# A Comparative Study of Multiple-Criteria Decision-Making Methods under Stochastic Inputs

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Review of Multiple-Criteria Decision-Making (MCDM) Methods

#### 2.2. Review of the Stochastic Expansion of Deterministic MCDM

## 3. Methodology

#### 3.1. An Overview of Selected MCDM Methods

#### 3.1.1. Weighted Sum Method (WSM) and Weighted Product Method (WPM)

#### 3.1.2. TOPSIS

#### 3.1.3. AHP

#### 3.1.4. Elimination Et Choix Traduisant la Realité (ELECTRE)

#### 3.1.5. Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE)

#### 3.2. Stochastic Expansion of Deterministic MCDM

_{i}probability that option X

_{i}will rank first. Figure 4. Stochastic expansion algorithm of deterministic MCDM methods illustrates the sequence of steps followed.

## 4. Case Study

#### Decision Criteria and Alternatives

- The compliance/maximum displacement of the rotor is considered to be a negative variable, and it represents the maximum displacement likely to be expected at the hub of the rotor that is affected by the support structure. It is treated in a different way for the floating and fixed structures; however, it does affect the rotor similarly for both structures.
- Dynamic performance is a positive variable, and it defines qualitatively the performance of a support structure in combination with the environmental effects and the operating loads. It is treated in a different way for the floating and fixed structures; the former has to combine the coupled effect of waves and turbine loads.
- Design redundancy is a positive variable, and it defines the capability to redistribute the load when a local failure is encountered.
- The cost of maintenance is a negative variable, and it reflects the qualitative assessment of the possible maintenance costs when, for example, any necessary equipment is involved or weather issues occur.
- The cost of installation is a negative variable, and it represents the qualitative assessment of the possible installation costs along with procedures, such as piling, etc.
- Environmental impact is a negative variable regarding the installation, operation and decommissioning impact of the foundation. Impacts on the natural environment can be considered as noise, visual, shadowing effects, disruption of the fish population’s routes, etc.
- Carbon footprint is a negative variable that takes into account the CO
_{2}emissions that were produced during all of the procedures needed for the support structure, such as the fabrication and installation processes. - Certification is a positive variable and reflects the confidence level against a range of engineering uncertainties. This covers a number of cases from existing installations related to the current application, to different applications or no applications at all.
- The likely cost is a negative variable. It represents the relative qualitative assessment of each of the concept’s costs, which, to some extent, could be quantified through the Net Present Value (NPV).
- Depth compatibility is a positive variable and represents the confidence levels when deploying a concept, which considers current installations for any applications with respect to a reference depth.

## 5. Results and Discussion

#### 5.1. Deterministic Results

- WSM: This has been the simplest method applied, and the result for the optimal solution is Alternative A3, the monopile design, followed by A1 (jacket) as the second option.
- WPM: WPM generates a matrix with pairwise comparison performance, as shown in Table 6. Hence, in this case, A1 (jacket) is superior to all of the other alternatives, because the ratio is higher than one in all cases. Following this, the monopile stands as the second best option.
- TOPSIS: According to this method, again, the jacket (A1) design achieves the highest score followed by the monopile concept.
- AHP: This method ranks the monopile (A3) design highest, followed by the jacket. The final ranking seems to be closer to the rest of the methods, and this can be explained due to the similarity of this method to the WSM.
- PROMETHEE I: Two different types of criteria were employed for the PROMETHEE I method. First, the Type I preference function was applied, and the monopile (A3) was found to be the best alternative in this case. Second, the results from the Type V preference function indicate that the jacket design achieves the highest score (A1).
- ELECTRE I: As a result, this method generates two matrices, which cumulatively qualify the tripod (A2) as the best option followed by the monopile and jacket.

#### 5.2. Stochastic Results

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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$n$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

$RI$ | 0 | 0.58 | 0.9 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 | 1.49 |

ID | Decision Criterion |
---|---|

A | Compliance/Max Displacement of Rotor |

B | Dynamic Performance |

C | Design Redundancy |

D | Cost of Maintenance |

E | Cost of Installation |

F | Environmental Impact |

G | Carbon Footprint |

H | Certification |

I | Likely Cost |

J | Depth Compatibility |

ID | Decision Alternative |
---|---|

A1 | Jacket |

A2 | Tripod |

A3 | Monopile |

A4 | Suction Bucket |

A5 | Jack-up |

A6 | Spar |

A7 | Barge |

A8 | TLP |

A9 | Semi-Submersible |

A10 | Tri-floater |

Alternatives/Criteria | Compliance/Max Displacement of Rotor | Dynamic Performance | Design Redundancy | Cost of Maintenance | Cost of Installation | Environmental Impact | Carbon Footprint | Certification | Likely Cost | Depth Compatibility |
---|---|---|---|---|---|---|---|---|---|---|

Jacket | 1.6 | 7.7 | 7.8 | 5.9 | 6.6 | 7.4 | 6.6 | 7.7 | 5.7 | 7.7 |

Tripod | 2 | 7.2 | 6.3 | 5.8 | 6.2 | 6.9 | 6.2 | 7.2 | 5.3 | 7 |

Monopile | 2.7 | 6.5 | 5.7 | 4.7 | 5.9 | 6.5 | 5.2 | 7.8 | 4.4 | 6.1 |

Suction Bucket | 3.3 | 6.1 | 5 | 5 | 5.3 | 6.5 | 5.1 | 5.9 | 4.5 | 5.3 |

Jack-up | 3.2 | 6.6 | 6.1 | 5.4 | 4.6 | 5 | 5.9 | 6.8 | 7 | 6.4 |

Spar | 5.8 | 5.9 | 5.1 | 4.8 | 4.8 | 3.6 | 5.3 | 5.4 | 6.5 | 3.9 |

Barge | 6.6 | 4.6 | 5.3 | 4.6 | 3.8 | 3.4 | 5.3 | 5.2 | 5.9 | 5.6 |

TLP | 4.2 | 6.6 | 4.4 | 5.7 | 5.6 | 5.2 | 6 | 5.5 | 7.3 | 5 |

Semi-Submersible | 5.6 | 5.8 | 5.3 | 4.6 | 4.2 | 3.7 | 5.9 | 5.6 | 6.7 | 5.9 |

Tri-floater | 5.5 | 5.7 | 4.9 | 5 | 3.9 | 3.5 | 5.7 | 4.3 | 6.4 | 5.7 |

Normalised weight values | 0.11 | 0.09 | 0.09 | 0.13 | 0.12 | 0.08 | 0.07 | 0.09 | 0.13 | 0.10 |

Alternatives | WSM | WPM | TOPSIS | AHP | PROMETHEE I Type I | PROMETHEE I Type V | ELECTRE I | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Score | Rank | Rank | Score | Rank | Score | Rank | Score | Rank | Score | Rank | Rank | |

A1 | −0.68968 | 2 | 1 | 0.6278 | 1 | −0.0191 | 2 | 0.0956 | 4 | 0.225556 | 1 | 3 |

A2 | −0.84274 | 3 | 3 | 0.6222 | 3 | −0.0218 | 3 | 0.1071 | 2 | 0.11 | 3 | 1 |

A3 | −0.67306 | 1 | 2 | 0.6237 | 2 | −0.019 | 1 | 0.3758 | 1 | 0.153333 | 2 | 2 |

A4 | −1.0571 | 5 | 5 | 0.5423 | 5 | −0.0258 | 5 | 0.0336 | 6 | 0.046667 | 5 | 4 |

A5 | −0.93839 | 4 | 4 | 0.5899 | 4 | −0.0233 | 4 | 0.1023 | 3 | 0.088889 | 4 | 5 |

A6 | −1.485 | 9 | 9 | 0.3662 | 10 | −0.0344 | 9 | −0.2051 | 9 | −0.17889 | 9 | 9 |

A7 | −1.27484 | 7 | 7 | 0.4108 | 8 | −0.0309 | 7 | 0.059 | 5 | −0.10111 | 8 | 7 |

A8 | −1.6779 | 10 | 10 | 0.3815 | 9 | −0.0372 | 10 | −0.4568 | 10 | −0.18111 | 10 | 6 |

A9 | −1.21339 | 6 | 6 | 0.4347 | 6 | −0.0293 | 6 | 0.0123 | 7 | −0.08 | 6 | 10 |

A10 | −1.33065 | 8 | 8 | 0.429 | 7 | −0.0312 | 8 | −0.1238 | 8 | −0.08333 | 7 | 8 |

A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 | |
---|---|---|---|---|---|---|---|---|---|---|

A1 | 1 | 1.0361 | 1.0188 | 1.0975 | 1.0750 | 1.1993 | 1.1391 | 1.2405 | 1.2405 | 1.1311 |

A2 | 0.9651 | 1 | 0.9833 | 1.0592 | 1.0375 | 1.1575 | 1.0994 | 1.1972 | 1.0916 | 1.1168 |

A3 | 0.9814 | 1.0169 | 1 | 1.0772 | 1.0551 | 1.1771 | 1.1180 | 1.2176 | 1.1101 | 1.1358 |

A4 | 0.9110 | 0.9440 | 0.9282 | 1 | 0.9794 | 1.0927 | 1.0378 | 1.1302 | 1.0305 | 1.0543 |

A5 | 0.9301 | 0.9638 | 0.9477 | 1.0209 | 1 | 1.1156 | 1.0596 | 1.1539 | 1.0521 | 1.0764 |

A6 | 0.8337 | 0.8639 | 0.8495 | 0.9151 | 0.8963 | 1 | 0.9498 | 1.0343 | 0.9431 | 0.9648 |

A7 | 0.8778 | 0.9095 | 0.8944 | 0.9635 | 0.9437 | 1.0528 | 1 | 1.0890 | 0.9929 | 1.0158 |

A8 | 0.8060 | 0.8352 | 0.8212 | 0.8847 | 0.8665 | 0.9667 | 0.9182 | 1 | 0.9117 | 0.9328 |

A9 | 0.8840 | 0.9160 | 0.9007 | 0.9703 | 0.9504 | 1.0603 | 1.0070 | 1.0967 | 1 | 1.0230 |

A10 | 0.8641 | 0.8953 | 0.8804 | 0.9484 | 0.9289 | 1.0363 | 0.9843 | 1.0720 | 0.9774 | 1 |

Alternatives | WSM | WPM | TOPSIS | AHP | PROMETHEE TYPE V | PROMETHEE I TYPE I | ELECTRE I |
---|---|---|---|---|---|---|---|

A1 | 21.64% | 21.94% | 18.12% | 21.62% | 47.75% | 22.09% | 27.33% |

A2 | 14.39% | 14.34% | 14.40% | 14.40% | 17.26% | 15.27% | 15.80% |

A3 | 25.62% | 23.59% | 24.09% | 25.16% | 6.07% | 23.07% | 6.24% |

A4 | 11.30% | 12.19% | 12.84% | 11.43% | 2.67% | 9.81% | 3.55% |

A5 | 10.35% | 8.73% | 11.58% | 11.13% | 9.62% | 10.41% | 14.59% |

A6 | 3.37% | 4.29% | 3.73% | 3.31% | 2.75% | 3.80% | 4.96% |

A7 | 4.65% | 6.04% | 5.11% | 4.28% | 1.09% | 5.65% | 3.01% |

A8 | 1.24% | 1.64% | 2.14% | 1.60% | 8.92% | 1.13% | 12.84% |

A9 | 4.83% | 4.27% | 4.88% | 4.57% | 2.94% | 5.56% | 7.90% |

A10 | 2.60% | 2.97% | 3.11% | 2.50% | 0.93% | 3.20% | 3.79% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kolios, A.; Mytilinou, V.; Lozano-Minguez, E.; Salonitis, K. A Comparative Study of Multiple-Criteria Decision-Making Methods under Stochastic Inputs. *Energies* **2016**, *9*, 566.
https://doi.org/10.3390/en9070566

**AMA Style**

Kolios A, Mytilinou V, Lozano-Minguez E, Salonitis K. A Comparative Study of Multiple-Criteria Decision-Making Methods under Stochastic Inputs. *Energies*. 2016; 9(7):566.
https://doi.org/10.3390/en9070566

**Chicago/Turabian Style**

Kolios, Athanasios, Varvara Mytilinou, Estivaliz Lozano-Minguez, and Konstantinos Salonitis. 2016. "A Comparative Study of Multiple-Criteria Decision-Making Methods under Stochastic Inputs" *Energies* 9, no. 7: 566.
https://doi.org/10.3390/en9070566