Figure 1.
Loading and unloading activities at a port
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Linear bilevel multiobjective optimization problem: Penalty approach
July
2019, 15(3): 1185-1211.
doi: 10.3934/jimo.2018091
Maritime inventory routing problem with multiple time windows
| 1. | Department of Industrial Engineering, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia |
| 2. | School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia |
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.
Keywords:
Maritime transportation,
inventory routing problem,
multiple time windows,
exact algorithm,
multi-heuristics,
genetic algorithm.
Mathematics Subject Classification:
Primary: 90B06, 90B10, 90B90; Secondary: 90C11, 68R99.
Citation:
Nurhadi Siswanto, Stefanus Eko Wiratno, Ahmad Rusdiansyah, Ruhul Sarker. Maritime inventory routing problem with multiple time windows. Journal of Industrial & Management Optimization,
2019, 15
(3)
: 1185-1211.
doi: 10.3934/jimo.2018091
![]()
References:
| [1] |
A. Agra, M. Christiansen, et al., A maritime inventory routing problem with stochastic sailing and port times, Computers & Operations Research, 61 (2015), 18-30
doi: 10.1016/j.cor.2015.01.008. |
| [2] |
F. Al-Khayyal and S. J. Hwang,
Discrete Optimization-Inventory constrained maritime routing and scheduling for multi-commodity liquid bulk, Part Ⅰ: Applications and model, European Journal of Operational Research, 176 (2007), 106-130.
doi: 10.1016/j.ejor.2005.06.047. |
| [3] |
K. Chakhlevitch and P. Cowling, Hyperheuristics: Recent developments, in Adaptive and Multilevel Metaheuristics (eds. C. Cotta, M. Sevaux, K. Sorensen), Springer-Verlag Berlin Heidelberg, (2008), 3-29 |
| [4] |
M. Christiansen and B. Nygreen,
A method for solving ship routing problems with inventory constraints, Annals of Operations Research, 81 (1998), 357-378.
doi: 10.1023/A:1018921527269. |
| [5] |
M. Christiansen and B. Nygreen,
Modeling path flows for a combined ship routing and inventory management problem, Annals of Operations Research, 82 (1998), 391-412.
doi: 10.1023/A:1018979107222. |
| [6] |
M. Christiansen,
Decomposition of a combined inventory and time constrained ship routing problem, Transportation Science, 33 (1999), 3-16.
doi: 10.1287/trsc.33.1.3. |
| [7] |
M. Christiansen and K. Fagerholt,
Robust ship scheduling with multiple time windows, Naval Research Logistics, 49 (2002), 611-625.
doi: 10.1002/nav.10033. |
| [8] |
M. Christiansen, K. Fagerholt, B. Nygreen and D. Ronen, Chapter 4 maritime transportation in handbooks in operations research and management science, (eds, C. Barnhart, G. Laporte), North-Holland, Amsterdam, 14 (2007), 189-284. |
| [9] |
M. Christiansen and K. Fagerholt, Maritime inventory routing problems, in Encyclopedia of Optimization, Second edition (eds C. A. Floudas, P. M. Pardalos), Springer-Verlag, (2009), 1947-1955 |
| [10] |
M. Christiansen, K. Fagerholt, T. Flatberg, Ø. Haugen, O. Kloster and E. H. Lund,
Maritime inventory routing with multiple products: A case study from the cement industry, European Journal of Operational Research, 208 (2011), 86-94.
doi: 10.1016/j.ejor.2010.08.023. |
| [11] |
K. F. Doerner, M. Gronalt, R. F. Hartl, G. Kiechle and M. Reimann,
Exact and heuristic algorithms for the vehicle routing problem with multiple interdependent time windows, Computers and Operations Research, 35 (2008), 3034-3048.
doi: 10.1016/j.cor.2007.02.012. |
| [12] |
D. Favaretto, E. Moretti and P. Pellegrini,
Ant colony system for a VRP with multiple time windows and multiple visits, Journal of Interdisciplinary Mathematics, 10 (2007), 263-284.
doi: 10.1080/09720502.2007.10700491. |
| [13] |
K. C. Furman, J. H. Song, G. R. Kocis, M. K. McDonald and P. H. Warrick,
Feedstock routing in the exxonmobil downstream sector, Interfaces, 41 (2011), 149-163.
doi: 10.1287/inte.1100.0508. |
| [14] |
A. Hemmati, L. M. Hvattum, et al., An iterative two-phase hybrid matheuristic for a multiproduct short sea inventory-routing problem, European Journal of Operational Research, 252 (2016), 775-788
doi: 10.1016/j.ejor.2016.01.060. |
| [15] |
S. J. Hwang, Inventory Constrained Maritime Routing and Scheduling for Multi-Commodity Liquid Bulk, Dissertation, School of Industrial and Systems Engineering, Georgia Institute of Technology, USA, 2005. |
| [16] |
Y. Jiang and I. E. Grossmann,
Alternative mixed-integer linear programming models of a maritime inventory routing problem, Computers & Chemical Engineering, 77 (2015), 147-161.
doi: 10.1016/j.compchemeng.2015.03.005. |
| [17] |
J. Lee and B. I. Kim,
Industrial ship routing problem with split delivery and two types of vessels, Expert Systems with Applications, 42 (2015), 9012-9023.
doi: 10.1016/j.eswa.2015.07.059. |
| [18] |
D. J. Papageorgiou, G. L. Nemhauser, J. Sokol, M. S. Cheon and A. B. Keha,
MIRPLib - A library of maritime inventory routing problem instances: Survey, core model, and benchmark results, European Journal of Operational Research, 235 (2014), 350-366.
doi: 10.1016/j.ejor.2013.12.013. |
| [19] |
D. J. Papageorgiou and A. B. Keha, et al., Two-stage decomposition algorithms for single product maritime inventory routing, INFORMS Journal on Computing, 26 (2014), 825-847
doi: 10.1287/ijoc.2014.0601. |
| [20] |
V. Rodrigues and R. Morabito, et al., Ship routing with pickup and delivery for a maritime oil transportation system: MIP model and heuristics, Systems, 4 (2016), p31. |
| [21] |
B. Santosa and R. Damayanti, et al., Solving multi-product inventory ship routing with a heterogeneous fleet model using a hybrid cross entropy-genetic algorithm: a case study in Indonesia, Production & Manufacturing Research, 4 (2016), 90-113
doi: 10.1080/21693277.2016.1204961. |
| [22] |
M. Savelsbergh and J. H. Song,
Inventory routing with continuous moves, Computers and Operations Research, 34 (2007), 1744-1763.
doi: 10.1016/j.cor.2005.05.036. |
| [23] |
N. Siswanto, D. Essam and R. Sarker,
Solving the ship inventory routing and scheduling problem with undedicated compartments, Computers and Industrial Engineering, 61 (2011), 289-299.
doi: 10.1109/ICCIE.2009.5223771. |
| [24] |
N. Siswanto, D. Essam and R. Sarker, Multi-heuristics based genetic algorithm for solving maritime inventory routing problem, IEEE International Conference on Industrial Engineering and Engineering Management, Singapore, (2011), 116-120
doi: 10.1109/IEEM.2011.6117890. |
| [25] |
J. Sokol, C. Zhang, G. Nemhauser, D. Papageorgiou and M. S. Cheon, Robust inventory routing with flexible time window allocation, Working paper, 2015. |
| [26] |
J. H. Song and K. C. Furman,
A maritime inventory routing problem: Practical approach, Computers and Operations Research, 40 (2011), 657-665.
doi: 10.1016/j.cor.2010.10.031. |
| [27] |
F. Tricoire, M. Romauch, K. F. Doerner and R. F. Hartl,
Heuristics for the multi-period orienteering problem with multiple time windows, Computers and Operations Research, 37 (2010), 351-367.
doi: 10.1016/j.cor.2009.05.012. |
show all references
References:
| [1] |
A. Agra, M. Christiansen, et al., A maritime inventory routing problem with stochastic sailing and port times, Computers & Operations Research, 61 (2015), 18-30
doi: 10.1016/j.cor.2015.01.008. |
| [2] |
F. Al-Khayyal and S. J. Hwang,
Discrete Optimization-Inventory constrained maritime routing and scheduling for multi-commodity liquid bulk, Part Ⅰ: Applications and model, European Journal of Operational Research, 176 (2007), 106-130.
doi: 10.1016/j.ejor.2005.06.047. |
| [3] |
K. Chakhlevitch and P. Cowling, Hyperheuristics: Recent developments, in Adaptive and Multilevel Metaheuristics (eds. C. Cotta, M. Sevaux, K. Sorensen), Springer-Verlag Berlin Heidelberg, (2008), 3-29 |
| [4] |
M. Christiansen and B. Nygreen,
A method for solving ship routing problems with inventory constraints, Annals of Operations Research, 81 (1998), 357-378.
doi: 10.1023/A:1018921527269. |
| [5] |
M. Christiansen and B. Nygreen,
Modeling path flows for a combined ship routing and inventory management problem, Annals of Operations Research, 82 (1998), 391-412.
doi: 10.1023/A:1018979107222. |
| [6] |
M. Christiansen,
Decomposition of a combined inventory and time constrained ship routing problem, Transportation Science, 33 (1999), 3-16.
doi: 10.1287/trsc.33.1.3. |
| [7] |
M. Christiansen and K. Fagerholt,
Robust ship scheduling with multiple time windows, Naval Research Logistics, 49 (2002), 611-625.
doi: 10.1002/nav.10033. |
| [8] |
M. Christiansen, K. Fagerholt, B. Nygreen and D. Ronen, Chapter 4 maritime transportation in handbooks in operations research and management science, (eds, C. Barnhart, G. Laporte), North-Holland, Amsterdam, 14 (2007), 189-284. |
| [9] |
M. Christiansen and K. Fagerholt, Maritime inventory routing problems, in Encyclopedia of Optimization, Second edition (eds C. A. Floudas, P. M. Pardalos), Springer-Verlag, (2009), 1947-1955 |
| [10] |
M. Christiansen, K. Fagerholt, T. Flatberg, Ø. Haugen, O. Kloster and E. H. Lund,
Maritime inventory routing with multiple products: A case study from the cement industry, European Journal of Operational Research, 208 (2011), 86-94.
doi: 10.1016/j.ejor.2010.08.023. |
| [11] |
K. F. Doerner, M. Gronalt, R. F. Hartl, G. Kiechle and M. Reimann,
Exact and heuristic algorithms for the vehicle routing problem with multiple interdependent time windows, Computers and Operations Research, 35 (2008), 3034-3048.
doi: 10.1016/j.cor.2007.02.012. |
| [12] |
D. Favaretto, E. Moretti and P. Pellegrini,
Ant colony system for a VRP with multiple time windows and multiple visits, Journal of Interdisciplinary Mathematics, 10 (2007), 263-284.
doi: 10.1080/09720502.2007.10700491. |
| [13] |
K. C. Furman, J. H. Song, G. R. Kocis, M. K. McDonald and P. H. Warrick,
Feedstock routing in the exxonmobil downstream sector, Interfaces, 41 (2011), 149-163.
doi: 10.1287/inte.1100.0508. |
| [14] |
A. Hemmati, L. M. Hvattum, et al., An iterative two-phase hybrid matheuristic for a multiproduct short sea inventory-routing problem, European Journal of Operational Research, 252 (2016), 775-788
doi: 10.1016/j.ejor.2016.01.060. |
| [15] |
S. J. Hwang, Inventory Constrained Maritime Routing and Scheduling for Multi-Commodity Liquid Bulk, Dissertation, School of Industrial and Systems Engineering, Georgia Institute of Technology, USA, 2005. |
| [16] |
Y. Jiang and I. E. Grossmann,
Alternative mixed-integer linear programming models of a maritime inventory routing problem, Computers & Chemical Engineering, 77 (2015), 147-161.
doi: 10.1016/j.compchemeng.2015.03.005. |
| [17] |
J. Lee and B. I. Kim,
Industrial ship routing problem with split delivery and two types of vessels, Expert Systems with Applications, 42 (2015), 9012-9023.
doi: 10.1016/j.eswa.2015.07.059. |
| [18] |
D. J. Papageorgiou, G. L. Nemhauser, J. Sokol, M. S. Cheon and A. B. Keha,
MIRPLib - A library of maritime inventory routing problem instances: Survey, core model, and benchmark results, European Journal of Operational Research, 235 (2014), 350-366.
doi: 10.1016/j.ejor.2013.12.013. |
| [19] |
D. J. Papageorgiou and A. B. Keha, et al., Two-stage decomposition algorithms for single product maritime inventory routing, INFORMS Journal on Computing, 26 (2014), 825-847
doi: 10.1287/ijoc.2014.0601. |
| [20] |
V. Rodrigues and R. Morabito, et al., Ship routing with pickup and delivery for a maritime oil transportation system: MIP model and heuristics, Systems, 4 (2016), p31. |
| [21] |
B. Santosa and R. Damayanti, et al., Solving multi-product inventory ship routing with a heterogeneous fleet model using a hybrid cross entropy-genetic algorithm: a case study in Indonesia, Production & Manufacturing Research, 4 (2016), 90-113
doi: 10.1080/21693277.2016.1204961. |
| [22] |
M. Savelsbergh and J. H. Song,
Inventory routing with continuous moves, Computers and Operations Research, 34 (2007), 1744-1763.
doi: 10.1016/j.cor.2005.05.036. |
| [23] |
N. Siswanto, D. Essam and R. Sarker,
Solving the ship inventory routing and scheduling problem with undedicated compartments, Computers and Industrial Engineering, 61 (2011), 289-299.
doi: 10.1109/ICCIE.2009.5223771. |
| [24] |
N. Siswanto, D. Essam and R. Sarker, Multi-heuristics based genetic algorithm for solving maritime inventory routing problem, IEEE International Conference on Industrial Engineering and Engineering Management, Singapore, (2011), 116-120
doi: 10.1109/IEEM.2011.6117890. |
| [25] |
J. Sokol, C. Zhang, G. Nemhauser, D. Papageorgiou and M. S. Cheon, Robust inventory routing with flexible time window allocation, Working paper, 2015. |
| [26] |
J. H. Song and K. C. Furman,
A maritime inventory routing problem: Practical approach, Computers and Operations Research, 40 (2011), 657-665.
doi: 10.1016/j.cor.2010.10.031. |
| [27] |
F. Tricoire, M. Romauch, K. F. Doerner and R. F. Hartl,
Heuristics for the multi-period orienteering problem with multiple time windows, Computers and Operations Research, 37 (2010), 351-367.
doi: 10.1016/j.cor.2009.05.012. |
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 |
| 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 |
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 | |||
| 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 | |||
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 |
| 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 |
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) |
||||||
| 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) |
||||||
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] |
| 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] |
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 | ||||||||||
| (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 | ||||||||||
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) | |||
| 1 | 10 | 55.8 | 20 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 50.3 |
| 50 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 105.6 | |||
| 100 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 222.7 | |||
| 15 | 91.4 | 20 | 91.4 | 0 | 108.7 | 94.3 | 5.4 | 0 | 166.0 | |
| 50 | 91.4 | 0 | 102.0 | 91.9 | 2.0 | 0 | 401.2 | |||
| 100 | 91.4 | 0 | 91.4 | 91.4 | 0 | 0 | 879.0 | |||
| 20 | 20 | 107.7 | - | 148.0 | 126.9 | 9.3 | 0 | 248.5 | ||
| 50 | 107.7 | - | 135.0 | 122.2 | 7.1 | 0 | 626.0 | |||
| 100 | 107.7 | - | 122.4 | 116.0 | 3.7 | 0 | 1,312.0 | |||
| 25 | 20 | 146.3 | - | 196.9 | 175.4 | 14.7 | 5 | 234.9 | ||
| 50 | 140.8 | - | 195.4 | 165.3 | 15.9 | 2 | 774.2 | |||
| 100 | 143.4 | - | 188.7 | 154.4 | 10.5 | 0 | 1,824.1 | |||
| 2 | 10 | 66.8 | 20 | 66.8 | 0 | 76.8 | 68.4 | 3.6 | 0 | 93.08 |
| 50 | 66.8 | 0 | 68.6 | 66.9 | 0.2 | 0 | 212.9 | |||
| 100 | 66.8 | 0 | 66.8 | 66.8 | 0 | 0 | 497.1 | |||
| 15 | 103.0 | 20 | 109.3 | 6.12 | 140.2 | 125.6 | 6.4 | 0 | 197.3 | |
| 50 | 103.0 | 0 | 130.5 | 120.2 | 6.5 | 0 | 535.2 | |||
| 100 | 105.2 | 2.14 | 124.2 | 116.6 | 4.4 | 0 | 1,207.4 | |||
| 20 | 149.7 | - | 191.8 | 170.3 | 9.8 | 3 | 208.3 | |||
| 50 | 142.3 | - | 184.0 | 166.2 | 7.5 | 5 | 694.3 | |||
| 100 | 149.7 | - | 191.0 | 163.7 | 10.4 | 3 | 1,520.2 | |||
| 25 | 20 | 177.5 | - | 231.2 | 202.8 | 12.5 | 12 | 295.8 | ||
| 50 | 171.6 | - | 207.3 | 190.9 | 10.4 | 6 | 938.4 | |||
| 100 | 173.6 | - | 208.7 | 187.0 | 8.6 | 4 | 1,771.2 | |||
| 3 | 10 | 99.0 | 20 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 43.9 |
| 50 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 109.0 | |||
| 100 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 239.9 | |||
| 15 | 216.0 | 20 | 216.0 | 0 | 241.0 | 222.4 | 5.0 | 0 | 147.0 | |
| 50 | 216.0 | 0 | 241.0 | 220.1 | 4.8 | 0 | 377.2 | |||
| 100 | 216.0 | 0 | 221.0 | 217.4 | 2.3 | 0 | 796.1 | |||
| 20 | 20 | 306.0 | - | 423.0 | 343.6 | 25.1 | 0 | 184.0 | ||
| 50 | 304.0 | - | 344.0 | 316.5 | 11.35 | 0 | 585.2 | |||
| 100 | 304.0 | - | 401.0 | 312.0 | 16.0 | 0 | 1,222.6 | |||
| 25 | 20 | 401.0 | - | 508.0 | 466.8 | 24.4 | 1 | 236.7 | ||
| 50 | 346.0 | - | 522.0 | 422.5 | 46.2 | 2 | 753.3 | |||
| 100 | 346.0 | - | 483.0 | 404.8 | 41.6 | 0 | 1,476.9 | |||
| 4 | 10 | 137.0 | 20 | 137.0 | 0 | 147.0 | 140.0 | 4.6 | 0 | 84.7 |
| 50 | 137.0 | 0 | 137.0 | 137.0 | 0 | 0 | 180.4 | |||
| 100 | 137.0 | 0 | 137.0 | 137.0 | 0 | 0 | 406.6 | |||
| 15 | 265.0 | 20 | 277.0 | 4.53 | 363.0 | 291.6 | 14.6 | 0 | 196.3 | |
| 50 | 275.0 | 3.77 | 294.0 | 284.6 | 5.4 | 0 | 530.0 | |||
| 100 | 265.0 | 0 | 287.0 | 281.4 | 4.8 | 0 | 1,294.4 | |||
| 20 | 20 | 354.0 | - | 479.0 | 423.7 | 24.7 | 1 | 217.5 | ||
| 50 | 407.0 | - | 454.0 | 423.2 | 15.2 | 5 | 676.3 | |||
| 100 | 350.0 | - | 454.0 | 396.3 | 29.5 | 0 | 1,498.9 | |||
| 25 | 20 | 484.0 | - | 632.0 | 543.2 | 31.9 | 14 | 304.6 | ||
| 50 | 431.0 | - | 567.0 | 517.9 | 34.3 | 7 | 894.2 | |||
| 100 | 431.0 | - | 558.0 | 505.3 | 31.8 | 6 | 1,818.1 | |||
| Note: (*) a feasible solution was not found before the time limit of 8.64E+4 seconds (24 hours) | ||||||||||
| 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) | |||
| 1 | 10 | 55.8 | 20 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 50.3 |
| 50 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 105.6 | |||
| 100 | 55.8 | 0 | 55.8 | 55.8 | 0 | 0 | 222.7 | |||
| 15 | 91.4 | 20 | 91.4 | 0 | 108.7 | 94.3 | 5.4 | 0 | 166.0 | |
| 50 | 91.4 | 0 | 102.0 | 91.9 | 2.0 | 0 | 401.2 | |||
| 100 | 91.4 | 0 | 91.4 | 91.4 | 0 | 0 | 879.0 | |||
| 20 | 20 | 107.7 | - | 148.0 | 126.9 | 9.3 | 0 | 248.5 | ||
| 50 | 107.7 | - | 135.0 | 122.2 | 7.1 | 0 | 626.0 | |||
| 100 | 107.7 | - | 122.4 | 116.0 | 3.7 | 0 | 1,312.0 | |||
| 25 | 20 | 146.3 | - | 196.9 | 175.4 | 14.7 | 5 | 234.9 | ||
| 50 | 140.8 | - | 195.4 | 165.3 | 15.9 | 2 | 774.2 | |||
| 100 | 143.4 | - | 188.7 | 154.4 | 10.5 | 0 | 1,824.1 | |||
| 2 | 10 | 66.8 | 20 | 66.8 | 0 | 76.8 | 68.4 | 3.6 | 0 | 93.08 |
| 50 | 66.8 | 0 | 68.6 | 66.9 | 0.2 | 0 | 212.9 | |||
| 100 | 66.8 | 0 | 66.8 | 66.8 | 0 | 0 | 497.1 | |||
| 15 | 103.0 | 20 | 109.3 | 6.12 | 140.2 | 125.6 | 6.4 | 0 | 197.3 | |
| 50 | 103.0 | 0 | 130.5 | 120.2 | 6.5 | 0 | 535.2 | |||
| 100 | 105.2 | 2.14 | 124.2 | 116.6 | 4.4 | 0 | 1,207.4 | |||
| 20 | 149.7 | - | 191.8 | 170.3 | 9.8 | 3 | 208.3 | |||
| 50 | 142.3 | - | 184.0 | 166.2 | 7.5 | 5 | 694.3 | |||
| 100 | 149.7 | - | 191.0 | 163.7 | 10.4 | 3 | 1,520.2 | |||
| 25 | 20 | 177.5 | - | 231.2 | 202.8 | 12.5 | 12 | 295.8 | ||
| 50 | 171.6 | - | 207.3 | 190.9 | 10.4 | 6 | 938.4 | |||
| 100 | 173.6 | - | 208.7 | 187.0 | 8.6 | 4 | 1,771.2 | |||
| 3 | 10 | 99.0 | 20 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 43.9 |
| 50 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 109.0 | |||
| 100 | 99.0 | 0 | 99.0 | 99.0 | 0 | 0 | 239.9 | |||
| 15 | 216.0 | 20 | 216.0 | 0 | 241.0 | 222.4 | 5.0 | 0 | 147.0 | |
| 50 | 216.0 | 0 | 241.0 | 220.1 | 4.8 | 0 | 377.2 | |||
| 100 | 216.0 | 0 | 221.0 | 217.4 | 2.3 | 0 | 796.1 | |||
| 20 | 20 | 306.0 | - | 423.0 | 343.6 | 25.1 | 0 | 184.0 | ||
| 50 | 304.0 | - | 344.0 | 316.5 | 11.35 | 0 | 585.2 | |||
| 100 | 304.0 | - | 401.0 | 312.0 | 16.0 | 0 | 1,222.6 | |||
| 25 | 20 | 401.0 | - | 508.0 | 466.8 | 24.4 | 1 | 236.7 | ||
| 50 | 346.0 | - | 522.0 | 422.5 | 46.2 | 2 | 753.3 | |||
| 100 | 346.0 | - | 483.0 | 404.8 | 41.6 | 0 | 1,476.9 | |||
| 4 | 10 | 137.0 | 20 | 137.0 | 0 | 147.0 | 140.0 | 4.6 | 0 | 84.7 |
| 50 | 137.0 | 0 | 137.0 | 137.0 | 0 | 0 | 180.4 | |||
| 100 | 137.0 | 0 | 137.0 | 137.0 | 0 | 0 | 406.6 | |||
| 15 | 265.0 | 20 | 277.0 | 4.53 | 363.0 | 291.6 | 14.6 | 0 | 196.3 | |
| 50 | 275.0 | 3.77 | 294.0 | 284.6 | 5.4 | 0 | 530.0 | |||
| 100 | 265.0 | 0 | 287.0 | 281.4 | 4.8 | 0 | 1,294.4 | |||
| 20 | 20 | 354.0 | - | 479.0 | 423.7 | 24.7 | 1 | 217.5 | ||
| 50 | 407.0 | - | 454.0 | 423.2 | 15.2 | 5 | 676.3 | |||
| 100 | 350.0 | - | 454.0 | 396.3 | 29.5 | 0 | 1,498.9 | |||
| 25 | 20 | 484.0 | - | 632.0 | 543.2 | 31.9 | 14 | 304.6 | ||
| 50 | 431.0 | - | 567.0 | 517.9 | 34.3 | 7 | 894.2 | |||
| 100 | 431.0 | - | 558.0 | 505.3 | 31.8 | 6 | 1,818.1 | |||
| Note: (*) a feasible solution was not found before the time limit of 8.64E+4 seconds (24 hours) | ||||||||||
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