#### Abstract

This paper investigates the geometric aggregation operators for aggregating the interval-valued complex fuzzy sets (IVCFSs) whose membership grades are a special set of complex numbers. We develop some geometric aggregation operators under the interval-valued complex fuzzy environment, namely, interval-valued complex fuzzy geometric (IVCFG), interval-valued complex fuzzy weighted geometric (IVCFWG), and interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operators. Then, we investigate the rotational and reflectional invariances of these operators. Further, a decision-making approach based on these operators is presented under the interval-valued complex fuzzy environment and an example is illustrated to demonstrate the efficiency of the proposed approach.

#### 1. Introduction

The aggregation operator is a powerful method for decision making, pattern recognition, and cluster analysis. In the past decades, in both theoretical and applied studies, aggregation operators have attained great advances. Many types of aggregation operators have been proposed under different environments, such as fuzzy environment [1–4], intuitionistic fuzzy environment [5–9], interval-valued intuitionistic fuzzy environment [10–14], Pythagorean fuzzy environment [15–17], neutrosophic fuzzy environment [18–20], and hesitant fuzzy environment [21–25].

In the above fuzzy environments, membership degrees are the subsets of real numbers. As a generalization of traditional fuzzy set [26], Ramot et al. [27] introduced the concept of complex fuzzy set (CFS), which is characterized by a complex-valued membership function. In many practical situations, complex fuzzy sets are useful [28–40]. Moreover, many researchers extended the concept of CFS to interval-valued complex fuzzy set (IVCFS) [41, 42] and complex intuitionistic fuzzy set (CIFS) [43]. Therefore, many researchers discussed how to aggregate CFSs. Ramot et al [32] introduced the concept of complex fuzzy aggregation. Ma et al. [38] proposed a product-sum aggregation operator under complex fuzzy environment. Bi et al. [39, 40] proposed several aggregation operators under complex fuzzy environment. Garg and Rani [44, 45] investigated the aggregation operators under complex intuitionistic fuzzy environment.

However, we still have the key question: why complex fuzzy aggregation? Moreover, why complex fuzzy sets? As mentioned in [46], from mathematical and practical viewpoints, complex fuzzy sets are natural and useful. But complex fuzzy sets (CFSs) remain a puzzle from the intuitive viewpoint. Fuzzy sets and other extensions give intuitively clear way to describe how humans deal with different types of uncertainty. So before we start to examine the information aggregation issue under interval-valued complex fuzzy environment, we first discuss some phenomena which maybe ignored in real life. When we ask the way, two persons may give the answers “it is about 1 km away,” and then we think that 1 km is a reasonable result. However, if their answers are not exclusively same about direction, as shown in Figure 1, 0.95 km also is a reasonable result since .

How does this phenomenon affect human decision making? For example, there are two hospitals and ; which one is the nearest hospital? Then, we get data about distance and direction from strangers. It is a very interesting case; two strangers both agree that hospital is nearer than hospital , but after data aggregation, the result that is nearer than hospital is also reasonable, as shown in Figure 2. The order only relying on the distance is reasonable since we want to go to the nearest hospital. The method based on complex fuzzy aggregation is reasonable since it is a center-based method.

Complex fuzzy aggregation operator can perfectly describe above phenomenon in human decision making since it does not satisfy the property of amplitude monotonicity [40]. Monotonicity is a basic property, which holds in many types of aggregation operators under different fuzzy environments [1–25]. Complex fuzzy aggregation operator as a nonmonotone average is very natural and gives an intuitive way to describe how humans deal with such type of uncertainty.

In this paper, we focus on the aggregation operator under interval-valued complex fuzzy environment. IVCFSs also are very appropriate for some applications in real life. For example, when we get lost, we often ask strangers for directions in our daily life. Then, we get some answers, such as “0.5-0.6 km, east” and “0.5–0.7 km, northeast.” These answers can be represented in terms of IVCFSs. We present the theory of the weighted geometric aggregation operators among the IVCFSs. First we review necessary concepts and some basic properties related to this paper in Section 2. In Section 3, we present an interval-valued complex fuzzy weighted geometric (IVCFWG) operator on CFSs. In Section 4, we present an interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operator. In Section 5, we present a decision-making approach based on the proposed operator under IVCFS environment. Finally, conclusions are given in Section 6.

#### 2. Preliminaries

In this paper, our discussion is based on interval-valued complex fuzzy set theory. Some basic concepts are recalled below, whereas for other concepts, refer to reports from Ref. [27, 28, 41, 42, 47].

Let be the set of complex numbers on complex unit disk, i.e., . Let be a fixed universe, and a mapping is called a complex fuzzy set on .

Let be the set of all closed subintervals of , i.e., . Let be the boundary set of i.e., . A mapping is called an IVCFS on . For any , its membership degree iswhere , is the dot product set of and , is the interval-valued amplitude part, and is the phase part.

For convenience, we only consider the values on , which are called interval-valued complex fuzzy values (IVCFVs). Let be an IVCFV, where the interval-valued amplitude part is and the phase part is . The modulus of is the interval-valued amplitude part , denoted by . An order of IVCFVs is defined by the interval-valued amplitude part, i.e.,

Let and be two IVCFVs and let the parameters be and ; four operators of IVCFVs include multiplication, power, rotation, and reflection which are defined as follows.(i)Multiplication of two IVCFVs :(ii)Power of an IVCFV :(iii)Rotation of an IVCFV :(iv)Reflection of an IVCFV :

When , is a complex intersection defined by Ramot et al. [27]. When , is a t-norm [48].

Theorem 1. *Suppose that are three IVCFVs, and the parameters are , and . Then, we have*(1)*(2)**(3)**(4)**(5)**(6)**(7)**(8)*

*Proof. * Let , , and .(1)By the definition of multiplication of two IVCFVs, we have(2)And(3)By the definition of power of two IVCFVs, we have(4)And(5)By the definition of reflection of two IVCFVs, we have(6)By the definition of rotation of two IVCFVs, we have(7)For any two IVCFVs and real value , we have(8)For any two IVCFVs , we haveMoreover, the multiplication and power operators have the following properties.

Theorem 2. *Suppose that , , , and are four IVCFVs; the parameters are , and . Then we have*(1)*(Amplitude monotonicity) If , then *(2)*(Amplitude boundedness) *(3)*If , then *(4)*If , then , , *

*Proof. * Since the amplitude terms of IVCFVs are interval values, we can obtain the above results from the properties of multiplication operator of interval values.

Rotational invariance and reflectional invariance are two important geometric properties of complex fuzzy operators [28, 29, 39]. They show that a complex fuzzy operator is invariant under a rotation and a reflection, respectively. We define the following two similar properties for interval-valued complex fuzzy operators as follows.(i)An interval-valued complex fuzzy operator is rotationally invariant if and only if, for any ,(ii)An interval-valued complex fuzzy operator is reflectionally invariant if and only ifThen, for the multiplication operator of IVCFVs, we have the following results.

Theorem 3. *The multiplication operator of IVCFVs is reflectionally invariant.*

*Proof. *Trivial from Theorem 1 (16).

Theorem 4. *The multiplication operator of IVCFVs is not rotationally invariant.*

*Proof. * For any two IVCFVs , and real value , we haveSince , the multiplication operator is not rotationally invariant.

#### 3. Interval-Valued Complex Fuzzy Weighted Geometric Operators

In this section, we introduce the weighted geometric operators in an interval-valued complex fuzzy environment and discuss their fundamental characteristics.

*Definition 1. *Let be a collection of IVCFVs; an interval-valued complex fuzzy weighted geometric (IVCFWG) operator is defined aswhere for all and .

When , then the IVCFWG operator is denoted by interval-valued complex fuzzy geometric (IVCFA) operator, i.e.,When , the IVCFWG operator can reduce to a traditional interval-valued fuzzy weighted geometric operator of values on unit interval .

When , the IVCFWG operator can reduce to a traditional weighted geometric operator [49] of real numbers on unit interval [0, 1].

When and , the CFWP operator can reduce to a traditional geometric mean operator [50] of real numbers on unit interval [0, 1].

Theorem 5. *Let be a collection of IVCFVs; then, the aggregated value is also an IVCFV andwhere for all and .*

*Proof. *It is easy to check that the aggregated value is also an IVCFV. Now, we prove equation (20) by mathematical induction method.

For , we have , ; then,If equation (20) holds for , i.e.,then for ,Therefore, equation (20) holds for all .

The working of the proposed IVCFWG operator is explained with a numerical example as follows.

*Example 1. *Let , , , and be four IVCFVs and be the associated weight vector. Then, by using equation (20),we obtain .

Based on Theorem 5, the proposed IVCFWG operator satisfies the following properties.

Theorem 6. *Let and be two collections of IVCFVs, the weights be , and . Then, we have the following properties:*(1)*(Idempotency) If , then**(2)**(Amplitude monotonicity) If , then**(3)**(Amplitude boundedness)**where are two interval values:*

*Proof. *(1)Let , ; then,(2)The property of the amplitude monotonicity is trivial from Theorem 2 (4).(3)The property of the amplitude boundedness is easily obtained from idempotency and amplitude monotonicity.Note that idempotency is concerned with both the phase part and amplitude part of IVCFVs. Amplitude boundedness and amplitude monotonicity are only concerned with the amplitude part of IVCFVs.

Theorem 7. *The IVCFWG operator is reflectionally invariant.*

*Proof. *For any collection of IVCFVs , from equation (20), we haveThen, is reflectionally invariant.

Theorem 8. *The IVCFWG operator is rotationally invariant.*

*Proof. *For any collection of IVCFVs and any , we haveThen, the IVCFWG operator is rotationally invariant.

#### 4. Interval-Valued Complex Fuzzy Ordered Weighted Geometric Operators

Based on the order of IVCFVs defined by equation (2) and the ordered weighted averaging (OWA) operator introduced by Yager [1], we define an interval-valued complex fuzzy ordered weighted geometric (IVCFOWG) operator as follows.

*Definition 2. *Let be a collection of CFVs; then, an IVCFOWG operator is defined aswhere and , is a permutation of such that for all .

Especially, when , the IVCFOWG operator is reduced to the IVCFWG operator.

Similar to the IVCFWG operator, the IVCFOWG operator has the following properties.

Theorem 9. *Let be a collection of IVCFVs; then, the aggregated value is also an IVCFV andwhere for all and , is a permutation of such that for all .*

*Proof. *The proof is similar to Theorem 5.

The working of the proposed IVCFOWG operator is explained with a numerical example as follows.

*Example 2. *Let , , , and be four IVCFVs and be the associated weight vector. By the order from equation (2), we have . Then, by using equation (33),we obtain . It is different from (see Example 1).

Theorem 10. *Let be a collection of CFVs, CFOWP weights be , and . Then, we have the following properties.*(1)*(Idempotency) If , then**(2)**(Amplitude monotonicity) If , then**(3)**(Amplitude boundedness)**where are two interval values:*

*Proof. *The proof is similar to Theorem 6.

Theorem 11. *The IVCFOWG operator is reflectionally invariant and rotationally invariant.*

*Proof. *The proof is similar to Theorems 7 and 8.

Note that the operator does not have the property of rotational invariance, but the IVCFWG and IVCFOWG operators defined based on operator have the property of rotational invariance.

Here, we investigate the aggregation operators based on special class of the IVCFVs, which belong to subsets of the upper-right quadrant of the complex unit disk. Let ; we consider the aggregation operator on .

Let us consider the closeness of IVCFVs on under the IVCFWG and IVCFOWG operations. For the IVCFWG operator, we have the following result.

Theorem 12. *Let . Then, the aggregated value*

*Proof. *Denoting , from Theorem 5, is an interval value. Since is a weighted arithmetic aggregation operator of real numbers on , then we have . Thus, .

Similar to the above theorem, we have the following.

Theorem 13. *Let . Then, the aggregated value*

The above theorems show us that the IVCFWG and the IVCFOWG operators are close under values on .

Consider other quadrants of the complex unit disk. Letfor to 4.

Now, we discuss the closeness of the IVCFWG and the IVCFOWG operators on values of other quadrants of the complex unit disk. Plainly, we have the following.

Theorem 14. *For any , if , then we have*

*Proof. *Similar to Theorem 12.

#### 5. An Approach to Decision Making with the IVCFWG Operator

In this section, we present an approach using the IVCFWG operator to a decision making with interval-valued complex fuzzy information.

We consider a target selection application of CFSs. Assume that our position is fixed, and then we can measure the distance and angle of the possible alternatives by using a position sensor and an angular sensor (or other attributes from expert opinions). Assume that we get a measurement . Here, we use an interval value to represent by setting and . To improve the target location accuracy, we repeatedly measure the alternatives. Then, the target is selected in the following approach according to aggregation theory.

Let be the possible alternatives and be the experts. Then, the decision maker provides a decision matrix , where is an interval-valued complex fuzzy value given by the expert for alternative . Further, assume that is the weight vector of the different experts such that and . The process can be summarized as follows: *Step 1*. Transform the decision matrix into the normalized interval-valued complex fuzzy decision matrix using where . *Step 2*. Aggregate all the interval-valued complex fuzzy values and get the overall interval-valued complex fuzzy value corresponding to the alternative by the IVCFWG operator, . *Step 3*. Rank the overall IVCFVs using (2).

Next, we give an example to illustrate the above approach.

*Example 3. *Suppose we want to go to the nearest bank, then we often use GPS navigation system or ask strangers for directions.

Suppose that there are three alternatives and three experts (GPS or stranger) with the weight vector . The three experts evaluate the three alternatives under the IVCFS environment, and their corresponding rating values are summarized in the decision matrix represented in Table 1, where is an interval-valued complex fuzzy value, represents the distance between km and km, and represents the direction for the alternative . *Step 1*. The values do not need normalization. *Step 2*. Aggregate the interval-valued complex fuzzy values of the alternatives by using the IVCFWG operator (see equation (20)): and we obtain . and we obtain . and we obtain . *Step 3*. Rank the interval-valued complex fuzzy values : , which shows that the alternative is the optimal choice, i.e., the nearest bank.An IVCFS is a generalization of complex fuzzy sers (CFS), interval-valued fuzzy sets (IVFS), and fuzzy sets (FS). IVCFS can handle two-dimensional information in a single set. Then, IVCFS contains much more information than CFS, IVFS, and FS.

It is revealed from the present study that the aggregation operators under IVFSs and FSs [5, 6] are the special cases of the proposed aggregation operators. Thus, the proposed aggregation operators can be equivalently utilized to solve the DM problem under these existing environments by setting phase term to be zero while the existing operators [5, 6] are unable to solve the problems under the IVCFSs environment considered in the present paper. Thus, interval-valued complex fuzzy aggregation operators are more general than some (interval-valued) fuzzy aggregation operators.

#### 6. Conclusion

As mentioned in [50], the basic feature of aggregation operators is their monotonicity property. However, aggregation operators under the complex fuzzy environment [40] are not monotone. Interestingly, such aggregation operators can explain some phenomena in our real life.

In this paper, we discussed the geometric aggregation operators under the interval-valued complex fuzzy environment. Two interval-valued complex fuzzy aggregation operators, the IVCFWG and the IVCFOWG operators, are developed and their properties are studied. It is also interesting to note that both the IVCFWG and the IVCFOWG operators are rotationally invariant and reflectionally invariant. Further, based on the proposed operators, we presented a decision-making approach under interval-valued complex fuzzy information. An illustrative example is given for illustrating the proposed approach.

It was observed that the aggregation operator based on nonadditive integrals (Choquet [51] and Sugeno integrals [52]) is one of the hot topics in this field [49]. As future work, we can consider the complex fuzzy aggregation operators based on complex integrals.

#### Data Availability

The data used to support the findings of this study are included in the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was funded by the Open Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (grant no. 2017CSOBDP0103).