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Maritime inventory routing problem with multiple time windows

  • * Corresponding author: Nurhadi Siswanto

    * Corresponding author: Nurhadi Siswanto 
The first author is supported by Ministry of Research, Technology and Higher Education, Republic of Indonesia through International Research Collaboration and Scientific Publication Research Grant No. 536/PKS/ITS/2017.
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  • This paper considers a maritime inventory routing problem with multiple time windows. The typical time windows considered that certain ports permit ships entering and leaving during the daytime only due to various operational limitations. We have developed an exact algorithm to represent this problem. However, due to the excessive computational time required for solving the model, we have proposed a multi-heuristics based genetic algorithm. The multi-heuristics are composed of a set of strategies that correspond to four decision points: ship selection, ship routing, the product type and the quantity of loading and unloading products. The experimental results show that the multi-heuristics can obtain acceptable solutions within a reasonable computational time. Moreover, the flexibility to add or remove the strategies means that the proposed method would not be difficult to implement for other variants of the maritime inventory routing problem.

    Mathematics Subject Classification: Primary: 90B06, 90B10, 90B90; Secondary: 90C11, 68R99.

    Citation:

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  • Figure 1.  Loading and unloading activities at a port

    Figure 2.  Daily multiple time windows at a port

    Figure 3.  Detailed activities of a ship during its time in a port

    Figure 4.  Several alternatives of a ship arriving and leaving a port when considering time windows

    Figure 5.  An Example of Chromosome

    Figure 6.  Chromosome in one and two steps

    Figure 7.  Changing states during every assignment in a chromosome

    Figure 8.  The values of the fitness functions for test problem 1 with the 15 day planning horizon from each of 40 runs

    Table 1.  An example of strategies for each decision point

    No Decision point Strategies
    1 Ship selection Based on the least ships current time
    2 Routing Visit two demand ports with the sequence based on the least CDik
    3 Loading Compartment [1] for product[1], compartment[2] for product[2] with loading quantities up to the maximum of compartments capacities
    4 Unloading Divide the same quantities for both ports
     | Show Table
    DownLoad: CSV

    Table 2.  Data of port and their storages

    No Description H1 H2 H3
    S11 S12 S21 S22 S31 S32
    1 Maximum capacity (unit) 160 180 55 41 68 51
    2 Minimum level (unit) 0 0 0 0 0 0
    3 Initial level (unit) 44 28 19 27 46 25
    4 Daily supply/demand rate (unit/day) 8 9 6 4 2 5
    5 Fixed setup loading time (day) 0.039 0.059 0.074 0.060 0.067 0.049
    6 Variable loading time (day/unit) 0.025 0.010 0.003 0.026 0.028 0.014
    7 Fixed setup loading cost ($) 10 8 6 9 8 10
    8 Variable loading cost ($/unit) 0 0 0 0 0 0
    9 Quantity penalty cost ($/day) 10 10 10 10 10 10
    10 Fixed setup port time (day) 0 0 0
    11 Fixed setup port cost ($) 0 0 0
    12 Daily starting time windows 7.12am 7.12am 7.12am
    13 Daily ending time windows 4.48pm 4.48am 4.48pm
     | Show Table
    DownLoad: CSV

    Table 3.  Data of ship and their compartments

    No Description V1 V2
    C11 C12 C21 C22
    1 Maximum compartment capacity 68 31 44 50
    2 Initial level 0 0 40 4
    3 Current product in the compartment - - P2 P1
     | Show Table
    DownLoad: CSV

    Table 4.  Data of travelling cost and time between ports

     | Show Table
    DownLoad: CSV

    Table 5.  The result of exact algorithm solved using Lingo

    Test Problem (TP) Planning Horizon (PH) Scenario 1 (Mp=3;Mc=2) Scenario 2 (Mp=3;Mc=2) Gap (%)
    Optimal Solution Run Time (in Second) Optimal Solution Run Time (in Second)
    1 10 55.8 1,329 - ﹥﹥8.64E + 4(*) -
    15 91.4 21,423 - ﹥﹥8.64E + 4(*) -
    2 10 66.8 1,012 66.8 582 0
    15 103, 0 25,451 103.0 74,166 0
    3 10 99.0 46 - ﹥﹥8.64E + 4(*) -
    15 216.0 34,827 - ﹥﹥8.64E + 4(*) -
    4 10 137.0 640 137.0 211 0
    15 265.0 47,210 265.0 ﹥﹥8.64E + 4(n) 0
    (n)the solution did not terminate before the time limit of 8.64E+4 seconds (24 hours)
    (*)a feasible solution was not obtained before the time limit of 8.64E+4 seconds (24 hours)
     | Show Table
    DownLoad: CSV

    Table 6.  The sequence of visiting demand ports

    Gene4 Gene3 The first visiting port The second visiting port
    0 0 CDP[0]
    1 CDP[0] CDP[1]
    1 0 CDP[1]
    1 CDP[1] CDP[0]
     | Show Table
    DownLoad: CSV

    Table 7.  An example of one ship's assignment output

    (A) Determining loading and unloading quantities
    Product # Quantity in supply port at the time a ship arrive(*) Remaining demands(*) Compartment Capacity(*) Loading Quantity(*) Unload for the first visited port(*) Unload for the second visited port(*)
    P1 40.8 28.6 44.0 28.6 0.0 28.6
    P1 50.4 43.8 50.0 43.8 16.0 27.8
    (*) in product units
    (B) Routing and schedule of the selected ship
    Source Port Departure Destination Port Port's last visit Arrival Waiting time to enter(n) Lay time
    Date Time Date Time Date Time Date Time
    H3 May 11 2.24pm H1 May 7 7.12am May 12 9.36am 0:00 May 12 9.36am
    H1 May 13 3.37pm H3 May 11 2.24am May 14 10.49am 0:00 May 14 10.49am
    H3 May 15 7.12am H2 May 11 7.12am May 16 7.12am 0:00 May 16 7.12am
    (n) in (hour:minutes)
    (C) Routing and schedule of the selected ship (continue)
    Destination Port Loading Time(n) Waiting time for supply/space(n) Ready to Leave Waiting time to leave(n) Leaving time
    Date Time Date Time
    H1 30:01 0:00 May 13 3.37pm 0:00 May 13 3.37pm
    H3 6:33 0:00 May 14 7.12pm 12:00 May 15 7.12am
    H2 22:38 0:00 May 17 5.50am 1:22 May 17 7.12am
    (n) in (hour:minutes)
    (D) Information of each visiting port's storage
    Destination Port Product# CDik Remaining Demand(*) Level of Storages(*) at ship's lay time Available space in storage(*) Loading/Unloading quantity(*)
    H1 P1 27.30 0.0 40.8 - 28.6
    H1 P2 26.80 0.0 50.4 - 43.8
    H3 P1 25.00 0.0 21.1 46.9 0.0
    H3 P2 21.80 16.0 36.7 14.3 16.0
    H2 P1 20.23 28.6 23.6 31.4 28.6
    H2 P2 18.05 27.8 7.0 34.0 27.8
    (*) in product units
     | Show Table
    DownLoad: CSV

    Table Appendix A.  The results of the multi-heuristics based GA in comparison to the results of the exact algorithm

    Test Problem (TP) Planning Horizon (PH) Exact Algorithm Solution Multi-Heuristics based GA (40 runs repetition)
    No. of Individuals in a population Best Solution (Min) Gap (%) Max. Solution Average Standart Deviation No. of infeasible solutions Average Running Time (in 2nd)
    11055.82055.8055.855.80050.3
    5055.8055.855.800105.6
    10055.8055.855.800222.7
    1591.42091.40108.794.35.40166.0
    5091.40102.091.92.00401.2
    10091.4091.491.400879.0
    20$(*)$20107.7-148.0126.99.30248.5
    50107.7-135.0122.27.10626.0
    100107.7-122.4116.03.701,312.0
    25$(*)$20146.3-196.9175.414.75234.9
    50140.8-195.4165.315.92774.2
    100143.4-188.7154.410.501,824.1
    21066.82066.8076.868.43.6093.08
    5066.8068.666.90.20212.9
    10066.8066.866.800497.1
    15103.020109.36.12140.2125.66.40197.3
    50103.00130.5120.26.50535.2
    100105.22.14124.2116.64.401,207.4
    20$(*)$20149.7-191.8170.39.83208.3
    50142.3-184.0166.27.55694.3
    100149.7-191.0163.710.431,520.2
    25$(*)$20177.5-231.2202.812.512295.8
    50171.6-207.3190.910.46938.4
    100173.6-208.7187.08.641,771.2
    31099.02099.0099.099.00043.9
    5099.0099.099.000109.0
    10099.0099.099.000239.9
    15216.020216.00241.0222.45.00147.0
    50216.00241.0220.14.80377.2
    100216.00221.0217.42.30796.1
    20$(*)$20306.0-423.0343.625.10184.0
    50304.0-344.0316.511.350585.2
    100304.0-401.0312.016.001,222.6
    25$(*)$20401.0-508.0466.824.41236.7
    50346.0-522.0422.546.22753.3
    100346.0-483.0404.841.601,476.9
    410137.020137.00147.0140.04.6084.7
    50137.00137.0137.000180.4
    100137.00137.0137.000406.6
    15265.020277.04.53363.0291.614.60196.3
    50275.03.77294.0284.65.40530.0
    100265.00287.0281.44.801,294.4
    20$(*)$20354.0-479.0423.724.71217.5
    50407.0-454.0423.215.25676.3
    100350.0-454.0396.329.501,498.9
    25$(*)$20484.0-632.0543.231.914304.6
    50431.0-567.0517.934.37894.2
    100431.0-558.0505.331.861,818.1
    Note: (*) a feasible solution was not found before the time limit of 8.64E+4 seconds (24 hours)
     | Show Table
    DownLoad: CSV
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