# Tugboat Scheduling with Multiple Berthing Bases under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

- (1)
- Most scholars model tugboat scheduling optimization problems from the perspectives of allocation and flow shop. Models established from these two perspectives can only be scheduled according to the predetermined order of ships and cannot be selected and scheduled from chaotic ship tasks. However, modeling from the perspective of vehicle routing can successfully solve this problem.
- (2)
- The existing literature rarely considers the mooring base of tugboats, while the literature that considers tugboat bases simply considers that tugboat bases are a fixed starting and ending point, without considering that tugboats can dock at any tugboat base.
- (3)
- A small amount of the literature considers the tugboat scheduling problem in uncertain scenarios, as well as the auxiliary service time, start location, end location, earliest start time, latest start time, and required tugboat demand of ships.
- (4)
- Existing literature has proposed multiple heuristic algorithms to solve the optimization problem of tugboat scheduling, but there is no calculation method for tugboat scheduling problems with multiple berthing bases and uncertain situations.

## 3. Problem Description and Uncertainty Modeling

#### 3.1. Problem Description

**Definition**

**1.**

**Definition**

**2.**

#### 3.2. Model Establishment

#### 3.2.1. Model Assumptions and Symbols

- (1)
- Set: $C$ is set of tasks, $i,j\in C=\{1,2,\dots NN\}$; $N$ is the union set of 0 tasks and other tasks, $i,j\in N=\{0,1,2,\dots ,NN\}$, 0 represents the task at time 0 before the start of the day’s task; $V$ is the set of tugboats, $k\in V=\{1,2,\dots NK\}$; and $L$ is the set of bases, $l\in L=\{1,2,\dots ,NL\}$.
- (2)
- Parameter: ${\tilde{w}}_{i}$ is the fuzzy service time for task $i$, $i\in N$; $\tilde{c}{1}_{li}$ is the fuzzy distance from the base $l$ to the starting point of the task $i$ operation, $i\in N$; $\tilde{c}{2}_{li}$ is the fuzzy distance from the base $l$ to the end of assignment location of the task $i$, $i\in N$; ${B}_{kl}$ is the position of the tugboat $k$ at the base $l$ at the end of the previous day’s mission; ${\tilde{e}}_{i}$ is the earliest fuzzy start time for the task $i$, $i\in N$; $l{\tilde{a}}_{i}$ is the latest fuzzy start time of the task $i$, $i\in N$; $M$ is a sufficiently large value; ${\tilde{D}}_{i}$ is the number of fuzzy tugboat requirements for the task $i$, $i\in C$; $vc$ is the speed of the tugboat; ${P}_{k}$ is the power of the tugboat $k$; ${m}_{i}$ is the required tugboat power for the task $i$; $\gamma $ is the unit time delay cost; and $Co$ is the unit cost per unit fuel consumed.
- (3)
- Decision variables: ${x}_{klij}$ is 0–1 variable. If the tugboat $k$ completes the task $i$ and returns to the base $l$ to complete the task $j$, it is 1, otherwise it is 0, $i,j\in N$. ${s}_{i}$ is the start time of the task $i$ processing.

#### 3.2.2. Objective Function and Constraints

#### 3.3. Defuzzified Model Formulation

## 4. Solution Method

#### 4.1. Grey Wolf Optimization Algorithm

#### 4.2. Individual Encoding and Decoding Processes

#### 4.3. Fitness and Selection Operation

#### 4.4. Crossover and Mutation Operations

## 5. Numerical Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | … | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.91 | 0.75 | 0.89 | 0.42 | 0.78 | 0.63 | 0.95 | 0.74 | 0.43 | 0.67 | 0.51 | 0.17 | 0.65 | 0.48 | 0.18 |

Task 5 (2) | Task 1 (1) | Task 3 (2) | Task 2 (2) | Task 4 (1) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Tug | Initial Base | Base | Task | Base | Task | Base | Task | Base | Task | Base | Task | Base |

1 | (2) | (2) | 0.91(5) | 0.42, 0.78, 0.63 | 0.93 | 0.22, 0.59, 0.64 | 0.79 | 0.74, 0.56, 0.58 (1) | 0.68(2) | 0.43, 0.73, 0.53 | 0.56 | 0.37, 0.69, 0.43 (2) |

2 | (1) | 0.75 | 0.95, 0.74, 0.59 (1) | 0.86(1) | 0.77, 0.23, 0.29 (1) | 0.87(3) | 0.36, 0.82, 0.76(2) | 0.71(2) | 0.07, 0.39, 0.08 | 0.84 | 0.67, 0.51, 0.17(1) | |

3 | (1) | (1) | 0.89(5) | 0.19, 0.38, 0.23 | 0.65 | 0.65, 0.41, 0.09 (1) | 0.97(3) | 0.49, 0.39, 0.94 | 0.83 | 0.55, 0.87, 0.32 (2) | 0.78(4) | 0.65, 0.48, 0.18(1) |

Example | Tasks | Tugs | Bases |
---|---|---|---|

1 | 5 | 3 | 3 |

2 | 15 | 9 | 9 |

3 | 25 | 15 | 15 |

4 | 35 | 21 | 21 |

5 | 45 | 27 | 27 |

6 | 60 | 36 | 36 |

7 | 70 | 42 | 42 |

8 | 80 | 48 | 48 |

9 | 90 | 54 | 54 |

10 | 100 | 60 | 60 |

11 | 110 | 66 | 66 |

12 | 120 | 72 | 72 |

Parameter | Range | Parameter | Range |
---|---|---|---|

${w}_{i}^{p}$ | Uniform (30, 50) | ${w}_{i}^{m}$ | Uniform (50, 70) |

${w}_{i}^{o}$ | Uniform (70, 90) | $c{1}_{li}^{p}$ | Uniform (9, 16) |

$c{1}_{li}^{m}$ | Uniform (10, 17) | $c{1}_{li}^{o}$ | Uniform (11, 20) |

$c{2}_{li}^{p}$ | Uniform (9, 16) | $c{2}_{li}^{m}$ | Uniform (10, 17) |

$c{2}_{li}^{o}$ | Uniform (11, 20) | ${e}_{i}^{p}$ | Uniform (0, 480) |

${e}_{i}^{m}$ | Uniform (0, 480) | ${e}_{i}^{o}$ | Uniform (0, 480) |

$l{a}_{i}^{p}$ | Uniform (0, 480) | $l{a}_{i}^{m}$ | Uniform (0, 480) |

$l{a}_{i}^{o}$ | Uniform (0, 480) | ${D}_{i}^{p}$ | Uniform (1, Total number of tugboats) |

${D}_{i}^{m}$ | Uniform (1, Total number of tugboats) | ${D}_{i}^{o}$ | Uniform (1, Total number of tugboats) |

$vc$ | Uniform (1, 8) | ${P}_{k}$ | Uniform (2600, 4600) |

${m}_{i}$ | Uniform (2600, 4600) | $\gamma $ | Uniform (100, 150) |

$Co$ | Uniform (30, 40) |

LINGO | pgGWO | PSO | GWO | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Example (Population Size, Number of Iterations) | Objective Value | CPU Time (seconds) | Objective Value | Average Objective value | CPU Time (seconds) | Objective Value | Average Objective Value | CPU Time (seconds) | Objective Value | Average Objective Value | CPU Time (seconds) |

1 (50, 100) | 12,180 | 1.83 | 12,180 | 12,833.38 | 39.78 | 12,180 | 13,021.29 | 39.26 | 12,180 | 12,945.87 | 38.67 |

2 (50, 100) | 36,536 | 13.67 | 36,536 | 37,142.62 | 80.65 | 36,628 | 37,347.42 | 79.34 | 36,597 | 37,253.73 | 78.69 |

3 (50, 150) | 60,873 | 99.82 | 60,873 | 61,828.26 | 170.24 | 61,051 | 62,146.79 | 168.75 | 60,912 | 61,917.59 | 167.53 |

4 (100, 150) | 85,197 | 192.37 | 85,267 | 86,138.36 | 267.26 | 85,637 | 86,527.45 | 249.95 | 85,468 | 86,227.79 | 238.35 |

5 (100, 200) | 109,581 | 385.68 | 109,976 | 110,217.23 | 350.03 | 110,923 | 112,571.62 | 332.38 | 110,157 | 111,256.71 | 323.17 |

6 (100, 300) | 146,206 | 983.57 | 146,923 | 147,510.57 | 500.43 | 148,165 | 149,267.25 | 483.49 | 147,681 | 148,049.36 | 462.34 |

7 (200, 300) | 170,612 | 2375.87 | 171,287 | 172,537.27 | 585.71 | 173,846 | 174,207.63 | 539.55 | 172,513 | 173,451.08 | 539.55 |

8 (200, 350) | 194,753 | 4219.53 | 195,632 | 196,135.52 | 686.62 | 197,675 | 198,852.26 | 654.13 | 196,247 | 197,247.63 | 635.83 |

9 (200, 400) | 219,265 * | >9867.82 | 219,213 | 220,246.51 | 778.53 | 223,549 | 224,527.04 | 752.56 | 221,564 | 222,276.76 | 717.48 |

10 (300, 350) | -- | -- | 243,529 | 243,067.56 | 896.37 | 245,105 | 246,023.75 | 817.14 | 244,276 | 245,207.36 | 805.25 |

11 (300, 400) | -- | -- | 267,953 | 268,210.26 | 965.37 | 269,358 | 271,652.31 | 928.26 | 268,247 | 269,514.28 | 896.24 |

12 (300, 450) | -- | -- | 292,337 | 293,967.37 | 1230.65 | 294,476 | 295,476.42 | 1195.67 | 293,275 | 294,571.23 | 1136.95 |

Tugboat | 1 | 2 | 3 |
---|---|---|---|

Fuel cost | 30 | 30 | 30 |

Power | 2600 | 4000 | 3600 |

Speed | 8 | 8 | 8 |

Task | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Number of tugs required | (0.8, 0.9, 1) | (1.8, 1.9, 2) | (1.8, 1.9, 2) | (0.8, 0.9, 1) | (1.8, 1.9, 2) |

Operation time | (50, 69, 79) | (49, 68, 78) | (49, 67, 79) | (50, 68, 68) | (49, 65, 81) |

Required power | 3600 | 2600 | 2600 | 3600 | 2600 |

Earliest start time (minutes) | (190, 195, 200) | (90, 95, 100) | (80, 85, 90) | (290, 295, 300) | (280, 285, 290) |

Latest start time (minutes) | (240, 245, 250) | (90, 145, 150) | (120, 125, 130) | (3300, 340, 350) | (3300, 340, 350) |

Confidence Level | 0% | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% |
---|---|---|---|---|---|---|---|---|---|---|---|

Objective value | 11,780 | 11,820 | 11,860 | 11,900 | 11,940 | 11,980 | 12,020 | 12,060 | 12,100 | 12,140 | 12,180 |

Task 3 | Task 2 | Task 1 | Task 5 | Task 4 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Tugboat | Base | Task | Base | Task | Base | Task | Base | Task | Base | Task | Base |

1 | 3 | 3 | 1 | 2 | - | - | 2 | 5 | - | - | 1 |

2 | 1 | 3 | 1 | 2 | - | - | - | - | 2 | 4 | 1 |

3 | - | - | - | - | 2 | 1 | 2 | 5 | - | - | 2 |

Task 3 | Task 2 | Task 1 | Task 5 | Task 4 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Tugboat | Base | Task | Base | Task | Base | Task | Base | Task | Base | Task | Base |

1 | 3 | 3 | 1 | 2 | - | - | 2 | 5 | - | - | 1 |

2 | - | - | - | - | 1 | 1 | 2 | 5 | - | - | 2 |

3 | 1 | 3 | 1 | 2 | - | - | - | - | 2 | 4 | 1 |

Task 3 | Task 2 | Task 1 | Task 5 | Task 4 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Tugboat | Base | Task | Base | Task | Base | Task | Base | Task | Base | Task | Base |

1 | - | - | 3 | 2 | - | - | - | - | 2 | 4 | 3 |

2 | 1 | 3 | - | - | 2 | 1 | 2 | 5 | - | - | 3 |

3 | 1 | 3 | 1 | 2 | - | - | 2 | 5 | - | - | 1 |

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## Share and Cite

**MDPI and ACS Style**

Li, B.; Chen, Q.; Lau, Y.-y.; Dulebenets, M.A.
Tugboat Scheduling with Multiple Berthing Bases under Uncertainty. *J. Mar. Sci. Eng.* **2023**, *11*, 2180.
https://doi.org/10.3390/jmse11112180

**AMA Style**

Li B, Chen Q, Lau Y-y, Dulebenets MA.
Tugboat Scheduling with Multiple Berthing Bases under Uncertainty. *Journal of Marine Science and Engineering*. 2023; 11(11):2180.
https://doi.org/10.3390/jmse11112180

**Chicago/Turabian Style**

Li, Botang, Qiong Chen, Yui-yip Lau, and Maxim A. Dulebenets.
2023. "Tugboat Scheduling with Multiple Berthing Bases under Uncertainty" *Journal of Marine Science and Engineering* 11, no. 11: 2180.
https://doi.org/10.3390/jmse11112180