# American Institute of Mathematical Sciences

April  2019, 15(2): 619-631. doi: 10.3934/jimo.2018061

## 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

* Corresponding author: Shi'an Wang

Received  November 2017 Revised  December 2017 Published  April 2019 Early access  April 2018

Fund Project: The authors are supported by NSERC grant A7101.

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.

Citation: Shi'an Wang, N. U. Ahmed. Optimum management of the network of city bus routes based on a stochastic dynamic model. Journal of Industrial and Management Optimization, 2019, 15 (2) : 619-631. doi: 10.3934/jimo.2018061
##### References:
 [1] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, U. K, co-published 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/beijingers-lead-chinas-pack-longest-daily-commute. [4] C. Jonathan and D. I. Wilson, OPTI: lowering the barrier between open source optimizers and the industrial MATLAB user, Foundations of Computer-Aided Process Operations, 24 (2012), p32. [5] D. Li and X. Sun, Nonlinear Integer Programming, Springer Science & Business Media, 2006. doi: 10.1007/0-387-32995-1. [6] 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. [7] A. T. Murray, R. Davis, R. J. Stimson and L. Ferreira, Public transportation access, Transportation Research Part D: Transport and Environment, 3 (1998), 319-328.  doi: 10.1016/S1361-9209(98)00010-8. [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/oc-transpo-cancellations-broken-down-across-the-system-.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), 809-817.  doi: 10.1007/978-981-10-3551-7_65.

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##### References:
 [1] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Longman Scientific and Technical, U. K, co-published 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/beijingers-lead-chinas-pack-longest-daily-commute. [4] C. Jonathan and D. I. Wilson, OPTI: lowering the barrier between open source optimizers and the industrial MATLAB user, Foundations of Computer-Aided Process Operations, 24 (2012), p32. [5] D. Li and X. Sun, Nonlinear Integer Programming, Springer Science & Business Media, 2006. doi: 10.1007/0-387-32995-1. [6] 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. [7] A. T. Murray, R. Davis, R. J. Stimson and L. Ferreira, Public transportation access, Transportation Research Part D: Transport and Environment, 3 (1998), 319-328.  doi: 10.1016/S1361-9209(98)00010-8. [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/oc-transpo-cancellations-broken-down-across-the-system-.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), 809-817.  doi: 10.1007/978-981-10-3551-7_65.
The $i$-th city bus route
Customer arrival rates of station 1 to 16 along route 1
Customer arrival rates of station 1 to 12 along route 2
Customer arrival rates of station 1 to 20 along route 3
Customer arrival rates of station 1 to 14 along route 4
Simulation result for each separate route over the whole day
Result for route 1 and 2 over the whole day
Parameters for Simulation
 Parameter Value Length of the $i$-th route $L_i$ $L_1$ = 24km, $L_2$ = 10km, $L_3$ = 22km, $L_4$ = 15km Total number of buses $M$ 10 Number of stations $s_i$ $s_1$ = 16, $s_2$ = 12, $s_3$ = 20, $s_4$ = 14 Average speed of city buses $v_i$ $v_1$ = 30, $v_2$ = 30, $v_3$ = 35, $v_4$ = 30 Coefficient of fuel cost $q_{i, k}$ $q_{1, k}$ = 20, $q_{2, k}$ = 20, $q_{3, k}$ = 15, $q_{4, k}$ = 20 $a_{1, 1} = a_{1, 2} = a_{1, 9} = a_{1, 10} = 30$ $a_{1, 3} = a_{1, 4} = a_{1, 11} = a_{1, 12} = 60$ $a_{1, 5} = a_{1, 6} = a_{1, 13} = a_{1, 14} = 70$ $a_{1, 7} = a_{1, 8} = a_{1, 15} = a_{1, 16} = 40$ $a_{2, 1} = a_{2, 2} = a_{2, 3} = 20$ $a_{2, 7} = a_{2, 8} = a_{2, 9} = 50$ $a_{2, 4} = a_{2, 5} = a_{2, 6} = 60$ Weight given to stations $a_{i, j}$ $a_{2, 10} = a_{2, 11} = a_{2, 12} = 30$ $a_{3, 1} = a_{3, 2} = a_{3, 11} = a_{3, 12} = 60$ $a_{3, 3} = a_{3, 4} = a_{3, 13} = a_{3, 14} = 80$ $a_{3, 5} = a_{3, 6} = a_{3, 15} = a_{3, 16} = 1000$ $a_{3, 7} = a_{3, 8} = a_{3, 17} = a_{3, 18} = 70$ $a_{3, 9} = a_{3, 10} = a_{3, 19} = a_{3, 20} = 50$ $a_{4, 1} = a_{4, 2} = a_{4, 13} = a_{4, 14} = 22$ $a_{4, 3} = a_{4, 4} = a_{4, 11} = a_{4, 12} = 52$ $a_{4, 5} = a_{4, 10} = 65$ $a_{4, 6} = a_{4, 7} = a_{4, 8} = a_{4, 9} = 35$ Ticket price $b$ 3 Time interval $\Delta$ 5mins
 Parameter Value Length of the $i$-th route $L_i$ $L_1$ = 24km, $L_2$ = 10km, $L_3$ = 22km, $L_4$ = 15km Total number of buses $M$ 10 Number of stations $s_i$ $s_1$ = 16, $s_2$ = 12, $s_3$ = 20, $s_4$ = 14 Average speed of city buses $v_i$ $v_1$ = 30, $v_2$ = 30, $v_3$ = 35, $v_4$ = 30 Coefficient of fuel cost $q_{i, k}$ $q_{1, k}$ = 20, $q_{2, k}$ = 20, $q_{3, k}$ = 15, $q_{4, k}$ = 20 $a_{1, 1} = a_{1, 2} = a_{1, 9} = a_{1, 10} = 30$ $a_{1, 3} = a_{1, 4} = a_{1, 11} = a_{1, 12} = 60$ $a_{1, 5} = a_{1, 6} = a_{1, 13} = a_{1, 14} = 70$ $a_{1, 7} = a_{1, 8} = a_{1, 15} = a_{1, 16} = 40$ $a_{2, 1} = a_{2, 2} = a_{2, 3} = 20$ $a_{2, 7} = a_{2, 8} = a_{2, 9} = 50$ $a_{2, 4} = a_{2, 5} = a_{2, 6} = 60$ Weight given to stations $a_{i, j}$ $a_{2, 10} = a_{2, 11} = a_{2, 12} = 30$ $a_{3, 1} = a_{3, 2} = a_{3, 11} = a_{3, 12} = 60$ $a_{3, 3} = a_{3, 4} = a_{3, 13} = a_{3, 14} = 80$ $a_{3, 5} = a_{3, 6} = a_{3, 15} = a_{3, 16} = 1000$ $a_{3, 7} = a_{3, 8} = a_{3, 17} = a_{3, 18} = 70$ $a_{3, 9} = a_{3, 10} = a_{3, 19} = a_{3, 20} = 50$ $a_{4, 1} = a_{4, 2} = a_{4, 13} = a_{4, 14} = 22$ $a_{4, 3} = a_{4, 4} = a_{4, 11} = a_{4, 12} = 52$ $a_{4, 5} = a_{4, 10} = 65$ $a_{4, 6} = a_{4, 7} = a_{4, 8} = a_{4, 9} = 35$ Ticket price $b$ 3 Time interval $\Delta$ 5mins
Simulation Result for the Network of City Bus Routes
 Time Optimal control $x^o$ 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 $x^o$ 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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