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Shipper collaboration in forward and reverse logistics

  • * Corresponding author: Nengmin Wang

    * Corresponding author: Nengmin Wang 
Abstract Full Text(HTML) Figure(5) / Table(21) Related Papers Cited by
  • In less than truckload transportation, shippers collaborate to reduce their logistics costs by consolidating their transportation requests in the procurement of transportation services from a carrier for serving the requests. In this paper, we study shipper collaboration in forward and reverse logistics, in which multiple shippers with forward or/and reverse logistics operations consolidate their transportation requests. In the forward and reverse logistics, manufacturers deliver new products to their customers and used products are collected from customers and transported to remanufacturers for repair or reproduction. This gives rise to a new vehicle routing problem with pickup and delivery requests and three different types of depots (product depots, vehicle depots and recycle depots). A hybrid approach combining greedy randomized adaptive search procedure (GRASP) and iterated local search (ILS) is proposed to find a near optimal solution of the problem. Numerical experiments on a large set of randomly generated instances with different problem sizes demonstrate that shipper collaboration in forward and reverse logistics can realize significant cost savings compared with the isolated operation of each shipper without cooperation, and the proposed approach is effective in the sense that it can find a high quality solution in a reasonable computation time.

    Mathematics Subject Classification: Primary: 90BXX; Secondary: 90B06.

    Citation:

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  • Figure 1.  Isolated transportation planning of each shipper in forward and reverse logistics

    Figure 2.  Collaborative transportation planning of all shippers in forward and reverse logistics

    Figure 3.  Illustration of the insertion positions for a product depot

    Figure 4.  Illustration of the insertion positions for a customer

    Figure 5.  Illustration of the insertion positions for a recycle depot

    Table 1.  The comparison of the first improvement and the best improvement

    Instance set Best improvement First improvement
    CPU Gap CPU Gap
    I40 827.63 5.20 29.22 5.18
    O40 796.54 0.62 35.18 0.58
     | Show Table
    DownLoad: CSV

    Table 2.  The comparison of two strategies

    Instance set General strategy Speedup strategy
    CPU Gap CPU Gap
    I40 327.63 5.18 29.22 5.18
    O40 296.54 0.61 35.18 0.58
     | Show Table
    DownLoad: CSV

    Table 3.  Parameter tuning according to instance size

    Description Parameter Value
    Small instances
    (20 requests)
    Medium instances
    (40 requests)
    Large instances
    (60 requests)
    Large instances
    (100 requests)
    Number of GRASP iterations np 10 10 15 15
    Number of ILS iterations ni 5 10 15 20
    RCL size nrcl 3 5 7 7
     | Show Table
    DownLoad: CSV

    Table 5.  Results for comparison of GRASP-ILS and CPLEX solver

    Instance set CPLEX GRASP-ILS
    ImpIni GapLB CPU ImpIni ImpCplex GapLB CPU
    I20 -- 0.00 2101.07 -- 0.00 0.00 3.17
    O20 -- 0.00 1077.42 -- 0.00 0.00 1.92
    A20 -- 0.00 1350.15 -- 0.00 0.00 2.49
    Average 0.00 1509.55 0.00 0.00 2.53
    I40 1.44 28.47 21600.00 6.57 5.18 24.58 29.22
    O40 6.02 10.78 21600.00 6.37 0.58 10.27 35.18
    A40 3.33 20.89 21600.00 6.61 3.58 17.91 29.94
    Average 3.60 20.05 21600.00 6.52 3.11 17.59 31.45
    I60 2.74 34.25 21600.00 12.45 9.99 26.73 125.42
    O60 4.77 28.38 21600.00 16.85 12.63 18.00 144.84
    A60 3.35 28.74 21600.00 14.78 11.81 19.18 148.35
    Average 3.62 30.46 21600.00 13.70 12.47 21.30 139.54
    I100 1.62 40.24 21600.00 16.85 13.00 31.27 246.32
    O100 2.57 38.25 21600.00 21.11 15.24 23.55 292.72
    A100 1.83 34.56 21600.00 19.52 14.68 23.17 282.45
    Average 2.01 37.68 21600.00 19.16 14.31 26.00 273.83
     | Show Table
    DownLoad: CSV

    Table 6.  Summarized results for comparison among GRASP-ILS, GRASP and ILS

    Instance set
    ImpGRASP-ILS, GRASP ImpGRASP-ILS, ILS
    I20 6.51 3.17
    O20 1.87 0.86
    A20 7.65 3.68
    Average 5.34 2.57
    I40 3.95 3.71
    O40 3.00 3.23
    A40 6.75 3.62
    Average 4.57 3.52
    I60 4.09 2.65
    O60 7.42 0.73
    A60 6.35 4.64
    Average 5.95 2.67
    I100 7.20 4.19
    O100 6.59 3.78
    A100 5.00 4.06
    Average 6.26 4.01
     | Show Table
    DownLoad: CSV

    Table 7.  Summarized results for deviations among GRASP-ILS, GRASP and ILS

    Instance set GRASP ILS GRASP-ILS
    Dev Dev Dev
    I20 1.93 4.31 0.37
    O20 0.26 0.43 0.01
    A20 1.21 5.23 0.76
    Average 1.13 3.32 0.38
    I40 4.46 7.73 0.42
    O40 3.81 7.01 0.44
    A40 3.68 8.65 0.89
    Average 3.98 7.80 0.58
    I60 6.92 11.58 0.51
    O60 4.80 6.72 0.81
    A60 4.09 8.66 0.92
    Average 5.27 8.99 0.75
    I100 7.04 12.38 0.88
    O100 5.25 8.96 1.16
    A100 5.40 9.31 1.12
    Average 5.90 10.22 1.05
     | Show Table
    DownLoad: CSV

    Table 8.  Summarized results of geographical distributions of customers on cost savings

    Geographical distributions of customers Non-collaboration Nv Collaboration Nv $\phi$
    I20 4.3 2.3 22.57
    O20 4.1 2 17.90
    A20 4.2 2.1 10.76
    I40 8 6 31.02
    O40 7.3 3.5 24.19
    A40 7.5 3.9 16.35
    I60 11 6.3 33.31
    O60 10.1 4.9 28.22
    A60 10.1 5.2 24.42
    I100 17.8 9.8 41.37
    O100 14.1 8.4 33.79
    A100 13.8 8.7 30.96
     | Show Table
    DownLoad: CSV

    Table 9.  Summarized results of number of requests on the cost savings

    Customer Requests Non-collaboration Nv Collaboration Nv $\phi$
    I20 4.3 2.3 22.57
    I40 8 6 31.02
    I60 11 6.3 33.31
    I100 17.8 9.8 41.37
    Average 4.2 2.1 17.08
    O20 4.1 2 17.90
    O40 7.3 3.5 24.19
    O60 10.1 4.9 28.22
    O100 14.1 8.4 33.79
    Average 7.6 4.4 23.85
    A20 4.2 2.1 10.76
    A40 7.5 3.9 16.35
    A60 10.1 5.2 24.42
    A100 13.8 8.7 30.96
    Average 10.4 5.5 28.65
     | Show Table
    DownLoad: CSV

    Table 10.  Detailed results of CPLEX solver and GRASP-ILS for instances with 20 requests

    Instance CPLEX GRASP-ILS
    Ini UB LB CPU GapLB Zmin CPU Zavg CPUavg ImpCplex GapLB
    I20-2-1-0 / 459.26 459.26 6727.61 0.00 459.26 6.24 459.26 7.25 0.00 0.00
    I20-2-1-1 / 366.66 366.66 142.44 0.00 366.66 2.23 366.66 4.47 0.00 0.00
    I20-2-1-2 / 478.19 478.19 1360.56 0.00 478.19 6.64 478.19 8.24 0.00 0.00
    I20-2-1-3 / 360.24 360.24 128.19 0.00 360.24 1.86 365.33 1.89 0.00 0.00
    I20-2-1-4 / 383.24 383.24 8933.10 0.00 383.24 1.67 383.24 2.01 0.00 0.00
    I20-2-1-5 / 343.70 343.70 369.21 0.00 343.70 1.41 350.76 1.81 0.00 0.00
    I20-2-1-6 / 344.51 344.51 366.51 0.00 344.51 1.59 344.51 1.77 0.00 0.00
    I20-2-1-7 / 354.59 354.59 266.03 0.00 354.59 1.75 355.59 3.11 0.00 0.00
    I20-2-1-8 / 351.37 351.37 2202.84 0.00 351.37 1.40 351.37 1.77 0.00 0.00
    I20-2-1-9 / 476.32 476.32 514.23 0.00 476.32 6.97 476.32 8.15 0.00 0.00
    Avg 2101.07 0.00 3.17
    O20-2-1-0 / 240.46 240.46 817.75 0.00 240.46 2.18 240.46 2.51 0.00 0.00
    O20-2-1-1 / 260.84 260.84 727.20 0.00 260.84 1.84 260.84 2.13 0.00 0.00
    O20-2-1-2 / 250.62 250.62 945.73 0.00 250.62 2.50 250.62 2.66 0.00 0.00
    O20-2-1-3 / 254.79 254.79 1213.25 0.00 254.79 1.50 254.79 1.83 0.00 0.00
    O20-2-1-4 / 273.79 273.79 563.98 0.00 273.79 2.03 273.85 1.99 0.00 0.00
    O20-2-1-5 / 269.61 269.61 2227.00 0.00 269.61 1.46 269.61 2.05 0.00 0.00
    O20-2-1-6 / 260.31 260.31 1456.89 0.00 260.31 1.64 260.41 2.33 0.00 0.00
    O20-2-1-7 / 245.95 245.95 1323.50 0.00 245.95 1.69 246.06 2.32 0.00 0.00
    O20-2-1-8 / 240.28 240.28 685.00 0.00 240.28 1.70 240.28 2.28 0.00 0.00
    O20-2-1-9 / 229.92 229.92 813.89 0.00 229.92 2.70 229.96 2.41 0.00 0.00
    Avg 1077.42 0.00 1.92 0.00 0.00
    A20-2-1-0 / 362.31 362.31 987.80 0.00 362.31 1.59 366.24 1.85 0.00 0.00
    A20-2-1-1 / 444.80 444.80 1188.06 0.00 444.80 7.26 444.80 8.29 0.00 0.00
    A20-2-1-2 / 329.67 329.67 2358.72 0.00 329.67 1.37 334.31 1.85 0.00 0.00
    A20-2-1-3 / 363.62 363.62 1206.36 0.00 363.62 4.06 372.23 5.38 0.00 0.00
    A20-2-1-4 / 343.77 343.77 2440.98 0.00 343.77 1.66 343.77 1.91 0.00 0.00
    A20-2-1-5 / 287.74 287.74 1266.59 0.00 287.74 1.99 294.17 1.55 0.00 0.00
    A20-2-1-6 / 338.23 338.23 821.69 0.00 338.23 1.56 338.23 1.70 0.00 0.00
    A20-2-1-7 / 290.21 290.21 985.11 0.00 290.21 1.40 292.05 1.59 0.00 0.00
    A20-2-1-8 / 314.20 314.20 1324.84 0.00 314.20 2.13 314.20 1.70 0.00 0.00
    A20-2-1-9 / 323.60 323.60 921.38 0.00 323.60 1.93 323.60 2.18 0.00 0.00
    Avg 1350.15 0.00 2.49 0.00 0.00
     | Show Table
    DownLoad: CSV

    Table 11.  Detailed results of CPLEX solver and GRASP-ILS for instances with 40 requests

    Instance CPLEX GRASP-ILS
    Ini UB LB CPU ImpIni GapLB Zmin CPU Zavg CPUavg ImpIni ImpCplex GapLB
    I40-2-1-0 764.73 733.02 554.38 21600.00 4.15 24.37 706.64 31.77 722.04 35.64 7.60 3.60 21.55
    I40-2-1-1 617.81 617.81 432.47 21600.00 0.00 30.00 600.90 30.07 607.03 33.09 2.74 2.74 28.03
    I40-2-1-2 776.04 776.04 526.08 21600.00 0.00 32.21 729.24 31.07 729.24 33.12 6.03 6.03 27.86
    I40-2-1-3 718.16 702.54 545.94 21600.00 2.18 22.29 651.55 26.00 651.55 28.16 9.28 7.26 16.21
    I40-2-1-4 720.41 716.43 519.91 21600.00 0.55 27.43 699.59 26.37 699.59 27.97 2.89 2.35 25.68
    I40-2-1-5 748.66 748.66 475.02 21600.00 0.00 36.55 647.36 27.67 647.36 31.16 13.53 13.53 26.62
    I40-2-1-6 615.92 615.92 435.58 21600.00 0.00 29.28 590.24 29.90 591.42 32.15 4.17 4.17 26.20
    I40-2-1-7 664.83 662.64 432.11 21600.00 0.33 34.79 612.68 28.53 613.44 29.29 7.84 7.54 29.47
    I40-2-1-8 634.10 594.85 447.33 21600.00 6.19 24.80 573.59 32.61 574.75 34.14 9.54 3.57 22.01
    I40-2-1-9 611.25 604.84 466.03 21600.00 1.05 22.95 598.44 28.16 601.82 31.40 2.10 1.06 22.13
    Avg 21600.00 1.44 28.47 29.22 31.61 6.57 5.18 24.58
    O40-2-1-0 590.88 494.73 486.52 21600.00 16.27 1.66 494.73 30.31 497.25 32.04 16.27 0.00 1.66
    O40-2-1-1 484.61 443.19 390.45 21600.00 9.99 11.90 434.04 36.80 439.23 35.30 10.44 2.06 10.04
    O40-2-1-2 452.71 445.80 401.58 21600.00 1.53 9.92 445.80 34.44 445.80 36.68 1.53 0.00 9.92
    O40-2-1-3 604.15 529.67 456.10 21600.00 12.33 13.89 523.35 36.44 523.35 37.24 13.37 1.19 12.85
    O40-2-1-4 551.18 550.47 458.60 21600.00 0.13 16.69 549.35 37.96 549.35 41.00 0.33 0.20 16.52
    O40-2-1-5 627.43 576.14 486.90 21600.00 8.17 15.49 573.11 34.70 573.11 36.25 8.66 0.53 15.04
    O40-2-1-6 496.89 492.06 422.04 21600.00 0.97 14.23 489.14 39.24 489.14 41.56 1.56 0.59 13.72
    O40-2-1-7 545.21 532.03 492.50 21600.00 2.78 7.43 528.80 31.72 542.87 36.13 3.01 0.61 6.86
    O40-2-1-8 441.62 427.55 427.55 21600.00 3.19 0.00 427.55 32.20 427.82 40.79 3.19 0.00 0.00
    O40-2-1-9 611.65 582.21 485.50 21600.00 4.81 16.61 578.85 37.98 579.14 37.03 5.36 0.58 16.13
    Avg 21600.00 6.02 10.78 35.18 37.40 6.37 0.58 10.27
    A40-2-1-0 595.16 581.93 463.33 21600.00 2.22 20.38 550.20 26.67 563.08 29.48 7.55 5.45 15.79
    A40-2-1-1 617.54 617.54 428.57 21600.00 0.00 30.60 597.17 29.64 601.48 32.31 3.30 3.30 28.23
    A40-2-1-2 685.71 593.58 508.05 21600.00 13.44 14.41 582.03 32.10 582.03 36.22 15.12 1.95 12.71
    A40-2-1-3 511.22 511.22 383.62 21600.00 0.00 24.96 505.21 31.86 505.21 34.47 1.18 1.18 24.07
    A40-2-1-4 574.55 569.49 460.72 21600.00 0.88 19.10 567.77 33.23 568.40 33.13 1.18 0.30 18.85
    A40-2-1-5 626.75 626.75 484.73 21600.00 0.00 22.66 617.34 28.18 617.38 30.09 1.50 1.50 21.48
    A40-2-1-6 574.52 572.59 457.56 21600.00 1.90 20.09 537.91 32.50 560.92 37.04 6.37 6.06 14.94
    A40-2-1-7 615.58 552.47 494.52 21600.00 10.90 10.49 542.06 30.23 548.20 28.19 11.94 1.88 8.77
    A40-2-1-8 642.09 630.61 480.02 21600.00 1.79 23.88 600.61 27.93 600.61 29.57 6.46 4.76 20.08
    A40-2-1-9 714.16 698.33 542.39 21600.00 2.22 22.33 632.33 27.06 635.86 30.86 11.46 9.45 14.22
    Avg 21600.00 3.33 20.89 29.94 32.14 6.61 3.58 17.91
     | Show Table
    DownLoad: CSV

    Table 12.  Detailed results of CPLEX solver and GRASP-ILS for instances with 60 requests

    Instance CPLEX GRASP-ILS
    Ini UB LB CPU ImpIni GapLB Zmin CPU Zavg CPUavg ImpIni ImpCplex GapLB
    I60-2-1-0 1075.15 1011.77 647.03 21600.00 5.89 36.05 911.77 99.40 914.95 114.83 15.20 9.88 29.04
    I60-2-1-1 1190.63 1187.76 774.42 21600.00 0.24 34.80 987.76 118.70 988.14 130.24 17.04 16.84 21.60
    I60-2-1-2 950.96 939.23 602.05 21600.00 1.23 35.90 877.23 131.62 879.37 139.26 7.75 6.60 31.37
    I60-2-1-3 1088.70 1082.44 690.16 21600.00 0.57 36.24 882.44 137.82 889.93 122.66 18.95 18.48 21.79
    I60-2-1-4 1047.69 1000.73 685.00 21600.00 4.48 31.55 896.87 108.94 900.72 119.33 14.40 10.38 23.62
    I60-2-1-5 1026.03 1026.03 632.75 21600.00 0.00 38.33 996.95 138.59 997.19 135.02 2.83 2.83 36.53
    I60-2-1-6 1214.28 1143.88 758.39 21600.00 5.80 33.70 1043.88 119.96 1046.20 126.73 14.03 8.74 27.35
    I60-2-1-7 993.02 993.02 642.58 21600.00 0.00 35.29 944.32 132.32 960.94 137.32 4.90 4.90 31.95
    I60-2-1-8 946.14 918.94 625.61 21600.00 2.87 31.92 818.94 132.87 826.45 119.74 13.44 10.88 23.61
    I60-2-1-9 1029.63 965.24 687.93 21600.00 6.25 28.73 865.24 133.94 867.75 135.43 15.97 10.36 20.49
    Avg 21600.00 2.74 34.25 125.42 12.45 9.99 26.73
    O60-2-1-0 778.81 742.35 556.24 21600.00 4.68 25.07 652.35 146.37 672.06 140.72 16.24 12.12 14.73
    O60-2-1-1 860.37 792.55 585.30 21600.00 7.88 26.15 746.22 144.23 750.48 150.35 13.27 5.85 21.56
    O60-2-1-2 883.39 832.49 606.55 21600.00 5.76 27.14 732.49 138.82 736.08 137.72 17.08 12.01 17.19
    O60-2-1-3 975.67 897.76 686.25 21600.00 7.99 23.56 787.76 128.96 795.54 144.19 19.26 12.25 12.89
    O60-2-1-4 889.25 832.01 565.60 21600.00 6.44 32.02 727.32 159.57 734.02 163.88 18.21 12.58 22.24
    O60-2-1-5 891.08 852.47 631.60 21600.00 4.33 25.91 752.74 150.60 755.68 145.87 15.53 11.70 16.09
    O60-2-1-6 813.00 813.00 542.76 21600.00 0.00 33.24 680.46 129.71 680.46 147.45 16.30 16.30 20.24
    O60-2-1-7 935.69 881.76 640.60 21600.00 5.76 27.35 752.96 162.61 761.11 172.49 19.53 14.61 14.92
    O60-2-1-8 850.19 835.58 571.37 21600.00 1.72 31.62 695.58 134.58 696.81 139.18 18.19 16.75 17.86
    O60-2-1-9 850.23 823.61 562.61 21600.00 3.13 31.69 723.61 152.90 727.63 150.04 14.89 12.14 22.25
    Avg 21600.00 4.77 28.38 144.84 16.85 12.63 18.00
    A60-2-1-0 936.24 936.24 594.70 21600.00 0.00 36.48 836.54 161.45 836.60 163.57 10.65 10.65 28.91
    A60-2-1-1 1072.31 1072.31 663.76 21600.00 0.00 38.1 911.33 161.93 941.39 158.79 15.01 15.01 27.17
    A60-2-1-2 859.84 803.79 620.37 21600.00 6.52 22.82 703.79 135.40 703.85 151.11 18.15 12.44 11.85
    A60-2-1-3 950.17 881.73 663.68 21600.00 7.20 24.73 781.73 122.15 782.94 123.85 17.73 11.34 15.10
    A60-2-1-4 912.08 902.08 622.80 21600.00 1.10 30.96 793.69 148.34 806.85 147.38 12.98 12.02 21.53
    A60-2-1-5 908.24 908.24 611.25 21600.00 0.00 32.7 740.28 148.01 741.38 148.81 18.49 18.49 17.43
    A60-2-1-6 919.06 854.71 676.50 21600.00 7.00 20.85 754.71 146.15 756.53 157.06 17.88 11.70 10.36
    A60-2-1-7 942.98 880.28 647.62 21600.00 6.65 26.43 780.28 155.59 796.40 155.86 17.25 11.36 17.00
    A60-2-1-8 799.61 799.61 594.75 21600.00 0.00 25.62 751.38 140.81 757.58 142.27 6.03 6.03 20.85
    A60-2-1-9 817.71 776.45 553.61 21600.00 5.05 28.7 705.99 163.67 712.95 157.17 13.66 9.07 21.58
    Avg 21600.00 3.35 28.74 148.35 14.78 11.81 19.18
     | Show Table
    DownLoad: CSV

    Table 13.  Detailed results of CPLEX solver and GRASP-ILS for instances with 100 requests

    Instance CPLEX GRASP-ILS
    Ini UB LB CPU ImpIni GapLB Zmin CPU Zavg CPUavg ImpIni ImpCplex GapLB
    I100-2-1-0 1735.62 1680.34 1041.47 21600.00 3.29 38.02 1500.16 242.56 1509.42 254.82 15.70 10.72 30.58
    I100-2-1-1 1651.57 1634.89 1015.27 21600.00 1.02 37.90 1399.61 244.23 1409.80 271.59 18.00 14.39 27.46
    I100-2-1-2 1668.99 1668.99 907.93 21600.00 0.00 45.60 1481.37 264.12 1491.31 270.40 12.67 11.24 38.71
    I100-2-1-3 1764.60 1689.42 1077.17 21600.00 4.45 36.24 1406.15 200.51 1412.52 228.29 25.49 16.77 23.40
    I100-2-1-4 1686.00 1686.00 985.47 21600.00 0.00 41.55 1471.77 231.05 1479.38 237.84 14.56 12.71 33.04
    I100-2-1-5 1700.66 1700.66 878.73 21600.00 0.00 48.33 1465.71 265.97 1488.23 273.44 16.03 13.82 40.05
    I100-2-1-6 1692.38 1691.53 1017.12 21600.00 0.05 39.87 1482.92 258.92 1503.80 271.70 14.12 12.33 31.41
    I100-2-1-7 1645.66 1645.66 900.34 21600.00 0.00 45.29 1439.77 227.63 1442.38 242.32 14.30 12.51 37.47
    I100-2-1-8 1505.41 1478.50 927.61 21600.00 1.82 37.26 1277.58 261.05 1283.99 274.56 17.83 13.59 27.39
    I100-2-1-9 1541.52 1460.33 987.62 21600.00 5.56 32.37 1286.29 267.15 1315.68 277.42 19.84 11.92 23.22
    Avg 21600.00 1.62 40.24 246.32 16.85 13.00 31.27
    O100-2-1-0 1467.93 1420.49 950.02 21600.00 3.34 33.12 1215.92 278.27 1225.50 288.86 20.73 14.40 21.87
    O100-2-1-1 1309.79 1309.79 836.30 21600.00 0.00 36.15 1113.13 316.97 1124.36 328.01 17.67 15.01 24.87
    O100-2-1-2 1283.48 1230.45 831.54 21600.00 4.31 32.42 1066.12 254.79 1068.68 263.26 20.39 13.36 22.00
    O100-2-1-3 1284.46 1321.90 810.99 21600.00 0.00 38.65 1166.47 300.23 1186.02 278.56 13.32 11.76 30.48
    O100-2-1-4 1382.26 1338.36 909.82 21600.00 3.28 32.02 1146.62 287.89 1151.50 298.72 20.55 14.33 20.65
    O100-2-1-5 1494.38 1431.12 903.18 21600.00 4.42 36.89 1223.53 294.71 1245.41 298.48 22.14 14.51 26.18
    O100-2-1-6 1318.96 1264.83 831.75 21600.00 4.28 34.24 1062.43 311.91 1064.81 321.69 24.15 16.00 21.71
    O100-2-1-7 1435.95 1432.51 894.75 21600.00 0.24 37.54 1224.31 306.67 1230.86 324.67 17.29 14.53 26.92
    O100-2-1-8 1254.70 1221.48 779.18 21600.00 2.72 36.21 971.56 296.99 1002.10 317.68 29.14 20.46 19.80
    O100-2-1-9 1381.18 1339.26 866.90 21600.00 3.13 35.27 1098.25 278.80 1119.88 283.29 25.76 18.00 21.06
    Avg 21600.00 2.57 35.25 292.72 21.11 15.24 23.55
    A100-2-1-0 1415.28 1404.33 863.94 21600.00 0.78 38.48 1212.68 235.67 1230.21 256.89 16.71 13.65 28.76
    A100-2-1-1 1696.07 1627.24 1168.36 21600.00 4.23 28.20 1271.73 269.87 1291.68 279.07 33.37 21.85 8.13
    A100-2-1-2 1328.88 1283.69 849.55 21600.00 3.52 33.82 1129.63 308.04 1132.34 325.42 17.64 12.00 24.79
    A100-2-1-3 1364.97 1360.34 865.59 21600.00 0.34 36.37 1162.85 263.527 1179.11 269.85 17.38 14.52 25.56
    A100-2-1-4 1433.15 1417.56 886.68 21600.00 1.10 37.45 1190.23 270.33 1197.46 281.37 20.41 16.04 25.50
    A100-2-1-5 1320.79 1320.79 802.64 21600.00 0.00 39.23 1093.48 267.46 1116.72 344.65 20.79 17.21 26.60
    A100-2-1-6 1450.78 1400.91 972.51 21600.00 3.56 30.58 1208.76 258.22 1212.17 356.84 20.02 13.72 19.54
    A100-2-1-7 1404.69 1371.23 920.23 21600.00 2.44 32.89 1174.24 364.28 1193.54 380.21 19.63 14.37 21.63
    A100-2-1-8 1334.8 1321.19 847.28 21600.00 1.03 35.87 1154.51 266.66 1177.15 345.79 15.62 12.62 26.61
    A100-2-1-9 1382.94 1365.60 918.36 21600.00 1.27 32.75 1217.18 320.42 1218.88 329.73 13.62 10.87 24.55
    Avg 21600.00 1.83 34.56 282.45 19.52 14.68 23.17
     | Show Table
    DownLoad: CSV

    Table 14.  Meta-heuristics results for instances with 20 requests

    Instance GRASP ILS GRASP-ILS
    Zmin Zavg Zmin Zavg Zmin ImpGRASP-ILS, GRASP ImpGRASP-ILS, ILS
    I20-2-1-0 459.26 459.26 459.26 459.26 459.26 0.00 0.00
    I20-2-1-1 449.88 449.88 366.66 366.66 366.66 22.70 0.00
    I20-2-1-2 478.19 478.19 478.19 478.19 478.19 0.00 0.00
    I20-2-1-3 449.29 459.44 360.24 415.28 360.24 24.72 0.00
    I20-2-1-4 387.33 413.78 387.33 412.08 383.24 1.07 1.07
    I20-2-1-5 364.89 370.22 364.89 373.59 343.70 6.17 6.17
    I20-2-1-6 344.51 344.51 344.51 344.51 344.51 0.00 0.00
    I20-2-1-7 379.50 407.70 426.47 479.74 354.59 7.03 20.27
    I20-2-1-8 363.34 368.28 366.19 376.44 351.37 3.41 4.22
    I20-2-1-9 476.32 476.32 476.32 494.04 476.32 0.00 0.00
    Avg 6.51 3.17
    O20-2-1-0 240.46 240.46 240.46 240.46 240.46 0.00 0.00
    O20-2-1-1 261.02 261.02 278.50 278.56 260.84 0.07 6.77
    O20-2-1-2 250.62 250.62 251.75 251.75 250.62 0.00 0.45
    O20-2-1-3 254.79 254.79 254.79 254.79 254.79 0.00 0.00
    O20-2-1-4 273.98 274.99 275.45 275.56 273.79 0.07 0.61
    O20-2-1-5 269.61 269.61 269.61 269.61 269.61 0.00 0.00
    O20-2-1-6 260.87 260.92 260.87 261.21 260.31 0.22 0.22
    O20-2-1-7 245.95 246.37 246.97 246.97 245.95 0.00 0.41
    O20-2-1-8 265.74 265.74 240.28 244.44 240.28 10.60 0.00
    O20-2-1-9 247.82 247.97 230.15 235.51 229.92 7.79 0.10
    Avg 1.87 0.86
    A20-2-1-0 367.97 367.97 367.91 379.79 362.31 1.56 1.55
    A20-2-1-1 444.8 444.80 444.80 444.80 444.80 0.00 0.00
    A20-2-1-2 364.77 375.31 364.77 404.68 329.67 10.65 10.65
    A20-2-1-3 412.44 439.21 423.93 456.66 363.62 13.43 16.59
    A20-2-1-4 415.4 415.40 343.77 397.64 343.77 20.84 0.00
    A20-2-1-5 295.37 297.82 299.91 305.22 287.74 2.65 4.23
    A20-2-1-6 340.74 340.74 338.23 338.74 338.23 0.74 0.00
    A20-2-1-7 298.51 304.03 301.26 304.94 290.21 2.86 3.81
    A20-2-1-8 314.20 314.20 314.20 314.20 314.20 0.00 0.00
    A20-2-1-9 400.40 400.40 323.60 361.33 323.60 23.73 0.00
    Avg 7.65 3.68
     | Show Table
    DownLoad: CSV

    Table 15.  Meta-heuristics results for instances with 40 requests

    Instance GRASP ILS GRASP-ILS
    Zmin Zavg Zmin Zavg Zmin ImpGRASP-ILS, GRASP ImpGRASP-ILS, ILS
    I40-2-1-0 733.03 769.61 741.58 773.84 706.64 3.73 4.94
    I40-2-1-1 617.81 617.81 610.54 675.38 600.90 2.81 1.60
    I40-2-1-2 756.99 806.65 755.90 772.53 729.24 3.81 3.66
    I40-2-1-3 651.55 681.20 718.16 718.16 651.55 0.00 10.22
    I40-2-1-4 699.59 731.00 720.42 720.42 699.59 0.00 2.98
    I40-2-1-5 718.38 761.48 672.90 771.28 647.36 10.97 3.95
    I40-2-1-6 605.91 639.66 627.27 700.22 590.24 4.35 6.27
    I40-2-1-7 653.35 653.35 612.68 686.02 612.68 6.64 0.00
    I40-2-1-8 601.37 635.83 582.26 645.55 573.59 4.84 1.51
    I40-2-1-9 612.71 654.01 610.28 677.78 598.44 2.38 1.98
    Avg 3.95 3.71
    O40-2-1-0 573.13 607.63 530.77 613.94 494.73 15.85 7.28
    O40-2-1-1 451.96 479.67 476.94 535.79 434.04 4.13 9.88
    O40-2-1-2 465.61 486.61 452.72 499.94 445.80 4.44 1.55
    O40-2-1-3 523.35 523.35 523.35 523.35 523.35 0.00 0.00
    O40-2-1-4 549.35 549.35 549.35 549.35 549.35 0.00 0.00
    O40-2-1-5 577.85 601.43 603.35 666.88 573.11 0.83 5.28
    O40-2-1-6 492.06 525.27 492.06 492.06 489.14 0.60 0.60
    O40-2-1-7 545.21 550.93 545.21 545.21 528.80 3.10 3.10
    O40-2-1-8 431.92 455.46 430.07 473.46 427.55 1.02 0.59
    O40-2-1-9 579.27 602.79 601.79 668.05 578.85 0.07 3.96
    Avg 3.00 3.23
    A40-2-1-0 636.15 636.28 551.33 631.55 550.20 15.62 0.21
    A40-2-1-1 606.64 644.56 616.86 622.47 597.17 1.59 3.30
    A40-2-1-2 587.75 587.75 582.03 583.37 582.03 0.98 0.00
    A40-2-1-3 506.85 506.85 508.34 559.68 505.21 0.32 0.62
    A40-2-1-4 573.67 573.78 568.42 570.30 567.77 1.04 0.11
    A40-2-1-5 683.98 697.73 617.34 703.95 617.34 10.79 0.00
    A40-2-1-6 587.71 660.17 569.77 572.96 537.91 9.26 5.92
    A40-2-1-7 615.47 645.69 605.92 703.23 542.06 13.54 11.78
    A40-2-1-8 624.30 670.44 624.30 702.84 600.61 3.94 3.94
    A40-2-1-9 697.84 724.57 697.60 816.89 632.33 10.36 10.32
    Avg 6.75 3.62
     | Show Table
    DownLoad: CSV

    Table 16.  Meta-heuristics results for instances with 60 requestss

    Instance GRASP ILS GRASP-ILS
    Zmin Zavg Zmin Zavg Zmin ImpGRASP-ILS, GRASP ImpGRASP-ILS, ILS
    I60-2-1-0 971.76 930.24 954.90 1040.17 911.77 6.58 4.73
    I60-2-1-1 997.55 990.98 997.55 1070.57 987.76 0.99 0.99
    I60-2-1-2 893.43 879.02 897.08 1033.44 877.23 1.85 2.26
    I60-2-1-3 907.53 902.23 901.81 1008.95 882.44 2.84 2.20
    I60-2-1-4 979.14 967.49 975.61 1121.17 896.87 9.17 8.78
    I60-2-1-5 1004.53 997.95 1004.53 1065.50 996.95 0.76 0.76
    I60-2-1-6 1061.43 1051.96 1056.81 1093.90 1043.88 1.68 1.24
    I60-2-1-7 1005.16 981.49 950.37 1110.89 944.32 6.44 0.64
    I60-2-1-8 882.39 854.26 854.36 965.17 818.94 7.75 4.33
    I60-2-1-9 890.19 877.62 870.52 1028.26 865.24 2.88 0.61
    Avg 4.09 2.65
    O60-2-1-0 713.15 746.03 703.50 758.65 652.35 9.32 3.73
    O60-2-1-1 820.25 872.50 782.50 820.53 746.22 9.92 0.55
    O60-2-1-2 782.26 799.70 761.74 868.31 732.49 6.79 0.15
    O60-2-1-3 847.18 875.98 820.05 853.67 787.76 7.54 0.52
    O60-2-1-4 776.48 845.20 771.15 817.65 727.32 6.76 0.64
    O60-2-1-5 820.66 851.19 763.93 844.07 752.74 9.02 0.40
    O60-2-1-6 708.48 714.50 680.46 680.46 680.46 4.12 0.00
    O60-2-1-7 816.27 830.39 781.70 811.56 752.96 8.41 1.16
    O60-2-1-8 723.94 759.78 715.02 806.47 695.58 4.08 0.00
    O60-2-1-9 783.28 871.56 747.26 771.70 723.61 8.25 0.18
    Avg 7.42 0.73
    A60-2-1-0 858.31 858.31 836.63 929.83 836.54 2.60 0.01
    A60-2-1-1 956.96 1024.04 952.71 1049.60 911.33 5.01 4.54
    A60-2-1-2 719.41 741.14 776.51 778.37 703.79 2.22 10.33
    A60-2-1-3 849.45 928.96 802.22 912.04 781.73 8.66 2.62
    A60-2-1-4 857.32 880.81 841.54 899.44 793.69 8.02 6.03
    A60-2-1-5 808.57 830.08 761.17 803.11 740.28 9.22 2.82
    A60-2-1-6 803.29 862.81 803.29 871.89 754.71 6.44 6.44
    A60-2-1-7 853.23 853.23 802.72 905.31 780.28 9.35 2.88
    A60-2-1-8 780.61 815.35 772.43 843.73 751.38 3.89 2.80
    A60-2-1-9 763.12 795.25 762.09 825.88 705.99 8.09 7.95
    Avg 6.35 4.64
     | Show Table
    DownLoad: CSV

    Table 17.  Meta-heuristics results for instances with 100 requests

    Instance GRASP ILS GRASP-ILS
    Zmin Zavg Zmin Zavg Zmin ImpGRASP-ILS, GRASP ImpGRASP-ILS, ILS
    I100-2-1-0 1638.35 1704.61 1609.09 1890.61 1500.16 9.21 7.26
    I100-2-1-1 1475.76 1550.51 1462.03 1720.88 1399.61 5.44 4.46
    I100-2-1-2 1549.56 1714.37 1519.43 1792.65 1481.37 4.60 2.57
    I100-2-1-3 1541.71 1630.81 1516.8 1799.58 1406.15 9.64 7.87
    I100-2-1-4 1586.41 1607.53 1501.97 1774.75 1471.77 7.79 2.05
    I100-2-1-5 1547.53 1591.87 1476.42 1763.20 1465.71 5.58 0.73
    I100-2-1-6 1594.07 1622.95 1501.05 1791.06 1482.92 7.50 1.22
    I100-2-1-7 1568.12 1653.99 1508.18 1741.59 1439.77 8.91 4.75
    I100-2-1-8 1340.87 1424.62 1358.42 1513.15 1277.58 4.95 6.33
    I100-2-1-9 1393.89 1430.99 1346.03 1604.42 1286.29 8.37 4.64
    Avg 7.20 4.19
    O100-2-1-0 1279.18 1347.70 1252.87 1352.97 1215.92 5.20 3.04
    O100-2-1-1 1169.86 1255.83 1149.36 1236.94 1113.13 5.10 3.25
    O100-2-1-2 1123.63 1191.98 1098.08 1156.11 1066.12 5.39 3.00
    O100-2-1-3 1272.18 1320.34 1209.93 1297.75 1166.47 9.06 3.73
    O100-2-1-4 1278.7 1376.42 1206.94 1273.12 1146.62 11.52 5.26
    O100-2-1-5 1282.2 1310.21 1264.78 1444.76 1223.53 4.80 3.37
    O100-2-1-6 1112.92 1195.06 1099.75 1266.59 1062.43 4.75 3.51
    O100-2-1-7 1271.97 1338.24 1231.68 1310.19 1224.31 3.89 0.60
    O100-2-1-8 1071.1 1141.54 1000.51 1190.17 971.56 10.25 2.98
    O100-2-1-9 1163.25 1211.45 1197.58 1339.77 1098.25 5.92 9.04
    Avg 6.59 3.78
    A100-2-1-0 1280.19 1361.66 1265.52 1359.89 1212.68 5.57 4.36
    A100-2-1-1 1362.77 1431.84 1362.04 1541.04 1271.73 7.16 7.10
    A100-2-1-2 1183.91 1240.90 1159.67 1326.68 1129.63 4.81 2.66
    A100-2-1-3 1214.65 1288.61 1198.17 1355.01 1162.85 2.73 3.04
    A100-2-1-4 1236.76 1325.19 1243.54 1400.64 1190.23 3.91 4.48
    A100-2-1-5 1177.71 1241.31 1144.83 1228.66 1093.48 7.70 4.70
    A100-2-1-6 1270.56 1328.54 1256.54 1336.74 1208.76 5.11 3.95
    A100-2-1-7 1227.98 1308.64 1221.57 1316.30 1174.24 4.58 4.03
    A100-2-1-8 1210.54 1298.73 1192.27 1288.85 1154.51 3.99 3.27
    A100-2-1-9 1271.11 1320.95 1253.95 1419.94 1217.18 4.43 3.02
    Avg 5.00 4.06
     | Show Table
    DownLoad: CSV

    Table 18.  Cost savings for instances with 20 requests

    Instance Non-collaboration Collaboration
    Cost NV Cost NV $\phi$
    I20-2-1-0 528.74 5 459.26 3 13.14
    I20-2-1-1 477.54 4 366.66 2 23.22
    I20-2-1-2 612.48 4 478.19 3 21.93
    I20-2-1-3 481.75 4 360.24 2 25.22
    I20-2-1-4 491.83 4 383.24 2 22.08
    I20-2-1-5 427.16 4 343.7 2 19.54
    I20-2-1-6 453.01 5 344.51 2 23.95
    I20-2-1-7 476.59 4 354.59 2 25.60
    I20-2-1-8 484.62 4 351.37 2 27.50
    I20-2-1-9 622.55 5 476.32 3 23.49
    Avg 4.3 2.3 22.57
    O20-2-1-0 283.63 4 240.46 2 15.22
    O20-2-1-1 299.27 4 260.84 2 12.84
    O20-2-1-2 336.31 4 250.62 2 25.48
    O20-2-1-3 316.26 4 254.79 2 19.44
    O20-2-1-4 329.42 5 273.79 2 16.89
    O20-2-1-5 323.26 4 269.61 2 16.60
    O20-2-1-6 293.28 4 260.31 2 11.24
    O20-2-1-7 348.76 4 245.95 2 29.48
    O20-2-1-8 296.46 4 240.28 2 18.95
    O20-2-1-9 263.93 4 229.92 2 12.89
    Avg 4.1 2 17.90
    A20-2-1-0 403.54 4 362.31 2 10.22
    A20-2-1-1 494.62 4 444.8 3 10.07
    A20-2-1-2 383.46 5 329.67 2 14.03
    A20-2-1-3 410.31 5 363.62 2 11.38
    A20-2-1-4 386.48 4 343.77 2 11.05
    A20-2-1-5 302.36 4 287.74 2 4.83
    A20-2-1-6 371.57 4 338.23 2 8.97
    A20-2-1-7 333.01 4 290.21 2 12.85
    A20-2-1-8 358.15 4 314.2 2 12.27
    A20-2-1-9 367.36 4 323.6 2 11.91
    Avg 4.2 10.76
     | Show Table
    DownLoad: CSV

    Table 19.  Cost savings for instances with 40 requests

    Instance Non- collaboration Collaboration
    Cost NV Cost NV $\phi$
    I40-2-1-0 1021.39 8 706.64 6 30.82
    I40-2-1-1 861.87 8 600.90 6 30.28
    I40-2-1-2 1006.32 8 729.24 6 27.53
    I40-2-1-3 943.17 8 651.55 6 30.92
    I40-2-1-4 1062.08 8 699.59 6 34.13
    I40-2-1-5 930.23 8 647.36 6 30.41
    I40-2-1-6 874.73 8 590.24 6 32.52
    I40-2-1-7 910.38 8 612.68 6 32.70
    I40-2-1-8 841.23 8 573.59 6 31.82
    I40-2-1-9 843.80 8 598.44 6 29.08
    Avg 8 6 31.02
    O40-2-1-0 650.32 7 494.73 3 23.93
    O40-2-1-1 574.51 7 434.04 3 24.45
    O40-2-1-2 578.02 7 445.80 3 22.87
    O40-2-1-3 692.40 8 523.35 4 24.42
    O40-2-1-4 749.51 7 549.35 4 26.71
    O40-2-1-5 750.72 8 573.11 4 23.66
    O40-2-1-6 636.65 7 492.06 4 22.71
    O40-2-1-7 734.12 7 528.80 3 27.97
    O40-2-1-8 553.40 7 427.55 3 22.74
    O40-2-1-9 746.28 8 578.85 4 22.43
    Avg 7.3 3.5 24.19
    A40-2-1-0 686.52 8 550.2 4 19.86
    A40-2-1-1 730.78 7 597.17 4 18.28
    A40-2-1-2 708.31 8 582.03 4 17.83
    A40-2-1-3 662.45 8 505.21 4 23.74
    A40-2-1-4 667.54 8 567.77 4 14.95
    A40-2-1-5 714.79 7 617.34 4 13.63
    A40-2-1-6 610.43 7 537.91 3 11.88
    A40-2-1-7 639.30 7 542.06 4 15.21
    A40-2-1-8 695.09 7 600.61 4 13.59
    A40-2-1-9 739.69 8 632.33 4 14.51
    Avg 7.5 3.9 16.35
     | Show Table
    DownLoad: CSV

    Table 20.  Cost savings for instances with 60 requests

    Instance Non- collaboration Collaboration
    Cost NV Cost NV $\phi$
    I60-2-1-0 1405.67 11 911.77 6 35.14
    I60-2-1-1 1466.10 10 987.76 7 32.63
    I60-2-1-2 1326.57 12 877.23 6 33.87
    I60-2-1-3 1381.98 12 882.44 6 36.15
    I60-2-1-4 1341.28 11 896.87 5 33.13
    I60-2-1-5 1438.62 11 996.95 7 30.70
    I60-2-1-6 1575.24 13 1043.88 7 33.73
    I60-2-1-7 1384.43 10 944.32 6 31.79
    I60-2-1-8 1245.97 10 818.94 5 34.27
    I60-2-1-9 1266.52 10 865.24 8 31.68
    Avg 11 6.3 33.31
    O60-2-1-0 920.63 8 652.35 5 29.14
    O60-2-1-1 1049.04 11 742.55 4 29.22
    O60-2-1-2 1092.15 11 732.49 5 32.93
    O60-2-1-3 1083.86 11 787.76 5 27.32
    O60-2-1-4 1031.74 11 727.32 5 29.51
    O60-2-1-5 981.97 10 752.74 5 23.34
    O60-2-1-6 956.39 9 680.46 5 28.85
    O60-2-1-7 1052.40 11 752.96 5 28.45
    O60-2-1-8 977.50 9 695.58 5 28.84
    O60-2-1-9 959.39 10 723.61 5 24.58
    Avg 10.1 4.9 4.9 28.22
    A60-2-1-0 1133.63 11 836.54 6 26.21
    A60-2-1-1 1187.66 10 911.33 6 23.27
    A60-2-1-2 956.53 10 703.79 5 26.42
    A60-2-1-3 969.58 10 781.73 5 19.37
    A60-2-1-4 1074.74 10 793.69 5 26.15
    A60-2-1-5 984.88 9 740.28 5 24.84
    A60-2-1-6 999.87 10 754.71 5 24.52
    A60-2-1-7 1061.10 11 780.28 5 26.46
    A60-2-1-8 962.43 11 751.38 5 21.93
    A60-2-1-9 941.98 9 705.99 5 25.05
    Avg 10.1 5.2 5.2 24.42
     | Show Table
    DownLoad: CSV

    Table 21.  Cost savings for instances with 100 requests

    Instance Non- collaboration Collaboration
    Cost NV Cost NV $\phi$
    I100-2-1-0 2741.02 20 1500.16 10 45.27
    I100-2-1-1 2308.06 17 1399.61 10 39.36
    I100-2-1-2 2634.95 19 1481.37 10 43.78
    I100-2-1-3 2517.73 18 1406.15 10 44.15
    I100-2-1-4 2275.46 17 1471.77 9 35.32
    I100-2-1-5 2393.39 17 1465.71 10 38.76
    I100-2-1-6 2635.37 19 1482.92 10 43.73
    I100-2-1-7 2628.76 19 1439.77 10 45.23
    I100-2-1-8 1910.26 15 1277.58 9 33.12
    I100-2-1-9 2337.01 17 1286.29 10 44.96
    Avg 17.8 9.8 41.37
    O100-2-1-0 1791.81 15 1215.92 9 32.14
    O100-2-1-1 1692.20 14 1113.13 8 34.22
    O100-2-1-2 1690.38 14 1066.12 8 36.93
    O100-2-1-3 1803.45 15 1166.47 9 35.32
    O100-2-1-4 1641.55 14 1146.62 9 30.15
    O100-2-1-5 1924.70 15 1223.53 9 36.43
    O100-2-1-6 1606.09 14 1062.43 8 33.85
    O100-2-1-7 1786.01 14 1224.31 9 31.45
    O100-2-1-8 1538.25 13 971.56 7 36.84
    O100-2-1-9 1582.04 13 1098.25 8 30.58
    Avg 14.1 8.4 33.79
    A100-2-1-0 1760.57 14 1212.68 9 31.12
    A100-2-1-1 1809.52 15 1271.73 9 29.72
    A100-2-1-2 1717.81 14 1129.63 8 34.24
    A100-2-1-3 1497.94 10 1162.85 8 22.37
    A100-2-1-4 1611.69 14 1190.23 9 26.15
    A100-2-1-5 1613.52 14 1093.48 8 32.23
    A100-2-1-6 1895.50 15 1208.76 9 36.23
    A100-2-1-7 1764.98 14 1174.24 8 33.47
    A100-2-1-8 1764.23 14 1154.51 9 34.56
    A100-2-1-9 1727.48 14 1217.18 10 29.54
    Avg 13.8 8.7 30.96
     | Show Table
    DownLoad: CSV

    Table 4.  Notations of performance indicators and their description

    Notations Description
    UB The best cost obtained by CPLEX solver in a preset running time.
    LB The lower bound obtained by CPLEX solver in a preset running time
    Ini The cost of the best initial solution obtained by GRASP-ILS
    Zmin The best cost of 10 runs for each meta-heuristic
    Zavg The average cost of 10 runs for each meta-heuristic
    Dev The standard deviation of the costs obtained in ten runs by each meta-heuristic.
    $Imp_{Ini}$ The improvement of best cost over Ini.
    For CPLEX solver $Imp_{Ini}$ is calculated by ${{(UB - Ini)} / {UB}}*100\%$
    For meta-heuristics, $Imp_{Ini}$ is calculated by$ {{({z_{min}} - Ini)} /{{z_{min}}}}*100\%$
    $Gap_{LB}$ The percentage gap between the best cost and LB.
    For CPLEX solver, $Gap_{LB}$ is calculated by $Gap_{LB} = {{(UB - Ini)} /{UB}}*100\%$
    For meta-heuristics, $Gap_{LB}$ is calculated by$ {{({z_{min}} - LB)} /{{z_{min}}}}*100\%$
    $Imp_{Cplex}$ The improvement of the best cost obtained by meta-heuristics over the best cost
    obtained by CPLEX solver.$Imp_{Cplex}$ is calculated as:${{(UB - {z_{min}})} /{UB}}*100\%$
    $Imp_{GRASP-ILS,GRASP}$ The improvement of over. It is calculated as:
    ${{({Z_{min,GRASP}} - {z_{min,GRASP - ILS}})} / {{z_{min,GRASP - ILS}}}}*100\%$
    $Imp_{GRASP-ILS,ILS}$ The improvement of over. It is calculated as:
    ${{({Z_{min,ILS}} - {z_{min,GRASP - ILS}})} / {{z_{min,GRASP - ILS}}}}*100\%$
    NV The average total number of vehicles used in the transportation
    $Cost_{N}$ The total transportation cost of the shippers without collaboration obtained by CPLEX solver.
    $Cost_{C}$ The total transportation cost of the shippers with collaboration, which is the best objective value of CPLEX solver and GRASP-ILS.
    $\phi$ The cost savings in percentage achieved by the collaboration among the shippers. is defined as: ${{({Cost_N} - {Cost_C})} /{{Cost_N}}}*100\%$
    $CPU_{avg}$ The average execution time in seconds for meta-heuristics
     | Show Table
    DownLoad: CSV
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