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A time-division distribution strategy for the two-echelon vehicle routing problem with demand blowout

  • * Corresponding author: Chao Meng

    * Corresponding author: Chao Meng

The first author is supported by China Postdoctoral Science Foundation (No.2020M673279), National Natural Science Foundation of China (NSFC) (No.51675450), Sichuan Science and Technology Program (No.2020JDTD0012) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No.18YJC630255)

Abstract Full Text(HTML) Figure(8) / Table(12) Related Papers Cited by
  • Based on the rapid development of e-commerce, major promotional events and holidays can lead to explosive growth in market demand and place significant pressure on distribution systems. In this study, we considered a distribution system in which products are first transported to transfer satellites from a central depot and then delivered to customers from the transfer satellites. We modeled this distribution problem as a two-echelon vehicle routing problem with demand blowout (2E-VRPDB). We adopt a time-division distribution strategy to address massive delivery demand in two phases by offering incentives to customers who accept flexible delivery dates. We propose a hybrid fireworks algorithm (HFWA) to solve the 2E-VRPDB model. This model fuses an optimal cutting algorithm with an improved fireworks algorithm. To demonstrate the effectiveness and efficiency of the proposed HFWA, we conducted comparative analysis on a genetic algorithm and ant colony algorithm using a VRP example set. Finally, we applied the proposed model and HFWA to solve a distribution problem for the Jingdong Mall in Chengdu, China. The computational results demonstrate that the proposed approach can effectively reduce logistical costs and maintain a high service level.

    Mathematics Subject Classification: Primary: 90B06, 62K05; Secondary: 68T37.

    Citation:

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  • Figure 1.  Schematic diagram of the 2E-VRPDB

    Figure 2.  Initial second level solution schematic diagram

    Figure 3.  Explosive operation Ⅰ (2-opt)

    Figure 4.  Explosive operation II

    Figure 5.  3-opt operation

    Figure 6.  Flow chart of the HFWA

    Figure 7.  Optimal results of three algorithms

    Figure 8.  Jindong distribution of self-pickup points in Jinniu District

    Table 1.  List of parameters and descriptions

    Sets and Parameters Description
    D Set of depots, $ D=\{d_0\} $
    S Set of satellites, $ S=\{s_1 $, $s _2 $$,…, s_{ns} $}, and the total number is $ ns $
    C Set of customers, $ C=\{c_1 $, $ c_2 $, …, $ c_{nc} \}$, and the total number is $ nc $
    G Set of first-level delivery vehicles, $G=\{g _1 $, $ g_2 $, …, $g _{ng} \}$, and the total number is $ ng $
    H Set of second-level delivery vehicles, $H=\{h _1 $, $h _2 $, …, $ h_{nh} \}$, and the total number is $ nh $
    M A large enough number
    T$ _0 $ Working hours per day
    d$ _{ij} $ The distance of the $ (i, j) $ edge
    q$ _i $ Demand of customer c$ _i $
    cap$ _1 $ The capacity of the first-level vehicle
    cap$ _2 $ The capacity of the second-level vehicle
    t$ _c $ The deadline of customer c
    b$ _1 $ Compensation per unit cargo for accepting flexible delivery
    b$ _2 $ Delay cost per delivery
    a$ _1 $ Fixed cost of the first-level delivery vehicle per delivery
    a$ _2 $ Fixed cost of the second-level delivery vehicle per delivery
    c$ _g $ Unit distance cost of the first-level delivery vehicle per delivery
    c$ _h $ Unit distance cost of the second-level delivery vehicle per delivery
    c$ _1 $ The labor cost of the first-level delivery vehicle per delivery
    c$ _2 $ The labor cost of the second -level delivery per delivery
    f$ _c $ If customer c chooses flexible delivery, fc =1; otherwise, fc =0
    T$ _s $ Time required to complete standard delivery
     | Show Table
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    Table 2.  List of variables and descriptions

    Variables Description
    x$ _{ijg} $ First-level distribution vehicle g travels the $ (i, j) $ edge, $x _{ijg} $$ \in \{0,1\}$; decision variable
    y$ _{ijg} $ Second -level distribution vehicle h travels the $ (i, j) $ edge, y$ _{ijg} $ $ \in \{0,1\}$; decision variable
    z$ _{cs} $ Customer c cargo comes from satellite s, z$ _{cs} $ $ \in \{0,1\}$; decision variable
    w$ _{sg} $ The actual load of first level vehicle g to satellite s; decision variable
    l$ _s $ The total demand of satellite s
    t$ _{sg} $ Time of first-level vehicle g arriving at satellite s
    t$ _{ch} $ Time of arrival of second-level vehicle h to satellite s
    t$ _1 $ The longest time for the first-level vehicle to complete the distribution task
    time$ _c $ The actual delivery time of the customer c
    dtime$ _c $ The delay time for customer c
    U1$ _{ig} $ Restrict the occurrence of sub-tour in the first-level vehicles
    U2$ _{ih} $ Restrict the occurrence of sub-tour in the second-level vehicles
    u$ _{sg} $ Intermediate variable, no actual meaning
    u$ _{cp} $ Intermediate variable, no actual meaning
     | Show Table
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    Table 3.  Parameter settings

    Algorithm Parameter Value
    HFWA Fireworks population size 5
    The number of explosion sparks 2
    Upper limit of the number of explosion sparks 50
    Variation spark number 2
    Number of iterations 1000
    ACO Number of ants 50
    Pheromone heuristic factor 1
    Fitness heuristic factor 9
    Pheromone volatile factor 0.1
    Constant coefficient 1
    Number of iterations 1000
    GA Population size 50
    Cross factor 0.8
    Mutation factor 0.2
    Number of iterations 1000
     | Show Table
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    Table 4.  The optimization results of ACO algorithm

    No Standard
    test
    n ACO Optimal
    solution (km)
    Optimal (km) Average (km) Time (s) GAP (%)
    1 Set2a_E-n22-k4-s6-17 22 422.93 422.93 50.7 1.40% 417.07
    2 Set2a_E-n22-k4-s8-14 22 387.84 387.84 50.5 0.75% 384.96
    3 Set2a_E-n22-k4-s9-19 22 479.05 484.18 49.1 1.80% 470.6
    4 Set2a_E-n22-k4-s10-14 22 377.56 377.56 49.1 1.63% 371.5
    5 Set2a_E-n33-k4-s1-9 33 753.75 768.28 74.9 3.23% 730.16
    6 Set2a_E-n33-k4-s2-13 33 761.76 776.57 75 6.60% 714.63
    7 Set2a_E-n33-k4-s3-17 33 745.38 759.23 74.9 5.36% 707.48
    8 Set2a_E-n33-k4-s7-25 33 790.55 804.92 75 4.45% 756.85
    9 Set2a_E-n33-k4-s14-22 33 797.87 802.72 75.7 2.42% 779.05
    10 Set2b_E-n51-k5-s2-4-17-46 51 618.69 637.24 124.9 16.57% 530.76
    11 Set2b_E-n51-k5-s2-17 51 665.23 684.06 120.8 11.34% 597.49
    12 Set2b_E-n51-k5-s4-46 51 613.78 627.29 120.2 15.64% 530.76
     | Show Table
    DownLoad: CSV

    Table 5.  The optimization results of GA algorithm

    No. Standard
    test
    n GA Optimal
    solution (km)
    Optimal (km) Average (km) Time (s) GAP (%)
    1 Set2a_E-n22-k4-s6-17 22 417.07 438.04 472.4 0.00% 417.07
    2 Set2a_E-n22-k4-s8-14 22 387.84 399.37 468.9 0.75 % 384.96
    3 Set2a_E-n22-k4-s9-19 22 475.62 492.41 474.9 1.07 % 470.6
    4 Set2a_E-n22-k4-s10-14 22 377.56 383.11 472.1 1.63 % 371.5
    5 Set2a_E-n33-k4-s1-9 33 730.16 764.25 502.2 0.00 % 730.16
    6 Set2a_E-n33-k4-s2-13 33 725.04 747.5 484.6 1.46 % 714.63
    7 Set2a_E-n33-k4-s3-17 33 732.37 760.82 488.5 3.52 % 707.48
    8 Set2a_E-n33-k4-s7-25 33 763.58 790.26 491.8 0.89 % 756.85
    9 Set2a_E-n33-k4-s14-22 33 782.04 792.21 510.7 0.38 % 779.05
    10 Set2b_E-n51-k5-s2-4-17-46 51 599.66 631.44 860.2 12.98 % 530.76
    11 Set2b_E-n51-k5-s2-17 51 641.66 671.12 695.5 7.39 % 597.49
    12 Set2b_E-n51-k5-s4-46 51 604.92 620.14 656.7 13.97 % 530.76
     | Show Table
    DownLoad: CSV

    Table 6.  The optimization results of HFWA algorithm

    No. Standard
    test
    n HFWA Optimal
    solution (km)
    Optimal (km) Average (km) Time (s) GAP (%)
    1 Set2a_E-n22-k4-s6-17 22 417.07 417.07 103.7 0.00 % 417.07
    2 Set2a_E-n22-k4-s8-14 22 384.96 386.69 106.2 0.00 % 384.96
    3 Set2a_E-n22-k4-s9-19 22 470.6 472.84 107.1 0.00 % 470.6
    4 Set2a_E-n22-k4-s10-14 22 371.5 376.35 104.2 0.00 % 371.5
    5 Set2a_E-n33-k4-s1-9 33 730.16 734.76 188.9 0.00 % 730.16
    6 Set2a_E-n33-k4-s2-13 33 714.63 724.6 191.1 0.00 % 714.63
    7 Set2a_E-n33-k4-s3-17 33 707.48 712.08 192.1 0.00 % 707.48
    8 Set2a_E-n33-k4-s7-25 33 756.85 765.18 192.1 0.00 % 756.85
    9 Set2a_E-n33-k4-s14-22 33 779.05 781.95 194.7 0.00 % 779.05
    10 Set2b_E-n51-k5-s2-4-17-46 51 530.76 557.82 593.6 0.00 % 530.76
    11 Set2b_E-n51-k5-s2-17 51 597.49 622.8 490 0.00 % 597.49
    12 Set2b_E-n51-k5-s4-46 51 530.76 549.47 499.2 0.00 % 530.76
     | Show Table
    DownLoad: CSV

    Table 7.  The results of existing literature

    No. Standard
    test
    n [18] and [11] Optimal
    solution (km)
    Optimal (km) GAP (%) Optimal (km) GAP (%)
    1 Set2a_E-n22-k4-s6-17 22 417.07 0.00 % 417.07 0.00 % 417.07
    2 Set2a_E-n22-k4-s8-14 22 384.96 0.00 % 384.96 0.00 % 384.96
    3 Set2a_E-n22-k4-s9-19 22 470.6 0.00 % 470.6 0.00 % 470.6
    4 Set2a_E-n22-k4-s10-14 22 371.5 0.00 % 371.5 0.00 % 371.5
    5 Set2a_E-n33-k4-s1-9 33 743.22 1.79 % 730.16 0.00 % 730.16
    6 Set2a_E-n33-k4-s2-13 33 710.48 -0.58 % 714.63 0.00 % 714.63
    7 Set2a_E-n33-k4-s3-17 33 - - 707.48 0.00 % 707.48
    8 Set2a_E-n33-k4-s7-25 33 756.85 0.00 % 756.85 0.00 % 756.85
    9 Set2a_E-n33-k4-s14-22 33 - - 779.05 0.00 % 779.05
    10 Set2b_E-n51-k5-s2-4-17-46 51 577.16 8.74 % 530.76 0.00 % 530.76
    11 Set2b_E-n51-k5-s2-17 51 - - 597.49 0.00 % 597.49
    12 Set2b_E-n51-k5-s4-46 51 - - 530.76 0.00 % 530.76
     | Show Table
    DownLoad: CSV

    Table 8.  Distribution network node coordinates

    Node X Y Node X Y Node X Y
    D 20639 18019 16 14533 5098 34 13728 8485
    S1 807 16768 17 13623 8242 35 12730 4622
    S2 33084 19137 18 13573 3064 36 12968 4261
    1 11066 5223 19 15874 7803 37 12611 3804
    2 10481 7350 20 11212 6558 38 15604 5195
    3 16356 5406 21 13396 3457 39 16340 4248
    4 14197 6453 22 11813 9258 40 15342 6701
    5 9591 5209 23 15606 5339 41 12902 3362
    6 12664 5236 24 17577 5196 42 12149 5210
    7 10772 5910 25 10496 5801 43 13221 5054
    8 15321 5178 26 17472 3701 44 13451 7415
    9 15239 5209 27 13839 7873 45 15543 7984
    10 13556 7147 28 16555 8635 46 13025 4248
    11 16660 4104 29 9347 6362 47 17460 4241
    12 12438 3987 30 9547 8857 48 10895 4818
    13 13850 6882 31 17942 3752 49 13704 10362
    14 15196 8050 32 11042 5921 50 12929 4844
    15 12864 4804 33 11387 5026
     | Show Table
    DownLoad: CSV

    Table 9.  Demand and delivery time of customer points

    Node Demand (packages) Time (days) Node Demand (packages) Time (days) Node Demand (packages) Time (days)
    1 91 3 18 46 3 35 302 6
    2 224 3 19 49 3 36 94 6
    3 215 3 20 85 3 37 249 6
    4 53 6 21 277 6 38 248 3
    5 39 6 22 84 6 39 125 3
    6 164 6 23 268 6 40 130 3
    7 316 6 24 80 3 41 25 3
    8 112 3 25 306 3 42 75 6
    9 192 3 26 115 3 43 175 6
    10 74 3 27 65 3 44 256 6
    11 247 6 28 83 6 45 307 6
    12 84 6 29 203 6 46 42 3
    13 166 6 30 156 6 47 186 3
    14 230 3 31 116 6 48 100 6
    15 293 3 32 273 3 49 57 6
    16 316 6 33 192 3 50 110 6
    17 180 6 34 181 3
     | Show Table
    DownLoad: CSV

    Table 10.  Computational results for the regular delivery model

    Level Vehicle NO. Standard delivery vehicle route
    First-level vehicle delivery 1S D-S1-D
    2S D-S1-D
    3S D-S1-D
    4S D-S1-D
    Second-level vehicle delivery 1 S1-21-18-41-S1
    2 S1-37-12-S1
    3 S1-35-6-S1
    4 S1-42-33-48-1-S1
    5 S1-32-20-S1
    6 S1-25-5-S1
    7 S1-29-2-S1
    8 S1-30-22-49-34-S1
    9 S1-17-27-S1
    10 S1-44-10-13-S1
    11 S1-4-40-14-S1
    12 S1-45-19-28-S1
    13 S1-24-47-31-26-S1
    14 S1-11-39-S1
    15 S1-3-23-S1
    16 S1-38-8-S1
    17 S1-7-S1
    18 S1-9-S1
    19 S1-16-43-S1
    20 S1-50-15-36-S1
    21 S1-46-S1
    Delivery time (days) 7
    Delay cost (yuan) 32240
    Compensation cost (yuan) 0
    Total cost (yuan) 2159128
     | Show Table
    DownLoad: CSV

    Table 11.  Computational results for the TDD model

    Delivery method Level Vehicle NO. TDD vehicle route
    Standard delivery First-level vehicle delivery 1S D-S1-D
    2S D-S1-D
    Second-level vehicle delivery 1 S1-26-47-24-8-9-S1
    2 S1-38-19-20-S1
    3 S1-32-1-S1
    4 S1-2-25-S1
    5 S1-33-41-18-S1
    6 S1-46-15-S1
    7 S1-10-27-S1
    8 S1-14-34-S1
    9 S1-40-3-S1
    10 S1-39-S1
    Flexible delivery First-level vehicle delivery 1S* D-S1-D
    2S* D-S1-D
    3S* D-S1-D
    Second-level vehicle delivery 11 S1-23-13-S1
    12 S1-17-44-S1
    13 S1-45-28-49-S1
    14 S1-22-31-11-S1
    15 S1-16-43-S1
    16 S1-6-37-12-S1
    17 S1-21-36-50-S1
    18 S1-42-35-48-S1
    19 S1-7-5-S1
    20 S1-29-30-S1
    Delivery time (days) 6
    Delay cost (yuan) 0
    Compensation cost (yuan) 2239.5
    Total cost (yuan) 1937381
     | Show Table
    DownLoad: CSV

    Table 12.  Delivery time sensitivity analysis

    Standard Penalty Standard TDD second- Penalty Compensation Total cost
    delivery (yuan) delivery level arrival (yuan) cost of delivery
    time (day) cost (yuan) time (day) (yuan) (yuan)
    2 40300 1881455 4 6250 2239.5 1783100
    5 3750 2239.5 1781518
    6 1250 2239.5 1779718
    3 32240 1868263 4 5000 2239.5 1782724
    5 2500 2239.5 1779725
    6 0 2239.5 1775722
    4 24180 1862604 5 2500 2239.5 1779704
    6 0 2239.5 1777541
    7 0 2239.5 1777541
     | Show Table
    DownLoad: CSV
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