# An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### Intuitionistic Fuzzy Sets

**Definition**

**1**

**.**An intuitionistic fuzzy set (IFS) on the universe $X$ is an object of the form

**Definition**

**2**

**.**Let ${I}_{1}=\left({\mu}_{1},{\nu}_{1}\right)$ and ${I}_{2}=\left({\mu}_{2},{\nu}_{2}\right)$ be two intuitionistic fuzzy numbers (IFNs); the operation formula can then be defined as:

**Definition**

**3**

**.**Let ${I}_{1}=\left({\mu}_{1},{\nu}_{1}\right)$ and ${I}_{2}=\left({\mu}_{2},{\nu}_{2}\right)$ be IFNs; the score and accuracy functions of ${I}_{1}$ and ${I}_{2}$ can then be expressed as:

**Definition**

**4.**

**.**Let ${I}_{j}=\left({\mu}_{{I}_{j}},{\nu}_{{I}_{j}}\right)\left(j=1,2,\cdots ,n\right)$ be a collection of IFNs; the intuitionistic fuzzy weighted averaging (IFWA) operator can then be defined as:

**Theorem**

**1.**

**Definition**

**5**

**.**Let ${I}_{j}\left(j=1,2,\cdots ,n\right)$ be a collection of IFNs; the intuitionistic fuzzy weighted geometric (IFWG) operator can then be defined as:

**Theorem**

**2.**

## 3. The EDAS Method with Intuitionistic Fuzzy Information

**Step 1.**Set up each decision maker’s intuitionistic fuzzy decision matrix ${Q}^{\left(k\right)}={\left({q}_{ij}^{k}\right)}_{m\times n}$ and calculate the overall intuitionistic fuzzy decision matrix $Q={\left({q}_{ij}\right)}_{m\times n}$.

**Step 2.**Normalize the overall intuitionistic fuzzy decision matrix $Q={\left({q}_{ij}\right)}_{m\times n}$ to ${Q}^{N}={\left[{q}_{ij}^{N}\right]}_{m\times n}$.

**Step 3.**Use the CRiteria Importance Through Intercriteria Correlation (CRITIC) method to determine the weighting matrix of attributes.

**(1)**- Depending on the normalized overall intuitionistic fuzzy decision matrix ${Q}^{N}={\left({q}_{ij}^{N}\right)}_{m\times n}$, the correlation coefficient between attributes can be calculated as:$$I{C}_{jt}=\frac{{\displaystyle \sum _{i=1}^{m}\left(S\left({q}_{ij}^{N}\right)-S\left({q}_{j}^{N}\right)\right)\left(S\left({q}_{it}^{N}\right)-S\left({q}_{t}^{N}\right)\right)}}{\sqrt{{\displaystyle \sum _{i=1}^{m}{\left(S\left({q}_{ij}^{N}\right)-S\left({q}_{j}^{N}\right)\right)}^{2}}}\sqrt{{\displaystyle \sum _{i=1}^{m}{\left(S\left({q}_{it}^{N}\right)-S\left({q}_{t}^{N}\right)\right)}^{2}}}},j,t=1,2,\dots ,n$$
**(2)**- Calculate the attributes’ standard deviation.$$I{S}_{j}=\sqrt{\frac{1}{m-1}{\displaystyle \sum _{i=1}^{m}{\left(S\left({q}_{ij}^{N}\right)-S\left({q}_{j}^{N}\right)\right)}^{2}}},j=1,2,\dots ,n$$
**(3)**- Calculate the attributes’ weights.$${z}_{j}=\frac{I{S}_{j}{\displaystyle \sum _{t=1}^{n}\left(1-I{C}_{jt}\right)}}{{\displaystyle \sum _{j=1}^{n}\left(I{S}_{j}{\displaystyle \sum _{t=1}^{n}\left(1-I{C}_{jt}\right)}\right)}},j=1,2\dots ,n$$

**Step 4.**Calculate the value of average solution (AV) regarding all proposed attributes.

**Step 5.**Depending on the AV results, the positive distance from average (PDA) and negative distance from average (NDA) can be calculated as:

**Step 6.**Calculate the values of $S{P}_{i}$ and $S{N}_{i}$ which denote the weighted sum of PDA and NDA.

**Step 7.**Depending on the above calculated results, $S{P}_{i}$ and $S{N}_{i}$ can be normalized as:

**Step 8.**Calculate the values of the appraisal score ($AS$) regarding each alternative’s $NS{P}_{i}$ and $NS{N}_{i}$:

**Step 9.**In terms of the calculated results of $AS$, all the alternatives can be ranked. The higher the value of $AS$, the higher the value of the optimal alternative that is selected.

## 4. The Empirical Example and Comparative Analysis

#### 4.1. An Empirical Example

**Step 1.**Set up each decision maker’s intuitionistic fuzzy evaluation matrix ${Q}^{\left(k\right)}={\left({q}_{ij}^{k}\right)}_{m\times n}$ $\left(i=1,2,\dots ,m,j=1,2,\dots ,n\right)$ as shown in Table 1, Table 2, Table 3, Table 4 and Table 5. From these tables and Equations (12)–(14), the overall intuitionistic fuzzy decision matrix can be calculated. The results are presented in Table 6.

**Step 2.**Normalize the evaluation matrix $Q={\left[{q}_{ij}^{}\right]}_{m\times n}$ to ${Q}^{N}={\left[{q}_{ij}^{N}\right]}_{m\times n}$ (See Table 7).

**Step 3.**Decide the attribute weights ${z}_{j}\left(j=1,2,\dots ,n\right)$ using the CRITIC method as presented in Table 8.

**Step 4.**Depending on the calculated results of Table 8, the value of the average solution (AV) can be obtained based on all proposed attributes using Equations (19) and (20) (see Table 9).

**Step 5.**Based on the results of AV, the positive distance from average (PDA) and negative distance from average (NDA) can be calculated using Equations (21) and (22) (see Table 10 and Table 11).

**Step 6.**Based on Equation (23) and the attributes’ weighting vector $\omega =\left(0.1410,0.2263,0.3234,0.3093\right)$, the values of $S{P}_{i}$ and $S{N}_{i}$ can be calculated:

**Step 7.**The results of Step 6 can be normalized using Equation (24):

**Step 8.**Based on each alternative’s $NS{P}_{i}$ and $NS{N}_{i}$, the values of AS can be calculated:

**Step 9.**Based on the calculated results of AS, all the alternatives can be ranked; the higher the value of AS, the higher the optimal alternative that is selected. Evidently, the rank of all alternatives is ${Y}_{2}>{Y}_{5}>{Y}_{3}>{Y}_{4}>{Y}_{1}$ and ${Y}_{2}$ is the best green building energy-saving design project.

#### 4.2. Comparative Analysis

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.35,0.65) | (0.52,0.48) | (0.24,0.76) | (0.43,0.57) |

${Y}_{2}$ | (0.39,0.61) | (0.66,0.34) | (0.75,0.25) | (0.61,0.39) |

${Y}_{3}$ | (0.40,0.60) | (0.33,0.67) | (0.56,0.44) | (0.28,0.72) |

${Y}_{4}$ | (0.67,0.33) | (0.58,0.42) | (0.41,0.59) | (0.47,0.53) |

${Y}_{5}$ | (0.26,0.74) | (0.42,0.58) | (0.52,0.48) | (0.62,0.38) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.38,0.62) | (0.43,0.57) | (0.29,0.71) | (0.55,0.45) |

${Y}_{2}$ | (0.63,0.37) | (0.34,0.66) | (0.48,0.52) | (0.52,0.48) |

${Y}_{3}$ | (0.50,0.50) | (0.27,0.73) | (0.41,0.59) | (0.16,0.84) |

${Y}_{4}$ | (0.46,0.54) | (0.62,0.38) | (0.57,0.43) | (0.29,0.71) |

${Y}_{5}$ | (0.60,0.40) | (0.46,0.54) | (0.42,0.58) | (0.33,0.67) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.44,0.56) | (0.58,0.42) | (0.31,0.69) | (0.40,0.60) |

${Y}_{2}$ | (0.58,0.42) | (0.65,0.35) | (0.42,0.58) | (0.74,0.26) |

${Y}_{3}$ | (0.35,0.65) | (0.48,0.52) | (0.18,0.82) | (0.62,0.38) |

${Y}_{4}$ | (0.27,0.73) | (0.26,0.74) | (0.62,0.38) | (0.31,0.69) |

${Y}_{5}$ | (0.46,0.54) | (0.44,0.56) | (0.34,0.66) | (0.65,0.35) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.52,0.48) | (0.37,0.63) | (0.25,0.75) | (0.22,0.78) |

${Y}_{2}$ | (0.51,0.49) | (0.64,0.36) | (0.77,0.23) | (0.42,0.58) |

${Y}_{3}$ | (0.43,0.57) | (0.58,0.42) | (0.41,0.59) | (0.66,0.34) |

${Y}_{4}$ | (0.68,0.32) | (0.32,0.68) | (0.64,0.36) | (0.15,0.85) |

${Y}_{5}$ | (0.37,0.63) | (0.63,0.37) | (0.52,0.48) | (0.27,0.73) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.63,0.37) | (0.45,0.55) | (0.39,0.61) | (0.53,0.47) |

${Y}_{2}$ | (0.53,0.47) | (0.37,0.63) | (0.60,0.40) | (0.59,0.41) |

${Y}_{3}$ | (0.47,0.53) | (0.29,0.71) | (0.52,0.48) | (0.27,0.73) |

${Y}_{4}$ | (0.41,0.59) | (0.53,0.47) | (0.56,0.44) | (0.19,0.81) |

${Y}_{5}$ | (0.33,0.67) | (0.48,0.52) | (0.54,0.46) | (0.21,0.79) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.4745,0.5255) | (0.4752,0.5248) | (0.2981,0.7019) | (0.4373,0.5627) |

${Y}_{2}$ | (0.5346,0.4654) | (0.5532,0.4468) | (0.6300,0.3700) | (0.5901,0.4099) |

${Y}_{3}$ | (0.4324,0.5676) | (0.4030,0.5970) | (0.4298,0.5702) | (0.4361,0.5639) |

${Y}_{4}$ | (0.5235,0.4765) | (0.4808,0.5192) | (0.5667,0.4333) | (0.2913,0.7087) |

${Y}_{5}$ | (0.4168,0.5832) | (0.4923,0.5077) | (0.4732,0.5268) | (0.4477,0.5523) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | (0.4745,0.5255) | (0.5248,0.4752) | (0.2981,0.7019) | (0.4373,0.5627) |

${Y}_{2}$ | (0.5346,0.4654) | (0.4468,0.5532) | (0.6300,0.3700) | (0.5901,0.4099) |

${Y}_{3}$ | (0.4324,0.5676) | (0.5970,0.4030) | (0.4298,0.5702) | (0.4361,0.5639) |

${Y}_{4}$ | (0.5235,0.4765) | (0.5192,0.4808) | (0.5667,0.4333) | (0.2913,0.7087) |

${Y}_{5}$ | (0.4168,0.5832) | (0.5077,0.4923) | (0.4732,0.5268) | (0.4477,0.5523) |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${z}_{j}$ | $0.1410$ | $0.2263$ | $0.3234$ | $0.3093$ |

Average Solution | |
---|---|

${Z}_{1}$ | $\left(0.4785,0.5215\right)$ |

${Z}_{2}$ | $\left(0.5216,0.4784\right)$ |

${Z}_{3}$ | $\left(0.4921,0.5079\right)$ |

${Z}_{4}$ | $\left(0.4487,0.5513\right)$ |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | 0.0000 | 0.0063 | 0.0000 | 0.0000 |

${Y}_{2}$ | 0.1173 | 0.0000 | 0.2802 | 0.3150 |

${Y}_{3}$ | 0.0000 | 0.1447 | 0.0000 | 0.0000 |

${Y}_{4}$ | 0.0941 | 0.0000 | 0.1517 | 0.0000 |

${Y}_{5}$ | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | |
---|---|---|---|---|

${Y}_{1}$ | 0.0084 | 0.0000 | 0.3942 | 0.0255 |

${Y}_{2}$ | 0.0000 | 0.1433 | 0.0000 | 0.0000 |

${Y}_{3}$ | 0.0963 | 0.0000 | 0.1266 | 0.0282 |

${Y}_{4}$ | 0.0000 | 0.0044 | 0.0000 | 0.3508 |

${Y}_{5}$ | 0.1289 | 0.0265 | 0.0384 | 0.0023 |

Methods | Ranking Order | The Optimal Alternative | The Worst Alternative |
---|---|---|---|

IFWA | ${Y}_{2}>{Y}_{4}>{Y}_{3}>{Y}_{5}>{Y}_{1}$ | ${Y}_{2}$ | ${Y}_{1}$ |

IFWG | ${Y}_{2}>{Y}_{3}>{Y}_{5}>{Y}_{4}>{Y}_{1}$ | ${Y}_{2}$ | ${Y}_{1}$ |

The modified VIKOR | ${Y}_{2}>{Y}_{3}>{Y}_{4}>{Y}_{5}>{Y}_{1}$ | ${Y}_{2}$ | ${Y}_{1}$ |

The GRA method | ${Y}_{2}>{Y}_{4}>{Y}_{5}>{Y}_{1}>{Y}_{3}$ | ${Y}_{2}$ | ${Y}_{3}$ |

The developed method | ${Y}_{2}>{Y}_{5}>{Y}_{3}>{Y}_{4}>{Y}_{1}$ | ${Y}_{2}$ | ${Y}_{1}$ |

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**MDPI and ACS Style**

Liang, Y.
An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects. *Symmetry* **2020**, *12*, 484.
https://doi.org/10.3390/sym12030484

**AMA Style**

Liang Y.
An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects. *Symmetry*. 2020; 12(3):484.
https://doi.org/10.3390/sym12030484

**Chicago/Turabian Style**

Liang, Yuan.
2020. "An EDAS Method for Multiple Attribute Group Decision-Making under Intuitionistic Fuzzy Environment and Its Application for Evaluating Green Building Energy-Saving Design Projects" *Symmetry* 12, no. 3: 484.
https://doi.org/10.3390/sym12030484