#
A Fuzzy Entropy-Based Group Consensus Measure for Financial Investments^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Continuous-Valued Logical Operations: Strict Triangular Norms, Strong Negations

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Definition**

**3.**

- (a)
- N is continuous (continuity);
- (b)
- N(0) = 1, N(1) = 0 (boundary conditions);
- (c)
- N(x) > N(y) for x < y (monotonicity);
- (d)
- N(N(x)) = x for any x ∈ [0, 1] (involution).

#### Fuzzy Entropies and Fuzziness Measures

- P
_{1} - $d\left(\mu \right)$ must be 0 if and only if $\mu $ takes on X the values 0 or 1.
- P
_{2} - $d\left(\mu \right)$ must assume the maximum value if and only if $\mu $ always has the value $\frac{1}{2}$.
- P
_{3} - $d\left(\mu \right)$ must be greater than or equal to $d\left({\mu}^{*}\right)$, where ${\mu}^{*}$ is any ‘sharpened’ version of $\mu $; that is, any fuzzy set such that ${\mu}^{*}\left(x\right)\ge \mu \left(x\right)$ if $\mu \left(x\right)\ge \frac{1}{2}$ and ${\mu}^{*}\left(x\right)\le \mu \left(x\right)$ if $\mu \left(x\right)\le \frac{1}{2}$.

**Definition**

**4.**

- (a)
- F is continuous on $[0,1]$.
- (b)
- $F\left(0\right)=0$ and $F\left(1\right)=0$.
- (c)
- F is strictly increasing on $\left(0,\frac{1}{2}\right)$, and F is strictly decreasing on $\left(\frac{1}{2},1\right)$.
- (d)
- $F\left(x\right)$ has a unique maximum at $x=\frac{1}{2}$, and $F\left(\frac{1}{2}\right)=1$.
- (e)
- $F\left(x\right)=F(1-x)$ for any $x\in [0,1]$.

- P
_{4} - $d\left(\mu \right)=d\left(\overline{\mu}\right)$, where $\overline{\mu}$ is the ‘complement’ of $\mu $, i.e., the function $\overline{\mu}\left(x\right)$ is given as $\overline{\mu}\left(x\right)=1-\mu \left(x\right)$, where $x\in X$.

## 3. A Fuzzy Entropy-Based Measure of Group Consensus Concerning a Financial Investment

- Invest in $I\left(r\right)$ if $p\ge \frac{1}{2}$.
- Do not invest in $I\left(r\right)$ if $p<\frac{1}{2}$.

- $S\left(\overline{x}\right)=0$ (with the convention $0\xb7ln\left(0\right)=0$) if and only if ${x}_{1}={x}_{2}=\cdots ={x}_{n}=0$ or ${x}_{1}={x}_{2}=\cdots ={x}_{n}=1$. That is, there is a maximal agreement among the group members concerning the investment I if and only if $S\left(\overline{x}\right)=0$.
- $S\left(\overline{x}\right)=1$ if and only if $n=2k$ for a $k\in \mathbb{N}$, $k\ge 1$ and k of the inputs are equal to zero and k of the inputs are equal to 1. That is, there is a maximal disagreement among the group members concerning the investment I if and only if $S\left(\overline{x}\right)=1$.

#### 3.1. Consensus Measures

- (C1)
- (Unanimity) For any $a\in [0,1]$, $C(a,a,\dots ,a)=1$.
- (C2)
- (Minimum consensus for $n=2$) For the special case with two inputs, it holds that $C(0,1)=C(1,0)=0$.

- (C3)
- (Symmetry) For any permutation $\pi :\{1,2,\dots ,n\}\to \{1,2,\dots ,n\}$ and input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})$, $\mathbf{x}\in {[0,1]}^{n}$, it holds that$$C({x}_{1},{x}_{2},\dots ,{x}_{n})=C\left({x}_{\pi \left(1\right)},{x}_{\pi \left(2\right)},\dots ,{x}_{\pi \left(n\right)}\right).$$
- (C4)
- (Maximum dissension) For $n=2k$, if k of the inputs are equal to zero and k of the inputs are equal to 1, then $C(0,0,\dots ,1,1)=0$ for all permutations of the input vector.
- (C5)
- (Reciprocity) For any input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})$, $\mathbf{x}\in {[0,1]}^{n}$, it holds that$$C({x}_{1},{x}_{2},\dots ,{x}_{n})=C\left(N\left({x}_{1}\right),N\left({x}_{2}\right),\dots ,N\left({x}_{n}\right)\right),$$
- (C6)
- (Replication invariance) For any input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})\in {[0,1]}^{n}$, replicating the inputs does not alter the degree of consensus, i.e.,$$C\left(\mathbf{x}\right)=C(\mathbf{x},\mathbf{x})=\dots =C(\mathbf{x},\mathbf{x}\dots ,\mathbf{x}).$$
- (C7)
- (Monotonicity with respect to the majority) For $n=2k$, let half of the inputs be equal and denoted by $\mathbf{a}=(a,a,\dots ,a)$, where $\mathbf{a}\in {[0,1]}^{k}$. Furthermore, let $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{k})$ and $\mathbf{y}=({y}_{1},{y}_{2},\dots ,{y}_{k})$ be two input vectors, where $\mathbf{x},\mathbf{y}\in {[0,1]}^{k}$. If $|a-{x}_{j}|\le |a-{y}_{j}|$ for all $j=1,2,\dots ,k$, then $C\left(\mathbf{a},{x}_{1},{x}_{2},\dots ,{x}_{k}\right)\ge C\left(\mathbf{a},{y}_{1},{y}_{2},\dots ,{y}_{k}\right)$ holds for any permutation of the inputs.

#### 3.1.1. Some Existing Consensus Measures

#### 3.1.2. Requirements for a Measure of Group Consensus on a Financial Investment

**Definition**

**5.**

- (C1${}^{*}$)
- (Unanimity) For any $a\in \{0,1\}$, ${C}^{\left(I\right)}(a,a,\dots ,a)=1$.
- (C2${}^{*}$)
- (Symmetry) For any permutation $\pi :\{1,2,\dots ,n\}\to \{1,2,\dots ,n\}$ and input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})$, $\mathbf{x}\in {\{0,1\}}^{n}$, it holds that$${C}^{\left(I\right)}({x}_{1},{x}_{2},\dots ,{x}_{n})={C}^{\left(I\right)}\left({x}_{\pi \left(1\right)},{x}_{\pi \left(2\right)},\dots ,{x}_{\pi \left(n\right)}\right).$$
- (C3${}^{*}$)
- (Maximum dissension) For $n=2k$, if k of the inputs are equal to zero and k of the inputs are equal to 1, then ${C}^{\left(I\right)}(0,0,\dots ,1,1)=0$ for all permutations of the input vector.
- (C4${}^{*}$)
- (Reciprocity) For any input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})$, $\mathbf{x}\in {\{0,1\}}^{n}$, it holds that$${C}^{\left(I\right)}({x}_{1},{x}_{2},\dots ,{x}_{n})={C}^{\left(I\right)}\left({N}_{s}\left({x}_{1}\right),{N}_{s}\left({x}_{2}\right),\dots ,{N}_{s}\left({x}_{n}\right)\right),$$
- (C5${}^{*}$)
- (Replication invariance) For any input vector $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{n})\in {\{0,1\}}^{n}$, replicating the inputs does not alter the degree of consensus, i.e., ${C}^{\left(I\right)}\left(\mathbf{x}\right)={C}^{\left(I\right)}(\mathbf{x},\mathbf{x})=\dots ={C}^{\left(I\right)}(\mathbf{x},\mathbf{x}\dots ,\mathbf{x})$.
- (C6${}^{*}$)
- (Monotonicity with respect to the majority) For $n=2k$, let half of the inputs be equal and denoted by $\mathbf{a}=(a,a,\dots ,a)$, where $\mathbf{a}\in {\{0,1\}}^{k}$. Furthermore, let $\mathbf{x}=({x}_{1},{x}_{2},\dots ,{x}_{k})$ and $\mathbf{y}=({y}_{1},{y}_{2},\dots ,{y}_{k})$ be two input vectors, where $\mathbf{x},\mathbf{y}\in {\{0,1\}}^{k}$. If $|a-{x}_{j}|\le |a-{y}_{j}|$ for all $j=1,2,\dots ,k$, then ${C}^{\left(I\right)}\left(\mathbf{a},{x}_{1},{x}_{2},\dots ,{x}_{k}\right)\ge {C}^{\left(I\right)}\left(\mathbf{a},{y}_{1},{y}_{2},\dots ,{y}_{k}\right)$ holds for any permutation of the inputs.

**Remark**

**1.**

## 4. A Novel, Fuzzy Entropy-Based Group Consensus Measure and Its Main Properties

**Theorem**

**2.**

**Proof.**

## 5. Connections with Known Consensus Measures

- (a)
- The standard deviation-based consensus measure ${C}_{\sigma}$ given in Equation (5).
- (b)
- The consensus measure ${C}_{SK}$ proposed by Szmidt and Kacprzyk in Equation (6).
- (c)
- The Bonferroni consensus measure with implication pairs ${C}_{B}^{({M}_{A},{I}_{L})}$ for the case where the aggregation function is the arithmetic mean ${M}_{A}$ and the fuzzy implication is the Łukasiewicz implication ${I}_{L}$ given in Equation (7).

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- The consensus measure ${C}_{SK}$ proposed by Szmidt and Kacprzyk.
- The Bonferroni consensus measure with implication pairs ${C}_{B}^{({M}_{A},{I}_{L})}$ for the case where the aggregation function is the arithmetic mean ${M}_{A}$ and the fuzzy implication is the Łukasiewicz implication ${I}_{L}$.

**Proof.**

**Remark**

**2.**

## 6. Group Consensus Map for Financial Investments

#### An Algorithm for Estimating the Group Consensus Map

- Inputs:
- –
- $r\in \mathbb{R}$.
- –
- ${r}_{01},{r}_{02},\dots ,{r}_{0n}$, where ${r}_{0i}$ denotes the investment rate-of-return threshold of the ith group member, $n\ge 2$, $i=1,2,\dots ,n$. ${r}_{0i}$ represents the rate of return at which the ith decision maker changes his/her decision from acceptance to rejection, and vice versa.

- Step 0: Consider dropping the extreme investment rate-of-return thresholds (i.e., those falling more than 3 standard deviations away from the mean), as they can seriously affect the shape of the map, which we demonstrate later.
- Step 1: Compute $\overline{x}\left(r\right)$ as$$\overline{x}\left(r\right)=\frac{{\sum}_{i=1}^{n}I({r}_{0i}\le r)}{n},$$$$I({r}_{0i}\le r)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{r}_{0i}\le r\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$
- Step 2: Compute the value of the group consensus map ${M}_{F}:\mathbb{R}\to [0,1]$ at r as$${M}_{F}\left(r\right)=1-F\left(\overline{x}\left(r\right)\right),$$
- Output: ${M}_{F}\left(r\right)$.

- Sample median of investment rate-of-return thresholds: At the median, the group consensus map has its global minimum, i.e., the median of the investment rate-of-return thresholds can be treated as the estimate of an ${r}_{0}$ value at which ${M}_{F}$ is minimal.
- Sample minimum and maximum of investment rate-of-return thresholds: For any $r<min\{{r}_{1},{r}_{2},\dots ,{r}_{n}\}$ ($r\ge max\{{r}_{1},{r}_{2},\dots ,{r}_{n}\}$, respectively), $\overline{x}\left(r\right)=0$ ($\overline{x}\left(r\right)=1$, respectively) and so ${M}_{F}\left(r\right)=1$. This means that at the furthest value of each tail of the investment thresholds’ distribution, the group consensus map reaches its limiting value, i.e., 1, which implies a full consensus.
- Shape parameters of the distribution of investment rate-of-return thresholds: The shape of the group consensus map largely depends on the shape of the distribution of the investment rate-of-return thresholds and the presence of outliers (this is why we suggested dropping these values at the very beginning of the procedure). Where long tails are present in the distribution of the thresholds, the increase in the consensus map is very slow toward its limiting value of 1. Conversely, on the dense side of the distribution, the consensus map increases quite sharply toward 1.

## 7. Conclusions and Future Research Plans

- The consensus measure ${C}_{SK}$ by Szmidt and Kacprzyk.
- The Bonferroni consensus measure with implication pairs ${C}_{B}^{({M}_{A},{I}_{L})}$ for the case where the aggregation function is the arithmetic mean ${M}_{A}$ and the fuzzy implication is the Łukasiewicz implication ${I}_{L}$.

- The rates of return that imply the lowest level of consensus and the highest level of dissension.
- The rates of return that imply the highest level of consensus and the lowest level of dissension.
- How the level of consensus evolves regarding other values of the rate of return.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Beliakov, G.; James, S.; Calvo, T. Aggregating fuzzy implications to measure group consensus. In Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, Edmonton, AB, Canada, 24–28 June 2013; pp. 1016–1021. [Google Scholar] [CrossRef]
- Bezdek, J.C.; Spillman, B.; Spillman, R. A fuzzy relation space for group decision theory. Fuzzy Sets Syst.
**1978**, 1, 255–268. [Google Scholar] [CrossRef] - Spillman, B.; Spillman, R.; Bezdek, J. A fuzzy analysis of consensus in small groups. In Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems; Springer: Boston, MA, USA, 1980; pp. 291–308. [Google Scholar] [CrossRef]
- Butler, C.L.; Rothstein, A. On Conflict and Consensus; Food Not Bombs: Takoma Park, MD, USA, 1988. [Google Scholar]
- Eklund, P.; Rusinowska, A.; De Swart, H. Consensus reaching in committees. Eur. J. Oper. Res.
**2007**, 178, 185–193. [Google Scholar] [CrossRef] - Zhang, H.; Dong, Y.; Chiclana, F.; Yu, S. Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design. Eur. J. Oper. Res.
**2019**, 275, 580–598. [Google Scholar] [CrossRef] - Li, Y.; Kou, G.; Li, G.; Peng, Y. Consensus reaching process in large-scale group decision making based on bounded confidence and social network. Eur. J. Oper. Res.
**2022**, 303, 790–802. [Google Scholar] [CrossRef] - Morente-Molinera, J.; Wu, X.; Morfeq, A.; Al-Hmouz, R.; Herrera-Viedma, E. A novel multi-criteria group decision-making method for heterogeneous and dynamic contexts using multi-granular fuzzy linguistic modelling and consensus measures. Inf. Fusion
**2020**, 53, 240–250. [Google Scholar] [CrossRef] - Labella, A.; Liu, H.; Rodríguez, R.M.; Martínez, L. A Cost Consensus Metric for Consensus Reaching Processes based on a comprehensive minimum cost model. Eur. J. Oper. Res.
**2020**, 281, 316–331. [Google Scholar] [CrossRef] - Cabrerizo, F.J.; Moreno, J.M.; Pérez, I.J.; Herrera-Viedma, E. Analyzing consensus approaches in fuzzy group decision making: Advantages and drawbacks. Soft Comput.
**2010**, 14, 451–463. [Google Scholar] [CrossRef] - Bordogna, G.; Fedrizzi, M.; Pasi, G. A linguistic modeling of consensus in group decision making based on OWA operators. IEEE Trans. Syst. Man Cybern.-Part A Syst. Hum.
**1997**, 27, 126–133. [Google Scholar] [CrossRef] - Bezdek, J.C.; Spillman, B.; Spillman, R. Fuzzy relation spaces for group decision theory: An application. Fuzzy Sets Syst.
**1979**, 2, 5–14. [Google Scholar] [CrossRef] - Guo, W.; Gong, Z.; Zhang, W.G.; Xu, Y. Minimum cost consensus modeling under dynamic feedback regulation mechanism considering consensus principle and tolerance level. Eur. J. Oper. Res.
**2023**, 306, 1279–1295. [Google Scholar] [CrossRef] - Gong, Z.; Guo, W.; Herrera-Viedma, E.; Gong, Z.; Wei, G. Consistency and consensus modeling of linear uncertain preference relations. Eur. J. Oper. Res.
**2020**, 283, 290–307. [Google Scholar] [CrossRef] - Gong, Z.; Guo, W.; Słowiński, R. Transaction and interaction behavior-based consensus model and its application to optimal carbon emission reduction. Omega
**2021**, 104, 102491. [Google Scholar] [CrossRef] - Tang, M.; Liao, H.; Mi, X.; Lev, B.; Pedrycz, W. A hierarchical consensus reaching process for group decision making with noncooperative behaviors. Eur. J. Oper. Res.
**2021**, 293, 632–642. [Google Scholar] [CrossRef] - Yuan, Y.; Cheng, D.; Zhou, Z. A minimum adjustment consensus framework with compromise limits for social network group decision making under incomplete information. Inf. Sci.
**2021**, 549, 249–268. [Google Scholar] [CrossRef] - Cheng, D.; Cheng, F.; Zhou, Z.; Wu, Y. Reaching a minimum adjustment consensus in social network group decision-making. Inf. Fusion
**2020**, 59, 30–43. [Google Scholar] [CrossRef] - Li, H.; Ji, Y.; Gong, Z.; Qu, S. Two-stage stochastic minimum cost consensus models with asymmetric adjustment costs. Inf. Fusion
**2021**, 71, 77–96. [Google Scholar] [CrossRef] - Herrera-Viedma, E.; Cabrerizo, F.J.; Kacprzyk, J.; Pedrycz, W. A review of soft consensus models in a fuzzy environment. Inf. Fusion
**2014**, 17, 4–13. [Google Scholar] [CrossRef] - Alcantud, J.; de Andrés Calle, R.; Cascón, J. On measures of cohesiveness under dichotomous opinions: Some characterizations of approval consensus measures. Inf. Sci.
**2013**, 240, 45–55. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; de Andrés Calle, R.; Cascón, J. Pairwise dichotomous cohesiveness measures. Group Decis. Negot.
**2015**, 24, 833–854. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Torrecillas, M.J.M. Consensus measures for various informational bases. Three new proposals and two case studies from political science. Qual. Quant.
**2017**, 51, 285–306. [Google Scholar] [CrossRef] - Beliakov, G.; Calvo, T.; James, S. Consensus measures constructed from aggregation functions and fuzzy implications. Knowl.-Based Syst.
**2014**, 55, 1–8. [Google Scholar] [CrossRef] - Klement, E.; Mesiar, R.; Pap, E. Triangular Norms; Trends in Logic; Springer: Dordrecht, The Netherlands, 2013. [Google Scholar]
- Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation functions: Means. Inf. Sci.
**2011**, 181, 1–22. [Google Scholar] [CrossRef] - Dombi, J.; Csiszár, O. Explainable Neural Networks Based on Fuzzy Logic and Multi-Criteria Decision Tools; Studies in Fuzziness and Soft Computing; Springer International Publishing: Cham, Switzerland, 2021. [Google Scholar]
- Fodor, J.C.; Roubens, M. Fuzzy Preference Modelling and Multicriteria Decision Support; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1994; Volume 14. [Google Scholar]
- De Luca, A.; Termini, S. A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Inf. Control
**1972**, 20, 301–312. [Google Scholar] [CrossRef] - Dombi, J.; Jónás, T. On a Parametric Measure of Vagueness. IEEE Trans. Fuzzy Syst.
**2022**, 31, 343–347. [Google Scholar] [CrossRef] - Dombi, J. A General Class of Fuzzy Operators, the De Morgan Class of Fuzzy Operators and Fuzziness Measures Included by Fuzzy Operators. Fuzzy Sets Syst.
**1982**, 8, 149–168. [Google Scholar] [CrossRef] - Tastle, W.J.; Wierman, M.J.; Dumdum, U.R. Ranking ordinal scales using the consensus measure. Issues Inf. Syst.
**2005**, 6, 96–102. [Google Scholar] - Szmidt, E.; Kacprzyk, J. Analysis of consensus under intuitionistic fuzzy preferences. In Proceedings of the EUSFLAT Conference, Leicester, UK, 5–7 September 2001; pp. 79–82. [Google Scholar]

r | ${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ | ${x}_{5}$ | ${x}_{6}$ | $\overline{x}$ |

0% | 0 | 0 | 0 | 0 | 0 | 0 | 0.0000 |

2% | 1 | 0 | 0 | 0 | 0 | 0 | 0.1667 |

3% | 1 | 1 | 0 | 0 | 0 | 0 | 0.3333 |

5% | 1 | 1 | 1 | 0 | 0 | 0 | 0.5000 |

8% | 1 | 1 | 1 | 1 | 0 | 0 | 0.6667 |

10% | 1 | 1 | 1 | 1 | 1 | 0 | 0.8333 |

15% | 1 | 1 | 1 | 1 | 1 | 1 | 1.0000 |

**Table 2.**Example: values of various consensus measures for the input vectors given in Table 1.

${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ | ${x}_{4}$ | ${x}_{5}$ | ${x}_{6}$ | $\overline{x}$ | ${C}_{{F}_{{g}_{D}}}^{\left(I\right)}\left(\mathit{x}\right)$ | ${C}_{SK}\left(\mathit{x}\right)$ | ${C}_{B}^{({M}_{A},{I}_{L})}\left(\mathit{x}\right)$ | ${C}_{{F}_{{g}_{p}}}^{\left(I\right)}\left(\mathit{x}\right)$ | ${C}_{\sigma}\left(\mathit{x}\right)$ | ${C}_{TWD}\left(\mathit{x}\right)$ |

0 | 0 | 0 | 0 | 0 | 0 | 0.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

1 | 0 | 0 | 0 | 0 | 0 | 0.1667 | 0.4444 | 0.4444 | 0.4444 | 0.2546 | 0.2546 | 0.3500 |

1 | 1 | 0 | 0 | 0 | 0 | 0.3333 | 0.1111 | 0.1111 | 0.1111 | 0.0572 | 0.0572 | 0.0817 |

1 | 1 | 1 | 0 | 0 | 0 | 0.5000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

1 | 1 | 1 | 1 | 0 | 0 | 0.6667 | 0.1111 | 0.1111 | 0.1111 | 0.0572 | 0.0572 | 0.0817 |

1 | 1 | 1 | 1 | 1 | 0 | 0.8333 | 0.4444 | 0.4444 | 0.4444 | 0.2546 | 0.2546 | 0.3500 |

1 | 1 | 1 | 1 | 1 | 1 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

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**MDPI and ACS Style**

Dombi, J.; Fáró, J.; Jónás, T.
A Fuzzy Entropy-Based Group Consensus Measure for Financial Investments. *Mathematics* **2024**, *12*, 4.
https://doi.org/10.3390/math12010004

**AMA Style**

Dombi J, Fáró J, Jónás T.
A Fuzzy Entropy-Based Group Consensus Measure for Financial Investments. *Mathematics*. 2024; 12(1):4.
https://doi.org/10.3390/math12010004

**Chicago/Turabian Style**

Dombi, József, Jenő Fáró, and Tamás Jónás.
2024. "A Fuzzy Entropy-Based Group Consensus Measure for Financial Investments" *Mathematics* 12, no. 1: 4.
https://doi.org/10.3390/math12010004