**2. Preliminaries**

In this section, we present some basic material, including the concepts of CFSs [1], rotational invariance [14], reflectional invariance, and complex fuzzy aggregation operators [1].

#### *2.1. Complex Fuzzy Sets*

Ramot et al. [1] defined the concept of CFSs as follows.

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

$$D = \{ a \in \mathcal{C} \, | \, |a| \le 1 \},$$

*a complex fuzzy set A defined on X is a mapping: X* → *D, which can be denoted as below:*

$$A = \left\{ <\mathbf{x}, \mathbf{t}\_A(\mathbf{x}) \cdot e^{j\nu\_A(\mathbf{x})} > |\mathbf{x} \in X \right\},\tag{1}$$

*where j* = √−<sup>1</sup>*, the amplitude term tA*(*x*) *and the phase term <sup>ν</sup>A*(*x*) *are both real-valued, and tA*(*x*) ∈ [0, 1]*.*

For convenience, we only consider the complex numbers on *D*, called complex fuzzy values (CFVs). Let *a* = *ta* · *ej<sup>ν</sup><sup>a</sup>* be a CFV, then the amplitude of *a* is denoted by *ta* and the phase of *a* is denoted by *νa*. They are both real-valued, *ta* ∈ [0, 1]. The modulus of *a* is *ta*, denoted by |*a*|.

Let *a* = *ta* · *ej<sup>ν</sup><sup>a</sup>* and *b* = *tb* · *ej<sup>ν</sup><sup>b</sup>* be two CFVs, then we have the following two commonly used binary operations.

(i) Algebraic product:

$$a \cdot b = t\_a \cdot t\_b \cdot e^{j(\nu\_a + \nu\_b)}.\tag{2}$$

(ii) Average:

$$\frac{a+b}{2} = \frac{t\_a \cos \nu\_a + t\_b \cos \nu\_b}{2} + j \frac{t\_a \sin \nu\_a + t\_b \sin \nu\_b}{2}. \tag{3}$$

The partial ordering of CFVs is the traditional partial ordering by the modulus of a complex number, that is, *a* ≤ *b* if and only if |*a*|≤|*b*|, equivalently, *ta* ≤ *tb*.

*2.2. Rotational Invariance and Reflectional Invariance*

Let *a* = *ta* · *ej<sup>ν</sup><sup>a</sup>* be a CFV, then we have the following two commonly used unary operations:

(i) the rotation of *a* by *θ* radians, denoted *Rotθ* (*a*), is defined as

$$Rot\_{\theta}(a) = t\_a \cdot e^{j(v\_{\theta} + \theta)};\tag{4}$$

(ii) the reflection of *a*, denoted *Ref*(*a*), is defined as

$$\text{Ref}(a) = t\_a \cdot e^{j - \nu\_a}.\tag{5}$$

Then, based on the rotation operation, Dick [14] introduced the concept of rotational invariance for complex fuzzy operators.

**Definition 2** ([14])**.** *A function f* : *D*<sup>2</sup> → *D is rotationally invariant if and only if*

$$f\left(\operatorname{Rot}\_{\theta}(a), \operatorname{Rot}\_{\theta}(b)\right) = \operatorname{Rot}\_{\theta}\left(f(a, b)\right),\tag{6}$$

*for any θ.*

> We extend the above concept to multivariate operators.

**Definition 3.** *Let f* : *Dn* → *D be an n-order function. f is rotationally invariant if and only if*

$$f(\operatorname{Rot}\_{\theta}(a\_1), \operatorname{Rot}\_{\theta}(a\_2), \dots, \operatorname{Rot}\_{\theta}(a\_n)) = \operatorname{Rot}\_{\theta}(f(a\_1, a\_2, \dots, a\_n)),\tag{7}$$

*for any θ.*

In particular, since the periodicity of complex-valued membership grade, that is, *Rot*2*π*(*x*) = *x* for any *x* ∈ *D*, we have *f* -*Rot*2*π*(*<sup>a</sup>*1), *Rot*2*π*(*<sup>a</sup>*2), ..., *Rot*2*π*(*an*). = *f*(*<sup>a</sup>*1, *a*2, ..., *an*) = *Rot*2*π*- *f*(*<sup>a</sup>*1, *a*2, ..., *an*).. This is a special case of rotational invariance.

Similar to the above definition, we introduce the concept of reflectional invariance for complex fuzzy operators based on the reflection operation.

**Definition 4.** *Let f* : *Dn* → *D be an n-order function. f is reflectionally invariant if and only if*

$$f\left(\text{Ref}\left(a\_1\right), \text{Ref}\left(a\_2\right), \dots, \text{Ref}\left(a\_n\right)\right) = \text{Ref}\left(f\left(a\_1, a\_2, \dots, a\_n\right)\right). \tag{8}$$

Rotational invariance and reflectional invariance are intuitive properties for complex fuzzy operators. It makes a grea<sup>t</sup> deal of sense that a operator is invariant under a rotation or a reflection. If we rotate two vectors by a common value, rotational invariance states that an aggregation of those vectors will be rotated by the same value, as shown in Figure 1a. If we reflect two vectors, reflectional invariance states that an aggregation of those vectors will be reflected as well, as shown in Figure 1b.

**Figure 1.** (**a**) Rotational invariance and (**b**) reflectional invariance.

Reflectional invariance and rotational invariance are two properties which are only concerned with the phase of CFVs.

These two properties of the algebraic product and average operators were examined, and the results are given as follows.

**Theorem 1** ([14])**.** *The algebraic product is not rotationally invariant.*

**Theorem 2.** *The algebraic product is reflectionally invariant.*

**Proof.** For any *a*, *b* ∈ *D*, we have

$$\operatorname{Ref}(a) \cdot \operatorname{Ref}(b) = t\_{\mathfrak{a}} \cdot \mathfrak{e}^{j - \nu\_{\mathfrak{a}}} \cdot t\_{\mathfrak{b}} \cdot \mathfrak{e}^{j - \nu\_{\mathfrak{b}}} = t\_{\mathfrak{a}} \cdot t\_{\mathfrak{b}} \cdot \mathfrak{e}^{j(-\nu\_{\mathfrak{a}} - \nu\_{\mathfrak{b}})},$$

$$\operatorname{Ref}(a \cdot b) = \operatorname{Ref}\left(t\_{\mathfrak{a}} \cdot t\_{\mathfrak{b}} \cdot \mathfrak{e}^{j(\nu\_{\mathfrak{a}} + \nu\_{\mathfrak{b}})}\right) = t\_{\mathfrak{a}} \cdot t\_{\mathfrak{b}} \cdot \mathfrak{e}^{j(-\nu\_{\mathfrak{a}} - \nu\_{\mathfrak{b}})},$$

then *Ref*(*a*) · *Ref*(*b*) = *Ref*(*a* · *b*).

**Theorem 3.** *The average operator is reflectionally invariant and rotationally invariant.*

**Proof.** (i) Let *a* = *ra* + *j<sup>ω</sup>a*, *b* = *rb* + *j<sup>ω</sup>b* ∈ *D*. We have

$$\frac{\operatorname{Ref}(a) + \operatorname{Ref}(b)}{2} = \frac{r\_a + r\_b}{2} + j\frac{-\omega\_a - \omega\_b}{2} = \frac{r\_a + r\_b}{2} - j\frac{\omega\_a + \omega\_b}{2} = \operatorname{Ref}(\frac{a+b}{2}).$$

Then, the average operator is reflectionally invariant.

(ii) For any *a*, *b* ∈ *D*, we have

$$\frac{a \cdot e^{j\theta} + b \cdot e^{j\theta}}{2} = \frac{(a+b) \cdot e^{j\theta}}{2} = \frac{(a+b)}{2} \cdot e^{j\theta}.$$

Then, the average operator is rotationally invariant.

*2.3. Complex Fuzzy Aggregation Operators*

> Ramot et al. [6] defined the aggregation operation on CFSs as vector aggregation:

**Definition 5** ([6])**.** *Let A*1, *A*2, ..., *An be CFSs defined on X. Vector aggregation on A*1, *A*2, ..., *An is defined by a function v.*

$$v: \quad D^{\mathfrak{n}} \to D. \tag{9}$$

*The function v produces an aggregate CFS A by operating on the membership grades of A*1, *A*2, ..., *An for each x* ∈ *X. For all x* ∈ *X, v is given by*

$$\begin{split} \mu\_{A}(\mathbf{x}) &= \upsilon \Big( t\_{A\_{1}} \cdot \mathbf{e}^{j\nu\_{A\_{1}}}, \mathbf{t}\_{A\_{2}} \cdot \mathbf{e}^{j\nu\_{A\_{2}}}, \dots, \mathbf{t}\_{A\_{n}} \cdot \mathbf{e}^{j\nu\_{A\_{n}}} \Big) \\ &= \sum\_{i=1}^{n} \left( \upsilon\_{i} \cdot \mathbf{t}\_{A\_{i}} \cdot \mathbf{e}^{j\nu\_{A\_{i}}} \right), \end{split} \tag{10}$$

*where wi* ∈ *D for all i, and* ∑*ni*=<sup>1</sup> |*wi*| = 1*.*

The complex weights are used in Ramot et al.'s definition for the purpose of maintaining a definition that is as general as possible. In this paper, we only discuss the complex fuzzy aggregation operations with real-valued weights.

We notice that the above definition of Ramot et al's [6] aggregation operator is a complex fuzzy weighted arithmetic (CFWA) operator. For convenience, let *a*1, *a*2, ..., *an* be CFVs. The CFWA operator is given as

$$CFWA(a\_1, a\_2, \dots, a\_n) = \sum\_{i=1}^{n} \left(w\_i \cdot a\_i\right),\tag{11}$$

where *wi* ∈ [0, 1] for all *i*, and ∑*ni*=<sup>1</sup> *wi* = 1.

When *a*1, *a*2, ..., *an* ∈ [0, 1], the CFWA operator can reduce to a traditional fuzzy weighted arithmetic operator.

When *wi* = 1/*n* for all *i*, then the CFWA operator is the arithmetic average of (*<sup>a</sup>*1, *a*2, ..., *an*), denoted by the complex fuzzy arithmetic average (CFAA) operator. That is,

$$
\mathbb{C}FAA(a\_1, a\_2, \dots, a\_n) = \frac{1}{n} \cdot \sum\_{i=1}^n a\_i. \tag{12}
$$

When *a*1, *a*2, ..., *an* ∈ [0, 1] and *wi* = 1/*n* for all *i*, the CFAA operator is the arithmetic mean of numbers on [0,1].

As a special case of the CFWA operator, note that the average operator on CFVs is reflectionally invariant and rotationally invariant (see Theorem 3). We show that the CFWA operator also possesses these two properties.

**Theorem 4.** *The CFWA operator is reflectionally invariant and rotationally invariant.* **Proof.** (i) Let *a*1 = *ra*1 + *j<sup>ω</sup>a*1 , *a*2 = *ra*2 + *j<sup>ω</sup>a*2 , ··· , *an* = *ran* + *j<sup>ω</sup>an* be CFVs. We have

$$\begin{aligned} \left(\operatorname{Ref}\left(\operatorname{CFWA}(a\_1, a\_2, \ldots, a\_n)\right)\right) &= \operatorname{Ref}\left(\sum\_{i=1}^n \left(w\_i \cdot r\_{a\_i}\right) + j \sum\_{i=1}^n \left(w\_i \cdot \omega\_{a\_i}\right)\right) \\ &= \sum\_{i=1}^n \left(w\_i \cdot r\_{a\_i}\right) - j \sum\_{i=1}^n \left(w\_i \cdot \omega\_{a\_i}\right) \\ &= \sum\_{i=1}^n \left(w\_i \cdot \left(r\_{a\_i} - \omega\_{a\_i}\right)\right) \\ &= \sum\_{i=1}^n \left(w\_i \cdot \operatorname{Ref}(a\_i)\right). \end{aligned}$$

Then, the CFWA operator is reflectionally invariant. (ii) For any CFVs *a*1, *a*2, ..., *an*, we have

$$\begin{aligned} \text{CFWA}(a\_1 \cdot e^{j\theta}, a\_2 \cdot e^{j\theta}, \dots, a\_{\text{Il}} \cdot e^{j\theta}) &= w\_1 \cdot a\_1 \cdot e^{j\theta} + w\_2 \cdot a\_2 \cdot e^{j\theta} + \dots + w\_{\text{Il}} \cdot a\_{\text{Il}} \cdot e^{j\theta} \\ &= (w\_1 \cdot a\_1 + w\_2 \cdot a\_2 + \dots + w\_{\text{Il}} \cdot a\_{\text{Il}}) \cdot e^{j\theta} \\ &= \text{CFWA}(a\_1, a\_2, \dots, a\_{\text{Il}}) \cdot e^{j\theta} .\end{aligned}$$

Then, the CFWA operator is rotationally invariant.

#### **3. Complex Fuzzy Weighted Geometric Operators**

In this section, we introduce the weighted geometric aggregation operators in a complex fuzzy environment and discuss their fundamental characteristics.

**Definition 6.** *Let a*1, *a*2, ..., *an be CFVs, a complex fuzzy weighted geometric (CFWG) operator is defined as:*

$$\text{CFWG}(a\_1, a\_2, \dots, a\_n) = \prod\_{i=1}^n a\_i^{w\_i} \tag{13}$$

*where wi* ∈ [0, 1] *for all i, and* ∑*n i*=1 *wi* = 1*.*

Denoting *CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*) = *t* · *<sup>e</sup>j<sup>ν</sup>*, we have a weighted geometric aggregation (WGA) operator on [0,1], that is, *t* = ∏*n i*=1 *t wi ai* and a weighted arithmetic aggregation (WAA) operator on R, that is, *ν* = ∑*n i*=1 *wi* · *<sup>ν</sup>ai* .

When *a*1, *a*2, ..., *an* ∈ [0, 1], the CFWG operator can reduce to a traditional fuzzy weighted geometric operator.

When *wi* = 1/*n* for all *i*, then *t* = L*n ta*1 · *ta*2 ··· *tan* is the geometric mean of real numbers on unit interval [0,1], *ν* = 1 *n*· ∑*n i*=1*<sup>ν</sup>ai*is the arithmetic mean of real numbers on R.

When *a*1, *a*2, ..., *an* ∈ [0, 1] and *wi* = 1/*n* for all *i*, the CFWG operator is the geometric mean of real numbers on unit interval [0,1].

**Theorem 5.** *Let a*1, *a*2, ..., *an be CFVs, then the aggregated value CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*) *is also a complex fuzzy value.*

**Proof.** Since |*CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*)| = ∏*n i*=1 *t wi ai* , which is a weighted arithmetic aggregation operator on unit interval [0,1], we have |*CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*)| ≤ 1.

Similar to Theorem 4, the CFWA operator is reflectionally invariant and rotationally invariant. We show that the CFWG operator also possesses these two properties.

**Theorem 6.** *The CFWG operator is reflectionally invariant and rotationally invariant.* **Proof.** (i) For any CFVs *a*1, *a*2, ..., *an*, we have

$$\begin{split} \operatorname{Ref}\left(\operatorname{CFWG}(a\_1, a\_2, \ldots, a\_n)\right) &= \operatorname{Ref}\left(\prod\_{i=1}^n t\_{a\_i}^{w\_i} \cdot e^{j\sum\_{i=1}^n w\_i \cdot u\_{i}}\right) \\ &= \prod\_{i=1}^n t\_{a\_i^w}^{w\_i} \cdot e^{j - \sum\_{i=1}^n w\_i \cdot u\_{i}} \\ &= \prod\_{i=1}^n t\_{a\_i^w}^{w\_i} \cdot e^{j\sum\_{i=1}^n w\_i \cdot (-v\_{a\_i})} \\ &= \prod\_{i=1}^n \operatorname{Ref}(a\_i)^{w\_i} \\ &= \operatorname{CFWG}\left(\operatorname{Ref}(a\_1), \operatorname{Ref}(a\_2), \ldots, \operatorname{Ref}(a\_n)\right); \end{split}$$

(ii) and

$$\begin{aligned} CFWG(a\_1 \cdot e^{j\theta}, a\_2 \cdot e^{j\theta}, \dots, a\_n \cdot e^{j\theta}) &= a\_1^{w\_1} \cdot e^{jw\_1\theta} \cdot a\_2^{w\_1} \cdot e^{jw\_2\theta} \cdot \dots \cdot a\_n^{w\_n} \cdot e^{jw\_n\theta} \\ &= \left(\prod\_{i=1}^n a\_i^{w\_i}\right) \cdot e^{j(w\_1\cdot\theta + w\_2\cdot\theta + \dots + w\_n\cdot\theta)} \\ &= CFWG(a\_1, a\_2, \dots, a\_n) \cdot e^{j\theta} .\end{aligned}$$

Idempotency, boundedness, and monotonicity are three important properties of aggregation operators. The CFWG operator satisfies the following properties.

**Theorem 7.** *Let a*1, *a*2, ..., *an, b*1, *b*2, ..., *bn be CFVs, CFWG weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup>*wi* = 1*. Then, we have the following:*

*(1) (Idempotency): If a*1 = *a*2 = ... = *an then*

$$
\Gamma^
\sigma \mathcal{W} G(a\_1, a\_2, \dots, a\_{\mathcal{U}}) = a\_1.
$$

*(2) (Amplitude boundedness):*

$$\left| CFWG(a\_1, a\_2, \ldots, a\_n) \right| \le a\_\prime$$

*where a* = max *i*|*ai*|*.*

 *(3) (Amplitude monotonicity): If* |*ai*|≤|*bi*| *i* = 1, 2, ..., *n, then*

$$|\text{CFWAA}(a\_1, a\_2, \dots, a\_n)| \le |\text{CFWAA}(b\_1, b\_2, \dots, b\_n)|.$$

**Proof.** (1) Trivial form the facts that both WAA operator on [0,1] and WGA operator on R satisfy the property of idempotency.

(2) Trivial form the facts that |*CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*)| = ∏*ni*=<sup>1</sup> *twi ai* and WGA operator on R satisfy the property of boundedness.

(3) Trivial form the facts that |*CFWG*(*<sup>a</sup>*1, *a*2, ..., *an*)| = ∏*ni*=<sup>1</sup> *twi ai* and WGA operator on R satisfy the property of monotonicity.

In this paper, for complex fuzzy aggregation operators, boundedness and monotonicity are restricted exclusively to the amplitude of CFVs. They are two properties which are only concerned with the amplitude of CFVs. Idempotency is a property that is concerned with both the phase and amplitude of CFVs.

It is easy to prove that the CFWA operator satisfies idempotency and amplitude boundedness, but it does not satisfy the property of amplitude monotonicity.

**Example 1.** *Let a*1 = 0.4*, a*2 = 0.4 · *<sup>e</sup>j*2*π*/3*, b*1 = *b*2 = 0.3 *and weights be w*1 = *w*2 = 0.5*. Then,*

$$\begin{aligned} \text{CFWA}(a\_1, a\_2) &= 0.5 \cdot 0.4 + 0.5 \cdot 0.4 \cdot e^{j2\pi/3} \\ &= 0.2 \cdot e^{j\pi/3} \end{aligned}$$

*and CFWA*(*b*1, *b*2) = 0.3*. Then,* |*<sup>a</sup>*1|≥|*b*1|*,* |*<sup>a</sup>*2|≥|*b*2|*, but* |*CFWA*(*<sup>a</sup>*1, *<sup>a</sup>*2)|≤|*CFWA*(*b*1, *b*2)|*.*

#### **4. Complex Fuzzy Ordered Weighted Geometric Operators**

Based on the partial ordering of complex numbers, we propose a complex fuzzy ordered weighted geometric (CFOWG) operator as follows:

**Definition 7.** *Let a*1, *a*2, ..., *an be CFVs, a CFOWG operator is defined as*

$$\text{CFOWG}(a\_1, a\_2, \dots, a\_n) = \prod\_{i=1}^n a\_{\sigma(i)'}^{w\_i} \tag{14}$$

*where wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup> *wi* = 1*,* (*σ*(1), *<sup>σ</sup>*(2), ..., *σ*(*n*)) *is a permutation of* (1, 2, ..., *n*) *such that* |*<sup>a</sup>σ*(*<sup>i</sup>*−<sup>1</sup>)|≥|*<sup>a</sup>σ*(*i*)| *for all i.*

Especially when *wi* = 1/*n* for all *i*, then the CFOWG operator is reduced to the CFWG operator. Similar to Theorem 5, we have the following.

**Theorem 8.** *Let a*1, *a*2, ..., *an be CFVs, then the aggregated value CFOWG*(*<sup>a</sup>*1, *a*2, ..., *an*) *is also a complex fuzzy value.*

Similar to Theorem 6, the CFWG operator is reflectionally invariant and rotationally invariant. The CFOWG operator also possesses these two properties.

**Theorem 9.** *The CFOWG operator is reflectionally invariant and rotationally invariant.*

Similar to Theorem 7, the CFOWG operator satisfies idempotency, amplitude boundedness, and amplitude monotonicity.

**Theorem 10.** *Let a*1, *a*2, ..., *an, b*1, *b*2, ..., *bn be CFVs, CFOWAA weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup>*wi* = 1*. Then, we have the following:*

*(1) (Idempotency): If a*1 = *a*2 = ... = *an, then*

$$
\Gamma \\
FOWG(a\_1, a\_2, \dots, a\_n) = a\_1 \dots$$

*(2) (Boundedness):*

$$|\mathcal{F}FOWG(a\_1, a\_2, \dots, a\_n)| \le a\_\prime$$

*where a* = max *i*|*ai*|*.*

*(3) (Monotonicity): If* |*ai*|≤|*bi*| *i* = 1, 2, ..., *n, then*

$$|\text{CFOWG}(a\_1, a\_2, \dots, a\_n)| \le |\text{CFVMAX}(b\_1, b\_2, \dots, b\_n)|.$$

Besides the above properties, the CFOWG operator has the following.

**Theorem 11.** *Let a*1, *a*2, ..., *an be CFVs, CFOWG weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup> *wi* = 1*. Then, we have the following:*

*(1) If w* = (1, 0, ..., <sup>0</sup>)*, then*

> |*CFOWG*(*<sup>a</sup>*1, *a*2, ..., *an*)| = max*i*|*ai*|;

*i*

*(2) If w* = (0, 0, ..., <sup>1</sup>)*, then*

H H*CFOWG*(*<sup>a</sup>*1, *a*2, ..., *an*) H H = min|*ai*|;

*(3) If wi* = 1, *wk* = 0, *k* = *i, then*

$$|\mathcal{F}FOWG(a\_1, a\_2, \dots, a\_n)| = |a\_{\sigma(i)}|\_{\prime \sigma}$$

*where <sup>a</sup>σ*(*i*) *is the i-th (modulus-based) largest of a*1*,a*2*, ...,an.*

Now we give a brief summary of the properties of the CFWG and CFOWG operators with real-valued weights. The results are summarized in Table 1, in which √ and × represent that the corresponding property holds and does not hold, respectively.

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


#### **5. Complex Fuzzy Values and Pythagorean Fuzzy Numbers**

Yager and Abbasov [25] showed that Pythagorean membership grades can be expressed using complex numbers, called Π − *i* numbers, which belong to the upper-right quadrant of the complex unit disk. Essentially, the CFWAA and CFOWAA operators are used to deal with special complex numbers, which belong to the complex unit disk.

In this section, we consider the CFWG and CFOWG operators in the domain of Π − *i* numbers. We first recall the concepts of Pythagorean fuzzy sets (PFSs) and Π − *i* numbers.

**Definition 8** ([25])**.** *Let X be a universe. A PFS A is defined by*

$$A = \left\{ <\mathbf{x}, p\_A(\mathbf{x}), \nu\_A(\mathbf{x}), > \, | \, \mathbf{x} \in X \right\},\tag{15}$$

*where pA*(*x*) ∈ [0, 1] *and <sup>ν</sup>A*(*x*) ∈ [0, 1] *respectively represent the membership grade and nonmembership grade of the element x to set A, such that*

$$0 \le \left(p\_A(\mathfrak{x})\right)^2 + \left(\nu\_A(\mathfrak{x})\right)^2 \le 1$$

*for all x* ∈ *X. The degree of indeterminacy of the element x to set A is <sup>π</sup>A*(*x*)*, defined by*

$$
\pi\_A(\mathbf{x}) = \sqrt{1 - \left(p\_A(\mathbf{x})\right)^2 - \left(\nu\_A(\mathbf{x})\right)^2}.
$$

For convenience, Zhang and Xu [49] referred to - *pA*(*x*), *<sup>ν</sup>A*(*x*) . as a Pythagorean fuzzy number (PFN) simply denoted by *a* = (*pa*, *<sup>ν</sup>a*), where *pa* ∈ [0, 1], *νa* ∈ [0, 1] and (*pa*)<sup>2</sup> + (*<sup>ν</sup>a*)<sup>2</sup> ≤ 1.

Yager and Abbasov [25] discussed the relationship between Pythagorean membership grades and complex numbers. They showed that the complex numbers of the form *z* = *r* · *ej<sup>θ</sup>* with conditions *r* ∈ [0, 1] and *θ* ∈ [0, *π*/2] can be interpretable as PFNs (*r* cos *θ*,*<sup>r</sup>* sin *<sup>θ</sup>*). They referred to these complex numbers as "Π − *i* numbers", which are complex numbers in the upper-right quadrant of the complex unit disk.

As explained in [25], we should consider which aggregation operation is closed under Π − *i* numbers.

Let us consider the closeness of Π − *i* numbers under the CFWG and CFOWG operations. For the CFWG operator, we have the following result.

**Theorem 12.** *Let z*1, *z*2, ..., *zn be* Π − *i numbers, and the CFWG weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup> *wi* = 1*. Then, the aggregated value CFWG*(*<sup>z</sup>*1, *z*2, ..., *zn*) *is also a* Π − *i number.*

**Proof.** Denoting *CFWG*(*<sup>z</sup>*1, *z*2, ..., *zn*) = *t* · *ej<sup>ν</sup>* = ∏*ni*=<sup>1</sup> *twi zi* · *ej* ∑*ni*=<sup>1</sup> *wi*·*νzi* , from Theorem 2, we have *t* = HH*CFWG*(*<sup>z</sup>*1, *z*2, ..., *zn*)HH ≤ 1.

Since *ν* = ∑*ni*=<sup>1</sup> *wi* · *<sup>ν</sup>zi* is a weighted geometric aggregation (WGA) operator of real numbers on [0, *<sup>π</sup>*/2], then we have *ν* ∈ [0, *<sup>π</sup>*/2]. Thus, *CFWG*(*<sup>z</sup>*1, *z*2, ..., *zn*) is also a Π − *i* number.

Similar to the above Theorem, we have the following.

**Theorem 13.** *Let z*1, *z*2, ..., *zn be* Π − *i numbers, and the CFOWG weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup>*wi* = 1*. Then, the aggregated value CFOWG*(*<sup>z</sup>*1, *z*2, ..., *zn*) *is also a* Π − *i number.*

The above theorems show us that the CFWG and the CFOWG operators are closed under Π − *i* numbers. When PFNs are interpreted as Π − *i* numbers, then we can aggregate these PFNs to a PFN by using the CFWG or CFOWG operator.

From the above theorems, the CFWG and the CFOWG operators are closed on the upper-right quadrant of complex unit disk.

Consider other quadrants of the complex unit disk. Let

$$D\_k = \left\{ z = r \cdot e^{j\theta} \, | \, r \in [0, 1], \theta \in \left[\frac{(k - 1)\pi}{2}, \frac{k\pi}{2}\right] \right\}$$

for *k* = 1 to 4. *D*1 is the set of Π − *i* numbers.

Now, we discuss the closeness of the CFWG and the CFOWG operators on other quadrants of the complex unit disk. Plainly, we have the following.

**Theorem 14.** *For any k* ∈ {1, 2, 3, <sup>4</sup>}*, if z*1, *z*2, ..., *zn* ∈ *Dk, and the weights are real values, that is, wi* ∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup> *wi* = 1*. Then, we have*

$$\begin{aligned} \text{CFWG}(z\_1, z\_2, \dots, z\_n) &\in D\_{k'} \\\\ \text{CFWG}(z\_1, z\_2, \dots, z\_n) &\in D\_k \end{aligned}$$

**Proof.** Similar to Theorem 13.

**Theorem 15.** *For any k* ∈ {1, 2, 3, <sup>4</sup>}*, if z*1, *z*2, ..., *zn*, *y*1, *y*2, ..., *yn* ∈ *Dk, and the weights are real values, that is, wi*∈ [0, 1] *for all i, and* ∑*ni*=<sup>1</sup>*wi*= 1*. Then, we have the following:*

*(1) (Idempotency): If z*1 = *z*2 = ... = *zn, then*

$$CFWG(z\_1, z\_2, \dots, z\_n) = z\_{1'}$$

$$CFWGG(z\_1, z\_2, \dots, z\_n) = z\_1.$$

*(2) (Amplitude boundedness):*

$$\left| CFWG(z\_1, z\_2, \ldots, z\_n) \right| \le z\_\prime$$

$$\left| CFOWG(z\_1, z\_2, \ldots, z\_n) \right| \le z\_\prime$$

*where z* = max *i*|*zi*|*.*

$$(3) \quad (A. \text{multi} \land \text{monotonicity}) \text{: If } |z\_i| \le |y\_i| \text{ i } = 1, 2, \dots, n \text{, then}$$

$$|\mathcal{C}FWG(a\_1, a\_2, \dots, a\_n)| \le |\mathcal{C}FWG(y\_1, y\_2, \dots, y\_n)|\_{\prime}$$

$$|\mathbb{C}FOWG(a\_1, a\_2, \dots, a\_n)| \le |\mathbb{C}FWGG(y\_1, y\_2, \dots, y\_n)|.$$

**Proof.** Similar to Theorem 7.

## **6. Example Application**

In this section, we consider a target location application of the complex fuzzy aggregation operator. We do not intend to show the potential advantages of using complex fuzzy aggregation methods in comparison with existing alternative aggregation approaches in this section.

Assume the observer position is fixed. Using a position sensor and an angular sensor, the observer measures the distance and angle of the fixed target. To improve the target location accuracy, the observer repeatedly measures the same target. Then, the target position is estimated according to aggregation theory.

Assume *n* measurements -(*d*1, *<sup>θ</sup>*1),(*d*2, *<sup>θ</sup>*2), ...,(*dn*, *<sup>θ</sup>n*).have been measured by the observer. The target position is estimated in the following five stages, as illustrated in Figure 2.


$$a = \prod\_{i=1}^{n} a\_i^{1/n} \,\prime \,\tag{16}$$

where *ta* = L*n ta*1 · *ta*2 ··· *tan* and *νa* = 1*n* · ∑*ni*=<sup>1</sup> *θi*.


**Figure 2.** A simple method of target location based on complex fuzzy aggregation.

Numerical Example:Assume that the observer obtains five measurements as follows:

> -(865, <sup>24</sup>◦),(867, <sup>25</sup>◦),(871, <sup>24</sup>◦),(866, <sup>25</sup>◦),(869, <sup>23</sup>◦).,

where (*d*, *θ*) means that the target lies on the *θ* degrees east of south of the observer and *d* metres from the observer. Then,

Step 1 Complexifications of the measured results are calculated as

> -(865 · *<sup>e</sup>j*2*π*336/360),(<sup>867</sup> · *<sup>e</sup>j*2*π*335/360),(<sup>871</sup> · *<sup>e</sup>j*2*π*336/360),(<sup>866</sup> · *<sup>e</sup>j*2*π*335/360),(<sup>869</sup> · *<sup>e</sup>j*2*π*337/360)..

Step 2 Normalizations of the amplitudes of all measurements are calculated as

> -0.9931 · *ej*2*π*336/360, 0.9954 · *ej*2*π*335/360, 1 · *ej*2*π*336/360, 0.9943 · *ej*2*π*335/360, 0.9977 · *<sup>e</sup>j*2*π*337/360..

Step 3 Aggregation of CFVs is calculated as

> 0.9961 · *ej*2*π*335.8/360,

where the weights are (1/5, 1/5, ..., 1/5).

Step 4 Defuzzification of the aggregate result is calculated as 867.6 · *ej*2*π*335.8/360.

Step 5 Decomplexification of the above result is calculated as (867.6, 24.2).

Then, the target position is estimated at (867.6, 24.2). That is, it lies 24.2 degrees east of south of the observer and 867.6 m from the observer.

Note that we do not discuss how to choose complex fuzzy aggregation functions, nor their weights.
