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Optimum management of the network of city bus routes based on a stochastic dynamic model

  • * Corresponding author: Shi'an Wang

    * Corresponding author: Shi'an Wang 
The authors are supported by NSERC grant A7101.
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  • 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.

    Mathematics Subject Classification: Primary: 93E20; Secondary: 90C30.

    Citation:

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  • Figure 1.  The $i$-th city bus route

    Figure 2.  Customer arrival rates of station 1 to 16 along route 1

    Figure 3.  Customer arrival rates of station 1 to 12 along route 2

    Figure 4.  Customer arrival rates of station 1 to 20 along route 3

    Figure 5.  Customer arrival rates of station 1 to 14 along route 4

    Figure 6.  Simulation result for each separate route over the whole day

    Figure 7.  Result for route 1 and 2 over the whole day

    Table 1.  Parameters for Simulation

    ParameterValue
    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
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    Table 2.  Simulation Result for the Network of City Bus Routes

    TimeOptimal 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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  • [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.
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    [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).
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