# A Hybrid Fuzzy MCDM Methodology for Optimal Structural System Selection Compatible with Sustainable Materials in Mass-Housing Projects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Literature

## 3. Materials and Methods

#### 3.1. Instruments and Techniques

#### 3.1.1. Sample of the Study

#### 3.1.2. Questionnaire and Validation

^{l}, a

^{m}, a

^{u}), is a number with the membership function of linear fractions A (Equation (2)), as shown in Figure 1.

^{l}, a

^{m}, and a

^{u}, respectively, represent the lowest, most likely (most promising), and highest values of a possible value that describes a fuzzy event. When a

^{l}= a

^{m}= a

^{u}, then A is a nonfuzzy number, according to the contract. References can be found for algebraic operations on two fuzzy triangular numbers [40]. It is easy to express the value of an object using fuzzy numbers to express the decision makers’ qualitative evaluations. As a result, the use of fuzzy numbers in decision-making methods has grown in popularity [42].

_{e}denotes the number of specialists chosen as the necessary option. The number of specialists at this phase is 12, and the acceptable validity is equal to 0.57, as determined by the minimum CVR and scoring specialists [44].

_{α}(according to Equation (4)) for the main criteria after collecting the results of the first part of the questionnaires.

_{j}

^{2}denotes variance of the j-th question scores, and σ

^{2}denotes variance of the total questions scores. The Cronbach’s alpha coefficient must be, at minimum, equal to 0.7 to be considered reliable [44].

#### 3.2. Problem-Solving Process

#### 3.2.1. Phase I: Preparing a Database

#### 3.2.2. Phase II: Screening of Factors Identified by Fuzzy Delphi Method (FDM)

#### 3.2.3. Phase III: Prioritization with Hybrid Fuzzy Method (Fuzzy SWARA–Fuzzy ARAS)

## 4. Findings

#### 4.1. Results of Phase I: Identification and Classification of Factors

_{2}), environmental indicator (C

_{3}), socio-cultural indicator (C

_{4})), and an additional indicator known as technical-executive indicator (C

_{1}). Since the present study was an attempt to apply the proposed support system to a real mass-housing project, the C

_{1}indicator was also taken into account in the study. The main indicators and subfactors associated with each indicator, as well as their codes and references, are listed in Table 7. The reliability and validity of the main indicators are presented in this table. Since R

_{α}and CVR values exceed the allowable limits considered for the main indicators, the designed questionnaire can be considered reliable and valid in terms of internal consistency of items.

#### 4.2. Results of Phase II: Factor Monitoring and Screening

_{1-11}, C

_{2-8}, C

_{3-7}, C

_{4-8}, C

_{3-11}, C

_{2-5}, C

_{2-2}, C

_{4-3}, C

_{3-9}, C

_{1-8}, C

_{1-5}, C

_{4-1}, C

_{1-2}, and C

_{3-4}were given higher comparative priority (respectively) in the process of sustainable material selection. In order to implement phase III, the extracted efficient factors were arranged in descending order and their type (benefit or cost) was specified.

#### 4.3. Results of Phase III: Prioritization of Factors and Selection of Alternatives

_{1-5}), life-cycle cost (C

_{2-5}), compatibility with sustainable certifications (C

_{3-9}), and human health and safety (C

_{4-1}) are ranked as first by C

_{1}, C

_{2}, C

_{3}, and C

_{4}indicators, respectively. The results also show that, adaptation with technical standards (C

_{1-11}), energy cost (C

_{2-8}), reusability (C

_{3-7}), and compatibility with identity (C

_{4-3}) are ranked as second by each of the mentioned indicators, respectively.

_{iα}, S

_{iβ}, and S

_{iγ}which represent the most pessimistic, probable, and optimistic values of the triangular fuzzy number, respectively. Finally, the obtained value was defuzzified using the COD method in order to obtain the relative degree of utility (Q

_{i}) of each alternative (structural system) with respect to each of the key factors. The final rank of the alternatives was determined as shown in Figure 6. The results could be used to evaluate different alternatives in terms of key factors affecting the sustainable material selection. For instance, in this case, the comparative priority of the alternatives was examined based on the most important key factors obtained from the previous phase (C

_{1-5}, C

_{2-5}, C

_{3-9}, and C

_{4-1}). The results showed that PRC, ICF, and LSF systems had respectively higher comparative priorities in terms of factor C

_{1-5}. However, when it came to factor C

_{2-5}, LSF, PRC, and ICF systems were more preferable, respectively. PRC, ICF, and 3DP systems were found to be among the top ranking priorities in terms of factor C

_{3-9}. LSF, ICF, and PRC systems were found to be most desirable (respectively) with respect to factor C

_{4-1}. Finally, the overall alternative ranking results showed that LSF, ICF, and PRC systems with utility degrees of 1.800, 1.614, and 1.536, could be identified as the most preferred systems in terms of meeting sustainable material selection goals in mass-housing projects.

## 5. Conclusions

_{1}), economic (C

_{2}), environmental (C

_{3}), socio-cultural (C

_{4})) in accordance with sustainable development aspects. The results of the fuzzy SWARA method were used to determine the key factors affecting the sustainable material selection. Based on the FDM implementation in phase II, the 14 identified factors were extracted as sufficient factors. These factors include: implementation (C

_{1-2}); operational flexibility (C

_{1-5}); resistance to weathering, humidity, water, and fire (C

_{1-8}); adaptation with technical standards (C

_{1-11}); updated technology (C

_{2-2}); life-cycle cost (C

_{2-5}); energy cost (C

_{2-8}); energy consumption (C

_{3-4}); reusability (C

_{3-7}); compatibility with sustainable certifications (C

_{3-9}); water savings (C

_{3-11}); human health and safety (C

_{4-1}); compatibility with identity (C

_{4-3}); and human satisfaction (C

_{4-8}). The results showed that operational flexibility, life-cycle cost, compatibility with sustainable certifications, and human health and safety were the most important four factors affecting to sustainable material selection in the related C

_{1}, C

_{2}, C

_{3}, and C

_{4}indicators. The results also showed that adaptation with technical standards, energy cost, reusability, and compatibility with identity were ranked as second by each of the mentioned indicators, respectively. The results of phase II based on the fuzzy SWARA method implementation were used to determine the key factors affecting the sustainable material selection. The results of this phase indicated that the factors associated with each main indicator can be comparatively prioritized as follows: indicator C

_{1}(C

_{1-4}> C

_{1-2}> C

_{1-5}), indicator C

_{2}(C

_{2-5}> C

_{2-3}> C

_{2-1}), indicator C

_{3}(C

_{3-8}> C

_{3-4}> C

_{3-6}), and indicator C

_{4}(C

_{4-4}> C

_{4-10}> C

_{4-5}). Finally, the results of the fuzzy ARAS method in phase III (during the process of ranking optimal structural systems based on 14 efficient key factors) indicated that the LSF, ICF, and PRC systems can be prioritized as the most preferred systems, respectively (in terms of fulfillment of sustainable development goals) in mass-housing projects. In other words, the aforementioned systems can play a more effective role (as compared to the identified subcriteria) in the implementation of the new building system in terms of sustainable material selection.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Triangular and fuzzy membership functions (adapted with permission from [41]).

**Figure 3.**Proposed options for building skeletons in mass construction projects, (

**a**) LSF, (

**b**) PRC, (

**c**) ICF, (

**d**) 3DP, and (

**e**) TRC.

**Figure 4.**Triangular fuzzy membership function (Reprinted with permission from [41]).

**Figure 6.**Degree of utility and ranking of the structural systems according to each of the key factors.

Authors (Ref.) | Methodologies | Goal |
---|---|---|

Rao and Davim, 2008 [18] | AHP and TOPSIS (Technique for order performance by similarity to ideal solution) | Material selection for a given engineering design |

Zhou et al., 2009 [16] | Artificial Neural Networks (ANN) and Genetic Algorithms (GA) | A decision support optimization system for sustainable material selection |

Chatterjee et al., 2009 [17] | VIKOR, ELECTRE | Selection of materials |

Onut et al., 2009 [11] | Fuzzy ANP (Analytic Network Process) and TOPSIS | Selection of the appropriate material handling equipment |

Tuzkaya et al., 2010 [13] | Fuzzy ANP and Fuzzy PROMETHEE | Selection of material handling equipment |

Akadiri et al., 2012 [6] | FEAHP | A decision-making model for building material selection |

Bakhoum and Brown, 2012 [3] | – | A sustainable scoring system for materials |

Rahman et al., 2012 [19] | TOPSIS | A decision support system to select the optimal roofing materials |

Liu et al., 2014 [20] | DANP and VIKOR | Material selection with target-based criteria |

Zhao et al., 2016 [21] | GRA | Commercially available materials selection in sustainable design |

Govindan et al., 2016 [14] | hybrid MCDM method | Proposed a model to select sustainable material |

Gul et al., 2018 [23] | presented a fuzzy logic-based PROMETHEE | Select the material for an automotive instrument panel |

Khoshnava et al., 2018 [24] | Hybrid MCDM method | Ranking the green building material criteria based on sustainability |

Kiani et al., 2018 [25] | VIKOR | Select the material for repair structural concrete |

Mousavi-Nasab and Sotoudeh-Anvari, 2018 [26] | COPRAS (Complex Proportional Assessment), VIKOR and TOPSIS | Suggestion a new MCDM-based model for sustainable material selection |

Mahmoudkelaye et al., 2019 [8] | ANP | Proposed a ranking model for sustainable material selection |

Chen et al., 2019 [27] | QFD (Quality Function Deployment) and ELECTRE | Sustainable building material selection |

Singh et al., 2020 [4] | Fuzzy AHP and M-TOPSIS | Choose the composite material based on mechanical and structural applications |

Rajeshkumar et al., 2020 [28] | Structural questionnaire survey | Material selection in high rise buildings |

Emovon and Oghenenyerovwho, 2020 [7] | A systematic review | Application of MCDM methods in material selection |

Mayhoub et al., 2021 [29] | AHP | Achieving the sustainable building façades |

Agrawal, 2021 [5] | SAW (Simple Additive Weighting), MOORA and TOPSIS | Sustainable material selection for additives manufacturing technologies |

Chen et al., 2021 [30] | QFD and TOPSIS | Sustainable building material selection |

Majer et al., 2022 [31] | WSM-weighted sum method | Selection of external walls based on user priority |

Sahlol et al., 2021 [2] | System dynamics and AHP | Sustainable building materials assessment |

Parameters | Component | Frequency | Frequency Percentage |
---|---|---|---|

Work position | Project Chief Supervising Engineer Site Manager Senior Project Manager | 3 7 2 | 25 58.33 16.67 |

Experience in the sustainable construction field | Between 5 and 10 years Between 10 and 20 years More than 20 years | 5 5 2 | 41.67 41.67 16.66 |

Field of work | Employer Contractor Consultant | 4 3 5 | 33.33 25.00 41.67 |

Skill and expertise related to sustainable construction | Number of workshops and training seminars related to materials Number of participations in sustainable construction projects Number of participations in mass-housing projects | At least 22 At least 19 At least 21 |

Structural Systems | Information |
---|---|

Light Steel Frame (LSF) | LSFs are used to construct buildings with a limited number of floors (usually up to 5 floors). LSF is composed of cold-rolled steel sheets to provide stability. The foundation thickness is very small in this system due to the small loads applied on the building; the system is highly resistance to earthquakes due to its light weight without need for traditional and heavy materials. Moreover, due to the uniform distribution of forces throughout the building, the system is recognized as a highly safe system. Environmental friendliness, flexibility, high durability, dimensional stability, and stiff sections are the main advantages of this system. |

Prefabricated Reinforced Concrete (PRC) | In this system, the concrete parts that are prefabricated according to the maps are transported from the factory to the construction site. Since concrete parts are prefabricated, there are no considerable dimensional and proportional limitations in the architectural design of this system. The system is characterized by its fast and easy implementation and, thus, short duration between investment and operation. In addition, short execution time and cost as well as high service life are among the advantages of this system. |

Insulating Concrete Framework (ICF) | This system consists of reinforced concrete as the load-bearing component and expanded polystyrene (EPS) panels as concrete formwork and thermal insulators. Earthquake resistance, acoustic and thermal insulation, low construction costs, lack of architectural form limitations, long durability and service life, easy operation of the building, integration into other systems, and fast execution, the possibility of implementation in different seasons of the year, formwork independence, allowing for embedding ducts and pipes in the walls, and easy transportation are among the advantages of this system. |

3D Sandwich Panels (3DP) | The 3DP is a suitable and effective system that can significantly simplify the building construction process. The system is characterized by lightness, strength, integrity, insulation, and fast and easy implementation, which make the system fully compliant with safety and other relevant standards. The advantages of this structural system include reduced weight, structural rigidity and limited displacement, decreased prime cost, reduced spaces occupied by walls, rapid implementation, and easy installation of electrical and mechanical ducts, etc. |

Tronco System (TRC) | TRC consists of simple frames with cold-rolled metal elements. This system is usually applicable in low-rise buildings. During operation and implementation of this system, the door frames, windows, and electrical and mechanical elements are installed within predetermined spaces. Since the empty spaces within pipes, walls, and ceiling are filled by EPS panels, this system is the best choice in terms of energy saving and light weight. Considering the advantages, this system resembles the LSF system but is constructed and implemented in a different and more expensive way. |

**Table 4.**Triangular fuzzy numbers based on 5-point Likert spectrum [41].

Significance (Verbal Phrase) | Triangular Fuzzy Number | Fuzzy Value |
---|---|---|

Very Low (VL) | $\tilde{1}$ | (0, 0, 0.25) |

Low (L) | $\tilde{2}$ | (0, 0.25, 0.5) |

Medium (M) | $\tilde{3}$ | (0.25, 0.5, 0.75) |

High (H) | $\tilde{4}$ | (0.5, 0.75, 1) |

Very High (VH) | $\tilde{5}$ | (0.75, 1, 1) |

Step | Description | Equations | |
---|---|---|---|

1 | Sort the descending criteria in order of importance | In this step, the efficient factors extracted from the previous phase are arranged in descending order based on the fuzzy Delphi method. | |

2 | Determining the relative importance of fuzzy (${\tilde{S}}_{j}$) factor j compared to the previous factor (j − 1) with more importance according to experts | In this step, the relative importance of each criterion compared to the previous criterion is determined using the verbal expressions in Table 3. | |

3 | Determination of fuzzy coefficient (${\tilde{k}}_{j}$) | ${\tilde{k}}_{j}=\{\begin{array}{ll}1& j=1\\ {\tilde{S}}_{j}+1& j>1\end{array}$ | (7) |

4 | Determining the initial fuzzy weight (${\tilde{q}}_{j}$) | ${\tilde{q}}_{j}=\{\begin{array}{ll}1& j=1\\ {\tilde{k}}_{j-1}/{\tilde{k}}_{j}& j>1\end{array}$ | (8) |

5 | Determining the relative fuzzy weight of evaluation criteria (${\tilde{w}}_{j}$) | ${\tilde{w}}_{j}={\tilde{q}}_{j}/{\displaystyle \sum _{k=1}^{n}{\tilde{q}}_{k}}$ | (9) |

6 | Defuzzification relative fuzzy weight of criterion j using region center method | ${w}_{j}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\tilde{w}}_{j}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left(\left({w}_{j\gamma}-{w}_{j\alpha}\right)+\left({w}_{j\beta}-{w}_{j\alpha}\right)+{w}_{j\alpha}\right)$ | (10) |

w_{jα}, w_{jβ} and w_{jγ} are the lower, middle, and upper bounds representing the relative fuzzy weights of the factors, respectively. |

Step | Description | Equations | |
---|---|---|---|

1 | Fuzzy decision matrix formation: criterion-option matrix | $\tilde{X}={\tilde{x}}_{ij}=\left[\begin{array}{cccc}{\tilde{x}}_{11}& \dots & {\tilde{x}}_{1j}& {\tilde{x}}_{0n}\\ \dots & \dots & \dots & \dots \\ {\tilde{x}}_{i1}& \dots & {\tilde{x}}_{ij}& {\tilde{x}}_{in}\\ {\tilde{x}}_{m1}& \dots & {\tilde{x}}_{mj}& {\tilde{x}}_{mn}\end{array}\right]i=0,1,\dots ,m;j=1,2,\dots ,n$ | (11) |

${\tilde{x}}_{ij}$ is a fuzzy number that represents the performance of the i-th option in the j-th criterion. m is the number of options, and n is the number of criteria. | |||

To form a fuzzy decision matrix, a row named the hypothetical ideal optimal value for the criteria (${\tilde{x}}_{0j}$ optimal value for the j-th criterion) is calculated as follows: | |||

$\begin{array}{l}{\tilde{x}}_{0j}=\underset{i}{Max}{\tilde{x}}_{ij}\text{}\mathit{if}\text{}\underset{i}{Max}{\tilde{x}}_{ij},\text{}\mathrm{is}\text{}\mathrm{prefreable}\\ {\tilde{x}}_{0j}=\underset{i}{Min}{\tilde{x}}_{ij}\text{}\mathit{if}\text{}\underset{i}{Min}{\tilde{x}}_{ij},\text{}\mathrm{is}\text{}\mathrm{prefreable}\end{array}$ | (12) | ||

Accordingly, positive criteria (such as profit: criteria whose increase improves the situation) with higher values and negative criteria (such as cost: criteria whose reduction is more economical) with lower values are preferred. | |||

2 | Normalization of the decision matrix | $\tilde{\overline{X}}=\left[\begin{array}{cccc}{\tilde{\overline{x}}}_{01}& \dots & {\tilde{\overline{x}}}_{0j}& {\tilde{\overline{x}}}_{0n}\\ \dots & \dots & \dots & \dots \\ {\tilde{\overline{x}}}_{i1}& \dots & {\tilde{\overline{x}}}_{ij}& {\tilde{\overline{x}}}_{in}\\ {\tilde{\overline{x}}}_{m1}& \dots & {\tilde{\overline{x}}}_{mj}& {\tilde{\overline{x}}}_{mn}\end{array}\right]i=0,1,\dots ,m;j=1,2,\dots ,n$ | (13) |

$\tilde{{\overline{x}}_{ij}}$ are normalized values of matrix $\tilde{\overline{X}}$ elements. | |||

At this stage, the positive and negative criteria are normalized separately according to the following equations: | |||

${\tilde{\overline{x}}}_{ij}={\tilde{x}}_{ij}/{\displaystyle \sum _{i=0}^{m}{\tilde{x}}_{ij}};{\tilde{x}}_{ij}=\frac{1}{{\tilde{x}}_{ij}^{*}},{\tilde{\overline{x}}}_{ij}={\tilde{x}}_{ij}/{\displaystyle \sum _{i=0}^{m}{\tilde{x}}_{ij}}$ | (14) | ||

3 | Formation of a normal balanced decision matrix | $\tilde{\widehat{X}}=\left[\begin{array}{cccc}{\tilde{\widehat{x}}}_{01}& \dots & {\tilde{\widehat{x}}}_{0j}& {\tilde{\widehat{x}}}_{0n}\\ \dots & \dots & \dots & \dots \\ {\tilde{\widehat{x}}}_{i1}& \dots & {\tilde{\widehat{x}}}_{ij}& {\tilde{\widehat{x}}}_{in}\\ {\tilde{\widehat{x}}}_{m1}& \dots & {\tilde{\widehat{x}}}_{mj}& {\tilde{\widehat{x}}}_{mn}\end{array}\right]i=0,1,\dots ,m;j=1,2,\dots ,n$ | (15) |

${\tilde{\widehat{x}}}_{ij}={w}_{j}{\tilde{\overline{x}}}_{ij}i=0,1,\dots ,m;j=1,2,\dots ,n$ | (16) | ||

It is sufficient to use the final weights obtained using methods such as Shannon entropy or FAHP to assign the initial weight to the criteria in the decision matrix. The weight obtained from the SWARA method is used as the initial weight of the criteria in this step. | |||

4 | Determining the ${\tilde{S}}_{i}$ value of the optimization function of the i-th option (the degree of utility of each option) | ${\tilde{S}}_{i}={\displaystyle \sum _{j=1}^{n}{\tilde{\widehat{x}}}_{ij}}i=0,1,\dots ,m;j=1,2,\dots ,n$ | (17) |

The higher the value ${\tilde{S}}_{i}$, the better the option. | |||

5 | Defuzzification of optimization function with region center method (as the simplest method) | ${S}_{i}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\tilde{S}}_{i}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left(\left({S}_{i\gamma}-{S}_{i\alpha}\right)+\left({S}_{i\beta}-{S}_{i\alpha}\right)+{S}_{i\alpha}\right)$ | (18) |

S_{iα}, S_{iβ}, and S_{iγ} represent the most pessimistic, probable, and optimistic values of the TFNs. | |||

6 | Prioritization of options by calculating the degree of desirability | ${Q}_{i}={S}_{i}/{S}_{0}i=0,1,\dots ,m$ | (19) |

Q_{i} is the degree of usefulness or relative degree of desirability of each option, and S_{0} is the most desirable option.The higher the degree of desirability for an option, the better it will be prioritized and ranked. |

**Table 7.**The effective main indicators and subfactors in the sustainable material selection process.

Indicators (Code) (CVR, R _{α}) | Factors (Code) | References |
---|---|---|

Technical-executive (C_{1}) (0.64, 0.84) | Manufacturability (C_{1-1}) | [6,16,18,20,27,31] |

Implementation (C_{1-2}) | [1,18,19,31] | |

Repairability and maintainability (C_{1-3}) | [20,29] | |

Easiness and speed in usability (C_{1-4}) | [21,30] | |

Operational flexibility (C_{1-5}) | [3,6,14,17,21,26,30] | |

Spatial scale (C_{1-6}) | [11,17,22,30,31] | |

Demolition (C_{1-7}) | [12,17,20] | |

Resistance to weathering, humidity, water, and fire (C_{1-8}) | [2,15,19,27] | |

Hardness and weight savings (C_{1-9}) | [1,4,8,14,22,29] | |

Compatibility with other material (C_{1-10}) | [4,13,18,24,28] | |

Adaptation with technical standards (C_{1-11}) | [11,23,28] | |

Resistance to erosion and corrosion (C_{1-12}) | [2,9,10,19,20,25,29] | |

Durability (C_{1-13}) | [1,9,19,28] | |

Expert labor (C_{1-14}) | [2,8,9,10,11,25,26,31] | |

Economics (C_{2})(0.51, 0.89) | Material cost (C_{2-1}) | [1,6,23,24,30,31] |

Updated technology (C_{2-2}) | [1,15,24,28,30] | |

Fabrication cost (C_{2-3}) | [3,5,8,16,17,23] | |

Transportation cost (C_{2-4}) | [3,5,8,12,17,19] | |

Life-cycle cost (C_{2-5}) | [7,13,18,24,29] | |

Competitiveness cost (C_{2-6}) | [1,5,8,19,22,23,31] | |

Repair and maintenance cost (C_{2-7}) | [1,3,14,15,23,24,25] | |

Energy cost (C_{2-8}) | [1,14,15,24,25,30] | |

Processing cost (C_{2-9}) | [1,12,13,14,21,23,28,29] | |

Recycle cost (C_{2-10}) | [5,10,12,23,25] | |

Environmental (C_{3})(0.62, 0.94) | Embodied energy (C_{3-1}) | [14,16,19,23,25] |

Acoustic resistance (C_{3-2}) | [12,15,17] | |

Source material extraction (C_{3-3}) | [4,6,20,24,30] | |

Energy consumption (C_{3-4}) | [2,3,8,26,31] | |

Polluting (C_{3-5}) | [10,16,17,24,25] | |

Environmental impacts (C_{3-6}) | [12,13,24,25,26,31] | |

Reusability (C_{3-7}) | [9,19,24,29] | |

Renewability (C_{3-8}) | [5,11,19,24,30] | |

Compatibility with sustainable certifications (C_{3-9}) | [9,23,27] | |

Disposal (C_{3-10}) | [1,15,25,30] | |

Water savings (C_{3-11}) | [8,18,27,28,29] | |

Climate change (C_{3-12}) | [14,15,22,29] | |

Socio-cultural (C_{4})(0.66, 0.88) | Human health and safety (C_{4-1}) | [11,14,17,19,20] |

Compatibility with ecology (C_{4-2)} | [3,13,22,29,31] | |

Compatibility with identity (C_{4-3}) | [5,10,22,28] | |

Flexibility about future plans (C_{4-4}) | [9,14,22,31] | |

Use of local material (C_{4-5}) | [1,11,16,17,24,28,31] | |

Productivity (C_{4-6}) | [1,7,17,19,21] | |

Convenience (C_{4-7}) | [16,21,23,24,27,31] | |

Human satisfaction (C_{4-8}) | [1,2,5,13,14,26,28,30] | |

Aesthetic appeal (C_{4-9}) | [1,12,17,23] |

**Table 8.**Fuzzy mean, deterministic number, class, and type of factors affecting the sustainable material selection.

Indicators | Key Factors | Fuzzy Average Comments | Defuzzification | Crisp Number | Type of Factors | Invoice Type (Cost/Benefit) | Descending Rank | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\overline{\mathit{a}}}^{\mathit{l}}$ | ${\overline{\mathit{a}}}^{\mathit{m}}$ | ${\overline{\mathit{a}}}^{\mathit{u}}$ | ${\mathit{x}}_{\mathbf{max}}^{\mathit{l}}$ | ${\mathit{x}}_{\mathbf{max}}^{\mathit{m}}$ | ${\mathit{x}}_{\mathbf{max}}^{\mathit{u}}$ | Group | General | |||||

C_{1} | C_{1}_{-1} | 0.229 | 0.417 | 0.667 | 0.438 | 0.427 | 0.438 | 0.438 | Inefficient | – | 9 | 28 |

C_{1-2} | 0.271 | 0.521 | 0.771 | 0.521 | 0.521 | 0.516 | 0.521 | Efficient | Benefit | 4 | 13 | |

C_{1-3} | 0.271 | 0.396 | 0.583 | 0.417 | 0.406 | 0.391 | 0.417 | Inefficient | – | 12 | 33 | |

C_{1-4} | 0.146 | 0.229 | 0.479 | 0.285 | 0.257 | 0.297 | 0.297 | Inefficient | – | 14 | 45 | |

C_{1-5} | 0.292 | 0.542 | 0.792 | 0.542 | 0.542 | 0.531 | 0.542 | Efficient | Cost | 3 | 11 | |

C_{1-6} | 0.104 | 0.292 | 0.542 | 0.313 | 0.302 | 0.344 | 0.344 | Inefficient | – | 13 | 43 | |

C_{1-7} | 0.229 | 0.458 | 0.667 | 0.451 | 0.455 | 0.448 | 0.455 | Inefficient | – | 7 | 25 | |

C_{1-8} | 0.375 | 0.563 | 0.729 | 0.556 | 0.559 | 0.505 | 0.559 | Efficient | Cost | 2 | 10 | |

C_{1-9} | 0.229 | 0.417 | 0.625 | 0.424 | 0.420 | 0.417 | 0.424 | Inefficient | – | 11 | 32 | |

C_{1-10} | 0.292 | 0.500 | 0.688 | 0.493 | 0.497 | 0.469 | 0.497 | Inefficient | – | 5 | 15 | |

C_{1-11} | 0.417 | 0.667 | 0.854 | 0.646 | 0.656 | 0.594 | 0.656 | Efficient | Benefit | 1 | 1 | |

C_{1-12} | 0.208 | 0.438 | 0.688 | 0.444 | 0.441 | 0.453 | 0.453 | Inefficient | – | 8 | 27 | |

C_{1-13} | 0.250 | 0.500 | 0.708 | 0.486 | 0.493 | 0.479 | 0.493 | Inefficient | – | 6 | 16 | |

C_{1-14} | 0.292 | 0.417 | 0.583 | 0.431 | 0.424 | 0.396 | 0.431 | Inefficient | – | 10 | 29 | |

C_{2} | C_{2-1} | 0.208 | 0.354 | 0.563 | 0.375 | 0.365 | 0.370 | 0.375 | Inefficient | – | 13 | 42 |

C_{2-2} | 0.375 | 0.583 | 0.792 | 0.583 | 0.583 | 0.542 | 0.583 | Efficient | Cost | 3 | 7 | |

C_{2-3} | 0.146 | 0.354 | 0.604 | 0.368 | 0.361 | 0.391 | 0.391 | Inefficient | – | 12 | 39 | |

C_{2-4} | 0.250 | 0.375 | 0.604 | 0.410 | 0.392 | 0.396 | 0.410 | Inefficient | – | 10 | 36 | |

C_{2-5} | 0.354 | 0.604 | 0.792 | 0.583 | 0.594 | 0.547 | 0.594 | Efficient | Benefit | 2 | 6 | |

C_{2-6} | 0.208 | 0.375 | 0.604 | 0.396 | 0.385 | 0.396 | 0.396 | Inefficient | – | 11 | 38 | |

C_{2-7} | 0.271 | 0.479 | 0.646 | 0.465 | 0.472 | 0.443 | 0.472 | Inefficient | – | 6 | 20 | |

C_{2-8} | 0.417 | 0.646 | 0.833 | 0.632 | 0.639 | 0.578 | 0.639 | Efficient | Cost | 1 | 2 | |

C_{2-9} | 0.229 | 0.458 | 0.667 | 0.451 | 0.455 | 0.448 | 0.455 | Inefficient | – | 7 | 25 | |

C_{2-10} | 0.292 | 0.479 | 0.688 | 0.486 | 0.483 | 0.464 | 0.486 | Inefficient | – | 5 | 17 | |

C_{3} | C_{3-1} | 0.229 | 0.458 | 0.708 | 0.465 | 0.462 | 0.469 | 0.469 | Inefficient | – | 5 | 21 |

C_{3-2} | 0.208 | 0.417 | 0.646 | 0.424 | 0.420 | 0.427 | 0.427 | Inefficient | – | 9 | 31 | |

C_{3-3} | 0.250 | 0.458 | 0.667 | 0.458 | 0.458 | 0.448 | 0.458 | Inefficient | – | 7 | 23 | |

C_{3-4} | 0.271 | 0.521 | 0.729 | 0.507 | 0.514 | 0.495 | 0.514 | Efficient | Benefit | 4 | 14 | |

C_{3-5} | 0.167 | 0.396 | 0.625 | 0.396 | 0.396 | 0.411 | 0.411 | Inefficient | – | 10 | 34 | |

C_{3-6} | 0.188 | 0.396 | 0.625 | 0.403 | 0.399 | 0.411 | 0.411 | Inefficient | – | 10 | 34 | |

C_{3-7} | 0.396 | 0.625 | 0.854 | 0.625 | 0.625 | 0.583 | 0.625 | Efficient | Benefit | 1 | 3 | |

C_{3-8} | 0.250 | 0.458 | 0.667 | 0.458 | 0.458 | 0.448 | 0.458 | Inefficient | – | 7 | 23 | |

C_{3-9} | 0.333 | 0.563 | 0.813 | 0.569 | 0.566 | 0.547 | 0.569 | Efficient | Cost | 3 | 9 | |

C_{3-10} | 0.292 | 0.458 | 0.646 | 0.465 | 0.462 | 0.438 | 0.465 | Inefficient | – | 6 | 22 | |

C_{3-11} | 0.375 | 0.604 | 0.792 | 0.590 | 0.597 | 0.547 | 0.597 | Efficient | Cost | 2 | 5 | |

C_{3-12} | 0.208 | 0.375 | 0.563 | 0.382 | 0.378 | 0.375 | 0.382 | Inefficient | – | 12 | 41 | |

C_{4} | C_{4-1} | 0.313 | 0.542 | 0.750 | 0.535 | 0.538 | 0.510 | 0.538 | Efficient | Benefit | 3 | 12 |

C_{4-2} | 0.104 | 0.271 | 0.521 | 0.299 | 0.285 | 0.328 | 0.328 | Inefficient | – | 9 | 44 | |

C_{4-3} | 0.375 | 0.583 | 0.771 | 0.576 | 0.580 | 0.531 | 0.580 | Efficient | Cost | 2 | 8 | |

C_{4-4} | 0.271 | 0.417 | 0.604 | 0.431 | 0.424 | 0.406 | 0.431 | Inefficient | – | 6 | 30 | |

C_{4-5} | 0.167 | 0.354 | 0.604 | 0.375 | 0.365 | 0.391 | 0.391 | Inefficient | – | 8 | 39 | |

C_{4-6} | 0.188 | 0.375 | 0.625 | 0.396 | 0.385 | 0.406 | 0.406 | Inefficient | – | 7 | 37 | |

C_{4-7} | 0.292 | 0.458 | 0.688 | 0.479 | 0.469 | 0.458 | 0.479 | Inefficient | – | 4 | 18 | |

C_{4-8} | 0.354 | 0.604 | 0.833 | 0.597 | 0.601 | 0.568 | 0.601 | Efficient | Benefit | 1 | 4 | |

C_{4-9} | 0.292 | 0.479 | 0.667 | 0.479 | 0.479 | 0.453 | 0.479 | Inefficient | – | 4 | 18 |

Indicators | Key Factors | ${\tilde{\mathit{S}}}_{\mathit{j}}$ | ${\tilde{\mathit{k}}}_{\mathit{j}}$ | ${\tilde{\mathit{q}}}_{\mathit{j}}$ | ${\tilde{\mathit{w}}}_{\mathit{j}}$ | w_{j} | Rank | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

S_{jα} | S_{jβ} | S_{jγ} | k_{jα} | k_{jβ} | k_{jγ} | q_{jα} | q_{jβ} | q_{jγ} | w_{jα} | w_{jβ} | w_{jγ} | Explicit | Normalized | Initial | Final | ||

C_{1} | C_{1-11} | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0.188 | 0.202 | 0.220 | 0.0565 | 0.2494 | 2 | 11 |

C_{1-8} | 0.250 | 0.438 | 0.646 | 1.250 | 1.438 | 1.646 | 1.250 | 1.438 | 1.646 | 0.234 | 0.290 | 0.362 | 0.0542 | 0.2393 | 4 | 13 | |

C_{1-5} | 0.273 | 0.521 | 0.682 | 1.273 | 1.521 | 1.682 | 1.018 | 1.058 | 1.022 | 0.191 | 0.214 | 0.225 | 0.0599 | 0.2646 | 1 | 8 | |

C_{1-2} | 0.300 | 0.542 | 0.700 | 1.300 | 1.542 | 1.700 | 1.277 | 1.457 | 1.664 | 0.239 | 0.294 | 0.366 | 0.0559 | 0.2468 | 3 | 12 | |

C_{2} | C_{2-8} | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0.269 | 0.293 | 0.304 | 0.0863 | 0.3412 | 2 | 3 |

C_{2-5} | 0.271 | 0.479 | 0.688 | 1.271 | 1.479 | 1.688 | 1.271 | 1.479 | 1.688 | 0.342 | 0.434 | 0.51 | 0.0878 | 0.3473 | 1 | 1 | |

C_{2-2} | 0.295 | 0.375 | 0.727 | 1.295 | 1.375 | 1.727 | 1.019 | 0.930 | 1.024 | 0.275 | 0.273 | 0.31 | 0.0787 | 0.3115 | 3 | 6 | |

C_{3} | C_{3–7} | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0.185 | 0.202 | 0.22 | 0.0553 | 0.2605 | 2 | 9 |

C_{3-11} | 0.292 | 0.521 | 0.729 | 1.292 | 1.521 | 1.729 | 1.292 | 1.521 | 1.729 | 0.239 | 0.308 | 0.383 | 0.0546 | 0.2574 | 3 | 10 | |

C_{3-9} | 0.273 | 0.604 | 0.705 | 1.273 | 1.604 | 1.705 | 0.985 | 1.055 | 0.986 | 0.182 | 0.214 | 0.218 | 0.0591 | 0.2786 | 1 | 7 | |

C_{3-4} | 0.225 | 0.438 | 0.675 | 1.225 | 1.438 | 1.675 | 1.243 | 1.363 | 1.699 | 0.230 | 0.276 | 0.376 | 0.0432 | 0.2035 | 4 | 14 | |

C_{4} | C_{4-8} | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0.269 | 0.280 | 0.305 | 0.0812 | 0.3256 | 3 | 5 |

C_{4-3} | 0.292 | 0.521 | 0.729 | 1.292 | 1.521 | 1.729 | 1.292 | 1.521 | 1.729 | 0.348 | 0.425 | 0.528 | 0.0818 | 0.3278 | 2 | 4 | |

C_{4-1} | 0.273 | 0.604 | 0.705 | 1.273 | 1.604 | 1.705 | 0.985 | 1.055 | 0.986 | 0.265 | 0.295 | 0.301 | 0.0865 | 0.3466 | 1 | 2 |

**Table 10.**Summary of the fuzzy ARAS computations and decision-matrix development process for each key factor and structural system.

Alternatives | Final Key Factors | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Decision Matrix | ||||||||||||

C_{1-11} | C_{1-8} | C_{1-5} | C_{4-1} | |||||||||

A0 | 0.188 | 0.202 | 0.22 | 0.234 | 0.29 | 0.362 | 0.191 | 0.214 | 0.225 | 0.265 | 0.295 | 0.301 |

LSF | 0.208 | 0.229 | 0.250 | 0.389 | 0.417 | 0.445 | 0.611 | 0.667 | 0.723 | 0.455 | 0.514 | 0.573 |

PRC | 0.458 | 0.479 | 0.500 | 0.701 | 0.729 | 0.757 | 0.882 | 0.938 | 0.994 | 0.521 | 0.580 | 0.639 |

ICF | 0.542 | 0.563 | 0.584 | 0.743 | 0.771 | 0.799 | 0.798 | 0.854 | 0.910 | 0.424 | 0.483 | 0.542 |

3DP | 0.167 | 0.188 | 0.209 | 0.243 | 0.271 | 0.299 | 0.423 | 0.479 | 0.535 | 0.399 | 0.458 | 0.517 |

TRC | 0.083 | 0.104 | 0.125 | 0.264 | 0.292 | 0.320 | 0.465 | 0.521 | 0.577 | 0.399 | 0.458 | 0.517 |

Normalized decision matrix | ||||||||||||

A0 | 0.099 | 0.114 | 0.133 | 0.078 | 0.104 | 0.140 | 0.048 | 0.058 | 0.066 | 0.085 | 0.105 | 0.122 |

LSF | 0.110 | 0.129 | 0.152 | 0.130 | 0.150 | 0.172 | 0.154 | 0.181 | 0.214 | 0.147 | 0.184 | 0.232 |

PRC | 0.242 | 0.271 | 0.304 | 0.235 | 0.263 | 0.294 | 0.222 | 0.255 | 0.295 | 0.168 | 0.208 | 0.259 |

ICF | 0.287 | 0.319 | 0.354 | 0.249 | 0.278 | 0.310 | 0.201 | 0.232 | 0.270 | 0.137 | 0.173 | 0.22 |

3DP | 0.088 | 0.106 | 0.126 | 0.081 | 0.098 | 0.116 | 0.107 | 0.130 | 0.159 | 0.129 | 0.164 | 0.210 |

TRC | 0.044 | 0.059 | 0.076 | 0.088 | 0.105 | 0.124 | 0.117 | 0.142 | 0.171 | 0.129 | 0.164 | 0.210 |

W | 0.056 | 0.056 | 0.056 | 0.054 | 0.054 | 0.054 | 0.06 | 0.056 | 0.056 | 0.086 | 0.086 | 0.086 |

Weighted normalized decision matrix | ||||||||||||

A0 | 0.005 | 0.006 | 0.007 | 0.004 | 0.005 | 0.007 | 0.003 | 0.003 | 0.004 | 0.007 | 0.009 | 0.010 |

LSF | 0.006 | 0.007 | 0.008 | 0.007 | 0.008 | 0.009 | 0.009 | 0.011 | 0.013 | 0.012 | 0.016 | 0.020 |

PRC | 0.013 | 0.015 | 0.017 | 0.012 | 0.014 | 0.016 | 0.013 | 0.015 | 0.017 | 0.014 | 0.018 | 0.022 |

ICF | 0.016 | 0.018 | 0.020 | 0.013 | 0.015 | 0.017 | 0.012 | 0.014 | 0.016 | 0.012 | 0.015 | 0.019 |

3DP | 0.005 | 0.006 | 0.007 | 0.004 | 0.005 | 0.006 | 0.006 | 0.007 | 0.009 | 0.011 | 0.014 | 0.018 |

TRC | 0.002 | 0.003 | 0.004 | 0.005 | 0.005 | 0.006 | 0.007 | 0.008 | 0.010 | 0.011 | 0.014 | 0.018 |

**Table 11.**Results of structural system ranking based on utility function (degree of desirability) in fuzzy ARAS method.

Alternatives | ${\tilde{\mathit{S}}}_{\mathit{i}}$ | Crisp S_{i} | Q_{i} | Final Rank | ||
---|---|---|---|---|---|---|

S_{iα} | S_{iβ} | S_{iγ} | ||||

A0 | 0.08336 | 0.10544 | 0.13607 | 0.0527 | – | – |

LSF | 0.17465 | 0.20860 | 0.25072 | 0.0949 | 1.800 | 1 |

PRC | 0.14132 | 0.17261 | 0.21163 | 0.0810 | 1.536 | 3 |

ICF | 0.15343 | 0.18485 | 0.22378 | 0.0851 | 1.614 | 2 |

3DP | 0.10394 | 0.13119 | 0.16529 | 0.0642 | 1.217 | 5 |

TRC | 0.11049 | 0.13857 | 0.17370 | 0.0673 | 1.276 | 4 |

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**MDPI and ACS Style**

Aghazadeh, E.; Yildirim, H.; Kuruoglu, M.
A Hybrid Fuzzy MCDM Methodology for Optimal Structural System Selection Compatible with Sustainable Materials in Mass-Housing Projects. *Sustainability* **2022**, *14*, 13559.
https://doi.org/10.3390/su142013559

**AMA Style**

Aghazadeh E, Yildirim H, Kuruoglu M.
A Hybrid Fuzzy MCDM Methodology for Optimal Structural System Selection Compatible with Sustainable Materials in Mass-Housing Projects. *Sustainability*. 2022; 14(20):13559.
https://doi.org/10.3390/su142013559

**Chicago/Turabian Style**

Aghazadeh, Ebrahim, Hasan Yildirim, and Murat Kuruoglu.
2022. "A Hybrid Fuzzy MCDM Methodology for Optimal Structural System Selection Compatible with Sustainable Materials in Mass-Housing Projects" *Sustainability* 14, no. 20: 13559.
https://doi.org/10.3390/su142013559