# The Recalculation of the Weights of Criteria in MCDM Methods Using the Bayes Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, A

_{2}, ..., A

_{n}or to arrange them according to their importance for the defined purpose. This may be the choice of the best technological process out of the suggested versions, the comparative evaluation of economic, social or ecological situations in particular states or their regions, as well as the performance of banks and enterprises, and the solution of many other similar problems. The MCDM methods are based on using a decision-making matrix

**R**= ‖r

_{ij}‖ of the values r

_{ij}of the criteria R

_{1}, R

_{2}, ..., R

_{m}, describing the considered process, and the vector

**Ω**= (ω

_{j}) of the significances of these criteria, i.e., their ω

_{j}, where i = 1, 2, ..., n; j = 1, 2, ..., m; m is the number of criteria and n is the number of the considered alternatives. The values of the criteria r

_{ij}can be represented by the statistical data, the estimates assigned by experts and the values of technological or technical characteristics of the considered process. The influence of the criteria on this process and their importance differ to some extent. However, the main idea of the criterion weight evaluation is that, in fact, the most important criterion is assigned the largest weight in any method used for criterion weight evaluation. The obtained weights are usually normalized as follows: ${{\displaystyle \sum}}_{j=1}^{m}{\omega}_{j}=1$.

_{ij}and their weights ω

_{j}for obtaining the standard of evaluation, which is the criterion of the method. This idea is successfully realized by using the SAW (Simple Additive Weighing) method. The alternatives performance level S

_{i}is calculated by [1]:

_{j}is the weight of the j-th criteria and $\tilde{{r}_{ij}}$ is the normalized value of the j-th criteria for the i-th alternative [4,5].

## 2. Integrating the Values of the Weights of the Criteria

## 3. The Methods Used for Determining the Weights of the Criteria

#### 3.1. The Method of Fuzzy Analytic Hierarchy Process (FAHP)

#### 3.2. The Method Based on Using the Aggregate Objective Weights (IDOCRIW)

_{j}and weights q

_{j}of the criterion impact loss methods as well as connecting them to the objective weights of criteria for assessing the weights ω

_{j}of the structure of the array:

## 4. The Applied MCDM Methods

## 5. Expert Evaluation of Distance Learning Courses of Studies

#### 5.1. Description of the Considered Criteria

#### 5.2. Determining the Subjective Weights

#### 5.3. The Calculation of the Objective Weights

#### 5.4. The Recalculation of the Values of the Subjective Weights, Assigned by the Teachers, When the Estimates of the Criteria Weights Awarded by the Students Are Obtained

## 6. Evaluating the Quality of the Course of Studies by MCDM Methods Based on the Comparative Analysis

#### 6.1. Method of Calculation 1

#### 6.2. Method of Calculation 2

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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Criterion | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

Criterion 1 | 1.0000 | 2.0000 | 0.5000 | 0.2500 | 3.0000 | 5.0000 | 4.0000 |

Criterion 2 | 0.5000 | 1.0000 | 0.3333 | 0.2500 | 2.0000 | 4.0000 | 3.0000 |

Criterion 3 | 2.0000 | 3.0000 | 1.0000 | 0.5000 | 4.0000 | 6.0000 | 5.0000 |

Criterion 4 | 4.0000 | 4.0000 | 2.0000 | 1.0000 | 5.0000 | 7.0000 | 6.0000 |

Criterion 5 | 0.3333 | 0.5000 | 0.2500 | 0.2000 | 1.0000 | 3.0000 | 2.0000 |

Criterion 6 | 0.2000 | 0.2500 | 0.1667 | 0.1429 | 0.3333 | 1.0000 | 0.5000 |

Criterion 7 | 0.2500 | 0.3333 | 0.2000 | 0.1667 | 0.5000 | 2.0000 | 1.0000 |

Criterion | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Criterion 1 | 1 | 1 | 1 | 0.3 | 1.5 | 3.0 | 0.3 | 1.7 | 4.0 | 0.2 | 0.9 | 3.0 | 1.0 | 3.3 | 7.0 | 2.0 | 4.8 | 6.0 | 0.3 | 3.4 | 5.0 |

Criterion 2 | 0.3 | 0.6 | 3 | 1 | 1 | 1 | 0.3 | 1.4 | 4.0 | 0.2 | 0.7 | 2.0 | 0.3 | 3.0 | 6.0 | 3.0 | 4.6 | 7.0 | 0.3 | 3.4 | 6.0 |

Criterion 3 | 0.3 | 0.6 | 4 | 0.3 | 0.7 | 3 | 1 | 1 | 1 | 0.3 | 0.6 | 2.0 | 0.3 | 3.2 | 5.0 | 3.0 | 4.8 | 7.0 | 0.5 | 3.3 | 6.0 |

Criterion 4 | 0.3 | 1.1 | 5 | 0.5 | 1.3 | 5 | 0.5 | 1.7 | 4 | 1 | 1 | 1 | 1.0 | 4.3 | 6.0 | 4.0 | 6.1 | 7.0 | 2.0 | 4.3 | 6.0 |

Criterion 5 | 0.1 | 0.3 | 1 | 0.2 | 0.3 | 3 | 0.2 | 0.3 | 3 | 0.2 | 0.2 | 1 | 1 | 1 | 1 | 0.3 | 2.8 | 6.0 | 0.2 | 1.9 | 5.0 |

Criterion 6 | 0.2 | 0.2 | 0.5 | 0.1 | 0.2 | 0.3 | 0.1 | 0.2 | 0.3 | 0.1 | 0.2 | 0.3 | 0.2 | 0.4 | 3 | 1 | 1 | 1 | 0.2 | 0.7 | 2.0 |

Criterion 7 | 0.2 | 0.3 | 3 | 0.2 | 0.3 | 4 | 0.2 | 0.3 | 2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.5 | 5 | 0.5 | 1.5 | 6 | 1 | 1 | 1 |

**Table 3.**The matrix of the values ${\tilde{S}}_{j\text{}}=({l}_{j}$,${m}_{j},{u}_{j})$ for teachers.

Criterion | ${\mathit{l}}_{\mathit{j}}$ | ${\mathit{m}}_{\mathit{j}}$ | ${\mathit{u}}_{\mathit{j}}$ |
---|---|---|---|

Criterion 1 | 0.0303 | 0.2089 | 0.9056 |

Criterion 2 | 0.0323 | 0.1883 | 0.9056 |

Criterion 3 | 0.0331 | 0.1781 | 0.8744 |

Criterion 4 | 0.0553 | 0.2504 | 1.0618 |

Criterion 5 | 0.0131 | 0.0865 | 0.6246 |

Criterion 6 | 0.0114 | 0.0360 | 0.2316 |

Criterion 7 | 0.0142 | 0.0520 | 0.6714 |

${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ | ${\mathit{\omega}}_{4}$ | ${\mathit{\omega}}_{5}$ | ${\mathit{\omega}}_{6}$ | ${\mathit{\omega}}_{7}$ |
---|---|---|---|---|---|---|

0.1647 | 0.1610 | 0.1587 | 0.1728 | 0.1341 | 0.0780 | 0.1307 |

Criterion | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

Criterion 1 | 1.00 | 0.50 | 2.00 | 0.20 | 3.00 | 0.33 | 0.25 |

Criterion 2 | 2.00 | 1.00 | 3.00 | 0.25 | 4.00 | 0.50 | 0.33 |

Criterion 3 | 0.50 | 0.33 | 1.00 | 0.17 | 2.00 | 0.25 | 0.20 |

Criterion 4 | 5.00 | 4.00 | 6.00 | 1.00 | 7.00 | 3.00 | 2.00 |

Criterion 5 | 0.33 | 0.25 | 0.50 | 0.14 | 1.00 | 0.20 | 0.17 |

Criterion 6 | 3.00 | 2.00 | 4.00 | 0.33 | 5.00 | 1.00 | 0.50 |

Criterion 7 | 4.00 | 3.00 | 5.00 | 0.50 | 6.00 | 2.00 | 1.00 |

Criterion | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Criterion 1 | 1 | 1 | 1 | 0.2 | 0.4 | 1.0 | 0.2 | 1.1 | 3.0 | 0.2 | 0.2 | 0.3 | 0.3 | 1.8 | 4.0 | 0.2 | 1.4 | 4.0 | 0.2 | 0.3 | 1.0 |

Criterion 2 | 1 | 2.7 | 6 | 1 | 1 | 1 | 1.0 | 2.9 | 6.0 | 0.3 | 0.6 | 1.0 | 0.3 | 4.0 | 6.0 | 0.2 | 3.3 | 7.0 | 0.3 | 1.0 | 3.0 |

Criterion 3 | 0.3 | 0.9 | 5 | 0.2 | 0.3 | 1 | 1 | 1 | 1 | 0.2 | 0.4 | 1.0 | 0.3 | 2.3 | 6.0 | 0.2 | 1.8 | 5.0 | 0.2 | 0.4 | 1.0 |

Criterion 4 | 3 | 4.5 | 6 | 1 | 1.6 | 4 | 1 | 2.7 | 6 | 1 | 1 | 1 | 2.0 | 5.0 | 7.0 | 0.5 | 4.2 | 7.0 | 0.3 | 1.6 | 4.0 |

Criterion 5 | 0.3 | 0.6 | 4 | 0.2 | 0.3 | 3 | 0.2 | 0.4 | 3 | 0.1 | 0.2 | 0.5 | 1 | 1 | 1 | 0.2 | 1.2 | 5.0 | 0.1 | 0.5 | 3.0 |

Criterion 6 | 0.3 | 0.7 | 5 | 0.1 | 0.3 | 5 | 0.2 | 0.6 | 5 | 0.1 | 0.2 | 2 | 0.2 | 0.8 | 5 | 1 | 1 | 1 | 0.1 | 0.7 | 5.0 |

Criterion 7 | 1 | 3.2 | 6 | 0.3 | 1 | 4 | 1 | 2.6 | 5 | 0.3 | 0.6 | 3 | 0.3 | 2 | 7 | 0.2 | 1.3 | 7 | 1 | 1 | 1 |

**Table 7.**The matrix of the values ${\tilde{S}}_{j\text{}}=({l}_{j}$,${m}_{j},{u}_{j})$ for students.

Criterion | ${\mathit{l}}_{\mathit{j}}$ | ${\mathit{m}}_{\mathit{j}}$ | ${\mathit{u}}_{\mathit{j}}$ |
---|---|---|---|

Criterion 1 | 0.0120 | 0.0884 | 0.5581 |

Criterion 2 | 0.0224 | 0.2215 | 1.1682 |

Criterion 3 | 0.0133 | 0.1020 | 0.7788 |

Criterion 4 | 0.0491 | 0.2966 | 1.3629 |

Criterion 5 | 0.0115 | 0.0602 | 0.7593 |

Criterion 6 | 0.0116 | 0.0630 | 1.0903 |

Criterion 7 | 0.0229 | 0.1683 | 1.2850 |

${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ | ${\mathit{\omega}}_{4}$ | ${\mathit{\omega}}_{5}$ | ${\mathit{\omega}}_{6}$ | ${\mathit{\omega}}_{7}$ |
---|---|---|---|---|---|---|

0.1201 | 0.1586 | 0.1336 | 0.1692 | 0.1270 | 0.1382 | 0.1533 |

Courses | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 | |
---|---|---|---|---|---|---|---|---|

Discrete mathematics | Estimate | 9 | 9.875 | 9.875 | 9.875 | 8.375 | 8.750 | 9.500 |

Place | 4.5 | 1 | 1 | 2 | 4.5 | 5 | 4 | |

Mathematics 2 | Estimate | 9 | 9.125 | 8.125 | 9.625 | 9.125 | 10 | 9.625 |

Place | 4.5 | 4.5 | 5 | 3.5 | 1 | 2 | 3 | |

Integral calculus | Estimate | 10 | 9.25 | 9.5 | 10 | 8.625 | 10 | 9.125 |

Place | 1 | 3 | 2 | 1 | 3 | 2 | 5 | |

Operational systems | Estimate | 9.75 | 9.75 | 9 | 9.5 | 8.75 | 10 | 9.875 |

Place | 2 | 2 | 3.5 | 5 | 2 | 2 | 1 | |

Information technology | Estimate | 9.125 | 9.125 | 9 | 9.625 | 8.375 | 9.75 | 9.75 |

Place | 3 | 4.5 | 3.5 | 3.5 | 4.5 | 4 | 2 |

Courses | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 | |
---|---|---|---|---|---|---|---|---|

Discrete mathematics | Estimate | 9.3 | 8.6 | 9.6 | 9 | 7.9 | 8.3 | 9.2 |

Place | 3 | 3 | 1 | 5 | 5 | 5 | 2 | |

Mathematics 2 | Estimate | 8.9 | 9 | 9.4 | 9.1 | 8 | 9.7 | 8.9 |

Place | 5 | 2 | 2 | 4 | 4 | 2.5 | 3 | |

Integral calculus | Estimate | 10 | 8.1 | 9 | 9.2 | 8.7 | 9.9 | 7.2 |

Place | 1 | 5 | 4 | 3 | 2 | 1 | 5 | |

Operational systems | Estimate | 9.4 | 9.1 | 9 | 9.3 | 8.8 | 9.6 | 9.8 |

Place | 2 | 1 | 4 | 1.5 | 1 | 4 | 1 | |

Information technology | Estimate | 9.1 | 8.5 | 9 | 9.3 | 8.4 | 9.7 | 8.8 |

Place | 4 | 4 | 4 | 1.5 | 3 | 2.5 | 4 |

**Table 11.**The objective weights of criteria calculated based on the estimates of the courses awarded by teachers and students.

Experts | Method | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|---|

The weights assigned by teachers | Entropy | 0.164 | 0.097 | 0.351 | 0.030 | 0.086 | 0.213 | 0.060 |

CILOS | 0.073 | 0.173 | 0.097 | 0.207 | 0.153 | 0.192 | 0.105 | |

IDOCRIW | 0.093 | 0.130 | 0.264 | 0.047 | 0.101 | 0.316 | 0.049 | |

The weights assigned by students | Entropy | 0.079 | 0.087 | 0.038 | 0.008 | 0.094 | 0.194 | 0.501 |

CILOS | 0.104 | 0.071 | 0.158 | 0.364 | 0.156 | 0.114 | 0.033 | |

IDOCRIW | 0.107 | 0.081 | 0.078 | 0.038 | 0.191 | 0.288 | 0.216 |

**Table 12.**The recalculation of the subjective weights’ values assigned by the experts based on using the Bayes’ method.

Criterion | Teachers | Students | $\mathit{\omega}\left({\mathit{R}}_{\mathit{j}}\right)\mathit{\omega}\left(\mathit{X}/{\mathit{R}}_{\mathit{j}}\right)$ | Recalculated Weights |
---|---|---|---|---|

$\mathit{\omega}\left({\mathit{R}}_{\mathit{j}}\right)$ | $\mathit{\omega}\left(\mathit{X}/{\mathit{R}}_{\mathit{j}}\right)$ | $\mathit{\omega}\left({\mathit{R}}_{\mathit{j}}/\mathit{X}\right)$ | ||

Criterion 1 | 0.16472 | 0.12010 | 0.01978 | 0.13777 |

Criterion 2 | 0.16100 | 0.15858 | 0.02553 | 0.17779 |

Criterion 3 | 0.15874 | 0.13360 | 0.02121 | 0.14770 |

Criterion 4 | 0.17276 | 0.16922 | 0.02923 | 0.20358 |

Criterion 5 | 0.13414 | 0.12697 | 0.01703 | 0.11861 |

Criterion 6 | 0.07796 | 0.13822 | 0.01078 | 0.07504 |

Criterion 7 | 0.13068 | 0.15331 | 0.02003 | 0.13951 |

- | - | - | 0.14360 | 1.00000 |

Experts | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

The weights assigned by the teachers | 0.093 | 0.130 | 0.264 | 0.047 | 0.101 | 0.316 | 0.049 |

The weights assigned by the students | 0.107 | 0.081 | 0.078 | 0.038 | 0.191 | 0.288 | 0.216 |

The recalculated weights | 0.061 | 0.064 | 0.126 | 0.011 | 0.118 | 0.556 | 0.065 |

**Table 14.**The recalculation of the values of the subjective criteria weights, taking into consideration the weights of the objective criteria, by using the Bayes’ method.

Experts | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

Subjective weights | 0.061 | 0.064 | 0.126 | 0.011 | 0.118 | 0.556 | 0.065 |

Objective weights | 0.13777 | 0.17779 | 0.14770 | 0.20358 | 0.11861 | 0.11861 | 0.13951 |

Recalculated weights | 0.0797 | 0.1079 | 0.1765 | 0.0212 | 0.1328 | 0.3958 | 0.0860 |

Courses | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 | |
---|---|---|---|---|---|---|---|---|

Discrete mathematics | Estimate | 9.15 | 9.238 | 9.738 | 9.438 | 8.138 | 8.525 | 9.350 |

Place | 3 | 3 | 1 | 5 | 5 | 5 | 2 | |

Mathematics 2 | Estimate | 8.950 | 9.063 | 8.763 | 9.363 | 8.563 | 9.850 | 9.263 |

Place | 5 | 2 | 2 | 4 | 4 | 2.5 | 3 | |

Integral calculus | Estimate | 10 | 8.675 | 9.250 | 9.600 | 8.663 | 9.95 | 8.538 |

Place | 1 | 5 | 4 | 3 | 2 | 1 | 5 | |

Operational systems | Estimate | 9.575 | 9.425 | 9.0 | 9.43 | 8.775 | 9.8 | 9.838 |

Place | 2 | 1 | 4 | 1.5 | 1 | 4 | 1 | |

Information technology | Estimate | 9.113 | 8.8135 | 9.0 | 9.463 | 8.388 | 9.725 | 9.275 |

Place | 4 | 4 | 4 | 1.5 | 3 | 2.5 | 4 |

**Table 16.**The estimates of five courses of studies obtained by using method of calculation 1 of MCDM methods.

Courses | TOPSIS | SAW-COPRAS | EDAS | The Average Estimates of the Courses | |
---|---|---|---|---|---|

Discrete mathematics | Estimate | 0.2619 | 0.1928 | 0.282 | 5 |

Place | 5 | 5 | 5 | ||

Mathematics 2 | Estimate | 0.7118 | 0.2003 | 0.637 | 3 |

Place | 3 | 4 | 3 | ||

Integral calculus | Estimate | 0.7734 | 0.2030 | 0.891 | 2 |

Place | 2 | 2 | 2 | ||

Operational systems | Estimate | 0.7787 | 0.2045 | 0.965 | 1 |

Place | 1 | 1 | 1 | ||

Information technology | Estimate | 0.7009 | 0.1994 | 0.535 | 4 |

Place | 4 | 3 | 4 |

**Table 17.**The recalculation of the values of the subjective weights of the criteria assigned by the teachers, using the Bayes’ approach, taking into consideration the values of the objective weights of criteria.

Experts | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

Subjective weights | 0.1647 | 0.1610 | 0.1587 | 0.1728 | 0.1341 | 0.0780 | 0.1307 |

Objective weights | 0.093 | 0.130 | 0.264 | 0.047 | 0.101 | 0.316 | 0.049 |

Recalculated weights | 0.117 | 0.159 | 0.320 | 0.062 | 0.104 | 0.188 | 0.049 |

**Table 18.**The evaluation of courses of studies by the teachers, using method of calculation 2 of the MCDM methods.

Courses | TOPSIS | SAW-COPRAS | EDAS | The Average Estimates of the Courses | |
---|---|---|---|---|---|

Discrete mathematics | Estimate | 0.6860 | 0.2018 | 0.700 | 3 |

Place | 2 | 3 | 3 | ||

Mathematics 2 | Estimate | 0.2886 | 0.1934 | 0.164 | 5 |

Place | 5 | 5 | 5 | ||

Integral calculus | Estimate | 0.7522 | 0.2048 | 0.849 | 1 |

Place | 1 | 1 | 1 | ||

Operational systems | Estimate | 0.5720 | 0.2028 | 0.706 | 2 |

Place | 3 | 2 | 2 | ||

Information technology | Estimate | 0.4998 | 0.1972 | 0.351 | 4 |

Place | 4 | 4 | 4 |

**Table 19.**The recalculation of the subjective weights of the criteria assigned by the students, using the Bayes’ equation, taking into consideration the objective criteria weights.

Experts | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 | Criterion 6 | Criterion 7 |
---|---|---|---|---|---|---|---|

Subjective weights | 0.1201 | 0.1586 | 0.1336 | 0.1692 | 0.1270 | 0.1382 | 0.1533 |

Objective weights | 0.107 | 0.081 | 0.078 | 0.038 | 0.191 | 0.288 | 0.216 |

Recalculated weights | 0.0920 | 0.0919 | 0.0746 | 0.0460 | 0.1736 | 0.2849 | 0.2370 |

**Table 20.**The evaluation of courses of studies by the students, using the MCDM methods, and their overall estimates obtained by using method of calculation 2.

Courses | TOPSIS | SAW-COPRAS | EDAS | The Average Estimates of the Courses | Overall Estimate (Place) of the Course | |
---|---|---|---|---|---|---|

Discrete mathematics | Estimate | 0.4977 | 0.1937 | 0.197 | 5 | 4 |

Place | 4 | 5 | 5 | |||

Mathematics 2 | Estimate | 0.6638 | 0.2008 | 0.551 | 2–3 | 5 |

Place | 2 | 2.5 | 3 | |||

Integral calculus | Estimate | 0.4239 | 0.1955 | 0.291 | 4 | 2 |

Place | 5 | 4 | 4 | |||

Operational systems | Estimate | 0.8753 | 0.2091 | 0.984 | 1 | 1 |

Place | 1 | 1 | 1 | |||

Information technology | Estimate | 0.6632 | 0.2008 | 0.547 | 2–3 | 3 |

Place | 3 | 2.5 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Vinogradova, I.; Podvezko, V.; Zavadskas, E.K.
The Recalculation of the Weights of Criteria in MCDM Methods Using the Bayes Approach. *Symmetry* **2018**, *10*, 205.
https://doi.org/10.3390/sym10060205

**AMA Style**

Vinogradova I, Podvezko V, Zavadskas EK.
The Recalculation of the Weights of Criteria in MCDM Methods Using the Bayes Approach. *Symmetry*. 2018; 10(6):205.
https://doi.org/10.3390/sym10060205

**Chicago/Turabian Style**

Vinogradova, Irina, Valentinas Podvezko, and Edmundas Kazimieras Zavadskas.
2018. "The Recalculation of the Weights of Criteria in MCDM Methods Using the Bayes Approach" *Symmetry* 10, no. 6: 205.
https://doi.org/10.3390/sym10060205