**1. Introduction**

Ramot et al. [1] introduced the innovative concept of complex fuzzy sets (CFSs), which is an extension of the traditional fuzzy sets [2] where traditional unit interval [0,1]-valued membership degrees are extended to the complex unit disk. CFSs are completely distinct from the fuzzy complex numbers discussed by Buckley [3–5]. The complex-valued membership grade has an amplitude term with the addition of a phase term. The phase term of complex-valued membership grade is the key feature which essentially distinguishes complex fuzzy sets from other extensions of fuzzy sets. Ramot et al. [1,6] then introduced several operators of CFSs and a novel framework for complex fuzzy reasoning. Hu et al. [7] introduced the orthogonality relation for CFSs. Bi et al. [8] proposed the parallelity of CFSs and the parallelity-preserving operators. Zhang et al. [9] proposed the *δ*-equalities for CFSs. Alkouri and Salleh [10] and Hu et al. [11] defined several distances between CFSs. Tamir et al. [12] proposed a new interpretation of complex membership degree. They [13] then proposed complex fuzzy propositional and first-order logics. Dick [14] proposed the concept of rotational invariance for complex fuzzy operators. Recently, several scholars have developed extensions of CFSs. Greenfield et.al [15,16] introduced interval-valued complex fuzzy sets. Alkouri and Saleh [17] proposed complex intuitionistic fuzzy sets. Ali and Smarandache [18] introduced complex neutrosophic sets. Recently, CFSs and their extensions have been successfully applied in many fields, such as time series prediction [19–22], decision making [23], signal processing [1,7,9], and image restoration [24].

Yager and Abbsocv [25] discussed the relationship between CFSs and Pythagorean fuzzy sets (PFSs), which was developed by Yager [26,27] as an extension of Atanssov's intuitionistic fuzzy sets [28]. They showed that Pythagorean fuzzy membership grades can be viewed as complex numbers on the upper-right quadrant of the complex unit disk, named Π − *i* numbers. Dick, Yager, and Yazdanbahksh [29] then discussed several lattice-theoretic properties of PFSs and CFSs. Quantum information processing also allows for meaningful aggregation using complex numbers. Since qubits can be represented by unit vectors in the two-dimensional complex Hilbert space, geometric information or vector aggregation are used for meaningful clustering [30,31].

The information aggregation operator plays an important role in many fields of decision making. In the past several decades, many aggregation techniques for decision making have been developed. The ordered weighted averaging (OWA) operator introduced by Yager [32] is one of the well-known aggregation operators. Many different aggregation techniques have been applied in many different fuzzy environments, such as intuitionistic [33–35], Pythagorean [36–38], neutrosophic [39–41], interval-valued intuitionistic [42–45], and hesitant fuzzy environments [46–48].

As mentioned in [19], CFSs are suitable to represent information with uncertainty and periodicity, and thus this information aggregation procedure needs to simultaneously process the uncertainty and periodicity in the data. However, comparatively few aggregation techniques have been made in the complex fuzzy environment. Ramot et al. [6] defined the complex fuzzy aggregation operations as vectors aggregation. In particular, the complex weights are used in their definition. Ma et al. [19] developed a product-sum aggregation operator for multiple periodic factor prediction problems. They proved the continuity of this operator. However, they did not focus on techniques for complex fuzzy information aggregation in these two articles.

In this paper, we study aggregation operators in the complex fuzzy environment. Dick's [14] concept of rotational invariance is an intuitive and desirable feature for complex fuzzy operators. This feature is examined for complex fuzzy aggregation operators. This paper proposes a novel feature for complex fuzzy aggregation operators called *reflectional invariance*. Moreover, we study the aggregation operators of complex numbers in the upper-right quadrant of the complex unit disk.

The main contributions of the study include: (1) A concept of reflectional invariance for complex fuzzy operators. (2) Several complex fuzzy weighted geometric operators; we also show that these operators can also be used in a Pythagorean fuzzy environment. (3) A target location method which involves the complex fuzzy aggregation operators.

This paper is organized as follows. In Section 2, we review some basic and fundamental concepts of CFSs, rotational invariance, reflectional invariance, and Ramot et al.'s [6] complex fuzzy aggregation operators. In Section 3 we study the complex fuzzy weighted geometric (CFWG) operator on CFSs and its properties. In Section 4, we develop the complex fuzzy ordered weighted geometric (CFOWG) operator based on the traditional partial ordering by the modulus of a complex number, and study its properties. In Section 5, we study these operators in the domain of Π − *i* numbers which belong to the upper-right quadrant of the complex unit disk. In Section 6, we present an application example in a target location. Conclusions are made in Section 7.
