1. Introduction
Multi-attribute decision making (MADM) issues are inescapable in the field of decision making. In numerous functional applications, MADM plays a significant role in the procedure of decision making. Many existing strategies tell us that the best way to pick the most appropriate elective depends on the decision makers’ (DMs) assessment data. Due to the progressively intricate outer decision-making condition and the abstract vulnerability of DMs themselves, it is hard for DMs to clarify their genuine inclination data plainly. In this manner, Zadeh [
1] characterized the idea of fuzzy sets (FSs) to clarify the imprecision and the doubt occurring during the assessment procedure. Until now, FSs have been examined and applied to different fields by a large number of scientists [
2,
3,
4]. Later, numerous researchers have concentrated on the most proficient method to characterize the appraisal inclinations communicated by DMs more extensively and precisely. Numerous categories of FSs have been proposed to adjust to different application conditions, for example, intuitionistic FSs (IFSs) explored by Atanassov [
5] contain supporting and non-supporting grades with a rule that the sum of both cannot be exceeded from a unit interval.
However, the condition of an IFS for a decision maker is somehow too restrictive for choosing the sum of supporting and non-supporting grades that is not exceeded from a unit interval. To resolve such issues, the theory of Pythagorean fuzzy sets (PFSs) was explored by Yager [
6], with a condition that the sum of the squares of both cannot be exceeded from a unit interval. IFSs and PFSs have had various applications [
7,
8,
9,
10,
11,
12]. Later, the theory of picture fuzzy sets (PiFSs) was presented by Cuong and Kreinovich [
13]. PiFSs are composed of the grades of truth, abstinence, and falsity with a condition that the sum of all grades cannot be exceeded from a unit interval. Similarly, the condition of PiFSs for a decision maker is also too restrictive for choosing the sum of truth, abstinence, and falsity grades that is not exceeded from a unit interval. Thus, the theory of spherical fuzzy set (SFS), proposed by Mahmood et al. [
14], with a condition that the sum of squares of all grades cannot be exceeded from a unit interval, was used to resolve these issues. By extending squares with q-powers, T-spherical fuzzy sets (TSFSs) were established by Ullah et al. [
15], in which the sum of q-powers of positive, abstinence, and negative grades belong to [0, 1] with various applications in different fields [
16,
17,
18,
19,
20].
From the above studies of decision maker processes, we can conclude that their introduction is constrained and can deal with only with vulnerability in information, yet at the same time neglects to manage changes at a given period of time. Be that as it may, data obtained, such as from a clinical examination, or a database for biometric and facial acknowledgment, consistently changes simultaneously with time. Along these lines, to manage such sorts of issue, the scope of a supporting grade is arrived at from a genuine subset to the unit plate of the mind boggling plane and thus Ramot et al. [
21] established the complex FS (CFS) which has had many applications [
22,
23,
24]. Additionally, the theory of complex IFSs (CIFSs) was presented by Alkouri and Salleh [
25] to provide a wide range of options to a decision maker for taking a decision. CIFSs compose the supporting grade and the non-supporting grade in the form of a complex number belonging to a unit disc in a complex plane. The limitations of CIFSs is that the sum of the real part (and the imaginary part) of both grades cannot be exceeded from a unit interval. However, a decision maker may give the grades of both real and imaginary parts whose sum is exceeded from a unit interval. The theory of complex PFSs (CPFSs), with a condition in which the sum of squares of the grades of both real and imaginary parts cannot be exceeded from a unit interval, was proposed by Ullah et al. [
26] for coping with this kind of issue. The theory of CIFS and CPFS have received lots of attention with applications in different fields [
27,
28,
29,
30].
When a decision maker faces more types of answer, such as truly, abstinence, no, or refusal in the form of complex numbers, casting a ballot can be a genuine case in such a circumstance, as voters might be separated into four categories of individuals, i.e., vote in favor of, abstinence, vote against, or refusal of the democratic process, in the form of polar coordinates. For instance, with , the IFS, PFS, PFS, CIFS, or CPFS are not able to investigate, because the conditions of all these notions are limited. For coping with such issues, the theory of complex spherical fuzzy sets (CSFSs) is explored in this paper to examine proficiency and ability. Thus, we summarize the contributions of the paper as follows:
To investigate the novelty of CSFS and their fundamental laws.
To investigate the Bonferroni mean (BM) operators based on CSFS and discuss their special cases.
To examine the TOPSIS method based on CSFS and propose a novel CSFS-TOPSIS method.
To resolve the MADM issues based on the proposed aggregation operators.
To give an application example of the proposed methods with comparative analysis and demonstrate the usefulness and effectiveness of the proposed methods.
The remainder of the paper is organized as follows. In
Section 2, we first review some basic definitions of CPFSs and BM operators, and then their score and accuracy function. We further consider their operational laws with some properties. In
Section 3, based on CSFS and BM operators, the complex spherical fuzzy Bonferroni mean (CSFBM) and complex spherical fuzzy weighted Bonferroni mean (CSFWBM) operators are proposed. The special cases of the explored operators are also discussed to improve the novelty of the presented work. In
Section 4, a MADM problem is chosen to be resolved based on the CSFBM operator and CSFWBM operator. Additionally, the TOPSIS method based on CSFS is also explored to construct a CSFS-TOPSIS method. An application example is given to demonstrate the effectiveness of the proposed methods with comparative analysis. Finally, we give conclusions in
Section 5.
3. The Bonferroni Mean Operators Based on CSFSs
In this section, we give two important Bonferroni mean (BM) operators based on CSFSs, called CSFBM and CSFWBM. Further, the specific cases of the CSFBM operator are also justified with some remarks.
Definition 6. For any CSFSs, the CSFBM operator is defined as: Theorem 1. The CSFBM operation result is still a CSFS, such that it has the following equation:=
Proof. For any two CSFNs and , based on the definition of CSFBM operator, we can get , and
Then, we have
And
.
Thus, we have
,
And
. □
We next investigate the properties of idempotency, monotonicity, and boundedness for the CSFBM operator.
Theorem 2 (Idempotency). For any CSFSs , we have .
Proof. Suppose First, we choose the supporting grade with . Let and . Then, and , and
. □
Similarly, we can examine for abstinence and non-supporting grades, such that and . Hence,
Theorem 3 (Monotonicity).For any two CSFSs
and
, with the conditions and , we have .
Proof. Let and
The proof of the truth grade is to show that the real part is with . If and , then we have
,
,
and
Hence, . Similarly, , for abstinence and falsity grades. Thus, we obtain the result with . □
Theorem 4 (Boundedness). For any two CSFSs
and
, we obtain.
Proof. Based on monotonicity, we have . By idempotency, we get and . Then, we obtain the result with . □
We previously have examined the three properties of idempotency, monotonicity, and boundedness for CSFSs. We next discuss more special cases with remarks.
Remark 1. Whenin Definition 6, we have
Remark 2. When in Definition 6, we have
Remark 3. When in Definition 6, we have
=
Remark 4. When in Definition 6, we have =
We next give the definition of the CSF-weighted-BM (CSFWBM) operator where the weight vector is expressed by and .
Definition7. For any CSFSs
, the CSFWBM operator is defined as:
Theorem 5. The aggregation result from Definition 7 is still a CSFS such that =
,
Proof. The proof of this theorem is similar to that of Theorem 1. □
Similarly, we can obtain the properties of idempotency, monotonicity, and boundedness for the CSFWBM operator.
Theorem 6 (Idempotency). For any CSFN , we have .
Proof. The proof of this theorem is similar to that of Theorem 2. □
Theorem 7 (Monotonicity). For any two CSFNs
and
, with the conditions and , we have
Proof. The proof of this theorem is similar to that of Theorem 3. □
Theorem 8 (Boundedness). For any two CSFNs and
, we have that .
Proof. The proof of this theorem is similar to that of Theorem 4. □