# A Hybrid Fuzzy Group Multi-Criteria Assessment of Structural Solutions of the Symmetric Frame Alternatives

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) emissions are the two critical indicators of sustainability in the construction industry. The primary source of adverse environmental impact on the life cycle of buildings is energy consumption at the stage of long-term building use [13]. The building sector accounts for about 40% of global primary energy consumption. When the service-life regarding the structural safety or serviceability of a deteriorating building does not meet the original target, the options for life cycle maintenance strategies need to be changed. The selection typically depends on costs and execution time [14] being redefined and developed throughout the early design stages. The structure produced by using only environmentally friendly materials is not necessary for a sustainable building. It is required to optimize the selection of materials for greater sustainability [15]. However, adequately selected materials and technologies, suppliers and contractors significantly improve the performance of the building [16,17,18].

_{2}[22]. While different materials can only be down-cycled, steel can be recycled countless times keeping its properties and quality (multi-cycling) [23]. Using scrap, the production of steel through EAF (electric arc furnace) instead of through BOF (basic oxygen furnace) can reduce about 32.14% up to 40.32% of the CO

_{2}emissions per ton of steel. According to Junichiro [24], the energy consumption through EAF is about 10.2 GJ per steel ton whereas through BOF it is 32.9 GJ/t. These values are in the range presented by Flues et al. [25]. Also, recent data from World Steel Association [26] shows that to recycling 1 ton of steel spares more than 1.4 t of iron ore, 1.4 t of CO

_{2}emissions, 120 of limestone, 740 kg of coal, and two-thirds of the amount of energy spent in the steel production process.

_{2}released during the entire cycle of life. Concrete is predominantly utilized in buildings and infrastructure worldwide by using ordinary Portland cement (OPC) as a binder. In recent years, the annual world cement production has grown from 1.0 billion tons to approximately 1.7 billion tons, which is enough to produce 1 m

^{3}of concrete per person.

_{2}) emissions. Some estimates suggest that the amount of CO

_{2}emitted from the global output of OPC may be as high as 7% of the total global CO

_{2}emissions. Furthermore, the production of OPC involves severe collateral environmental impacts, such as environmental pollution caused by dust and the enormous energy consumption required from having a plasticity temperature over 1300 °C. For these reasons, the cement industry has been challenged in the past 10 years to effectively reduce and control CO

_{2}emissions effectively.

## 2. Multi-Criteria Decision-Making

#### 2.1. Available MCDM Methods for Problem-Solving

#### 2.2. A Hybrid MCDM Model for Problem Solution

#### 2.2.1. Fuzzy Number

_{1}, x

_{2}, …, x

_{n}, the fuzzy set A can be represented as

#### 2.2.2. Defuzzification

#### 2.3. The DHP (Delphi Hierarchy Process)

_{1}, C

_{2}, …, C

_{n}denote the set of elements, while a

_{ij}represents a quantified judgment on a pair of elements C

_{i}and C

_{j}. The relative importance of the two aspects is rated using a scale (Table 1). These scales yield an n × n matrix A as C

_{1}, C

_{2}, …, C

_{n}where a

_{ij}= 1 and a

_{ij}= 1/a

_{ij}, i, j = 1, 2, …, n. In matrix A, the problem becomes one of assigning to the n elements C

_{1}, C

_{2}, …, C

_{n}a set of numerical weights w

_{1}, w

_{2}, …, w

_{n}that reflect the recorded judgments.

#### 2.3.1. Fuzzy Group Criterion Weight is Determined as Follows:

#### 2.3.2. Additive Ratio Assessment Method (ARAS) with Fuzzy Criteria Values (ARAS-F)

_{i}and S

_{0}are the optimal criterion values, obtained from Equation (24).

#### 2.3.3. The fuzzy Multiplicative Utility Function

_{i}and U

_{0}are the optimal criterion values, obtained from Equation (27).

#### 2.3.4. Integrated Utility Function

_{Ai}are K

_{i}values obtained from Equations (25) and (27) respectively.

## 3. Description of the Problem

## 4. Problem Solution

_{1}, representing the beam cross-section of the IPE300 and columns cross-sections of the HEA160 profiles was ranked second. According to the MULT-F method, the alternative A

_{1}, representing the beam cross-section of the IPE300 and columns cross-sections of the HEA160 profiles, was best suited; while the alternative A

_{4}, representing the frame, was made from precast concrete beam and columns representing the beam cross-section of the IPE300 and columns cross-sections of the HEA160 profiles was ranked fourth. The alternative A

_{1}was the best suited according to the integrated assessment of alternative performances.

## 5. Results and Conclusions

_{1}—Costs, x

_{2}—Impact on the environment, x

_{3}—Installment time, x

_{4}—Weight (tons), x

_{5}—Consumption of steel, x

_{6}—Consumption of concrete. The criteria set listed from the most important in decreasing importance order. The problem solution using the ARAS-F method result shows that the best method is to use alternative A

_{4}and after that A

_{1}alternative. The worst alternative is A

_{5}and the second worst is A

_{3}. The difference among scores of the best and worst alternatives is 44%. The MULT-F method shows that the best alternative is A

_{1}alternative, the A

_{2}alternative ranks as the second best, while A

_{5}is the worst alternative, and A

_{4}is the second worst alternative. The alternatives rank is as follows: ${\mathrm{A}}_{4}\succ {\mathrm{A}}_{1}\succ {\mathrm{A}}_{2}\succ {\mathrm{A}}_{3}\succ {\mathrm{A}}_{5}.$ The MULT-F method is not sensitive to the criteria weights values, and the ratio of the best and worst scores is 3.25. The alternatives rank as follows ${\mathrm{A}}_{1}\succ {\mathrm{A}}_{2}\succ {\mathrm{A}}_{3}\succ {\mathrm{A}}_{4}\succ {\mathrm{A}}_{5}.$ We offer to rank alternatives according to integrated utility values of investigated alternatives. In this case, the best alternative is the first, and the worst one is the fifth alternative. The ratio of the best alternative score to the worst alternative score equals 89 percent. The final ranking of alternatives is as follows: ${\mathrm{A}}_{1}\succ {\mathrm{A}}_{2}\succ {\mathrm{A}}_{3}\succ {\mathrm{A}}_{4}\succ {\mathrm{A}}_{5}$.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Structure of the hybrid decision support system, based on the ARAS-F, fuzzy multiplicative utility method and Delphi Hierarchy Process.

Meaning | Diagonal Elements i = j | C_{i} and C_{j} Are Equally Important | C_{i} Is Weakly More Important Than C_{j} | C_{i} Is Strongly More Important Than C_{j} | C_{i} Is Demonstratively More Important Than C_{j} | C_{i} Is Absolutely More Important Than C_{j} | Compromise between Two Judgments | If Element C_{j} Dominates over Element C_{i} |
---|---|---|---|---|---|---|---|---|

a(i,j) | 1 | 1 | 3 | 5 | 7 | 9 | 2, 4, 6, 8 | a(i,j) = 1/a(j,i) |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

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

Abbreviation | Criterion Name |
---|---|

x_{1} | Costs, € |

x_{2} | Impact on the environment |

x_{3} | Installment time, hours |

x_{4} | Weight, tons |

x_{5} | Consumption of steel, tons |

x_{6} | Consumption of concrete, m^{3} |

**Table 4.**The opinion of experts regarding criteria importance according to Saaty’s nine-point scale.

The Opinion of Expert E_{1} | ||||||

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |

x_{1} | 1.00 | 3.00 | 5.00 | 6.00 | 7.00 | 8.00 |

x_{2} | 0.33 | 1.00 | 3.00 | 4.00 | 5.00 | 6.00 |

x_{3} | 0.20 | 0.33 | 1.00 | 2.00 | 3.00 | 4.00 |

x_{4} | 0.17 | 0.25 | 0.50 | 1.00 | 2.00 | 3.00 |

x_{5} | 0.14 | 0.20 | 0.33 | 0.50 | 1.00 | 2.00 |

x_{6} | 0.13 | 0.17 | 0.25 | 0.33 | 0.50 | 1.00 |

The Opinion of Expert E_{2} | ||||||

⁞ | ||||||

The Opinion of Expert E_{5} | ||||||

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |

x_{1} | 1.00 | 2.00 | 5.00 | 6.00 | 7.00 | 8.00 |

x_{2} | 0.50 | 1.00 | 2.00 | 4.00 | 5.00 | 7.00 |

x_{3} | 0.20 | 0.50 | 1.00 | 4.00 | 5.00 | 3.00 |

x_{4} | 0.17 | 0.25 | 0.25 | 1.00 | 2.00 | 3.00 |

x_{5} | 0.14 | 0.20 | 0.20 | 0.50 | 1.00 | 2.00 |

x_{6} | 0.13 | 0.14 | 0.33 | 0.33 | 0.50 | 1.00 |

Criteria | Criteria Weights Determined by Expert | w_{j} | ||||||
---|---|---|---|---|---|---|---|---|

E_{1} | E_{2} | E_{3} | E_{4} | E_{5} | α | γ | β | |

x_{1} | 0.464 | 0.406 | 0.412 | 0.422 | 0.437 | 0.406 | 0.428 | 0.464 |

x_{2} | 0.249 | 0.274 | 0.276 | 0.276 | 0.257 | 0.249 | 0.266 | 0.276 |

x_{3} | 0.121 | 0.159 | 0.156 | 0.142 | 0.152 | 0.121 | 0.145 | 0.159 |

x_{4} | 0.079 | 0.090 | 0.086 | 0.080 | 0.071 | 0.071 | 0.081 | 0.090 |

x_{5} | 0.052 | 0.042 | 0.041 | 0.046 | 0.048 | 0.041 | 0.046 | 0.052 |

x_{6} | 0.035 | 0.030 | 0.029 | 0.034 | 0.036 | 0.029 | 0.033 | 0.036 |

CI | 0.050 | 0.126 | 0.138 | 0.045 | 0.055 | |||

RI | 1.24 | |||||||

CR | 0.041 | 0.101 | 0.111 | 0.037 | 0.045 |

A_{i} | x_{j} | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |||||||||||||

α | γ | β | α | γ | β | α | γ | β | α | γ | β | α | γ | β | α | γ | β | |

w_{j} | 0.406 | 0.428 | 0.464 | 0.249 | 0.266 | 0.276 | 0.121 | 0.145 | 0.159 | 0.071 | 0.081 | 0.090 | 0.041 | 0.046 | 0.052 | 0.029 | 0.033 | 0.036 |

Opt. | min | min | min | min | min | min | min | min | min | min | min | min | min | min | min | min | min | min |

A_{1} | 33,300 | 33,300 | 33,300 | 74,195 | 81,614.7 | 89,776 | 227 | 272 | 326 | 27.2 | 27.2 | 27.2 | 27.2 | 27.2 | 27.2 | 0.1 | 0.1 | 0.1 |

A_{2} | 34,100 | 34,100 | 34,100 | 75,832 | 83,414.7 | 91,756 | 232 | 278 | 334 | 27.8 | 27.8 | 27.8 | 27.8 | 27.8 | 27.8 | 0.1 | 0.1 | 0.1 |

A_{3} | 38,000 | 38,000 | 38,000 | 84,559 | 93,014.7 | 102,316 | 258 | 310 | 372 | 31 | 31 | 31 | 31 | 31 | 31 | 0.1 | 0.1 | 0.1 |

A_{4} | 29,900 | 29,900 | 29,900 | 34,405 | 37,845 | 41,630 | 328 | 394 | 473 | 141 | 141 | 141 | 6 | 6 | 6 | 135 | 135 | 135 |

A_{5} | 37,200 | 37,200 | 37,200 | 71,107 | 78,218.1 | 86,040 | 303 | 363 | 436 | 66.3 | 66.3 | 66.3 | 24 | 24 | 24 | 42.3 | 42.3 | 42.3 |

A_{0} | 24,917 | 24,917 | 24,917 | 28,670 | 31,538 | 34,691 | 189 | 227 | 272 | 23 | 23 | 23 | 5 | 5 | 5 | 0.08 | 0.08 | 0.08 |

A_{i} | x_{j} | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |||||||||||||

α | γ | β | α | γ | β | α | γ | β | α | γ | β | α | γ | β | α | γ | β | |

Opt | max | max | max | max | max | max | max | max | max | max | max | max | max | max | max | max | max | max |

A_{1} | 0.066 | 0.069 | 0.075 | 0.024 | 0.031 | 0.039 | 0.015 | 0.026 | 0.042 | 0.015 | 0.017 | 0.019 | 0.003 | 0.003 | 0.004 | 0.007 | 0.008 | 0.009 |

A_{2} | 0.064 | 0.067 | 0.073 | 0.023 | 0.030 | 0.038 | 0.015 | 0.026 | 0.041 | 0.015 | 0.017 | 0.019 | 0.003 | 0.003 | 0.004 | 0.007 | 0.008 | 0.009 |

A_{3} | 0.057 | 0.060 | 0.066 | 0.021 | 0.027 | 0.034 | 0.013 | 0.023 | 0.037 | 0.013 | 0.015 | 0.017 | 0.003 | 0.003 | 0.003 | 0.007 | 0.008 | 0.009 |

A_{4} | 0.073 | 0.077 | 0.083 | 0.051 | 0.066 | 0.083 | 0.011 | 0.018 | 0.029 | 0.003 | 0.003 | 0.004 | 0.013 | 0.015 | 0.017 | 0.000 | 0.000 | 0.000 |

A_{5} | 0.059 | 0.062 | 0.067 | 0.025 | 0.032 | 0.040 | 0.011 | 0.020 | 0.031 | 0.006 | 0.007 | 0.008 | 0.003 | 0.004 | 0.004 | 0.000 | 0.000 | 0.000 |

A_{0} | 0.088 | 0.092 | 0.100 | 0.062 | 0.080 | 0.100 | 0.018 | 0.032 | 0.050 | 0.018 | 0.021 | 0.023 | 0.016 | 0.018 | 0.020 | 0.008 | 0.009 | 0.010 |

Solution Methods | ||||||||
---|---|---|---|---|---|---|---|---|

A_{i} | ARAS-F | MULT-F | Integrated | |||||

S | K | R | U | K | R | K | R | |

A_{1} | 0.157 | 0.616 | 2 | 0.0170 | 0.560 | 1 | 0.587 | 1 |

A_{2} | 0.154 | 0.602 | 3 | 0.0167 | 0.549 | 2 | 0.576 | 2 |

A_{3} | 0.139 | 0.543 | 4 | 0.0152 | 0.502 | 3 | 0.522 | 3 |

A_{4} | 0.182 | 0.714 | 1 | 0.0054 | 0.179 | 4 | 0.402 | 4 |

A_{5} | 0.127 | 0.496 | 5 | 0.0051 | 0.169 | 5 | 0.311 | 5 |

A_{0} | 0.255 | 1.000 | 0.0303 |

© 2019 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**

Turskis, Z.; Urbonas, K.; Daniūnas, A.
A Hybrid Fuzzy Group Multi-Criteria Assessment of Structural Solutions of the Symmetric Frame Alternatives. *Symmetry* **2019**, *11*, 261.
https://doi.org/10.3390/sym11020261

**AMA Style**

Turskis Z, Urbonas K, Daniūnas A.
A Hybrid Fuzzy Group Multi-Criteria Assessment of Structural Solutions of the Symmetric Frame Alternatives. *Symmetry*. 2019; 11(2):261.
https://doi.org/10.3390/sym11020261

**Chicago/Turabian Style**

Turskis, Zenonas, Kęstutis Urbonas, and Alfonsas Daniūnas.
2019. "A Hybrid Fuzzy Group Multi-Criteria Assessment of Structural Solutions of the Symmetric Frame Alternatives" *Symmetry* 11, no. 2: 261.
https://doi.org/10.3390/sym11020261