# Distributionally Robust Model and Metaheuristic Frame for Liner Ships Fleet Deployment

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- formulation of distributional robust model;
- demonstration that this approximation is more general than the most modern method.
- development of metaheuristic frame.
- development of computer-based program for solving the fleet deployment problem under uncertainty within acceptable CPU time.

## 3. Model Construction

- it is possible to develop an innovative mathematical model of robust fleet deployment that will offer feasible solution(s) fast and cost-efficiently;
- it is possible to create efficient metaheuristic algorithm(s) in the decision-support system in solving the problem of fleet deployment.

#### 3.1. Problem Description

^{3}of volume) for Ship k. An operator should determine the finite number of ships k ∈ K and deploy them on routes r ∈ R, while following the sailing schedule. Beside their own ships, an operator’s broker may charter the available ships from the market for the duration of the planning horizon. The charter rate for Ship k is ${C}_{k}^{in}$ (USD/ship). ${N}_{k}^{max}$ and $NC{I}_{k}^{max}$ denote the number of ships k, either owned or chartered. Such a number is often limited due to expenses involved or the ships availability on the open charter market.

#### 3.2. Scenario for Container Shipment Demand

#### 3.3. Model Buildup

#### 3.3.1. Decision Variables

#### 3.3.2. Parameters

#### 3.4. Optimization Model

#### 3.5. Robust Optimization Model

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Results

## 5. Discussion

**C**has the cost of fuel, cost of daily labor and harbor and light dues. Based on this, the ship OPEX plus charter fees were also calculated in the mode. Based on all the formulations set above, we were able to calculate an optimization model with the objective of maximizing the income, which is shown in Equation (5). This is how an expected value model was obtained. In Equations (6) and (7) it is made sure that the number of ships, both owned and chartered ones, did not exceed the available number of vessels.

_{kr}- Add a long-term vessel: Add to the solution a single viable vessel for the long-term charter.
- Remove a long-term vessel: Remove from the solution a single long-term chartered vessel.
- Add a vessel for a short time charter: Add to the solution a single viable vessel for the long-term charter, but for a strictly specified charter period.
- Remove vessel short-term: Remove from the solution a single viable vessel for the long-term charter, but for a strictly specified charter period.
- Replace vessels for a short-term 1: Remove one vessel with short-term charter during the charter period and replace it with another type of vessel chartered on a short-term.
- Replace vessels for a short-term 2: Remove one vessel for short-term charter during the charter period and insert it in another charter period.
- Replace vessels: Remove from the solution a vessel with a long-term charter and replace it with another type of vessel on a long-term charter.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Global Containerization Trade (1996–2021), Source [2].

**Table 1.**Dataset Source [7].

Types of Ships | |||||

t = 1 | t = 2 | t = 3 | t = 4 | t = 5 | |

${V}_{k}$ | 2808 | 3218 | 4500 | 5714 | 8063 |

${c}_{kr}$ | 19.8 | 22.5 | 30.9 | 38.8 | 54.2 |

${c}_{k}^{out}$ | 1.82 | 2.34 | 3.21 | 4.32 | 5.12 |

${c}_{k}^{in}$ | 2 | 2.6 | 3.5 | 4.7 | 6 |

${N}_{k}^{max}$ | 2 | 2 | 9 | 2 | 12 |

$NC{I}_{k}^{max}$ | 10 | 10 | 10 | 6 | 6 |

${t}_{k1}$ | 25.2 | 24.1 | 21.9 | 21.6 | 21.0 |

${t}_{k2}$ | 20.7 | 19.7 | 17.9 | 17.6 | 17.2 |

${t}_{k3}$ | 15.1 | 14.4 | 13.1 | 12.9 | 12.6 |

${t}_{k4}$ | 38.9 | 37.1 | 33.8 | 33.2 | 32.4 |

${t}_{k5}$ | 63.8 | 60.9 | 55.4 | 54.5 | 53.2 |

${t}_{k6}$ | 22.8 | 21.7 | 19.8 | 19.4 | 19.0 |

${t}_{k7}$ | 58.0 | 55.4 | 50.4 | 49.5 | 48.4 |

${t}_{k8}$ | 2.1 | 2.0 | 1.8 | 1.8 | 1.8 |

Ship Type | Route 1 | Route 2 | Route 3 | Route 4 | Route 5 | Route 6 | Route 7 | Route 8 |
---|---|---|---|---|---|---|---|---|

t = 1 (2808 TEU) | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |

t = 2 (3218 TEU) | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |

t = 3 (4500 TEU) | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

t = 4 (5714 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

t = 5 (8063 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Ship Type | Route 1 | Route 2 | Route 3 | Route 4 | Route 5 | Route 6 | Route 7 | Route 8 |
---|---|---|---|---|---|---|---|---|

t = 1 (2808 TEU) | 0 | 2 | 2 | 3 | 0 | 0 | 0 | 0 |

t = 2 (3218 TEU) | 3 | 2 | 0 | 0 | 5 | 0 | 0 | 0 |

t = 3 (4500 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

t = 4 (5714 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

t = 5 (8063 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Ship Type | Route 1 | Route 2 | Route 3 | Route 4 | Route 5 | Route 6 | Route 7 | Route 8 |
---|---|---|---|---|---|---|---|---|

t = 1 (2808 TEU) | 0 | 14 | 24 | 25 | 8 | 26 | 26 | 26 |

t = 2 (3218 TEU) | 26 | 12 | 0 | 0 | 18 | 0 | 0 | 0 |

t = 3 (4500 TEU) | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 |

t = 4 (5714 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

t = 5 (8063 TEU) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Bukljaš, M.; Rogić, K.; Jerebić, V.
Distributionally Robust Model and Metaheuristic Frame for Liner Ships Fleet Deployment. *Sustainability* **2022**, *14*, 5551.
https://doi.org/10.3390/su14095551

**AMA Style**

Bukljaš M, Rogić K, Jerebić V.
Distributionally Robust Model and Metaheuristic Frame for Liner Ships Fleet Deployment. *Sustainability*. 2022; 14(9):5551.
https://doi.org/10.3390/su14095551

**Chicago/Turabian Style**

Bukljaš, Mihaela, Kristijan Rogić, and Vladimir Jerebić.
2022. "Distributionally Robust Model and Metaheuristic Frame for Liner Ships Fleet Deployment" *Sustainability* 14, no. 9: 5551.
https://doi.org/10.3390/su14095551