# An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information

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^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Literature survey

#### 2.1. Material Selection in Construction Industry and Sustainability

#### 2.2. MSP and Various MCDMs

## 3. Preliminaries

**Definition**

**1.**

- If$S\left({\beta}_{1}\right)<S\left({\beta}_{2}\right),$then${\beta}_{1}<{\beta}_{2};$
- If$S\left({\beta}_{1}\right)=S\left({\beta}_{2}\right)$, then
- If$H\left({\beta}_{1}\right)=H\left({\beta}_{2}\right)$, then${\beta}_{1}={\beta}_{2}$;
- If$H\left({\beta}_{1}\right)<H\left({\beta}_{2}\right)$, then${\beta}_{1}<{\beta}_{2};$

_{1}= ([p

_{1}, q

_{1}], [r

_{1}, s

_{1}]) and β

_{2}= ([p

_{2}, q

_{2}], [r

_{2}, s

_{2}]) defined as

_{1}= ([p

_{1},q

_{1}], [r

_{1},s

_{1}]), β

_{2}= ([p

_{2},q

_{2}], [r

_{2},s

_{2}]) be three IVIFNs and λ > 0. Some of their basic operational laws [34,37] are given below.

- (1)
- 1 − β
_{1}= β_{1}^{c}= ([r_{1},s_{1}], [p_{1},q_{1}]) - (2)
- β
_{1}∩ β_{2}= ([min(p_{1},p_{2}), min(q_{1},q_{2})], [max(r_{1},r_{2}), max(s_{1},s_{2})]) - (3)
- β
_{1}∪ β_{2}= ([max(p_{1},p_{2}), max(q_{1},q_{2})], [min(r_{1},r_{2}), min(s_{1},s_{2})]) - (4)
- β
_{1}+ β_{2}= ([p_{1}+ p_{2}− p_{1}p_{2}, q_{1}+ q_{2}− q_{1}q_{2}], [r_{1}r_{2}, s_{1}s_{2}]) - (5)
- β
_{1}· α_{2}= ([p_{1}p_{2}, q_{1}q_{2}], [r_{1}+ r_{2}− r_{1}r_{2}, s_{1}+ s_{2}− s_{1}s_{2}]) - (6)
- λβ = ([1 − (1 − p)
^{λ},1 − (1 − q)^{λ}], [r^{λ}, s^{λ}]) - (7)
- β
^{λ}= ([p^{λ}, q^{λ}], [1 − (1 − r)^{λ},1 − (1 − s)^{λ}])

**Definition**

**2.**

_{j}= ([p

_{j}, q

_{j}], [r

_{j}, s

_{j}]) (j = 1,2,…,n) be a collection of IVIFNs, and w = (w

_{1},w

_{2},…,w

_{n})

^{T}be their associated weight vector, with 0 ≤ w

_{j}≤ 1 and ${\sum}_{j=1}^{n}{w}_{j}=1,$ then he interval valued intuitionistic fuzzy weighted geometric (IVIFWG) operator is defined as

**Definition**

**3.**

_{1}= ([p

_{1}, q

_{1}], [r

_{1}, s

_{1}]) and β

_{2}= ([p

_{2}, q

_{2}], [r

_{2}, s

_{2}]) are computed as:

**Definition**

**4.**

_{1}, x

_{2}, …, x

_{n}}, then the distance measure between${\beta}_{1}$and${\beta}_{2}$is defined as follows:

## 4. Proposed CODAS Method Using IVIFNs

_{1}, d

_{2}, …, d

_{l}} be the group of decision makers, C = {c

_{1}, c

_{2}, …, c

_{n}} be the set of criteria, and A = {a

_{1},a

_{2},…,a

_{m}} be the set of alternatives. The group of experts/decision makers D = {d

_{1}, d

_{2}, …, d

_{l}} provide their opinions regarding the criteria C = {c

_{1}, c

_{2}, …, c

_{n}} corresponding to each alternative A = {a

_{1}, a

_{2}, …, a

_{m}} using linguistic terms which are presented by IVIFNs. In this algorithm, we consider that significance of individual decision makers are different and the weights of the decision makers are expressed using fuzzy membership grades. We also consider that opinions of individual decision makers about the importance of various criteria are different. A flow chart of the proposed approach is given in Figure 1.

**Step 1.**Opinion of each expert is expressed using decision matrix given below.

**Step 2.**Interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) aggregation operator is used to aggregate the opinion of individual decision makers. The aggregated/collective decision matrix is formed as

**Step 3.**Calculate the weights of evaluation criteria

- (1)
- A weak ranking: ${\mathrm{H}}_{1}=\left\{{\mathrm{w}}_{\mathrm{i}}\ge {\mathrm{w}}_{\mathrm{j}}\right\}$;
- (2)
- A strict ranking: ${\mathrm{H}}_{2}=\left\{{\mathrm{w}}_{\mathrm{i}}-{\mathrm{w}}_{\mathrm{j}}\ge {\mathsf{\beta}}_{\mathrm{j}}|{\mathsf{\beta}}_{\mathrm{j}}\rangle 0\right\}$;
- (3)
- A ranking of differences: ${\mathrm{H}}_{3}=\{{\mathrm{w}}_{\mathrm{i}}-{\mathrm{w}}_{\mathrm{j}}\ge {\mathrm{w}}_{\mathrm{l}}-{\mathrm{w}}_{\mathrm{l}}|\mathrm{j}\ne \mathrm{k}\ne \mathrm{l}\}$;
- (4)
- A ranking with multiples: ${\mathrm{H}}_{4}=\{{\mathrm{w}}_{\mathrm{i}}\ge {\mathsf{\beta}}_{\mathrm{j}}{\mathrm{w}}_{\mathrm{j}}|0\le {\mathsf{\beta}}_{\mathrm{j}}\le 1\}$;
- (5)
- An interval form: ${\mathrm{H}}_{5}=\{{\mathsf{\beta}}_{\mathrm{i}}\ge {\mathrm{w}}_{\mathrm{i}}\le {\mathsf{\beta}}_{\mathrm{j}}+{\mathsf{\u03f5}}_{\mathrm{i}}|0\le {\mathsf{\beta}}_{\mathrm{i}}\le {\mathsf{\beta}}_{\mathrm{i}}+{\mathsf{\u03f5}}_{\mathrm{i}}\}$;

**Step 4.**The collective decision matrix is normalized by determining the highest IVIFN under each criterion for all the alternatives and then performing the division operation of between the highest IVIFN and the corresponding IVIFN as given below.

**Step 5.**The weighted normalized decision matrix is determined by performing product operation between the aggregated criteria weights and the normalized performance values of the criteria corresponding to the alternatives, which is give below. It is noted that both the aggregated criteria weights and the evaluating values are expressed as IVIFNs.

**Step 6.**The interval-valued intuitionistic fuzzy negative ideal solution is computed as

**Step 7.**IVIFN-based Hamming (HD) and Euclidean distances (ED) of the alternatives $\mathrm{i}\in \left\{1,\text{}2,\text{}\dots ,\mathrm{m}\right\}$ from the interval-valued intuitionistic fuzzy negative ideal solution ($\tilde{\mathrm{NS}}$) are computed as

**Step 8.**The relative assessment matrix (RA) is computed as

**Step 9.**The assessment score ($A{S}_{i}$) of each alternative is computed, which is given below.

## 5. Application of the IVIF--CODAS in MSPs

#### 5.1. Illustrative Example

- The collective decision matrix is shown in Table 3 which is computed by aggregating the opinions of the decision makers (DM1, DM2, …, DM5).
- The consensus making model [39] is used to determine the weights of the evaluation criteria. In our study, the criteria weights are unknown while the weights of experts (${\sum}_{k=1}^{5}{\lambda}_{k}=1$) are known in advance. Hence, applying model ($M-1$) and assuming that the criteria weights are partially known as follows: $H=\{{w}_{1}\le 0.08,{w}_{1}={w}_{2},0.10\le {w}_{3}\le 0.20,{w}_{4}+{w}_{2}\ge 0.3,0.13\le {w}_{5}\le 0.20,0.12\le {w}_{6}\le 0.17,0.12\le {w}_{7}\le 0.16,{w}_{8}={w}_{6},{w}_{5}-{w}_{6}\ge 0.05,{w}_{5}-{w}_{7}\ge 0.03,{w}_{j}\ge 0,j=1,2,\dots ,8,{\sum}_{j=1}^{8}{w}_{j}=1\}$. To compute the criteria priorities, Equations (16)–(18) and model (M − 1) are used to develop the linear programming model given below:$$\{\begin{array}{c}\mathrm{max}\mathrm{D}\left(\mathrm{w}\right)=0.450{\mathrm{w}}_{1}+0.590{\mathrm{w}}_{2}+0.550{\mathrm{w}}_{3}+0.421{\mathrm{w}}_{4}\\ +0.550{\mathrm{w}}_{5}+0.530{\mathrm{w}}_{6}+0.640{\mathrm{w}}_{7}+0.605{\mathrm{w}}_{8}\\ \mathrm{subject\; to}\mathrm{to}\mathrm{w}\in \mathrm{H}\end{array}$$
- The final criteria weights (${\mathrm{w}}^{*}$) are obtained by solving the above model and they are represented by the following weight vector ${\mathrm{w}}^{*}={\left(0.050,\text{}0.050,\text{}0.100,\text{}0.250,\text{}0.170,\text{}0.120,\text{}0.140,\text{}0.120\right)}^{\mathrm{T}}$.
- Weighted normalized decision matrix, shown in Table 4, is computed by finding the normalized decision matrix and then combining the criteria weights obtained in the previous step with the normalized decision matrix.
- Interval-valued intuitionistic fuzzy negative ideal solution for each of the criteria is shown in Table 5.
- Hamming distance and Euclidean distance of the alternatives from the Interval-valued intuitionistic fuzzy negative ideal solution is shown in Table 6.
- Relative assessment matrix, assessment scores, and rank of the alternatives are given in Table 7.
- Table 8 shows the ranking orders by the proposed method along with four other existing methods for result comparisons.

#### 5.2. A Real Case of MSP in Sustainable Construction Projects

**Step 1.**Data collection

**Step 2.**Calculation of criteria weights

**Step 3.**Evaluating the best sustainable material using IVIF-CODAS method

## 6. Result Discussion

#### 6.1. Comparisons

- (1)
- Crisp ratings is used to evaluate the classical CODAS method but this ratings often fail in real life scenario as real life problems are much uncertain. For example, the construction company may consider some criteria as highly important and to signify the importance, the corresponding rating scale need to be more flexible. This “highly important” term can be preferably expressed using as an IVIF number $\left(\left[8,10\right],\left[0,0\right]\right)$ rather than a single crisp number $9$. However, in this paper, we use IVIF numbers to assess the alternative bricks and criteria importance since DMs can flexibly express their opinions using IVIF numbers.
- (2)
- Compared with fuzzy CODAS, the proposed IVIF-CODAS has an advantage. Grattan-Guinness [52] argued that it is a difficult task for decision makers to represent linguistic ratings in the form of a single membership degree in classical fuzzy set theory. In response, Atanassov [53] introduced IFS as an extension of fuzzy sets. In IFS, hesitation margin is introduced as a new concept and the sum of membership and non-membership degree may be less than one. However, both in fuzzy set and IFS, the membership values are exact and crisp in nature. To present the membership and non-membership values are in intervals, Atanassov and Gargov [8] extended IFS in IVIFS. Thus, in group decision making problems the extended IVIF-CODAS method offers a better treatment in handling uncertainty in the decision making process.
- (3)
- As it is difficult to show the applicability and trustworthiness of a newly proposed method, hence it is necessary to assess it in solving several MCDM problems. Wang et al. [54] asserted a comparison which is only way to apprehend the validity of newly proposed MCDM model (here, IVIF-CODAS). To justify any proposed approach, one has to compare it with several related approaches for the same problem. Accordingly, we have presented two illustrative examples and found encouraging results that show the similarity of IVIF-CODAS to other methods. One can consider this to be one of the advantages of the novel approach that is reckoned to be applicable irrespective of its case studies.

#### 6.2. Sensitivity Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Linguistic Terms | Corresponding IVIFNs |
---|---|

Very High (VH) | ([0.9,1], [0,0]) |

High (H) | ([0.8,0.8], [0.1,0.1]) |

Medium High (MH) | ([0.6,0.7], [0.2,0.3]) |

Medium (M) | ([0.5,0.5], [0.4,0.5]) |

Medium Low (ML) | ([0.3,0.4], [0.5,0.6]) |

Low (L) | ([0.2,0.2], [0.7,0.7]) |

Very Low (VL) | ([0,0.1], [0.8,0.9]) |

Max | Max | Max | Max | Max | Max | Min | Min | ||
---|---|---|---|---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | ||

DM1 | A1 | MH | H | MH | MH | H | MH | MH | MH |

A2 | M | MH | H | H | H | MH | H | MH | |

A3 | VH | H | VH | H | H | VH | H | VH | |

A4 | H | VH | H | H | VH | H | VH | H | |

DM2 | A1 | MH | H | MH | MH | MH | H | MH | MH |

A2 | MH | M | MH | M | MH | M | M | M | |

A3 | H | VH | H | VH | H | VH | H | H | |

A4 | H | H | VH | H | VH | H | VH | VH | |

DM3 | A1 | MH | H | MH | H | MH | MH | H | MH |

A2 | MH | MH | H | H | H | H | MH | H | |

A3 | VH | VH | H | H | VH | VH | H | H | |

A4 | H | VH | H | H | H | VH | H | H | |

DM4 | A1 | H | MH | H | MH | MH | H | MH | H |

A2 | MH | MH | M | MH | M | MH | M | MH | |

A3 | H | VH | H | VH | H | H | VH | H | |

A4 | H | H | VH | H | H | H | VH | VH | |

DM5 | A1 | MH | H | MH | H | MH | H | MH | MH |

A2 | H | MH | H | MH | H | MH | M | M | |

A3 | VH | H | VH | H | VH | H | VH | H | |

A4 | H | VH | H | H | H | VH | H | M |

C1 | C2 | C3 | C4 | |

A1 | ([0.636, 0.719], [0.181, 0.264]) | ([0.755, 0.779], [0.121, 0.144]) | ([0.636, 0.719], [0.181, 0.264]) | ([0.645, 0.724], [0.176, 0.255]) |

A2 | ([0.668, 0.716], [0.204, 0.284]) | ([0.687, 0.753], [0.169, 0.247]) | ([0.775, 0.787], [0.136, 0.171]) | ([0.782, 0.800], [0.124, 0.157]) |

A3 | ([0.931, 0.956], [0.021, 0.021]) | ([0.943, 0.978], [0.010, 0.010]) | ([0.915, 0.925], [0.036, 0.036]) | ([0.926, 0.946], [0.026, 0.026]) |

A4 | ([0.925, 0.925 ], [0.036, 0.036]) | ([0.947, 0.967], [0.016, 0.016]) | ([0.941, 0.956], [0.021, 0.021]) | ([0.925, 0.925], [0.036, 0.036]) |

C5 | C6 | C7 | C8 | |

A1 | ([0.654, 0.729], [0.171, 0.245]) | ([0.713, 0.758], [0.141, 0.186]) | ([0.618, 0.709], [0.191, 0.282]) | ([0.636, 0.719], [0.181, 0.264]) |

A2 | ([0.775, 0.787], [0.136, 0.171]) | ([0.727, 0.774], [0.149, 0.208]) | ([0.718, 0.741], [0.180, 0.237]) | ([0.727, 0.774], [0.149, 0.208]) |

A3 | ([0.921, 0.935], [0.031, 0.031]) | ([0.954, 1.000], [0.000, 0.000]) | ([0.904, 0.904], [0.046, 0.046]) | ([0.915, 0.925], [0.036, 0.036]) |

A4 | ([0.964, 1.000], [0.000, 0.000]) | ([0.925, 0.925], [0.036, 0.036]) | ([0.964, 1.000], [0.000, 0.000]) | ([0.941, 0.956], [0.021, 0.021]) |

C1 | C2 | C3 | C4 | |

A1 | ([0.042,0.049], [0.028,0.035]) | ([0.056,0.056], [0.014, 0.021]) | ([0.090,0.105], [0.030, 0.045]) | ([0.042,0.049], [0.014, 0.021]) |

A2 | ([0.035,0.035], [0.028, 0.035]) | ([0.042,0.049], [0.014, 0.021]) | ([0.120,0.120], [0.030, 0.045]) | ([0.056,0.056], [0.014, 0.021]) |

A3 | ([0.063,0.070], [0.028, 0.035]) | ([0.056,0.056], [0.014, 0.021]) | ([0.135,0.150], [0.030, 0.045]) | ([0.056,0.056], [0.014, 0.021]) |

A4 | ([0.056,0.056], [0.028, 0.035]) | ([0.063,0.070], [0.014, 0.021]) | ([0.120,0.120], [0.030, 0.045]) | ([0.056,0.056], [0.014, 0.021]) |

C5 | C6 | C7 | C8 | |

A1 | ([0.144,0.144], [0.018, 0.018]) | ([0.087,0.102], [0.029, 0.044]) | ([0.102,0.119], [0.034, 0.051]) | ([0.087,0.102], [0.029, 0.044]) |

A2 | ([0.144,0.144], [0.018, 0.018]) | ([0.087,0.102], [0.029, 0.044]) | ([0.102,0.119], [0.017, 0.017]) | ([0.087,0.102], [0.029, 0.044]) |

A3 | ([0.144,0.144], [0.018, 0.018]) | ([0.087,0.102], [0.000, 0.000]) | ([0.102,0.119], [0.017, 0.017]) | ([0.087,0.102], [0.000, 0.000]) |

A4 | ([0.162,0.180], [0.018, 0.018]) | ([0.087,0.102], [0.015, 0.015]) | ([0.102,0.119], [0.000, 0.000]) | ([0.087,0.102], [0.015, 0.015]) |

Criteria | IVIF Negative Ideal Solutions |
---|---|

C1 | ([0.1407 0.1703], [0.7286 0.7692)] |

C2 | ([0.1520 0.1791], [0.7167 0.7572)] |

C3 | ([0.4668 0.5526], [0.2885 0.3857)] |

C4 | ([0.1427 0.1722], [0.7191 0.7598)] |

C5 | ([0.5886 0.7290], [0.1710 0.2450)] |

C6 | ([0.5158 0.5787], [0.2650 0.3472)] |

C7 | ([0.5401 0.6705], [0.2120 0.3007)] |

C8 | ([0.4668 0.5526], [0.2885 0.3857)] |

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

ED | 0.0182 | 0.1899 | 0.767 | 0.8317 |

HD | 0.0117 | 0.1264 | 0.5361 | 0.5824 |

Relative Assessment Matrix | Appraisal Scores | Ranking | ||||
---|---|---|---|---|---|---|

A1 | A2 | A3 | A4 | |||

A1 | 0 | −0.2864 | −1.2731 | −1.3842 | −2.9437 | 4 |

A2 | 0.2864 | 0 | −0.9867 | −1.0977 | −1.798 | 3 |

A3 | 1.2731 | 0.9867 | 0 | −0.1111 | 2.1487 | 2 |

A4 | 1.3842 | 1.0977 | 0.1111 | 0 | 2.593 | 1 |

MCDM Methods | Ranking Order |
---|---|

Classical CODAS | A4 > A3 > A2 >A1 |

Fuzzy CODAS | A4 > A3 > A2 >A1 |

IVIF-VIKOR | A3 > A4 > A2 >A1 |

IVIF-TOPSIS | A4 > A3 > A2 >A1 |

The proposed IVIF-CODAS | A4 > A3 > A2 >A1 |

Decision Makers | Expertise |
---|---|

DM1 | Head of establishing standards and techniques with 21 years of work experience |

DM2 | Health, Safety and Environment (HSE) management employee and the head of operations evaluation with 20 years of work experience |

DM3 | Expert supervisor of construction project implementation with 21 years of work experience |

DM4 | Project manager with 17 years of work experience |

DM5 | Financial manager with 18 years of work experience |

Dimension | Criteria | Description |
---|---|---|

Economical | Initial cost (C1) | Cost considered for purchasing/manufacturing of materials |

Maintenance cost (C2) | Cost considered for maintaining in its lifetime | |

Disposal cost (C3) | Cost considered for material disposal | |

Tax contribution (C4) | Tax regarding the materials | |

Environmental | Raw material extraction (C5) | It is necessary to manufacture the final material |

Land acquisition (C6) | Land required for the material construction | |

Soil consumption (C7) | It is required by the material at the time of manufacturing and operation | |

Production and transportation (C8) | Comfortable transportation and production is important | |

Social | Fire resistance (C9) | Necessary arrangements to resist fire |

Esthetics (C10) | Looking of the material | |

Use of local material (C11) | To develop society, more use of local material is needed | |

Labor availability (C12) | Quality labor is vital for production |

Dimension | Criteria | Local Weights | Global Weights | Rank |
---|---|---|---|---|

EC | - | 0.4268 | - | - |

C1 | 0.3487 | 0.1488 | 2 | |

C2 | 0.1857 | 0.0793 | 6 | |

C3 | 0.4031 | 0.1720 | 1 | |

C4 | 0.0625 | 0.0267 | 11 | |

EN | - | 0.3568 | - | - |

C5 | 0.3838 | 0.1369 | 3 | |

C6 | 0.3396 | 0.1212 | 4 | |

C7 | 0.0143 | 0.0051 | 12 | |

C8 | 0.2623 | 0.0936 | 5 | |

SO | - | 0.2164 | - | - |

C9 | 0.2387 | 0.0517 | 9 | |

C10 | 0.2180 | 0.0472 | 10 | |

C11 | 0.3000 | 0.0649 | 7 | |

C12 | 0.2434 | 0.0527 | 8 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DM1 | A1 | MH | MH | MH | MH | MH | MH | M | MH | MH | H | H | H |

A2 | H | H | H | MH | H | H | VH | MH | M | M | MH | M | |

A3 | H | MH | H | MH | H | H | H | H | MH | H | H | H | |

DM2 | A1 | VH | MH | MH | VH | MH | MH | M | MH | VH | H | H | H |

A2 | H | VH | H | MH | H | VH | VH | MH | M | M | VH | VH | |

A3 | H | MH | VH | MH | VH | H | VH | VH | MH | VH | H | H | |

DM3 | A1 | MH | MH | MH | MH | MH | MH | M | MH | MH | MH | H | H |

A2 | H | MH | H | MH | MH | H | MH | MH | M | M | MH | M | |

A3 | H | MH | MH | MH | H | H | H | H | MH | H | H | MH | |

DM4 | A1 | M | MH | MH | MH | MH | M | M | M | MH | H | H | H |

A2 | H | M | H | M | M | H | M | MH | M | M | MH | M | |

A3 | H | MH | M | MH | H | H | H | H | M | H | M | H | |

DM5 | A1 | VH | MH | MH | VH | MH | MH | M | MH | VH | H | H | H |

A2 | H | VH | H | MH | H | VH | VH | MH | M | M | VH | VH | |

A3 | H | MH | VH | MH | VH | H | VH | VH | MH | VH | H | H |

C1 | C2 | C3 | C4 | |

A1 | ([0.673, 0.754], [0.174, 0.264]) | ([0.600, 0.700], [0.200, 0.144]) | ([0.600, 0.700], [0.200, 0.264]) | ([0.691, 0.793], [0.125, 0.255]) |

A2 | ([0.800, 0.800], [0.100, 0.343]) | ([0.733, 0.785], [0.145, 0.356]) | ([0.800, 0.800], [0.100, 0.249]) | ([0.584, 0.666], [0.245, 0.288]) |

A3 | ([0.800, 0.800], [0.100, 0.046]) | ([0.600, 0.700], [0.200, 0.046]) | ([0.733, 0.785], [0.145, 0.056]) | ([0.600, 0.700], [0.200, 0.056]) |

C5 | C6 | C7 | C8 | |

A1 | ([0.600, 0.700], [0.200, 0.245]) | ([0.584, 0.666], [0.245, 0.186]) | ([0.500, 0.500], [0.400, 0.282]) | ([0.584, 0.666], [0.245, 0.264]) |

A2 | ([0.704, 0.726], [0.180, 0.249]) | ([0.834, 0.865], [0.061, 0.340]) | ([0.760, 0.839], [0.117, 0.383]) | ([0.600, 0.700], [0.200, 0.373]) |

A3 | ([0.834, 0.865], [0.061, 0.076]) | ([0.800, 0.800], [0.100, 0.036]) | ([0.834, 0.865], [0.061, 0.066]) | ([0.834, 0.865], [0.061, 0.071]) |

C9 | C10 | C11 | C12 | |

A1 | ([0.691, 0.793], [0.125, 0.264]) | ([0.755, 0.779], [0.111, 0.264]) | ([0.800, 0.800], [0.100, 0.264]) | ([0.800, 0.800], [0.100, 0.264]) |

A2 | ([0.500, 0.500], [0.400, 0.373]) | ([0.500, 0.500], [0.400, 0.373]) | ([0.691, 0.793], [0.125, 0.373]) | ([0.614, 0.637], [0.264, 0.373]) |

A3 | ([0.584, 0.666], [0.245, 0.071]) | ([0.854, 0.904], [0.051, 0.071]) | ([0.746, 0.746], [0.170, 0.071]) | ([0.755, 0.779], [0.111, 0.071]) |

C1 | C2 | C3 | C4 | |

A1 | ([0.149, 0.180], [0.718, 0.763]) | ([0.133, 0.167], [0.727, 0.724]) | ([0.448, 0.557], [0.294, 0.376]) | ([0.129, 0.159], [0.702, 0.760]) |

A2 | ([0.149, 0.180], [0.693, 0.788]) | ([0.133, 0.167], [0.708, 0.793]) | ([0.448, 0.557], [0.205, 0.363]) | ([0.129, 0.159], [0.743, 0.771]) |

A3 | ([0.149, 0.180], [0.693, 0.693]) | ([0.133, 0.167], [0.727, 0.693]) | ([0.448, 0.557], [0.245, 0.200]) | ([0.129, 0.159], [0.727, 0.696]) |

C5 | C6 | C7 | C8 | |

A1 | ([0.540, 0.700], [0.200, 0.245]) | ([0.422, 0.508], [0.348, 0.329]) | ([0.437, 0.473], [0.416, 0.301]) | ([0.429, 0.512], [0.344, 0.477]) |

A2 | ([0.540, 0.700], [0.180, 0.249]) | ([0.422, 0.508], [0.189, 0.456]) | ([0.437, 0.473], [0.140, 0.399]) | ([0.440, 0.538], [0.344, 0.477]) |

A3 | ([0.540, 0.700], [0.061, 0.076]) | ([0.422, 0.508], [0.222, 0.206]) | ([0.437, 0.473], [0.085, 0.090]) | ([0.612, 0.665], [0.344, 0.477]) |

C9 | C10 | C11 | C12 | |

A1 | ([0.166, 0.217], [0.773, 0.786]) | ([0.181, 0.213], [0.773, 0.786]) | ([0.554, 0.598], [0.295, 0.502]) | ([0.192, 0.218], [0.722, 0.786]) |

A2 | ([0.120, 0.137], [0.773, 0.786]) | ([0.120, 0.137], [0.773, 0.786]) | ([0.479, 0.593], [0.295, 0.502]) | ([0.147, 0.174], [0.722, 0.786]) |

A3 | ([0.140, 0.182], [0.773, 0.786]) | ([0.205, 0.247], [0.773, 0.786]) | ([0.517, 0.558], [0.295, 0.502]) | ([0.181, 0.213], [0.722, 0.786]) |

NIS | IVIF Negative Ideal Solutions |
---|---|

C1 | ([0.149, 0.180], [0.718, 0.788]) |

C2 | ([0.133, 0.167], [0.727, 0.793]) |

C3 | ([0.448, 0.557], [0.294, 0.376]) |

C4 | ([0.129, 0.159], [0.743, 0.771]) |

C5 | ([0.540, 0.700], [0.200, 0.249]) |

C6 | ([0.422, 0.508], [0.348, 0.456]) |

C7 | ([0.437, 0.473], [0.416, 0.399]) |

C8 | ([0.429, 0.512], [0.344, 0.477]) |

C9 | ([0.120, 0.137], [0.773, 0.786]) |

C10 | ([0.120, 0.137], [0.773, 0.786]) |

C11 | ([0.479, 0.558], [0.295, 0.502]) |

C12 | ([0.147, 0.174], [0.722, 0.786]) |

A1 | A2 | A3 | |
---|---|---|---|

ED | 0.3519 | 0.3258 | 0.9647 |

HD | 0.2107 | 0.1682 | 0.6426 |

Relative Assessment Matrix | Appraisal Scores | Ranking | |||
---|---|---|---|---|---|

A_{1} | A_{2} | A_{3} | |||

A_{1} | 0.000 | 0.069 | −1.045 | −0.976 | 2 |

A_{2} | −0.069 | 0.000 | −1.113 | −1.182 | 3 |

A_{3} | 1.045 | 1.113 | 0.000 | 2.158 | 1 |

MCDM Methods | Ranking Order |
---|---|

Classical CODAS | A_{3} > A_{2} > A_{1} |

Fuzzy CODAS | A_{3} > A_{1} > A_{2} |

IVIF-VIKOR | A_{3} > A_{2} > A_{1} |

IVIF-TOPSIS | A_{3} > A_{1} > A_{2} |

The proposed IVIF-CODAS | A_{3} > A_{1} > A_{2} |

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

A_{1} | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 1 |

>A_{2} | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

>A_{3} | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 |

© 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**

Roy, J.; Das, S.; Kar, S.; Pamučar, D.
An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information. *Symmetry* **2019**, *11*, 393.
https://doi.org/10.3390/sym11030393

**AMA Style**

Roy J, Das S, Kar S, Pamučar D.
An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information. *Symmetry*. 2019; 11(3):393.
https://doi.org/10.3390/sym11030393

**Chicago/Turabian Style**

Roy, Jagannath, Sujit Das, Samarjit Kar, and Dragan Pamučar.
2019. "An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information" *Symmetry* 11, no. 3: 393.
https://doi.org/10.3390/sym11030393