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Immediate schedule adjustment and semidefinite relaxation
Optimum management of the network of city bus routes based on a stochastic dynamic model
School of EECS, University of Ottawa, 800 King Edward Ave. Ottawa, ON K1N 6N5, Canada 
In this paper, we develop a stochastic dynamic model for the network of city bus routes subject to resource and other practical constraints. We define an objective function on the basis of four terms: fuel cost, operating cost, customers waiting time, and revenue of the bus company. Hereafter, an optimization problem is formulated and solved by use of nonlinear integer programming. If the technique presented here is implemented, it is expected to boost the bus company's revenue, reduce waiting time and therefore promote customer satisfaction. A series of numerical experiments is carried out and the corresponding optimization problems are addressed giving the optimal number of buses allocated to each of the bus routes in the network. Since the dynamic model proposed here can be applied to any network of bus routes, it is believed that the procedure developed in this paper is of great potential for both the city bus company and the customers.
References:
[1] 
N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, U. K, copublished by John Wiley & Sons, New York, 1988. 
[2] 
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd, 2006. 
[3] 
S. Chen, Beijing workers have longest daily commute in China at 52 minutes each way, in South China Morning Post, 2015. Available from: http://www.scmp.com/news/china/article/1692839/beijingersleadchinaspacklongestdailycommute. 
[4] 
C. Jonathan and D. I. Wilson, OPTI: lowering the barrier between open source optimizers and the industrial MATLAB user, Foundations of ComputerAided Process Operations, 24 (2012), p32. 
[5] 
D. Li and X. Sun, Nonlinear Integer Programming, Springer Science & Business Media, 2006. doi: 10.1007/0387329951. 
[6] 
C. E. Mandl, Evaluation and optimization of urban public transportation networks, European Journal of Operational Research, 5 (1980), 396404. doi: 10.1016/03772217(80)901265. 
[7] 
A. T. Murray, R. Davis, R. J. Stimson and L. Ferreira, Public transportation access, Transportation Research Part D: Transport and Environment, 3 (1998), 319328. doi: 10.1016/S13619209(98)000108. 
[8] 
R. Tumilty, Every day OC Transpo cancels about 57 trips: Metro analysis, May 14,2017. Available from: http://www.metronews.ca/news/ottawa/2017/05/14/octranspocancellationsbrokendownacrossthesystem.html. 
[9] 
S. Wang and N. U. Ahmed, Stochastic dynamic model of city bus routes and their optimum management, To appear, Control Science and Systems Engineering (ICCSSE), 2018 4th International Conference on. IEEE, (2018). 
[10] 
L. Wu, Comparative analysis of the public transit modes based on urban area location theory, International Conference on Green Intelligent Transportation System and Safety, (2016), 809817. doi: 10.1007/9789811035517_65. 
show all references
References:
[1] 
N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, U. K, copublished by John Wiley & Sons, New York, 1988. 
[2] 
N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd, 2006. 
[3] 
S. Chen, Beijing workers have longest daily commute in China at 52 minutes each way, in South China Morning Post, 2015. Available from: http://www.scmp.com/news/china/article/1692839/beijingersleadchinaspacklongestdailycommute. 
[4] 
C. Jonathan and D. I. Wilson, OPTI: lowering the barrier between open source optimizers and the industrial MATLAB user, Foundations of ComputerAided Process Operations, 24 (2012), p32. 
[5] 
D. Li and X. Sun, Nonlinear Integer Programming, Springer Science & Business Media, 2006. doi: 10.1007/0387329951. 
[6] 
C. E. Mandl, Evaluation and optimization of urban public transportation networks, European Journal of Operational Research, 5 (1980), 396404. doi: 10.1016/03772217(80)901265. 
[7] 
A. T. Murray, R. Davis, R. J. Stimson and L. Ferreira, Public transportation access, Transportation Research Part D: Transport and Environment, 3 (1998), 319328. doi: 10.1016/S13619209(98)000108. 
[8] 
R. Tumilty, Every day OC Transpo cancels about 57 trips: Metro analysis, May 14,2017. Available from: http://www.metronews.ca/news/ottawa/2017/05/14/octranspocancellationsbrokendownacrossthesystem.html. 
[9] 
S. Wang and N. U. Ahmed, Stochastic dynamic model of city bus routes and their optimum management, To appear, Control Science and Systems Engineering (ICCSSE), 2018 4th International Conference on. IEEE, (2018). 
[10] 
L. Wu, Comparative analysis of the public transit modes based on urban area location theory, International Conference on Green Intelligent Transportation System and Safety, (2016), 809817. doi: 10.1007/9789811035517_65. 
Parameter  Value 
Length of the 

Total number of buses 
10 
Number of stations 

Average speed of city buses 

Coefficient of fuel cost 

Weight given to stations 

Ticket price 
3 
Time interval 
5mins 
Parameter  Value 
Length of the 

Total number of buses 
10 
Number of stations 

Average speed of city buses 

Coefficient of fuel cost 

Weight given to stations 

Ticket price 
3 
Time interval 
5mins 
Time  Optimal control 
Optimal cost 
Whole day  [3,1,4,2]  7976343.4179 
00:00 AM to 6:00 AM  [2,1,2,1]  1317212.4488 
6:00 AM to 20:00 PM  [3,1,4,2]  5406920.1899 
20:00 PM to 24:00 PM  [2,1,3,2]  1088617.3315 
Time  Optimal control 
Optimal cost 
Whole day  [3,1,4,2]  7976343.4179 
00:00 AM to 6:00 AM  [2,1,2,1]  1317212.4488 
6:00 AM to 20:00 PM  [3,1,4,2]  5406920.1899 
20:00 PM to 24:00 PM  [2,1,3,2]  1088617.3315 
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