Decision support system based on complex T-Spherical fuzzy power aggregation operators

1.   Introduction

Zadeh ^[61] defined fuzzy set (FS) theory. Fuzzy set theory is a incredible achievement with applications in a variety of fields. In every element of $X$ (universal set) on $[0, 1]$ (closed interval), an FS is defined by $\sigma$, a membership degree (MD). After that, fuzzy set theory was extended to intuitionistic fuzzy sets by Atanassov ^[1] (IFS). An IFS is characterized by two functions, MD and non-membership degree (NMD), on the closed interval $0$ to $1.$ Moreover, their sum must belong to $[0, 1],$ i.e., (membership degree ($\sigma$) + non-membership degree ($\upsilon$)) $\in \lbrack 0, 1]$. If we take $\sigma = 0.7$ and $\upsilon = 0.4$ correspondingly, then their sum is greater than one. As a result of this constraint, Yager ^[57,58] developed the Pythagorean fuzzy set (PFS) concept of, expanding the domain of the IF set. A PFS is characterized by two functions, MD and NMD, on the closed interval $0$ to $1,$ and the sum of the squares of MD and NMD is less than or equal to one, i.e., sum $\left(\sigma ^{2}, \upsilon ^{2}\right) \in \lbrack 0, 1]$. As a result, Pythagorean fuzzy sets are the generalized form of IFSs, as the domain of PFSs is greater than that of IFSs. Khan et al. ^[21] defined Pythagorean fuzzy Dombi aggregation operators and their application in decision support system. Abdullah et al. ^[7] proposed Pythagorean cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem.

While IFSs and PFSs can accurately define the ambiguous details, there are still challenges that IFSs and PFSs are unable to overcome. For example, when the expert selects $0.6$ for MD and $0.9$ for NMD, the condition of PFSs is such that $0.6^{2}+0.9^{2} = 1.17 > 1$. So, with regard to this condition, Yager ^[59] defined the definition of q-rung orthopair fuzzy sets (q-ROFSs), as being more effective than intuitionistic fuzzy sets and envisioning fuzzy sets to deal with challenging and ambiguous data. In addition, Liu & Wang ^[25] have developed q-ROF aggregation operators (AOs) to calculate the assessment details. Peng et al. ^[35] introduced the exponential operational laws and AOs for q-ROFSs. Qiyas et al. ^[48] defined similarity measures based on q-rung linear Diophantine fuzzy sets and discussed their application in multiple attribute decision making (MADM).

Ullah et al. ^[47] presented the concept of a T-Spherical fuzzy set (T-SPS) and defined some AOs. Ullah et al. ^[60] defined the Hamacher AOs for T-SFSs. Munir et al. ^[31] used Einstein hybrid AOs based on T-SFSs for solving MCGDM problems. Garg et al. ^[12] proposed several new T-SFS operators. The MADM problem was solved by Liu et al. ^[27] by defining the Muirhead mean operators for the T-SFS information. However, some algorithms are described in the literature (Ullah et al. ^[45,49]) to address pattern recognition and other problems using information measures such as similarity and correlation. Quek et al. ^[37] introduced some novel T-Spherical fuzzy set operational laws and obtained some of their properties. The Einstein interactive averaging AOs and the Einstein interactive geometric AOs are two types of Einstein AOs based on these new procedures. Ju et al. ^[19] created the T-SF interaction AOs and built the TODIM method in a T-Spherical fuzzy information using these operators. The MADM approach was used by Chen et al. ^[9] to investigate certain generalized T-Spherical and group-generalized fuzzy geometric aggregation operators. Guleria and Bajaj ^[17] defined various T-Spherical fuzzy soft set aggregation operations.

Inspired by the performance of the power operator, several forms of power aggregation operators (PAOs) have been introduced for different fuzzy settings, such as the generalized PAOs ^[52,56], intuitionistic fuzzy power AO ^[65], the interval-valued intuitionistic fuzzy power AO ^[23], Pythagorean fuzzy power AO ^[24,51] and hesitant fuzzy PAO ^[66]. Garg & Rani ^[13,41] provided weighted and powerful AOs to solve the MADM problem. Zhou, Chen & Liu suggested generalizing PAOs ^[63], and Zhou & Chen also introduced linguistic generalized power aggregation operators ^[64]. Xu ^[53] has defined the IF power AOs and interval-valued intuitionistic fuzzy power AOs. In addition, Xu & Cai ^[54] established an uncertain power ordered weighted averaging operator utilizing the PA (power averaging) operator and the uncertain ordered weighted averaging operator. Xu & Yager ^[52] have implemented an uncertain ordered weighted geometric operator using the power geometric operator and the UOW averaging operator. Xiong et al. ^[55] proposed an extended power average operator for decision-making, a case study in the emergency response plan selection of civil aviation. Chen et al. ^[10] defined a power-average operator based hybrid multi-attribute online product recommendation model for consumer decision-making.

Zhang et al. ^[62], mixed complex numbers and fuzzy sets and defined the concept of complex fuzzy sets (CFSs). Further, Ramot et al. ^[40] defined a new notion that is somewhat different from other studies, in which they expanded the range of MD to the unit circle in the complex plane, unlike the others that were limited, the generalized form of Zadeh fuzzy sets (FSs) ^[61]. A CFS is represented by a complex valued function, such as $\sigma _{\Im }(x) = \kappa _{\Im }(x).e^{2\pi i\Omega _{\kappa _{\Im }}(x)}$ satisfying the condition: $0\leq \kappa _{\Im }(x), \Omega _{\kappa _{\Im }}(x)\leq 1$. The distinction between CFSs and FSs is that the CFSs range is not limited to $[0, 1]$ but spread in a complex plane to a unit disk. In FS theory, the CFSs information has earned more attention. Bi et al. ^[8] recently defined complex fuzzy geometrical AOs. Due to their merits and benefits, CFSs are extensively tested for problems in DM and other fields ^[32]. Since FSs and CFSs can only describe the MD and their complex-valued grade, they are unable to express the complex-valued negative degree of membership. Alkouri et al. ^[44] proposed the complex intuitionistic fuzzy set (CIFS) structure, which essentially consists of two complex membership functions that represent the positive and negative membership grades of an element. Ma et al. ^[30] defined the CFS idea for resolving the problems under multi-periodic factors. Dick et al. ^[11] discussed several CFSs, and Liu & Zhang ^[26] developed the results of Dick et al. ^[11]. Hu et al. ^[18] suggested some new distance steps for the CFS and investigated the consistency of CFS operations.

However, in some theories like fuzzy sets ^[61], intuitionistic fuzzy sets ^[1], and complex fuzzy sets ^[40], each element in these types of sets is represented as an ordered pair of MD and NMD with the ultimate aim of limiting their total to one ^[20,44]. Garg and Rani ^[13] suggested some CIFS aggregation operators and used these operators to solve a multi-attribute decision making (MADM) problem. The theory of power AOs for CIFSs was defined by Rani and Garg ^[41], which was later applied in MADM. Singh et al. ^[43] defined the interval-valued lattices of CFSs and their granular decomposition. Selvachandran et al. ^[42] suggested several similarity tests of CFSs, and their applications studied in pattern recognition. In ^[36], Quek and Selvachandran researched group-associated CIFS algebraic structures. Garg and Rani ^[14,15] developed Bonferroni mean operators and robust average and geometric AOs for CIFSs. Liu et al. ^[39] defined an approach for the MAGDM problem under Cq-ROFs linguistic information using the Heronian mean operators. Liu et al. ^[28] defined some AOs for complex q-rung orthopair fuzzy sets (Cq-ROFSs) and discussed their applications in MCGDM. Garg et al. ^[16] investigated the innovative approach of Cq-ROFSs, which are a combination of q-ROFSs and CFSs, for dealing with difficult and complicated information. Naeem et al. ^[34] defined complex Spherical fuzzy sets and applied them to a decision support system using entropy measure and a power operator. Karaaslan and Dawood ^[22] proposed CT-SF Dombi AOs and studied their applications in MCDM. Ali et al. ^[2] defined some AOs for CT-SFSs and discussed their application in the MADM problem. Akram et al. ^[3] presented a hybrid DM framework under complex Spherical fuzzy (CSF) prioritized weighted aggregation operators. Akram et al. ^[4] defined some extension of the TOPSIS model to the DM under CSF information. Akram et al. ^[5] proposed hybrid DM frameworks under complex Spherical fuzzy soft sets. Akram et al. ^[6] proposed complex Spherical Dombi fuzzy aggregation operators for decision-making.

In the CT-SFS theory, membership degree (MD), neutral membership degree (NuMD) and non-membership degree (NMD) are complex valued and are represented in polar coordinates. The amplitude term connected with the MD, NuMD and NMD gives the extent of an object's belongings in a CIFS, whereas the phase term associated with MD, NuMD and NMD gives extra information, which is generally related to periodicity. The phase terms are novel MD, NuMD and NMD parameters, and they are the parameters that distinguish the traditional T-SFS and CT-SFS theories. T-SFS theory deals with only one dimension at a time, while the CT-SFS deals with two dimensions at a time. CT-SFSs are very effective in representing the two dimensions of certain objects, as the phase term of a CT-SFS is used to capture the second dimension. In addition, T-SFSs are quite remarkable as a general model for fuzzy information, which becomes indispensable when neutral opinions occur. Thus, by introducing this second dimension, all of the information can be projected in a single set, avoiding information loss. The features of CFS theory and the T-SFS theory are combined with the CT-SFS theory.

We present the theory of PA operators among the CT-SFSs, maintaining the benefits of this collection and focusing on the value of aggregation operators. Based on our knowledge, operators cannot be used in the aforementioned studies to manage CT-SFS information. In order to do this, we first describe certain operation laws for CT-SFSs and analyze their basic properties. Next, some PA operators called CT-SFS power average, CT-SFS power geometric, CT-SFS weighted power average and geometric, as well as their respective ordered weighted operators, are proposed to aggregate the various complex T-Spherical fuzzy numbers (CT-SFNs). The basic features of these operators will be discussed in detail. In addition, we suggest a MCGDM method based on current CT-SFSs operators. Both the viability and the effectiveness of the strategy have been demonstrated by an illustrative example.

The rest of the manuscript is organized as follows: In Section 2, we summarize briefly the definitions of CFSs, PFSs, CPFSs, Cq-ROFSs and CT-SFSs. In Section 3, we propose some simple operational laws for CT-SFNs, and a series of averaging and geometric aggregation operators are developed based on the defined operational laws in Section 4. In Section 5, we describe a MCGDM approach using the developed operators with the CT-SFS information, where CT-SFNs are characterized by each element of the set. An example is described in Section 6, to show the functionality of the proposed method. Also, we compare our results with other results of current approaches, and finally this analysis is summarized in Section 7.

2.   Preliminaries

In this section, we give a brief literature review of existing concepts such as CFSs, CIFSs, CPFSs, Cq-ROFSs.

Definition 2.1. ^[40] A CFS $C$ on $X$ (universal set) is of the form:

where $\sigma _{C}:U\rightarrow \{z:z\in C, \left\vert z\right\vert \leq 1\}$ and $\sigma _{C}(x) = a+ib = \kappa _{C}(x).e^{2\pi i\Omega _{\kappa _{C}}(x)}.$ Here, $\kappa _{C}(x) = \sqrt{a^{2}+b^{2}}\in R$ and $\kappa _{C}(x), \Omega _{C(x)}\in \left[0, 1\right],$ where $i = \sqrt{-1}.$

Definition 2.2. ^[44] A CIFS $I$ on $X$ is of the form:

where $\sigma _{I}:U\rightarrow \{z_{1}:z_{1}\in I, \left\vert z_{1}\right\vert \leq 1\},$ $\upsilon _{I}:U\rightarrow \{z_{2}:z_{2}\in I, \left\vert z_{2}\right\vert \leq 1\},$ such as $\sigma _{I}(x) = z_{1} = a_{1}+ib_{1}$ and $\upsilon _{I}(x) = z_{2} = a_{2}+ib_{2}$ on the condition that $0\leq \left\vert z_{1}\right\vert +\left\vert z_{2}\right\vert \leq 1$ or $\sigma _{I}(x) = \kappa _{I}(x).e^{2\pi i\Omega _{\kappa _{I}(x)}}$ and $\upsilon _{i}(x) = \xi _{i}(x).e^{2\pi i\Omega _{\xi _{I}(x)}}$ satisfying the conditions $0\leq \kappa _{I}(x)+\xi _{I}(x)\leq 1$ and $0\leq \Omega _{\kappa _{I}(x)}+\Omega _{\xi _{I}(x)}\leq 1.$ The term $H_{I}(x) = R.e^{2\pi i\Omega _{R}},$ such that $R = 1-(\left\vert z_{1}\right\vert +\left\vert z_{2}\right\vert)$, and $\Omega _{R}(x) = 1-\left(\Omega _{\kappa _{I}(x)}+\Omega _{\xi _{I}(x)}\right)$ is considered a hesitancy degree of $(x).$ Furthermore, $I = \left(\kappa.e^{2\pi i\Omega _{\kappa }}, \xi.e^{2\pi i\Omega _{\xi }}\right)$ is called the complex intuitionistic fuzzy number (CIFN).

Definition 2.3. ^[46] A CPFS $P$ on $X$ is of the form:

where $\sigma _{P}:U\rightarrow \{z_{1}:z_{1}\in P, \left\vert z_{1}\right\vert \leq 1\},$ $\upsilon _{P}:U\rightarrow \{z_{2}:z_{2}\in P, \left\vert z_{2}\right\vert \leq 1\},$ such as $\sigma _{P}(x) = z_{1} = a_{1}+ib_{1}$ and $\upsilon _{P}(x) = z_{2} = a_{2}+ib_{2}$ on the condition that $0\leq \left\vert z_{1}\right\vert ^{2}+\left\vert z_{2}\right\vert ^{2}\leq 1$ or $\sigma _{P}(x) = \kappa _{P}(x).e^{2\pi i\Omega _{\kappa _{P}(x)}}$ and $\upsilon _{P}(x) = \xi _{P}(x).e^{2\pi i\Omega _{\xi _{P}(x)}}$ satisfy the condition $0\leq \kappa _{P}^{2}(x)+\xi _{P}^{2}(x)\leq 1$, and $0\leq \Omega _{\kappa _{P}(x)}^{2}+\Omega _{\xi _{P}(x)}^{2}\leq 1.$ The term $H_{P}(x) = R.e^{2\pi i\Omega _{R}},$ such that $R = \sqrt{1-(\kappa _{P}^{2}(x)+\xi _{P}^{2}(x))}$ and $\Omega _{R}(x) = \sqrt{1-\left(\Omega _{\kappa _{P}(x)}^{2}+\Omega _{\xi _{P}(x)}^{2}\right) }$ is considered a hesitancy degree of $x.$ Furthermore, $P = \left(\kappa.e^{2\pi i\Omega _{\kappa }}, \xi.e^{2\pi i\Omega _{\xi }}\right)$ is called the complex Pythagorean fuzzy number (CPFN). CPFS positive and negative membership grades are clearly polar or Cartesian numbers in complex form.

Definition 2.4. ^[28] A Cq-ROFS $S$ on $X$ is of the form

where $\sigma _{S}:U\rightarrow \{z_{1}:z_{1}\in S, \left\vert z_{1}\right\vert \leq 1\},$ and $\upsilon _{S}:U\rightarrow \{z_{2}:z_{2}\in S, \left\vert z_{2}\right\vert \leq 1\},$ such that $\sigma _{S}(x) = z_{1} = a_{1}+ib_{1}$ and $\upsilon _{S}(x) = z_{2} = a_{2}+ib_{2}$, on the condition that $0\leq \left\vert z_{1}\right\vert ^{q}+\left\vert z_{2}\right\vert ^{q}\leq 1$ or $\sigma _{S}(x) = \kappa _{S}(x).e^{2\pi i\Omega _{\kappa _{S}(x)}}$, and $\upsilon _{S}(x) = \xi _{S}(x).e^{2\pi i\Omega _{\xi _{S}(x)}}$ satisfying the condition $0\leq \kappa _{S}^{q}(x)+\xi _{S}^{q}(x)\leq 1$ and $0\leq \Omega _{\kappa _{S}(x)}^{q}+\Omega _{\xi _{S}(x)}^{q}\leq 1.$ The term $H_{S}(x) = R.e^{2\pi i\Omega _{R}},$ such that $R = \left(1-\kappa _{S}^{q}(x)-\xi _{S}^{q}(x)\right) ^{\frac{1}{q}}$ and $\Omega _{R}(x) = \left(1-\left(\Omega _{\kappa _{S}(x)}^{q}+\Omega _{\xi _{S}(x)}^{q}\right) \right) ^{ \frac{1}{q}}$ is considered a hesitancy degree of $x.$ Furthermore, $S = \left(\kappa.e^{2\pi i\Omega _{\kappa }}, \xi.e^{2\pi i\Omega _{\xi }}\right)$ is called the complex q-rung orthopair fuzzy number (Cq-ROFN).

Definition 2.5. ^[56] For a family of $\Phi _{i}(i = 1, ..., n),$ the power average operator is defined as:

where $T(\Phi _{i}) = \sum_{i = 1}^{n}$Sup$(\Phi _{i}, \Phi _{s})$, and Sup$(\Phi _{i}, \Phi _{s})$ is the support of $\Phi _{i}$ from $\Phi _{s}$ and known as the similarity index, which satisfies the below properties:

(1) Sup$\left(\Phi _{i}, \Phi _{s}\right) \in \lbrack 0, 1];$

(2) Sup$\left(\Phi _{i}, \Phi _{s}\right) =$ Sup$\left(\Phi _{s}, \Phi _{i}\right);$

(3) Sup$\left(\Phi _{i}, \Phi _{s}\right) \geq$ Sup$\left(\Phi _{k}, \Phi _{l}\right),$ if $\left\vert \Phi _{i}-\Phi _{s}\right\vert \leq \left\vert \Phi _{k}-\Phi _{l}\right\vert.$

3.   Complex T-Spherical fuzzy set

CT-SFSs and their basic operation laws for the collection of CT-SFNs, as well as their corresponding PA operators, are introduced in this section.

Definition 3.1. ^[2] A CT-SFS $S$ on $X$ is of the form

where $\sigma _{S}:U\rightarrow \{z_{1}:z_{1}\in S, \left\vert z_{1}\right\vert \leq 1\}, \upsilon _{S}:U\rightarrow \{z_{2}:z_{2}\in S, \left\vert z_{2}\right\vert \leq 1\},$ and $\chi _{S}:U\rightarrow \{z_{3}:z_{3}\in S, \left\vert z_{3}\right\vert \leq 1\}$ such that $\sigma _{S}(x) = z_{1} = a_{1}+ib_{1}$, $\upsilon _{S}(x) = z_{2} = a_{2}+ib_{2}$, $\chi _{S}(x) = z_{3} = a_{3}+ib_{3}$, on the condition that $0\leq \left\vert z_{1}\right\vert ^{q}+\left\vert z_{2}\right\vert ^{q}+\left\vert z_{3}\right\vert ^{q}\leq 1$ or $\sigma _{S}(x) = \kappa _{S}(x).e^{2\pi i\Omega _{\kappa _{S}(x)}}$, $\upsilon _{S}(x) = \xi _{S}(x).e^{2\pi i\Omega _{\xi _{S}(x)}}$ and $\chi (x) = \phi _{S}(x).e^{2\pi i\Omega _{\phi _{S}(x)}}$ satisfying the conditions; $0\leq \kappa _{S}^{q}(x)+\xi _{S}^{q}(x)+\phi _{S}^{q}(x)\leq 1$ and $0\leq \Omega _{\kappa _{S}(x)}^{q}+\Omega _{\xi _{S}(x)}^{q}+\Omega _{\phi _{S}(x)}^{q}\leq 1.$ The term $H_{S}(x) = R.e^{2\pi i\Omega _{R}},$ such that $R = \left(1-\kappa _{S}^{q}(x)-\xi _{S}^{q}(x)-\phi _{S}^{q}(x)\right) ^{\frac{1}{q}}$, and $\Omega _{R}(x) = \left(1-\left(\Omega _{\kappa _{S}(x)}^{q}+\Omega _{\xi _{S}(x)}^{q}+\Omega _{\phi _{S}(x)}^{q}\right) \right) ^{\frac{1}{q}}$ is considered a hesitancy degree of $x.$ Furthermore, $S = \left(\kappa.e^{2\pi i\Omega _{\kappa }}, \xi.e^{2\pi i\Omega _{\xi }}, \phi.e^{2\pi i\Omega _{\phi }}\right)$ is called the complex T-Spherical fuzzy number (CT-SFN).

3.1.   Operational laws of CT-SFNs

Definition 3.2. For CT-SFN $\Upsilon = \left\{ \left(\kappa _{\Upsilon }, \Omega _{\kappa _{\Upsilon }}\right), (\xi _{\Upsilon }, \Omega _{\xi _{\Upsilon }}), (\phi _{\Upsilon }, \Omega _{\phi _{\Upsilon }})\right\},$ the score function $Sc^{\ast }$ of $\Upsilon$ is defined as

and the accuracy function $Hc^{\ast }$ of $\Upsilon$ is defined as:

where $Sc^{\ast }\left(\Upsilon \right) \in \lbrack 0, 1]$ and $Hc^{\ast }\left(\Upsilon \right) \in \lbrack 0, 1].$

Definition 3.3. The following comparison rules between two CT-SFNs $\Upsilon _{1}$ and $\Upsilon _{2}$ are satisfied:

(1) If $Sc^{\ast }\left(\Upsilon _{1}\right) > Sc^{\ast }\left(\Upsilon _{2}\right),$ then $\Upsilon _{1} > \Upsilon _{2};$

(2) If $Sc^{\ast }\left(\Upsilon _{1}\right) = Sc^{\ast }\left(\Upsilon _{2}\right),$

(3) If $Hc^{\ast }\left(\Upsilon _{1}\right) > Hc^{\ast }\left(\Upsilon _{2}\right),$ then $\Upsilon _{1} > \Upsilon _{2};$

(4) If $Hc^{\ast }\left(\Upsilon _{1}\right) = Hc^{\ast }\left(\Upsilon _{2}\right),$ then $\Upsilon _{1} = \Upsilon _{2}.$

Definition 3.4. For any two CT-SFNs $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\}$ and

$\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\},$ the distance measure is defined as:

Definition 3.5. For any two CT-SFNs $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\}$ and

$\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\}$ and $\lambda > 0$, the basic operation laws are defined as:

(1) $\Upsilon _{1}\oplus \Upsilon _{2} = \left\{ \begin{array}{c} \left(\sqrt[q]{1-\prod\limits_{i = 1}^{2}\left(1-\kappa _{\Upsilon _{i}}^{q}\right), 2\pi \left(1-\prod\limits_{i = 1}^{2}\left(1-\frac{\Omega _{\kappa _{\Upsilon _{i}}}^{q}}{2\pi }\right) \right) }\right), \\ \left(\prod\limits_{i = 1}^{2}\xi _{\Upsilon _{i}}, 2\pi \left(\prod\limits_{i = 1}^{2}\frac{\Omega _{\xi _{\Upsilon _{i}}}}{2\pi }\right) \right), \left(\prod\limits_{i = 1}^{2}\phi _{\Upsilon _{i}}, 2\pi \left(\prod\limits_{i = 1}^{2}\frac{\Omega _{\phi _{\Upsilon _{i}}}}{2\pi }\right) \right) \end{array} \right\};$

(2) $\Upsilon _{1}\otimes \Upsilon _{2} = \left\{ \begin{array}{c} \left(\prod\limits_{i = 1}^{2}\kappa _{\Upsilon _{i}}, 2\pi \left(\prod\limits_{i = 1}^{2}\frac{\Omega _{\kappa _{\Upsilon _{i}}}}{2\pi }\right) \right), \\ \left(\sqrt[q]{1-\prod\limits_{i = 1}^{2}\left(1-\xi _{\Upsilon _{i}}^{q}\right), 2\pi \left(1-\prod\limits_{i = 1}^{2}\left(1-\frac{\Omega _{\xi _{\Upsilon _{i}}}^{q}}{2\pi }\right) \right) }\right), \\ \left(\sqrt[q]{1-\prod\limits_{i = 1}^{2}\left(1-\phi _{\Upsilon _{i}}^{q}\right), 2\pi \left(1-\prod\limits_{i = 1}^{2}\left(1-\frac{\Omega _{\phi _{\Upsilon _{i}}}^{q}}{2\pi }\right) \right) }\right) \end{array} \right\};$

(3) $\lambda \Upsilon _{1} = \left\{ \begin{array}{c} \left(\sqrt[q]{1-\left(1-\kappa _{\Upsilon _{1}}^{q}\right) ^{\lambda }, 2\pi \left(1-\left(1-\frac{\Omega _{\kappa _{\Upsilon _{1}}}^{q}}{2\pi } \right) ^{\lambda }\right) }\right), \\ \left(\xi _{\Upsilon _{1}}^{\lambda }, 2\pi \left(\frac{\Omega _{\xi _{\Upsilon _{1}}}}{2\pi }\right) ^{\lambda }\right), \left(\phi _{\Upsilon _{1}}^{\lambda }, 2\pi \left(\frac{\Omega _{\phi _{\Upsilon _{1}}}}{2\pi } \right) ^{\lambda }\right) \end{array} \right\};$

(4) $\Upsilon _{1}^{\lambda } = \left\{ \begin{array}{c} \left(\kappa _{\Upsilon _{1}}^{\lambda }, 2\pi \left(\frac{\Omega _{\kappa _{\Upsilon _{1}}}}{2\pi }\right) ^{\lambda }\right), \left(\sqrt[q]{ 1-\left(1-\xi _{\Upsilon _{1}}^{q}\right) ^{\lambda }, 2\pi \left(1-\left(1-\frac{\Omega _{\xi _{\Upsilon _{1}}}^{q}}{2\pi }\right) ^{\lambda }\right) }\right), \\ \left(\sqrt[q]{1-\left(1-\phi _{\Upsilon _{1}}^{q}\right) ^{\lambda }, 2\pi \left(1-\left(1-\frac{\Omega _{\phi _{\Upsilon _{1}}}^{q}}{2\pi }\right) ^{\lambda }\right) }\right) \end{array} \right\}.$

Theorem 3.1. Let $\Upsilon _{1}, \Upsilon _{2}, \Upsilon _{3}$ be any three CT-SFNs. Then, the following properties are satisfied.

(1) $\Upsilon _{1}\oplus \Upsilon _{2} = \Upsilon _{2}\oplus \Upsilon _{1};$

(2) $\Upsilon _{1}\otimes \Upsilon _{2} = \Upsilon _{2}\otimes \Upsilon _{1};$

(3) $\left(\Upsilon _{1}\oplus \Upsilon _{2}\right) \oplus \Upsilon _{3} = \Upsilon _{1}\oplus \left(\Upsilon _{2}\oplus \Upsilon _{3}\right);$

(4) $\left(\Upsilon _{1}\otimes \Upsilon _{2}\right) \otimes \Upsilon _{3} = \Upsilon _{1}\otimes \left(\Upsilon _{2}\otimes \Upsilon _{3}\right).$

Proof. Here, we prove parts (1) and (3), and the proofs of the others are similar.

(1) As $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\}$ and

$\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\}$ are the two CT-SFNs, then we have

(3) As $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\},$

$\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\}$ and

$\Upsilon _{3} = \left\{ \left(\kappa _{\Upsilon _{3}}, \Omega _{\kappa _{\Upsilon _{3}}}\right), \left(\xi _{\Upsilon _{3}}, \Omega _{\xi _{\Upsilon _{3}}}\right), \left(\phi _{\Upsilon _{3}}, \Omega _{\phi _{\Upsilon _{3}}}\right) \right\}$ are the three CT-SFNs, we have

Theorem 3.2. For any two CT-SFNs $\Upsilon _{1}$ and $\Upsilon _{2}$ and $\lambda, \lambda _{1}, \lambda _{2}$ (positive real numbers), we have

(1) $\lambda \left(\Upsilon _{1}\oplus \Upsilon _{2}\right) = \lambda \Upsilon _{1}\oplus \lambda \Upsilon _{2};$

(2) $\left(\Upsilon _{1}\otimes \Upsilon _{2}\right) ^{\lambda } = \Upsilon _{1}^{\lambda }\oplus \Upsilon _{2}^{\lambda };$

(3) $\left(\lambda _{1}+\lambda _{2}\right) \Upsilon _{1} = \lambda _{1}\Upsilon _{1}\oplus \lambda _{2}\Upsilon _{1};$

(4) $\Upsilon _{1}^{\lambda _{1}+\lambda _{2}} = \Upsilon _{1}^{\lambda _{1}}\otimes \Upsilon _{1}^{\lambda _{2}}.$

Proof. Here, we have proven only parts (1) and (3), and the proofs of other parts are similar.

As given that $\Upsilon _{1}$ and $\Upsilon _{2}$ are CT-SFNs, we have

Hence, $\lambda \left(\Upsilon _{1}\oplus \Upsilon _{2}\right) = \lambda \Upsilon _{1}\oplus \lambda \Upsilon _{2}.$

(3) From the operational laws we can write given,

4.   Power aggregation operators

Several power averaging (PA) operators are introduced with the CT-SF information in this section, using the defined operation laws of the CT-SFNs, in order to aggregate the CT-SFNs.

4.1.   Complex T-Spherical fuzzy power averaging operator

Definition 4.1. For a family of CT-SFNs $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n),$ the complex T-Spherical fuzzy power averaging (CT-SFPA) aggregation operator is a function $CT$-$SFPA:\Omega ^{n}\rightarrow \Omega$ defined by

where $\rho _{i} = \frac{1+T\left(\Upsilon _{i}\right) }{\sum_{i = 1}^{n}\left(1+T\left(\Upsilon _{i}\right) \right) }$ and $T\left(\Upsilon _{i}\right) = \sum\limits_{\underset{s\neq i}{s = i}}^{n}\left({\rm{Sup}}\left(\Upsilon _{i}, \Upsilon _{s}\right) \right) (i = 1, ..., n).$ Here, Sup$\left(\Upsilon _{i}, \Upsilon _{s}\right)$ is the support of $\Upsilon _{i}$ from $\Upsilon _{s}$ satisfying the aforementioned properties, and Sup$\left(\Upsilon _{i}, \Upsilon _{s}\right) = 1-d\left(\Upsilon _{i}, \Upsilon _{s}\right)$, where $d$ is the distance measure defined in Definition $\left(3.1 \right)$.

Theorem 4.1. Let a family of CT-SFNs be $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n).$ Then, the aggregated value obtained by using the CT-SFPA operator is also a CT-SFN and is given as follows.

Proof. To prove Eq $\left(4.2\right)$, we use mathematical induction. For each $i, \Upsilon _{i}$ is a CT-SFN and $\Upsilon _{i} > 0$. Therefore, we have $\rho _{i}\Upsilon _{i}$ is again a CT-SFN. Then, applying the mathematical induction steps on $n,$ we obtain the following.

(1) For $n = 2,$ we get $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\}$ and $\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\}.$ As a result of the CT-SFNs operating law, we have the following.

and

As a result, using the CT-SFNs addition law, we have the following.

Thus, the result holds for $n = 2$.

(2) Let Eq $\left(4.2\right)$ be true for $n = k$, i.e., the following.

Then, $n = k+1,$ and we get the following.

Thus, Eq $\left(4.2\right)$ is true for all positive natural numbers $n.$

The CT-SFPA operator satisfies some of the properties listed below, according to Theorem $\left(4.1\right)$.

Property 1 (Idempotency). Let $\Upsilon _{0}$ be a CT-SFN, and if $\Upsilon _{i} = \Upsilon _{0}$ for all $i = 1, ..., n,$ then

Proof. Let $\Upsilon _{0} = \left\{ \left(\kappa _{\Upsilon _{0}}, \Omega _{\kappa _{\Upsilon _{0}}}\right), \left(\xi _{\Upsilon _{0}}, \Omega _{\xi _{\Upsilon _{0}}}\right), \left(\phi _{\Upsilon _{0}}, \Omega _{\phi _{\Upsilon _{0}}}\right) \right\}$ and

$\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ be the CT-SFNs, such that $\Upsilon _{i} = \Upsilon _{0}$ for all $i,$ where $\kappa _{\Upsilon _{i}} = \kappa _{\Upsilon _{0}}, \xi _{\Upsilon _{i}} = \xi _{\Upsilon _{0}}, \phi _{\Upsilon _{i}} = \phi _{\Upsilon _{0}}, \Omega _{\kappa _{\Upsilon _{i}}} = \Omega _{\kappa _{\Upsilon _{0}}}, \Omega _{\xi _{\Upsilon _{i}}} = \Omega _{\xi _{\Upsilon _{0}}}$ and $\Omega _{\phi _{\Upsilon _{i}}} = \Omega _{\phi _{\Upsilon _{0}}}$ for all $i.$ Then, based on the $\rho _{i}$, we have $\sum_{i = 1}^{n}\rho _{i} = 1.$ So, by Theorem $\left(4.1 \right)$, we get

Property 2 (Boundedness). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} \left(i = 1, ..., n\right)$ be the family of CT-SFNs, and $\Upsilon ^{-} = \left\{ \underset{i}{\min }\left(\kappa _{\Upsilon _{i}}\right), \underset{i}{\min } \left(\Omega _{\kappa _{\Upsilon _{i}}}\right), \underset{i}{\min }\left(\xi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\xi _{\Upsilon _{i}}}\right), \underset{i}{\max }\left(\phi _{\Upsilon _{i}}\right), \underset{i}{\max }\left(\Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon ^{+} = \left\{ \underset{i}{\max }\left(\kappa _{\Upsilon _{i}}\right), \underset{i}{\max }\left(\Omega _{\kappa _{\Upsilon _{i}}}\right), \underset{i}{\min }\left(\xi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\xi _{\Upsilon _{i}}}\right), \underset{i} {\min }\left(\phi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\phi _{\Upsilon _{i}}}\right) \right\}.$ Then,

Proof. Take $\Upsilon \leq CT-SFPA(\Upsilon _{1}, ..., \Upsilon _{n}),$ and hence by Theorem $\left(4.1 \right)$, we get $\Upsilon = \left\{ \left(\kappa _{\Upsilon }, \Omega _{\kappa _{\Upsilon }}\right), (\xi _{\Upsilon }, \Omega _{\xi _{\Upsilon }}), (\phi _{\Upsilon }, \Omega _{\phi _{\Upsilon }})\right\}.$ For a CT-SFN $\Upsilon _{i},$ we have

and

Similarly, we obtain $\underset{i}{\min }\left(\Omega _{\kappa _{\Upsilon _{i}}}\right) \leq \Omega _{\kappa _{\Upsilon }}\leq \underset{i}{\max } \left(\Omega _{\kappa _{\Upsilon _{i}}}\right),$ $\underset{i}{\min } \left(\Omega _{\xi _{\Upsilon _{i}}}\right) \leq \Omega _{\xi _{\Upsilon }}\leq \underset{i}{\max }\left(\Omega _{\xi _{\Upsilon _{i}}}\right)$ and $\underset{i}{\min }\left(\Omega _{\phi _{\Upsilon _{i}}}\right) \leq \Omega _{\phi _{\Upsilon }}\leq \underset{i}{\max }\left(\Omega _{\phi _{\Upsilon _{i}}}\right).$ Now, by using Definition $\left(3.1 \right)$, we get the following.

Thus, $Sc^{\ast }\left(\Upsilon ^{-}\right) \leq Sc^{\ast }\left(\Upsilon \right) \leq Sc^{\ast }\left(\Upsilon ^{+}\right).$

Hence, by the ranking order, we have

Property 3 (Commutivity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} \left(i = 1, ..., n\right)$ be the family of CT-SFNs, if $\left(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast }\right)$ are the permutations of $\left(\Upsilon _{1}, ..., \Upsilon _{n}\right).$ Then,

Proof. $\left(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast }\right)$ is an arbitrary arrangement of $\left(\Upsilon _{1}, ..., \Upsilon _{n}\right).$ Therefore,

Thus, $CT-SFPA\left(\Upsilon _{1}, ..., \Upsilon _{n}\right) = CT-SFPA\left(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast }\right).$

Property 4 (Monotonicity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon _{i}^{\ast } = \left\{ \left(\kappa _{\Upsilon _{i}}^{\ast }, \Omega _{\kappa _{\Upsilon _{i}}}^{\ast }\right), \left(\xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}^{\ast }\right), \left(\phi _{\Upsilon _{i}}^{\ast }, \Omega _{\phi _{\Upsilon _{i}}}^{\ast }\right) \right\} \left(i = 1, ..., n\right)$ be the families of CT-SFNs, such that $\Upsilon _{i}\leq \Upsilon _{i}^{\ast }.$ Then,

Proof. It's given that $\kappa _{\Upsilon _{i}}\leq \kappa _{\Upsilon _{i}}^{\ast }, \Omega _{\kappa _{\Upsilon _{i}}}\leq \Omega _{\kappa _{\Upsilon _{i}}}^{\ast },$ $\xi _{\Upsilon _{i}}\leq \xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}\leq \Omega _{\xi _{\Upsilon _{i}}}^{\ast },$ $\phi _{\Upsilon _{i}}\leq \phi _{\Upsilon _{i}}^{\ast }$ and $\Omega _{\phi _{\Upsilon _{i}}}\leq \Omega _{\phi _{\Upsilon _{i}}}^{\ast }$ for all $i.$ Then,

and

Similarly, we can show that $\xi _{\Upsilon _{i}}\leq \xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}\leq \Omega _{\xi _{\Upsilon _{i}}}^{\ast },$ $\phi _{\Upsilon _{i}}\leq \phi _{\Upsilon _{i}}^{\ast }$ and $\Omega _{\phi _{\Upsilon _{i}}}\leq \Omega _{\phi _{\Upsilon _{i}}}^{\ast }.$ Thus, we obtain

Hence, from the above Equation we prove that, $CT-SFPA\left(\Upsilon _{1}, ..., \Upsilon _{n}\right) \leq CT-SFPA\left(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast }\right).$

4.2.   Complex T-Spherical fuzzy weighted power averaging operator

In this part, we consider the distinct weighting vector of the CT-SFNs $\Upsilon _{i}(i = 1, ..., n)$ during the aggregation process, as opposed to the above-mentioned CT-SFPA operator, and developed a modified CT-SF weighted power averaging (CT-SFWPA) aggregation operator as follows.

Definition 4.2. For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ the CT-SFWPA operator is a function $CT-SFWPA:\Omega ^{n}\rightarrow \Omega$ defined by:

where $\phi _{i} = \frac{\psi _{i}\left(1+T^{/}\left(\Upsilon _{i}\right) \right) }{\sum_{i = 1}^{n}\psi _{i}\left(1+T^{/}\left(\Upsilon _{i}\right) \right) }, T^{/}\left(\Upsilon _{i}\right) = \sum\limits_{\underset{s\neq i}{ s = 1}}^{n}\psi _{s}\left({\rm{Sup}}\left(\Upsilon _{i}, \Upsilon _{s}\right) \right)$ and $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T}$ be the weight vector of CT-SFNs $\Upsilon _{i},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1.$

Theorem 4.2. Let a family of CT-SFNs $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n),$ with their corresponding weight vector $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T}$, such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1.$ Then, the aggregated value obtained by using the CT-SFWPA operator is again a CT-SFN, and it is given as follows.

Proof. The proof is the same as that of Theorem $\left(4.1 \right)$.

For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n)$ with weight vector $\psi = \left(\psi _{1}, ..., \psi _{n}\right)^{T},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1,$ the CT-SFWPA operator satisfies the same properties as the CT-SFPA operator, such as the following.

Property 1 (Idempotency). Let $\Upsilon _{0}$ be a CT-SFN, and if $\Upsilon _{i} = \Upsilon _{0}$ for all $i = 1, ..., n,$ then,

Property 2 (Boundedness). Let $\Upsilon ^{-}$ and $\Upsilon ^{+}$ be the lower bound and upper bound of the CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ respectively. Then, we have,

Property 3 (Commutivity). For a permutation $(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast })$ of CT-SFNs $(\Upsilon _{1}, ..., \Upsilon _{n})$ and their corresponding permutation weights $\psi ^{\ast } = \left(\psi _{1}^{\ast }, ..., \psi _{n}^{\ast }\right) ^{T}$, $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T},$ we have

Property 4 (Monotonicity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon _{i}^{\ast } = \left\{ \left(\kappa _{\Upsilon _{i}}^{\ast }, \Omega _{\kappa _{\Upsilon _{i}}}^{\ast }\right), \left(\xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}^{\ast }\right), \left(\phi _{\Upsilon _{i}}^{\ast }, \Omega _{\phi _{\Upsilon _{i}}}^{\ast }\right) \right\} \left(i = 1, ..., n\right)$ be the families of CT-SFNs, such that $\Upsilon _{i}\leq \Upsilon _{i}^{\ast }.$ Then,

4.3.   Complex T-Spherical fuzzy ordered weighted power averaging operator

The defined aggregation operator has been extended to its ordered weighted form in this subsection.

Definition 4.3. For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ a complex T-Spherical fuzzy ordered weighted power averaging (CT-SFOWPA) operator is a function $CT-SFOWPA:\Omega ^{n}\rightarrow \Omega$ defined as:

where $\Omega$ denotes the set of CT-SFNs, and $\sigma (1), ..., \sigma (n)$ are the permutations of $(1, ..., n)$ satisfies that $\Upsilon _{\sigma (i-1)}\geq \Upsilon _{\sigma (i)}$ for $i = 2, ..., n.$ Also, $\zeta _{i}$ is defined as

where $B_{i} = \sum_{s = 1}^{i}V_{\sigma (s)}, V_{\sigma (i)} = 1+\sum\limits_{ \underset{s\neq i}{s = 1}}^{n}($Sup$(\Upsilon _{i}, \Upsilon _{s})), TV = \sum_{i = 1}^{n}V_{\sigma (i)}$ and the mapping $g:[0, 1]\rightarrow \lbrack 0, 1]$ is a basic unit-interval monotonic function that satisfies the three properties $g(0) = 0, g(1) = 1$, and if $x\leq y$ then $g(x)\leq g(y).$

Theorem 4.3. Let a family of CT-SFNs be $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n).$ Then, the aggregated value calculated by using the CT-SFOWPA operator is again a CT-SFN and given as

where $\zeta _{i}$ is defined as in Eq $\left(4.14\right)$.

In addition, the CT-SFOWPA operator satisfies the properties of idempotency, boundedness, commutativity and monotonicity.

4.4.   Complex T-Spherical fuzzy power geometric aggregation operator

The AOs mentioned above are expanded in this subsection to geometric AOs with complex T-Spherical fuzzy information, such as CSF power geometric (CT-SFPG), CT-SF weighted power geometric (CT-SFWPG) and CT-SF ordered weighted power geometric (CT-SFOWPG).

Definition 4.4. For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ the complex T-Spherical fuzzy power geometric (CT-SFPG) operator is a function $CT-SFPG:\Omega ^{n}\rightarrow \Omega$ defined as:

where $\rho _{i} = \frac{1+T\left(\Upsilon _{i}\right) }{\sum_{i = 1}^{n}\left(1+T\left(\Upsilon _{i}\right) \right) }$ and $T\left(\Upsilon _{i}\right) = \sum\limits_{\underset{s\neq i}{s = 1}}^{n}\left({\rm{Sup}}\left(\Upsilon _{i}, \Upsilon _{s}\right) \right) (i = 1, ..., n).$ Here, Sup$\left(\Upsilon _{i}, \Upsilon _{s}\right)$ is the support of $\Upsilon _{i}$ from $\Upsilon _{s}$ satisfying the aforementioned properties, and Sup$\left(\Upsilon _{i}, \Upsilon _{s}\right) = 1-d\left(\Upsilon _{i}, \Upsilon _{s}\right),$ while $d$ is the distance measure.

Theorem 4.4. Let a family of CT-SFNs $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n).$ Then, the aggregated value obtained by using the CT-SFPG operator is again a CT-SFN and given as

Proof. We'll show that Eq $\left(4.17\right)$ holds by using mathematical induction. For each $i, \Upsilon _{i}$ is a CT-SFN, and $\Upsilon _{i} > 0$; therefore, we have that $\Upsilon _{i}^{\rho _{i}}$ is still CT-SFN. Then, using the steps of mathematical induction, we have the following.

(1) For $n = 2,$ we get $\Upsilon _{1} = \left\{ \left(\kappa _{\Upsilon _{1}}, \Omega _{\kappa _{\Upsilon _{1}}}\right), \left(\xi _{\Upsilon _{1}}, \Omega _{\xi _{\Upsilon _{1}}}\right), \left(\phi _{\Upsilon _{1}}, \Omega _{\phi _{\Upsilon _{1}}}\right) \right\}$ and $\Upsilon _{2} = \left\{ \left(\kappa _{\Upsilon _{2}}, \Omega _{\kappa _{\Upsilon _{2}}}\right), \left(\xi _{\Upsilon _{2}}, \Omega _{\xi _{\Upsilon _{2}}}\right), \left(\phi _{\Upsilon _{2}}, \Omega _{\phi _{\Upsilon _{2}}}\right) \right\}.$ Thus, by the operational law of CT-SFNs, we have

and

Hence, using the addition law of CT-SFNs, we obtain

Thus, the resuls holds for $n = 2$.

(2) Let Eq $\left(4.17\right)$ be true for $n = k,$ ($k$ is a positive natural number), i.e.,

Then, $n = k+1,$ and we get

Thus, Eq $\left(4.17\right)$ is true for all positive natural numbers $n.$

For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n)$ with weight vector $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1,$ the CT-SFPG operator satisfies certain properties, such as the following.

Property 1 (Idempotency). Let $\Upsilon _{0}$ be a CT-SFN, and if $\Upsilon _{i} = \Upsilon _{0}$ for all $i = 1, ..., n,$ then

Property 2 (Boundedness). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} \left(i = 1, ..., n\right)$ be the family of CT-SFNs, and $\Upsilon ^{-} = \left\{ \underset{i}{\min }\left(\kappa _{\Upsilon _{i}}\right), \underset{i}{\min } \left(\Omega _{\kappa _{\Upsilon _{i}}}\right), \underset{i}{\min }\left(\xi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\xi _{\Upsilon _{i}}}\right), \underset{i}{\max }\left(\phi _{\Upsilon _{i}}\right), \underset{i}{\max }\left(\Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon ^{+} = \left\{ \underset{i}{\max }\left(\kappa _{\Upsilon _{i}}\right), \underset{i}{\max }\left(\Omega _{\kappa _{\Upsilon _{i}}}\right), \underset{i}{\min }\left(\xi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\xi _{\Upsilon _{i}}}\right), \underset{i} {\min }\left(\phi _{\Upsilon _{i}}\right), \underset{i}{\min }\left(\Omega _{\phi _{\Upsilon _{i}}}\right) \right\}.$ Then,

Property 3 (Commutivity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} \left(i = 1, ..., n\right)$ be family of CT-SFNs. If $\left(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast }\right)$ are the permutation of $\left(\Upsilon _{1}, ..., \Upsilon _{n}\right),$ then

Property 4 (Monotonicity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon _{i}^{\ast } = \left\{ \left(\kappa _{\Upsilon _{i}}^{\ast }, \Omega _{\kappa _{\Upsilon _{i}}}^{\ast }\right), \left(\xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}^{\ast }\right), \left(\phi _{\Upsilon _{i}}^{\ast }, \Omega _{\phi _{\Upsilon _{i}}}^{\ast }\right) \right\} \left(i = 1, ..., n\right)$ be the families of CT-SFNs, such that $\Upsilon _{i}\leq \Upsilon _{i}^{\ast }.$ Then,

4.5.   Complex T-Spherical fuzzy weighted power geometric operator

In this section, we consider the distinct weighting factor of the CT-SFNs $\Upsilon _{i}(i = 1, ..., n)$ during the aggregation process, as opposed to the above CT-SFPG operator, and suggest a new CT-SF weighted power geometric (CT-SFWPG) aggregation operator.

Definition 4.5. For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ the CT-SFWPG operator is a function $CT-SFWPG:\Omega ^{n}\rightarrow \Omega$ defined by;

where $\phi _{i} = \frac{\psi _{i}\left(1+T^{/}\left(\Upsilon _{i}\right) \right) }{\sum_{i = 1}^{n}\left(1+T^{/}\left(\Upsilon _{i}\right) \right) }, T^{/}\left(\Upsilon _{i}\right) = \sum\limits_{\underset{s\neq i}{s = 1} }^{n}\psi _{s}\left({\rm{Sup}}\left(\Upsilon _{i}, \Upsilon _{s}\right) \right)$ and $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T}$ are the weights of CT-SFNs $\Upsilon _{i},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1.$

Theorem 4.5. Let a family of CT-SFNs $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n),$ with the respective weight vector $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T}$, such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1.$ Then, the aggregated value obtained by using the CT-SFWPG operator is again a CT-SFN, as follows.

Proof. The proof of the theorem is same as that of Theorem $\left(\ref{th2}\right)$.

For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n)$ with weight vector $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1,$ the CT-SFWPG operator also satisfies the same properties as the CT-SFPG operator such as the following.

Property 1 (Idempotency). Let $\Upsilon _{0}$ be a CT-SFN, and if $\Upsilon _{i} = \Upsilon _{0}$ for all $i = 1, ..., n,$ then

Property 2 (Boundedness). Let $\Upsilon ^{-}$ and $\Upsilon ^{+}$ be the lower bound and upper bound of the CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ respectively. Then, we have,

Property 3 (Commutivity). For a permutation $(\Upsilon _{1}^{\ast }, ..., \Upsilon _{n}^{\ast })$ of CT-SFNs $(\Upsilon _{1}, ..., \Upsilon _{n})$ and their corresponding permutation weights $\psi ^{\ast } = \left(\psi _{1}^{\ast }, ..., \psi _{n}^{\ast }\right) ^{T}$ of $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T},$ we have

Property 4 (Monotonicity). Let $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ and

$\Upsilon _{i}^{\ast } = \left\{ \left(\kappa _{\Upsilon _{i}}^{\ast }, \Omega _{\kappa _{\Upsilon _{i}}}^{\ast }\right), \left(\xi _{\Upsilon _{i}}^{\ast }, \Omega _{\xi _{\Upsilon _{i}}}^{\ast }\right), \left(\phi _{\Upsilon _{i}}^{\ast }, \Omega _{\phi _{\Upsilon _{i}}}^{\ast }\right) \right\} \left(i = 1, ..., n\right)$ be the families of CT-SFNs, such that $\Upsilon _{i}\leq \Upsilon _{i}^{\ast }.$ Then,

4.6.   Complex T-Spherical fuzzy ordered weighted power geometric operator

The existing AO is expanded to an ordered weighted AO in this segment.

Definition 4.6. For a family of CT-SFNs $\Upsilon _{i}(i = 1, ..., n),$ a complex T-Spherical fuzzy ordered weighted power geometric (CT-SFOWPG) operator is a mapping $CT-SFOWPG:\Omega ^{n}\rightarrow \Omega$ defined by

where $\Omega$ represent the CT-SFNs, and $\sigma (1), ..., \sigma (n)$ are the permutations of $(1, ..., n)$ satisfying that $\Upsilon _{\sigma (i-1)}\geq \Upsilon _{\sigma (i)}$ for $i = 2, ..., n.$ Also, $\zeta _{i}$ is defined as

where $B_{i} = \sum_{s = 1}^{i}V_{\sigma (s)}, V_{\sigma (i)} = 1+\sum_{\underset{ s\neq i}{s = 1}}^{n}($Sup$(\Upsilon _{i}, \Upsilon _{s})), TV = \sum_{i = 1}^{n}V_{\sigma (i)}$, and the mapping $g:[0, 1]\rightarrow \lbrack 0, 1]$ is a basic unit-interval monotonic function satisfying these three properties: $g(0) = 0, g(1) = 1,$ if $x\leq y$ then $g(x)\leq g(y).$

Theorem 4.6. Let a family of CT-SFNs $\Upsilon _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., n).$ Then, the aggregated value calculated by using the CT-SFOWPG operator is also a CT-SFN and given as

where $\zeta _{i}$ is defined as in Eq $\left(4.29\right)$.

Also, the CT-SFOWPA operator meets the properties of idempotency, commutativity and boundedness when applied to the set of CT-SFNs.

5.   MCGDM approach using complex T-Spherical fuzzy power aggregation operators

Using the given operators and the CT-SFS information, an MCGDM algorithm is created in this section. Suppose that a DM problem with the $m$ alternatives $\wp _{1}, ..., \wp _{m},$ is evaluated with the $n$ criteria $C_{1}, ..., C_{n}$. Let us have $p$ experts $E = \left(E_{1}, ..., E_{p}\right)$ who evaluate the different alternatives with the different criteria. Every expert evaluates every alternative with the CT-SF information and assigns its rating values to CT-SFNs $\Upsilon _{ij}^{(k)} = \left\{ \left(\kappa _{\Upsilon _{ij}}^{\left(k\right) }, \Omega _{\kappa _{\Upsilon _{ij}}}^{\left(k\right) }\right), \left(\xi _{\Upsilon _{ij}}^{\left(k\right) }, \Omega _{\xi _{\Upsilon _{ij}}}^{\left(k\right) }\right), \left(\phi _{\Upsilon _{ij}}^{\left(k\right) }, \Omega _{\phi _{\Upsilon _{ij}}}^{\left(k\right) }\right) \right\},$ where $k = 1, ..., p; i = 1, ..., m;$ and $j = 1, ..., n,$ $0\leq \kappa _{\Upsilon _{ij}}^{\left(k\right) }+\xi _{\Upsilon _{ij}}^{\left(k\right) }+\phi _{\Upsilon _{ij}}^{\left(k\right) }\leq 1$ and $0\leq \Omega _{\kappa _{\Upsilon _{ij}}}^{\left(k\right) }+\Omega _{\xi _{\Upsilon _{ij}}}^{\left(k\right) }+\Omega _{\phi _{\Upsilon _{ij}}}^{\left(k\right) }\leq 1.$ Further, assume that the weights of the criteria are $\psi = \left(\psi _{1}, ..., \psi _{n}\right) ^{T},$ such that $\psi _{i} > 0$ and $\sum_{i = 1}^{n}\psi _{i} = 1.$ Then, to find the most desirable alternatives, the defined operators are used to define an MCGDM approach under the CT-SF information, and we have the below steps.

Step 1. Define a CT-SFNs matrix $R^{(k)} = \left(\Upsilon _{ij}^{(k)}\right) _{m\times n}$ with the values of every alternative assigned by expert $E^{(k)}(k = 1, ..., p)$ as follows.

Step 2a. Aggregate the expert's rating values $R^{(k)}(k = 1, ..., p)$ into the total collective CT-SF decision matrix $R = \left(\Xi _{ij}\right)$ where $\Xi _{ij} = \left\{ \left(\kappa _{\Upsilon _{ij}}, \Omega _{\kappa _{\Upsilon _{ij}}}\right), \left(\xi _{\Upsilon _{ij}}, \Omega _{\xi _{\Upsilon _{ij}}}\right), \left(\phi _{\Upsilon _{ij}}, \Omega _{\phi _{\Upsilon _{ij}}}\right) \right\}$ by using the CT-SFOWPA operator as follows.

Step 2b. Aggregate the expert's rating values $R^{(k)}(k = 1, ..., p)$ into the total collective CT-SF decision matrix $R = \left(\Xi _{ij}\right)$ where $\Xi _{ij} = \left\{ \left(\kappa _{\Upsilon _{ij}}, \Omega _{\kappa _{\Upsilon _{ij}}}\right), \left(\xi _{\Upsilon _{ij}}, \Omega _{\xi _{\Upsilon _{ij}}}\right), \left(\phi _{\Upsilon _{ij}}, \Omega _{\phi _{\Upsilon _{ij}}}\right) \right\}$ by using the CT-SFOWPG operator as follows:

where $\zeta _{ij}^{\left(1\right) }, ..., \zeta _{ij}^{\left(p\right) }$ are the weights obtained by using Eq $\left(4.14\right)$, and $\sigma$ is the permutation mapping from $\left(1, ..., p\right)$ to $\left(1, ..., p\right).$

Step 3a. Aggregate the collected values $R = \left(\Xi _{ij}\right)$ of alternatives $\wp _{i}(i = 1, ..., m)$ into the total assessment value $\Xi _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ using the power averaging operator as defined in Eq $\left(4.2\right).$

For example, if we aggregate every value of the alternative using the CT-SFWPA, we will obtain the evaluation value of the alternative $\Xi _{i}(i = 1, ..., m)$ as:

Step 3b. Aggregate the collected values $R = \left(\Xi _{ij}\right)$ of alternatives $\wp _{i}(i = 1, ..., m)$ into the total assessment value $\Xi _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\}$ using the power geometric operator as defined in Eq $\left(4.8\right).$

For example, if we aggregate every value of the alternative using the CT-SFWPA, we will obtain the evaluation value of the alternative $\Xi _{i}(i = 1, ..., m)$ as;

Step 4. Find the score value from the total aggregated values of

$\Xi _{i} = \left\{ \left(\kappa _{\Upsilon _{i}}, \Omega _{\kappa _{\Upsilon _{i}}}\right), \left(\xi _{\Upsilon _{i}}, \Omega _{\xi _{\Upsilon _{i}}}\right), \left(\phi _{\Upsilon _{i}}, \Omega _{\phi _{\Upsilon _{i}}}\right) \right\} (i = 1, ..., m)$ by using the following equation:

Step 5. Allow ranking of the alternatives $\wp _{i}(i = 1, ..., m)$ and select the appropriate one (s).

In Figure 1, we show the proposed algorithm steps.

6.   Illustrative example

Emergency management is a term that refers to dealing with the possibility of major incidents and disasters, as well as involving government and other public bodies in the emergency response, prevention, disposal and recovery process, as well as developing an effective response plan to take a variety of necessary steps. The use of research, technology, planning, and management implies assuring the security of public safety, health, and property-related emergency operations, as well as facilitating society's harmonious and long-term growth. Natural disasters have been causing enormous injury and damage to human life and the global economy in recent years. The Emergency Management Center (EMC) will construct a range of emergency choices based on the types of incidents and ask specialists from other disciplines to assess alternative emergency plans in order to effectively limit the damages sustained by significant incidents and catastrophes. Alternative emergency assessment is a critical component of emergency management. The traditional DM problem is in the core, and it has attracted a lot of attention from a lot of people. As a result, we will apply the defined technique to the stated problem, to determine the best emergency solution for the EMC. Four better alternatives would be investigated further after a series of screenings. There are five alternatives set as follows; $\wp = \left\{ \wp _{1}, ..., \wp _{5}\right\}$. Expert discussions take into account the following four requirements for proper modeling of the characteristics of alternatives: $(C_{1})$ preparation ability, $(C_{2})$ rescuing ability, $C_{3}$ Restoration ability, $(C_{4})$ Reaction capacity. $\psi = (0.20, 0.25, 0.25, 0.30)^{T}$ are the weights of the criteria, such that $\sum_{i = 1}^{4}\Theta _{i} = 1.$ The information given by experts $E_{1}, E_{2}$ and $E_{3}$ is shown in Tables 1–3 with the weight vector $\varpi = (0.3, 0.4, 0.3)$.

Step 1. The CT-SFN information is used by the three experts to evaluate options, and their individual rating values are provided in Tables 1–3 in the decision matrix.

Step 2a. Different assessments of the experts $\Upsilon _{ij}^{(k)}(k = 1, 2, 3)$ are aggregated into a collective one $\Xi _{ij}(i = 1, ..., 5;j = 1, ..., 4)$, taking the function $g(x) = x^{2}$ and using the CT-SFOWPA operator. The obtained values are given in Table 4.

Step 2b. We use the CT-SFOWPA operator to aggregate the different assessments of the experts $\Upsilon _{ij}^{(k)}(k = 1, 2, 3),$ into a collective one $\Xi _{ij}(i = 1, ..., 5;j = 1, ..., 4)$, taking $g(x) = x^{2}$. The values obtained using this operator are described in Table 5.

Step 3a. Now, we use the CT-SFWPA operator to aggregate the different values $\Xi _{ij}(i = 1, ..., 5;j = 1, ..., 4),$ obtained from Table $4,$ with the corresponding weights are $\psi = (0.20, 0.25, 0.25, 0.30)^{T}$. The total values of each alternative $\wp _{i}(i = 1, ..., 5)$ are,

Step 3b. We use the CT-SFWPG operator to aggregate the different values $\Xi _{ij}(i = 1, ..., 5;j = 1, ..., 4),$ obtained from Table $5,$ with the corresponding weights $\psi = (0.20, 0.25, 0.25, 0.30)^{T}$. The total values of each alternative $\wp _{i}(i = 1, ..., 5)$ are,

Step 4. The scores of the alternatives $\wp _{i}(i = 1, ..., 5)$ are computed on the basis of the overall assessment values of $\Xi _{i}(i = 1, ..., 5;j = 1, ..., 4)$ as follows:

Meanwhile, the scores of the alternatives $\wp _{i}(i = 1, ..., 5)$ are based on the aggregated values of Step 3b.

Step 5. The ranking of all feasible alternatives $\wp _{i}(i = 1, ..., 5)$ is shown as follows.

Thus, the best alternative is $\wp _{1}$, that is, $\wp _{1}$ is the most optimal choice.

In Figure 2, we show graphically the ranking order of the alternatives.

Now, we take a transferred alternative Table 6, as shown as:

6.1.   Validity test

When various MCGDM techniques are used to solve the same DM problem, they produce different outcomes (rankings), which contributes to ambiguous results. In order to investigate the reliability and validity of the MCGDM techniques, Wang & Triantaphyllou ^[50] offered the following test conditions.

Test criteria on 1. The MCGDM strategy is effective if the better option remains the same and the nonoptimal alternative is converted to a worse alternative while the relative values of the decision criteria remains constant.

Test criteria on 2. A successful MCGDM strategy should follow transitive properties.

Test criteria on 3. When dividing the MCGDM problem into sub-problems and applying the proposed MCGDM approach to these sub-problems for ranking alternatives, the MCGDM approach is effective. The ranking of the alternatives is identical to the ranking of the problem.

The above criteria are used to determine the correctness of the suggested solution.

6.2.   Validity check with criteria on $1$

In order to evaluate the validity of the developed method using criteria on $1$, the non-optimal alternative $\wp _{5}$ is substituted by the worst alternative $\wp _{5}^{/}$ in the original decision matrix for each expert, and the rating values are reported in Table $4.$

We now compute the scores of the alternatives using the CT-SFOWPA operator in Step $2$ and the CT-SFWPA operator in Step $3$ for these modified results, which are; $Sc^{\ast }(\wp _{1}) = 0.822,$ $Sc^{\ast }(\wp _{2}) = 0.641,$ $Sc^{\ast }(\wp _{3}) = 0.683,$ $Sc^{\ast }(\wp _{4}) = 0.435,$ $Sc^{\ast }(\wp _{5}) = 0.263$. As a result, the final ranking of the alternatives indicates that $\wp _{1}$ remains the best choice and that the method developed fulfills test criteria on 1.

6.3.   Validity check with criteria $2$ and $3$

We divided the original decision making problem into sub DM problems, which contained the options $\left(\wp _{1}, \wp _{2}, \wp _{3}, \wp _{4}\right),$ $\left(\wp _{2}, \wp _{3}, \wp _{4}, \wp _{5}\right)$ and $\left(\wp _{1}, \wp _{3}, \wp _{4}, \wp _{5}\right)$, in order to test the defined MCGDM approach with criteria $2$ and $3.$ When we apply the suggested MCGDM technique to these subproblems, we get the ratings as $\wp _{1} > \wp _{3} > \wp _{2} > \wp _{4},$ $\wp _{3} > \wp _{2} > \wp _{4} > \wp _{5}$ and $\wp _{1} > \wp _{3} > \wp _{4} > \wp _{5}$. We get the final ranking order as $\wp _{1} > \wp _{3} > \wp _{2} > \wp _{4} > \wp _{5}$ by introducing the ranking of alternatives to these smaller problems. This is a non-decomposed problem that discloses a transitive property. As a result, the defined MCGDM approach is consistent with criteria 2 and 3.

6.4.   Comparative study

In order to demonstrate the benefits of the defined aggregation operators, we compare the proposed approach to previous methods in this section. First the priorities considered by the experts are translated into SFNs, taking the phase term relevant to the CT-SFN as zero. On the basis of this information, we tested the current methods, as follows.

(1) If we apply the T-SFS aggregation operators, proposed by Garg et al. ^[12], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.672,$ $Sc^{\ast }(\wp _{2}) = 0.529,$ $Sc^{\ast }(\wp _{3}) = 0.632,$ $Sc^{\ast }(\wp _{4}) = 0.347,$ $Sc^{\ast }(\wp _{5}) = 0.274.$ Therefore, their corresponding ranking order is $\wp _{1} > \wp _{3} > \wp _{2} > \wp _{4} > \wp _{5}$, which means that $\wp _{1}$ is the appropriate option.

(2) If we apply the T-SFS aggregation operators, proposed by Liu et al. ^[27], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.794, Sc^{\ast }(\wp _{2}) = 0.653,$ $Sc^{\ast }(\wp _{3}) = 0.812,$ $Sc^{\ast }(\wp _{4}) = 0.645,$ $Sc^{\ast }(\wp _{5}) = 0.573.$ Therefore, their corresponding ranking order is $\wp _{3} > \wp _{1} > \wp _{2} > \wp _{4} > \wp _{5},$ which means that $\wp _{3}$ is the appropriate option.

(3) If we apply T-SFS power Maclaurin symmetric mean operators, proposed by Munir et al. ^[31], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.434,$ $Sc^{\ast }(\wp _{2}) = 0.424,$ $Sc^{\ast }(\wp _{3}) = 0.326,$ $Sc^{\ast }(\wp _{4}) = 0.223,$ $Sc^{\ast }(\wp _{5}) = 0.238.$ Therefore, their corresponding ranking order is $\wp _{1} > \wp _{2} > \wp _{3} > \wp _{5} > \wp _{4}$, which means that $\wp _{1}$ is the appropriate option.

(4) If we apply T-SFS operators, proposed by Ullah et al. ^[49], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.742,$ $Sc^{\ast }(\wp _{2}) = 0.463,$ $Sc^{\ast }(\wp _{3}) = 0.572,$ $Sc^{\ast }(\wp _{4}) = 0.651,$ $Sc^{\ast }(\wp _{5}) = 0.552.$ Therefore, their corresponding ranking order is $\wp _{1} > \wp _{4} > \wp _{3} > \wp _{5} > \wp _{2}$, which means that $\wp _{1}$ is the appropriate option.

(5) If we apply CT-SFS aggregation operators, proposed by Ali et al. ^[2], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.532,$ $Sc^{\ast }(\wp _{2}) = 0.413,$ $Sc^{\ast }(\wp _{3}) = 0.628,$ $Sc^{\ast }(\wp _{4}) = 0.611,$ $Sc^{\ast }(\wp _{5}) = 0.537.$ Therefore, their corresponding ranking order is $\wp _{3} > \wp _{4} > \wp _{5} > \wp _{1} > \wp _{2}$, which means that $\wp _{3}$ is the appropriate option.

(6) If we apply CT-SF Dombi aggregation operators, proposed by Karaaslan and Dawood ^[22], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.574,$ $Sc^{\ast }(\wp _{2}) = 0.462,$ $Sc^{\ast }(\wp _{3}) = 0.382,$ $Sc^{\ast }(\wp _{4}) = 0.283,$ $Sc^{\ast }(\wp _{5}) = 0.299.$ Therefore, their corresponding ranking order is $\wp _{1} > \wp _{2} > \wp _{3} > \wp _{5} > \wp _{4}$, which means that $\wp _{1}$ is the appropriate option.

(7) If we apply CT-SF 2-tuple linguistic Muirhead mean AOs, proposed by Liu et al. ^[29], on the stated problem, then the alternatives $\wp _{i}(i = 1, ..., 5)$ score values are $Sc^{\ast }(\wp _{1}) = 0.630,$ $Sc^{\ast }(\wp _{2}) = 0.625,$ $Sc^{\ast }(\wp _{3}) = 0.651,$ $Sc^{\ast }(\wp _{4}) = 0.563, Sc^{\ast }(\wp _{5}) = 0.517.$ Therefore, their corresponding ranking order is $\wp _{3} > \wp _{1} > \wp _{2} > \wp _{4} > \wp _{5},$ which means that $\wp _{3}$ is the appropriate option.

In Figure 3, we show graphically the ranking of alternatives using different approaches.

Figure 3 shows the ranking order of alternatives. The example of the operators studied in this paper shows that the best alternative is $\wp _{1}$ whereas the best alternatives are $\wp _{1}$ and $\wp _{3}$ when we apply the existing operators. It's worth noting that the existing operators can only work with one dimension at a time. As a result, the operators provided here are more generic, as they may capture two dimensions and convey more full data. Hence, the CT-SFS model is an extension of current theories that simultaneously deal with imprecision and periodicity.

7.   Conclusions

The CT-SFS theory combines the features of both the CFS and T-SFS theories. CT-SFSs have three degrees (MD, NuMD and NMD) in polar coordinates, in such a way that the range of the degrees is extended from $[0, 1]$ to the unit disk in a complex plane. The concept that prioritized AOs for CT-SFNs would enable us to tackle a new class of decision-making problems motivated us to write this research paper.

The purpose of this article is to provide information in the form of CT-SFSs and using PA operators involved in the DM process. First, we've established some fundamental operational laws as well as a new score function for ranking the CT-SFNs. We offer a variety of power averaging and geometric AOs, such as CT-SFPA, CT-SFWPA, CT-SFOWPA, CT-SFPG, CT-SFWPG and CT-SFOWPG; and we investigate the basic properties of these operators, keeping these points in mind with the CT-SFS details. We have provided the mathematical description of the MCGDM problem, and then an algorithm we initiated for the MCGDM problem. To demonstrate the validity of the approach, a practical example we used to describe the stated method, and the results we compared to some of the other current methods. A validity test was also used to assess the effectiveness and validity of our proposed technique. Using the proposed operators, we conducted a comparison with existing operators in the literature to demonstrate the superiority of the proposed operators. It should also be mentioned that under the CT-SFSs information, existing operators can be considered a special case of the developed method.

Other aggregation operators such as Hamacher, Bonferroni mean, similarity measures, Dombi, Maclaurin's symmetric mean, Banzhaf-Choquet copula, Heronian mean and fractional orthotriple fuzzy operators will be studied in the future for this structure.

Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by grant (Code: 22UQU4310396DSR15).

Conflicts of interest

The authors declares that they have no conflict of interest.