# Eliciting Correlated Weights for Multi-Criteria Group Decision Making with Generalized Canonical Correlation Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Criteria Group Decision Making

#### 2.1. Extended TOPSIS for GDM

**Step 1:****Step 2:**- compute the normalized decision matrix ${\mathcal{R}}^{k}=\left[{r}_{ij}^{k}\right],i=1,\dots ,M,j=1,\dots ,{n}_{k}$, for the k-th DM. For this purpose, the vector normalization scheme is usually employed [29]:$${r}_{ij}^{k}=\frac{{x}_{ij}^{k}}{\sqrt{{\sum}_{i=1}^{M}{\left({x}_{ij}^{k}\right)}^{2}}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M;\phantom{\rule{0.277778em}{0ex}}j=1,\dots ,{n}_{k}.$$
**Step 3:**- calculate the positive ideal solution ${V}^{k+}$ and the negative ideal solution ${V}^{k-}$ for the k-th DM, as follows:$$\begin{array}{c}\hfill {V}^{k+}=\left\{{v}_{1}^{k+},\dots ,{v}_{{n}_{k}}^{k+}\right\}=\left\{\left(\underset{i}{\mathrm{max}}\left\{{r}_{ij}^{k}\right\}\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}j\in {J}_{1})\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\left(\underset{i}{\mathrm{min}}\left\{{r}_{ij}^{k}\right\}\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}j\in {J}_{2})\right\},\end{array}$$$$\begin{array}{c}\hfill {V}^{k-}=\left\{{v}_{1}^{k-},\dots ,{v}_{{n}_{k}}^{v-}\right\}=\left\{\left(\underset{i}{\mathrm{min}}\left\{{r}_{ij}^{k}\right\}\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}j\in {J}_{1})\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\left(\underset{i}{\mathrm{max}}\left\{{r}_{ij}^{k}\right\}\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}j\in {J}_{2})\right\},\end{array}$$
**Step 4:**- assign a weight vector ${w}^{k}=({w}_{1}^{k},{w}_{2}^{k},\dots ,{w}_{{n}_{k}}^{k})$ to the criteria set of the k-th DM, such that ${\sum}_{j=1}^{{n}_{k}}{w}_{j}^{k}=1$.
**Step 5:**- compute the overall separation of a given alternative from the set of positive and negative ideal solutions. Here, two substeps should be performed. The first considers the PIS and NIS that are associated with each DM separately, while the second aggregates the measurements for the whole group.
**Substep 5a:**- compute the distances ${S}_{i}^{k+}$ and ${S}_{i}^{k-}$ of the i-th alternative, $i=1,\dots ,M$, to the pair of PIS and NIS associated with the k-th DM, $k=1,\dots ,K$. Here, we have considered the Euclidean distance:$${S}_{i}^{k+}=\sqrt{{\displaystyle \sum _{j=1}^{{n}_{k}}}{w}_{j}^{k}{({r}_{ij}^{k}-{v}_{j}^{k+})}^{2}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M,$$$${S}_{i}^{k-}=\sqrt{{\displaystyle \sum _{j=1}^{{n}_{k}}}{w}_{j}^{k}{({r}_{ij}^{k}-{v}_{j}^{k-})}^{2}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M,$$
**Substep 5b:**- compute the overall separation measures $\overline{{S}_{i}^{+}}$ and $\overline{{S}_{i}^{-}}$ for each alternative. For this purpose, one should calculate the geometric mean over the K values of ${S}_{i}^{k+}$ (5) and ${S}_{i}^{k-}$ (6) to yield:$$\overline{{S}_{i}^{+}}={\left({\displaystyle \prod _{k=1}^{K}}{S}_{i}^{k+}\right)}^{\frac{1}{K}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M,$$$$\overline{{S}_{i}^{-}}={\left({\displaystyle \prod _{k=1}^{K}}{S}_{i}^{k-}\right)}^{\frac{1}{K}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M.$$

**Step 6:**- compute $\overline{{C}_{i}^{*}}$, the overall relative closeness of the i-th alternative ${A}_{i}$, $i=1,\dots ,M$, to the K positive ideal solutions, which can be expressed as:$$\overline{{C}_{i}^{*}}=\frac{\overline{{S}_{i}^{-}}}{{S}_{i}^{+}+{S}_{i}^{-}},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M,$$

#### 2.2. Objective Methods for Criteria Weighting

#### 2.2.1. Entropy Method

#### 2.2.2. Statistical Variance Method

#### 2.2.3. Standard Deviation Method

#### 2.2.4. Criteria Importance through Inter-Criteria Correlation

**Step 1:**- the score values associated with benefit/cost criteria are first normalized using Equations (17) and (18), respectively.$${r}_{ij}^{k}=\frac{{x}_{ij}^{k}-\mathrm{min}\left({x}_{ij}^{k}\right)}{\mathrm{max}\left({x}_{ij}^{k}\right)-\mathrm{min}\left({x}_{ij}^{k}\right)},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M;\phantom{\rule{1.em}{0ex}}j=1,\dots ,{n}_{k},\phantom{\rule{1.em}{0ex}}(\mathrm{benefit}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria})$$$${r}_{ij}^{k}=\frac{\mathrm{max}\left({x}_{ij}^{k}\right)-{x}_{ij}^{k}}{\mathrm{max}\left({x}_{ij}^{k}\right)-\mathrm{min}\left({x}_{ij}^{k}\right)},\phantom{\rule{1.em}{0ex}}i=1,\dots ,M;\phantom{\rule{1.em}{0ex}}j=1,\dots ,{n}_{k},\phantom{\rule{1.em}{0ex}}(\mathrm{cos}\mathrm{t}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria})$$
**Step 2:**- the correlation between each pair of criteria is calculated via Equation (19).$${\rho}_{j{j}^{\prime}}^{k}=\frac{{\displaystyle \sum _{i=1}^{M}}({r}_{ij}^{k}-\overline{{r}_{j}^{k}})({r}_{i{j}^{\prime}}^{k}-\overline{{r}_{{j}^{\prime}}^{k}})}{\sqrt{{\displaystyle \sum _{i=1}^{M}}{({r}_{ij}-\overline{{r}_{j}^{k}})}^{2}}\xb7\sqrt{{\displaystyle \sum _{i=1}^{M}}{({r}_{i{j}^{\prime}}^{k}-\overline{{r}_{{j}^{\prime}}^{k}})}^{2}}},\phantom{\rule{1.em}{0ex}}j,{j}^{\prime}=1,\dots ,{n}_{k}$$
**Step 3:**- finally, Equations (20) and (21) are employed for producing the weights.$${w}_{j}^{k}=\frac{{c}_{j}^{k}}{{\displaystyle \sum _{{j}^{\prime}=1}^{{n}_{k}}}{c}_{{j}^{\prime}}},\phantom{\rule{1.em}{0ex}}j=1,\dots ,{n}_{k},$$$${c}_{j}^{k}={\sigma}_{j}^{k}{\displaystyle \sum _{j=1}^{{n}_{k}}}(1-{\rho}_{jk}^{k}),\phantom{\rule{1.em}{0ex}}j=1,\dots ,{n}_{k}.$$

#### 2.3. DEMATEL

#### 2.4. DEMATEL-Based ANP (DANP)

## 3. Generalized Canonical Correlation Analysis

## 4. A Canonical Multi-Criteria Group Decision Making Approach

## 5. Example on Human Resource Selection

**Test #1:**- the best alternative selected should not alter when a non-optimal alternative is added or removed from the problem (assuming that the relative importance of each criterion remains unchanged) [54].
**Test #2:****Test #3:**

## 6. Example on Machine Acquisition

## 7. Simulated Cases

- number of DMs: $\{5,7,10\}$;
- number of criteria: 7;
- number of alternatives: $\{17,19,21,23,25,27,30\}$;
- scores of the alternatives: randomly generated by a uniform distribution in the range [0–100];
- criteria weighting approach: GCCA and the methods described in Section 2.2;
- number of trials: 100 for each parameter configuration, thereby yielding 6300 different decision problem instances; and,
- performance criteria: the three irregularity tests described in Section 5.

## 8. Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Fictitious multi-criteria group decision making (MCGDM) problem—non-normalized generalized version of canonical correlation analysis (GCCA) weights.

Criteria | DM1 | DM2 | DM3 | DM4 |
---|---|---|---|---|

C1 | −0.0116 | −0.0081 | −0.0105 | 0.0071 |

C2 | −0.0122 | −0.0063 | 0.0037 | 0.0033 |

C3 | −0.0049 | −0.0123 | −0.0053 | −0.0184 |

C4 | −0.0123 | −0.0132 | 0.0015 | −0.0167 |

C5 | 0.0148 | 0.0157 | −0.0229 | 0.0004 |

Criteria | DM1 | DM2 | DM3 | DM4 |
---|---|---|---|---|

C1 | 0.0218 | 0.1227 | 0.1528 | 0.3769 |

C2 | 0.0025 | 0.1652 | 0.3284 | 0.3201 |

C3 | 0.2107 | 0.0206 | 0.2173 | 0.0000 |

C4 | 0.0000 | 0.0000 | 0.3014 | 0.0250 |

C5 | 0.7650 | 0.6915 | 0.0000 | 0.2780 |

**Table 3.**Decision matrix (objective criteria) for the first example, adapted from ([26] [Table 6a]).

No. | Candidates | Objective Criteria | ||||
---|---|---|---|---|---|---|

Knowledge Tests | Skill Tests | |||||

Language Test | Professional Test | Safety Rule Test | Professional Skills | Computer Skills | ||

1 | James B. Wang | 80 | 70 | 87 | 77 | 76 |

2 | Carol L. Lee | 85 | 65 | 76 | 80 | 75 |

3 | Kenney C. Wu | 78 | 90 | 72 | 80 | 75 |

4 | Robert M. Liang | 75 | 84 | 69 | 85 | 65 |

5 | Sophia M. Cheng | 84 | 67 | 60 | 75 | 85 |

6 | Lily M. Pai | 85 | 78 | 82 | 81 | 79 |

7 | Abon C. Hsieh | 77 | 83 | 74 | 70 | 71 |

8 | Frank K. Yang | 78 | 82 | 72 | 80 | 78 |

9 | Ted C. Yang | 85 | 90 | 80 | 88 | 90 |

10 | Sue B. Ho | 89 | 75 | 79 | 67 | 77 |

11 | Vincent C. Chen | 65 | 55 | 68 | 62 | 70 |

12 | Rosemary I. Lin | 70 | 64 | 65 | 65 | 60 |

13 | Ruby J. Huang | 95 | 80 | 70 | 75 | 70 |

14 | George K. Wu | 70 | 80 | 79 | 80 | 85 |

15 | Philip C. Tsai | 60 | 78 | 87 | 70 | 66 |

16 | Michael S. Liao | 92 | 85 | 88 | 90 | 85 |

17 | Michelle C. Lin | 86 | 87 | 80 | 70 | 72 |

**Table 4.**Decision matrix (subjective criteria) for the first example, adapted from ([26] [Table 6b]).

No. | Subjective Criteria | |||||||
---|---|---|---|---|---|---|---|---|

DM #1 | DM #2 | DM #3 | DM #4 | |||||

Panel Interview | 1-on-1 Interview | Panel Interview | 1-on-1 Interview | Panel Interview | 1-on-1 Interview | Panel Interview | 1-on-1 Interview | |

1 | 80 | 75 | 85 | 80 | 75 | 70 | 90 | 85 |

2 | 65 | 75 | 60 | 70 | 70 | 77 | 60 | 70 |

3 | 90 | 85 | 80 | 85 | 80 | 90 | 90 | 95 |

4 | 65 | 70 | 55 | 60 | 68 | 72 | 62 | 72 |

5 | 75 | 80 | 75 | 80 | 50 | 55 | 70 | 75 |

6 | 80 | 80 | 75 | 85 | 77 | 82 | 75 | 75 |

7 | 65 | 70 | 70 | 60 | 65 | 72 | 67 | 75 |

8 | 70 | 60 | 75 | 65 | 75 | 67 | 82 | 85 |

9 | 80 | 85 | 95 | 85 | 90 | 85 | 90 | 92 |

10 | 70 | 75 | 75 | 80 | 68 | 78 | 65 | 70 |

11 | 50 | 60 | 62 | 65 | 60 | 65 | 65 | 70 |

12 | 60 | 65 | 65 | 75 | 50 | 60 | 45 | 50 |

13 | 75 | 75 | 80 | 80 | 65 | 75 | 70 | 75 |

14 | 80 | 70 | 75 | 72 | 80 | 70 | 75 | 75 |

15 | 70 | 65 | 75 | 70 | 65 | 70 | 60 | 65 |

16 | 90 | 95 | 92 | 90 | 85 | 80 | 88 | 90 |

17 | 80 | 85 | 70 | 75 | 75 | 80 | 70 | 75 |

**Table 5.**Original criteria weights for the first example, adapted from ([26] [Table 6b]).

No. | Criteria | The Weights of the Group | |||
---|---|---|---|---|---|

DM #1 | DM #2 | DM #3 | DM #4 | ||

Knowledge tests | |||||

1 | Language test | 0.066 | 0.042 | 0.060 | 0.047 |

2 | Professional test | 0.196 | 0.112 | 0.134 | 0.109 |

3 | Safety rule test | 0.066 | 0.082 | 0.051 | 0.037 |

Skill tests | |||||

4 | Professional skills | 0.130 | 0.176 | 0.167 | 0.133 |

5 | Computer skills | 0.130 | 0.118 | 0.100 | 0.081 |

Interviews | |||||

6 | Panel interview | 0.216 | 0.215 | 0.203 | 0.267 |

7 | 1-on-1 interview | 0.196 | 0.215 | 0.285 | 0.326 |

Sum | 1 | 1 | 1 | 1 |

**Table 6.**Criteria weights (non-normalized values in parentheses) for the first example, as elicited by GCCA.

No. | Criteria | The Weights of the Group | |||
---|---|---|---|---|---|

DM #1 | DM #2 | DM #3 | DM #4 | ||

Knowledge tests | |||||

1 | Language test | 0.2665 (0.0003) | 0.1661 (−0.0001) | 0.1668 (0.0014) | 0.1251 (0.0005) |

2 | Professional test | 0.0000 (−0.0119) | 0.0383 (−0.0092) | 0.0156 (−0.0107) | 0.0104 (−0.0109) |

3 | Safety rule test | 0.2368 (−0.0011) | 0.0626 (−0.0075) | 0.1927 (0.0035) | 0.1827 (0.0062) |

Skill tests | |||||

4 | Professional skills | 0.2057 (−0.0025) | 0.1885 (0.0015) | 0.0875 (−0.0049) | 0.1470 (0.0027) |

5 | Computer skills | 0.0503 (−0.0096) | 0.2984 (0.0093) | 0.0798 (−0.0056) | 0.2033 (0.0083) |

Interviews | |||||

6 | Panel interview | 0.1266 (−0.0061) | 0.2101 (0.0030) | 0.2985 (0.0119) | 0.2157 (0.0095) |

7 | 1-on-1 interview | 0.1142 (−0.0067) | 0.0360 (−0.0094) | 0.1590 (0.0008) | 0.1158 (−0.0004) |

Sum | 1 | 1 | 1 | 1 |

**Table 7.**Assessment of the extended TOPSIS [26] on the ranking irregularity tests—first example.

Extended TOPSIS | Test #1–Addition | Test #1–Removal | Test #2 | ||||
---|---|---|---|---|---|---|---|

Rank | Score | Rank | Score | Rank | Score | Rank | Score |

A16 | 0.8960 | A16 | 0.8956 | A16 | 0.8963 | A16 | 0.8960 |

A9 | 0.8797 | A9 | 0.8797 | A9 | 0.8797 | A9 | 0.8797 |

A3 | 0.7860 | A3 | 0.7862 | A3 | 0.7859 | A3 | 0.7860 |

A6 | 0.6611 | A6 | 0.6611 | A6 | 0.6611 | A6 | 0.6611 |

A1 | 0.6272 | A1 | 0.6259 | A14 | 0.5925 | A14 | 0.5924 |

A14 | 0.5924 | A1 | 0.6259 | A17 | 0.5915 | A17 | 0.5919 |

A17 | 0.5920 | A17 | 0.5925 | A8 | 0.5700 | A8 | 0.5701 |

A8 | 0.5701 | A14 | 0.5924 | A13 | 0.5565 | A13 | 0.5568 |

A13 | 0.5568 | A8 | 0.5701 | A10 | 0.5079 | A10 | 0.5080 |

A10 | 0.5080 | A13 | 0.5571 | A5 | 0.4660 | A5 | 0.4660 |

A5 | 0.4660 | A10 | 0.5082 | A7 | 0.4516 | A4 | 0.4524 |

A4 | 0.4527 | A5 | 0.4659 | A4 | 0.4514 | A7 | 0.4522 |

A7 | 0.4523 | A4 | 0.4538 | A2 | 0.4401 | A2 | 0.4404 |

A2 | 0.4404 | A7 | 0.4530 | A15 | 0.4092 | A1 | 0.4224 |

A15 | 0.4091 | A2 | 0.4406 | A11 | 0.2101 | A15 | 0.4091 |

A11 | 0.2097 | A15 | 0.4091 | A12 | 0.1673 | A11 | 0.2097 |

A12 | 0.1678 | A11 | 0.2093 | ---- | ---- | A12 | 0.1677 |

---- | ---- | A12 | 0.1682 | ---- | ---- | ---- | ---- |

CMCGDM | Test #1–Addition | Test #1–Removal | Test #2 | ||||
---|---|---|---|---|---|---|---|

Rank | Score | Rank | Score | Rank | Score | Rank | Score |

A16 | 0.9016 | A16 | 0.9013 | A16 | 0.9018 | A16 | 0.9016 |

A9 | 0.8797 | A9 | 0.8799 | A9 | 0.8794 | A9 | 0.8796 |

A3 | 0.7230 | A3 | 0.7231 | A3 | 0.7228 | A3 | 0.7230 |

A6 | 0.6720 | A6 | 0.6719 | A6 | 0.6722 | A6 | 0.6721 |

A14 | 0.6608 | A14 | 0.6608 | A14 | 0.6608 | A14 | 0.6608 |

A1 | 0.5964 | A1 | 0.5954 | A8 | 0.5854 | A8 | 0.5855 |

A8 | 0.5855 | A1 | 0.5954 | A5 | 0.5491 | A5 | 0.5495 |

A5 | 0.5495 | A8 | 0.5855 | A2 | 0.5170 | A2 | 0.5173 |

A2 | 0.5173 | A5 | 0.5499 | A17 | 0.4813 | A17 | 0.4811 |

A17 | 0.4811 | A2 | 0.5176 | A13 | 0.4778 | A13 | 0.4779 |

A13 | 0.4779 | A17 | 0.4810 | A4 | 0.4699 | A4 | 0.4705 |

A4 | 0.4706 | A13 | 0.4781 | A10 | 0.4634 | A10 | 0.4632 |

A10 | 0.4632 | A4 | 0.4713 | A7 | 0.3956 | A7 | 0.3957 |

A7 | 0.3957 | A10 | 0.4630 | A15 | 0.3627 | A1 | 0.3627 |

A15 | 0.3620 | A7 | 0.3958 | A11 | 0.2259 | A15 | 0.3621 |

A11 | 0.2254 | A15 | 0.3615 | A12 | 0.1408 | A11 | 0.2255 |

A12 | 0.1409 | A11 | 0.2250 | ---- | ---- | A12 | 0.1409 |

---- | ---- | A12 | 0.1410 | ---- | ---- | ---- | ---- |

**Table 9.**Decision matrix associated with DM #1 for the second example, adapted from ([21] [Table 2]).

C1 | C2 | C3 | C4 | C5 | C6 | |
---|---|---|---|---|---|---|

A1 | 0.05 | 500 | 850 | 8.6 | 30,500 | 1.5 |

A2 | 0.01 | 550 | 925 | 8.2 | 26,500 | 2.0 |

A3 | 0.008 | 600 | 960 | 9.0 | 28,500 | 3.0 |

A4 | 0.008 | 450 | 720 | 9.2 | 25,800 | 2.0 |

A5 | 0.015 | 400 | 650 | 8.0 | 24,000 | 1.5 |

A6 | 0.012 | 480 | 710 | 8.4 | 23,500 | 1.0 |

A7 | 0.017 | 505 | 691 | 8.0 | 30,100 | 1.3 |

**Table 10.**Decision matrix associated with DM #2 for the second example, adapted from ([21] [Table 2]).

C5 | C6 | C7 | C8 | |
---|---|---|---|---|

A1 | 30,500 | 1.5 | 530,000 | 50 |

A2 | 26,500 | 2.0 | 420,000 | 40 |

A3 | 28,500 | 3.0 | 450,000 | 35 |

A4 | 25,800 | 2.0 | 480,000 | 40 |

A5 | 24,000 | 1.5 | 380,000 | 30 |

A6 | 23,500 | 1.0 | 40,000 | 60 |

A7 | 30,100 | 1.3 | 392,000 | 57 |

**Table 11.**Assessment of the extended TOPSIS [26] on the ranking irregularity tests—second example.

Extended TOPSIS | Test #1–Addition | Test #1–Removal | Test #2 | ||||
---|---|---|---|---|---|---|---|

Rank | Score | Rank | Score | Rank | Score | Rank | Score |

A1 | 1.0000 | A1 | 1.0000 | A1 | 1.0000 | A1 | 1.0000 |

A3 | 0.5892 | A3 | 0.5816 | A3 | 0.5984 | A3 | 0.5908 |

A4 | 0.5265 | A4 | 0.5189 | A4 | 0.5357 | A4 | 0.5281 |

A2 | 0.4741 | A7 | 0.4682 | A7 | 0.4704 | A7 | 0.4694 |

A7 | 0.4691 | A2 | 0.4667 | A5 | 0.4011 | A5 | 0.4004 |

A5 | 0.4002 | A2 | 0.4667 | A6 | 0.0000 | A2 | 0.3748 |

A6 | 0.0000 | A5 | 0.3996 | ---- | ---- | A6 | 0.0000 |

---- | ---- | A6 | 0.0000 | ---- | ---- | ---- | ---- |

CMCGDM | Test #1–Addition | Test #1–Removal | Test #2 | ||||
---|---|---|---|---|---|---|---|

Rank | Score | Rank | Score | Rank | Score | Rank | Score |

A3 | 0.8455 | A3 | 0.8425 | A3 | 0.8521 | A3 | 0.8478 |

A4 | 0.5798 | A4 | 0.5784 | A4 | 0.5800 | A4 | 0.5798 |

A2 | 0.5582 | A2 | 0.5581 | A1 | 0.4529 | A2 | 0.4631 |

A1 | 0.4578 | A2 | 0.5581 | A5 | 0.3574 | A1 | 0.4577 |

A5 | 0.3600 | A1 | 0.4627 | A7 | 0.2919 | A5 | 0.3581 |

A7 | 0.2965 | A5 | 0.3581 | A6 | 0.0000 | A7 | 0.2936 |

A6 | 0.0000 | A7 | 0.2942 | ---- | ---- | A6 | 0.0000 |

---- | ---- | A6 | 0.0000 | ---- | ---- | ---- | ---- |

**Table 13.**Assessment of the different criteria weighting methods on the ranking irregularity tests—simulated cases.

Test #3 | Test #3 | Test #3 | |||||
---|---|---|---|---|---|---|---|

Dist. | Method | Addition | Removal | Replacement | Test #1 | Test #2 | Total Test #3 |

uniform | GCCA | 11,737 | 13,226 | 2727 | 231 | 11 | 27,690 |

Entropy | 12,504 | 12,753 | 2640 | 246 | 31 | 27,897 | |

Std | 12,504 | 12,753 | 2640 | 246 | 31 | 27,897 | |

DEMATEL | 12,741 | 12,837 | 2626 | 252 | 17 | 28,204 | |

DEMATEL-ANP | 12,145 | 13,592 | 2766 | 283 | 25 | 28,503 | |

CRITIC | 12,411 | 13,276 | 2869 | 218 | 16 | 28,556 | |

VarProc | 12,366 | 13,313 | 3059 | 183 | 23 | 28,738 |

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## Share and Cite

**MDPI and ACS Style**

Santos, F.J.d.; Coelho, A.L.V.
Eliciting Correlated Weights for Multi-Criteria Group Decision Making with Generalized Canonical Correlation Analysis. *Symmetry* **2020**, *12*, 1612.
https://doi.org/10.3390/sym12101612

**AMA Style**

Santos FJd, Coelho ALV.
Eliciting Correlated Weights for Multi-Criteria Group Decision Making with Generalized Canonical Correlation Analysis. *Symmetry*. 2020; 12(10):1612.
https://doi.org/10.3390/sym12101612

**Chicago/Turabian Style**

Santos, Francisco J. dos, and André L. V. Coelho.
2020. "Eliciting Correlated Weights for Multi-Criteria Group Decision Making with Generalized Canonical Correlation Analysis" *Symmetry* 12, no. 10: 1612.
https://doi.org/10.3390/sym12101612