Complex fuzzy arithmetic aggregation operators

Abstract

A complex fuzzy set, characterized by complex-valued membership functions, is a generalization of a fuzzy set. In this paper, we present complex fuzzy arithmetic aggregation (CFAA) operators, complex fuzzy weighted arithmetic aggregation (CFWAA) operators. Based on the partial ordering of complex fuzzy values, we develop complex fuzzy ordered weighted arithmetic aggregation (CFOWAA) operators. Highly pertinent to complex fuzzy operations is the concept of rotational. This concept has been developed for complex fuzzy aggregation operators. We show that CFAA, CFWAA and CFOWAA operators possess the property of rotational invariance. Moreover, based on the relationship between Pythagorean membership grades and complex numbers, these operators are studied under Pythagorean fuzzy values.

Keywords

Complex fuzzy sets complex fuzzy arithmetic aggregation (CFAA) operators complex fuzzy ordered weight arithmetic aggregation (CFOWAA) operators pythagorean fuzzy sets

1. Introduction

Ramot et al. [1] introduced the concept of complex fuzzy sets (CFS), which is an extension of the concept of fuzzy sets [2]. Recently, the CFS has received more and more scholars’ attention. Ramot et al. [3] then introduced several basic complex fuzzy operations and a method of complex fuzzy inference. Dick [4] discussed complex fuzzy logic based on the concept of rotational invariance, which is an intuitive and desirable property. Hu et al. [5 –7] established the orthogonality and parallelity relations of CFSs. Zhang et al. [8] proposed the δ-equalities of CFSs. Alkouri and Salleh [9] defined several distances between CFSs, and discussed complex linguistic variables. Tamir et al. [10] gave a new interpretation of complex membership grade. Tamir et al. [11, 12] then defined a complex fuzzy propositional logic and a first-order logic. Recently, Greenfield et.al [13] developed the interval-valued complex fuzzy sets and interval-valued complex fuzzy operations. They [14] also discussed the property of rotational invariance for interval-valued complex fuzzy operations. CFSs have a relationship with Pythagorean fuzzy sets (PFS), which is developed by Yager [15, 16] as a generalization of Atanssov’s intuitionistic fuzzy sets [17]. Yager and Abbsocv [18] showed that Pythagorean fuzzy grades can be viewed as a subclass of complex numbers called Π - i numbers. This is a set of the upper-right quadrant of the unit disc of complex plane. Dick et al. [19] examined the lattice-theoretic properties of PFSs and then extended them to CFSs.

Information aggregation plays an important role in many different fields of decision making. In the past decades, many aggregation techniques have been developed. The ordered weighted averaging (OWA) operator [20] is one of the well-known aggregation functions. These aggregation techniques have been used in many different fuzzy environments, such as intuitionistic [21 –23] and Pythagorean fuzzy environments [24, 25].

However, comparatively little investigation has been made on the aggregation operators in complex fuzzy environment. In Ref. [3], Ramot et al. defined the complex fuzzy aggregation operations with complex weights, which termed as vector aggregation. But the research on complex fuzzy aggregation operations has not been studied in details. In this paper, we study the complex fuzzy aggregation operations and their properties. First, we review some basic concepts of CFSs in Section 2. In Section 3, we study the complex fuzzy weighted arithmetic aggregation (CFWAA) operator on CFSs and its properties. In Section 4, we developed the complex fuzzy ordered weighted arithmetic aggregation (CFOWAA) operator based on the modulus values of the complex numbers, and study its properties. In Section 5, we study these operators in the domain of Π - i numbers. Conclusions are made in Section 6.

2. Preliminaries

In this section, we review some concepts of the CFSs and complex fuzzy operators. In addition, we also review the rotational invariance of complex fuzzy operator.

2.1. Complex fuzzy sets

Ramot et al. [1] generalized the concept of fuzzy set and defined the concept of CFS as follows.

Definition 1. [1] Let X be a fixed set, a CFS A is defined in the following form: $A = {< x, μ_{A} (x) > | x \in X}$ (1) where the complex-valued membership function μ_A (x) is of the form r_A (x) · e^jθ_A(x), $j = \sqrt{- 1}$ , the amplitude term r_A (x) and the phase term θ_A (x) are both real-valued, and r_A (x) ∈ [0, 1].

For convenience, let a = r_a · e^{jθ
_a} be a complex fuzzy value (CFV), where r_a and θ_a are both real-valued, and r_a ∈ [0, 1], |a| = r_a, $ℂ$ be the set of complex numbers and let $D = {a \in ℂ | | a | \leq 1}$ be the unit disc of the complex plane.

Let a = r_a · e^{jθ
_a} and b = r_b · e^{jθ
_b} be two CFVs, then we have the following operators:

(i) (Algebraic product) $\begin{matrix} a \cdot b = r_{a} \cdot r_{b} \cdot e^{j (θ_{a} + θ_{b})} \end{matrix}$

(ii) (Average) $\begin{matrix} \frac{a + b}{2} = & \frac{r_{a} \cdot cos θ_{a} + r_{b} \cdot cos θ_{b}}{2} \\ + j \frac{r_{a} \cdot sin θ_{a} + r_{b} \cdot sin θ_{b}}{2} . \end{matrix}$

The partial ordering of CFVs is given by the modulus of a complex number, i.e., a ≤ b if and only if |a| ≤ |b|.

2.2. Rotational invariance

Dick [4] introduced the concept of rotational invariance for complex fuzzy operator as follows.

Definition 2. [4] A function v : D² → D is rotationally invariant if and only if $v (a \cdot e^{j θ}, b \cdot e^{j θ}) = v (a, b) \cdot e^{j θ},$ (2) for any a, b ∈ D.

The partial ordering of CFVs is rotationally invariant, i.e., if |a| ≤ |b|, then |a · e^jθ| ≤ |b · e^jθ|. The algebraic product is not rotationally invariant [4]. But, we have the following.

Theorem 1. The average operator is rotationally invariant.

Proof. For any a, b ∈ D, we have $\frac{a \cdot e^{j θ} + b \cdot e^{j θ}}{2} = \frac{(a + b) \cdot e^{j θ}}{2} = \frac{(a + b)}{2} \cdot e^{j θ} .$ □

3. Complex fuzzy weighted arithmetic aggregation operators

In this section, we discuss some basic aggregation techniques on CFSs and some of their fundamental characteristics. Ramot et al. [3] defined the aggregation operation on complex fuzzy sets as vector aggregation:

Definition 3. [3] Let A₁, A₂, …, A_n be CFSs defined on the universe of discourse X. Vector aggregation on A₁, A₂, …, A_n is defined by a function v. $v : D^{n} \to D,$ (3)

The function v produces an aggregate complex fuzzy set A by operating on the membership grades of A₁, A₂, …, A_n for each x ∈ X. For all x ∈ X, v is given by $\begin{matrix} μ_{A} (x) & = v (μ_{A_{1}} (x), μ_{A_{2}} (x), . . ., μ_{A_{n}} (x)) \\ = \sum_{i = 1}^{n} (w_{i} \cdot μ_{A_{i}} (x)), \end{matrix}$ (4) where w_i ∈ D for all i, and $\sum_{i = 1}^{n} | w_{i} | = 1$ .

In Ref. [3], the complex weights are used in Definition 3 for the purpose of maintaining a definition that is as general as possible. In this paper, we also discuss the complex fuzzy aggregation operations with real-valued weights.

For convenience, let a₁, a₂, …, a_n be CFVs, based on Definition 3, a complex fuzzy weighted arithmetic aggregation (CFWAA) operator is defined as $CFWAA (a_{1}, a_{2}, . . ., a_{n}) = \sum_{i = 1}^{n} (w_{i} \cdot a_{i}),$ (5) where w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1$ .

If w_i = 1/n for all i, then the CFWAA operator is the arithmetic average of (a₁, a₂, …, a_n), denoted by complex fuzzy arithmetic average (CFAA) operator i.e., $CFAA (a_{1}, a_{2}, . . ., a_{n}) = \frac{1}{n} \cdot \sum_{i = 1}^{n} a_{i},$ (6)

Theorem 2. Let a₁, a₂, …, a_n be CFVs, then the aggregated value CFWAA (a₁, a₂, …, a_n) is also a complex fuzzy value.

Proof. Since |a_i|≤1 for all i, and $\sum_{i = 1}^{n} w_{i} = 1$ , then $\begin{matrix} | w_{1} a_{1} + w_{2} a_{2} + . . . + w_{n} a_{n} | \\ \leq w_{1} | a_{1} | + w_{2} | a_{2} | + . . . . + w_{n} | a_{n} | \\ \leq w_{1} + w_{2} + . . . + w_{n} = 1 . \end{matrix}$ Thus the aggregated value |CFWAA (a₁, a₂, …, a_n) |≤1, so it is also a complex fuzzy value. □ Moreover, based on Definition 2, we define the following.

Definition 4. A complex fuzzy aggregation operator v : Dⁿ → D is rotationally invariant if $v (a_{1} \cdot e^{j θ}, a_{2} \cdot e^{j θ}, . . ., a_{n} \cdot e^{j θ}) = v (a_{1}, a_{2}, . . ., a_{n}) \cdot e^{j θ}$ (7) for any CFVs a₁, a₂, …, a_n.

Theorem 3. The CFWAA operator is rotationally invariant.

Proof. For any CFVs a₁, a₂, …, a_n, we have $\begin{matrix} CFWAA (a_{1} \cdot e^{j θ}, a_{2} \cdot e^{j θ}, . . ., a_{n} \cdot e^{j θ}) \\ = w_{1} \cdot a_{1} \cdot e^{j θ} + w_{2} \cdot a_{2} \cdot e^{j θ} + . . . + w_{n} \cdot a_{n} \cdot e^{j θ} \\ = (w_{1} \cdot a_{1} + w_{2} \cdot a_{2} + . . . + w_{n} \cdot a_{n}) \cdot e^{j θ} \\ = CFWAA (a_{1}, a_{2}, . . ., a_{n}) \cdot e^{j θ} . \end{matrix}$ □ Idempotency, boundedness and monotonicity are three important properties for aggregation operators, the CFWAA operator satisfies idempotency and boundedness.

Theorem 4. Let a₁, a₂, …, a_n be CFVs, CFWAA weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1$ . Then we have the following.

(1) (Idempotency): If a₁ = a₂ = … = a_n then $CFWAA (a_{1}, a_{2}, . . ., a_{n}) = a_{1} .$

(2) (Boundedness): $| CFWAA (a_{1}, a_{2}, . . ., a_{n}) | \leq r,$ for every real-valued weight vector, where $r = max_{i} | a_{i} |$ .

Proof. (1). Since a₁ = a₂ = … = a_n and $\sum_{i = 1}^{n} w_{i} = 1$ , then $\begin{matrix} w_{1} a_{1} + w_{2} a_{2} + . . . + w_{n} a_{n} \\ = w_{1} a_{1} + w_{2} a_{1} + . . . + w_{n} a_{1} \\ = (\sum_{i = 1}^{n} w_{i}) \cdot a_{1} = a_{1} \end{matrix}$

(2). Let $a = max_{i} | a_{i} |$ , since w_i ∈ [0, 1] for all i, then $\begin{matrix} | w_{1} a_{1} + w_{2} a_{2} + . . . + w_{n} a_{n} | \\ \leq | w_{1} a_{1} | + | w_{2} a_{2} | + . . . + | w_{n} a_{n} | \\ = w_{1} | a_{1} | + w_{2} | a_{2} | + . . . + w_{n} | a_{n} | \\ \leq w_{1} a + w_{2} a + . . . + w_{n} a = a . \end{matrix}$ □ However, the CFWAA operator does not satisfy the property of monotonicity.

Example 1. Let a₁ = 0.6, a₂ = 0.6e^j2π/3, b₁ = b₂ = 0.5 and w₁ = w₂ = 0.5. Using Equation (5), we have $\begin{matrix} CFWAA (a_{1}, a_{2}) = 0.5 \cdot 0.6 + 0.5 \cdot 0.6 \cdot e^{j 2 π / 3} \\ = 0.3 \cdot e^{j π / 3} \end{matrix}$ and CFWAA (b₁, b₂) =0.5. So |a₁| ≥ |b₁|,|a₂| ≥ |b₂|, but |CFWAA (a₁, a₂) | ≤ |CFWAA (b₁, b₂) |.

From above example, the CFAA operator does not satisfy the property of monotonicity.

Note that idempotency is concerned with both the phase and amplitude of CFVs. But, boundedness and monotonicity are restricted exclusively to the amplitude of CFVs. They are only concerned with the amplitude of CFVs.

Table 1

Properties of the complex fuzzy aggregation operators

Idempotency	Boundedness	Monotonicity	Commutativity	Rotational invariance
CFAA	√	√	×	√	√
CFWAA	√	√	×	×	√
CFOWAA	√	√	×	×	√

√ and × represent the corresponding property holds and does not hold respectively.

4. Complex fuzzy ordered weighted arithmetic aggregation operators

Based on the partial ordering of CFVs, we proposed a complex fuzzy ordered weighted arithmetic aggregation (CFOWAA) operator as follow:

Definition 5. Let a₁, a₂, …, a_n be CFVs, a CFOWAA operator is defined as $CFOWAA (a_{1}, a_{2}, . . ., a_{n}) = \sum_{i = 1}^{n} w_{i} a_{σ (i)}$ (8) where w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1$ , (σ (1), σ (2), …, σ (n)) is a permutation of (1, 2, …, n) such that |a_σ(i-1)| ≥ |a_σ(i)| for all i.

Especially, when w_i = 1/n for all i, then the CFOWAA operator is reduced to the CFAA operator.

Similar to Theorem 2 and 3, we have the following.

Theorem 5. Let a₁, a₂, …, a_n be CFVs, then the aggregated value CFOWAA (a₁, a₂, …, a_n) is also a complex fuzzy value.

Theorem 6. The CFOWAA operator is rotationally invariant.

Similar to the CFWAA operator, the CFOWAA operator satisfies idempotency and boundedness.

Theorem 7. {Let a₁, a₂, …, a_n be CFVs, CFOWAA weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1$ . Then we have the following. (1) (Idempotency): If a₁ = a₂ = … = a_n then $CFOWAA (a_{1}, a_{2}, . . ., a_{n}) = a_{1} .$ (2) (Boundedness): $| CFOWAA (a_{1}, a_{2}, . . ., a_{n}) | \leq / a r,$ for every real-valued weight vector, where $a r = \max_{i} | a_{i} | .$

However, the CFOWAA operator does not satisfy the property of commutativity.

Example 2. Let a₁ = e^jπ, a₂ = e^jπ/2, a₃ = 1, and w₁ = 0.5, w₂ = 0.4, w₃ = 0.1. Then

\begin{array}{l} w_{1} a_{1} + w_{2} a_{2} + w_{3} a_{3} = 0.4 \cdot \sqrt{2} \\ w_{1} a_{1} + w_{2} a_{3} + w_{3} a_{2} = 0.1 \cdot \sqrt{2} \end{array}

Thus we have

| w_{1} a_{1} + w_{2} a_{2} + w_{3} a_{3} | \neq | w_{1} a_{1} + w_{2} a_{3} + w_{3} a_{2} |

Similarly, the CFWAA operator does not satisfy the property of commutativity. However, the CFAA operator satisfies the property of commutativity.

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

Theorem 8. Let a₁, a₂, …, an be CFVs, CFOWAA weights are real values, i.e., i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1.$ Then we have the following.

(1) If w = (1, 0, …, 0). Then

$| C F O W A A (a_{1}, a_{2}, …, a_{n}) | = \max_{i} | a_{i} | .$

(2) If w = (0, 0, …, 1). Then

$| C F O W A A (a_{1}, a_{2}, …, a_{n}) | = \min_{i} | a_{i} | .$

(3) If w_i = 1,w_k = 0, k ≠ i. Then

$| C F O W A A (a_{1}, a_{2}, …, a_{n}) | = | a_{σ} (i) |,,$

where a_σ(i) is i-th (modulus based) largest of a₁, a₂,…, a_n.

Now we give a brief summary of properties of the CFWAA and CFOWAA operators with real-valued weights. The results can be summarized as in Table 1, in which ✓ and × represent the corresponding property holds and does not hold respectively.

5. Complex fuzzy valuesand Pythagorean fuzzy numbers

In Ref. [18], Pythagorean membership grades can be expressed using complex numbers, called ∏ - i numbers. Essentially, the CFWAA and CFOWAA operators are used to deal with special complex num224 bers. In this section, we consider the CFWAA and CFOWAAoperators in the domain of∏ - i numbers. We first recall the concepts of Pythagorean fuzzy sets (PFS) and ∏ - i numbers.

Definition 6. [18] Let X be a fixed set, a PFS A is defined in the following form:

A = {< x, Y_{A} (x), N_{A} (x), > | x \in X}

(9)

where Y_A(x) : X → [0, 1] and N_A(x) : X → [0, 1] represent, respectively, the membership degree and nonmembership degree of the element x to set A, such that

0 \leq {(Y_{A} (x))}^{2} + {(N_{A} (x))}^{2} \leq 1,

for all x ∈ X. The degree of indeterminacy of the element x to set A is complex-valued membership function π_A(x) is

π_{A} (x) = \sqrt{1 - {(Y_{A} (x))}^{2} - {(N_{A} (x))}^{2} .}

For convenience, Zhang and Xu [4] referred to Y_A(x),N_A(x) as a Pythagorean fuzzy number (PFN) simply denoted by a = (Y_a,N_a), where Y_a ∈ [0, 1], N_a ∈ [0, 1] and (Y_a)² + (N_a)² ≤ 1.

Yager and Abbasov [18] presented a relation ship between Pythagorean membership grades and complex numbers. They showed that the complex numbers of the form z = r · e^jθ with conditions r ∈[0, 1] and θ ∈[0, π/2] can interpretable as Pythagorean membership grades (r cos θ, r sin θ). They referred to these complex numbers as "∏ - i numbers".

As mentioned in [18], we should consider which operation allow us to start with ∏ - i numbers and end with ∏ - i numbers, i.e., which operations are closed under ∏ - i numbers.

It is easy to know that algebraic product of ∏ - i numbers are not closed. Since for two ∏ - i numbers a, b, their algebraic product a · b = r_a · _{r_b} · e^j(θ_a+θ_b) is not a ∏ — i number when θ_a + θ_b < π/2. But for the average operator, we have following result.

Theorem 9. The average operator is closed under ∏ - i numbers.

Proof. For any two ∏ — i numbers a = r_a · e ^jθ_a and b = r_b · e ^jθ_b , from Theorem 2, we have $| \frac{a + b}{2} | \leq 1.$

Since a, b are two ∏ — i numbers then we have r_i cos(θ_i) ≥ 0 and r_i sin(θ_i) ≥ 0 for all i ∈ {a, b}. Then

\begin{array}{l} \frac{r_{a} \cos (θ_{a}) + r_{b} \cos (θ_{b})}{2} \geq 0, \\ \frac{r_{a} \sin (θ_{a}) + r_{b} \sin (θ_{b})}{2} \geq 0. \end{array}

Thus $\frac{a + b}{2}$ is also a ∏ - i number. □

Let us further consider the closeness of ∏ - i numbers under the complex fuzzy arithmetic aggregation perations. First, we have the following.

Theorem 10. Let z₁, z₂, …, z_n be ∏ - i numbers, and the CFWAA weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1.$ Then the aggregated value CFWAA (z₁, z₂, …, z_n) also is a ∏ - i number.

Proof. From Theorem 2, we have

| C F W A A (z_{1}, z_{2}, …, z_{n}) | \leq 1.

(10)

Assume z₁ = r₁ cos(θ₁) + jr₁ sin(θ₁), z₁ = r₂ cos (θ₂) + jr₂ sin (θ₂), …, z₁ = r_n cos (θ_n) + jr_n sin (θ_n).

Since z₁, z₂, …, z_n are ∏ - i numbers, then we have r_i cos(θ_i) ≥ 0 and r_i sin(θ_i) ≥ 0 for all i = 1, 2, …, n. Then

\begin{array}{l} w_{1} r_{1} \cos (θ_{1}) + w_{2} r_{2} \cos (θ_{2}) \\ + … + w_{n} r_{n} \cos (θ_{n}) \geq 0, \end{array}

(11)

\begin{array}{l} w_{1} r_{1} \cos (θ_{1}) + w_{2} r_{2} \cos (θ_{2}) \\ + … + w_{n} r_{n} \cos (θ_{n}) \geq 0 . \end{array}

(12)

From (10)-(12), CFWAA (z₁, z₂, …, z_n) also is a ∏ - i number. □

Similar to Theorem 10, we have the following.

Theorem 11. Let z₁, z₂, …, z_n be ∏ - i numbers, and the CFOWAA weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1.$ Then the aggregated value CFOWAA (z₁, z₂, …, z_n) also is a ∏ - i number.

Above Theorems show us that the CFWAA and the CFOWAA operators with real-valued weights are closed under ∏ - i numbers. When PFNs are interpreted as ∏ - i numbers, then we can aggregate these PFNs to a PFN by the CFWAA operator or the CFOWAA operator. Above Theorems show that the CFWAA and the CFOWAA operators are closed on the upper-right quadrant of the unit disc of complex plane.

Consider other quadrants of the unit disc. Let

D_{k} = {z = r e^{j θ} | r \in [0, 1], θ \in [\frac{(k - 1) π}{2}], \frac{k π}{2} |},

for k = 1 to 4. D₁ is the set of ∏ - i numbers. Plainly, we have the following.

Theorem 12. For any k ∈ {1, 2, 3, 4}, if z₁, z₂, …, z_n ∈ D_k, and the weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1.$ Then we have

\begin{array}{l} C F W A A (z_{1}, z_{2}, …, z_{n}) \in D_{k}, \\ C F O W A A (z_{1}, z_{2}, …, z_{n}) \in D_{k} . \end{array}

Moreover, similar to Theorems 7 and 8, we have the following.

Theorem 13. For any k ∈ {1, 2, 3, 4}, if z₁, z₂, …, z_n ∈ D_k, and the weights are real values, i.e., w_i ∈ [0, 1] for all i, and $\sum_{i = 1}^{n} w_{i} = 1.$ Then we have the following.

(1) (Idempotency):

\begin{array}{l} I f Z_{1} = Z_{2} = ... = Z_{n} t h e n \\ C F W A A (Z_{1}, Z_{2}, ..., Z_{n}) = ... = Z_{1} . \\ C F O W A A (Z_{1}, Z_{2}, ..., Z_{n}) = ... = Z_{1} \end{array}

(2) (Boundedness):

\begin{array}{l} | C F W A A (z_{1}, z_{2}, …, z_{n}) | \leq z, \\ | C F O W A A (z_{1}, z_{2}, …, z_{n}) | \leq z, \end{array}

for every real-valued weight vector, where z = max_i |z_i|.

6. Example application

We consider a target location application of the complex fuzzy aggregation operator. The problem statement of the example is taken from Ref. [26]. Let (r₁, θ₁), (r₂, θ₂), …, (r_n, θ_n) be n measurements, where r_i and θ_i represent the distance and direction of the target, respectively. We apply the following method supported by complex fuzzy aggregation operator theory to estimate the target position.

Step 1. Transform the measurements into the complex fuzzy values, (a₁, a₂, …, a_n) by a_i = r_i · e^jθ_i/d where $\sum_{i}^{n} d_{i} .$

Step 2. Aggregate all the complex fuzzy values by the CFAA, r · e^jθ = CFAA(a₁, a₂, …, a_n).

Step 3. Transform complex fuzzy value into the measurement (r · d, θ).

Next, we give an example to illustrate the proposed method:

Suppose that four measurements are taken:

((667, 14°), (671, 15°), (667, 15°), (669, 13°)),

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

Step 1. Transform the measurements into the complex fuzzy values: (0.249 · e^j2π14/360, 0.251 · e^j2π15/360, 0.249 · e^j2π15/360, 0.25 · e^j2π13/360).

Step 2. Utilize the CFAAto aggregate all the complex fuzzy values:

CFAA(a₁, a₂, a₃, a₄) = 0.2497 · e^j0.2487.

Step 3. Calculate the estimated location: (667.7, 14.2).

Then the target position is estimated at 14.2 degrees east of north of the observer and 667.7 m from the observer.

7. Conclusion

In this paper, we discussed two complex fuzzy aggregation operators, the CFWAA and the CFOWAA operators. Their main properties for real-valued weights can be summarized as in Table 1. In particular, both the CFWA and the CFOWA operators are rotationally invariant.We specifically showed that the CFWA and the CFOWA operators are closed under ∏ - i numbers.

Of course, we should note that the CFAA, CFWAA and CFOWAA operators are not monotone. In our point of view, operators employed for some applications might be non-monotone. However, we may require some weaker type of monotonicity for these applications. This is a problem left for further investigation.

Finally, except the target location application, complex data are frequently encountered in image processing and signal processing, complex fuzzy aggregation operators in these applications are possible topics for future consideration.

Footnotes

Acknowledgments

The work is supported by the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No. 2017CSOBDP0103).

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