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Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization

  • * Corresponding author: Xia Zhao

    * Corresponding author: Xia Zhao 
This work is supported by the Programs of National Natural Science Foundation of China(No.51575108 and No.71403114), China Special Fund for Grain-scientific Research in the Public Interest(No.201513004), the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD) and Qinglan Project.
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  • Motivated by observing the importance of logistics cost in the cost structure of some products, this paper aims at multi-objective optimization of integrating supply chain network design with the selection of transportation modes (TMs) for a single-product four-echelon supply chain composed of suppliers, production plants, distribution centers (DCs) and customer zones. The key design decisions are the number, capacity and location of plants and DCs, the flow of products through the network, and the selection of TMs for each flow path. A bi-objective mixed integer linear programming model is first formulated. The two incompatible objectives are minimizing the total cost and maximizing the demand fill rate. The model is validated by applying to the case of the design of fresh apple supply chain. Then, a new metaheuristic, called multi-objective modified particle swarm optimization (MMPSO), is presented to find non-dominated solutions. A new modified binary PSO for updating binary variables along with the adaptive mutation is incorporated into the MMPSO. The MMPSO is compared with a multi-objective basic PSO (MBPSO) and the NSGA-Ⅱ against three small cases and six randomly generated medium and large size problems. The comparative results indicate that the MMPSO is better than the NSGA-Ⅱ and the MBPSO with respect to solution quality and computation efficiency for the problem.

    Mathematics Subject Classification: 90B80.

    Citation:

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  • Figure 1.  The studied single product four-echelon supply chain network

    Figure 2.  The solution representation of binary and continuous variables by a particle

    Figure 3.  Solution procedure of the MILP model by PSO and LINGO

    Figure 8.  Location map of an apple supply chain in Shanxi Province of China

    Figure 4.  Solution report of case 1 by LINGO given DFR as one

    Figure 5.  Non-dominated solutions of case 1 obtained by three algorithms

    Figure 6.  Non-dominated solutions of problem 4 obtained by three algorithms

    Figure 7.  Average and best values of ERs and ∆ versus nine problems

    Table 1.  Key decision variables of five typical non-dominated solutions for case 1

    # Plant locations DC locations
    YA TC BO XY WN XA BJ SH SZ WH ZZ CD XA
    1 1 0 0 1 1 0 0 0 0 1 1 0 1
    2 1 1 0 0 1 0 0 0 0 1 1 0 1
    3 1 0 0 1 0 1 0 0 0 1 1 0 1
    4 1 0 0 1 1 1 0 0 0 1 1 1 1
    5 1 1 1 1 1 0 0 0 0 1 1 1 1
    Demand BJ SH SZ WH CS ZZ CD CQ XA
    1 24000 32000 16000 32000 40000 24000 32000 24000 40000
    2 24000 32000 16287.6 33855.8 41188 27999.6 36140.1 24000 40000
    3 24000 32000 19765.1 35230.7 48442 25584.1 33739.3 27232.1 50000
    4 30000 32687.3 20000 40000 46619.4 30000 39736.1 24000 50000
    5 30000 40000 20000 40000 50000 30000 40000 30000 50000
    TM Supplier-to-Plant Plant-to-DC DC-to-CZ
    1 truck rail rail+truck
    2 truck rail rail+truck
    3 truck rail+truck rail+truck
    4 truck rail+truck rail+truck
    5 truck rail+truck rail+truck
    # 1 2 3 4 5
    Cost/1E8 4.7783 5.0033 5.3648 5.6808 5.9819
    DFR 0.8 0.835 0.897 0.949 1.0
     | Show Table
    DownLoad: CSV

    Table 6.  Expected maximal sale amounts of fresh apples for nine CZs(K tons)

    CZs BJ SH SZ WH CS ZZ CD CQ XA
    Apples 30 40 20 40 50 30 40 30 50
     | Show Table
    DownLoad: CSV

    Table 7.  Capacities and procumbent costs for five suppliers

    Locations YA TC BO XY WN
    Capacity (K tons) 120 60 60 160 100
    cost (RMB/ton) 610 620 630 620 630
     | Show Table
    DownLoad: CSV

    Table 8.  Variable costs of producing fresh apples for potential plants (RMB/ton)

    Sites YA TC BO XY WN XA
    Cost 500 520 540 530 530 520
     | Show Table
    DownLoad: CSV

    Table 9.  Variable costs of warehousing for nine sites (RMB/ton)

    DCs BJ SH SZ WH CS ZZ CD CQ XA
    Cost 420 430 440 410 400 400 400 400 400
     | Show Table
    DownLoad: CSV

    Table 10.  Unit transportation costs for supplier-to-plant (RMB/ton)

    cost YA TC BO XY WN XA
    YA 0 103.2 212.4 156 139.8 157.2
    TC 103.2 0 135 53.4 48 53.4
    BO 212.4 135 0 99.6 153 120
    XY 156 53.4 99.6 0 53.4 21
    WN 139.8 48 153 53.4 0 35.4
     | Show Table
    DownLoad: CSV

    Table 11.  Unit transportation costs of truck and rail (RMB/ton)

    cost BJ SH SZ WH CS ZZ CD CQ XA
    YA 127.6 150.8 199.6 116.1 143.3 75.9 100.8 93.9 37.6
    516 877.2 984.6 517.8 607.8 298.2 533.4 508.8 157.2
    TC 110.2 133.4 182.2 98.7 125.9 58.5 83.4 78.5 20.2
    588 866.4 899.4 467.4 529.2 300.6 444 406.2 53.4
    BO 114.2 137.4 186.2 102.7 129.9 62.5 61.4 82.5 24.2
    720.6 976.8 912.6 543 566.4 427.2 322.2 324 120
    XY 102.9 126.1 174.9 91.5 118.6 51.3 72.6 67.8 12.9
    636.6 879.6 866.4 454.8 503.4 327.6 393.6 352.2 21
    WN 97 120.2 187.7 85.5 112.7 45.3 78.6 73.7 15.4
    589.8 829.2 852 416.4 481.2 273.6 440.4 383.4 35.4
    XA 101.2 124.4 173.2 90 116.9 49.5 74.4 69.5 11.2
    624 858.6 852.6 435 486.6 308.4 405.6 355.2 0
    BJ 0 120.3 189.1 102.7 130.7 62.9 164.4 167 101.2
    0 670.2 1167 645 816 391.8 1030.2 958.2 624
    SH 120.3 0 137.6 72.8 99.6 86.1 172.5 173.8 124.4
    670.2 0 757.2 481.8 600.6 578.4 1159.8 1005.6 858.6
    SZ 189.1 137.6 0 103.9 75.3 153.5 187.9 164.3 173.2
    1167 757.2 0 528 381 816 858.6 689.4 852.6
    WH 102.7 72.8 103.9 0 38.3 51.4 113.1 102.7 90
    645 481.8 528 0 176.4 291 679.8 523.2 435
    CS 130.7 99.6 75.3 38.3 0 78.6 112.9 93.3 116.9
    816 600.6 381 176.4 0 442.8 616.2 442.8 486.6
    ZZ 62.9 86.1 153.5 51.4 78.6 0 112.7 115.5 49.5
    391.8 578.4 816 291 442.8 0 690 586.8 308.4
    CD 164.4 172.5 187.9 113.1 112.9 112.7 0 34.7 74.4
    1030.2 1159.8 858.6 679.8 616.2 690 0 175.8 405.6
    CQ 167 173.8 164.3 102.7 93.3 115.5 34.7 0 69.5
    958.2 1005.6 689.4 523.2 442.8 586.8 175.8 0 355.2
    Note: the italic numbers denote unit cost of rail, and non-italic numbers denote unit cost of truck.
     | Show Table
    DownLoad: CSV

    Table 5.  Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three medium and three large randomly generated problems

    Pro. Item ER CT/seconds $\Delta$
    MMPSO MBPSO NSGA-Ⅱ MMPSO MBPSO NSGA-Ⅱ MMPSO MBPSO NSGA-Ⅱ
    Pro
    4
    avg. 0.367 0.96 0.52 767.1 784.9 798.3 0.303 0.455 0.436
    best 0.3 0.933 0.13 737.6 762.3 773.1 0.199 0.394 0.373
    std. 0.0782 0.0279 0.288 35.8 20 29.3 0.049 0.047 0.051
    Pro
    5
    avg. 0.513 0.9 0.547 1147.9 1193.8 1174.6 0.318 0.714 0.453
    best 0.433 0.8 0.433 1091.6 1138.7 1131 0.266 0.695 0.408
    std. 0.0506 0.085 0.0989 36.7 41.2 41.6 0.039 0.014 0.043
    Pro
    6
    avg. 0.487 0.953 0.52 1487.6 1472.1 1491.9 0.315 0.502 0.461
    best 0.433 0.9 0.3 1424.8 1421.8 1404.4 0.232 0.358 0.412
    std. 0.0581 0.04 0.1609 46.4 38.69 100.8 0.062 0.084 0.041
    Pro
    7
    avg. 0.488 0.958 0.519 1511.4 1529.3 1538.9 0.304 0.436 0.423
    best 0.4 0.85 0.35 1484.8 1494.3 1526.6 0.239 0.356 0.378
    std. 0.0598 0.054 0.165 16.29 38.03 12.52 0.059 0.042 0.037
    Pro
    8
    avg. 0.48 0.946 0.48 1947.2 1953.6 1951.6 0.346 0.506 0.435
    best 0.375 0.9 0.4 1933.2 1932.8 1944.6 0.292 0.421 0.302
    std. 0.087 0.025 0.043 13.7 14.5 6.43 0.041 0.046 0.057
    Pro
    9
    avg. 0.472 0.958 0.544 2351.2 2383 2355.8 0.323 0.509 0.413
    best 0.35 0.95 0.325 2343 2316.3 2348.1 0.239 0.432 0.303
    std. 0.078 0.013 0.188 7.62 9.21 6.74 0.079 0.053 0.069
     | Show Table
    DownLoad: CSV

    Table 2.  Characteristics of all randomly generated SCND problems

    Problem # I J K S P M
    Medium size 4 8 10 12 19 3 3
    5 10 12 14 23 3 3
    6 12 14 15 25 3 3
    Large size 7 14 15 16 29 5 4
    8 15 18 18 33 5 4
    9 17 19 20 35 5 4
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison of location decisions for case 3 with and without TM selection

    Plant locations DC locations
    YA TC BO XY WN BJ SH SZ ZZ WH XA
    1 1 0 1 1 1 1 1 1 1 1
    1 1 0 1 1 0 1 0 1 1 1
    TM Selection
    Optional TMs Supplier-to-Plant Plant-to-DC DC-to-CZ
    Truck only Truck Truck Truck
    Truck and Rail Truck Rail Rail
    Note: the first row with italic numbers is binary decision without TM selection, and the second row is with TM selection
     | Show Table
    DownLoad: CSV

    Table 4.  Three metrics for MMPSO, MBPSO and NSGA-Ⅱ against three small cases

    Case Metric Algorithm Run 1 Run 2 Run 3 Run 4 Run 5 Average Best
    Case1 ER MMPSO 15/20 12/20 8/20 3/20 7/20 $45\%$ $15\%$
    MBPSO 19/20 11/20 12/20 14/20 16/20 72% 55%
    NSGA-Ⅱ 11/20 8/20 10/20 12/20 10/20 51% 40%
    CT/seconds MMPSO 46.550 49.901 45.615 41.948 48.298 46.462 41.948
    MBPSO 53.029 54.990 51.532 54.416 52.358 53.265 51.532
    NSGA-Ⅱ 56.441 53.788 52.416 52.791 54.148 53.917 52.416
    $\Delta$ MMPSO 0.478 0.467 0.313 0.322 0.462 0.408 0.313
    MBPSO 0.290 0.326 0.357 0.339 0.315 0.326 0.290
    NSGA-Ⅱ 0.580 0.470 0.626 0.426 0.645 0.549 0.426
    Case2 ER MMPSO 11/20 8/20 9/20 8/20 11/20 47% 40%
    MBPSO 20/20 15/20 18/20 16/20 16/20 85% 75%
    NSGA-Ⅱ 11/20 12/20 12/20 10/20 9/20 54% 50%
    CT/seconds MMPSO 54.413 52.953 54.179 59.585 59.405 56.107 52.953
    MBPSO 50.104 53.325 55.099 60.025 54.380 54.587 50.104
    NSGA-Ⅱ 56.471 58.846 60.847 55.968 57.705 57.967 55.968
    $\Delta$ MMPSO 0.283 0.369 0.326 0.275 0.341 0.319 0.275
    MBPSO 0.329 0.353 0.269 0.391 0.357 0.339 0.269
    NSGA-Ⅱ 0.631 0.578 0.455 0.517 0.503 0.537 0.455
    Case3 ER MMPSO 8/20 9/20 8/20 14/20 8/20 47% 40%
    MBPSO 17/20 14/20 17/20 13/20 16/20 77% 65%
    NSGA-Ⅱ 7/20 10/20 12/20 12/20 10/20 51% 35%
    CT/seconds MMPSO 43.607 43.898 37.106 39.952 42.460 41.405 37.106
    MBPSO 43.612 42.369 43.306 42.728 48.301 44.063 42.369
    NSGA-Ⅱ 45.038 48.282 44.382 45.383 46.448 45.907 44.382
    $\Delta$ MMPSO 0.365 0.257 0.373 0.338 0.286 0.324 0.257
    MBPSO 0.412 0.379 0.408 0.292 0.238 0.346 0.238
    NSGA-Ⅱ 0.536 0.544 0.562 0.498 0.571 0.542 0.498
     | Show Table
    DownLoad: CSV
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