# Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview

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## Abstract

**:**

## 1. Introduction

- Which consensus problems can be handled by each of these aggregating techniques?
- What are the limitations of each technique?

## 2. Basic Definitions and Properties

**Definition**

**1.**

- An aggregation function of dimension $n\in \mathbb{N}$ is an n-ary function ${A}^{\left(n\right)}:{[0,1]}^{n}\to [0,1]$ satisfying
**A1.**- $A\left(x\right)=x$, for $n=1$ and any $x\in [0,1]$;
**A2.**- ${A}^{\left(n\right)}({x}_{1},\cdots ,{x}_{n})\le {A}^{\left(n\right)}({y}_{1},\cdots ,{y}_{n})$ if $({x}_{1},\cdots ,{x}_{n})\le ({y}_{1},\cdots ,{y}_{n})$;
**A3.**- ${A}^{\left(n\right)}(0,0,\cdots ,0)=0$ and ${A}^{\left(n\right)}(1,1,\cdots ,1)=1$.

- An extended aggregation function is the function $A:{\bigcup}_{n\in \mathbb{N}}{[0,1]}^{n}\to [0,1]$ whose restriction ${A|}_{{\mathbb{I}}^{n}}:={A}^{\left(n\right)}$ to ${\mathbb{I}}^{n}$ is the n-ary aggregation function ${A}^{\left(n\right)}$ for any $n\in \mathbb{N}$.

**Example**

**1.**

- 1.
- median Med defined by $Med({x}_{1},\cdots ,{x}_{n})={x}_{\frac{n+1}{2}}$ if n is odd and $Med({x}_{1},\cdots ,{x}_{n})=\frac{1}{2}[{x}_{\frac{n}{2}}+{x}_{\frac{n}{2}+1}]$ if n is even where ${x}_{1}\le {x}_{2}\le \cdots \le {x}_{n}$;
- 2.
- arithmetic mean $AM\left({\mathbf{x}}_{\left(n\right)}\right)=\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}$;
- 3.
- weighted arithmetic mean $WAM\left({\mathbf{x}}_{\left(n\right)}\right)={\sum}_{i=1}^{n}{w}_{i}{x}_{i}$ where ${w}_{i}\in [0,1]$ and ${\sum}_{i=1}^{n}{w}_{i}=1$;
- 4.
- geometric mean $GM\left({\mathbf{x}}_{\left(n\right)}\right)={\left({\prod}_{i=1}^{n}{x}_{i}\right)}^{\frac{1}{n}}$;
- 5.
- harmonic mean $HM\left({\mathbf{x}}_{\left(n\right)}\right)=\frac{n}{{\sum}_{i=1}^{n}\frac{1}{{x}_{i}}}$;
- 6.
- minimum $Min\left({\mathbf{x}}_{\left(n\right)}\right)={min}_{i=1}^{n}{x}_{i}$ and maximum $Max\left({\mathbf{x}}_{\left(n\right)}\right)={max}_{i=1}^{n}{x}_{i}$;
- 7.
- product function $\mathsf{\Pi}\left({\mathbf{x}}_{\left(n\right)}\right)={\prod}_{i=1}^{n}{x}_{i}$;
- 8.
- projection function to the kth coordinate ${P}_{k}\left({\mathbf{x}}_{\left(n\right)}\right)={x}_{k}$

**Remark**

**1.**

**Definition**

**2.**

- has a neutral element $e\in [0,1]$ if$$\begin{array}{c}\hfill A({x}_{1},\cdots ,{x}_{i-1},e,{x}_{i+1},\cdots ,{x}_{n})=A({x}_{1},\cdots ,{x}_{i-1},{x}_{i+1},\cdots ,{x}_{n})\end{array}$$
- has an annihilator element (absorbing element or zero element) $a\in [0,1]$ if$$\begin{array}{c}\hfill A({x}_{1},\cdots ,{x}_{i-1},a,{x}_{i+1},\cdots ,{x}_{n})=a\end{array}$$
- has no zero divisors if it has the zero element a, and$$\begin{array}{c}\hfill A({x}_{1},\cdots ,{x}_{n})=a\Rightarrow {\exists}_{1\le s\le n}s:{x}_{s}=a\end{array}$$

**Example**

**2.**

**Example**

**3.**

#### 2.1. Classification of Aggregation Functions

**Definition**

**3.**

- Conjunctive if for every ${\mathbf{x}}_{\left(n\right)}\in {\mathbb{I}}^{n}$, $A\left({\mathbf{x}}_{\left(n\right)}\right)$ is bounded by the minimum function:$$\begin{array}{c}\hfill A\left({\mathbf{x}}_{\left(n\right)}\right)\le Min\left({\mathbf{x}}_{\left(n\right)}\right)=min({x}_{1},\cdots ,{x}_{n})\end{array}$$
- Disjunctive if for every ${\mathbf{x}}_{\left(n\right)}\in {\mathbb{I}}^{n}$, $A\left({\mathbf{x}}_{\left(n\right)}\right)$ is bounded by the maximum function:$$\begin{array}{c}\hfill A\left({\mathbf{x}}_{\left(n\right)}\right)\ge Max\left({\mathbf{x}}_{\left(n\right)}\right)=max({x}_{1},\cdots ,{x}_{n})\end{array}$$
- Average whenever for all ${\mathbf{x}}_{\left(n\right)}\in {\mathbb{I}}^{n}$ we have:$$\begin{array}{c}\hfill Min\left({\mathbf{x}}_{\left(n\right)}\right)\le A\left({\mathbf{x}}_{\left(n\right)}\right)\le Max\left({\mathbf{x}}_{\left(n\right)}\right)\end{array}$$

**Example**

**4.**

**Remark**

**2.**

**Example**

**5.**

**Remark**

**3.**

#### Triangular Norms and Conorms

**Definition**

**4.**

**Example**

**6.**

#### 2.2. Properties of Aggregation Functions

**Definition**

**5.**

- 1.
- Additive if $A({\mathbf{x}}_{\left(n\right)}+{\mathbf{y}}_{\left(n\right)})=A\left({\mathbf{x}}_{\left(n\right)}\right)+A\left({\mathbf{y}}_{\left(n\right)}\right)$ for all ${\mathbf{x}}_{\left(n\right)},{\mathbf{y}}_{\left(n\right)}\in {\mathbb{I}}^{n}$ such that ${\mathbf{x}}_{\left(n\right)}+{\mathbf{y}}_{\left(n\right)}\in {\mathbb{I}}^{n}$;
- 2.
- Idempotent if $A(x,\cdots ,x)=x$ for all $x\in [0,1]$;
- 3.
- Symmetric if $A({x}_{1},\cdots ,{x}_{n})=A({x}_{\sigma \left(1\right)},\cdots ,{x}_{\sigma \left(n\right)})$ for all ${\mathbf{x}}_{\left(n\right)}=({x}_{1},\cdots ,{x}_{n})\in {\mathbb{I}}^{n}$ where σ is any permutation of $\{1,\cdots ,n\}$;
- 4.
- Bisymmetric if for all ${x}_{ij}\in [0,1]$ where $i,j\in \{1,\cdots ,n\}$ we have $A(A({x}_{11},\cdots ,{x}_{1n}),\cdots ,A({x}_{n1},\cdots ,{x}_{nn}))=A(A({x}_{11},\cdots ,{x}_{n1}),\cdots ,A({x}_{1n},\cdots ,{x}_{nn}))$;
- 5.
- Strongly bisymmetric if $A\left({\mathbf{x}}_{\left(n\right)}\right)={\mathbf{x}}_{\left(n\right)}$ for all ${\mathbf{x}}_{\left(n\right)}\in {\mathbb{I}}^{n}$ and $A(A({x}_{11},\cdots ,{x}_{1n}),\cdots ,A({x}_{m1},\cdots ,{x}_{mn}))=A(A({x}_{11},\cdots ,{x}_{m1}),\cdots ,A({x}_{1n},\cdots ,{x}_{mn}))$ for any $m,n\in \mathbb{N}$;
- 6.
- Associative if for all $({x}_{1},{x}_{2},{x}_{3})\in {\mathbb{I}}^{3}$ we have $A(A({x}_{1},{x}_{2}),{x}_{3})=A({x}_{1},A({x}_{2},{x}_{3}))$;
- 7.
- Continuous if for any ${\mathbf{x}}_{\left(n\right)}\in {\mathbb{I}}^{n}$ and ${\left({x}_{{i}_{j}}\right)}_{j\in \mathbb{N}}\in {\mathbb{I}}^{\mathbb{N}}$ where $i\in \{1,\cdots ,n\}$, if ${lim}_{j\to \infty}{x}_{{i}_{j}}={x}_{i}$ then ${lim}_{j\to \infty}A({x}_{{1}_{j}},\cdots ,{x}_{{n}_{j}})=A({x}_{1},\cdots ,{x}_{n})$ or equivalently $\forall \u03f5>0$, $\exists \delta >0$: if $\mid {x}_{i}-{y}_{i}\mid <\delta $ where $i\in \{1,\cdots ,n\}$ then $\mid A({x}_{1},\cdots ,{x}_{n})-A({y}_{1},\cdots ,{y}_{n})\mid <\u03f5$;
- 8.
- c-Lipschitz with respect to the norm $\parallel .\parallel :\mathbb{R}\to [0,+\infty )$, if for some constant $c\in (0,+\infty )$ we have: $\mid A\left({\mathbf{x}}_{\left(n\right)}\right)-A\left({\mathbf{y}}_{\left(n\right)}\right)\mid \le c\parallel {\mathbf{x}}_{\left(n\right)}-{\mathbf{y}}_{\left(n\right)}\parallel $ for all ${\mathbf{x}}_{\left(n\right)},{\mathbf{y}}_{\left(n\right)}\in {\mathbb{I}}^{n}$.

**Example**

**7.**

**Remark**

**4.**

**Example**

**8.**

**Remark**

**5.**

**Example**

**9.**

#### 2.3. Construction Methods of Aggregation Functions

**Problem I.**Constructing an aggregation function $A\in \mathcal{A}$, possibly with some additional properties, to find the best output.

**Problem I**, several construction methods have been discussed in literature to create new aggregation functions. In the following sections, we review three commonly used construction methods of aggregation functions, namely transformation, composition, and convex combination, which answer

**Problem I**. We also compare the type of consensus problems that can be solved by each of them.

## 3. Transformation of Aggregation Functions

**Proposition**

**1.**

#### 3.1. Duality of Aggregation Functions

**Remark**

**6.**

**Proposition**

**2.**

- If A is average (or alternatively idempotent), then ${A}^{d}$ is also average (idempotent).
- If A is conjunctive (disjunctive), then ${A}^{d}$ is disjunctive (conjunctive).

**Remark**

**7.**

**Proposition**

**3.**

**Remark**

**8.**

#### 3.2. Quasi-Arithmetic Means

**Problem II.**Constructing an averaging aggregation function $A\in \mathcal{A}$ such that $Min\le A\le Max$.

**Example**

**10.**

- 1.
- if $f\left(x\right)=x$ then ${AM}_{f}\left({\mathbf{x}}_{\left(n\right)}\right)=AM\left({\mathbf{x}}_{\left(n\right)}\right)=\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}$ (arithmetic mean),
- 2.
- if $f\left(x\right)={x}^{2}$ then ${AM}_{f}\left({\mathbf{x}}_{\left(n\right)}\right)=QM\left({\mathbf{x}}_{\left(n\right)}\right)={\left(\frac{1}{n}{\sum}_{i=1}^{n}{{x}_{i}}^{2}\right)}^{1/2}$ (quadratic mean mean),
- 3.
- if $f\left(x\right)=logx$ then ${AM}_{f}\left({\mathbf{x}}_{\left(n\right)}\right)=GM\left({\mathbf{x}}_{\left(n\right)}\right)={\left({\prod}_{i=1}^{n}{x}_{i}\right)}^{1/n}$ (geometric mean),
- 4.
- if $f\left(x\right)=\frac{1}{x}$ then ${AM}_{f}\left({\mathbf{x}}_{\left(n\right)}\right)=HM\left({\mathbf{x}}_{\left(n\right)}\right)=\frac{1}{\frac{1}{n}{\sum}_{i=1}^{n}\frac{1}{{x}_{i}}}$ (harmonic mean),
- 5.
- if $f\left(x\right)={e}^{\alpha x}$ where $0\ne \alpha \in \mathbb{R}$ then ${AM}_{f}\left({\mathbf{x}}_{\left(n\right)}\right)={EM}_{\alpha}\left({\mathbf{x}}_{\left(n\right)}\right)=\frac{1}{\alpha}ln\left(\frac{1}{n}{\sum}_{i=1}^{n}{e}^{\alpha {x}_{i}}\right)$ (exponential mean).

**Theorem**

**1.**

- 1.
- ${AM}_{f}\le {AM}_{g}$ if and only if $g\circ {f}^{-1}$ is convex.
- 2.
- ${AM}_{f}={AM}_{g}$ if and only if $g\circ {f}^{-1}$ is linear, i.e., $g\left(x\right)=af\left(x\right)+b$ where $a,b\in \mathbb{R}$ and $a\ne 0$.

**Theorem**

**2.**

## 4. Composite Aggregation Functions

**Problem III.**Constructing an aggregation function $A\in \mathcal{A}$ to combine the input data that have been merged by different techniques.

**Proposition**

**4.**

**Remark**

**9.**

#### 4.1. Composition over Different Source of Data

**Problem IV.**Constructing an aggregation function $A\in \mathcal{A}$ to combine two different types of input data.

**Proposition**

**5.**

**Remark**

**10.**

#### 4.2. Composition over Sub-Groups of Data

**Problem V.**Constructing an alternative aggregating approach in multi-criteria decision-making problems with n arguments where any possible list of $\alpha \le n$ arguments (not necessarily all n arguments) can affect the final decision at the consensus level $\alpha $.

**Theorem**

**3.**

**Example**

**11.**

## 5. Aggregating of Weighted Input

**Problem VI.**Constructing an aggregation operators ${A}_{\mathbf{w}}\in \mathcal{A}$ that permit to consider different weights of the sources or data where in fact,

#### 5.1. Weighted Aggregation Functions

**Remark**

**11.**

- If ${w}_{1}=1$ and ${w}_{i}=0$ else, then $OWA:=Max$ and $OWG:=Max$;
- If ${w}_{n}=1$ and ${w}_{i}=0$ else, then $OWA:=Min$ and $OWG:=Min$;
- If ${w}_{i}=\frac{1}{n}$, then $OWA:=AM$ and $OWG:=GM$;
- If n is odd, ${w}_{\frac{n+1}{2}}=1$ and ${w}_{i}=0$ else, then $OWA:=Med$ and $OWG:=Med$;
- If n is even, ${w}_{\frac{n}{2}}={w}_{\frac{n}{2}+1}=\frac{1}{2}$, and ${w}_{i}=0$ else, then $OWA:=Med$.

#### 5.2. Weighted Quasi–Arithmetic Means

**Theorem**

**4.**

- $p\left(x\right)\ge {p}^{\prime}\left(x\right)\xb7l\left(I\right)$ for all $x\in I$ where $l\left(I\right)$ is the length of the interval I;
- $p\left(x\right)\ge {p}^{\prime}\left(x\right)\xb7(x-infI)$ for all $x\in I$.

**Definition**

**6.**

**Theorem**

**5.**

**Example**

**12.**

#### 5.3. Weighted Rule Based on a Convex Combination

**Problem VII.**Constructing aggregation operators ${A}_{\mathbf{w}}\in \mathcal{A}$ based on a convex combination of unweighted aggregation operators.

**Theorem**

**6.**

**Corollary**

**1.**

**Remark**

**12.**

**Example**

**13.**

## 6. Discussion

**Example**

**14.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- 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] - Xu, G.l.; Wan, S.P.; Wang, F.; Dong, J.V.; Zeng, Y.F. Mathematical programming methods for consistency and consensus in group decision-making with intuitionistic fuzzy preference relations. Knowl.-Based Syst.
**2016**, 98, 30–43. [Google Scholar] [CrossRef] - Zahedi Khameneh, A.; Kilicman, A. Multi-attribute decision-making based on soft set theory: A systematic review. Soft Comput.
**2019**, 23, 6899–6920. [Google Scholar] [CrossRef] - De Baets, B.; Mesiar, R. Triangular norms on product lattices. Fuzzy Sets Syst.
**1999**, 104, 61–75. [Google Scholar] [CrossRef] - Choquet, G. Theory of capacities. Ann. Inst. Fourier.
**1953**, 5, 131–295. [Google Scholar] [CrossRef] [Green Version] - Sugeno, M. Theory of Fuzzy Integral and Its Application. Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, Japan, 1974. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Comput.
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Molodtsov, D. Soft set theory-first results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] [Green Version] - Yager, R.R. On ordered weighted averaging aggregation operators in multicriteria decisionmaking. In Readings in Fuzzy Sets for Intelligent Systems; Elsevier: New York, NY, USA, 1993; pp. 80–87. [Google Scholar]
- Chiclana, F.; Herrera, F.; Herrera-Viedma, E. The Ordered Weighted Geometric Operator: Properties and Application in MCDM Problems. In Proceedings of the 8th Conference on Information Processing and Management of Uncertainty in Knowledge Based Systems (IPMU), Madrid, Spain, 15–19 July 2002; pp. 985–991. [Google Scholar]
- Xu, Z.S.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst.
**2006**, 35, 417–433. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**2007**, 15, 1179–1187. [Google Scholar] - Xu, Z.S.; Chen, J. On geometric aggregation over interval-valued intuitionistic fuzzy information. In Proceedings of the 4th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), Haikou, China, 24–27 August 2007; pp. 466–471. [Google Scholar]
- Wang, J.Q.; Zhang, Z. Multi-criteria decision-making method with incomplete certain information based on intuitionistic fuzzy number. Control Decis.
**2009**, 24, 226–230. [Google Scholar] - Wei, G. Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision-making. Appl. Soft Comput.
**2010**, 10, 423–431. [Google Scholar] [CrossRef] - Zhao, H.; Xu, Z.S.; Ni, M.F.; Liu, S.S. Generalized aggregation operators for intuionistic fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 1–30. [Google Scholar] [CrossRef] - Merigo, J.M.; Gil-Lafuente, A.M. Fuzzy induced generalized aggregation operators and its application in multi-person decision-making. Expert Syst. Appl.
**2011**, 38, 9761–9772. [Google Scholar] [CrossRef] - Xu, Y.; Wang, H. The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision-making. Appl. Soft Comput.
**2012**, 12, 1168–1179. [Google Scholar] [CrossRef] - Shakeel, M.; Abdullah, S.; Sajjad Ali Khan, M.; Rahman, K. Averaging aggregation operators with interval pythagorean trapezoidal fuzzy numbers and their application to group decision-making. Punjab Univ. J. Math.
**2018**, 50, 147–170. [Google Scholar] - Zahedi Khameneh, A.; Kilicman, A. m-polar fuzzy soft weighted aggregation operators and their applications in group decision-making. Symmetry
**2018**, 10, 636. [Google Scholar] [CrossRef] [Green Version] - Xu, Z.S.; Da, Q.L. An Overview of operators for aggregating information. Int. J. Intell. Syst.
**2003**, 18, 953–969. [Google Scholar] [CrossRef] - Calvo, T.; Kolesárová, A.; Komorníková, M.; Mesiar, R. Aggregation operators: Properties, classes and construction methods. In Aggregation Operators. Studies in Fuzziness and Soft Computing; Calvo, T., Mayor, G., Mesiar, R., Eds.; Physica: Heidelberg, Germany, 2002; Volume 97, pp. 3–104. [Google Scholar]
- Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes. Inf. Sci.
**2011**, 181, 23–43. [Google Scholar] [CrossRef] [Green Version] - Martinez, D.L.L.R.; Acosta, J.C. Aggregation operators review—Mathematical properties and behavioral measures. Int. J. Intell. Syst. Tech. Appl.
**2015**, 10, 63–76. [Google Scholar] [CrossRef] [Green Version] - Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation functions: Means. Inf. Sci.
**2011**, 181, 1–22. [Google Scholar] [CrossRef] [Green Version] - Rosanisah, M.; Abdullah, L. Aggregation methods in group decision-making: A decade survey. Informatica
**2017**, 41, 71–86. [Google Scholar] - Calvo, T.; Pradera, A. Double aggregation operators. Fuzzy Sets Syst.
**2004**, 142, 15–33. [Google Scholar] [CrossRef] - García-Lapresta, J.L.; Pereira, R.A.M. The self-dual core and the anti-self-dual remainder of an aggregation operator. Fuzzy Sets Syst.
**2008**, 159, 47–62. [Google Scholar] [CrossRef] - Aczél, J. On mean values. Bull. Am. Math. Soc.
**1984**, 54, 392–400. [Google Scholar] [CrossRef] [Green Version] - Mesiar, R.; Špirkova, J.; Vavrıkova, L. Weighted aggregation operators based on minimization. Inf. Sci.
**2008**, 178, 1133–1140. [Google Scholar] [CrossRef] - Špirková, J.; Beliakov, G.; Bustince, H.; Fernandez, J. Mixture functions and their monotonicity. Inf. Sci.
**2019**, 481, 520–549. [Google Scholar] [CrossRef] - Del Moral, M.J.; Chiclana, F.; Tapia, J.M.; Herrera-Viedma, E. A comparative study on consensus measures in group decision-making. Int. J. Intell. Syst.
**2018**, 33, 1624–1638. [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. A
**1997**, 27, 126–133. [Google Scholar] [CrossRef] - Yager, R.; Rybalov, A. Bipolar aggregation using the Uninorms. Fuzzy Optim. Decis. Mak.
**2011**, 10, 59–70. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. On the use of aggregation operations in information fusion processes. Fuzzy Sets Syst.
**2004**, 142, 143–161. [Google Scholar] [CrossRef] [Green Version] - Komornikova, M.; Mesiar, R. Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst.
**2011**, 175, 48–56. [Google Scholar] [CrossRef] - Cauchy, A.L. Cours D’analyse de L’Ecole Royale Polytechnique. In Analyse Algebrique; Debure: Paris, France, 1821; Volume I. [Google Scholar]
- Zahedi Khameneh, A.; Kilicman, A. A fuzzy majority-based construction method for composed aggregation functions by using combination operator. Inf. Sci.
**2019**, 505, 367–387. [Google Scholar] [CrossRef] - Fagin, R.; Wimmers, E.L. A formula for incorporating weights into scoring rules. Theor. Comput. Sci.
**2000**, 239, 309–338. [Google Scholar] [CrossRef] [Green Version] - Yager, R.R. A new methodology for ordinal multiobjective decisions based on fuzzy sets. In Readings in Fuzzy Sets for Intelligent Systems; Elsevier: New York, NY, USA, 1993; pp. 751–756. [Google Scholar]
- Yager, R.R.; Filev, D.P. Induced ordered weighted averaging operators. IEEE Trans. Syst. Man. Cybern. B Cybern.
**1999**, 29, 141–150. [Google Scholar] [CrossRef] [PubMed] - Kacprzyk, J. Group decision-making with a fuzzy linguistic majority. Fuzzy Sets Syst.
**1986**, 18, 105–118. [Google Scholar] [CrossRef] - Kacprzyk, J.; Fedrizzi, M.; Nurmi, H. Group decision-making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets Syst.
**1992**, 49, 21–31. [Google Scholar] [CrossRef] - Sang, X.; Liu, X. Parametric extension of the most preferred OWA operator and its application in search engine’s rank. J. Appl. Math.
**2013**. [Google Scholar] [CrossRef] - Liu, X.; Han, B.; Chen, H.; Zhou, L. The probabilistic ordered weighted continuous OWA operator and its application in group decision-making. Int. J. Mach. Learn. Cybern.
**2019**, 10, 705–715. [Google Scholar] [CrossRef] - Yager, R.R. OWA aggregation with an uncertainty over the arguments. Inf. Sci.
**2019**, 52, 206–212. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. Weighted minimum and maximum operations in fuzzy set Theory. Inf. Sci.
**1986**, 39, 205–210. [Google Scholar] [CrossRef] - Ribeiro, R.A.; Pereira, R.A.M. Weights as functions of attribute satisfaction values. In Proceedings of the Workshop on Preference Modelling and Applications (EUROFUSE), Granada, Spain, 25–27 April 2001. [Google Scholar]
- Ribeiro, R.A.; Pereira, R.A.M. Generalized mixture operators using weighting functions: A comparative study with WA and OWA. Eur. J. Oper. Res.
**2003**, 145, 329–342. [Google Scholar] [CrossRef]

Course 1 | Course 2 | Course 3 | AM | |
---|---|---|---|---|

Student 1 | 0.9 | 0.5 | 0.7 | 0.7 |

Student 2 | 0.7 | 0.7 | 0.7 | 0.7 |

Student 3 | 0.5 | 0.6 | 0.4 | 0.5 |

Student 4 | 0.85 | 0.7 | 0.5 | 0.68 |

Course 1 | Course 2 | Course 3 | WAM | |
---|---|---|---|---|

0.3 | 0.5 | 0.2 | ||

Student 1 | 0.9 | 0.5 | 0.7 | 0.66 |

Student 2 | 0.7 | 0.7 | 0.7 | 0.7 |

Student 3 | 0.5 | 0.6 | 0.4 | 0.53 |

Student 4 | 0.85 | 0.7 | 0.5 | 0.705 |

Course 1 | Course 2 | Course 3 | IOWA | |
---|---|---|---|---|

First Exam | Third Exam | Second Exam | ||

0.3 | 0.2 | 0.5 | ||

Student 1 | 0.9 | 0.5 | 0.7 | 0.72 |

Student 2 | 0.7 | 0.7 | 0.7 | 0.7 |

Student 3 | 0.5 | 0.6 | 0.4 | 0.47 |

Student 4 | 0.85 | 0.7 | 0.5 | 0.645 |

Course 1 | Course 2 | Course 3 | ${\mathit{F}}_{2;\mathit{Max},\mathit{WAM}}^{\left(3\right)}$ | |
---|---|---|---|---|

0.3 | 0.5 | 0.2 | ||

Student 1 | 0.9 | 0.5 | 0.7 | 0.82 |

Student 2 | 0.7 | 0.7 | 0.7 | 0.7 |

Student 3 | 0.5 | 0.6 | 0.4 | 0.56 |

Student 4 | 0.85 | 0.7 | 0.5 | 0.75 |

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Zahedi Khameneh, A.; Kilicman, A.
Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview. *Symmetry* **2020**, *12*, 694.
https://doi.org/10.3390/sym12050694

**AMA Style**

Zahedi Khameneh A, Kilicman A.
Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview. *Symmetry*. 2020; 12(5):694.
https://doi.org/10.3390/sym12050694

**Chicago/Turabian Style**

Zahedi Khameneh, Azadeh, and Adem Kilicman.
2020. "Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview" *Symmetry* 12, no. 5: 694.
https://doi.org/10.3390/sym12050694