# Impact of Normalization on Entropy-Based Weights in Hellwig’s Method: A Case Study on Evaluating Sustainable Development in the Education Area

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## Abstract

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## 1. Introduction

- Compare the performance of two variants of entropy-based weight methods in assessing sustainable development in education.
- Evaluate the effectiveness of three normalization formulas in Hellwig’s method for assessing sustainable development in education.
- Investigate and compare the combined performance of entropy-based weight methods and normalization within Hellwig’s method for assessing sustainable development in education.
- Conduct the simulation study by modifying Eurostat data to discuss and investigate the sensitivity of the obtained results and provide more general conclusions regarding the influence of normalization on entropy-based weights in Hellwig’s approach.

## 2. The Preliminaries and Literature Review

#### 2.1. Entropy-Based Weight Method

- Step 1. Determination of decision matrix.

- Step 2 (optional). Normalization of decision matrix.

- Step 3. Calculation of the information entropy of each criterion.

- Step 4. Calculation of weights.

#### 2.2. Literature Review

## 3. The Hellwig’s Method

## 4. A Case Study: Evaluation of Sustainable Development in the Education Area by Hellwig’s Framework

#### 4.1. The Source of Data

- ${C}_{1}:$Early leavers from education and training (%) [sdg_04_10a] (destimulant)
- ${C}_{2}$: Tertiary educational attainment (%) [sdg_04_20] (stimulant)
- ${C}_{3}$: Participation in early childhood education (%) [sdg_04_31] (stimulant)
- ${C}_{4}$: Adult participation in learning in the past four weeks (%) [sdg_04_60] (stimulant)
- ${C}_{5}$: Share of individuals having at least basic digital skills (%) [sdg_04_70] (stimulant)

#### 4.2. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AHP | analytic hierarchy process |

CMI | combination mode I: none in EWM and S in Hellwig’s method |

CMII | combination mode II: MM in EWM and S in Hellwig’s method |

CMIII | combination mode III: none in EWM and MM in Hellwig’s method |

CMIV | combination mode IV: MM in EWM and MM in Hellwig’s method |

CMV | combination mode V: none in EWM and VN in Hellwig’s method |

CMVI | combination mode VI: MM in EWM and VN in Hellwig’s method |

CRITIC | CRiteria Importance Through Inter-criteria Correlation |

DM | decision maker |

EWM | entropy weight method |

EWMn | entropy weight method without max–min normalization |

EWMMM | entropy weight method with max–min normalization |

MCDM | multi-criteria decision-making |

MM | max–min normalization |

MORRA | multi-objective optimization based on the ratio analysis |

S | standardization |

SD | standard deviation |

SDG | Sustainable Development Goal |

SN | sum normalization |

TODIM | An acronym in Portuguese for Interactive and Multi-criteria Decision Making |

TOPSIS | Technique for Ordering Preferences by Similarity to Ideal Solution |

VN | vector normalization |

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**Figure 3.**Box plots for correlation coefficients between CMI and CMII results in simulation studies with Ghaziri indexes of quality of sampling.

EU Country | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ |
---|---|---|---|---|---|

Austria | 8.0 | 42.4 | 89.0 | 14.6 | 63.33 |

Belgium | 6.7 | 50.9 | 97.9 | 10.2 | 54.23 |

Bulgaria | 12.2 | 33.6 | 79.4 | 1.8 | 31.18 |

Croatia | 2.4 | 35.7 | 77.8 | 5.1 | 63.37 |

Cyprus | 10.2 | 58.3 | 85.8 | 9.7 | 50.21 |

Czechia | 6.4 | 34.9 | 84.2 | 5.8 | 59.69 |

Denmark | 9.8 | 49.7 | 97.0 | 22.3 | 68.65 |

Estonia | 9.8 | 43.2 | 91.5 | 18.4 | 56.37 |

Finland | 8.2 | 40.1 | 90.6 | 30.5 | 79.18 |

France | 7.8 | 50.3 | 100.0 | 11.0 | 61.96 |

Germany | 12.5 | 36.9 | 93.1 | 7.7 | 48.92 |

Greece | 3.2 | 44.2 | 68.8 | 3.5 | 52.48 |

Hungary | 12.0 | 32.9 | 93.4 | 5.9 | 49.09 |

Ireland | 3.3 | 61.7 | 96.4 | 13.6 | 70.49 |

Italy | 12.7 | 28.3 | 91.0 | 9.9 | 45.60 |

Latvia | 7.3 | 45.5 | 94.5 | 8.6 | 50.80 |

Lithuania | 5.3 | 57.5 | 92.1 | 8.5 | 48.84 |

Luxembourg | 9.3 | 62.6 | 88.9 | 17.9 | 63.79 |

Malta | 10.7 | 42.5 | 86.2 | 13.9 | 61.23 |

Netherlands | 5.1 | 55.6 | 93.0 | 26.6 | 78.94 |

Poland | 5.9 | 40.6 | 90.4 | 5.4 | 42.93 |

Portugal | 5.9 | 47.5 | 90.5 | 12.9 | 55.31 |

Romania | 15.3 | 23.3 | 75.6 | 4.9 | 27.82 |

Slovakia | 7.8 | 39.5 | 77.4 | 4.8 | 55.18 |

Slovenia | 3.1 | 47.9 | 92.3 | 18.9 | 49.67 |

Spain | 13.3 | 48.7 | 96.0 | 14.4 | 64.16 |

Sweden | 8.4 | 49.3 | 96.1 | 34.7 | 66.52 |

Min | 2.40 | 23.30 | 68.80 | 1.80 | 27.82 |

Max | 15.30 | 62.60 | 100.00 | 34.70 | 79.18 |

Mean | 8.24 | 44.58 | 89.22 | 12.65 | 56.29 |

Standard deviation | 3.37 | 9.68 | 7.50 | 8.19 | 11.88 |

Coefficient of variation | 40.84 | 21.72 | 8.40 | 64.73 | 21.10 |

Variants of EWM | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ |
---|---|---|---|---|---|

EWMn | 0.263 | 0.072 | 0.011 | 0.584 | 0.070 |

EWMMM | 0.184 | 0.168 | 0.124 | 0.374 | 0.150 |

**Table 3.**The values and rank-ordering of EU countries obtained by the combination modes of Hellwig’s measures.

Country | CMI | Rank CMI | CMII | Rank CMII | CMII | Rank CMII | CMIV | Rank CMIV | CMV | Rank CMV | CMVI | Rank CMVI |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Austria | 0.454 | 9 | 0.451 | 9 | 0.452 | 9 | 0.449 | 9 | 0.462 | 8 | 0.459 | 8 |

Belgium | 0.353 | 14 | 0.379 | 14 | 0.353 | 14 | 0.378 | 14 | 0.353 | 14 | 0.354 | 14 |

Bulgaria | 0.097 | 27 | 0.066 | 26 | 0.095 | 27 | 0.067 | 26 | 0.116 | 27 | 0.106 | 27 |

Croatia | 0.231 | 21 | 0.232 | 22 | 0.233 | 21 | 0.232 | 22 | 0.225 | 22 | 0.225 | 22 |

Cyprus | 0.315 | 16 | 0.331 | 16 | 0.313 | 16 | 0.329 | 16 | 0.327 | 15 | 0.326 | 15 |

Czechia | 0.240 | 20 | 0.244 | 18 | 0.241 | 19 | 0.243 | 18 | 0.239 | 20 | 0.238 | 20 |

Denmark | 0.615 | 4 | 0.616 | 4 | 0.610 | 4 | 0.610 | 4 | 0.643 | 4 | 0.637 | 4 |

Estonia | 0.525 | 6 | 0.502 | 7 | 0.521 | 6 | 0.499 | 7 | 0.548 | 6 | 0.539 | 6 |

Finland | 0.794 | 2 | 0.712 | 3 | 0.788 | 2 | 0.708 | 3 | 0.839 | 2 | 0.813 | 2 |

France | 0.367 | 13 | 0.400 | 13 | 0.367 | 13 | 0.397 | 13 | 0.370 | 13 | 0.372 | 13 |

Germany | 0.241 | 19 | 0.238 | 20 | 0.237 | 20 | 0.234 | 21 | 0.264 | 19 | 0.257 | 19 |

Greece | 0.190 | 25 | 0.182 | 25 | 0.192 | 25 | 0.185 | 25 | 0.183 | 25 | 0.185 | 25 |

Hungary | 0.202 | 24 | 0.197 | 24 | 0.199 | 24 | 0.193 | 24 | 0.221 | 23 | 0.214 | 23 |

Ireland | 0.454 | 8 | 0.498 | 8 | 0.455 | 8 | 0.496 | 8 | 0.448 | 9 | 0.453 | 9 |

Italy | 0.287 | 18 | 0.243 | 19 | 0.283 | 18 | 0.240 | 19 | 0.318 | 16 | 0.302 | 18 |

Latvia | 0.308 | 17 | 0.324 | 17 | 0.308 | 17 | 0.324 | 17 | 0.309 | 18 | 0.309 | 17 |

Lithuania | 0.315 | 15 | 0.342 | 15 | 0.316 | 15 | 0.343 | 15 | 0.312 | 17 | 0.314 | 16 |

Luxembourg | 0.523 | 7 | 0.538 | 6 | 0.520 | 7 | 0.534 | 6 | 0.539 | 7 | 0.538 | 7 |

Malta | 0.410 | 11 | 0.405 | 12 | 0.407 | 11 | 0.400 | 11 | 0.431 | 10 | 0.425 | 10 |

Netherlands | 0.777 | 3 | 0.777 | 1 | 0.776 | 3 | 0.776 | 1 | 0.781 | 3 | 0.781 | 3 |

Poland | 0.231 | 22 | 0.234 | 21 | 0.232 | 22 | 0.236 | 20 | 0.229 | 21 | 0.227 | 21 |

Portugal | 0.425 | 10 | 0.432 | 10 | 0.425 | 10 | 0.433 | 10 | 0.425 | 12 | 0.424 | 11 |

Romania | 0.135 | 26 | 0.050 | 27 | 0.130 | 26 | 0.050 | 27 | 0.177 | 26 | 0.154 | 26 |

Slovakia | 0.209 | 23 | 0.210 | 23 | 0.209 | 23 | 0.210 | 23 | 0.210 | 24 | 0.209 | 24 |

Slovenia | 0.585 | 5 | 0.556 | 5 | 0.586 | 5 | 0.560 | 5 | 0.585 | 5 | 0.579 | 5 |

Spain | 0.390 | 12 | 0.409 | 11 | 0.383 | 12 | 0.400 | 12 | 0.426 | 11 | 0.421 | 12 |

Sweden | 0.824 | 1 | 0.775 | 2 | 0.817 | 1 | 0.771 | 2 | 0.879 | 1 | 0.857 | 1 |

Mean | 0.389 | 0.383 | 0.387 | 0.381 | 0.402 | 0.397 | ||||||

SD | 0.194 | 0.192 | 0.193 | 0.191 | 0.201 | 0.198 | ||||||

Min | 0.097 | 0.050 | 0.095 | 0.050 | 0.116 | 0.106 | ||||||

Max | 0.824 | 0.777 | 0.817 | 0.776 | 0.879 | 0.857 |

**Table 4.**Kendall tau coefficients between rankings obtained by six combination modes of Hellwig’s measures.

Kendall Tau Coefficient | Rank CMI | Rank CMII | Rank CMIII | Rank CMIV | Rank CMV | Rank CMVI |
---|---|---|---|---|---|---|

Rank CMI | 1.000 | |||||

Rank CMII | 0.954 | 1.000 | ||||

Rank CMIII | 0.994 | 0.960 | 1.000 | |||

Rank CMIV | 0.954 | 0.989 | 0.960 | 1.000 | ||

Rank CMV | 0.954 | 0.920 | 0.949 | 0.920 | 1.000 | |

Rank CMVI | 0.972 | 0.937 | 0.966 | 0.937 | 0.983 | 1.000 |

**Table 5.**Pearson coefficients between rankings obtained by six combination modes of Hellwig’s measures.

Pearson Coefficient | CMI | CMII | CMIII | CMIV | CMV | CMVI |
---|---|---|---|---|---|---|

CMI | 1.0000 | |||||

CMII | 0.9874 | 1.0000 | ||||

CMIII | 0.9999 | 0.9884 | 1.0000 | |||

CMIV | 0.9874 | 0.9999 | 0.9885 | 1.0000 | ||

CMV | 0.9968 | 0.9757 | 0.9958 | 0.9751 | 1.0000 | |

CMVI | 0.9987 | 0.9831 | 0.9980 | 0.9825 | 0.9993 | 1.0000 |

**Table 6.**The values and rank-ordering of EU countries obtained by the equal weights of Hellwig’s measures.

Country | H_S | Rank H_S | H_MM | Rank H_MM | H_VN | Rank H_VN |
---|---|---|---|---|---|---|

Austria | 0.467 | 11 | 0.462 | 11 | 0.445 | 9 |

Belgium | 0.476 | 10 | 0.475 | 10 | 0.374 | 13 |

Bulgaria | 0.022 | 26 | 0.026 | 26 | 0.056 | 26 |

Croatia | 0.278 | 20 | 0.279 | 20 | 0.263 | 18 |

Cyprus | 0.363 | 17 | 0.360 | 17 | 0.310 | 17 |

Czechia | 0.303 | 18 | 0.301 | 18 | 0.259 | 19 |

Denmark | 0.595 | 4 | 0.582 | 4 | 0.570 | 5 |

Estonia | 0.456 | 12 | 0.451 | 12 | 0.475 | 8 |

Finland | 0.588 | 5 | 0.580 | 5 | 0.693 | 3 |

France | 0.506 | 8 | 0.500 | 8 | 0.387 | 11 |

Germany | 0.245 | 21 | 0.236 | 22 | 0.202 | 24 |

Greece | 0.179 | 25 | 0.186 | 24 | 0.226 | 21 |

Hungary | 0.214 | 23 | 0.206 | 23 | 0.170 | 25 |

Ireland | 0.647 | 3 | 0.644 | 3 | 0.501 | 7 |

Italy | 0.179 | 24 | 0.172 | 25 | 0.213 | 23 |

Latvia | 0.400 | 15 | 0.400 | 14 | 0.321 | 16 |

Lithuania | 0.439 | 13 | 0.445 | 13 | 0.346 | 14 |

Luxembourg | 0.548 | 6 | 0.540 | 6 | 0.509 | 6 |

Malta | 0.388 | 16 | 0.379 | 16 | 0.379 | 12 |

Netherlands | 0.776 | 1 | 0.773 | 1 | 0.773 | 1 |

Poland | 0.294 | 19 | 0.300 | 19 | 0.244 | 20 |

Portugal | 0.480 | 9 | 0.482 | 9 | 0.437 | 10 |

Romania | −0.107 | 27 | −0.107 | 27 | 0.024 | 27 |

Slovakia | 0.239 | 22 | 0.240 | 21 | 0.226 | 22 |

Slovenia | 0.530 | 7 | 0.540 | 7 | 0.570 | 4 |

Spain | 0.415 | 14 | 0.395 | 15 | 0.341 | 15 |

Sweden | 0.672 | 2 | 0.664 | 2 | 0.729 | 2 |

Mean | 0.392 | 0.389 | 0.372 | |||

SD | 0.196 | 0.195 | 0.186 | |||

Min | −0.107 | −0.107 | 0.024 | |||

Max | 0.776 | 0.773 | 0.773 |

**Table 7.**Kendall tau coefficients between rankings obtained by Hellwig’s measures with equal weights.

Kendal Tau Coefficient | Rank H_S | Rank H_MM | Rank H_VN |
---|---|---|---|

Rank H_S | 1.000 | ||

Rank H_MM | 0.983 | 1.000 | |

Rank H_VN | 0.840 | 0.846 | 1.000 |

Pearson Coefficient | H_S | H_MM | H_VN |
---|---|---|---|

H_S | 1.000 | ||

H_MM | 0.999 | 1.000 | |

H_VN | 0.946 | 0.946 | 1.000 |

Pearson Coefficient | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ |
---|---|---|---|---|---|

${C}_{1}$ | 1.000 | ||||

${C}_{2}$ | −0.437 | 1.000 | |||

${C}_{3}$ | 0.037 | 0.452 | 1.000 | ||

${C}_{4}$ | −0.075 | 0.411 | 0.506 | 1.000 | |

${C}_{5}$ | −0.383 | 0.520 | 0.393 | 0.706 | 1.000 |

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**MDPI and ACS Style**

Roszkowska, E.; Wachowicz, T.
Impact of Normalization on Entropy-Based Weights in Hellwig’s Method: A Case Study on Evaluating Sustainable Development in the Education Area. *Entropy* **2024**, *26*, 365.
https://doi.org/10.3390/e26050365

**AMA Style**

Roszkowska E, Wachowicz T.
Impact of Normalization on Entropy-Based Weights in Hellwig’s Method: A Case Study on Evaluating Sustainable Development in the Education Area. *Entropy*. 2024; 26(5):365.
https://doi.org/10.3390/e26050365

**Chicago/Turabian Style**

Roszkowska, Ewa, and Tomasz Wachowicz.
2024. "Impact of Normalization on Entropy-Based Weights in Hellwig’s Method: A Case Study on Evaluating Sustainable Development in the Education Area" *Entropy* 26, no. 5: 365.
https://doi.org/10.3390/e26050365