# A Fuzzy Multi-Criteria Method for Sustainable Ferry Operator Selection: A Case Study

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## Abstract

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## 1. Introduction

- (1)
- To the best of our knowledge, previous literature focused on: (i) ferry transport pricing [7] (ii) ferry transport safety [1], (iii) ferry transport fleet [2], and (iv) ferry network design problem [39], etc., but ignored the evaluation of ferry operators despite this being a complex MCDM issue. Naturally, this is the first paper that investigates the evaluation of ferry operators from a sustainability perspective.
- (2)
- In view of the superior performance of FTOPSIS in alternative solutions, it has been widely used in practice [40,41,42,43,44,45]. Again to the best of our knowledge, this is the first FAHP-EW-FTOPSIS-based MCDM technique proposed to select the best ferry operator in an attempt to extent the field of application of the FTOPSIS method. Moreover, we also build an effective integrated framework model for ferry operator evaluation for better implementation.
- (3)
- Evaluation criteria are significantly important for ferry operator selection. Given that there is currently no literature report on the evaluation index system for ferry operators. Thus, this is the first paper that constructs an evaluation index system for ferry operators based on the perspective of sustainability.

## 2. Literature Review

#### 2.1. Method of Determining Criteria Weights

#### 2.2. Application of TOPSIS Method

#### 2.3. Ferry Operator Evaluation

## 3. Methodology

#### 3.1. Criteria Weight Determination Method

#### 3.1.1. Fuzzy Analytic Hierarchy Process

- (1)
- Build the hierarchy analysis structure of criteria and sub-criteria respectively. The expert’s linguistic terms were transformed with the TFNs in Table 1 [81]. This paper uses the triangular fuzzy scale in the literature [81], because: (1) the scale has been widely used; (2) in the implementation process, the evaluation experts also unanimously selected this triangular fuzzy scale; (3) the triangular fuzzy scale can reflect the difference between the criteria degree of importance. Aggregate evaluation term of experts thus is given by Equation (1).

- (2)
- The fuzzy synthetic extent value of the $i$-th ${P}_{i}$ object is given by$${P}_{i}={\displaystyle \sum}_{j=1}^{m}{Y}_{ij}\times {\left[{\displaystyle \sum}_{i=1}^{n}{\displaystyle \sum}_{j=1}^{m}{Y}_{ij}\right]}^{-1},$$$$\sum}_{j=1}^{m}{Y}_{ij}=\left({\displaystyle \sum}_{j=1}^{m}{a}_{ij},{\displaystyle \sum}_{j=1}^{m}{b}_{ij},{\displaystyle \sum}_{j=1}^{m}{c}_{ij}\right),$$$${\left[{\displaystyle \sum}_{i=1}^{n}{\displaystyle \sum}_{j=1}^{m}{Y}_{ij}\right]}^{-1}=\left(\frac{1}{{{\displaystyle \sum}}_{i=1}^{n}{{\displaystyle \sum}}_{j=1}^{m}{c}_{ij}},\frac{1}{{{\displaystyle \sum}}_{i=1}^{n}{{\displaystyle \sum}}_{j=1}^{m}{b}_{ij}},\frac{1}{{{\displaystyle \sum}}_{i=1}^{n}{{\displaystyle \sum}}_{j=1}^{m}{a}_{ij}}\right).$$

- (3)
- The degree of possibility of $\left({a}_{1},{b}_{1},{c}_{1}\right)={P}_{1}\le {P}_{2}=\left({a}_{2},{b}_{2},{c}_{2}\right)$ is given by$$V\left({P}_{2}\ge {P}_{1}\right)=\left\{\begin{array}{c}0,\mathrm{if}{a}_{1}\ge {\mathrm{c}}_{2}\\ 1,\mathrm{if}{\mathrm{b}}_{2}\ge {\mathrm{b}}_{1}\\ \frac{{a}_{1}-{c}_{2}}{\left({b}_{2}-{c}_{2}\right)-\left({b}_{1}-{a}_{1}\right)},\mathrm{otherwise}\end{array}\right..$$

- (4)
- Let ${\mathrm{d}}^{\prime}\left({z}_{i}\right)=\mathrm{min}V\left({P}_{i}\ge {P}_{l}\right),l=1,2,\cdots ,n,i\ne k$, then the weight vector is given by$${w}^{\prime}={\left[{\mathrm{d}}^{\prime}\left({z}_{1}\right),{\mathrm{d}}^{\prime}\left({z}_{2}\right),\cdots ,{\mathrm{d}}^{\prime}\left({z}_{n}\right)\right]}^{T}.$$$$\mathrm{min}V\left(P\ge {P}_{i}\right)=V\left[\left(P\ge {P}_{1}\right)\mathrm{and}(P\ge {P}_{2})\mathrm{and}\cdots \mathrm{and}(P\ge {P}_{k})\right].$$

- (5)
- Let ${w}_{j}^{s}$ denote subjective criteria weight, and the normalized subjective criteria weight vector is given by$${w}_{j}^{s}=\left({w}_{1}^{s},{w}_{2}^{s},\cdots ,{w}_{j}^{s}\right)=(\frac{{\mathrm{d}}^{\prime}\left({z}_{1}\right)}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{d}}^{\prime}\left({z}_{n}\right)},\frac{{\mathrm{d}}^{\prime}\left({z}_{2}\right)}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{d}}^{\prime}\left({z}_{n}\right)},\cdots ,\frac{{\mathrm{d}}^{\prime}\left({z}_{n}\right)}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{d}}^{\prime}\left({z}_{n}\right)})$$

#### 3.1.2. Entropy Weight Method

- (1)
- Experts give the evaluation matrix $\overline{U}$ according to the criteria performance of the alternatives, then de-fuzzify the obtained fuzzy matrix, and finally aggregate the evaluation information of all experts. Finally, a matrix $U$ with crisp values is obtained.$$\overline{U}=\left[\begin{array}{c}{u}_{11}^{k}\\ {u}_{21}^{k}\\ \vdots \\ {u}_{m1}^{k}\end{array}\begin{array}{c}{u}_{12}^{k}\\ {u}_{22}^{k}\\ \vdots \\ {u}_{m2}^{k}\end{array}\begin{array}{c}\cdots \\ \cdots \\ \ddots \\ \cdots \end{array}\begin{array}{c}{u}_{1n}^{k}\\ {u}_{2n}^{k}\\ \vdots \\ {u}_{mn}^{k}\end{array}\right]$$$${u}_{ij}^{k}=\frac{{a}_{ij}^{k}+4{b}_{ij}^{k}+{c}_{ij}^{k}}{6},$$$${u}_{ij}={\displaystyle \sum}_{k=1}^{K}{w}^{k}{u}_{ij}^{k}.$$

- (2)
- Normalize the matrix $U$ by the following equation:$${u}_{ij}^{\ast}=\left\{\begin{array}{c}\frac{{u}_{ij}-\underset{j}{\mathrm{min}}{u}_{ij}}{\underset{j}{\mathrm{max}}{u}_{ij}-\underset{j}{\mathrm{min}}{u}_{ij}}Benefitcriteria\\ \frac{\underset{j}{\mathrm{max}}{u}_{ij}-{u}_{ij}}{\underset{j}{\mathrm{max}}{u}_{ij}-\underset{j}{\mathrm{min}}{u}_{ij}}Costcriteria\end{array}\right.,$$

- (3)
- The entropy of each criterion ${E}_{j}$ is given by$${E}_{j}=-\frac{1}{\mathrm{ln}m}{\displaystyle \sum}_{i=1}^{m}\frac{{u}_{ij}^{\ast}}{{{\displaystyle \sum}}_{i=1}^{m}{u}_{ij}^{\ast}}\mathrm{ln}\frac{{u}_{ij}^{\ast}}{{{\displaystyle \sum}}_{i=1}^{m}{u}_{ij}^{\ast}}.$$

- (4)
- Let ${w}_{j}^{o}$ denote objective criteria $j$’s weight, which is determined by the following equation:$${w}_{j}^{o}=\frac{1-{E}_{j}}{n-{{\displaystyle \sum}}_{j=1}^{n}{E}_{j}}.$$

#### 3.1.3. Comprehensive Criteria Weight

#### 3.2. Fuzzy TOPSIS

- (1)
- Build initial fuzzy evaluation matrix ${U}^{\prime}$ of the alternatives.$${U}^{\prime}=\left[\begin{array}{c}{u}_{11}^{\prime}\\ {u}_{21}^{\prime}\\ \vdots \\ {u}_{m1}^{\prime}\end{array}\begin{array}{c}{u}_{12}^{\prime}\\ {u}_{22}^{\prime}\\ \vdots \\ {u}_{m2}^{\prime}\end{array}\begin{array}{c}\cdots \\ \cdots \\ \ddots \\ \cdots \end{array}\begin{array}{c}{u}_{1n}^{\prime}\\ {u}_{2n}^{\prime}\\ \vdots \\ {u}_{mn}^{\prime}\end{array}\right]=\left[\begin{array}{c}\left({a}_{11}^{\prime},{b}_{11}^{\prime},{c}_{11}^{\prime}\right)\\ \left({a}_{21}^{\prime},{b}_{21}^{\prime},{c}_{21}^{\prime}\right)\\ \vdots \\ \left({a}_{m1}^{\prime},{b}_{m1}^{\prime},{c}_{m1}^{\prime}\right)\end{array},\begin{array}{c}\left({a}_{12}^{\prime},{b}_{12}^{\prime},{c}_{12}^{\prime}\right)\\ \left({a}_{22}^{\prime},{b}_{22}^{\prime},{c}_{22}^{\prime}\right)\\ \vdots \\ \left({a}_{m2}^{\prime},{b}_{m2}^{\prime},{c}_{m2}^{\prime}\right)\end{array}\begin{array}{c}\cdots \\ \cdots \\ \ddots \\ \cdots \end{array}\begin{array}{c}\left({a}_{1n}^{\prime},{b}_{1n}^{\prime},{c}_{1n}^{\prime}\right)\\ \left({a}_{2n}^{\prime},{b}_{2n}^{\prime},{c}_{2n}^{\prime}\right)\\ \vdots \\ \left({a}_{mn}^{\prime},{b}_{mn}^{\prime},{c}_{mn}^{\prime}\right)\end{array}\right],$$

- (2)
- Normalize initial fuzzy evaluation matrix ${U}^{\prime}$$${u}_{ij}^{\u2033}=\left\{\begin{array}{c}\left(\frac{{a}_{ij}^{\prime}}{\underset{i}{\mathrm{max}}{c}_{ij}^{\prime}},\frac{{b}_{ij}^{\prime}}{\underset{i}{\mathrm{max}}{c}_{ij}^{\prime}},\frac{{c}_{ij}^{\prime}}{\underset{i}{\mathrm{max}}{c}_{ij}^{\prime}}\right)Benefitcriteria\\ \left(\frac{\underset{i}{\mathrm{min}}{a}_{ij}^{\prime}}{{c}_{ij}^{\prime}},\frac{\underset{i}{\mathrm{min}}{a}_{ij}^{\prime}}{{b}_{ij}^{\prime}},\frac{\underset{i}{\mathrm{min}}{a}_{ij}^{\prime}}{{a}_{ij}^{\prime}}\right)Costcriteria\end{array}\right.,$$

- (3)
- Determine the weighted normalize matrix $\overline{C}$$$\overline{C}=\left[\begin{array}{c}\left({w}_{1}{a}_{11}^{\u2033},{w}_{1}{b}_{11}^{\u2033},{w}_{1}{c}_{11}^{\u2033}\right)\\ \left({w}_{1}{a}_{21}^{\u2033},{w}_{1}{b}_{21}^{\u2033},{w}_{1}{c}_{21}^{\u2033}\right)\\ \vdots \\ \left({w}_{1}{a}_{m1}^{\u2033},{w}_{1}{b}_{m1}^{\u2033},{w}_{1}{c}_{m1}^{\u2033}\right)\end{array},\begin{array}{c}\left({w}_{2}{a}_{12}^{\u2033},{w}_{2}{b}_{12}^{\u2033},{w}_{2}{c}_{12}^{\u2033}\right)\\ \left({w}_{2}{a}_{22}^{\u2033},{w}_{2}{b}_{22}^{\u2033},{w}_{2}{c}_{22}^{\u2033}\right)\\ \vdots \\ \left({w}_{2}{a}_{m2}^{\u2033},{w}_{2}{b}_{m2}^{\u2033},{w}_{2}{c}_{m2}^{\u2033}\right)\end{array}\begin{array}{c}\cdots \\ \cdots \\ \ddots \\ \cdots \end{array},\begin{array}{c}\left({w}_{n}{a}_{1n}^{\u2033},{w}_{n}{b}_{1n}^{\u2033},{w}_{n}{c}_{1n}^{\u2033}\right)\\ \left({w}_{n}{a}_{2n}^{\u2033},{w}_{n}{b}_{2n}^{\u2033},{w}_{n}{c}_{2n}^{\u2033}\right)\\ \vdots \\ \left({w}_{n}{a}_{mn}^{\u2033},{w}_{n}{b}_{mn}^{\u2033},{w}_{n}{c}_{mn}^{\u2033}\right)\end{array}\right],.$$

- (4)
- Defuzzification by the following equation:$${\mathrm{h}}_{\mathrm{ij}}=\frac{{\mathrm{w}}_{\mathrm{j}}{\mathrm{a}}_{\mathrm{ij}}^{\u2033}+4{\mathrm{w}}_{\mathrm{j}}{\mathrm{b}}_{\mathrm{ij}}^{\u2033}+{\mathrm{w}}_{\mathrm{j}}{\mathrm{c}}_{\mathrm{ij}}^{\u2033}}{6}.$$

- (5)
- Determine the fuzzy positive ideal solution ${h}_{j}^{+}$ and negative ideal solution ${h}_{j}^{-}$, respectively.$$\left\{\begin{array}{c}{h}_{j}^{+}=\left\{(\underset{i}{\mathrm{max}}{h}_{ij}\left|jisbenefitcriteria\right.),(\underset{i}{\mathrm{min}}{h}_{ij}\left|jiscostcriteria\right.)\right\}\\ {h}_{j}^{-}=\left\{(\underset{i}{\mathrm{min}}{h}_{ij}\left|jisbenefitcriteria\right.),(\underset{i}{\mathrm{max}}{h}_{ij}\left|jiscostcriteria\right.)\right\}\end{array}\right.,$$

- (6)
- The distance of alternative from positive and negative ideal solution ${d}_{i}^{+}$,${d}_{i}^{-}$ is computed by$$\left\{\begin{array}{c}{d}_{i}^{+}={\left\{{\displaystyle {\displaystyle \sum}_{j=1}^{m}}{\left({h}_{j}^{+}-{h}_{ij}\right)}^{2}\right\}}^{\frac{1}{2}}\\ {d}_{i}^{-}={\left\{{\displaystyle {\displaystyle \sum}_{j=1}^{m}}{\left({h}_{j}^{-}-{h}_{ij}\right)}^{2}\right\}}^{\frac{1}{2}}\end{array}\right..$$

- (7)
- Compute the closeness coefficient of each alternative ${S}_{i}$ by$${S}_{i}=\frac{{d}_{i}^{-}}{{d}_{i}^{+}+{d}_{i}^{-}},$$

#### 3.3. Integrated Framework of Ferry Operator Evaluation

## 4. Discussion

#### 4.1. Social Criteria (C1)

#### 4.2. Flexibility Criteria (C2)

#### 4.3. Economic Criteria (C3)

#### 4.4. Management Criteria (C4)

#### 4.5. Environmental Criteria (C5)

## 5. Case Study

#### 5.1. Case Description

#### 5.2. Criteria Weight

#### 5.2.1. Determine Subjective Criteria Weight by FAHP

#### 5.2.2. Determine Objective Criteria Weight by EW Method

#### 5.2.3. Combine FAHP and EW Method

#### 5.3. Determine the Alternative Rankings by Fuzzy TOPSIS

#### 5.4. Sensitive Analysis

#### 5.5. Management Recommendations

- (1)
- For decision makers, how to choose the decision preference parameter $\beta $ is still an important issue. Because decision parameter $\beta $ has an impact on the outcome, how to reduce the risk of decision failure due to inconsistent decision preference parameters requires further scrutiny. The decision maker can consider the best alternative that has little effect on the outcome due to changes in the decision preference parameter $\beta $, because the most criteria performance of the best alternative is superior. The criteria performance of an alternative where weight changes have a large impact on the results can vary widely, and it is not recommended to choose such an alternative.

- (2)
- Regarding the evaluation index system of ferry operators, this paper constructs 5 criteria and 15 sub-criteria from the perspective of relevant literature and expert opinions. Since there are no previous literature reports on ferry operator evaluations, our criteria system may need to be justified in future studies.
- (3)
- In the MCDM approach, since experts have ambiguity about the criteria assessment, reducing this ambiguity will reduce the risk of failing to select the best ferry operator.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Criteria | Experts | Social (C_{1}) | Flexibility (C_{2}) | Economic (C_{3}) | Management (C_{4}) | Environment (C_{5}) |
---|---|---|---|---|---|---|

Social (C_{1}) | E_{1} | (1, 1, 1) | (2, 5/2, 3) | (1, 1, 1) | (1/3, 2/5, 1/2) | (1/3, 2/5, 1/2) |

E_{2} | (1, 1, 1) | (2/3, 1, 2) | (2/3, 1, 2) | (1/3, 2/5, 1/2) | ||

E_{3} | (2/3, 1, 2) | (2/3, 1, 2) | (2/3, 1, 2) | (1, 3/2, 2) | ||

E_{4} | (1, 1, 1) | (2/5, 1/2, 2/3) | (1/3, 2/5, 1/2) | (1/2, 1, 3/2) | ||

E_{5} | (1/3, 2/5, 1/2) | (2/5, 1/2, 2/3) | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

Flexibility (C_{2}) | E_{1} | (1/3, 2/5, 1/2) | (1, 1, 1) | (1/2, 2/3, 1) | (1/2, 2/3, 1) | (2, 5/2, 3) |

E_{2} | (1, 1, 1) | (1/2, 2/3, 1) | (1/2, 2/3, 1) | (1/2, 1, 3/2) | ||

E_{3} | (1/2, 1, 3/2) | (1, 1, 1) | (1/2, 2/3, 1) | (1/2, 1, 3/2) | ||

E_{4} | (1,1,1) | (1, 1, 1) | (2/5, 1/2, 2/3) | (1/2, 1, 3/2) | ||

E_{5} | (2, 5/2, 3) | (2/3, 1, 2) | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

Economic (C_{3}) | E_{1} | (1, 1, 1) | (1, 3/2, 2) | (1, 1, 1) | (3/2, 2, 5/2) | (2/5, 1/2, 2/3) |

E_{2} | (1/2, 1, 3/2) | (1, 3/2, 2) | (3/2, 2, 5/2) | (1, 1, 1) | ||

E_{3} | (1/2, 1, 3/2) | (1, 1, 1) | (1, 3/2, 2) | (2/5, 1/2, 2/3) | ||

E_{4} | (3/2, 2, 5/2) | (1, 1, 1) | (1/2, 1, 3/2) | (3/2, 2, 5/2) | ||

E_{5} | (3/2, 2, 5/2) | (1/2, 1, 3/2) | (1/2, 1, 3/2) | (3/2, 2, 5/2) | ||

Management (C_{4}) | E_{1} | (2, 5/2, 3) | (1, 3/2, 2) | (2/5, 1/2, 2/3) | (1, 1, 1) | (2/3, 1, 2) |

E_{2} | (1/2, 1, 3/2) | (1, 3/2, 2) | (2/5, 1/2, 2/3) | (2/3, 1, 2) | ||

E_{3} | (1/2, 1, 3/2) | (1, 3/2, 2) | (1/2, 2/3, 1) | (1/2, 1, 3/2) | ||

E_{4} | (2, 5/2, 3) | (3/2, 2, 5/2) | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

E_{5} | (1/2, 1, 3/2) | (1/2, 1, 3/2) | (2/3, 1, 2) | (1, 1, 1) | ||

Environment (C_{5}) | E_{1} | (2, 5/2, 3) | (1/3, 2/5, 1/2) | (3/2, 2, 5/2) | (1/2, 1, 3/2) | (1, 1, 1) |

E_{2} | (2, 5/2, 3) | (2/3, 1, 2) | (1, 1, 1) | (1/2, 1, 3/2) | ||

E_{3} | (1/2, 2/3, 1) | (2/3, 1, 2) | (3/2, 2, 5/2) | (2/3, 1, 2) | ||

E_{4} | (2/3, 1, 2) | (2/3, 1, 2) | (2/5, 1/2, 2/3) | (2/3, 1, 2) | ||

E_{5} | (2/3, 1, 2) | (2/3, 1, 2) | (2/5, 1/2, 2/3) | (1, 1, 1) |

Sub-Criteria | Experts | Customer Satisfaction (C_{11}) | Operator Reputation (C_{12}) | Operator Development Plan (C_{13}) |
---|---|---|---|---|

Customer satisfaction (C_{11}) | E_{1} | (1, 1, 1) | (1/2, 1, 3/2) | (1, 1, 1) |

E_{2} | (1/2, 1, 3/2) | (2/3, 1, 2) | ||

E_{3} | (2/3, 1, 2) | (2/3, 1, 2) | ||

E_{4} | (2/3, 1, 2) | (1/2, 2/3, 1) | ||

E_{5} | (1, 3/2, 2) | (2/3, 1, 2) | ||

Operator reputation (C_{12}) | E_{1} | (2/3, 1,2) | (1, 1, 1) | (1/2, 1, 3/2) |

E_{2} | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

E_{3} | (1/2, 1, 3/2) | (1, 3/2, 2) | ||

E_{4} | (1/2, 1, 3/2) | (1, 3/2, 2) | ||

E_{5} | (1/2, 2/3, 1) | (1/3, 2/5, 1/2) | ||

Operator development plan (C_{13}) | E_{1} | (1, 1, 1) | (2/3, 1, 2) | (1, 1, 1) |

E_{2} | (1/2, 1, 3/2) | (2/3, 1, 2) | ||

E_{3} | (1/2, 1, 3/2) | (1/2, 2/3, 1) | ||

E_{4} | (1, 3/2, 2) | (1/2, 2/3, 1) | ||

E_{5} | (1/2, 1, 3/2) | (2, 5/2, 3) |

Sub-Criteria | Experts | Responsiveness (C_{21}) | Ship Diversity (C_{22}) | Informatization (C_{23}) |
---|---|---|---|---|

Responsiveness (C_{21}) | E_{1} | (1, 1, 1) | (1, 3/2, 2) | (2/3, 1, 2) |

E_{2} | (1, 3/2, 2) | (2/3, 1, 2) | ||

E_{3} | (1/2, 1, 3/2) | (2, 5/2, 3) | ||

E_{4} | (1/2, 1, 3/2) | (2/7, 1/3, 2/5) | ||

E_{5} | (2/5, 1/2, 2/3) | (2/3, 1, 2) | ||

Ship diversity (C_{22}) | E_{1} | (1/2, 2/3, 1) | (1, 1, 1) | (2/3, 1, 2) |

E_{2} | (1/2, 2/3, 1) | (1/2, 2/3, 1) | ||

E_{3} | (2/3, 1, 2) | (1/2, 2/3, 1) | ||

E_{4} | (2/3, 1, 2) | (2, 5/2, 3) | ||

E_{5} | (3/2, 2, 5/2) | (5/2, 3, 7/2) | ||

Informatization (C_{23}) | E_{1} | (1/2, 2/3, 1) | (2, 5/2, 3) | (1, 1, 1) |

E_{2} | (1/2, 2/3, 1) | (1, 3/2, 2) | ||

E_{3} | (1/3, 2/5, 1/2) | (1, 3/2, 2) | ||

E_{4} | (5/2, 3, 7/2) | (1/3, 2/5, 1/2) | ||

E_{5} | (1/2, 1, 3/2) | (2/7, 1/3, 2/5) |

Sub-Criteria | Experts | Unit Transportation Cost (C_{31}) | Occupancy Rate (C_{32}) | Carrying Capacity (C_{33}) |
---|---|---|---|---|

Unit transportation cost (C_{31}) | E_{1} | (1, 1, 1) | (2/3, 1, 2) | (2/3,1, 2) |

E_{2} | (2/3, 1, 2) | (2/3,1, 2) | ||

E_{3} | (2/3, 1, 2) | (2/3, 1, 2) | ||

E_{4} | (2/3, 1, 2) | (1/2, 2/3, 1) | ||

E_{5} | (1, 3/2, 2) | (2/3, 1, 2) | ||

Occupancy rate (C_{32}) | E_{1} | (1/2, 1, 3/2) | (1, 1, 1) | (1/2, 1, 3/2) |

E_{2} | (1/2, 1, 3/2) | (1/2, 1, 3/2) | ||

E_{3} | (1/2, 1, 3/2) | (1, 3/2, 2) | ||

E_{4} | (1/2, 1, 3/2) | (1, 3/2, 2) | ||

E_{5} | (1/2, 2/3, 1) | (3/2, 2, 5/2) | ||

Carrying capacity (C_{33}) | E_{1} | (1/2, 1, 3/2) | (2/3, 1, 2) | (1, 1, 1) |

E_{2} | (1/2, 1, 3/2) | (2/3, 1, 2) | ||

E_{3} | (1/2, 1, 3/2) | (1/2, 2/3, 1) | ||

E_{4} | (1, 3/2, 2) | (1/2, 2/3, 1) | ||

E_{5} | (1/2, 1, 3/2) | (2/5, 1/2, 2/3) |

Sub-Criteria | Experts | Emergency Management (C_{41}) | Safety Management (C_{42}) | Service Management (C_{43}) |
---|---|---|---|---|

Emergency management (C_{41}) | E_{1} | (1, 1, 1) | (1/2, 1, 3/2) | (2, 5/2, 3) |

E_{2} | (1/2, 1, 3/2) | (2/5, 1/2, 2/3) | ||

E_{3} | (2/3, 1, 2) | (2/3, 1, 2) | ||

E_{4} | (2/3, 1, 2) | (2/3, 1, 2) | ||

E_{5} | (2/3, 1, 2) | (1, 1, 1) | ||

Safety management (C_{42}) | E_{1} | (2/3, 1, 2) | (1, 1, 1) | (2/5, 1/2, 2/3) |

E_{2} | (2/3, 1, 2) | (2/5, 1/2, 2/3) | ||

E_{3} | (1/2, 1, 3/2) | (2, 5/2, 3) | ||

E_{4} | (1/2, 1, 3/2) | (2, 5/2, 3) | ||

E_{5} | (1/2, 1, 3/2) | (1/2, 2/3, 1) | ||

Service management (C_{43}) | E_{1} | (3/2, 2, 5/2) | (1/3, 2/5, 1/2) | (1, 1, 1) |

E_{2} | (3/2, 2, 5/2) | (3/2, 2, 5/2) | ||

E_{3} | (1/2, 1, 3/2) | (1, 3/2, 2) | ||

E_{4} | (1/2, 1, 3/2) | (1/3, 2/5, 1/2) | ||

E_{5} | (1, 1, 1) | (1/3, 2/5, 1/2) |

Sub-Criteria | Experts | Carbon Emissions (C_{51}) | The Impact of Ships on the Environment (C_{52}) | Energy Saving Measures (C_{53}) |
---|---|---|---|---|

Carbon emissions (C_{51}) | E_{1} | (1, 1, 1) | (2/3, 1, 2) | (1, 3/2, 2) |

E_{2} | (1/2, 2/3, 1) | (1, 3/2, 2) | ||

E_{3} | (2/5, 1/2, 2/3) | (1/3, 2/5, 1/2) | ||

E_{4} | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

E_{5} | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

The impact of ships on the environment (C_{52}) | E_{1} | (1/2, 1, 3/2) | (1, 1, 1) | (1, 1, 1) |

E_{2} | (1, 3/2, 2) | (1/2, 1, 3/2) | ||

E_{3} | (3/2, 2, 5/2) | (1/2, 1, 3/2) | ||

E_{4} | (1/2, 1, 3/2) | (2/3, 1, 2) | ||

E_{5} | (1/2, 1, 3/2) | (2/3, 1, 2) | ||

Energy saving measures (C_{53}) | E_{1} | (1/2, 2/3, 1) | (1, 1, 1) | (1, 1, 1) |

E_{2} | (1/2, 2/3, 1) | (2/3, 1, 2) | ||

E_{3} | (2, 5/2, 3) | (2/3, 1, 2) | ||

E_{4} | (2/3, 1, 2) | (1/2, 1, 3/2) | ||

E_{5} | (2/3, 1, 2) | (1/2, 1, 3/2) |

Experts | Ai | C11 | C12 | C13 | C21 | C22 | C23 | C31 | C32 | C33 | C41 | C42 | C43 | C51 | C52 | C53 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E1 | A1 | 93% | H | VH | H | G | VH | 359.57 | 62% | H | G | H | H | L | G | VH |

A2 | 96% | VH | G | H | H | L | 342.48 | 58% | L | G | H | G | L | L | H | |

A3 | 94% | VH | H | G | H | H | 374.23 | 53% | H | H | G | H | G | VL | H | |

A4 | 98% | H | H | VH | G | G | 328.85 | 61% | H | G | L | H | VL | L | H | |

E2 | A1 | 93% | H | H | H | G | G | 359.57 | 62% | G | VH | G | VH | L | G | VH |

A2 | 96% | G | H | L | G | H | 342.48 | 58% | L | L | VH | G | L | L | VH | |

A3 | 94% | L | VH | G | G | VH | 374.23 | 53% | G | H | G | G | L | G | H | |

A4 | 98% | H | H | H | G | G | 328.85 | 61% | H | L | H | H | VL | G | G | |

E3 | A1 | 93% | VH | G | G | H | G | 359.57 | 62% | G | G | H | H | L | VL | G |

A2 | 96% | VH | H | H | VH | L | 342.48 | 58% | L | G | H | G | VL | L | H | |

A3 | 94% | H | G | G | H | H | 374.23 | 53% | L | H | H | G | G | VL | G | |

A4 | 98% | VH | H | G | VH | G | 328.85 | 61% | H | G | L | G | VL | VL | H | |

E4 | A1 | 93% | G | VH | VH | G | L | 359.57 | 62% | L | G | H | H | L | G | VH |

A2 | 96% | VH | G | G | H | L | 342.48 | 58% | L | L | L | H | L | G | H | |

A3 | 94% | G | G | G | G | H | 374.23 | 53% | G | H | G | VH | L | VL | VH | |

A4 | 98% | H | H | G | VH | L | 328.85 | 61% | G | H | L | VH | VL | L | H | |

E5 | A1 | 93% | G | G | VH | G | L | 359.57 | 62% | VH | VH | VH | G | VL | G | G |

A2 | 96% | VH | G | H | L | G | 342.48 | 58% | G | H | G | G | L | L | H | |

A3 | 94% | G | H | VH | G | H | 374.23 | 53% | H | H | G | H | VL | G | G | |

A4 | 98% | H | G | G | G | H | 328.85 | 61% | G | H | H | G | L | G | G |

Ai | C1 | C2 | C3 | C4 | C5 | Ai | C1 | C2 | C3 | C4 | C5 | Ai | C1 | C2 | C3 | C4 | C5 | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E1 | A1 | H | H | H | L | H | E2 | A1 | H | H | G | L | H | E3 | A1 | G | VH | G | VH | VH |

A2 | H | G | VH | G | VH | A2 | H | G | VH | H | L | A2 | H | L | G | G | G | |||

A3 | VH | H | G | G | H | A3 | G | L | G | G | H | A3 | G | H | G | VH | L | |||

A4 | VH | G | H | G | G | A4 | G | VH | H | H | G | A4 | L | L | H | G | H | |||

E4 | A1 | L | VH | G | H | VH | E5 | A1 | H | G | VH | H | VH | |||||||

A2 | G | H | G | H | G | A2 | VH | L | G | H | G | |||||||||

A3 | G | G | VL | H | L | A3 | H | L | VH | G | G | |||||||||

A4 | H | H | L | L | L | A4 | G | VL | H | VH | VH |

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Linguistic Scales | Triangular Fuzzy Scale | Triangular Fuzzy Reciprocal Scale |
---|---|---|

Equal importance | (1, 1, 1) | (1, 1, 1) |

Weak importance | (1/2, 1, 3/2) | (2/3, 1, 2) |

Weakly more importance | (1, 3/2, 2) | (1/2, 2/3, 1) |

More importance | (3/2, 2, 5/2) | (2/5, 1/2, 2/3) |

Strongly more importance | (2, 5/2, 3) | (1/3, 2/5, 1/2) |

Absolutely more importance | (5/2, 3, 7/2) | (2/7, 1/3, 2/5) |

Linguistic Terms | Very High (VH) | High (H) | General (G) | Low (L) | Very low (VL) |
---|---|---|---|---|---|

TFNs | (0.8, 0.9, 1.0) | (0.6, 0.7, 0.9) | (0.4, 0.5, 0.7) | (0.2, 0.3, 0.5) | (0, 0.1, 0.3) |

Experts | Fields | Age | Education | Work Experience | Professional Title | Experts’ Weight |
---|---|---|---|---|---|---|

E1 | Economy | 60 | Ph. D. | 36 | Senior engineer | 0.25 |

E2 | Economy | 39 | Master | 17 | Engineer | 0.15 |

E3 | College | 56 | Ph. D. | 30 | Professor | 0.25 |

E4 | College | 43 | Ph. D. | 15 | Assistant professor | 0.15 |

E5 | Environment | 57 | Master | 35 | Senior engineer | 0.2 |

Criteria | Society (C_{1}) | Flexibility (C_{2}) | Economic (C_{3}) | Management (C_{4}) | Environment (C_{5}) |
---|---|---|---|---|---|

Society (C_{1}) | (1, 1, 1) | (1.03, 1.26, 1.65) | (0.66, 0.83, 1.28) | (0.53, 0.76, 1.4) | (0.56, 0.89, 1.23) |

Flexibility (C_{2}) | (0.91, 1.15, 1.40) | (1, 1, 1) | (0.73, 0.87, 1.2) | (0.52, 0.71, 1.15) | (0.88, 1.38, 1.88) |

Economic (C_{3}) | (0.98, 1.35, 1.73) | (0.9, 1.2, 1.5) | (1, 1, 1) | (1.03, 1.53, 2.03) | (0.88, 1.1, 1.36) |

Management (C_{4}) | (1.10, 1.6, 2.10) | (0.98, 1.48, 1.98) | (0.52, 0.72, 1.22) | (1, 1, 1) | (0.67, 1, 1.6) |

Environment (C_{5}) | (1.16, 1.52, 2.15) | (0.58, 0.85, 1.63) | (1.04, 1.33, 1.63) | (0.67, 1, 1.6) | (1, 1, 1) |

Sub-Criteria | Customer Satisfaction (C_{11}) | Operator Reputation (C_{12}) | Operator Development Plan (C_{13}) |
---|---|---|---|

Customer satisfaction (C_{11}) | (1, 1, 1) | (0.67, 1.10, 1.80) | (0.73, 0.95, 1.60) |

Operator reputation (C_{12}) | (0.57, 0.93, 1.60) | (1, 1, 1) | (0.67, 1.08, 1.50) |

Operator development plan (C_{13}) | (0.70, 1.08, 1.45) | (0.87, 01.17, 1.80) | (1, 1, 1) |

Sub-Criteria | Responsiveness (C_{21}) | Ship Diversity (C_{22}) | Informatization (C_{23}) |
---|---|---|---|

Responsiveness (C_{21}) | (1, 1, 1) | (0.68, 1.1, 1.53) | (0.94, 1.28, 2.01) |

Ship diversity (C_{22}) | (0.77, 1.07, 1.7) | (1, 1, 1) | (1.17, 1.49, 2.05) |

Informatization (C_{23}) | (0.76, 1.02, 1.35) | (1.01, 1.35, 1.71) | (1, 1, 1) |

Sub-Criteria | Unit Transportation Cost (C_{31}) | Occupancy Rate (C_{32}) | Carrying Capacity (C_{33}) |
---|---|---|---|

Unit transportation cost (C_{31}) | (1, 1, 1) | (0.73, 1.1, 2) | (0.64, 0.95, 1.85) |

Occupancy rate (C_{32}) | (0.5, 0.93, 1.4) | (1, 1, 1) | (0.9, 1.4, 1.9) |

Carrying capacity (C_{33}) | (0.58, 1.08, 1.58) | (0.55, 0.77, 1.33) | (1, 1, 1) |

Sub-Criteria | Emergency Management (C_{41}) | Safety Management (C_{42}) | Service Management (C_{43}) |
---|---|---|---|

Emergency management (C_{41}) | (1, 1, 1) | (0.6, 1, 1.8) | (1.03, 1.3, 1.85) |

Safety management (C_{42}) | (0.57, 1,1.7) | (1, 1, 1) | (1.06, 1.33, 1.67) |

Service management (C_{43}) | (1, 1.4, 1.8) | (0.68, 0.92, 1.18) | (1, 1, 1) |

Sub-Criteria | Carbon Emissions (C_{51}) | The Impact of Ships on Environment (C_{52}) | Energy Saving Measures (C_{53}) |
---|---|---|---|

Carbon emissions (C_{51}) | (1, 1, 1) | (0.58, 0.83, 1.52) | (0.66, 1.05, 1.45) |

The impact of ships on the environment (C_{52}) | (0.83, 1.33, 1.83) | (1, 1, 1) | (0.68, 1, 1.55) |

Energy saving measures (C_{53}) | (0.93, 1.24, 1.85) | (0.68, 1, 1.58) | (1, 1, 1) |

Criteria | Weight | Sub-Criteria | Local Weight | Global Weight |
---|---|---|---|---|

C1 | 0.1724 | C11 | 0.3295 | 0.0568 |

C12 | 0.3245 | 0.0560 | ||

C13 | 0.3460 | 0.0597 | ||

C2 | 0.1839 | C21 | 0.3288 | 0.0605 |

C22 | 0.3468 | 0.0638 | ||

C23 | 0.3244 | 0.0596 | ||

C3 | 0.2236 | C31 | 0.3346 | 0.0748 |

C32 | 0.3544 | 0.0792 | ||

C33 | 0.3111 | 0.0696 | ||

C4 | 0.2114 | C41 | 0.3321 | 0.0702 |

C42 | 0.3350 | 0.0708 | ||

C43 | 0.3329 | 0.0704 | ||

C5 | 0.2087 | C51 | 0.3073 | 0.0641 |

C52 | 0.3501 | 0.0731 | ||

C53 | 0.3426 | 0.0715 |

Criteria | Weight | Sub-Criteria | Local Weight | Global Weight |
---|---|---|---|---|

C1 | 0.2603 | C11 | 0.3905 | 0.1017 |

C12 | 0.346 | 0.0901 | ||

C13 | 0.2635 | 0.0686 | ||

C2 | 0.1886 | C21 | 0.3874 | 0.0731 |

C22 | 0.27 | 0.0509 | ||

C23 | 0.3426 | 0.0646 | ||

C3 | 0.1981 | C31 | 0.3805 | 0.0754 |

C32 | 0.3183 | 0.0631 | ||

C33 | 0.3012 | 0.0596 | ||

C4 | 0.1651 | C41 | 0.3818 | 0.0630 |

C42 | 0.3308 | 0.0546 | ||

C43 | 0.2874 | 0.0474 | ||

C5 | 0.1879 | C51 | 0.2985 | 0.0561 |

C52 | 0.2753 | 0.0517 | ||

C53 | 0.4262 | 0.0801 |

$\mathit{\beta}.$ | Criteria | Weight | Sub-Criteria | Local Weight | Global Weight |
---|---|---|---|---|---|

0.6 | C1 | 0.2075 | C11 | 0.3539 | 0.0735 |

C12 | 0.3331 | 0.0691 | |||

C13 | 0.3130 | 0.0650 | |||

C2 | 0.1858 | C21 | 0.3522 | 0.0654 | |

C22 | 0.3160 | 0.0587 | |||

C23 | 0.3317 | 0.0616 | |||

C3 | 0.2134 | C31 | 0.3530 | 0.0753 | |

C32 | 0.3400 | 0.0725 | |||

C33 | 0.3071 | 0.0655 | |||

C4 | 0.1929 | C41 | 0.3520 | 0.0679 | |

C42 | 0.3333 | 0.0643 | |||

C43 | 0.3147 | 0.0607 | |||

C5 | 0.2004 | C51 | 0.3038 | 0.0609 | |

C52 | 0.3202 | 0.0642 | |||

C53 | 0.3760 | 0.0754 |

Ai | C11 | C12 | C13 | C21 | C22 | C23 | C31 | C32 | C33 | C41 | C42 | C43 | C51 | C52 | C53 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 0.070 | 0.050 | 0.052 | 0.054 | 0.041 | 0.036 | 0.069 | 0.073 | 0.048 | 0.049 | 0.052 | 0.049 | 0.008 | 0.022 | 0.059 |

A2 | 0.072 | 0.060 | 0.045 | 0.046 | 0.046 | 0.028 | 0.072 | 0.068 | 0.028 | 0.037 | 0.046 | 0.038 | 0.008 | 0.027 | 0.061 |

A3 | 0.071 | 0.045 | 0.050 | 0.044 | 0.044 | 0.050 | 0.066 | 0.062 | 0.044 | 0.054 | 0.041 | 0.046 | 0.006 | 0.039 | 0.054 |

A4 | 0.074 | 0.055 | 0.051 | 0.047 | 0.047 | 0.035 | 0.075 | 0.071 | 0.051 | 0.042 | 0.033 | 0.045 | 0.014 | 0.028 | 0.053 |

Ai | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ |
---|---|---|---|---|

${d}_{i}^{+}$ | 0.0004 | 0.0016 | 0.0009 | 0.0010 |

${d}_{i}^{-}$ | 0.0017 | 0.0007 | 0.0013 | 0.0011 |

${S}_{i}$ | 0.8113 | 0.3026 | 0.5789 | 0.5023 |

Ranking | 1 | 4 | 2 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Cheng, H.; Zheng, S.; Feng, J.
A Fuzzy Multi-Criteria Method for Sustainable Ferry Operator Selection: A Case Study. *Sustainability* **2022**, *14*, 6135.
https://doi.org/10.3390/su14106135

**AMA Style**

Cheng H, Zheng S, Feng J.
A Fuzzy Multi-Criteria Method for Sustainable Ferry Operator Selection: A Case Study. *Sustainability*. 2022; 14(10):6135.
https://doi.org/10.3390/su14106135

**Chicago/Turabian Style**

Cheng, Huibing, Shanshui Zheng, and Jianghong Feng.
2022. "A Fuzzy Multi-Criteria Method for Sustainable Ferry Operator Selection: A Case Study" *Sustainability* 14, no. 10: 6135.
https://doi.org/10.3390/su14106135