# A Modified CRITIC Method to Estimate the Objective Weights of Decision Criteria

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## Abstract

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## 1. Introduction

#### 1.1. Literature and Motivation

- Contrast intensity of decision criteriaThe contrast intensity reflects the degree of variability associated with the local scores of each criterion. The original CRITIC method uses standard deviation to measure the contrast intensity of each criterion [32]. The method ensures that a criterion with a higher contrast intensity or standard deviation is assigned with a higher weight. The logic of this scenario can be explained as follows. If a criterion’s scores show more variance from one alternative to another, this criterion is expected to provide more exciting or meaningful information [33]. Thus, from a decision-making viewpoint, more attention or weight should be given to such a criterion than to criteria with homogeneous scores.
- Conflicting relationships between decision criteriaThe alternatives considered in an MCDM problem are usually characterized by conflicting criteria [34]. Thus, it is sometimes impossible for an alternative to perfectly satisfy all the predetermined criteria [25]. For instance, it is difficult for a buyer to purchase a brand new car that has a higher engine capacity and is cheaper at the same time: generally, the higher the engine capacity, the more expensive the car. In short, a conflict between criteria represents a type of relationship that can be present between decision criteria. The CRITIC method considers such conflicting relationships by utilizing the Pearson correlation coefficient [35], which ranges between −1 and 1. When the coefficient is zero, it implies that the two criteria, ${c}_{j}$ and ${c}_{{j}^{\prime}}$, are independent of each other. Meanwhile, a negative coefficient indicates that both criteria move in an opposite direction. To be precise, as the coefficient approaches −1, the conflict between the two criteria becomes stronger. On the other hand, a positive coefficient indicates a parallel direction between both criteria. It means that two criteria with a high positive coefficient share too much redundant information. Hence, a criterion that holds high positive correlations with other criteria does not deliver any extra information [36] and is considered to play a minor role in the entire decision system. By adhering to this principle, based on certain formulas, the CRITIC method ensures that a criterion with a higher degree of conflict or a lower degree of redundancy, is assigned with a higher weight.

#### 1.2. Statement on Contributions

## 2. The Proposed D-CRITIC Method

- Normalization of the decision matrix (Step 1)The scores of different criteria are incommensurable as they are expressed in different measurement units or scales. Normalization is a process of transforming the scores into standard scales, which range between 0 and 1. In the proposed method, as a first step, we use Equation (2) for normalizing the scores available in the decision matrix.$$\overline{{x}_{ij}}=\frac{{x}_{ij}-{x}_{j}^{worst}}{{x}_{j}^{best}-{x}_{j}^{worst}},$$
- Calculate the standard deviation of each criterion (Step 2)In the second step, the standard deviation of each criterion, ${s}_{j}$, is calculated using Equation (3). Note that $\overline{{x}_{j}}$ in Equation (2) is the mean score of criterion $j$ and that $m$ is the total number of alternatives.$${s}_{j}=\sqrt{\frac{{\left({{\displaystyle \sum}}_{i=1}^{m}{x}_{ij}-\overline{{x}_{j}}\right)}^{2}}{m-1}},$$
- Calculate the distance correlation of every pair of criteria (Step 3)The main difference between the proposed D-CRITIC and the original CRITIC method can be observed in the third step. In the original CRITIC method, the conflicting relationships between criteria are captured with the help of the Pearson correlation. However, as explained in Section 1.1, the Pearson correlation has the risk of inaccurately capturing the actual relationships between criteria. More precisely, two criteria with a zero Pearson correlation coefficient may not be completely independent. Accordingly, Székely et al. [43] introduced a new correlation measure, called distance correlation, that is zero if, and only if, the criteria are independent. Therefore, in the modified D-CRITIC method, the distance correlation is used as an alternative way to model the relationships, with the aim of minimizing the possible error in the final weights. Equation (4) defines the distance correlation between ${c}_{j}$ and ${c}_{{j}^{\prime}}$.$$dCor\left({c}_{j},{c}_{{j}^{\prime}}\right)=\frac{dCov\left({c}_{j},{c}_{{j}^{\prime}}\right)}{sqrt\left(dVar\left({c}_{j}\right)dVar\left({c}_{{j}^{\prime}}\right)\right)},$$
- Step 3.1—Construct the Euclidean distance matrix of ${c}_{j}$ based on its scores associated with all the alternatives under consideration. Construct a similar matrix for ${c}_{{j}^{\prime}}$.
- Step 3.2—Perform the following double-centring steps on each matrix, so that the row means, column means, and the overall mean of the elements in each matrix become zero: Deduct the row mean from each element; in the result, deduct the column mean from each element; in the result, add the matrix mean to each element.
- Step 3.3—Multiply the double-centred matrices elementwise and calculate the average value of the elements from the resulting matrix, that is, the sum of elements divided by the total number of elements. The square root of this average value is the distance covariance of ${c}_{j}$ and ${c}_{{j}^{\prime}}$, that is, $dCov\left({c}_{j},{c}_{{j}^{\prime}}\right)$.
- Step 3.4—Compute the distance variance of ${c}_{j}$, $dVar\left({c}_{j}\right),$ and the distance variance of ${c}_{{j}^{\prime}}$, $dVar\left({c}_{{j}^{\prime}}\right)$. Since $dVar\left({c}_{j}\right)=dCov\left({c}_{j},{c}_{j}\right)$ and $dVar\left({c}_{{j}^{\prime}}\right)=dCov\left({c}_{{j}^{\prime}},{c}_{{j}^{\prime}}\right)$, these two values can be computed by repeating Steps 3.1–3.4.
- Step 3.5—The available $dCov\left({c}_{j},{c}_{{j}^{,}}\right)$, $dVar\left({c}_{j}\right)$, and aaaaa are substituted into Equation (4) to determine the distance correlation between ${c}_{j}$ and ${c}_{{j}^{\prime}}$, that is, $dCor\left({c}_{j},{c}_{{j}^{\prime}}\right)$.

- d.
- Compute the information content (Step 4)The amount of information contained in criterion $j$ is calculated by applying Equation (5).$${I}_{j}={s}_{j}{{\displaystyle \sum}}_{{j}^{\prime}=1}^{n}(1-dCor\left({c}_{j},{c}_{{j}^{\prime}}\right)),$$
- e.
- Determine the objective weights (Step 5)The objective weight of criterion $j$ is determined using Equation (6).$${w}_{j}=\frac{{I}_{j}}{{{\displaystyle \sum}}_{j=1}^{n}{I}_{j}},$$

## 3. Application of D-CRITIC to a Decision Problem

- The standard deviation values, which are relatively close to each other, do not show a clear distinction in terms of their contrast intensity, so we are unable to make a concrete decision about the importance of the criteria.
- The relationships held by the criteria are yet to be considered.

## 4. Comparison Analysis

#### 4.1. Distance Correlation Test

#### 4.2. Spearman Rank-Order Correlation Test

#### 4.3. sMAPE Test

## 5. Sensitivity Analysis

## 6. Discussion and Conclusions

## 7. Limitations and Recommendations

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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Model/Criterion | ${\mathit{c}}_{1}$ (Nonbeneficial) | ${\mathit{c}}_{2}$ (Beneficial) | ${\mathit{c}}_{3}$ (Beneficial) | ${\mathit{c}}_{4}$ (Nonbeneficial) | ${\mathit{c}}_{5}$ (Nonbeneficial) |
---|---|---|---|---|---|

Model A | 649 | 4.7 | 326 | 7.1 | 143 |

Model B | 749 | 5.5 | 401 | 7.3 | 192 |

Model C | 740 | 5.7 | 520 | 7.6 | 171 |

Model D | 400 | 5.7 | 520 | 11.1 | 179 |

Model E | 600 | 5.5 | 538 | 8.9 | 152 |

Model/Criterion | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ |
---|---|---|---|---|---|

A | 0.2865 | 0 | 0 | 1 | 1 |

B | 0 | 0.8000 | 0.3538 | 0.9500 | 0 |

C | 0.0258 | 1 | 0.9151 | 0.8750 | 0.4286 |

D | 1 | 1 | 0.9151 | 0 | 0.2653 |

E | 0.4269 | 0.8000 | 1 | 0.5500 | 0.8163 |

Standard deviation | 0.4062 | 0.4147 | 0.4394 | 0.4161 | 0.4063 |

Criterion | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ |
---|---|---|---|---|---|

${c}_{1}$ | 1 | 0.4777 | 0.5114 | 0.9437 | 0.6229 |

${c}_{2}$ | 0.4777 | 1 | 0.8465 | 0.5499 | 0.7564 |

${c}_{3}$ | 0.5114 | 0.8465 | 1 | 0.6957 | 0.6043 |

${c}_{4}$ | 0.9437 | 0.5499 | 0.6957 | 1 | 0.5027 |

${c}_{5}$ | 0.6229 | 0.7564 | 0.6043 | 0.5027 | 1 |

Criterion | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ |
---|---|---|---|---|---|

Information content | 0.5867 | 0.5680 | 0.5898 | 0.5442 | 0.6149 |

Weight | 0.2021 | 0.1956 | 0.2031 | 0.1874 | 0.2118 |

Criterion | Entropy | CILOS | IDOCRIW | CRITIC | D-CRITIC | |||||
---|---|---|---|---|---|---|---|---|---|---|

Weight | Rank | Weight | Rank | Weight | Rank | Weight | Rank | Weight | Rank | |

${c}_{1}$ | 0.3481 | 1 | 0.0738 | 5 | 0.1465 | 3 | 0.1872 | 3 | 0.2021 | 3 |

${c}_{2}$ | 0.1360 | 5 | 0.3864 | 1 | 0.2996 | 2 | 0.1838 | 4 | 0.1956 | 4 |

${c}_{3}$ | 0.1690 | 3 | 0.0997 | 4 | 0.0960 | 5 | 0.1691 | 5 | 0.2031 | 2 |

${c}_{4}$ | 0.1463 | 4 | 0.1467 | 3 | 0.1223 | 4 | 0.2599 | 1 | 0.1874 | 5 |

${c}_{5}$ | 0.2006 | 2 | 0.2934 | 2 | 0.3355 | 1 | 0.2000 | 2 | 0.2118 | 1 |

Method | Entropy | CILOS | IDOCRIW | CRITIC | D-CRITIC |
---|---|---|---|---|---|

Entropy | 1 | 0.5897 | 0.4609 | 0.4658 | 0.6336 |

CILOS | 0.5897 | 1 | 0.9527 | 0.4968 | 0.5975 |

IDOCRIW | 0.4609 | 0.9527 | 1 | 0.5057 | 0.5919 |

CRITIC | 0.4658 | 0.4968 | 0.5057 | 1 | 0.8186 |

D-CRITIC | 0.6336 | 0.5975 | 0.5919 | 0.8186 | 1 |

Average | 0.6300 | 0.7273 | 0.7022 | 0.6574 | 0.7283 |

Method | Entropy | CILOS | IDOCRIW | CRITIC | D-CRITIC |
---|---|---|---|---|---|

Entropy | 1 | −0.7000 | 0.1000 | 0.1000 | 0.6000 |

CILOS | −0.7000 | 1 | 0.6000 | 0.1000 | −0.1000 |

IDOCRIW | 0.1000 | 0.6000 | 1 | 0.3000 | 0.3000 |

CRITIC | 0.1000 | 0.1000 | 0.3000 | 1 | −0.3000 |

D-CRITIC | 0.6000 | −0.1000 | 0.3000 | −0.3000 | 1 |

Average | 0.2200 | 0.1800 | 0.4600 | 0.2400 | 0.3000 |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ |
---|---|---|---|---|

0.1802 | 0.2375 | 0.1493 | 0.1763 | 0.2568 |

Entropy | CILOS | IDOCRIW | CRITIC | D-CRITIC |
---|---|---|---|---|

34.6994% | 40.6001% | 29.9940% | 20.9825% | 17.3267% |

Scenario | Amendment Done | Decision Matrix | Normalized Decision Matrix |
---|---|---|---|

Scenario 1 (Sc1) | Removed the data of Model B | ||

Scenario 2 (Sc2) | Removed the data of Model C | ||

Scenario 3 (Sc3) | Removed the data of Model D | ||

Scenario 4 (Sc4) | Removed the data of Model E | ||

Scenario 5 (Sc5) | Removed the data of Model A | ||

Scenario 6 (Sc6) | Duplicated the data of Model A | ||

Scenario 7 (Sc7) | Duplicated the data of Model B | ||

Scenario 8 (Sc8) | Duplicated the data of Model C | ||

Scenario 9 (Sc9) | Duplicated the data of Model D | ||

Scenario 10 (Sc10) | Duplicated the data of Model E |

Criterion | Sc1 | Sc2 | Sc3 | Sc4 | Sc5 | Sc6 | Sc7 | Sc8 | Sc9 | Sc10 |
---|---|---|---|---|---|---|---|---|---|---|

${c}_{1}$ | 0.2242 | 0.1657 | 0.2071 | 0.2309 | 0.1836 | 0.1898 | 0.1896 | 0.2125 | 0.2245 | 0.1880 |

${c}_{2}$ | 0.1897 | 0.2018 | 0.2021 | 0.1849 | 0.3058 | 0.2039 | 0.2008 | 0.1925 | 0.1805 | 0.2015 |

${c}_{3}$ | 0.2227 | 0.1868 | 0.2091 | 0.1898 | 0.1826 | 0.2114 | 0.1901 | 0.2110 | 0.1947 | 0.2146 |

${c}_{4}$ | 0.1905 | 0.1710 | 0.1999 | 0.2180 | 0.1669 | 0.1825 | 0.1857 | 0.1955 | 0.2031 | 0.1719 |

${c}_{5}$ | 0.1729 | 0.2747 | 0.1818 | 0.1765 | 0.1611 | 0.2123 | 0.2338 | 0.1886 | 0.1973 | 0.2240 |

$\mathbf{Scenarios}\text{}\mathbf{with}\text{}\mathit{m}=4$ | $\mathbf{Actual}\text{}\mathbf{scenario}\text{}\mathbf{with}\text{}\mathit{m}=5$ | $\mathbf{Scenarios}\text{}\mathbf{with}\text{}\mathit{m}=6$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Criteria | Sc1 | Sc2 | Sc3 | Sc4 | Sc5 | Actual Estimation | Sc6 | Sc7 | Sc8 | Sc9 | Sc10 |

${c}_{1}$ | 1 | 5 | 2 | 1 | 2 | 3 | 4 | 4 | 1 | 1 | 4 |

${c}_{2}$ | 4 | 2 | 3 | 4 | 1 | 4 | 3 | 2 | 4 | 5 | 3 |

${c}_{3}$ | 2 | 3 | 1 | 3 | 3 | 2 | 2 | 3 | 2 | 4 | 2 |

${c}_{4}$ | 3 | 4 | 4 | 2 | 4 | 5 | 5 | 5 | 3 | 2 | 5 |

${c}_{5}$ | 5 | 1 | 5 | 5 | 5 | 1 | 1 | 1 | 5 | 3 | 1 |

Total unaffected ranks = 4 | Total unaffected ranks = 10 |

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## Share and Cite

**MDPI and ACS Style**

Krishnan, A.R.; Kasim, M.M.; Hamid, R.; Ghazali, M.F.
A Modified CRITIC Method to Estimate the Objective Weights of Decision Criteria. *Symmetry* **2021**, *13*, 973.
https://doi.org/10.3390/sym13060973

**AMA Style**

Krishnan AR, Kasim MM, Hamid R, Ghazali MF.
A Modified CRITIC Method to Estimate the Objective Weights of Decision Criteria. *Symmetry*. 2021; 13(6):973.
https://doi.org/10.3390/sym13060973

**Chicago/Turabian Style**

Krishnan, Anath Rau, Maznah Mat Kasim, Rizal Hamid, and Mohd Fahmi Ghazali.
2021. "A Modified CRITIC Method to Estimate the Objective Weights of Decision Criteria" *Symmetry* 13, no. 6: 973.
https://doi.org/10.3390/sym13060973