, doi: 10.1017/S1446181117000384.  Google Scholar

[18]

W. Liang W, B. Çatay and R. Eglese, Finding a minimum cost path between a pair of nodes in a time-varying road network with a congestion charge, European Journal of Operational Research, 236 (2014), 915-923.  doi: 10.1016/j.ejor.2013.10.044.  Google Scholar

[19]

W. Liang and R. Eglese, Minimum cost VRP with time-dependent speed data and congestion charge, Computers and Operations Research, 56 (2015), 41-50.  doi: 10.1016/j.cor.2014.10.007.  Google Scholar

[20]

W. Maden, R. Eglese and D. Black, Vehicle routing and scheduling with time-varying data: A case study, Journal of the Operational Research Society, 61 (2010), 515-522.   Google Scholar

[21]

M. Maiti M, S. Maity and A. Roy, An improved genetic algorithm and its application in constrained solid TSP in uncertain environments. Facets of Uncertainties and Applications, Springer, New Delhi, 2015,177-200. Google Scholar

[22]

D. Männel and A. Bortfeldt, A hybrid algorithm for the vehicle routing problem with pickup and delivery and three-dimensional loading constraints, European Journal of Operational Research, 254 (2016), 840-858.  doi: 10.1016/j.ejor.2016.04.016.  Google Scholar

[23]

E. Mardaneh, R. Loxton, Q. Lin and P. Schmidli, A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations, Journal of Industrial and Management Optimization, 13 (2017), 1601-1623.  doi: 10.3934/jimo.2017009.  Google Scholar

[24]

A. McKinnon, CO$-2$ emissions from freight transport: An analysis of UK data, Logistics Research Network-2007 Conference Global Supply Chains: Developing Skills, Capabilities and Networks, 2007. Google Scholar

[25]

A. McKinnon, M. Browne, A. Whiteing and M. Piecyk (Eds.), Green Logistics: Improving the environmental sustainability of logistics, Kogan Page Publishers, 2015 (2015), 215-228. Google Scholar

[26]

S. Mima and P. Criqui, The future of fuel cells in a long term inter-technology competition framework, The Economic Dynamics of Fuel Cell Technologies, Springer Berlin Heidelberg, 2003, 43–78. Google Scholar

[27]

D. M. Miranda and S. V. Conceic$\tilde{a}$o, The vehicle routing problem with hard time windows and stochastic travel and service time, Expert Systems with Applications, 64 (2016), 104-116.   Google Scholar

[28]

A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D. thesis, Cranfield University, School of Management, 2007. Google Scholar

[29]

C. Prins, A simple and effective evolutionary algorithm for the vehicle routing problem, Computers and Operations Research, 31 (2004), 1985-2002.  doi: 10.1016/S0305-0548(03)00158-8.  Google Scholar

[30]

C. Schreyer, C. Schneider, M. Maibach, et al. External Costs of Transport: Update Study, final report, Infras, 2004. Google Scholar

[31]

P. Shaw, Using constraint programming and local search methods to solve vehicle routing problems, Principles and Practice of Constraint Programming-CP98, Springer, Berlin Heidelberg, 1998,417–431. Google Scholar

[32]

S. R. Thangiah, K. E. Nygard and J. P. L. Gideon, A genetic algorithm system for vehicle routing with time windows, Artificial Intelligence Applications, 1991. Proceedings., Seventh IEEE Conference on IEEE, 1 (1991), 322-328.   Google Scholar

[33]

P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973594.  Google Scholar

[34]

X. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Appl. Math. Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

[35]

X. Zhang, S. Huang and Z. Wan, Stochastic Programming Approach to Global Supply Chain Management Under Random Additive Demand, Operational Research, 2017. doi: 10.1007/s12351-016-0269-2.  Google Scholar

show all references

References:
[1]

R. Akcelik and M. Besley, Operating cost, fuel consumption, and emission models in aaSIDRA and aaMOTION, 25th Conference of Australian Institutes of Transport Research (CAITR 2003), 2003 (2003), 1-15.   Google Scholar

[2]

J. P. Arnaout, G. Arnaout and J. EIKhoury, Simulation and optimization of ant colony optimization algorithm for the stochastic uncapacitated location-allocation problem, Journal of Industrial and Management Optimization, 12 (2016), 1215-1225.  doi: 10.3934/jimo.2016.12.1215.  Google Scholar

[3]

R. Baldacci, N. Christofides and A. Mingozzi, An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts, Mathematical Programming, 115 (2008), 351-385.  doi: 10.1007/s10107-007-0178-5.  Google Scholar

[4]

M. Barth and K. Boriboonsomsin, Real-world carbon dioxide impacts of traffic congestion, Transportation Research Record: Journal of the Transportation Research Board, 2058 (2008), 163-171.   Google Scholar

[5]

T. Bektas and G. Laporte, The pollution-routing problem, Transportation Research Part B: Methodological, 45 (2011), 1232-1250.   Google Scholar

[6]

J. F. Cordeau, G. Laporte, M. W. P. Savelsbergh, et al. Vehicle routing, Transportation, Handbooks in Operations Research and Management Science, 14 (2006), 367-428. Google Scholar

[7]

G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 80-91.  doi: 10.1287/mnsc.6.1.80.  Google Scholar

[8]

J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich., 1975  Google Scholar

[9]

J. Homberger and H. Gehring, Two evolutionary metaheuristics for the vehicle routing problem with time windows, INFOR: Information Systems and Operational Research, 37 (1999), 297-318.   Google Scholar

[10]

J. Homberger and H. Gehring, A two-phase hybrid metaheuristic for the vehicle routing problem with time windows, European Journal of Operational Research, 162 (2005), 220-238.   Google Scholar

[11]

S. Huang and Z. Wan, A new nonmonotone spectral residual method for nonsmooth nonlinear equations, Journal of Computational and Applied Mathematics, 313 (2017), 82-101.  doi: 10.1016/j.cam.2016.09.014.  Google Scholar

[12]

S. Huang, Z. Wan and X. H. Chen, A new nonmonotone line search technique for unconstrained optimization, Numerical Algorithms, 68 (2015), 671-689.  doi: 10.1007/s11075-014-9866-4.  Google Scholar

[13]

S. Huang, Z. Wan and J. Zhang, An extended nonmonotone line search technique for large-scale unconstrained optimization, Journal of Computational and Applied Mathematics, 330 (2018), 586-604.  doi: 10.1016/j.cam.2017.09.026.  Google Scholar

[14]

O. Jabali, T. Woensel and A. G. de Kok, Analysis of travel times and CO$_2$ emissions in time-dependent vehicle routing, Production and Operations Management, 21 (2012), 1060-1074.   Google Scholar

[15]

I. Kara, B. Y. Kara and M. K. Yetis, Energy minimizing vehicle routing problem, Combinatorial Optimization and Applications, Springer Berlin Heidelberg, 2007, 62-71. doi: 10.1007/978-3-540-73556-4_9.  Google Scholar

[16]

M. Lai, H. Yang, S. Yang, et al., Cyber-physical logistics system-based vehicle routing optimization, Journal of Industrial and Management Optimization, 10 (2014), 701-715. doi: 10.3934/jimo.2014.10.701.  Google Scholar

[17]

Y. X. Li and Z. Wan, Bi-level programming approach to optimal strategy for VMI problems under random demand, ANZIAM Journal, 59 (2017), 247-270, doi: 10.1017/S1446181117000384.  Google Scholar

[18]

W. Liang W, B. Çatay and R. Eglese, Finding a minimum cost path between a pair of nodes in a time-varying road network with a congestion charge, European Journal of Operational Research, 236 (2014), 915-923.  doi: 10.1016/j.ejor.2013.10.044.  Google Scholar

[19]

W. Liang and R. Eglese, Minimum cost VRP with time-dependent speed data and congestion charge, Computers and Operations Research, 56 (2015), 41-50.  doi: 10.1016/j.cor.2014.10.007.  Google Scholar

[20]

W. Maden, R. Eglese and D. Black, Vehicle routing and scheduling with time-varying data: A case study, Journal of the Operational Research Society, 61 (2010), 515-522.   Google Scholar

[21]

M. Maiti M, S. Maity and A. Roy, An improved genetic algorithm and its application in constrained solid TSP in uncertain environments. Facets of Uncertainties and Applications, Springer, New Delhi, 2015,177-200. Google Scholar

[22]

D. Männel and A. Bortfeldt, A hybrid algorithm for the vehicle routing problem with pickup and delivery and three-dimensional loading constraints, European Journal of Operational Research, 254 (2016), 840-858.  doi: 10.1016/j.ejor.2016.04.016.  Google Scholar

[23]

E. Mardaneh, R. Loxton, Q. Lin and P. Schmidli, A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations, Journal of Industrial and Management Optimization, 13 (2017), 1601-1623.  doi: 10.3934/jimo.2017009.  Google Scholar

[24]

A. McKinnon, CO$-2$ emissions from freight transport: An analysis of UK data, Logistics Research Network-2007 Conference Global Supply Chains: Developing Skills, Capabilities and Networks, 2007. Google Scholar

[25]

A. McKinnon, M. Browne, A. Whiteing and M. Piecyk (Eds.), Green Logistics: Improving the environmental sustainability of logistics, Kogan Page Publishers, 2015 (2015), 215-228. Google Scholar

[26]

S. Mima and P. Criqui, The future of fuel cells in a long term inter-technology competition framework, The Economic Dynamics of Fuel Cell Technologies, Springer Berlin Heidelberg, 2003, 43–78. Google Scholar

[27]

D. M. Miranda and S. V. Conceic$\tilde{a}$o, The vehicle routing problem with hard time windows and stochastic travel and service time, Expert Systems with Applications, 64 (2016), 104-116.   Google Scholar

[28]

A. Palmer, The Development of an Integrated Routing and Carbon Dioxide Emissions Model for Goods Vehicles, Ph.D. thesis, Cranfield University, School of Management, 2007. Google Scholar

[29]

C. Prins, A simple and effective evolutionary algorithm for the vehicle routing problem, Computers and Operations Research, 31 (2004), 1985-2002.  doi: 10.1016/S0305-0548(03)00158-8.  Google Scholar

[30]

C. Schreyer, C. Schneider, M. Maibach, et al. External Costs of Transport: Update Study, final report, Infras, 2004. Google Scholar

[31]

P. Shaw, Using constraint programming and local search methods to solve vehicle routing problems, Principles and Practice of Constraint Programming-CP98, Springer, Berlin Heidelberg, 1998,417–431. Google Scholar

[32]

S. R. Thangiah, K. E. Nygard and J. P. L. Gideon, A genetic algorithm system for vehicle routing with time windows, Artificial Intelligence Applications, 1991. Proceedings., Seventh IEEE Conference on IEEE, 1 (1991), 322-328.   Google Scholar

[33]

P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973594.  Google Scholar

[34]

X. Zhang, S. Huang and Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Appl. Math. Modelling, 40 (2016), 10105-10130.  doi: 10.1016/j.apm.2016.06.054.  Google Scholar

[35]

X. Zhang, S. Huang and Z. Wan, Stochastic Programming Approach to Global Supply Chain Management Under Random Additive Demand, Operational Research, 2017. doi: 10.1007/s12351-016-0269-2.  Google Scholar

Figure 1.  Example of crossover
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Figure 2.  Minimal total costs by Algorithm 5
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Figure 3.  Sensitivity analysis on speed of vehicles
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Table 1.  Numerical results in case study
NC MVN $v$ TD TPV WTV VC TFC TCE TC CPU
1 19 6 50 1544.2 5.14 0.33 600 1160.5 7.09 1767.6 68.5
2 39 10 50 2208.7 3.40 0.28 1000 1654.2 10.1 2664.3 79.4
3 59 19 50 4368.9 4.60 0.32 1900 3145.7 19.2 5064.9 92.9
4 79 26 50 6070.6 4.66 0.32 2600 4379.8 26.8 7006.6 125.7
5 99 35 50 8636.1 4.93 0.27 3500 6133.3 37.5 9670.8 163.0
6 119 38 50 9743.4 5.12 0.25 3800 6970.8 42.6 10813.4 219.3
7 139 45 50 11470.4 5.09 0.43 4500 8213.9 50.2 12764.1 244.3
8 159 49 50 13913.2 5.68 0.44 4900 9938.1 60.8 14898.9 259.0
9 179 60 50 14883.3 4.96 0.50 6000 10673.7 65.3 16739.0 274.4
10 199 71 50 17599.0 4.96 0.62 7100 12544.3 76.7 19721.0 309.7
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NC MVN $v$ TD TPV WTV VC TFC TCE TC CPU
1 19 6 50 1544.2 5.14 0.33 600 1160.5 7.09 1767.6 68.5
2 39 10 50 2208.7 3.40 0.28 1000 1654.2 10.1 2664.3 79.4
3 59 19 50 4368.9 4.60 0.32 1900 3145.7 19.2 5064.9 92.9
4 79 26 50 6070.6 4.66 0.32 2600 4379.8 26.8 7006.6 125.7
5 99 35 50 8636.1 4.93 0.27 3500 6133.3 37.5 9670.8 163.0
6 119 38 50 9743.4 5.12 0.25 3800 6970.8 42.6 10813.4 219.3
7 139 45 50 11470.4 5.09 0.43 4500 8213.9 50.2 12764.1 244.3
8 159 49 50 13913.2 5.68 0.44 4900 9938.1 60.8 14898.9 259.0
9 179 60 50 14883.3 4.96 0.50 6000 10673.7 65.3 16739.0 274.4
10 199 71 50 17599.0 4.96 0.62 7100 12544.3 76.7 19721.0 309.7
Table 2.  Routes of all the vehicles in the case $n = 19$
LV LNC DT AT WT LW LL DD TFC TCEt
1 0 9:00 - - - - - - -
6 12:08 11:48 20 0.32 1.12 130.2 86.6 0.53
4 13:32 13:32 0 0.18 0.48 59.0 38.3 0.23
0 - 15:20 - 0 0 1654.2 50.2 0.31
2 0 9:00 - - - - - - -
5 10:19 10:19 0 1.26 0.52 56.4 43.7 0.27
1 11:20 11:20 0 1.05 0.31 40.8 30.6 0.19
10 13:29 12:28 61 0.59 0.05 46.0 32.0 0.20
0 - 14:24 - 0 0 1654.2 22.6 0.14
3 0 9:00 - - - - - - -
9 11:54 11:54 0 2.57 1.62 135.3 125.7 0.77
17 13:11 13:10 0 1.91 1.35 53.7 45.7 0.28
7 14:35 14:35 0 1.00 0.64 60.5 45.0 0.28
0 - 15:13 0 0 0 1654.2 13.5 0.08
4 0 9:00 - - - - - - -
2 10:57 10:37 20 1.14 1.47 71.1 54.1 0.33
13 13:07 12:07 60 0.91 0.97 48.4 35.5 0.22
19 14:22 14:22 0 0.38 0.42 51.8 34.8 0.21
0 - 15:19 - 0 0 37.7 23.6 0.14
5 0 9:00 - - - - - - -
18 11:55 11:15 39 2.31 2.03 120.9 92.4 0.57
3 12:53 12:53 0 1.93 1.81 38.2 32.6 0.20
12 13:44 13:44 0 1.28 1.54 33.2 25.9 0.16
16 14:23 14:23 0 0.85 0.60 22.4 16.3 0.10
0 - 15:11 - 0 0 29.7 18.6 0.11
6 0 9:00 - - - - - - -
15 11:30 11:30 0 1.98 2.58 115.3 99.1 0.61
8 13:03 13:03 0 1.74 1.82 67.2 55.9 0.35
11 14:14 14:14 0 0.92 0.82 49.5 36.4 0.22
14 15:55 15:55 0 0.33 0.55 73.6 49.0 0.30
0 - 16:48 0 0 0 83.8 52.5 0.32
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LV LNC DT AT WT LW LL DD TFC TCEt
1 0 9:00 - - - - - - -
6 12:08 11:48 20 0.32 1.12 130.2 86.6 0.53
4 13:32 13:32 0 0.18 0.48 59.0 38.3 0.23
0 - 15:20 - 0 0 1654.2 50.2 0.31
2 0 9:00 - - - - - - -
5 10:19 10:19 0 1.26 0.52 56.4 43.7 0.27
1 11:20 11:20 0 1.05 0.31 40.8 30.6 0.19
10 13:29 12:28 61 0.59 0.05 46.0 32.0 0.20
0 - 14:24 - 0 0 1654.2 22.6 0.14
3 0 9:00 - - - - - - -
9 11:54 11:54 0 2.57 1.62 135.3 125.7 0.77
17 13:11 13:10 0 1.91 1.35 53.7 45.7 0.28
7 14:35 14:35 0 1.00 0.64 60.5 45.0 0.28
0 - 15:13 0 0 0 1654.2 13.5 0.08
4 0 9:00 - - - - - - -
2 10:57 10:37 20 1.14 1.47 71.1 54.1 0.33
13 13:07 12:07 60 0.91 0.97 48.4 35.5 0.22
19 14:22 14:22 0 0.38 0.42 51.8 34.8 0.21
0 - 15:19 - 0 0 37.7 23.6 0.14
5 0 9:00 - - - - - - -
18 11:55 11:15 39 2.31 2.03 120.9 92.4 0.57
3 12:53 12:53 0 1.93 1.81 38.2 32.6 0.20
12 13:44 13:44 0 1.28 1.54 33.2 25.9 0.16
16 14:23 14:23 0 0.85 0.60 22.4 16.3 0.10
0 - 15:11 - 0 0 29.7 18.6 0.11
6 0 9:00 - - - - - - -
15 11:30 11:30 0 1.98 2.58 115.3 99.1 0.61
8 13:03 13:03 0 1.74 1.82 67.2 55.9 0.35
11 14:14 14:14 0 0.92 0.82 49.5 36.4 0.22
14 15:55 15:55 0 0.33 0.55 73.6 49.0 0.30
0 - 16:48 0 0 0 83.8 52.5 0.32
Table 3.  Comparison of the optimal costs by different models
Case NC TD TFC TCE CPU
Test1$^a$ 20 1544.2 1160.5 7.1 68.5
Test1$^b$ 20 1442.2 1214.1 7.5 102.8
Test2$^a$ 40 2208.7 1654.2 10.1 79.4
Test2$^b$ 40 2304.6 1815.2 11.1 111.2
Test3$^a$ 60 4368.9 3145.7 19.2 92.9
Test3$^b$ 60 4262.7 3351.5 20.5 148.6
Test4$^a$ 80 6070.6 4379.8 26.8 125.7
Test4$^b$ 80 6118.4 4811.0 29.4 188.6
Test5$^a$ 100 8636.1 6133.3 37.5 163.0
Test5$^b$ 100 8288.8 6517.7 39.8 252.7
Test6$^a$ 120 9743.4 6970.8 42.6 219.3
Test6$^b$ 120 9758.8 7693.3 46.9 346.5
Test7$^a$ 140 11470.4 8213.9 50.2 244.3
Test7$^b$ 140 11853.7 9320.9 57.0 359.1
Test8$^a$ 160 13913.2 9938.1 60.8 259.0
Test8$^b$ 160 13011.9 10230.9 62.5 383.3
Test9$^a$ 180 14883.3 10673.7 65.3 274.4
Test9$^b$ 180 15488.2 12178.9 74.4 466.5
Test10$^a$ 200 17599.0 12544.3 76.7 309.7
Test10$^b$ 200 17378.1 13664.8 83.5 526.5
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Case NC TD TFC TCE CPU
Test1$^a$ 20 1544.2 1160.5 7.1 68.5
Test1$^b$ 20 1442.2 1214.1 7.5 102.8
Test2$^a$ 40 2208.7 1654.2 10.1 79.4
Test2$^b$ 40 2304.6 1815.2 11.1 111.2
Test3$^a$ 60 4368.9 3145.7 19.2 92.9
Test3$^b$ 60 4262.7 3351.5 20.5 148.6
Test4$^a$ 80 6070.6 4379.8 26.8 125.7
Test4$^b$ 80 6118.4 4811.0 29.4 188.6
Test5$^a$ 100 8636.1 6133.3 37.5 163.0
Test5$^b$ 100 8288.8 6517.7 39.8 252.7
Test6$^a$ 120 9743.4 6970.8 42.6 219.3
Test6$^b$ 120 9758.8 7693.3 46.9 346.5
Test7$^a$ 140 11470.4 8213.9 50.2 244.3
Test7$^b$ 140 11853.7 9320.9 57.0 359.1
Test8$^a$ 160 13913.2 9938.1 60.8 259.0
Test8$^b$ 160 13011.9 10230.9 62.5 383.3
Test9$^a$ 180 14883.3 10673.7 65.3 274.4
Test9$^b$ 180 15488.2 12178.9 74.4 466.5
Test10$^a$ 200 17599.0 12544.3 76.7 309.7
Test10$^b$ 200 17378.1 13664.8 83.5 526.5
Table 4.  Vehicle parameters
V VC VL VW VH WV VP $LC$
V1 Light 4.0 1.9 2.3 2.6 3 100
V2 Light 6.2 2 2 3.5 5 150
V3 Medium 9.6 2.3 2.7 25 25 300
V4 Medium 12.0 2.4 2.7 28 28 400
V5 Heavy 17.5 2.4 2.7 35 35 500
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V VC VL VW VH WV VP $LC$
V1 Light 4.0 1.9 2.3 2.6 3 100
V2 Light 6.2 2 2 3.5 5 150
V3 Medium 9.6 2.3 2.7 25 25 300
V4 Medium 12.0 2.4 2.7 28 28 400
V5 Heavy 17.5 2.4 2.7 35 35 500
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