# A New Approach to the Viable Ranking of Zero-Carbon Construction Materials with Generalized Fuzzy Information

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## Abstract

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## 1. Introduction

#### Motivation and Contributions

- The generalized fuzzy structure (GFS) [10] was adopted in this study for the decision process, which can effectively represent uncertainty in three dimensions–such as the degree of truthfulness, the degree of falsity, and the degree of hesitation. Besides, the structure can flexibly allow experts to share their preference by increasing or shrinking the window size of preference articulation. It may also be noted that the orthopair structure allows the mitigation of subjective randomness during the decision process.
- Criteria importance–through the inter-criteria correlation (CRITIC) technique [11], which comes under the objective weight calculation category–was extended to the GFS for the methodical determination of criteria weights. As claimed by Kao [12], it is clear that the estimation of weights by using a method reduces biases and inaccuracies, which motivated the authors to propose a stepwise procedure for weight calculation. Besides, the claim from Kao [12] towards the variability in the preference distribution mimics the hesitation of the experts, which is also considered in the CRITIC approach and supports the rational calculation of the weights of the criteria.
- Furthermore, the popular complex proportional assessment (COPRAS) technique was extended to the GFS for ranking zero- and low-carbon materials, which could support the construction industry in expediting their sustainability goals.
- Finally, a real case example has been demonstrated to illustrate the usefulness of the integrated model and a comparative study, from both the theoretical and numerical perspectives, is presented to realize both the superiority and the limitations of the model.

- The GFS [10] is a generalized structure for preference elicitation that mitigates subjective randomness and provides a flexible window for sharing the degree of preference and the degree of non-preference. In the GFS, an adjustable factor (q) is considered and is used to expand or shrink the rating window, allowing experts to flexibly share their views.
- Moreover, the CRITIC technique [11] is an objective weight-calculation approach that not only allows the methodical estimation of weights, but also captures the interaction among criteria and the variability in the preference distribution, which models the hesitation of experts during preference articulation. In this way, it can be intuitively inferred that a criterion with a high interaction with other criteria and a higher variability in the distribution will have a high importance or weight. This indicates that the criterion contains potential information or semantics that are essential or useful for the decision process.
- COPRAS [13] is a popular and powerful ranking technique that effectively considers the nature of the criteria during the ordering of alternatives. Furthermore, the COPRAS method offers ranking from different angles and considers the complex proportionality of the opinions in its formulation of ranking alternatives [14].

## 2. Literature Review

#### 2.1. CRITIC Technique

#### 2.2. COPRAS Method

#### 2.3. Material Selection with Decision Approaches

#### 2.4. Research Insights

## 3. Research Methodology

#### 3.1. Preliminaries

**Definition**

**1 [10].**

**Definition**

**2 [106].**

- (i)
- If $\mathbb{S}\left({\phi}_{1}\right)>\mathbb{S}\left({\phi}_{2}\right),$ then ${\phi}_{1}>{\phi}_{2},$
- (ii)
- If $\mathbb{S}\left({\phi}_{1}\right)=\mathbb{S}\left({\phi}_{2}\right),$ then
- (iii)
- if $\hslash \left({\phi}_{1}\right)>\hslash \left({\phi}_{2}\right),$ then ${\phi}_{1}<{\phi}_{2},$
- (iv)
- if $\hslash \left({\phi}_{1}\right)=\hslash \left({\phi}_{2}\right),$ then ${\phi}_{1}={\phi}_{2}.$

**Definition**

**3 [107].**

#### 3.2. Q-ROF-CRITIC-COPRAS Framework

**Step 1:**Create a linguistic decision matrix (LDM).

**Step 2:**Estimate the weights of the DMs.

**Step 3:**Aggregate all the q-ROF-DMs.

**Step 4:**Use the CRITIC tool for the estimation of criteria weights.

**Step 5:**The sum of the cost- and benefit-type criteria ratings.

**Step 6:**Computation of the relative degree of alternatives.

**Step 7:**Prioritize the alternatives.

**Step 8:**Evaluate the utility degree.

## 4. Real Case Example

**Steps 1–2:**Table 1 depicts the significance of the DEs and the criteria in the form of LVs, which are then converted into q-ROFNs. Table 2 presents the DEs weight, based on Table 1 and Equation (3). Table 3 describes the importance of the DEs in evaluating the options and the assessments of the options, concerning each attribute.

**Step 3:**The LDM provided by four DEs have been combined by Equation (4) for each option, over diverse criteria of zero-carbon construction material selection into an A-q-ROF-DM $A={\left({\xi}_{ij}\right)}_{m\times n},$ and is depicted in Table 4.

**Step 4:**To estimate the criteria weights, the CRITIC tool was used on q-ROFSs. Using Equation (5) and Table 4, firstly we obtained the score-matrix $S={\left({\xi}_{ij}\right)}_{p\times q}.$ After, we computed the standard q-ROF-matrix $\tilde{S}={\left({\tilde{\chi}}_{ij}\right)}_{p\times q}$ by Equation (6). By using Equations (7)–(9), the SD, CRC, and quantity of information of each criterion were estimated. The weights of criteria were estimated by using Equation (10) and referred to in Table 5.

_{12}) with the weight value 0.0974, have come out to be the most significant criteria of zero- and low-carbon construction material selection. The financial risk (b

_{13}) with the weight value 0.0926, was the second most important criteria of zero- and low-carbon construction material selection. Safety (b

_{1}) was in third position (significance), with the value 0.0891. Water pollution (b

_{7}) was placed fourth, with the weight value 0.0843. Soil/land contamination (b

_{9}), with the significance value 0.0799, was the fifth most important criteria of zero- and low-carbon construction material selection. This ranking follows for the other criteria as well, which are also considered crucial criteria of zero- and low-carbon construction material selection.

**Steps 5–8:**In the process of the assessment of the criteria of zero-carbon construction material selection, all risk factors are the maximum type. Using Equations (11)–(16), the values of ${\tau}_{i},$${\iota}_{i},$ ${\gamma}_{i}$ and ${\delta}_{i}$ of ${O}_{i}\left(i=1\left(1\right)5\right)$ were computed over the criteria ${b}_{j}\left(j=1\left(1\right)13\right),$ and specified in Table 6. As seen in Table 7, the ranking order of the material alternatives is ${O}_{1}\succ {O}_{3}\succ {O}_{5}\succ {O}_{4}\succ {O}_{2}$ and, thus, O

_{1}is the best material, based on the ratings of the different criteria for the zero- and low-carbon construction material selection problem.

#### 4.1. Comparative Discussion

#### 4.1.1. Q-ROF-TOPSIS Approach

**Steps 1–4:**The same as the aforementioned model.

**Step 5:**Assess the “q-ROF-ideal solution (q-ROF-IS)” and the “q-ROF-anti-ideal solution (q-ROF-AIS)”.

**Step 6:**Obtain the distances of the options from q-ROF-IS and q-ROF-AIS.

**Step 7:**Assess the closeness index (CI).

**Step 8:**Choose the maximum degree, $RC\left({O}_{k}\right),$ among the degrees $RC\left({O}_{i}\right).$ This validates that ${O}_{k}$ is the optimal choice.

_{1}.

#### 4.1.2. Q-ROF-WASPAS Model

**Steps 1–4:**Similar to the aforesaid model.

**Step 5:**For each option, we estimate the degrees of the “weighted-sum method (WSM)” ${C}_{i}^{\left(1\right)}$, as follows:

**Step 6:**For each option, we compute the degrees of the “weighted-product method (WPM)” ${C}_{i}^{\left(2\right)}$, as follows:

**Step 7:**For each option, we obtain the degree of WASPAS measure as

**Step 8:**Rank the alternatives according to the decreasing ratings (i.e., score values) of ${C}_{i}$.

**Steps 5–8:**By applying Equations (22)–(24), the WSM $\left({C}_{i}^{\left(1\right)}\right),$ the WPM $\left({C}_{i}^{\left(2\right)}\right)$, and the WASPAS $\left({C}_{i}\right)$ measures for each option were obtained and are depicted in Table 8. Therefore, the prioritization of the material was assessed as follows: ${O}_{1}\succ {O}_{5}\succ {O}_{3}\succ {O}_{4}\succ {O}_{2}$, with O

_{1}being the best option. On the other hand, the outcomes were slightly different between the developed and the extant models. Therefore, the q-ROF-CRITIC-COPRAS method is more robust and is steadier than the q-ROF-TOPSIS and q-ROF-WASPAS tools and, thus, has wider applicability.

- The q-ROFSs can reflect the DE’s hesitancy more objectively than other classical extensions of FS. Therefore, the use of the developed q-ROF-CRITIC-COPRAS approach gives a more flexible way to express the uncertainty when evaluating the criteria of zero-carbon construction material selection.
- The CRITIC method is employed to evaluate the objective weights of each criterion in the evaluation of the criteria of zero-carbon construction material selection, which makes the introduced q-ROF-CRITIC-COPRAS method a more reliable, efficient, and sensible tool.
- The proposed q-ROF-CRITIC-COPRAS method can process the information in a more useful and a more suitable way and from different perspectives, such as benefit-type and cost-type criteria.

#### 4.2. Sensitivity Investigation

_{1}, has the maximum rating when $\phi =0.0$ to 0.5, while the option, O

_{3}, has the maximum rating when $\phi =0.6$ to 1.0. Moreover, the option, O

_{2}, has the minimum rating when $\phi =0.0$ to 0.6, and the option, O

_{4}, has the minimum rating when $\phi =0.7$to 1.0. Therefore, the q-ROF-CRITIC-COPRAS approach has more stability for diverse parameter $\phi $ values. Furthermore, the criteria objective weights obtained by CRITIC were preserved to improve the sensitivity of the developed approach. In the aforesaid discussion, we observed that the utilization of different parameter ($\phi $) values would provide more stability in the developed method.

- q-ROFN was considered as the preferred style for this study It is not only flexible but also represents uncertainty from three degrees–membership, non-membership, and hesitancy. The factor, clearly controls the preference window by aiding experts in sharing their opinions flexibly.
- The criteria weights were methodically determined to properly model the competition and conflicts among the criteria. Unlike in the extant models, the interrelationships that the criteria implicitly incur were well captured by the proposed work.
- Furthermore, the hesitation of the experts during preference articulation was captured via the variability in the preference distribution. Specifically, if all experts provide the same preference for a criterion, the variability is zero, indicating that the experts have no considerable difference of opinion towards that particular criterion. A higher variability signifies a high dispersion of preferences, indicating some sense of hesitation towards a particular criterion.
- During the rank estimation, the type of criteria is actively considered, which plays a crucial role in the decision process. Unlike the extant models, the proposed work followed utility measures and ranked the alternatives from the benefit- and cost-type criteria separately. It can be seen that the proposed rank scheme is simple and elegant, with the ability to determine ranks from different angles, based on the complex proportions. Cumulatively, based on the strategy values, the rank of the alternatives is determined using the different weights for the benefit criteria and the cost criteria.

#### 4.3. Results and Discussion

- The framework is a supportive tool that considers qualitative rating information and aids in the selection of a rational material for a construction project, which is of zero- or low-carbon content. The concept of sustainable construction is fundamentally supported by the proposed framework.
- The framework can be used by a customer who is planning a construction activity, the contractor who helps the customer in the construction project, and the material designer who manufactures such low-carbon materials for sustainable construction. Each of these entities can use the model for validating their pros and cons and can effectively refine their strategies to compete with the global market.
- The framework can be used as a ready-made tool to assess the performance of zero- and low-carbon materials and it can be seen that the framework can be extended to different decision applications. Furthermore, the tool attempts to reduce the subjectivity and human intervention that affects the rationality of the decision process.
- For the effective utilization of the framework in different decision problems, the stakeholders must be trained so that they gain a sense and a feel of the rationality and the mathematical support that aids their decision-making process.
- The model primarily focuses on handling uncertainty effectively by adopting three grades–namely membership, hesitancy, and non-membership–that could effectively model uncertainty, with a flexible window for adjusting the preference zone.

## 5. Conclusions

_{2}), with an overall utility degree of 0.2261; blended cement (O

_{1}), with an overall utility degree of 0.2229; and bamboo (O

_{3}), with an overall utility degree of 0.2217, achieved a higher overall performance, compared to the other low-carbon materials. Some limitations of the proposed work are as follows: the data are assumed to be complete so if there is non-availability then the present system cannot handle the situation; though experts’ weights are methodically determined, their interdependency is not captured effectively; and the conversion of qualitative data to qROF numbers uses predetermined values that might restrict experts in using different grades flexibly.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Significance values/weight of the different criteria of zero-carbon construction material selection.

**Figure 4.**Variation in the utility degree of options over diverse parameters $\left(\phi \right)$ values.

LVs | q-ROFNs |
---|---|

Absolutely high (AH)/Extremely significant (ES) | (0.95, 0.20, 0.240) |

Very high (VH)/Very significant (VS) | (0.80, 0.35, 0.487) |

High (H)/Significant (S) | (0.70, 0.45, 0.554) |

Moderate high (MH)/Moderate significant (MS) | (0.60, 0.55, 0.581) |

Moderate (M)/Average (A) | (0.50, 0.60, 0.624) |

Moderate low (ML)/Moderate insignificant (MI) | (0.40, 0.70, 0.592) |

Low (L)/Very insignificant (VI) | (0.30, 0.75, 0.589) |

Very low (VL)/Very very insignificant (VVI) | (0.20, 0.85, 0.487) |

Absolutely low (AL)/Extremely insignificant (EI) | (0.10, 0.95, 0.296) |

DEs | LVs | q-ROFNs | Weights |
---|---|---|---|

d_{1} | Significant (S) | (0.70, 0.45, 0.554) | 0.2321 |

d_{2} | Very very significant (VVS) | (0.80, 0.35, 0.487) | 0.2753 |

d_{3} | Very significant (VS) | (0.95, 0.20, 0.240) | 0.3143 |

d_{4} | Moderate significant (MS) | (0.60, 0.55, 0.581) | 0.1783 |

O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | |
---|---|---|---|---|---|

b_{1} | (ML, MH, L, L) | (H, MH, A, A) | (AL, VL, L, A) | (VL, MH, A, A) | (A, L, L, VL) |

b_{2} | (H, A, VL, MH) | (ML, AL, L, MH) | (L, A, ML, ML) | (ML, VL, AL, ML) | (AL, L, L, MH) |

b_{3} | (MH, AL, ML, AL) | (AL, VL, A, L) | (ML, L, H, VL) | (ML, ML, A, MH) | (MH, L, H, H) |

b_{4} | (MH, ML, AL, A) | (ML, ML, ML, A) | (H, MH, A, H) | (ML, L, L, VL) | (H, AL, A, A) |

b_{5} | (A, L, ML, L) | (A, ML, MH, A) | (VL, MH, VL, MH) | (L, A, H, ML) | (VL, L, ML, ML) |

b_{6} | (ML, A, AL, A) | (ML, VL, VL, MH) | (L, ML, ML, H) | (L, A, ML, ML) | (L, A, VL, L) |

b_{7} | (A, MH, H, A) | (ML, H, A, MH) | (ML, ML, ML, H) | (MH, L, L, L) | (H, VL, VL, MH) |

b_{8} | (AL, L, L, VL) | (ML, MH, L, H) | (H, L, A, ML) | (VL, L, H, A) | (MH, A, H, VL) |

b_{9} | (ML, VL, A, AL) | (AL, L, ML, MH) | (MH, VL, MH, A) | (ML, MH, VL, A) | (AL, L, L, MH) |

b_{10} | (VL, ML, MH, L) | (H, ML, VL, ML) | (A, VL, A, ML) | (MH, ML, A, VL) | (A, VL, L, L) |

b_{11} | (VL, VL, VL, A) | (A, VL, A, ML) | (H, VL, MH, ML) | (A, ML, A, MH) | (VL, A, VL, ML) |

b_{12} | (VL, VL, H, H) | (L, A, ML, L) | (L, L, MH, H) | (AL, MH, A, ML) | (H, VL, MH, A) |

b_{13} | (AL, MH, VL, MH) | (MH, MH, ML, MH) | (A, A, AL, AL) | (VL, A, ML, VL) | (A, A, MH, A) |

**Table 4.**A-q-ROF-DM for options over different criteria of zero-carbon construction material selection.

O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | |
---|---|---|---|---|---|

b_{1} | (0.435, 0.678, 0.593) | (0.586, 0.548, 0.597) | (0.322, 0.761, 0.563) | (0.490, 0.635, 0.597) | (0.349, 0.728, 0.590) |

b_{2} | (0.529, 0.617, 0.583) | (0.376, 0.745, 0.551) | (0.413, 0.682, 0.604) | (0.287, 0.813, 0.507) | (0.358, 0.750, 0.556) |

b_{3} | (0.388, 0.760, 0.521) | (0.337, 0.765, 0.549) | (0.498, 0.643, 0.582) | (0.477, 0.639, 0.603) | (0.608, 0.543, 0.580) |

b_{4} | (0.432, 0.709, 0.558) | (0.420, 0.681, 0.600) | (0.622, 0.521, 0.585) | (0.313, 0.755, 0.576) | (0.510, 0.637, 0.578) |

b_{5} | (0.389, 0.697, 0.602) | (0.514, 0.609, 0.604) | (0.449, 0.698, 0.559) | (0.539, 0.593, 0.597) | (0.338, 0.746, 0.573) |

b_{6} | (0.400, 0.718, 0.569) | (0.366, 0.752, 0.548) | (0.466, 0.657, 0.592) | (0.413, 0.682, 0.604) | (0.350, 0.734, 0.582) |

b_{7} | (0.603, 0.535, 0.591) | (0.572, 0.566, 0.594) | (0.481, 0.647, 0.592) | (0.402, 0.698, 0.593) | (0.478, 0.679, 0.557) |

b_{8} | (0.251, 0.810, 0.530) | (0.518, 0.619, 0.590) | (0.512, 0.613, 0.601) | (0.508, 0.632, 0.585) | (0.575, 0.572, 0.585) |

b_{9} | (0.366, 0.743, 0.560) | (0.387, 0.734, 0.558) | (0.514, 0.630, 0.583) | (0.451, 0.677, 0.581) | (0.358, 0.750, 0.556) |

b_{10} | (0.439, 0.687, 0.578) | (0.469, 0.672, 0.574) | (0.425, 0.679, 0.599) | (0.470, 0.653, 0.594) | (0.342, 0.737, 0.583) |

b_{11} | (0.285, 0.799, 0.530) | (0.425, 0.679, 0.599) | (0.536, 0.618, 0.575) | (0.498, 0.616, 0.610) | (0.352, 0.746, 0.565) |

b_{12} | (0.545, 0.621, 0.563) | (0.398, 0.690, 0.604) | (0.515, 0.621, 0.591) | (0.468, 0.670, 0.577) | (0.549, 0.601, 0.581) |

b_{13} | (0.442, 0.716, 0.540) | (0.550, 0.593, 0.587) | (0.374, 0.752, 0.542) | (0.374, 0.727, 0.576) | (0.535, 0.584, 0.610) |

**Table 5.**The standard q-ROF-matrix $\tilde{S}={\left({\tilde{\xi}}_{ij}\right)}_{m\times n},$ SD, quantity of information, and weight values.

Criteria | O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | ${\mathit{\sigma}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{j}}$ | ${\mathit{w}}_{\mathit{j}}$ |
---|---|---|---|---|---|---|---|---|

b_{1} | 0.397 | 1.000 | 0.000 | 0.602 | 0.130 | 0.355 | 5.325 | 0.0891 |

b_{2} | 1.000 | 0.344 | 0.595 | 0.000 | 0.303 | 0.334 | 3.756 | 0.0628 |

b_{3} | 0.079 | 0.000 | 0.560 | 0.532 | 1.000 | 0.363 | 4.417 | 0.0739 |

b_{4} | 0.264 | 0.314 | 1.000 | 0.000 | 0.554 | 0.336 | 4.018 | 0.0672 |

b_{5} | 0.715 | 0.121 | 0.588 | 0.000 | 1.000 | 0.373 | 3.979 | 0.0666 |

b_{6} | 0.346 | 0.000 | 1.000 | 0.634 | 0.072 | 0.370 | 4.716 | 0.0789 |

b_{7} | 0.000 | 0.175 | 0.656 | 1.000 | 0.767 | 0.374 | 5.041 | 0.0843 |

b_{8} | 1.000 | 0.198 | 0.198 | 0.244 | 0.000 | 0.346 | 4.137 | 0.0692 |

b_{9} | 0.948 | 0.849 | 0.000 | 0.408 | 1.000 | 0.382 | 4.779 | 0.0799 |

b_{10} | 0.334 | 0.116 | 0.337 | 0.000 | 1.000 | 0.346 | 3.896 | 0.0652 |

b_{11} | 1.000 | 0.401 | 0.000 | 0.082 | 0.731 | 0.380 | 4.353 | 0.0728 |

b_{12} | 0.114 | 1.000 | 0.236 | 0.661 | 0.000 | 0.373 | 5.823 | 0.0974 |

b_{13} | 0.711 | 0.000 | 1.000 | 0.899 | 0.014 | 0.433 | 5.535 | 0.0926 |

Option | ${\mathit{\tau}}_{\mathit{i}}$ | $\mathbb{S}\left({\mathit{\tau}}_{\mathit{i}}\right)$ | ${\mathit{\iota}}_{\mathit{i}}$ | $\mathbb{S}\left({\mathit{\iota}}_{\mathit{i}}\right)$ | ${\mathit{\gamma}}_{\mathit{i}}$ | ${\mathit{\delta}}_{\mathit{i}}$ | Ranking |
---|---|---|---|---|---|---|---|

O_{1} | (0.275, 0.874, 0.400) | 0.1559 | (0.361, 0.792, 0.492) | 0.2516 | 0.2261 | 100.00 | 1 |

O_{2} | (0.278, 0.869, 0.409) | 0.1609 | (0.395, 0.758, 0.520) | 0.2910 | 0.2086 | 92.24 | 5 |

O_{3} | (0.301, 0.854, 0.425) | 0.1807 | (0.389, 0.768, 0.509) | 0.2812 | 0.2229 | 98.58 | 2 |

O_{4} | (0.261, 0.873, 0.412) | 0.1529 | (0.375, 0.771, 0.514) | 0.2728 | 0.2131 | 94.25 | 4 |

O_{5} | (0.285, 0.864, 0.415) | 0.1675 | (0.376, 0.775, 0.507) | 0.2703 | 0.2217 | 98.02 | 3 |

Options | $\mathit{D}\left({\mathit{O}}_{\mathit{i}},{\mathit{\zeta}}^{+}\right)$ | $\mathit{D}\left({\mathit{O}}_{\mathit{i}},{\mathit{\zeta}}^{-}\right)$ | $\mathit{R}\mathit{C}\left({\mathit{O}}_{\mathit{i}}\right)$ | Ranking |
---|---|---|---|---|

O_{1} | 0.103 | 0.127 | 0.5540 | 1 |

O_{2} | 0.149 | 0.096 | 0.3901 | 5 |

O_{3} | 0.124 | 0.116 | 0.4836 | 3 |

O_{4} | 0.146 | 0.110 | 0.4296 | 4 |

O_{5} | 0.126 | 0.124 | 0.4972 | 2 |

Options | WSM | WPM | $\mathbf{WASPAS}{\mathit{C}}_{\mathit{i}}\left(\mathit{\lambda}\right)$ | Ranking | ||
---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{i}}^{\left(1\right)}$ | $\mathbb{S}\left({\mathit{C}}_{\mathit{i}}^{\left(1\right)}\right)$ | ${\mathit{C}}_{\mathit{i}}^{\left(2\right)}$ | $\mathbb{S}\left({\mathit{C}}_{\mathit{i}}^{\left(2\right)}\right)$ | |||

O_{1} | (0.637, 0.499, 0.588) | 0.5782 | (0.579, 0.570, 0.5831) | 0.5054 | 0.542 | 1 |

O_{2} | (0.589, 0.544, 0.598) | 0.5258 | (0.546, 0.588, 0.5962) | 0.4762 | 0.501 | 5 |

O_{3} | (0.606, 0.533, 0.590) | 0.5417 | (0.568, 0.566, 0.5971) | 0.5014 | 0.522 | 3 |

O_{4} | (0.596, 0.533, 0.601) | 0.5356 | (0.547, 0.580, 0.6040) | 0.4811 | 0.508 | 4 |

O_{5} | (0.616, 0.515, 0.596) | 0.5569 | (0.561, 0.570, 0.5997) | 0.4947 | 0.526 | 2 |

φ | O_{1} | O_{2} | O_{3} | O_{4} | O_{5} | Ranking Order |
---|---|---|---|---|---|---|

φ = 0.0 | 0.2964 | 0.2563 | 0.2652 | 0.2733 | 0.2758 | ${O}_{1}\succ {O}_{5}\succ {O}_{4}\succ {O}_{3}\succ {O}_{2}$ |

φ = 0.1 | 0.2823 | 0.2467 | 0.2567 | 0.2613 | 0.2650 | ${O}_{1}\succ {O}_{5}\succ {O}_{4}\succ {O}_{3}\succ {O}_{2}$ |

φ = 0.2 | 0.2683 | 0.2372 | 0.2483 | 0.2492 | 0.2542 | ${O}_{1}\succ {O}_{5}\succ {O}_{4}\succ {O}_{3}\succ {O}_{2}$ |

φ = 0.3 | 0.2542 | 0.2277 | 0.2398 | 0.2372 | 0.2433 | ${O}_{1}\succ {O}_{5}\succ {O}_{3}\succ {O}_{4}\succ {O}_{2}$ |

φ = 0.4 | 0.2402 | 0.2181 | 0.2314 | 0.2252 | 0.2325 | ${O}_{1}\succ {O}_{5}\succ {O}_{3}\succ {O}_{4}\succ {O}_{2}$ |

φ = 0.5 | 0.2261 | 0.2086 | 0.2229 | 0.2131 | 0.2217 | ${O}_{1}\succ {O}_{3}\succ {O}_{5}\succ {O}_{4}\succ {O}_{2}$ |

φ = 0.6 | 0.2121 | 0.1991 | 0.2145 | 0.2011 | 0.2108 | ${O}_{3}\succ {O}_{1}\succ {O}_{5}\succ {O}_{4}\succ {O}_{2}$ |

φ = 0.7 | 0.1980 | 0.1895 | 0.2060 | 0.1891 | 0.2000 | ${O}_{3}\succ {O}_{5}\succ {O}_{1}\succ {O}_{2}\succ {O}_{4}$ |

φ = 0.8 | 0.1840 | 0.1800 | 0.1976 | 0.1770 | 0.1892 | ${O}_{3}\succ {O}_{5}\succ {O}_{1}\succ {O}_{2}\succ {O}_{4}$ |

φ = 0.9 | 0.1699 | 0.1705 | 0.1891 | 0.1650 | 0.1783 | ${O}_{3}\succ {O}_{5}\succ {O}_{1}\succ {O}_{2}\succ {O}_{4}$ |

φ = 1.0 | 0.1559 | 0.1609 | 0.1807 | 0.1529 | 0.1675 | ${O}_{3}\succ {O}_{5}\succ {O}_{1}\succ {O}_{2}\succ {O}_{4}$ |

Factors | Proposed | [86] | [88] | [89] |
---|---|---|---|---|

Data | q-ROFN | Fuzzy | Fuzzy | Interval-valued intuitionistic fuzzy |

Criteria weights | Calculated | Directly assigned | Calculated | Calculated |

Apriori information | Not needed | Not needed | Not needed | Needed |

Flexible preference window | Provided | Not provided | Not provided | Not provided |

Criteria interrelationship | Captured | Not captured | Not captured | Not captured |

Criteria type | Considered | Considered | Considered | Considered |

Total preorder | Yes | Yes | Yes | Yes |

Solution measure | Utility-driven | Compromise-driven | Compromise-driven | Compromise-driven |

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## Share and Cite

**MDPI and ACS Style**

Krishankumar, R.; Mishra, A.R.; Cavallaro, F.; Zavadskas, E.K.; Antuchevičienė, J.; Ravichandran, K.S.
A New Approach to the Viable Ranking of Zero-Carbon Construction Materials with Generalized Fuzzy Information. *Sustainability* **2022**, *14*, 7691.
https://doi.org/10.3390/su14137691

**AMA Style**

Krishankumar R, Mishra AR, Cavallaro F, Zavadskas EK, Antuchevičienė J, Ravichandran KS.
A New Approach to the Viable Ranking of Zero-Carbon Construction Materials with Generalized Fuzzy Information. *Sustainability*. 2022; 14(13):7691.
https://doi.org/10.3390/su14137691

**Chicago/Turabian Style**

Krishankumar, Raghunathan, Arunodaya Raj Mishra, Fausto Cavallaro, Edmundas Kazimieras Zavadskas, Jurgita Antuchevičienė, and Kattur Soundarapandian Ravichandran.
2022. "A New Approach to the Viable Ranking of Zero-Carbon Construction Materials with Generalized Fuzzy Information" *Sustainability* 14, no. 13: 7691.
https://doi.org/10.3390/su14137691