# Complex Fuzzy Geometric Aggregation Operators

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Complex Fuzzy Sets

**Definition**

**1**

**.**Let X be a universe, D be the set of complex numbers whose modulus is less than or equal to 1, i.e.,

- (i)
- Algebraic product:$$a\xb7b={t}_{a}\xb7{t}_{b}\xb7{e}^{j({\nu}_{a}+{\nu}_{b})}.$$
- (ii)
- Average:$$\frac{a+b}{2}=\frac{{t}_{a}cos{\nu}_{a}+{t}_{b}cos{\nu}_{b}}{2}+j\frac{{t}_{a}sin{\nu}_{a}+{t}_{b}sin{\nu}_{b}}{2}.$$

#### 2.2. Rotational Invariance and Reflectional Invariance

- (i)
- the rotation of a by $\theta $ radians, denoted $Ro{t}_{\theta}\left(a\right)$, is defined as$$Ro{t}_{\theta}\left(a\right)={t}_{a}\xb7{e}^{j({\nu}_{a}+\theta )};$$
- (ii)
- the reflection of a, denoted $Ref\left(a\right)$, is defined as$$Ref\left(a\right)={t}_{a}\xb7{e}^{j-{\nu}_{a}}.$$

**Definition**

**2**

**.**A function $f:{D}^{2}\to D$ is rotationally invariant if and only if

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.3. Complex Fuzzy Aggregation Operators

**Definition**

**5**

**.**Let ${A}_{1},{A}_{2},\dots ,{A}_{n}$ be CFSs defined on X. Vector aggregation on ${A}_{1},{A}_{2},\dots ,{A}_{n}$ is defined by a function v.

**Theorem**

**4.**

**Proof.**

## 3. Complex Fuzzy Weighted Geometric Operators

**Definition**

**6.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

- (1)
- (Idempotency): If ${a}_{1}={a}_{2}=\dots ={a}_{n}$ then$$CFWG({a}_{1},{a}_{2},\dots ,{a}_{n})={a}_{1}.$$
- (2)
- (Amplitude boundedness):$$\left|CFWG\right({a}_{1},{a}_{2},\dots ,{a}_{n}\left)\right|\le a,$$
- (3)
- (Amplitude monotonicity): If $|{a}_{i}|\le |{b}_{i}|$ $i=1,2,\dots ,n$, then$$|CFWAA({a}_{1},{a}_{2},\dots ,{a}_{n})|\le |CFWAA({b}_{1},{b}_{2},\dots ,{b}_{n})|.$$

**Proof.**

**Example**

**1.**

## 4. Complex Fuzzy Ordered Weighted Geometric Operators

**Definition**

**7.**

**Theorem**

**8.**

**Theorem**

**9.**

**Theorem**

**10.**

- (1)
- (Idempotency): If ${a}_{1}={a}_{2}=\dots ={a}_{n}$, then$$CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})={a}_{1}.$$
- (2)
- (Boundedness):$$\left|CFOWG\right({a}_{1},{a}_{2},\dots ,{a}_{n}\left)\right|\le a,$$
- (3)
- (Monotonicity): If $|{a}_{i}|\le |{b}_{i}|$ $i=1,2,\dots ,n$, then$$|CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})|\le |CFWAA({b}_{1},{b}_{2},\dots ,{b}_{n})|.$$

**Theorem**

**11.**

- (1)
- If $w=(1,0,\dots ,0)$, then$$|CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})|=\underset{i}{max}\left|{a}_{i}\right|;$$
- (2)
- If $w=(0,0,\dots ,1)$, then$$|CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})|=\underset{i}{min}\left|{a}_{i}\right|;$$
- (3)
- If ${w}_{i}=1,{w}_{k}=0,k\ne i$, then$$\left|CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})\right|=\left|{a}_{\sigma \left(i\right)}\right|,$$

## 5. Complex Fuzzy Values and Pythagorean Fuzzy Numbers

**Definition**

**8**

**.**Let X be a universe. A PFS A is defined by

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Theorem**

**14.**

**Proof.**

**Theorem**

**15.**

- (1)
- (Idempotency): If ${z}_{1}={z}_{2}=\dots ={z}_{n}$, then$$CFWG({z}_{1},{z}_{2},\dots ,{z}_{n})={z}_{1},$$$$CFOWG({z}_{1},{z}_{2},\dots ,{z}_{n})={z}_{1}.$$
- (2)
- (Amplitude boundedness):$$\left|CFWG\right({z}_{1},{z}_{2},\dots ,{z}_{n}\left)\right|\le z,$$$$\left|CFOWG\right({z}_{1},{z}_{2},\dots ,{z}_{n}\left)\right|\le z,$$
- (3)
- (Amplitude monotonicity): If $|{z}_{i}|\le |{y}_{i}|$ $i=1,2,\dots ,n$, then$$|CFWG({a}_{1},{a}_{2},\dots ,{a}_{n})|\le |CFWG({y}_{1},{y}_{2},\dots ,{y}_{n})|,$$$$|CFOWG({a}_{1},{a}_{2},\dots ,{a}_{n})|\le |CFWG({y}_{1},{y}_{2},\dots ,{y}_{n})|.$$

**Proof.**

## 6. Example Application

- Step 1
- Complexification of the measured results; each measurement is represented as ${c}_{i}={d}_{i}\xb7{e}^{{\theta}_{i}}$.
- Step 2
- (Fuzzification) Normalize the amplitudes of all measurements. Let $d={max}_{i}{d}_{i}$, for each ${c}_{i}$, the normalized result is ${a}_{i}={c}_{i}/d,$ where ${t}_{{a}_{i}}={d}_{i}/d$.
- Step 3
- (Aggregation) Produce an aggregate result. For simplicity, using the CFWG operator with weights $(1/n,1/n,\dots ,1/n)$. We obtain$$a=\prod _{i=1}^{n}{a}_{i}^{1/n},$$
- Step 4
- (Defuzzification) Calculate $c=a\xb7d$, where ${t}_{c}={t}_{a}\xb7d$.
- Step 5
- Decomplexification (or sometimes “realification”) of c. We get the target position $(p,\nu )$, where $p={t}_{c},\nu ={\nu}_{a}$.

- Step 1
- Complexifications of the measured results are calculated as$$\left((865\xb7{e}^{j2\pi 336/360}),(867\xb7{e}^{j2\pi 335/360}),(871\xb7{e}^{j2\pi 336/360}),(866\xb7{e}^{j2\pi 335/360}),(869\xb7{e}^{j2\pi 337/360})\right).$$
- Step 2
- Normalizations of the amplitudes of all measurements are calculated as$$\left(0.9931\xb7{e}^{j2\pi 336/360},0.9954\xb7{e}^{j2\pi 335/360},1\xb7{e}^{j2\pi 336/360},0.9943\xb7{e}^{j2\pi 335/360},0.9977\xb7{e}^{j2\pi 337/360}\right).$$
- Step 3
- Aggregation of CFVs is calculated as$$0.9961\xb7{e}^{j2\pi 335.8/360},$$
- Step 4
- Defuzzification of the aggregate result is calculated as $867.6\xb7{e}^{j2\pi 335.8/360}.$
- Step 5
- Decomplexification of the above result is calculated as $(867.6,24.2).$

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ramot, D.; Milo, R.; Friedman, M.; Kandel, A. Complex fuzzy sets. IEEE Trans. Fuzzy Syst.
**2002**, 10, 171–186. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Buckley, J.J. Fuzzy complex numbers. Fuzzy Sets Syst.
**1989**, 33, 333–345. [Google Scholar] [CrossRef] - Buckley, J.J. Fuzzy complex analysis I: Definition. Fuzzy Sets Syst.
**1991**, 41, 269–284. [Google Scholar] [CrossRef] - Buckley, J.J. Fuzzy complex analysis II: Integration. Fuzzy Sets Syst.
**1992**, 49, 171–179. [Google Scholar] [CrossRef] - Ramot, D.; Friedman, M.; Langholz, G.; Kandel, A. Complex fuzzy logic. IEEE Trans. Fuzzy Syst.
**2003**, 11, 450–461. [Google Scholar] [CrossRef] - Hu, B.; Bi, L.; Dai, S. The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection. Symmetry
**2017**, 9, 175. [Google Scholar] [CrossRef] - Bi, L.; Hu, B.; Li, S.; Dai, S. The Parallelity of Complex Fuzzy sets and Parallelity Preserving Operators. J. Intell. Fuzzy Syst.
**2018**, 34. [Google Scholar] [CrossRef] - Zhang, G.; Dillon, T.S.; Cai, K.-Y.; Ma, J.; Lu, J. Operation properties and delta-equalities of complex fuzzy sets. Int. J. Approx. Reason.
**2009**, 50, 1227–1249. [Google Scholar] [CrossRef] - Alkouri, A.U.M.; Salleh, A.R. Linguistic variables, hedges and several distances on complex fuzzy sets. J. Intell. Fuzzy Syst.
**2014**, 26, 2527–2535. [Google Scholar] - Hu, B.; Bi, L.; Dai, S.; Li, S. Distances of Complex Fuzzy Sets and Continuity of Complex Fuzzy Operations. J. Intell. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Tamir, D.E.; Jin, L.; Kandel, A. A new interpretation of complex membership grade. Int. J. Intell. Syst.
**2011**, 26, 285–312. [Google Scholar] [CrossRef] - Tamir, D.E.; Kandel, A. Axiomatic theory of complex fuzzy logic and complex fuzzy classes. Int. J. Comput. Commun. Control
**2011**, 4, 562–576. [Google Scholar] [CrossRef] - Dick, S. Towards Complex Fuzzy Logic. IEEE Trans. Fuzzy Syst.
**2005**, 13, 405–414. [Google Scholar] [CrossRef] - Greenfield, S.; Chiclana, F.; Dick, S. Interval-valued complex fuzzy logic. In Proceedings of the 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Vancouver, BC, Canada, 24–29 July 2016. [Google Scholar]
- Greenfield, S.; Chiclana, F.; Dick, S. Join and meet operations for interval-valued complex fuzzy logic. In Proceedings of the 2016 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS), El Paso, TX, USA, 31 October–4 November 2016; pp. 1–5. [Google Scholar]
- Alkouri, A.; Salleh, A. Complex intuitionistic fuzzy sets. In Proceedings of the AIP Conference International Conference on Fundamental and Applied Sciences, Kuala Lumpur, Malaysia, 12–14 June 2012; pp. 464–470. [Google Scholar]
- Ali, M.; Smarandache, F. Complex neutrosophic set. Neural Comput. Appl.
**2017**, 28, 1–18. [Google Scholar] [CrossRef] - Ma, J.; Zhang, G.; Lu, J. A method for multiple periodic factor prediction problems using complex fuzzy sets. IEEE Trans. Fuzzy Syst.
**2012**, 20, 32–45. [Google Scholar] - Chen, Z.; Aghakhani, S.; Man, J.; Dick, S. ANCFIS: A Neuro-Fuzzy Architecture Employing Complex Fuzzy Sets. IEEE Trans. Fuzzy Syst.
**2011**, 19, 305–322. [Google Scholar] [CrossRef] - Li, C. Complex Neuro-Fuzzy ARIMA Forecasting. A New Approach Using Complex Fuzzy Sets. IEEE Trans. Fuzzy Syst.
**2013**, 21, 567–584. [Google Scholar] [CrossRef] - Li, C.; Chiang, T.-W.; Yeh, L.-C. A novel self-organizing complex neuro-fuzzy approach to the problem of time series forecasting. Neurocomputing
**2013**, 99, 467–476. [Google Scholar] [CrossRef] - Al-Quran, A.; Hassan, N. The Complex Neutrosophic Soft Expert Relation and Its Multiple Attribute Decision-Making Method. Entropy
**2018**, 20, 101. [Google Scholar] [CrossRef] - Li, C.; Wu, T.; Chan, F.-T. Self-learning complex neuro-fuzzy system with complex fuzzy sets and its application to adaptive image noise canceling. Neurocomputing
**2012**, 94, 121–139. [Google Scholar] [CrossRef] - Yager, R.R.; Abbasov, A.M. Pythagorean membership grades, complex numbers and decision-making. Int. J. Intell. Syst.
**2013**, 28, 436–452. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst.
**2014**, 22, 958–965. [Google Scholar] [CrossRef] - Yager, R.R. Pythagorean fuzzy subsets. In Proceedings of the Joint IFSA Congress and NAFIPS Meeting, Edmonton, AB, Canada, 24–28 June 2013; pp. 57–61. [Google Scholar]
- Atanassov, K.T. Intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Dick, S.; Yager, R.R.; Yazdanbahksh, O. On Pythagorean and Complex Fuzzy Set Operations. IEEE Trans. Fuzzy Syst.
**2016**, 24, 1009–1021. [Google Scholar] [CrossRef] - Mayburov, S.N. Fuzzy Geometry of Commutative Spaces and Quantum Dynamics. EPJ Web Conf.
**2016**, 126, 05009. [Google Scholar] [CrossRef] [Green Version] - Bagarello, F.; Cinà, M.; Gargano, F. Projector operators in clustering. Appl. Sci.
**2017**, 40, 49–59. [Google Scholar] [CrossRef] - Yager, R.R. On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man. Cybern.
**1988**, 18, 183–190. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst.
**2007**, 15, 1179–1187. [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 and Clustering; Springer: Heidelberg/Berlin, Germany, 2013. [Google Scholar]
- Zeng, S.; Mu, Z.; Baležentis, T. A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making. Int. J. Intell. Syst.
**2018**, 33, 573–585. [Google Scholar] [CrossRef] - Zhang, X.L. A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int. J. Intell. Syst.
**2016**, 31, 593–611. [Google Scholar] [CrossRef] - Garg, H. A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making. Int. J. Intell. Syst.
**2016**, 31, 886–920. [Google Scholar] [CrossRef] - Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 2459–2466. [Google Scholar] - Tu, A.; Ye, J.; Wang, B. Multiple Attribute Decision-Making Method Using Similarity Measures of Neutrosophic Cubic Sets. Symmetry
**2018**, 10, 215. [Google Scholar] [CrossRef] - Tu, A.; Ye, J.; Wang, B. Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems. Symmetry
**2018**, 10, 144. [Google Scholar] [CrossRef] - Liu, P. Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making. IEEE Trans. Fuzzy Syst.
**2014**, 22, 83–97. [Google Scholar] [CrossRef] - Liu, P. Multiple Attribute Decision-Making Methods Based on Normal Intuitionistic Fuzzy Interaction Aggregation Operators. Symmetry
**2017**, 9, 261. [Google Scholar] [CrossRef] - Yu, D.; Wu, Y.; Lu, T. Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl. Based Syst.
**2012**, 30, 57–66. [Google Scholar] [CrossRef] - Hussain, S.A.I.; Mandal, U.K.; Mondal, S.P. Decision Maker Priority Index and Degree of Vagueness Coupled Decision Making Method: A Synergistic Approach. Int. J. Fuzzy Syst.
**2018**, 20, 1551–1566. [Google Scholar] [CrossRef] - Xia, M.; Xu, Z. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason.
**2011**, 52, 395–407. [Google Scholar] [CrossRef] [Green Version] - Zhu, B.; Xu, Z.; Xia, M. Hesitant fuzzy geometric Bonferroni means. Inf. Sci.
**2012**, 205, 72–85. [Google Scholar] [CrossRef] - Song, C.; Xu, Z.; Zhao, H. A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making. Symmetry
**2018**, 10, 177. [Google Scholar] [CrossRef] - Zhang, X.; Xu, Z. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int. J. Intell. Syst.
**2014**, 29, 1061–1078. [Google Scholar] [CrossRef]

**Table 1.**Properties of the complex fuzzy aggregation operators. √ and × represent that the corresponding property holds and does not hold, respectively.

Idempotency | Amplitude Boundedness | Amplitude Monotonicity | Reflectional Invariance | Rotational Invariance | |
---|---|---|---|---|---|

CFAA | √ | √ | × | √ | √ |

CFWA | √ | √ | × | √ | √ |

CFWG | √ | √ | √ | √ | √ |

CFOWG | √ | √ | √ | √ | √ |

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**MDPI and ACS Style**

Bi, L.; Dai, S.; Hu, B.
Complex Fuzzy Geometric Aggregation Operators. *Symmetry* **2018**, *10*, 251.
https://doi.org/10.3390/sym10070251

**AMA Style**

Bi L, Dai S, Hu B.
Complex Fuzzy Geometric Aggregation Operators. *Symmetry*. 2018; 10(7):251.
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**Chicago/Turabian Style**

Bi, Lvqing, Songsong Dai, and Bo Hu.
2018. "Complex Fuzzy Geometric Aggregation Operators" *Symmetry* 10, no. 7: 251.
https://doi.org/10.3390/sym10070251