# A Multi-Objective Tri-Level Algorithm for Hub-and-Spoke Network in Short Sea Shipping Transportation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Problem Description

- the islands that are assigned as hub nodes within several potential alternatives;
- the islands that are to be served by each route (vessel) originating either at a central port or at a hub node;
- the island service priority within each route.

## 4. Proposed Model

#### 4.1. Model Assumptions

- All passengers initiate their trips from a central port on the mainland.
- Two types of routes are considered, direct connection from a central port or transit travel through a hub node connection.
- Hub islands are served directly from a central port.
- All islands are served once.
- Several alternatives for placing hub nodes can be considered.
- Ships serving different routes can have different velocities.
- Time-varied intermediate stops are considered.

_{1}, w

_{2}, and w

_{3}are weight coefficients of the sub-objective functions. The first component closely relates to the traveling cost, while the second considers the number of passengers by destination, the average vessel speed, and the necessary time interval for passenger boarding and disembarking. These components are calculated as follows:

#### 4.2. Parameters

I | set of islands to be served |

i | island index of an initial (random) island list, ∀ i ∈ {1, 2, …, I} |

j | island index associated with the island service sequence, ∀ j ∈ {1, 2, …, I} |

K | set of routes (lines) |

k | route (line) index, k ∈ {1, 2, …, K} |

L | set of alternative ports for route initiation (L ≥ K) |

l | port index of route initiation, l ∈ {1, 2, …, L} |

#### 4.3. Data

${\left[{\mathrm{V}}_{\mathrm{l}}\right]}_{1\times \mathrm{L}}$ | vessel speed of journey initiating at port l, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{X}}_{0,\mathrm{l}}\right]}_{1\times \mathrm{L}}$ | X coordinate of the port where vessel l initiates, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{Y}}_{0,\mathrm{l}}\right]}_{1\times \mathrm{L}}$ | Y coordinate of the port where vessel l initiates, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{P}}_{\mathrm{i}}\right]}_{\mathrm{I}\times 1}$ | number of passengers disembarking in island I, ∀ i ∈ {1, 2, …, I} |

${\left[{\mathrm{T}}_{\mathrm{i}}\right]}_{\mathrm{I}\times 1}$ | passenger transfer time in island I, ∀ i ∈ {1, 2, …, I} |

${\left[{\mathrm{X}}_{\mathrm{i}}\right]}_{\mathrm{I}\times 1}$ | X coordinate of island i port, ∀ I ∈ {1, 2, …, I} |

${\left[{\mathrm{Y}}_{\mathrm{i}}\right]}_{\mathrm{I}\times 1}$ | Y coordinate of island i port, ∀ i ∈ {1, 2, …, I} |

#### 4.4. Decision Variables

${\left[{\mathsf{\delta}}_{\mathrm{j},\mathrm{i}}\right]}_{\mathrm{I}\times \mathrm{I}}$ | sequence permutation matrix for assigning island i to priority sequence j, ∀ i, j ∈ {1, 2, …, I} |

${\left[{\mathsf{\epsilon}}_{\mathrm{i},\mathrm{k}}\right]}_{\mathrm{I}\times \mathrm{K}}$ | island-to-route assignment matrix for assigning island i to route k, ∀ i ∈ {1, 2, …, I}, k ∈ {1, 2, …, K} |

${\left[{\mathsf{\zeta}}_{\mathrm{k},\mathrm{l}}\right]}_{\mathrm{K}\times \mathrm{L}}$ | route-to-origin node assignment matrix for assigning route k at initiation node l, ∀ k ∈ {1, 2, …, K}, l ∈ {1, 2, …, L} |

#### 4.5. Auxiliary Matrix

${\left[{\mathsf{\theta}}_{\mathrm{i},\mathrm{k}}\right]}_{\mathrm{I}\times \mathrm{K}}$ | Island-to-Hub Node Assignment Matrix for assigning island i as hub node of route k, ∀ i ∈ {1, 2, …, I}, k ∈ {1, 2, …, K} |

#### 4.6. Calculations

#### 4.6.1. Distances

${\left[{\mathrm{D}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the distance traveled between successive islands j − 1 and j of the vessel journey originating at port l, ∀ j ∈ {1, 2, …, I}, ∀ l ∈ {1, 2, …, L} |

#### 4.6.2. Passenger Hours

${\left[{\mathrm{A}\mathrm{D}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the cumulative distance of the vessel journey originating at port l up to island j, ∀ j ∈ {1, 2, …, I}, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{T}\mathrm{T}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the travel time between successive islands j − 1 and j of the vessel journey originating at port l, ∀ j ∈ {1, 2, …, I}, ∀ l ∈ {1, 2, …, L}. |

${\left[{\mathrm{A}\mathrm{T}}_{0,\mathrm{l}}\right]}_{1\times \mathrm{L}}$ | the arrival time at hub node connection l of vessel journey originating at a central port, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{A}\mathrm{T}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the arrival time at port j of the vessel journey departing from node port l, ∀ j ∈ {1, 2, …, I}, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{D}\mathrm{T}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the departure time from port j of the vessel journey originating at port l, ∀ j ∈ {1, 2, …, I}, ∀ l ∈ {1, 2, …, L} |

${\left[{\mathrm{P}\mathrm{H}}_{\mathrm{j},\mathrm{l}}\right]}_{\mathrm{I}\times \mathrm{L}}$ | the passenger hours spent on board by passengers in vessel departing from port l and disembarking at island j, ∀ j ∈ {1, 2, …, I},∀ l ∈ {1, 2, …, L} |

#### 4.7. Constraints and Preferences

- the travel time of each vessel does not exceed a specified value;
- the passenger travel time does not exceed a specified value;
- the service time of a particular island does not exceed a specified value;
- a desired priority for serving a particular island can be set;
- a specific island may be served directly from the central port;
- a minimum and/or maximum number of islands served by a single vessel can be set;
- constraints regarding islands’ (operational) capability to serve as hub nodes can be set.

#### 4.8. Model Structure

## 5. Case Study Applications

- C1: One vessel departs from a central port (Piraeus) and serves all islands without hub nodes.
- C2: Two vessels depart from central ports (one from Piraeus and one from Rafina) and serve all islands without hub nodes.
- C3: One vessel departs from a central port (Piraeus) with one hub connector.
- C4: Two vessels depart from central ports (Piraeus and Rafina) with one hub connector to be assigned by the algorithm.

_{max-trip}) among all passengers.

_{1}and w

_{2}of Equation (1) are altered (after scalarizing the two parameters), the produced point clouds are moved to an up-left and down-right direction (Figure 15a). A full-range Pareto front is formed based on individual points corresponding to different decision parameter weighting (Figure 15b).

^{®}Core ™ i7-7500, CPU 2.90 GHz, RAM 8.00 Gb) progressively increases and ranges from about a minute (case C1) up to seven minutes for the most demanding case (C4).

- C5: Four vessels depart from central ports (three from Piraeus and one from Rafina);
- C6: Four vessels depart from central ports and one from a hub connector (selected by algorithm among several alternatives);
- C7: Four vessels depart from central ports and two from hub connectors (selected by algorithm among several alternatives).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- COM (Commission of the European Communities). Communication from the Commission to the Council, the European Parliament, the Economic and Social Committee and the Committee of the Regions—The Development of Short Sea Shipping in Europe: A Dynamic Alternative in a Sustainable Transport Chain—Second Two-Yearly Progress Report; COM (Commission of the European Communities): Luxembourg, 1999. [Google Scholar]
- Ghaderi, H. Autonomous technologies in short sea shipping: Trends, feasibility and implications. Transp. Rev.
**2019**, 39, 152–173. [Google Scholar] [CrossRef] - O’Kelly, M.E. The location of interacting hub facilities. Transp. Sci.
**1986**, 20, 92–106. [Google Scholar] [CrossRef] - O’Kelly, M.E. Activity levels at hub facilities in interacting networks. Geogr. Anal.
**1986**, 18, 343–356. [Google Scholar] [CrossRef] - O’Kelly, M.E. A quadratic integer program for the location of interacting hub facilities. Eur. J. Oper. Res.
**1987**, 32, 393–404. [Google Scholar] [CrossRef] - O’Kelly, M.E. Hub facility location with fixed costs. Reg. Sci.
**1992**, 71, 293–306. [Google Scholar] [CrossRef] - Campbell, J.F. Integer programming formulations of discrete hub location problems. Eur. J. Oper. Res.
**1994**, 72, 387–405. [Google Scholar] [CrossRef] - Campbell, J.F. Hub location and the p-hub median problem. Oper. Res.
**1996**, 44, 923–935. [Google Scholar] [CrossRef] - Skorin-Kapov, D.; Skorin-Kapov, J.; O’Kelly, M. Tight linear programming relaxations of uncapacitated p-hub median problems. Eur. J. Oper. Res.
**1996**, 94, 582–593. [Google Scholar] [CrossRef] - Ernst, A.T.; Krishnamoorthy, M. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Locat. Sci.
**1996**, 4, 139–154. [Google Scholar] [CrossRef] - Ernst, A.T.; Krishnamoorthy, M. Exact and heuristic algorithms for the uncapacitated multiple allocation p-hub median problem. Eur. J. Oper. Res.
**1998**, 104, 100–112. [Google Scholar] [CrossRef] - Ernst, A.T.; Krishnamoorthy, M. Solution algorithms for the capacitated single allocation hub location problem. Ann. Oper. Res.
**1999**, 86, 141–159. [Google Scholar] [CrossRef] - Ebery, J. Solving large single allocation p-hub problems with two or three hubs. Eur. J. Oper. Res.
**2001**, 128, 447–458. [Google Scholar] [CrossRef] - Kratica, J.; Stanimirović, Z.; Tošić, D.; Filipović, V. Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem. Eur. J. Oper. Res.
**2007**, 182, 15–28. [Google Scholar] [CrossRef] - Kara, B.Y.; Tansel, B.C. On the single-assignment p-hub center problem. Eur. J. Oper. Res.
**2000**, 125, 648–655. [Google Scholar] [CrossRef] - Ernst, A.T.; Hamacher, H.; Jiang, H.; Krishnamoorthy, M.; Woeginger, G. Uncapacitated single and multiple allocation p-hub center problems. Comput. Oper. Res.
**2009**, 36, 2230–2241. [Google Scholar] [CrossRef] - Meyer, T.; Ernst, A.T.; Krishnamoorthy, M. A 2-phase algorithm for solving the single allocation p-hub center problem. Comput. Oper. Res.
**2009**, 36, 3143–3151. [Google Scholar] [CrossRef] - Ilić, A.; Urošević, D.; Brimberg, J.; Mladenović, N. A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem. Eur. J. Oper. Res.
**2010**, 206, 289–300. [Google Scholar] [CrossRef] - Kara, B.Y.; Tansel, B.C. The single-assignment hub covering problem: Models and linearizations. J. Oper. Res. Soc.
**2003**, 54, 59–64. [Google Scholar] [CrossRef] - Ernst, A.T.; Jiang, H.; Krishnamoorthy, M.; Baatar, D.; Judge, C. Reformulations and computational results for uncapacitated single and multiple allocation hub covering problems. CSIRO Math. Inf. Sci. 2005; unpublished Report. [Google Scholar]
- Contreras, I.; Cordeau, J.F.; Laporte, G. Stochastic uncapacitated hub location. Eur. J. Oper. Res.
**2011**, 212, 518–528. [Google Scholar] [CrossRef] - Correia, I.; Nickel, S.; Saldanha-da-Gama, F. Hub and spoke network design with single-assignment, capacity decisions and balancing requirements. Appl. Math. Model.
**2011**, 35, 4841–4851. [Google Scholar] [CrossRef] - An, Y.; Zhang, Y.; Zeng, B. The reliable hub-and-spoke design problem: Models and algorithms. Transp. Res. Part B Methodol.
**2015**, 77, 103–122. [Google Scholar] [CrossRef] [Green Version] - Rabbani, M.; Kazemi, S. Solving uncapacitated multiple allocation p-hub center problem by Dijkstra’s algorithm-based genetic algorithm and simulated annealing. Int. J. Ind. Eng. Comput.
**2015**, 6, 405–418. [Google Scholar] [CrossRef] - Ghaffari-Nasab, N.; Ghazanfari, M.; Teimoury, E. Robust optimization approach to the design of hub-and-spoke networks. Int. J. Adv. Manuf. Technol.
**2015**, 76, 1091–1110. [Google Scholar] [CrossRef] - Azizi, N.; Chauhan, S.; Salhi, S.; Vidyarthi, N. The impact of hub failure in hub-and-spoke networks: Mathematical formulations and solution techniques. Comput. Oper. Res.
**2016**, 65, 174–188. [Google Scholar] [CrossRef] - Zetina, C.A.; Contreras, I.; Cordeau, J.F.; Nikbakhsh, E. Robust uncapacitated hub location. Transp. Res. Part B Methodol.
**2017**, 106, 393–410. [Google Scholar] [CrossRef] - Yang, K.; Yang, L.; Gao, Z. Planning and optimization of intermodal hub-and-spoke network under mixed uncertainty. Transp. Res. Part E Logist. Transp. Rev.
**2016**, 95, 248–266. [Google Scholar] [CrossRef] - Mokhtar, H.; Krishnamoorthy, M.; Ernst, A.T. The 2-allocation p-hub median problem and a modified Benders decomposition method for solving hub location problems. Comput. Oper. Res.
**2019**, 104, 375–393. [Google Scholar] [CrossRef] [Green Version] - Parsa, M.; Nookabadi, A.S.; Flapper, S.D.; Atan, Z. Green hub-and-spoke network design for aviation industry. J. Clean. Prod.
**2019**, 229, 1377–1396. [Google Scholar] [CrossRef] - Tran, T.H.; O’Hanley, J.R.; Scaparra, M.P. Reliable hub network design: Formulation and solution techniques. Transp. Sci.
**2017**, 51, 358–375. [Google Scholar] [CrossRef] [Green Version] - Azizi, N.; Vidyarthi, N.; Chauhan, S.S. Modelling and analysis of hub-and-spoke networks under stochastic demand and congestion. Ann. Oper. Res.
**2018**, 264, 1–40. [Google Scholar] [CrossRef] - Alkaabneh, F.; Diabat, A.; Elhedhli, S. A Lagrangian heuristic and GRASP for the hub-and-spoke network system with economies-of-scale and congestion. Transp. Res. Part C Emerg. Technol.
**2019**, 102, 249–273. [Google Scholar] [CrossRef] - Karimi-Mamaghan, M.; Mohammadi, M.; Pirayesh, A.; Karimi-Mamaghan, A.M.; Irani, H. Hub-and-spoke network design under congestion: A learning based metaheuristic. Transp. Res. Part E Logist. Transp. Rev.
**2020**, 142, 102069. [Google Scholar] [CrossRef] - Karlaftis, M.G.; Kepaptsoglou, K.; Sambracos, E. Containership routing with time deadlines and simultaneous deliveries and pickups. Transp. Res. Part E Logist. Transp. Rev.
**2009**, 45, 210–221. [Google Scholar] [CrossRef] - Imai, A.; Shintani, K.; Papadimitriou, S. Multi-port vs. Hub-and-Spoke port calls by containerships. Transp. Res. Part E Logist. Transp. Rev.
**2009**, 45, 740–757. [Google Scholar] [CrossRef] [Green Version] - Wang, C.; Wang, J. Spatial pattern of the global shipping network and its hub-and-spoke system. Res. Transp. Econ.
**2011**, 32, 54–63. [Google Scholar] [CrossRef] - Gelareh, S.; Nickel, S.; Pisinger, D. Liner shipping hub network design in a competitive environment. Transp. Res. Part E Logist. Transp. Rev.
**2010**, 46, 991–1004. [Google Scholar] [CrossRef] [Green Version] - Gelareh, S.; Pisinger, D. Fleet deployment, network design and hub location of liner shipping companies. Transp. Res. Part E Logist. Transp. Rev.
**2011**, 47, 947–964. [Google Scholar] [CrossRef] - Zheng, J.; Meng, Q.; Sun, Z. Liner hub-and-spoke shipping network design. Transp. Res. Part E Logist. Transp. Rev.
**2015**, 75, 32–48. [Google Scholar] [CrossRef] - Wei, H.; Sheng, Z.; Lee, P.T.W. The role of dry port in hub-and-spoke network under Belt and Road Initiative. Marit. Policy Manag.
**2018**, 45, 370–387. [Google Scholar] [CrossRef] - Bai, B.; Fan, W. Research on strategic liner ship fleet planning with regard to hub-and-spoke network. Oper. Manag. Res.
**2023**, 16, 363–376. [Google Scholar] [CrossRef] - Fadda, P.; Fancello, G.; Mancini, S.; Pani, C.; Serra, P. Design and optimisation of an innovative Two-Hub-and-Spoke network for the Mediterranean Short-Sea-Shipping market. Comput. Ind. Eng.
**2020**, 149, 106847. [Google Scholar] [CrossRef] - Martínez-López, A. A multi-objective mathematical model to select fleets and maritime routes in short sea shipping: A case study in Chile. J. Mar. Sci. Technol.
**2020**, 26, 673–692. [Google Scholar] [CrossRef] - Medbøen, C.A.B.; Holm, M.B.; Msakni, M.K.; Fagerholt, K.; Schütz, P. Combining Optimization and Simulation for Designing a Robust Short-Sea Feeder Network. Algorithms
**2020**, 13, 304. [Google Scholar] [CrossRef] - Stevenson, T.C.; Davies, J.; Huntington, H.P.; Sheard, W. An examination of trans-Arctic vessel routing in the Central Arctic Ocean. Mar. Policy
**2019**, 100, 83–89. [Google Scholar] [CrossRef] - Chen, X.; Liu, S.; Liu, R.W.; Wu, H.; Han, B.; Zhao, J. Quantifying Arctic oil spilling event risk by integrating an analytic network process and a fuzzy comprehensive evaluation model. Ocean Coast. Manag.
**2022**, 228, 106326. [Google Scholar] [CrossRef]

**Figure 8.**Optimization results for case study C1 (single vessel scenario). (

**a**) C1a: D = 645 nm, T = 26:13 h, PH = 21,574 h, T

_{max-trip}= 26:13 h. (

**b**) C1b: D = 705 nm, T = 28:26 h, PH = 17,032 h, T

_{max-trip}= 28:26 h. (

**c**) C1c: D = 684 nm, T = 27:40 h, PH = 17,121 h, T

_{max-trip}= 27:40 h.

**Figure 9.**Optimization results for case study C2 (two vessels from central ports) (the colours in the numbers below correspond to those of the lines in the maps). (

**a**) C2a: D

_{1}/D

_{2}/D

_{tot}= 315/339/654 nm, T

_{1}/T

_{2}/T

_{tot}= 13:00/13:23/26:23 h, PH = 12,265 h, T

_{max-trip}= 13:23 h. (

**b**) C2b: D

_{1}/D

_{2}/D

_{tot}= 370/421/791 nm, T

_{1}/T

_{2}/T

_{tot}= 14:52/16:35/31:27 h, PH = 10,369 h, T

_{max-trip}= 16:35 h. (

**c**) C2c:D

_{1}/D

_{2}/D

_{tot}= 427/231/658 nm, T

_{1}/T

_{2}/T

_{tot}= 17:08/9:23/26:31 h, PH = 10,732 h, T

_{max-trip}= 17:08 h.

**Figure 10.**Optimization results of combined objectives D and PH for case study C3 (one vessel from the central port and one from a hub node) (the colours in the numbers below correspond to those of the lines in the maps). (

**a**) C3a: D

_{1}/D

_{2}/D

_{tot}= 443/160/603 nm, T

_{1}/T

_{2}/T

_{tot}= 17:44/15:38/33:22 h, PH = 16,627 h, T

_{max-trip}= 20:40 h. (

**b**) C3b: D

_{1}/D

_{2}/D

_{tot}= 500/85/585 nm, T

_{1}/T

_{2}/T

_{tot}= 20:31/15:21/35:52 h, PH = 19,852 h, T

_{max-trip}= 20:31 h. (

**c**) C3c: D

_{1}/D

_{2}/D

_{tot}= 486/106/592 nm, T

_{1}/T

_{2}/T

_{tot}= 19:30/15:16/34:46 h, PH = 15,438 h, T

_{max-trip}= 19:30 h. (

**d**) C3d: D

_{1}/D

_{2}/D

_{tot}= 441/170/611 nm, T

_{1}/T

_{2}/T

_{tot}= 17:30/16:44/34:14 h, PH = 16,894 h, T

_{max-trip}= 21:45 h.

**Figure 11.**Optimization results for case study C4 (up to two vessels from central ports and one from a hub node) (the colours in the numbers below correspond to those of the lines in the maps). (

**a**) C4a: D

_{1}/D

_{2}/D

_{3}/D

_{tot}= 0/347/228/575 nm, T

_{1}/T

_{2}/T

_{3}/T

_{tot}= 0:00/14:11/21:56/36:07 h, PH = 18,683 h, T

_{max-trip}= 29:33 h. (

**b**) C4b: D

_{1}/D

_{2}/D

_{3}/D

_{tot}= 390/233/97/720 nm, T

_{1}/T

_{2}/T

_{3}/T

_{tot}= 15:18/9:23/9:18/34:09 h, PH = 9863 h, T

_{max-trip}= 15:18 h. (

**c**) C4c: D

_{1}/D

_{2}/D

_{3}/D

_{tot}= 153/233/261/617 nm, T

_{1}/T

_{2}/T

_{3}/T

_{tot}= 6:00/9:23/22:24/37:47 h, PH = 12,012 h, T

_{max-trip}= 26:16 h. (

**d**) C4d: (max trip time ≤ 15h), D

_{1}/D

_{2}/D

_{3}/D

_{tot}= 290/22/53/615 nm, T

_{1}/T

_{2}/T

_{3}/T

_{tot}= 11:34/11:04/5:04/27:48 h, PH = 12,138 h, T

_{max-trip}= 11:34 h.

**Figure 15.**Traveled distance–cumulative passenger-hours point cloud across different runs: (

**a**) full set of values; (

**b**) subset of best values embodied in the circle of (

**a**) (different colors correspond to different values of w

_{1}and w

_{2}of the objective function).

**Figure 17.**Optimization results for case study C7b (up to four vessels from central ports and two from a hub node).

**Figure 18.**Traveled distance–cumulative passenger hours point clouds at different weights of the objective function (1) (w

_{1}>> w

_{2}in blue, w

_{1}≈ w

_{2}in orange, w

_{1}<< w

_{2}in green) (

**a**) full set of values; (

**b**) subset of best values embodied in the circle of (

**a**).

DISTANCES (Nautical Miles) | PIRAEUS | RAFINA | LESVOS | CHIOS | LIMNOS | SAMOS | IKARIA | FOYRNOI | PSARA | AG. EYSTRATIOS | PATMOS | SKIROS | KALYMNOS | KOS | INOUSES | THASSOS | SAMOTHRAKI | SHIPS DEPART FROM | SPEED (knots) |

Central Ports | 27.00 | ||||||||||||||||||

Hub Nodes | 10.80 | ||||||||||||||||||

PIRAEUS | 170 | 131 | 180 | 156 | 125 | 141 | 120 | 155 | 145 | 114 | 169 | 179 | 145 | 225 | 217 | ||||

RAFINA | 135 | 100 | 149 | 133 | 119 | 103 | 86 | 124 | 125 | 81 | 152 | 165 | 115 | 193 | 184 | Passengers | |||

LESVOS | 170 | 135 | 51 | 67 | 93 | 97 | 98 | 51 | 63 | 116 | 85 | 137 | 151 | 41 | 118 | 82 | LESVOS | 400 | |

CHIOS | 131 | 100 | 51 | 98 | 56 | 49 | 54 | 25 | 81 | 70 | 76 | 97 | 111 | 12 | 157 | 133 | CHIOS | 250 | |

LIMNOS | 180 | 149 | 67 | 98 | 152 | 147 | 155 | 84 | 27 | 168 | 72 | 196 | 208 | 99 | 54 | 39 | LIMNOS | 85 | |

SAMOS | 156 | 133 | 93 | 56 | 152 | 32 | 20 | 78 | 137 | 27 | 129 | 46 | 59 | 54 | 206 | 174 | SAMOS | 165 | |

IKARIA | 125 | 119 | 97 | 49 | 147 | 32 | 16 | 64 | 127 | 25 | 109 | 54 | 68 | 56 | 200 | 175 | IKARIA | 40 | |

FOYRNOI | 141 | 103 | 98 | 54 | 155 | 20 | 16 | 74 | 136 | 16 | 122 | 43 | 55 | 56 | 205 | 176 | FOYRNOI | 20 | |

PSARA | 120 | 86 | 51 | 25 | 84 | 78 | 64 | 74 | 62 | 87 | 53 | 116 | 130 | 32 | 132 | 111 | PSARA | 15 | |

AG. EYSTRATIOS | 155 | 124 | 63 | 81 | 27 | 137 | 127 | 136 | 62 | 151 | 43 | 179 | 194 | 85 | 71 | 60 | AG. EYSTRATIOS | 10 | |

PATMOS | 145 | 125 | 116 | 70 | 168 | 27 | 25 | 16 | 87 | 151 | 133 | 29 | 43 | 71 | 219 | 192 | PATMOS | 80 | |

SKIROS | 114 | 81 | 85 | 76 | 72 | 129 | 109 | 122 | 53 | 43 | 133 | 162 | 178 | 73 | 109 | 106 | SKIROS | 40 | |

KALYMNOS | 169 | 152 | 137 | 97 | 196 | 46 | 54 | 43 | 116 | 179 | 29 | 162 | 14 | 89 | 151 | 121 | KALYMNOS | 80 | |

KOS | 179 | 165 | 151 | 111 | 208 | 59 | 68 | 55 | 130 | 194 | 43 | 178 | 14 | 111 | 159 | 230 | KOS | 170 | |

INOUSES | 145 | 115 | 41 | 12 | 99 | 54 | 56 | 56 | 32 | 85 | 71 | 73 | 89 | 111 | 149 | 120 | INOUSES | 20 | |

THASSOS | 225 | 193 | 118 | 157 | 54 | 206 | 200 | 205 | 132 | 71 | 219 | 109 | 151 | 159 | 149 | 46 | THASSOS | 65 | |

SAMOTHRAKI | 217 | 184 | 82 | 133 | 39 | 174 | 175 | 176 | 111 | 60 | 192 | 106 | 121 | 230 | 120 | 46 | SAMOTHRAKI | 20 |

MODEL | Case | Performance Measure | Traveling Distance (nm) | Passengers’ Hours (hh:mm) | Vessels Travel Time (hh:mm) | Islands per Vessel | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Islands/Routes/ Potential Hub-Ports | Total | Min | Max | Total | Min | Max | Min | Max | |||

49/4/0 | C5a | D | 1666 | 158 | 564 | 29,399 | 7:00 | 23:44 | 73:30 | 9 | 17 |

C5b | PH | 1823 | 179 | 669 | 26,347 | 7:46 | 26:37 | 81:40 | 8 | 18 | |

C5c | D–PH | 1733 | 272 | 565 | 27,437 | 12:05 | 22:45 | 75:01 | 10 | 15 | |

49/5/1 | C6a | D | 1508 | 133 | 543 | 28,379 | 6:14 | 21:56 | 70:34 | 7 | 12 |

C6b | PH | 1617 | 98 | 586 | 22,920 | 8:11 | 23:31 | 72:41 | 6 | 12 | |

C6c | D–PH | 1552 | 95 | 685 | 25,134 | 7:00 | 22:45 | 70:05 | 8 | 11 | |

49/6/2 | C7a | D | 1383 | 60 | 425 | 26,070 | 5:52 | 17:04 | 69:44 | 3 | 10 |

C7b | PH | 1731 | 60 | 518 | 22,591 | 5:52 | 20:30 | 81:30 | 3 | 11 | |

C7c | D–PH | 1462 | 60 | 554 | 23,097 | 5:52 | 21:51 | 74:22 | 3 | 11 |

MODEL | Run | Performance Measure | Fitness Function | Traveling Distance (nm) | Passenger Hours (hh:mm) | Number of Trials Until Best | Time until Best Trial (h:mm:ss) |
---|---|---|---|---|---|---|---|

Size (Islands/Routes/ Potential Hub-Ports) | |||||||

Small (15/2/3) | 1 | D | 378 | 378 | 6141:58 | 5191 | 0:01:26 |

PH | 4080:00 | 441 | 4080:00 | 20,371 | 0:05:47 | ||

D–PH | 2.11 | 383 | 4406:20 | 6905 | 0:02:24 | ||

2 | D | 375 | 375 | 5382:42 | 14,574 | 0:04:13 | |

PH | 4078:36 | 398 | 4078:36 | 4291 | 0:01:16 | ||

D–PH | 2.12 | 408 | 4132:21 | 10,193 | 0:03:00 | ||

3 | D | 375 | 375 | 5382:42 | 4202 | 0:01:16 | |

PH | 4096:25 | 471 | 4096:25 | 25,868 | 0:06:56 | ||

D–PH | 2.12 | 408 | 4132:21 | 19,055 | 0:05:10 | ||

Medium (50/6/8) | 1 | D | 897 | 897 | 37,126:24 | 41,568 | 0:18:59 |

PH | 23,249:44 | 1435 | 23,249:44 | 56,717 | 0:21:25 | ||

D–PH | 1.99 | 927 | 24,264:18 | 39,187 | 0:21:52 | ||

2 | D | 848 | 848 | 36,485:48 | 37,992 | 0:15:41 | |

PH | 24,435:08 | 1384 | 24,435:08 | 49,749 | 0:18:06 | ||

D–PH | 2.09 | 1055 | 23,350:57 | 46,157 | 0:18:14 | ||

3 | D | 796 | 796 | 31,020:57 | 100,776 | 0:38:45 | |

PH | 24,844:08 | 1488 | 24,844:08 | 66,203 | 0:24:32 | ||

D–PH | 1.99 | 998 | 22,566:19 | 49,420 | 0:18:50 | ||

Large (100/12/15) | 1 | D | 2669 | 2669 | 108,844:36 | 47,855 | 0:58:45 |

PH | 53,924:45 | 3488 | 53,924:45 | 87,502 | 1:17:49 | ||

D–PH | 2.07 | 2983 | 52,702:46 | 85,832 | 1:14:09 | ||

2 | D | 2365 | 2365 | 72,932:05 | 76,791 | 1:09:58 | |

PH | 53,416:51 | 3881 | 53,416:51 | 66,742 | 1:00:31 | ||

D–PH | 2.25 | 3199 | 58,120:33 | 113,030 | 1:41:45 | ||

3 | D | 2775 | 2775 | 117,085:35 | 41,820 | 0:36:30 | |

PH | 50,638:18 | 3442 | 50,638:18 | 108,163 | 1:41:49 | ||

D–PH | 2.23 | 3040 | 60,926:07 | 55,112 | 0:47:36 |

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## Share and Cite

**MDPI and ACS Style**

Farmakis, P.; Chassiakos, A.; Karatzas, S.
A Multi-Objective Tri-Level Algorithm for Hub-and-Spoke Network in Short Sea Shipping Transportation. *Algorithms* **2023**, *16*, 379.
https://doi.org/10.3390/a16080379

**AMA Style**

Farmakis P, Chassiakos A, Karatzas S.
A Multi-Objective Tri-Level Algorithm for Hub-and-Spoke Network in Short Sea Shipping Transportation. *Algorithms*. 2023; 16(8):379.
https://doi.org/10.3390/a16080379

**Chicago/Turabian Style**

Farmakis, Panagiotis, Athanasios Chassiakos, and Stylianos Karatzas.
2023. "A Multi-Objective Tri-Level Algorithm for Hub-and-Spoke Network in Short Sea Shipping Transportation" *Algorithms* 16, no. 8: 379.
https://doi.org/10.3390/a16080379