# Speed Optimization for Container Ship Fleet Deployment Considering Fuel Consumption

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Mathematical Model Formulation

#### 3.1. Parameter and Variable Definition

#### 3.2. Fuel Consumption Cost

#### 3.3. Container Ships’ Fleet Deployment Model

## 4. Linearization of the Model

#### 4.1. Linearization of the Reciprocal of Sailing Speeds

#### 4.2. Linearization of the Objective Function of Fuel Consumption Cost

#### 4.3. Underestimating Bilinear Terms

## 5. Approximation Algorithm

#### 5.1. Linear Outer-Approximation Algorithm

**Proposition**

**1.**

**Proof.**

Algorithm 1. The procedure of linear outer-approximation algorithm to obtain tangent point sets. | |

Input: | Convex function ${\left({u}_{ri}\right)}^{1-{c}_{2}^{r}}$, the tangent point set ${\mathsf{\Omega}}_{ri}$. The lower limit and upper limit of ${u}_{ri}$ is $\left. [{u}_{ri}^{min},{u}_{ri}^{max}\right]$, the approximation relative error ${\epsilon}_{1}$ |

Output: | the tangent point set ${\mathsf{\Omega}}_{ri}\left({\epsilon}_{1}\right)$ |

Step 1 | $\mathrm{Calculate}\text{}\mathrm{the}\text{}\mathrm{midpoint}\text{}{u}_{ri}^{mid}$ of the interval $\left. [\text{}{u}_{ri}^{min},{u}_{ri}^{max}\right]$; ${u}_{ri}^{mid}$ can be carried out by the bisection search method |

Step 2 | At tangent point ${u}_{ri}^{mid}$, the tangent line is defined as ${\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}+{\left({\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}\right)}^{\prime}\times \left({u}_{ri}-{u}_{ri}^{mid}\right)$ |

Step 3 | Calculate the relative approximation error of points ${u}_{ri}^{min}$ and ${u}_{ri}^{max}$ according to ${\epsilon}_{l}={\left({u}_{ri}^{min}\right)}^{1-{c}_{2}^{r}}-{\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}-{\left({\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}\right)}^{\prime}\times \left({u}_{ri}^{min}-{u}_{ri}^{mid}\right)$ ${\epsilon}_{r}={\left({u}_{ri}^{max}\right)}^{1-{c}_{2}^{r}}-{\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}-{\left({\left({u}_{ri}^{mid}\right)}^{1-{c}_{2}^{r}}\right)}^{\prime}\times \left({u}_{ri}^{max}-{u}_{ri}^{mid}\right)$ |

Step 4 | If ${\epsilon}_{l}>\epsilon $ or ${\epsilon}_{r}>\epsilon $, then branch the feasible range of ${u}_{ri}$ is divided into two ranges: $\left. [{u}_{ri}^{min},{u}_{ri}^{mid}\right]$ and $\left. [{u}_{ri}^{mid},{u}_{ri}^{max}\right]$, and the tangent point set ${\mathsf{\Omega}}_{ri}\left({\epsilon}_{1}\right)={\mathsf{\Omega}}_{ri}\left({\epsilon}_{1}\right)\cup \left. {{u}_{ri}^{mid}\right\}$ |

In one branch ${u}_{ri}\in \left. [{u}_{ri}^{min},{u}_{ri}^{mid}\right]$, repeat the above step 1 to e step 4 until the stop criterion is reached | |

In the other branch ${u}_{ri}\in \left. [{u}_{ri}^{mid},{u}_{ri}^{max}\right]$, repeat the above step 1 to e step 4 until the stop criterion is reached | |

Stop criterion check: if ${\epsilon}_{\mathrm{l}}<\epsilon $ and ${\epsilon}_{r}<\epsilon $, stop and output the current solution. Otherwise, go to Step4. | |

Step 5 | Return output ${\mathsf{\Omega}}_{ri}\left({\epsilon}_{1}\right)$ |

#### 5.2. Improved Piecewise Linear Approximation Algorithm

**Proposition**

**2.**

**Proof.**

**Function Generate approximate**

**straight**

**lines**${l}_{K}\left({w}_{ri}\right)$ two endpoints of the curve $f\left({w}_{ri}\right)={\left({w}_{ri}\right)}^{{c}_{3}^{r}}$ in interval $\left. [{w}_{ri}^{min},{w}_{ri}^{max}\right]$ are $A=\left({w}_{ri}^{min},{\left({w}_{ri}^{min}\right)}^{{c}_{3}^{r}}\right)$ and $B=\left({w}_{ri}^{max},{\left({w}_{ri}^{max}\right)}^{{c}_{3}^{r}}\right)$, the equation of a straight line is found through these two points. Assuming the point $C=\left({w}_{ri},{l}_{K}\left({w}_{ri}\right)\right)$ is on a straight line, hence the slope of the line AC and the line AB are the same, expressed as:

**Proposition**

**3.**

**Proof.**

#### 5.3. Mixed Integer Linear Programming Model

## 6. Numerical Experiments

#### 6.1. Parameter Setting

#### 6.2. Sensitivity Analysis of Various Fleet Costs

#### 6.3. Analyze the Relationship between Ship Deployment and Sailing Speed

#### 6.4. Analysis of the Relationship between Loading Rate and Sailing Speed

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Sets | Description |
---|---|

$R$ | $\mathrm{Ship}\text{}\mathrm{routes}\text{}\mathrm{set}\text{}\left(r\in R\right)$ |

$P$ | Ports set $\left(p,o,d\in P\right)$ |

$V$ | $\mathrm{Ship}\text{}\mathrm{types}\text{}\left(v\in V\right)$ |

Parameters | |

${V}_{r}$ | $\mathrm{Candidate}\text{}\mathrm{ship}\text{}\mathrm{types}\text{}\mathrm{set}\text{}\mathrm{of}\text{}\mathrm{route}r$ |

${I}_{r}$ | $\mathrm{Port}\text{}\mathrm{indices}\text{}\mathrm{for}\text{}\mathrm{route}\text{}r$ |

${P}_{r,i}$ | The ith port of call |

${I}_{r,p}$ | $\mathrm{Port}\text{}p$$\text{}\mathrm{indices}\text{}\mathrm{of}\text{}\mathrm{route}\text{}\mathrm{r}\text{}\left(p\in P,{I}_{r,p}\subseteq {I}_{r}\right)$ |

${R}_{p}$ | $\mathrm{Routes}\text{}\mathrm{set}\text{}\mathrm{of}\text{}\mathrm{berthing}\text{}\mathrm{port}\text{}p$ |

${C}_{p}^{tran}$ | Unit container transshipment cost |

${C}_{p}^{load}$ | $\mathrm{Unit}\text{}\mathrm{loading}\text{}\mathrm{cos}\mathrm{t}\text{}\mathrm{at}\text{}\mathrm{port}\text{}p$ |

${C}_{p}^{disc}$ | $\mathrm{Unit}\text{}\mathrm{discharge}\text{}\mathrm{cos}\mathrm{t}\text{}\mathrm{at}\text{}\mathrm{port}\text{}p$ |

${D}_{o,d}$ | $\mathrm{Transport}\text{}\mathrm{demand}\text{}\mathrm{between}\text{}\mathrm{port}\text{}o$$\text{}\mathrm{and}\text{}\mathrm{port}\text{}d$ |

${C}_{v}^{opr}$ | $\mathrm{Fixed}\text{}\mathrm{cos}\mathrm{ts}\text{}\mathrm{of}\text{}\mathrm{operating}\text{}\mathrm{a}\text{}\mathrm{ship}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ |

${C}_{p,v}^{ber}$ | $\mathrm{Unit}\text{}\mathrm{cos}\mathrm{t}\text{}\mathrm{of}\text{}\mathrm{berthing}\text{}\mathrm{operation}\text{}\mathrm{at}\text{}\mathrm{port}\text{}p$$\text{}\mathrm{for}\text{}\mathrm{ship}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ |

${Q}_{v}$ | $\mathrm{Ship}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ capacity |

${C}_{v}^{out}$ | $\mathrm{Chartering}-\mathrm{out}\text{}\mathrm{profit}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ ship |

${C}_{r,v}^{fix}$ | Costs related to the voyage |

${T}_{p,v}^{cont}$ | $\mathrm{Per}-\mathrm{container}\text{}\mathrm{operating}\text{}\mathrm{time}\text{}\mathrm{at}\text{}\mathrm{port}\text{}p$$\text{}\mathrm{for}\text{}\mathrm{ship}\text{}\mathrm{type}\text{}v$ |

Decision Variables | |

${n}_{v}^{in}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{ships}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$$\text{}\mathrm{charted}\text{}\mathrm{for}\text{}\mathrm{the}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ ships |

${n}_{v}^{out}$ | $\mathrm{Chartering}-\mathrm{out}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$ ships |

${m}_{r,v}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$$\text{}\mathrm{ships}\text{}\mathrm{deployed}\text{}\mathrm{on}\text{}\mathrm{ship}\text{}\mathrm{route}\text{}r$ |

${z}_{r,v,i}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{containers}\text{}\mathrm{handled}\text{}\mathrm{for}\text{}\mathrm{a}\text{}\mathrm{ship}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$$\text{}\mathrm{at}\text{}\mathrm{the}\text{}\mathrm{ith}\text{}\mathrm{port}\text{}\mathrm{of}\text{}\mathrm{call}\text{}\mathrm{on}\text{}\mathrm{ship}\text{}\mathrm{route}\text{}r$ |

${x}_{r,v}$ | $1,\text{}\mathrm{if}\text{}\mathrm{deployed}\text{}\mathrm{ships}\text{}\mathrm{of}\text{}\mathrm{type}\text{}v$$\text{}\mathrm{on}\text{}\mathrm{route}\text{}r$, and 0 otherwise |

${z}_{o,r,i}^{load}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{containers}\text{}\mathrm{loaded}\text{}\mathrm{at}\text{}\mathrm{the}\text{}\mathrm{i}-\mathrm{th}\text{}\mathrm{port}\text{}\mathrm{of}\text{}\mathrm{call}\text{}\mathrm{on}\text{}\mathrm{route}\text{}r$$\text{}\mathrm{originating}\text{}\mathrm{from}\text{}\mathrm{port}\text{}o$ |

${z}_{o,r,i}^{disc}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{containers}\text{}\mathrm{discharged}\text{}\mathrm{at}\text{}\mathrm{the}\text{}\mathrm{i}-\mathrm{th}\text{}\mathrm{port}\text{}\mathrm{of}\text{}\mathrm{call}\text{}\mathrm{on}\text{}\mathrm{route}\text{}r$ originating from port o |

${z}_{o,r,i}^{flow}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{containers}\text{}\mathrm{originating}\text{}\mathrm{from}\text{}\mathrm{port}\text{}o$$\mathrm{sailing}\text{}\mathrm{on}\text{}\mathrm{the}\text{}\mathrm{ith}\text{}\mathrm{leg}\text{}\mathrm{of}\text{}\mathrm{route}\text{}r$ |

${z}_{p}^{tran}$ | $\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{containers}\text{}\mathrm{transshipped}\text{}\mathrm{at}\text{}\mathrm{a}\text{}\mathrm{particular}\text{}\mathrm{port}\text{}p$ |

${e}_{r,i}$ | Ballast water weight required for ship stability sailing on the i-th leg of route r |

${s}_{ri}$ | $\mathrm{Sailing}\text{}\mathrm{speed}\text{}\mathrm{during}\text{}\mathrm{leg}\text{}i$$\text{}\mathrm{of}\text{}\mathrm{route}\text{}r$ |

${w}_{ri}$ | $\mathrm{Ship}\text{}\mathrm{payload}\text{}\mathrm{during}\text{}\mathrm{leg}\text{}i$$\text{}\mathrm{of}\text{}\mathrm{route}\text{}r$ |

Ship Type | ||||
---|---|---|---|---|

Small | Medium | Large | Giant | |

Capacity of different types of ships ${Q}_{v}$ (TEU) | 1500 | 3000 | 5000 | 10,000 |

Fixed operating costs of different types of ships ${C}_{v}^{opr}$(week) | 51,923 | 76,923 | 115,384 | 173,076 |

Unit cost of berthing operation of different types of ships ${C}_{p,v}^{ber}$(h) | 500 | 1000 | 1666 | 3333 |

Per-container operating time of different types of ships ${T}_{p,v}^{cont}$(h) | 0.025 | 0.012 | 0.011 | 0.008 |

Number of ships owned by the shipping lines ${N}_{v}^{own}$ | 20 | 20 | 20 | 20 |

Chartering-out profit of different types of ships ${C}_{v}^{out}$ | 52,500 | 77,000 | 98,000 | 140,000 |

Chartering-in cost of different types of ships ${C}_{v}^{in}$ | 66,500 | 94,500 | 122,500 | 175,000 |

No. | Ports Corresponding Number, Ports Order (Distance between Ports) |
---|---|

1 | 1Yokohama (15)$\to $2Tokyo (177)$\to $3Nagoya (201)$\to $4Kobe (734)$\to $5Shanghai (745)$\to $6Hong Kong (1568) $\to $ 1Yokohama |

2 | 7Ho Chi Minh (589)$\text{}\to \text{}$8Laem Chabang (755)$\text{}\to \text{}$9Singapore (187)$\text{}\to \text{}$10Port Klang (830) $\to \text{}$7Ho Chi Minh |

3 | 11Brisbane (419)$\text{}\to \text{}$12Sydney (512)$\text{}\to \text{}$13Melbourne (470)$\text{}\to \text{}$14Adelaide (1325)$\text{}\to \text{}$15Fremantle (1733)$\text{}\to \text{}$16Jakarta (483)$\text{}\to \text{}$9Singapore (3649)$\text{}\to \text{}$11Brisbane |

4 | 17Manila (527)$\text{}\to $ 18Kaohsiung (164)$\text{}\to \text{}$19Xiamen (260) $\to \text{}$6Hong Kong (15)$\text{}\to \text{}$20Yantian (19)$\text{}\to \text{}$21Chiwan (17)$\text{}\to \text{}$6Hong Kong (620)$\text{}\to \text{}$17Manila |

5 | 22Dalian (187)$\text{}\to \text{}$23Xingang (379)$\text{}\to \text{}$24Qingdao (303)$\text{}\to \text{}$5Shanghai (93)$\text{}\to \text{}$25Ningbo (93)$\text{}\to \text{}$5Shanghai (383)$\text{}\to \text{}$26Kwangyang (72)$\text{}\to \text{}$27Busan (487)$\text{}\to \text{}$22Dalian |

6 | 28Chittagong (872)$\text{}\to \text{}$29Chennai (573)$\text{}\to \text{}$30Colombo (306)$\text{}\to \text{}$31Cochin (723)$\text{}\to \text{}$32Nhava Sheva (723)$\text{}\to \text{}$31Cochin (306) $\to \text{}$30Colombo (573)$\text{}\to \text{}$29Chennai (872)$\text{}\to \text{}$28Chittagong |

7 | 33Sokhna (265)$\text{}\to \text{}$34Aqabah (554)$\text{}\to \text{}$35Jeddah (1268)$\text{}\to \text{}$36Salalah (885)$\text{}\to \text{}$37Karachi (688)$\text{}\to $ 38Jebel Ali (862)$\text{}\to \text{}$36Salalah (1878)$\text{}\to \text{}$33Sokhna |

8 | 39Southampton (165)$\to $40Thamesport (386)$\to $41Hamburg (82) $\to $42Bremerhaven (196)$\to $43Rotterdam (42)$\to $44Antwerp (51)$\to $45Zeebrugge (168)$\to $46Le Havre (103)$\to $39Southampton |

9 | 10Port Klang (187)$\to $9Singapore (483)$\to $16Jakarta (1917)$\to $18Kaohsiung (904)$\to $27Busan (904) $\to $18Kaohsiung (342)$\to $6Hong Kong (17)$\to $21Chiwan (1597)$\to $10Port Klang |

10 | 39Southampton (3162)$\text{}\to \text{}$33Sokhna (1878)$\text{}\to \text{}$36Salalah (1643)$\text{}\to \text{}$30Colombo (1560)$\text{}\to \text{}$9Singapore (1415)$\text{}\to \text{}$6Hong Kong (260) $\to \text{}$19Xiamen (486)$\text{}\to \text{}$5Shanghai (448)$\text{}\to \text{}$27Busan (487)$\text{}\to \text{}$22Dalian (187)$\text{}\to \text{}$23Xingang (379)$\text{}\to \text{}$24Qingdao (303) $\to \text{}$5Shanghai (745)$\text{}\to \text{}$6Hong Kong (1415)$\text{}\to \text{}$9Singapore (1560)$\text{}\to \text{}$30Colombo (1643)$\text{}\to \text{}$36Salalah (5029)$\text{}\to \text{}$39Southampton |

11 | 11Brisbane (419)$\text{}\to $12Sydney (512)$\text{}\to $13Melbourne (470)$\text{}\to $14Adelaide (1325)$\text{}\to $15Fremantle (3148)$\text{}\to $30Colombo (1643)$\text{}\to $36Salalah (5244)$\text{}\to $43Rotterdam (5244)$\text{}\to $36Salalah (1643)$\text{}\to $30Colombo (5191)$\text{}\to $11Brisbane |

12 | 20Yantian (9956)$\to $41Hamburg (3621)$\to $33Sokhna (620)$\to $35Jeddah (4156)$\to $10Port Klang (187)$\to $9Singapore (1309)$\to $17Manila (629)$\to $20Yantian |

${\mathit{c}}_{\mathit{r},\mathit{v}}^{\mathit{f}}$ | Ship Type | Ship Type | |||||||
---|---|---|---|---|---|---|---|---|---|

No. | Small | Medium | Large | Giant | No. | Small | Medium | Large | Giant |

1 | 226,198 | 280,542 | − | 404,000 | 7 | 404,711 | 499,139 | − | 710,000 |

2 | 154,791 | 191,900 | − | 276,100 | 8 | − | 129,712 | 149,622 | 199,300 |

3 | 533,980 | 656,891 | − | 929,100 | 9 | − | 501,088 | 551,946 | 715,100 |

4 | − | 155,123 | 176,013 | 232,200 | 10 | − | − | 1,883,007 | 2,430,000 |

5 | 148,807 | 187,600 | − | 279,700 | 11 | 1,504,231 | 1,843,178 | − | 2,583,900 |

6 | 322,916 | 400,072 | − | 574,800 | 12 | 1,235,313 | 1,512,755 | − | 2,117,800 |

**Table 5.**Four scenarios composed of the different approximation error ${\epsilon}_{1}$ and ${\epsilon}_{2}$ combinations.

${\mathit{\epsilon}}_{1}$ | Segments | ${\mathit{\epsilon}}_{2}$ | Segments | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
---|---|---|---|---|---|---|---|

1.6 × 10^{−3} | 68 | 7.6 × 10^{−3} | 11 | 68 | 68 | 135 | 135 |

4.2 × 10^{−4} | 135 | 9.6 × 10^{−4} | 22 | 11 | 22 | 11 | 22 |

$\mathit{\beta}$ | Scenario 1 | Scenario 2 | ${\mathbf{\Delta}}_{1}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | ${\mathbf{\Delta}}_{\mathbf{2}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | ||||||||

$\mathit{f}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{7}}\right)$ | ${\mathit{f}}^{\mathit{F}\mathit{C}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | ${\mathit{f}}^{\mathit{C}\mathit{T}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | Gap | Time | $\mathit{f}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{7}}\right)$ | ${\mathit{f}}^{\mathit{F}\mathit{C}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | ${\mathit{f}}^{\mathit{C}\mathit{T}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | Gap | Time | |||

100 | 2.26345 | 1.456898 | 5.4227 | 4.5785% | 157s | 2.25179 | 1.305053 | 5.4227 | 4.1103% | 1320s | −0.5 | −1.0 |

200 | 2.38397 | 2.291148 | 5.4082 | 4.7686% | 131s | 2.37363 | 2.203396 | 5.4082 | 4.7209% | 1693s | −0.4 | −3.8 |

300 | 2.50588 | 2.978536 | 5.397 | 4.3182% | 116s | 2.4736 | 2.757697 | 5.397 | 4.7709% | 1455s | −1.2 | −7.4 |

400 | 2.61124 | 3.153422 | 5.3957 | 4.9317% | 132s | 2.55396 | 2.924008 | 5.3957 | 4.3785% | 1791s | −2.1 | −7.2 |

500 | 2.65745 | 3.470209 | 5.3562 | 4.9863% | 170s | 2.67637 | 3.780855 | 5.3562 | 4.7166% | 1422s | −0.7 | 8.9 |

600 | 2.71938 | 3.725877 | 5.4509 | 4.8223% | 176s | 2.6914 | 3.686778 | 5.4509 | 4.7034% | 1534s | −1.0 | −1.0 |

$\mathit{\beta}$ | Scenario 3 | Scenario 4 | ${\mathbf{\Delta}}_{\mathbf{3}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | ${\mathbf{\Delta}}_{\mathbf{4}}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | ||||||||

$\mathit{f}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{7}}\right)$ | ${\mathit{f}}^{\mathit{F}\mathit{C}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | ${\mathit{f}}^{\mathit{C}\mathit{T}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | Gap | Time | $\mathit{f}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{7}}\right)$ | ${\mathit{f}}^{\mathit{F}\mathit{C}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | ${\mathit{f}}^{\mathit{C}\mathit{T}}\left(\mathbf{\times}{\mathbf{10}}^{\mathbf{6}}\right)$ | Gap | Time | |||

100 | 2.25775 | 1.53476 | 5.4208 | 4.2802% | 219s | 2.2547 | 1.285647 | 5.3908 | 4.4730% | 1663s | −0.1 | −16 |

200 | 2.39083 | 2.230419 | 5.3724 | 4.8466% | 182s | 2.37366 | 2.208508 | 5.3239 | 4.6335% | 1807s | −0.7 | −0.9 |

300 | 2.50871 | 3.146133 | 5.408999 | 4.7001% | 160s | 2.47303 | 2.589513 | 5.352 | 4.9580% | 2006s | −1.4 | −18 |

400 | 2.59337 | 3.204796 | 5.3449 | 4.7656% | 146s | 2.56488 | 3.067922 | 5.3726 | 4.8849% | 1514s | −1.1 | −4.2 |

500 | 2.66613 | 3.596815 | 5.3561 | 4.9071% | 159s | 2.64155 | 3.536123 | 5.3642 | 4.9157% | 1514s | −0.9 | −1.6 |

600 | 2.6916 | 3.704831 | 5.346299 | 4.8691% | 167s | 2.69072 | 3.677827 | 5.3619 | 4.8956% | 1606s | −0.0 | −0.7 |

$\mathit{\beta}$ | Ship Deployment | Each Legs Speed of Route 2 | Ship Deployment | Each Legs Speed of Route 6 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Type | Num | 1 | 2 | 3 | 4 | Type | Num | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

100 | 3000 | 1 | 25.98 | 25.62 | 25.98 | 23.83 | 1500 | 3 | 16.26 | 16.00 | 16.00 | 16.26 | 16.21 | 16.26 | 16.26 | 24.45 |

200 | 1500 | 2 | 10.36 | 10.42 | 10.36 | 23.83 | 3000 | 2 | 21.06 | 20.62 | 20.62 | 20.89 | 20.62 | 20.62 | 20.62 | 24.45 |

300 | 3000 | 1 | 25.51 | 25.98 | 25.98 | 23.83 | 3000 | 2 | 20.62 | 21.06 | 20.62 | 20.89 | 20.62 | 21.06 | 20.62 | 24.45 |

400 | 1500 | 2 | 10.41 | 10.36 | 10.47 | 23.83 | 3000 | 2 | 21.06 | 20.62 | 20.62 | 20.62 | 20.71 | 21.06 | 20.62 | 24.45 |

500 | 3000 | 1 | 25.51 | 25.98 | 25.98 | 23.83 | 1500 | 3 | 16.26 | 16.07 | 16.00 | 16.26 | 16.26 | 16.00 | 16.26 | 24.45 |

600 | 1500 | 2 | 10.58 | 10.61 | 10.58 | 23.83 | 1500 | 3 | 11.24 | 11.17 | 11.17 | 11.17 | 11.17 | 11.17 | 11.17 | 24.45 |

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## Share and Cite

**MDPI and ACS Style**

Gao, C.-F.; Hu, Z.-H. Speed Optimization for Container Ship Fleet Deployment Considering Fuel Consumption. *Sustainability* **2021**, *13*, 5242.
https://doi.org/10.3390/su13095242

**AMA Style**

Gao C-F, Hu Z-H. Speed Optimization for Container Ship Fleet Deployment Considering Fuel Consumption. *Sustainability*. 2021; 13(9):5242.
https://doi.org/10.3390/su13095242

**Chicago/Turabian Style**

Gao, Chao-Feng, and Zhi-Hua Hu. 2021. "Speed Optimization for Container Ship Fleet Deployment Considering Fuel Consumption" *Sustainability* 13, no. 9: 5242.
https://doi.org/10.3390/su13095242