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A spatio-temporal deep learning model for short-term bike-sharing demand prediction

  • Bike-sharing systems are widely operated in many cities as green transportation means to solve the last mile problem and reduce traffic congestion. One of the critical challenges in operating high-quality bike-sharing systems is rebalancing bike stations from being full or empty. However, the complex characteristics of spatiotemporal dependency on usage demand may lead to difficulties for traditional statistical models in dealing with this complex relationship. To address this issue, we propose a graph-based neural network model to learn the representation of bike-sharing demand spatial-temporal graph. The model has the ability to use graph-structured data and takes both spatial- and temporal aspects into consideration. A case study about bike-sharing systems in Nanjing, a large city in China, is conducted based on the proposed method. The results show that the algorithm can predict short-term bike demand with relatively high accuracy and low computing time. The predicted errors for the hourly station level usage demand prediction are often within 20 bikes. The results provide helpful tools for short-term usage demand prediction of bike-sharing systems and other similar shared mobility systems.

    Citation: Ruo Jia, Richard Chamoun, Alexander Wallenbring, Masoomeh Advand, Shanchuan Yu, Yang Liu, Kun Gao. A spatio-temporal deep learning model for short-term bike-sharing demand prediction[J]. Electronic Research Archive, 2023, 31(2): 1031-1047. doi: 10.3934/era.2023051

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  • Bike-sharing systems are widely operated in many cities as green transportation means to solve the last mile problem and reduce traffic congestion. One of the critical challenges in operating high-quality bike-sharing systems is rebalancing bike stations from being full or empty. However, the complex characteristics of spatiotemporal dependency on usage demand may lead to difficulties for traditional statistical models in dealing with this complex relationship. To address this issue, we propose a graph-based neural network model to learn the representation of bike-sharing demand spatial-temporal graph. The model has the ability to use graph-structured data and takes both spatial- and temporal aspects into consideration. A case study about bike-sharing systems in Nanjing, a large city in China, is conducted based on the proposed method. The results show that the algorithm can predict short-term bike demand with relatively high accuracy and low computing time. The predicted errors for the hourly station level usage demand prediction are often within 20 bikes. The results provide helpful tools for short-term usage demand prediction of bike-sharing systems and other similar shared mobility systems.



    The vehicle routing problem (VRP) is a class of problems that entails finding an optimal set of routes for a fleet of vehicles to serve a set of customers. The objective of the VRP is to minimize vehicle routes costs while originating and terminating at a depot. The VRP was first proposed by Dantzig and Ramser [1] before being extended with different variants: the VRP with time window (VRPTW) [2,3,4,5,6,7], the capacitated VRP (CVRP) [8,9,10,11,12] and other VRPs [13,14,15,16,17]. In recent years, the VRPs have attracted much interest [18]. Exact solutions of the VRPs generally result in branch-and-price [13,19], branch-and-cut [20,21,22] and branch-cut-and-price algorithms [22,23]. Ben Ticha et al. [24,25,26] proposed well-performing branch-and-price algorithms for VRPs on multigraphs with multiple time-related attributes and a homogeneous vehicle fleet. In such approaches, the linear relaxation in each branch-and-bound node is by solved column generation, which has proved to be a powerful approach [9,27,28]. Column generation has been widely used to solve a variety of large mathematical programs such as vehicle routing and crew scheduling problems [3,29,30].

    Many authors have suggested solving a shortest path problem with resource constraints (SPPRC) introduced by Desrochers [31] as a multi-dimensional generalization of the shortest path problem with time windows. Resolutions methods and applications of the SPPRC have been extensively discussed in the literature [3,29,31,32,33]. These strategies range from exact methods to heuristics and meta-heuristics.

    Exact SPPRC resolution techniques usually use the dynamic programming which has a pseudopolynomial complexity. Desrochers and Soumis [32] proposed a label correcting reaching algorithm that extends the Ford-Bellman algorithm to take resource constraints into account. The algorithm has been shown to be successful for tight resource constraints. Feillet et al. [34] adapted the Desrochers algorithm to solve exactly the ESPPRC (SPPRC with elementary path) pricing problems in the context of the VRPTW. Since that, this algorithm has been the backbone of a number of algorithms based on column generation applied to several important problems such as vehicle routing and crew management [16,26]. Table 1 summarizes some reviewed literature that applied column-generation for VRPs as classified based on solution approaches used for solving the sub-problem (SPPRC or ESPPRC).

    Table 1.  Summary of some papers based on the sub-problem solution method.
    Authors Year Elementary Non-Elementary Class of VRP
    Desrosiers et al. [3] 1995 * VRPTW
    Kohl et al. [4] 1999 * VRPTW
    Feillet et al. [34] 2004 * VRPTW
    Irnich and Villeneuve [5] 2006 * VRPTW
    Righini and Salani [35] 2006 * VRPTW
    Nagih and Soumis [14] 2006 * Other VRPs
    Fukasawa et al. [10] 2006 * CVRP
    Chabrier [36] 2006 * VRPTW
    Feillet et al. [37] 2007 * VRPTW
    Tagmouti et al. [38] 2007 * VRPTW
    Jepsen et al. [39] 2008 * VRPTW
    Baldacci et al. [11] 2008 * * CVRP
    Righini and Salani [40] 2008 * VRPTW
    Desaulniers et al. [41] 2008 * VRPTW
    Qureshi et al. [42] 2009 * Other VRPs
    Baldacci et al. [7] 2010 * VRPTW
    Baldacci et al. [12] 2011 * * CVRP
    Bettinelli et al. [43] 2011 * Other VRPs
    Liberatore et al. [44] 2011 * VRPTW
    Dabia et al. [13] 2013 * * Other VRPs
    Duque et al. [45] 2015 * Other VRPs
    Fukasawa et al. [8] 2015 * CVRP
    Lozano et al. [46] 2016 * VRPTW
    Pecin et al. [28] 2017 * VRPTW
    Pecin et al. [9] 2017 * * CVRP
    Lera-Romero and Miranda-Bront [47] 2018 * Other VRPs
    Ben Ticha et al. [26] 2019 * VRPTW
    Himmich et al. [15] 2020 * Other VRPs
    Sadykov et al. [6] 2021 * VRPTW
    Behnke et al. [16] 2021 * Other VRPs
    Dalmeijer and Desaulniers [48] 2021 * Other VRPs
    Mathlouthi et al. [17] 2021 * * Other VRPs
    Taş [49] 2021 * Other VRPs

     | Show Table
    DownLoad: CSV

    The main drawback of exact resolution techniques is the computational time which increases with the number of resources, making them suitable for small-scale problems only. To speed up the computations and obtain solutions in a reasonable time, some authors proposed approximate, heuristics and meta-heuristics techniques [14,18,50,51,52,53,54,55]. These algorithms yield near-optimal solutions and are more suitable for real-life larger-scale applications (e.g., thousands of customers from, dozens of depots with numerous vehicles subject to a variety of constraints). Nagih and Soumis [14] proposed a basic heuristic to quickly produce feasible solutions. But, this approach has been limited to acyclic graphs and its speed has not been proved. Some authors have used simulated annealing [56], genetic algorithms [52,53], non-dominated sorting genetic algorithms [54] and simplified swarm optimization [55]. Meta-heuristic methods allows to avoid getting trapped in local optima but this comes at a price of slower convergence.

    In this paper, we propose a Lagrangian relaxation based algorithm to approximately solve the SPPRC for arbitrary, acyclic and cyclic graphs. In our approach, called "dominance on Lagrangian cost for the SPPRC" (DLC-SPPRC), a dominance is expressed on a subset of the resources while the rest of the resources are dualized in the objective. Optimized parameter update schemes are used to ensure better performances (descent steps, Lagrange multipliers, subgradients).

    The rest of the paper is organized as follows. Section 2 describes formally the SPPRC. Section 3 presents our approximate solution method that speeds up the computations while keeping a good approximate solution. In Section 4, we present an application to column generation and show the results obtained for various VRP datasets. The paper ends with the final conclusions.

    Consider a graph G=(V,A), where V=N{o,d} is a set of nodes N={v1,,vn} that would be visited from an origin o=v0 to a destination d=vn+1 and a set of arcs AV×V. We denote by R a set of resources of cardinality |R|. An arc (i,j)A costs cij and consumes a quantity trij0 of each resource rR. We suppose that the resources satisfy the triangle inequality. If (i,j)A, then the node i is called a predecessor of j and j is called a successor of i. A path is a finite sequence of nodes

    P=(vk0=v0,vk1,,vkm),where(vki,vki+1)A,i=0,,m1.

    To each node vj of P, we associate CPj which is the sum of the costs of its composite arcs up to vj. We denote by Tr,Pj the amount of resource rR used to reach the node vj. For the sake of simplification, we drop the index P from these notations and write Cj and Trj. The path P is said to be feasible if it satisfies the resource constraints

    Trki[arki,brki],rR,i{0,..,m}.

    A path P can also be represented by X=(xij)(i,j)A{0,1}, where xij=1 if the nodes vi and vj belong to P, and 0 otherwise.

    The SPPRC seeks a feasible path from the origin to the destination with a minimal cost. Similar to [3], one can formulate the SPPRC as follows:

    minimizeX(i,j)Acijxij (2.1)

    subject to

    (i,j)Axij(i,j)Axji=0,iV{o,d}, (2.2)
    (o,j)Axoj=1;(i,d)Axid=1, (2.3)
    xij(Tri+trijTrj)0,(i,j)A,rR, (2.4)
    arjTrjbrj,jV,rR, (2.5)
    xij{0,1},(i,j)A, (2.6)

    where X=(xij)i,j represents a path solution. Eqs (2.2) and (2.3) define the flow constraints on the graph; Eq (2.4) encodes the compatibility requirements between flow and time variables, and Eq (2.5) is the time windows constraint. If the problem is feasible, then there exists an optimal solution [14].

    To solve the SPPRC, labeling algorithms have been efficiently applied by many authors, e.g., Desrochers and Soumis [32], Dabia et al. [13] and Sun et al. [57,58]. Labels are defined for each node to encode partial paths, along with their cost and resource consumption. Each node receives labels and then extend them toward every possible successor node. The algorithm iteratively treats the nodes until no new labels are created.

    To limit the proliferation of labels and reduce the computational time, dominance rules are introduced. Dominance relation is a partial ordering applied to eliminate non optimal solutions and retain only non dominated labels. The number of retained labels and, thus, the computation time, increase with the number of resources.

    In the next section, we propose an algorithm to determine approximate solutions. The size of the search space is reduced by eliminating some labels with a heuristic dominance procedure.

    With each path Pi arriving to a node vi, we associate a label (or state) vector

    Li=(Ci,T1i;;T|R|i). (3.1)

    The path associated to a label L is denoted by XL. The path P is extended to each successor vj with

    Cj=Ci+cijandTrj=max{arj,Tri+trij},rR. (3.2)

    Let Pi and Pi be two feasible paths arriving to a node vi. We say that Pi is dominated by Pi and we write PiPi if the label vector of Pi is component-wise less than the one of Pi. On the other hand, we say that PiPi if the first component of Pi is smaller than the one of Pi or by comparing recursively the following ones if the first components are equal. At least one of the inequalities should be strict. This defines a partial order relation.

    To speed up the resolution of the SPPRC (2.1)–(2.6), we propose to relax a subset of resources R1R and apply dominance on the remaining resources R2=RR1 only. This is equivalent to consider Eqs (2.4) and (2.5) on R2 and replace Eq (2.1) by the equation

    minimizeXL(λ,X), (1)

    where the Lagrangian function L is defined by

    L(λ,X)=(i,j)Axij(cij+rR1λri,j(max(aj,Tri+trij)brj)). (3.3)

    Since the flow is supported by at most one incoming arc at each node j from its predecessors i, the multipliers λri,j=λrj are independent of i. The vector of all multipliers λj= (λrj)rR1 can be obtained by solving the following maximization problem

    {maxλminXL(λ,X),s.t.λrj0,rR1. (3.4)

    To solve this optimization sub-problem, one can use a projected sub-gradient algorithm which proved to be efficient in practice. The updating scheme is

    λk+1j=max{λkj+τkjgkj,0} (3.5)

    where gkjL(λ,X) is a subgradient vector, τkj>0 is a step size, and the max operator is taken for each element. We choose

    gkj=max(aj,Tri+trij)brj, (3.6)
    τkj=θk(ZUBL(λk,X))||gkj||2, (3.7)

    where ZUB is the best-known-smallest (upper)-bound of the solution of the problem described by (2.1)–(2.6) and θk(0,2] is a scalar. One iteration for the estimation of λ proved to be sufficient (precision and speed) when integrated into column generation.

    The dominance at each node j corresponds to the determination of the Pareto optima of the multicriteria problem applied to a set of labels. With the adopted Lagrangian approach, the label vector becomes

    Lj=(Cj+r1R1λr1j(Tr1jbr1j);Tr2j,r2R2;). (3.8)

    When the obtained final solution is not feasible (due to relaxation), we need to solve again the relaxed SPPRC by dominating on the Lagrangian cost determined in the first phase and on the resources R2 and by checking the resource windows in the original space R.

    Assuming that all predecessors of the node jN have been considered, the dominance at the node j can be interpreted as the determination of the Pareto optima for the multicriteria problem of |R2|+1 functions:

    min(i;(i,j)A)(Ci+cij  +r1R1λr1j(Tr1jbr1j); max{ar2j,(Tr2i+tr2ij)},r2R2;)              st.Tri+trijbrj,rR. (3.9)

    First, we introduce the following notations.

    Ei: List of labels at a node vi.

    Fi: The list of paths associated to the labels of Ei.

    Si: Set of the successors of the node vi indices.

    Q: List of unprocessed nodes indices.

    Pareto(L,Ej): Add a new label L to a set of non-dominated labels Ej while keeping the non-dominance property.

    ZL is the cost associated with a label L (first component of Eq (3.9)).

    Our improved label-correcting procedure is detailed in Algorithms 1 and 2.

    Algorithm 1: Modified label-correcting algorithm
    input  : An arbitrary directed graph G=(V,A), unnecessarily acyclic; R a set of constraints; R1R subset of constraints to be included in the objective function; Lagrange multipliers λiR|R1|+ where viV{o}; R2 a set of resources
    output  : All non-dominated paths from the source to the destination En+1, and the associated list of paths Fn+1.
    // Initialization;
    R2=RR1;
    for viV do
     | Ei
    end
    E0{(0,0,,0)}; // The size of the labels is: |Eo|=|R2|+1
    Q{0}
    // Main loop;
    while: Q do

    Algorithm 2: DLC-SPPRC: Dominance on Lagrangian cost for the SPPRC.
    Input  : An arbitrary directed graph G=(V,A), unnecessarily acyclic; R=R1R2 a set of resources constraints composed of two types.
    Output  : ˉE, all non-dominated feasible paths from the source to the destination node d.
    // Parameters choice;
    Choose θ(0, 2].
    kmax=10.
    // Initialization;
    for viV{o} do
    λ(0)i=(0,,0) a vector of size |R1|.
    // Initial value of the Lagrangian corresponding to the solution of (3.4)
    ˉZmax.
    // Choose an initial solution of the problem (2.1)–(2.6).
    X0=(x0ij)ij.
    // Compute the objective function value (1) (upper bound).
    Zu(i,j)Acijx0ij.
    ˉE.
    k=0.
    // Step 1;
    while (k<kmax) do

    To solve the VRPTW, we use the Dantzig-Wolfe decomposition which defines K independent sub-problems and a global master problem. Applying a column generation, we alternatively solve the master problem and the K sub-problems. We addressed the VRPTW instances with a column generation approach by using the SPPRC as a sub-problem.

    We used the Solomon data sets [59] and Homberger 200 customer instances. Each instance contains customers locations, resources (two for each costumer, i.e., |R|=2) and constraints. These data sets are classified in three categories: the r-instances (customers are located randomly), the c-instances (customers are located in clusters) and the rc-instances (mixed random and clustered structures). Furthermore, each family of instances is divided into two types. The first type (r1xx, c1xx, rc1xx) has a short scheduling horizon, i.e., small time windows, that allows only a few customers per route (approximately 5 to 10). The second type (r2xx, c2xx, rc2xx) has a long scheduling horizon permitting many customers (more than 30) to be serviced by the same vehicle. We applied our approximate DLC-SPPRC algorithm and the classical SPPRC [32].

    We implemented our algorithm using Java programming language. For the simulation, we used a CPU Intel Core i9-9900KF (8 cores), 3.60 GHz, RAM 32 GB, running under Windows 10 (64 bit). For the SPPRC, we implemented the Desrochers and Soumis algorithm [32]. Linear programs for restricted master problems are solved with ILOG CPLEX 20.1.

    Tables 25 report the iteration number (Ni), the lower bound (Lbi), the computational time in seconds (Ti) and the number of generated columns (Ci), where i=1 for the SPPRC and i=2 for the DLC-SPPRC. A gap is computed as

    Gap=100×Lb1Lb2Lb1. (4.1)
    Table 2.  Comparison of two approaches for solving the VRPTW for Solomon's instances with 25 customers: approximate DLC-SPPRC algorithm (2) and the classical SPPRC [32].
    SPPRC DLC-SPPRC Comparison
    Instance Lb1 T1 N1 C1 Lb2 T2 N2 C2 Gap T1/T2
    c101 191.3 0.1163 33 579 191.3 0.1045 69 396 0.00 1.11
    c102 189.2 0.285 40 1452 189.175 0.2817 82 976 0.01 1.01
    c103 187.857 0.4005 43 1578 183.214 0.3721 66 807 2.47 1.08
    c104 184.339 0.4012 45 1896 183.457 0.2618 44 787 0.48 1.53
    c105 191.3 0.0651 36 718 191.3 0.0319 48 368 0.00 2.04
    c106 191.3 0.0438 38 698 191.3 0.0372 73 386 0.00 1.18
    c107 191.3 0.0476 27 749 191.3 0.0397 48 416 0.00 1.20
    c108 187.84 0.1046 30 1090 185.809 0.0734 41 564 1.08 1.43
    c109 181.927 0.1442 29 1132 185.314 0.0752 32 578 -1.86 1.92
    c201 214.7 0.2174 89 2240 214.7 0.1531 237 1019 0.00 1.42
    c202 214.7 0.3528 86 2637 214.7 0.2835 175 1266 0.00 1.24
    c203 213.775 2.0617 76 3820 213.775 1.0133 153 1562 0.00 2.03
    c204 207.174 4.3672 80 4880 207.172 2.0133 125 2083 0.00 2.17
    c205 196.525 0.2943 43 3064 196.525 0.1322 88 1089 0.00 2.23
    c206 194.018 0.5541 48 3472 194.015 0.2823 85 1875 0.00 1.96
    c207 200.443 1.6901 67 4840 200.442 0.5865 84 2237 0.00 2.88
    c208 183.863 1.0371 50 4124 183.861 0.3695 69 1751 0.00 2.81
    r101 617.1 0.0065 10 108 617.1 0.0053 17 73 0.00 1.23
    r102 546.333 0.0172 13 314 546.333 0.017 29 201 0.00 1.01
    r103 454.067 0.0464 22 538 454.067 0.0342 35 280 0.00 1.36
    r104 414.85 0.0715 25 735 414.85 0.0544 39 374 0.00 1.31
    r105 530.5 0.014 19 254 530.5 0.011 25 145 0.00 1.27
    r106 457.3 0.0439 22 543 457.3 0.023 27 265 0.00 1.91
    r107 415.125 0.0573 25 618 415.125 0.0446 35 329 0.00 1.28
    r108 389.424 0.0846 22 796 389.424 0.0623 32 422 0.00 1.36
    r109 439.425 0.0247 19 341 439.425 0.0215 30 243 0.00 1.15
    r110 419.072 0.0402 19 502 419.072 0.034 31 331 0.00 1.18
    r111 412.815 0.044 19 554 412.815 0.036 31 302 0.00 1.22
    r112 365.03 0.0994 27 818 365.03 0.07 35 464 0.00 1.42
    r201 448.5 0.0706 16 877 448.5 0.0468 43 510 0.00 1.51
    r202 374.092 0.1505 28 1479 374.092 0 122 53 754 0.00 1.23
    r203 337.517 0.4222 37 2044 337.51 0.2901 62 897 0.00 1.46
    r204 303.991 1.737 51 3900 303.993 0.615 74 1454 0.00 2.82
    r205 365.475 0.2141 31 1744 365.475 0.1885 74 1043 0.00 1.14
    r206 317.984 1.2275 42 3775 317.983 0.3397 59 1119 0.00 3.61
    r207 309.623 3.0179 53 3413 309.612 0.7511 67 1377 0.00 4.02
    r208 291.203 51.794 88 9718 291.217 3.4682 100 2952 0.00 14.93
    r209 327.256 0.6002 34 2590 327.258 0.2808 53 1133 0.00 2.14
    r210 340.505 0.3063 31 2033 340.505 0.182 49 875 0.00 1.68
    r211 299.456 7.8134 57 5667 299.46 1.5388 71 1590 0.00 5.08
    rc101 390.15 0.0187 18 284 390.15 0.0178 31 203 0.00 1.05
    rc102 347.08 0.0591 20 467 347.08 0.0489 32 284 0.00 1.21
    rc103 313.976 0.1227 21 751 309.907 0.08 25 330 1.30 1.53
    rc104 287.539 0.255 29 888 292.458 0.1133 22 326 -1.71 2.25
    rc105 408.525 0.0433 17 516 408.525 0.0299 32 202 0.00 1.45
    rc106 314.274 0.0795 23 660 306.044 0.0618 33 343 2.62 1.29
    rc107 281.289 0.258 31 1043 274.389 0.1383 33 460 2.45 1.87
    rc108 270.573 0.4552 35 939 250.412 0.2489 40 596 7.45 1.83
    rc201 315.853 0.1049 30 1325 315.853 0.0673 59 464 0.00 1.56
    rc202 233.586 7.4873 57 3545 233.588 1.404 68 1043 0.00 5.33
    rc203 178.264 31.379 93 8566 178.283 2.9895 77 1756 -0.01 10.50
    rc204 153.909 90.1709 141 12217 153.554 4.7891 107 2803 0.23 18.83
    rc205 258.915 0.3965 30 1960 258.915 0.1849 50 562 0.00 2.14
    rc206 188.287 4.0061 71 6013 188.249 0.7239 70 1096 0.02 5.53
    rc207 169.161 11.7082 79 7492 169.19 0.9203 68 1812 -0.02 12.72
    rc208 138.287 145.445 163 16490 139.544 4.3513 121 3469 -0.91 33.43

     | Show Table
    DownLoad: CSV
    Table 3.  Comparison of two approaches for solving the VRPTW for Solomon's instances with 50 customers: approximate DLC-SPPRC algorithm (2) and the classical SPPRC [32].
    SPPRC DLC-SPPRC Comparison
    Instance Lb1 T1 N1 C1 Lb2 T2 N2 C2 Gap T1/T2
    c101 362.4 0.4482 60 1527 362.4 0.4382 155 1052 0.00 1.02
    c102 360.25 1.2188 83 2922 360.275 0.9699 136 1792 -0.01 1.26
    c103 360.25 2.5992 106 3599 353.341 1.7055 182 2264 1.92 1.52
    c104 352.266 6.1074 119 5660 346.952 2.8453 127 2480 1.51 2.15
    c105 362.4 0.5268 81 2172 361.2 0.3185 166 1259 0.33 1.65
    c106 362.4 0.3838 88 2169 362.4 0.2125 125 973 0.00 1.81
    c107 362.4 0.4258 73 2435 361.2 0.3272 150 1433 0.33 1.30
    c108 359.81 0.6627 62 2595 356.31 0.539 106 1469 0.97 1.23
    c109 354.316 0.9185 57 2784 340.938 0.6901 94 1528 3.78 1.33
    c201 360.2 6.6625 213 13828 360.2 1.6169 489 3335 0.00 4.12
    c202 360.2 8.1671 200 12216 360.2 2.5508 259 3440 0.00 3.20
    c203 359.8 38.6973 252 15108 359.8 14.1948 333 5850 0.00 2.73
    c204 347.404 114.6872 243 18375 347.679 35.7323 320 8201 -0.08 3.21
    c205 341.763 9.5662 145 13681 341.769 3.5832 223 4713 0.00 2.67
    c206 338.519 19.4161 180 15798 338.379 4.8996 222 5187 0.04 3.96
    c207 348.631 38.0489 203 18358 348.613 7.9878 235 7176 0.01 4.76
    c208 331.276 20.6723 180 15147 331.278 3.7519 174 4892 0.00 5.51
    r101 1043.367 0.0447 23 463 1043.367 0.0421 44 276 0.00 1.06
    r102 909 0.1435 29 987 909 0.1272 54 534 0.00 1.13
    r103 756.117 0.3896 43 1468 756.117 0.293 59 790 0.00 1.33
    r104 608.521 1.2382 66 2600 608.491 0.8123 87 1255 0.00 1.52
    r105 890.187 0.0958 26 821 890.187 0.0834 43 395 0.00 1.15
    r106 789.433 0.3278 38 1255 789.433 0.2506 59 690 0.00 1.31
    r107 697.767 0.504 46 1793 697.767 0.3672 60 855 0.00 1.37
    r108 578.482 1.6057 68 2761 578.482 1.2553 106 1613 0.00 1.28
    r109 727.515 0.2537 36 1403 727.515 0.1712 53 614 0.00 1.48
    r110 675.457 0.4694 42 1578 675.246 0.3234 59 761 0.03 1.45
    r111 658.752 0.592 47 1927 658.752 0.4077 65 913 0.00 1.45
    r112 582.715 1.3422 63 2502 582.611 0.7894 82 1170 0.02 1.70
    r201 754.098 0.5013 42 2332 754.098 0.4006 91 1243 0.00 1.25
    r202 637.556 1.9978 62 4231 637.562 1.1956 106 1717 0.00 1.67
    r203 538.526 5.4656 78 4791 538.529 2.7795 140 2696 0.00 1.97
    r204 441.005 329.0178 229 23629 441.021 28.7788 272 7887 0.00 11.43
    r205 595.538 2.0053 66 4505 595.55 1.0479 115 1970 0.00 1.91
    r206 530.527 8.3613 87 7378 530.512 3.2432 142 3182 0.00 2.58
    r207 467.282 34.0113 145 11323 467.287 7.7842 177 4139 0.00 4.37
    r208 423.781 2234.4085 302 34624 423.799 120.1348 309 9984 0.00 18.60
    r209 535.282 8.0839 85 6534 535.284 2.9226 139 2811 0.00 2.77
    r210 532.1 5.5001 86 6296 532.096 2.3795 135 2803 0.00 2.31
    r211 457.426 31.1272 129 10024 457.43 6.4031 172 4082 0.00 4.86
    rc101 826.613 0.1158 31 716 826.613 0.0998 47 428 0.00 1.16
    rc102 706.606 0.4957 47 1160 706.556 0.3879 62 652 0.01 1.28
    rc103 612.24 1.2763 58 1756 605.145 0.9114 77 1008 1.16 1.40
    rc104 524.119 4.7657 88 3146 511.583 1.7867 86 1417 2.39 2.67
    rc105 746.314 0.2826 38 1091 745.928 0.2183 52 625 0.05 1.29
    rc106 633.228 0.5801 43 1407 628.517 0.4113 53 741 0.74 1.41
    rc107 570.669 2.1353 73 2734 561.863 1.1098 87 1273 1.54 1.92
    rc108 525.123 3.43 79 2630 501.695 1.984 96 1477 4.46 1.73
    rc201 530.505 1.1744 64 3001 530.505 0.5838 79 1015 0.00 2.01
    rc202 416.517 25.8833 132 8997 416.518 5.6157 147 2801 0.00 4.61
    rc203 326.285 216.7017 264 20712 326.294 15.1477 184 5153 0.00 14.31
    rc204 263.83 18,964.6011 431 47037 264.301 70.3698 333 12067 -0.18 269.50
    rc205 481.609 6.5427 96 7435 481.621 2.5118 126 2633 0.00 2.60
    rc206 363.888 56.1137 189 15075 363.892 7.3344 173 3830 0.00 7.65
    rc207 333.071 282.0786 239 23198 333.071 18.2862 196 5016 0.00 15.43
    rc208 265.107 1974.0373 456 41459 264.912 86.3844 399 10,952 0.07 22.85

     | Show Table
    DownLoad: CSV
    Table 4.  Comparison of two approaches for solving the VRPTW for Solomon's instances with 100 customers: approximate DLC-SPPRC algorithm (2) and the classical SPPRC [32].
    SPPRC DLC-SPPRC Comparison
    Instance Lb1 T1 N1 C1 Lb2 T2 N2 C2 Gap T1/T2
    c101 827.3 2.2196 122 3553 827.3 2.4001 383 2852 0.00 0.92
    c102 827.3 9.7928 191 7359 820.3 7.3079 277 4279 0.85 1.34
    c103 826.3 21.8133 223 9480 817.879 17.6085 344 5663 1.02 1.24
    c104 821.597 55.5135 327 13285 793.653 47.0154 544 8789 3.40 1.18
    c105 827.3 2.9878 142 4392 821.2 2.9093 308 3648 0.74 1.03
    c106 827.3 4.0281 134 4928 827.4 3.5591 258 3438 -0.01 1.13
    c107 827.3 3.0508 132 4626 819.6 4.128 331 4435 0.93 0.74
    c108 817.352 6.5019 160 6311 801.696 5.2622 219 3318 1.92 1.24
    c109 809.265 12.6714 231 9430 778.527 7.4906 256 4669 3.80 1.69
    c201 589.1 44.9491 418 28845 589.1 11.6671 528 6526 0.00 3.85
    c202 589.1 179.6573 552 44694 589.1 61.4668 722 12,701 0.00 2.92
    c203 585.767 1223.024 995 75,865 585.767 446.2332 1488 32,852 0.00 2.74
    c204 582.383 4893.6566 1630 136,556 582.226 990.1879 2056 58,070 0.03 4.94
    c205 582.369 151.9166 472 42,390 582.363 44.6547 617 15,218 0.00 3.40
    c206 575.993 346.5503 614 56,564 575.845 61.8254 652 16,738 0.03 5.61
    c207 570.524 367.8769 544 52,839 570.52 74.188 691 18,018 0.00 4.96
    c208 570.255 423.4152 635 59,277 570.278 65.6802 614 17,867 0.00 6.45
    r101 1631.15 0.416 44 1458 1631.15 0.3951 92 829 0.00 1.05
    r102 1466.6 1.5553 63 2627 1466.6 1.5157 126 1469 0.00 1.03
    r103 1203.241 5.3805 104 4549 1203.2 3.7518 154 2123 0.00 1.43
    r104 937.062 28.4824 190 9368 936.742 22.0167 310 4715 0.03 1.29
    r105 1341.194 1.2316 64 2714 1341.194 0.8896 92 1270 0.00 1.38
    r106 1212.355 6.0697 116 4719 1212.312 3.0527 143 2155 0.00 1.99
    r107 1036.956 13.5413 148 6645 1037.263 7.2842 192 3094 -0.03 1.86
    r108 891.646 44.4008 249 12269 890.074 24.6322 336 5525 0.18 1.80
    r109 1097.456 4.8182 101 4941 1097.343 2.2239 114 1915 0.01 2.17
    r110 1021.333 11.5028 135 6226 1021.138 5.6338 171 2783 0.02 2.04
    r111 1006.032 12.7601 153 7009 1005.671 7.4647 200 3273 0.04 1.71
    r112 892.576 35.3526 201 9372 887.714 18.0717 283 4819 0.54 1.96
    r201 1080.749 9.8496 119 9102 1080.771 6.4248 231 3947 0.00 1.53
    r202 933.446 93.7052 241 15273 933.458 23.7589 353 6624 0.00 3.94
    r203 756.739 451.4435 402 28,667 756.731 87.1933 565 11,958 0.00 5.18
    r204 640.238 7638.2746 638 56,850 640.272 559.2558 863 22660 -0.01 13.66
    r205 838.773 76.568 242 19,494 838.772 29.6386 372 8314 0.00 2.58
    r206 749.068 474.0996 370 28,002 749.066 84.8317 538 12161 0.00 5.59
    r207 668.711 3609.1966 568 50,978 668.711 281.5144 703 19,200 0.00 12.82
    r208 610.278 1197.6882 1001 31,544 0.00
    r209 750.455 445.0778 292 28162 750.454 82.0649 484 11,179 0.00 5.42
    r210 753.985 205.3728 309 23,599 753.993 63.3126 494 11,025 0.00 3.24
    r211 650.834 1793.735 573 47,048 650.845 167.9773 741 17,532 0.00 10.68
    rc101 1567.449 1.0734 61 2250 1567.282 0.8967 92 1159 0.01 1.20
    rc102 1380.209 2.9682 84 3319 1378.326 2.6881 124 1776 0.14 1.10
    rc103 1170.318 12.2669 155 5901 1164.407 7.3091 177 2656 0.51 1.68
    rc104 1052.551 38.1889 190 8215 1027.508 19.5054 250 4057 2.38 1.96
    rc105 1453.894 2.213 72 2938 1453.147 1.6459 105 1565 0.05 1.34
    rc106 1248.958 4.2376 97 3982 1242.824 2.694 124 1847 0.49 1.57
    rc107 1117.371 12.9665 146 5703 1096.177 7.1789 179 2847 1.90 1.81
    rc108 1035.93 32.3093 198 7553 1019.784 13.1855 225 3831 1.56 2.45
    rc201 1107.012 13.2285 149 9682 1107.011 6.554 227 3656 0.00 2.02
    rc202 880.343 127.7105 265 1, 7952 880.329 29.5938 339 6574 0.00 4.32
    rc203 693.53 902.8656 475 3, 5439 693.1 146.6249 591 13829 0.06 6.16
    rc204 607.663 14,066.2658 815 7, 8145 606.758 587.2997 854 25540 0.15 23.95
    rc205 967.105 59.4491 230 15595 967.097 19.7616 304 5936 0.00 3.01
    rc206 852.167 130.5895 305 24,025 852.178 30.1325 348 7607 0.00 4.33
    rc207 767.951 567.5535 417 29,098 767.982 80.8772 456 10,815 0.00 7.02
    rc208 627.276 3223.8542 638 44904 627.155 223.5683 726 17,074 0.02 14.42

     | Show Table
    DownLoad: CSV
    Table 5.  Comparison of two approaches for solving the VRPTW for Homberger's instances with 200 customers: approximate DLC-SPPRC algorithm (2) and the classical SPPRC [32].
    SPPRC DLC-SPPRC Comparison
    Instance Lb1 T1 N1 C1 Lb2 T2 N2 C2 Gap T1/T2
    c1_2_1 2698.6 21.6988 258 9936 2694.3 18.046 422 6025 0.2 1.20
    c1_2_2 2681.19 268.0735 636 19,085 2674.933 136.1003 732 10,144 0.2 1.97
    c1_2_3 2619.191 1091.5661 855 25,103 2718.196 364.9952 940 14,092 -3.8 2.99
    c1_2_4 2504.595 2681.358 1096 33,006 3018.105 297.099 1045 15,950 -20.5 9.03
    c1_2_5 2694.9 40.977 347 13,060 2693.35 30.2351 490 8212 0.1 1.36
    c1_2_6 2694.9 56.4944 354 13,018 2691.1 39.2523 481 7749 0.1 1.44
    c1_2_7 2694.9 57.3833 404 14,065 2690.6 40.9054 513 9006 0.2 1.40
    c1_2_8 2595.243 157.6131 506 17311 2570.971 65.4297 458 8383 0.9 2.41
    c1_2_9 2512.459 336.6065 662 21,390 2468.483 134.5933 667 11,797 1.8 2.50
    c1_2_10 2456.881 849.7886 743 23,092 2418.042 273.4165 829 14281 1.6 3.11
    c2_2_1 1915.035 586.6113 1042 70,794 1915.035 142.9373 1122 17930 0.0 4.10
    c2_2_2 1830.676 11,210.2029 2147 133,228 1830.262 3136.7113 2505 41,482 0.0 3.57
    c2_2_3 1703.946 32,211.3313 2903 168,434 1704.568 7460.8854 3318 55,403 0.0 4.32
    c2_2_4 1556.237 134,817.7847 4424 294,220 1555.633 27,365.9701 5969 132,146 0.0 4.93
    c2_2_5 1759.722 1413.2772 997 67,540 1759.242 368.3524 1098 22,931 0.0 3.84
    c2_2_6 1681.695 3003.6015 1155 79,219 1681.713 742.8865 1250 29,383 0.0 4.04
    c2_2_7 1709.103 3694.6209 1506 100,137 1709.081 906.8754 1533 34,220 0.0 4.07
    c2_2_8 1626.857 7029.1749 1330 98,417 1626.946 1292.0355 1445 37,099 0.0 5.44
    c2_2_9 1635.193 11,642.7703 1584 108,311 1634.903 2111.9977 1840 43,214 0.0 5.51
    c2_2_10 1572.073 13,937.5185 1602 113,454 1571.286 2543.9178 1752 46,140 0.1 5.48
    r1_2_1 4654.913 26.9629 199 10,342 4651.425 16.4504 363 4431 0.1 1.64
    r1_2_2 3905.961 338.0128 445 22,615 3908.449 110.3577 581 8513 -0.1 3.06
    r1_2_3 3282.964 2155.3814 676 31,966 3528.17 216.1976 859 12,419 -7.5 9.97
    r1_2_4 2925.629 4974.6947 1081 56,343 3749.5 83.8418 678 11,857 -28.2 59.33
    r1_2_5 3978.831 68.9078 261 14,902 3945.239 26.7507 321 5028 0.8 2.58
    r1_2_6 3383.356 605.1315 458 23,455 3442.907 110.0345 539 8589 -1.8 5.50
    r1_2_7 2971.763 2556.5618 672 31,896 3462.465 128.5044 654 10,654 -16.5 19.89
    r1_2_8 2777.372 5019.4831 1357 74,400 3701.82 135.8771 864 15,700 -33.3 36.94
    r1_2_9 3592.027 134.3127 300 17795 3594.651 35.4428 310 5635 -0.1 3.79
    r1_2_10 3080.993 611.0459 529 28032 3149.732 86.7837 544 9846 -2.2 7.04
    r2_2_1 3284.637 252.7638 500 38,393 3284.626 116.7495 803 12,874 0.0 2.17
    r2_2_2 2438.04 5482.703 1069 82,918 2438.042 964.8113 1512 26,275 0.0 5.68
    r2_2_3 1851.098 10,488.4304 3301 60,935
    r2_2_4 2181.11 4187.5913 5641 134,920
    r2_2_5 2702.26 1143.0031 784 65,349 2702.254 314.7569 1007 20,114 0.0 3.63
    r2_2_6 2051.403 4754.4525 2050 41,661
    r2_2_7 1781.771 6469.1145 3047 68,141
    r2_2_8 1524.816 25,399.6232 6141 151,513
    r2_2_9 2372.809 4375.8383 1069 92,700 2372.895 872.3769 1425 28,674 0.0 5.02
    r2_2_10 2057.437 51,478.3753 1533 135,588 2054.675 2828.0145 1790 35,968 0.1 18.20
    rc1_2_1 3408.547 77.1215 254 14,080 3335.566 28.9713 251 4772 2.1 2.66
    rc1_2_2 3079.317 701.3758 477 22,792 3064.343 170.516 610 10,268 0.5 4.11
    rc1_2_3 2851.48 2135.8783 649 31,119 3171.093 203.1931 745 12,443 -11.2 10.51
    rc1_2_4 2704.62 3337.8092 1016 44,426 3499.367 94.3266 670 11,472 -29.4 35.39
    rc1_2_5 3175.998 371.7829 291 18201 3040.87 108.5267 353 7519 4.3 3.43
    rc1_2_6 3137.163 276.9911 296 18,063 3060.842 69.2983 308 6950 2.4 4.00
    rc1_2_7 3002.62 686.0828 351 22,094 2798.384 225.8202 458 9610 6.8 3.04
    rc1_2_8 2889.813 982.34 453 26,034 3005.031 143.7637 526 11,799 -4.0 6.83
    rc1_2_9 2870.361 985.5621 427 25,897 2906.32 161.3079 474 10,730 -1.3 6.11
    rc1_2_10 2797.848 1378.7379 516 30,122 2765.18 261.1241 658 14,439 1.2 5.28
    rc2_2_1 2418.139 1220.3385 636 59,964 2418.217 408.3209 1094 16,622 0.0 2.99
    rc2_2_2 1910.279 4305.9053 2020 35,924
    rc2_2_3 2248.714 1358.3973 1925 42,749
    rc2_2_4 2010.182 12,228.8797 9810 314,665
    rc2_2_5 2159.079 2153.8087 1429 40,285
    rc2_2_6 2085.187 2778.6479 1376 35,622
    rc2_2_7 1927.925 4422.8466 1680 51,136
    rc2_2_8 1902.738 3276.8221 1726 63,367
    rc2_2_9 1860.267 3542.0999 1719 64,506
    rc2_2_10 1875.748 3580.6746 2317 86,154

     | Show Table
    DownLoad: CSV

    Figure 1 shows the total number of generated columns.

    Figure 1.  Number of generated columns for the Solomon and Homberger instances.

    For most of the cases, we obtained the optimal solutions in reasonable times. As in [34], the column generation process is initiated with an adaptation of the Clarke and Wright algorithm [60]. The sub-problem is stopped at each iteration when 500 labels have been extended to the depot with a negative cost.

    From Tables 25, we can draw the following conclusions. Our algorithm was faster in all cases (228). In 15 instances (7%), the SPPRC algorithm could not obtain a solution in reasonable time (more than 10 hours for one iteration). Our algorithm was 10 times faster in 37 cases (16% of the cases) and 1.5 times faster in 166 case (73% of the cases). On average, we achieved a gain of 564% of the computational time. For the precision, in 188 cases (82%), we obtained similar precisions (a gap less than 1%) whereas the difference was more than 1% only in 25 cases (11%) and greater than 5% only in 2 cases (0.88%).

    This paper proposes an approximate solution algorithm to solve the SPPRC, which relies on a Lagrangian relaxation. Our method applies to both acyclic and cyclic graphs. Dominance is applied on a subset of the resources only. Optimized parameters updates schemes are used to ensure fast convergence. When applied to the VRP, our approach offers a good compromise between precision and computational time, and thus, can be applied to large-scale practical problems. In the future, we plan to apply our algorithm to solve various problems such as branch-and-price, branch-and-cut and branch-and-price-and-cut. In addition, we will explore more advanced optimization algorithms (heuristics, meta-heuristics, etc.) that have been successfully applied in other domains, such as, online learning, scheduling, multi-objective optimization, transportation, medicine, and data classification.

    The authors declare that there is no conflicts of interest.



    [1] K. Gao, Y. Yang, X. Qu, Diverging effects of subjective prospect values of uncertain time and money, Commun. Transp. Res., 1 (2021), 100007. https://doi.org/10.1016/j.commtr.2021.100007 doi: 10.1016/j.commtr.2021.100007
    [2] J. Ke, X. Qin, H. Yang, Z. Zheng, Z. Zhu, J. Ye, Predicting origin-destination ride-sourcing demand with a spatio-temporal encoder-decoder residual multi-graph convolutional network, Transp. Res. Part C Emerging Technol., 122 (2021), 102858. https://doi.org/10.1016/j.trc.2020.102858 doi: 10.1016/j.trc.2020.102858
    [3] J. G. Jin, H. Nieto, L. Lu, Robust bike-sharing stations allocation and path network design: a two-stage stochastic programming model, Transp. Lett., 12 (2020), 682–691. https://doi.org/10.1080/19427867.2019.1691299 doi: 10.1080/19427867.2019.1691299
    [4] H. I. Ashqar, M. Elhenawy, H. A. Rakha, M. Almannaa, L. House, Network and station-level bike-sharing system prediction: a San Francisco bay area case study, J. Intell. Transp. Syst., 26 (2022), 602–612. https://doi.org/10.1080/15472450.2021.1948412 doi: 10.1080/15472450.2021.1948412
    [5] G. E. Box, D. A. Pierce, Distribution of residual autocorrelations in autoregressive-integrated moving average time series models, J. Am. Stat. Assoc., 65 (1970), 1509–1526. https://doi.org/10.1080/01621459.1970.10481180 doi: 10.1080/01621459.1970.10481180
    [6] H. Drucker, C. J. Burges, L. Kaufman, A. Smola, V. Vapnik, Support vector regression machines, Adv. Neural Inf. Process. Syst., 9 (1996), 155–161.
    [7] K. Davoian, W. M. Lippe, Including phenotype information in mutation to evolve artificial neural networks, in International Joint Conference on Neural Networks, 2007, 2782–2787. https://doi.org/10.1109/IJCNN.2007.4371400
    [8] Y. Wang, B. Chaib-Draa, A KNN based Kalman filter Gaussian process regression, in Twenty-Third International Joint Conference on Artificial Intelligence, 2013, 1771–1777. https://dl.acm.org/doi/10.5555/2540128.2540382
    [9] U. Johansson, H. Boström, T. Löfström, H. Linusson, Regression conformal prediction with random forests, Mach. Learn., 97 (2014), 155–176. https://doi.org/10.1007/s10994-014-5453-0 doi: 10.1007/s10994-014-5453-0
    [10] X. Li, R. Bai, Freight vehicle travel time prediction using gradient boosting regression tree, in 2016 15th IEEE International Conference on Machine Learning and Applications (ICMLA), 2016, 1010–1015. https://doi.org/10.1109/ICMLA.2016.0182
    [11] D. E. Rumelhart, G. E. Hinton, R. J. Williams, Learning representations by back-propagating errors, Nature, 323 (1986), 533–536. https://doi.org/10.1038/323533a0 doi: 10.1038/323533a0
    [12] K. Cho, B. Van Merriënboer, D. Bahdanau, Y. Bengio, On the properties of neural machine translation: Encoder-decoder approaches, arXiv preprint, 2014, arXiv: 1409.1259. https://doi.org/10.48550/arXiv.1409.1259
    [13] Y. Liang, S. Ke, J. Zhang, X. Yi, Y. Zheng, Geoman: Multi-level attention networks for geo-sensory time series prediction, in Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18), 1 (2018), 3428–3434. https://www.ijcai.org/proceedings/2018/0476.pdf.
    [14] Y. Li, Z. Zhu, D. Kong, M. Xu, Y. Zhao, Learning heterogeneous spatial-temporal representation for bike-sharing demand prediction, in Proceedings of the AAAI Conference on Artificial Intelligence, 33 (2019), 1004–1011. https://doi.org/10.1609/aaai.v33i01.33011004
    [15] C. Guo, F. Berkhahn, Entity embeddings of categorical variables, arXiv preprint, 2022, arXiv: 1604.06737.
    [16] Y. Yan, Y. Tao, J. Xu, S. Ren, H. Lin, Visual analytics of bike-sharing data based on tensor factorization, J. Visualization, 21 (2018), 495–509. https://doi.org/10.1007/s12650-017-0463-1 doi: 10.1007/s12650-017-0463-1
    [17] Y. Guo, J. A. Kelly, J. P. Clinch, Variability in total cost of vehicle ownership across vehicle and user profiles, Commun. Transp. Res., 2 (2022), 100071. https://doi.org/10.1016/j.commtr.2022.100071 doi: 10.1016/j.commtr.2022.100071
    [18] K. Gao, H. Wang, S. Wang, X. Qu, Data and code disclosure and sharing policy of communications in transportation research, Commun. Transp. Res., 2 (2022), 100055. https://doi.org/10.1016/j.commtr.2022.100055 doi: 10.1016/j.commtr.2022.100055
    [19] L. Lin, Z. He, S. Peeta, Predicting station-level hourly demand in a large-scale bike-sharing network: A graph convolutional neural network approach, Transp. Res. Part C Emerging Technol., 97 (2018), 258–276. https://doi.org/10.1016/j.trc.2018.10.011 doi: 10.1016/j.trc.2018.10.011
    [20] P. Goyal, N. Kamra, X. He, Y. Liu, Dyngem: Deep embedding method for dynamic graphs, arXiv preprint, 2018, arXiv: 1805.11273. https://doi.org/10.48550/arXiv.1805.11273
    [21] B. Yu, H. Yin, Z. Zhu, Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting, arXiv preprint, 2018, arXiv: 1709.04875. https://doi.org/10.48550/arXiv.1709.04875
    [22] K. Gao, Y. Yang, X. Qu, Examining nonlinear and interaction effects of multiple determinants on airline travel satisfaction, Transp. Res. Part D Transp. Environ., 97 (2021), 102957. https://doi.org/10.1016/j.trd.2021.102957 doi: 10.1016/j.trd.2021.102957
    [23] K. Gao, Y. Yang, A. Li, X. Qu, Spatial heterogeneity in distance decay of using bike sharing: An empirical large-scale analysis in Shanghai, Transp. Res. Part D Transp. Environ., 94 (2021), 102814. https://doi.org/10.1016/j.trd.2021.102814 doi: 10.1016/j.trd.2021.102814
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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