A Multi-MOORA decision making method based on Muirhead mean operators and complex spherical fuzzy uncertain linguistic setting

Abstract

The Muirhead mean (MM) operators offer a flexible arrangement with its modifiable factors because of Muirhead’s general structure. On the other hand, MM aggregation operators perform a significant role in conveying the magnitude level of options and characteristics. In this manuscript, the complex spherical fuzzy uncertain linguistic set (CSFULS), covering the grade of truth, abstinence, falsity, and their uncertain linguistic terms is proposed to accomplish with awkward and intricate data in actual life dilemmas. Furthermore, by using the MM aggregation operators with the CSFULS, the complex spherical fuzzy uncertain linguistic MM (CSFULMM), complex spherical fuzzy uncertain linguistic weighted MM (CSFULWMM), complex spherical fuzzy uncertain linguistic dual MM (CSFULDMM), complex spherical fuzzy uncertain linguistic dual weighted MM (CSFULDWMM) operators, and their important results are also elaborated with the help of some remarkable cases. Additionally, multi-attribute decision-making (MADM) based on the Multi-MOORA (Multi-Objective Optimization Based on a Ratio Analysis plus full multiplicative form), and proposed operators are developed. To determine the rationality and reliability of the elaborated approach, some numerical examples are illustrated. Finally, the supremacy and comparative analysis of the elaborated approaches with the help of graphical expressions are also developed.

Keywords

Complex spherical fuzzy uncertain linguistic sets Muirhead mean Aggregation operators Dual Muirhead mean Aggregation operators Multi-attribute decision-making methods.

1 Introduction

MADM is the critical sense of the decision-making troubles whose aim is to identify the extremely profitable choice(s) from the set of possible ones. In actual decision-making, the person needs to calculate the offered choices by various groups like lone, time, etc., of assessment principles. Nevertheless, in several unstable situations, it is typically difficult for the boss to deliver their options as a crisp number. To organize it, Zadeh [1] referred to the principle of fuzzy sets (FSs), to manage the worries in the knowledge, which operates the TG (“Truth grade”) to clarify the understanding. Subsequent on, Atanassov [2] extended FSs to the intuitionistic FSs (IFSs) by adding the FG (“Falsity grade”) along with TG to signify the data. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [3 –6]. But the variety of IFS data is actual slight with the restraint form that TG $T_{C_{TG}}$ and FG $F_{C_{FG}}$ should not be superior to one. To accomplish the decision-making troubles additional specifically, Yager [7] stretched the IFS to Pythagorean FSs (PFSs) by spreading the term of $T_{C_{TG}} + F_{C_{FG}} ⩽ 1$ to $T_{C_{TG}}^{2} + F_{C_{FG}}^{2} ⩽ 1$ . IFS is a certain case of PFS and consequently, PFS can contract the additional expertise with the fuzzy natural environment. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [8 –11]. The geometrical expressions of the IFSs and PFSs are discussed in the form of Fig. 1.

Fig. 1

Graphical expressions of the IFSs and PFSs.

In some genuine life troubles, the principle of PFS has been failed for instance, when an individual faces such sorts of troubles which consist of four possibilities such that: yes, abstain, no, and refusal. Then the principle of PFS is not able to manage with it. To investigated such sorts of troubles, the theory of picture FS (PiFS) was utilized by Cuong [12] which covers the TG $T_{C_{TG}}$ , abstinence grade $A_{C_{AG}}$ (AG), and FG $F_{C_{FG}}$ with a rule that is $T_{C_{TG}} + A_{C_{AG}} + F_{C_{FG}} ⩽ 1$ . IFS is a certain case of PiFS and consequently, PiFS can contract the additional expertise with the fuzzy natural environment. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [13 –15]. But the variety of PiFS data is actual slight with the restraint form that TG $T_{C_{TG}}$ , AG $A_{C_{AG}}$ , and FG $F_{C_{FG}}$ should not be superior to one. To accomplish the decision-making troubles additional specifically, Mahmood et al. [16] stretched the PFS to spherical FSs (SFSs) by spreading the term of $T_{C_{TG}} + A_{C_{AG}} + F_{C_{FG}} ⩽ 1$ to $T_{C_{TG}}^{2} + A_{C_{AG}}^{2} + F_{C_{FG}}^{2} ⩽ 1$ . PiFS is a certain case of SFS and consequently, SFS can contract the additional expertise with the fuzzy natural environment. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [17, 18]. The geometrical expressions of the PiFSs and SFSs are discussed in the form of Fig. 2.

Fig. 2

Geometrical expressions of the picture and spherical fuzzy sets.

Ramot et al. [19] referred to the principle of complex FSs (CFSs), to manage the worries in the knowledge, which operates the TG (“Truth grade”) to clarify the understanding in the shape of complex number belonging to unit disc. The geometrical expression of the unit disc is discussed in the form of Fig. 3. Subsequent on, Alkouri and Salleh [20] extended CFSs to the complex IFSs (CIFSs) by adding the FG (“Falsity grade”) along with TG to signify the data. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [21 –25]. But the variety of CIFS data is actual slight with the restraint form that TG $T_{C_{RP}} e^{i 2 π (T_{C_{IP}})}$ and FG $F_{C_{RP}} e^{i 2 π (F_{C_{IP}})}$ should not be superior to one. To accomplish the decision-making troubles additional specifically, Ullah et al. [26] stretched the CIFS to complex PFSs (CPFSs) by spreading the term of $T_{C_{RP}} + F_{C_{RP}} ⩽ 1, T_{C_{IP}} + F_{C_{IP}} ⩽ 1$ to $T_{C_{RP}}^{2} + F_{C_{RP}}^{2} ⩽ 1, T_{C_{IP}}^{2} + F_{C_{IP}}^{2} ⩽ 1$ . CIFS is a certain case of CPFS and consequently, CPFS can contract the additional expertise with the fuzzy natural environment. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [27].

Fig. 3

Graphical expressions of the unit disc in complex plane.

In some genuine life troubles, the principle of CPFS has been failed for instance, when an individual faces such sorts of troubles which consist of four possibilities such that: yes, abstain, no, and refusal. Then the principle of CPFS is not able to manage with it. To investigated such sorts of troubles, the theory of complex PiFS (CPiFS) was utilized by Akram et al. [28] which covers the TG $T_{C_{RP}} e^{i 2 π (T_{C_{IP}})}$ , AG $A_{C_{RP}} e^{i 2 π (A_{C_{IP}})}$ , and FG $F_{C_{RP}} e^{i 2 π (F_{C_{IP}})}$ with rules that are $T_{C_{RP}} + A_{C_{RP}} + F_{C_{RP}} ⩽ 1, T_{C_{IP}} + A_{C_{IP}} + F_{C_{IP}} ⩽ 1$ . CIFS is a certain case of CPiFS and consequently, CPiFS can contract the additional expertise with the fuzzy natural environment.

But the variety of CPiFS data is actual slight with the restraint form that TG $T_{C_{RP}} e^{i 2 π (T_{C_{IP}})}$ , AG $A_{C_{RP}} e^{i 2 π (A_{C_{IP}})}$ , and FG $F_{C_{RP}} e^{i 2 π (F_{C_{IP}})}$ should not be superior to one. To accomplish the decision-making troubles additional specifically, Ali et al. [29] stretched the CPFS to complex SFSs (CSFSs) by spreading the term of $T_{C_{RP}} + A_{C_{RP}} + F_{C_{RP}} ⩽ 1, T_{C_{IP}} + A_{C_{IP}} + F_{C_{IP}} ⩽ 1$ to $T_{C_{RP}}^{2} + A_{C_{RP}}^{2} + F_{C_{RP}}^{2} ⩽ 1, T_{C_{IP}}^{2} + A_{C_{IP}}^{2} + F_{C_{IP}}^{2} ⩽ 1$ . CPiFS is a certain case of CSFS and consequently, CSFS can contract the additional expertise with the fuzzy natural environment. Later their existence, distinct investigators started to analyze it broadly and sincerely to resolve MADM troubles [30].

Nevertheless, in several actual dilemmas, it is not uncomplicated for decision-makers to provide their thoughts in quantifiable interpretations. For illustration, when a consultant assesses a claimant’s concentration of specific knowledge, he may contemplate additionally suitable or beyond comfortable to employing linguistic phrases, such as “very good”, “good”, or “medium”, to express his or her estimation. To manage with such sort out of worries, Zadeh [31] probed the principle of linguistic variable (LV) to illustrate the inclinations of decision-makers. The previous linguistic models had an important weakness in the loss of information during the computational procedure. To avoid this loss, the principle of uncertain linguistic sets was proposed by Xu [32]. Xu [33] elaborated the ordered weighted aggregation operators for uncertain linguistic sets. Zavadskas et al. [34] explored the interval-valued intuitionistic fuzzy MULTIMOORA method and its applications in decision making. Li [35] elaborated the MULTIMOORA method based on hesitant fuzzy sets and their application in decision-making. On other hand certain shortcomings in CFSs and CIFSs analogous to the TG and FG. Mutual limitations indicate that there exists several reliance among the evaluations. When a person provides such sorts of information which contain the TG, AG, FG, and ULT with a rule that is the sum of the squares of the TG, AG, and FG is exceeded from the unit interval, then the prevailing ideas such as FSs, CFSs, IFSs, CIFSs, CPiFSs, SFSs, and their special cases are not able to manage with it. To eliminate these confines, the principle of CSFULSs is a useful technique to cope with awkward and inconsistent information in guanine life troubles, which covers the $T_{C_{CLU}} (X_{EL}) = T_{C_{RP}} (X_{EL}) e^{i 2 π (T_{C_{IP}} (X_{EL}))}, A_{C_{CLU}} (X_{EL}) = A_{C_{RP}}$ $(X_{EL}) e^{i 2 π (A_{C_{IP}} (X_{EL}))}$ , $F_{C_{CLU}} (X_{EL}) = F_{C_{RP}}$ $(X_{EL}) e^{i 2 π (F_{C_{IP}} (X_{EL}))}$ and $[Ξ_{α}, Ξ_{β}], Ξ_{α}, Ξ_{β} \in \bar{\bar{Ξ}}$ are expressed the TG, AG, FG, and ULT with some rules that are $0 ⩽ T_{ℂ_{RP}}^{2} (X_{EL}) + A_{ℂ_{RP}}^{2} (X_{EL}) + F_{ℂ_{RP}}^{2} (X_{EL}) ⩽ 1$ and $[Ξ_{α}, Ξ_{β}], Ξ_{α}, Ξ_{β} \in \bar{\bar{Ξ}}$ . Moreover, Muirhead means (MM) is a well-known aggregation operator which can consider interrelationships among any number of arguments assigned by a variable vector. Besides, it is a universal operator since it can contain other general operators by assigning some special parameter values. However, the MM can only process the crisp numbers. Inspired by the MM’s advantages, this paper aims to extend MM to process the complex spherical fuzzy uncertain linguistic numbers (CSFULNs) and then to solve the multi-attribute decision making (MADM) problems. The advantages of the proposed approaches are discussed below.

If we choose the value of imaginary is zero in the proposed CSFULSs, then the proposed work is converted for spherical fuzzy uncertain linguistic sets.

If we choose the value of abstinence is zero in the proposed CSFULSs, then the proposed work is converted for complex pythagorean fuzzy uncertain linguistic sets.

If we choose the value of Ξ_α = Ξ_β, then the proposed CSFULSs is converted for complex spherical fuzzy linguistic sets.

If we choose the value of parameter 2 = 1, then the proposed CSFULSs is converted for complex picture fuzzy uncertain linguistic sets.

If we choose the value of Ξ_α = Ξ_β = 0, in the proposed work, then the proposed CSFULSs is converted for complex spherical fuzzy sets.

If we choose the value of Ξ_α = Ξ_β = 0, and the value of parameter 2 = 1, then the proposed CSFULSs is converted for complex picture fuzzy sets.

By using the advantages of the elaborated ideas, the main proposes of this manuscript are summarized as follows:

To elaborate on the CSFULSs and developed their operational laws.

To explore the CSFULMM, CSFULWMM, CSFULDMM, CSFULDWMM operators and their properties.

To utilize the elaborated aggregation operators based on CSFULSs and discuss their special cases.

To develop a MADM technique by using the elaborated operators with CSFULSs to ascertain the consistency and authenticity of the elaborated approaches.

To find the supremacy and consistency of the elaborated operators with the help of comparative analysis and their graphical expressions.

The goal of this manuscript is following as In section 2, we briefly assessment some prevailing concepts like CSFSs, ULSs, and their operational laws. The idea of the MM operator is also revised. In section 3, we elaborate on the principle of CSFULSs. CSFULSs cover the grade of truth, falsity, and their uncertain linguistic terms to accomplish with awkward and intricate data in actual life dilemmas. Some operational laws by using the elaborated CSFULSs are also settled. In section 4, by using the MM operators based on CSFULSs to elaborate the CSFULMM, CSFULWMM, CSFULDMM, CSFULDWMM operators, and the important results are also elaborated with the help of some remarkable cases. In section 5, by using the elaborated operators based on CSFULMM, a MADM dilemma is solved. To determine the rationality and reliability of the elaborated operators, some numerical examples are illustrated. Finally, the supremacy and comparative analysis of the elaborated approaches with the help of graphical expressions are also developed. In section 6, we discussed the conclusion of this manuscript.

2 Preliminaries

In this study, we briefly assessment some prevailing concepts like CSFSs, ULSs, and their operational laws. The idea of the MM operator is also revised. The symbol $X^{UNI}$ expressed the fixed set with the grade of truth $T$ , falsity $F$ , reference parameters $A$ and $B$ . The geometrical expressions of the elaborated approaches are discussed in the form of Fig. 4.

Fig. 4

Geometrical expressions of the proposed works in this manuscript.

Definition 1 [29] A CSFS $ℂ_{CP}$ is initiated by:

$\begin{matrix} ℂ_{CP} = {(X_{EL}, T_{ℂ_{CP}} (X_{EL}), A_{ℂ_{CP}} (X_{EL}), \\ F_{ℂ_{CP}} (X_{EL})) : X_{EL} \in X^{UNI}} \end{matrix}$ (1) where $T_{C_{CP}} (X_{EL}) = T_{C_{RP}} (X_{EL}) e^{i 2 π (T_{C_{IP}} (X_{EL}))}, A_{C_{CP}} (X_{EL}) = A_{C_{RP}} (X_{EL}) e^{i 2 π (A_{C_{IP}} (X_{EL}))}$ and $F_{C_{CP}} (X_{EL}) = F_{C_{RP}} (X_{EL}) e^{i 2 π (F_{C_{IP}} (X_{EL}))}$ with the certain rule that is $0 ⩽ T_{ℂ_{RP}}^{2} (X_{EL}) + A_{ℂ_{RP}}^{2} (X_{EL}) + F_{ℂ_{RP}}^{2} (X_{EL}) ⩽ 1$ and $0 ⩽ T_{ℂ_{IP}}^{2} (X_{EL}) + A_{ℂ_{IP}}^{2} (X_{EL}) + F_{ℂ_{IP}}^{2} (X_{EL}) ⩽ 1$ . The complex spherical fuzzy

numbers (CSFNs) are dented by $ℂ_{CP - i} = (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})}), i = 1, 2, \dots, n_{NE}$ . By using any two CSFNs $ℂ_{CP - 1} = (T_{C_{RP - 1}} e^{i 2 π (T_{C_{IP - 1}})}, A_{C_{RP - 1}} e^{i 2 π (A_{C_{IP - 1}})}, F_{C_{RP - 1}} e^{i 2 π (F_{C_{IP - 1}})})$ and $ℂ_{CP - 2} = (T_{C_{RP - 2}} e^{i 2 π (T_{C_{IP - 2}})}, A_{C_{RP - 2}} e^{i 2 π (A_{C_{IP - 2}})}, F_{C_{RP - 2}} e^{i 2 π (F_{C_{IP - 2}})})$ , with constant δ_SC, then

$ℂ_{C P - 1} \oplus ℂ_{C P - 2} = (\begin{matrix} {(\begin{matrix} T_{ℂ R P - 1}^{2} + T_{ℂ R P - 2}^{2} \\ - T_{ℂ R P - 1}^{2} T_{R P - 2}^{2} \end{matrix})}^{\frac{1}{2}} e^{i 2 π}^{{(\begin{matrix} T_{ℂ I P - 1}^{2} + T_{ℂ I P - 2}^{2} \\ - T_{ℂ I P - 1}^{2} T_{ℂ I P - 2}^{2} \end{matrix})}^{\frac{1}{2}}} \\ A_{ℂ R P - 1} A_{ℂ R P - 2} e i 2 π (A_{ℂ I P - 1} A_{ℂ I P - 2}), ℱ_{ℂ_{R P - 1}} ℱ_{ℂ_{R P - 2}} e^{i 2 π (ℱ_{ℂ_{I P - 1}} ℱ_{ℂ_{I P - 2}})} \end{matrix})$ (2) $ℂ_{CP - 1} \otimes ℂ_{CP - 2} = (\begin{matrix} (T_{C_{RP - 1}} T_{C_{RP - 2}}) e^{i 2 π} (T_{C_{IP - 1}} + T_{C_{IP - 2}}), (\begin{matrix} A_{ℂ_{RP - 1}}^{2} + A_{ℂ_{RP - 2}}^{2} \\ - A_{ℂ_{RP - 1}}^{2} A_{ℂ_{RP - 2}}^{2} \end{matrix}) \frac{1}{2} e^{i 2 π {(A_{ℂ_{IP - 1}}^{2} + A_{ℂ_{IP - 2}}^{2} - A_{ℂ_{IP - 1}}^{2} A_{ℂ_{IP - 2}}^{2})}^{\frac{1}{2}},} \\ {(\begin{matrix} F_{ℂ_{RP - 1}}^{2} + F_{ℂ_{RP - 2}}^{2} \\ - F_{ℂ_{RP - 1}}^{2} F_{ℂ_{RP - 2}}^{2} \end{matrix})}^{\frac{1}{2}} e^{i 2 π {(F_{ℂ IP - 1}^{2} + F_{ℂ_{IP - 2}}^{2} - F_{ℂ_{IP - 1}}^{2} F_{ℂ_{IP - 2}}^{2})}^{\frac{1}{2}}} \end{matrix})$ (3) $δ_{SC} ℂ_{CP - 1} = (\begin{matrix} {(1 - {(1 - T_{ℂ_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - T_{ℂ_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}}, \\ A_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (A_{C_{IP - 1}}^{δ_{SC}})}, F_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (F_{C_{IP - 1}}^{δ_{SC}})} \end{matrix})$ (4) $ℂ_{CP - 1}^{δ_{SC}} = (\begin{matrix} T_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (T_{C_{IP - 1}}^{δ_{SC}})}, {(1 - {(1 - A_{C_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - A_{C_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}}, \\ {(1 - {(1 - F_{ℂ_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - F_{ℂ_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}} \end{matrix})$ (5)

By using any CSFN $ℂ_{CP - 1} = (T_{C_{RP - 1}} e^{i 2 π (T_{C_{IP - 1}})}, A_{C_{RP - 1}} e^{i 2 π (A_{C_{IP - 1}})}, F_{C_{RP - 1}} e^{i 2 π (F_{C_{IP - 1}})})$ , we initiated the score value and accuracy value as the form: $S_{SV} (ℂ_{CP - 1}) = \frac{1}{2} ((T_{C_{RP - 1}}^{2} - A_{C_{RP - 1}}^{2} - F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} - A_{C_{IP - 1}}^{2} - F_{C_{IP - 1}}^{2})), S_{SV} (ℂ_{CP - 1}) \in [- 1, 1]$ (6) $ℍ_{AV} (ℂ_{CP - 1}) = \frac{1}{2} ((T_{C_{RP - 1}}^{2} + A_{C_{RP - 1}}^{2} + F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} + A_{C_{IP - 1}}^{2} + F_{C_{IP - 1}}^{2})), ℍ_{SV} (ℂ_{CP - 1}) \in [0, 1]$ (7)

To determine the relations among any numbers of arrangements, we determine some rules such as:

When $S_{SV} (ℂ_{CP - 1}) > S_{SV} (ℂ_{CP - 2}) \Rightarrow ℂ_{CP - 1} > ℂ_{CP - 2}$ ;

When $S_{SV} (ℂ_{CP - 1}) < S_{SV} (ℂ_{CP - 2}) \Rightarrow ℂ_{CP - 1} < ℂ_{CP - 2}$ ;

When $S_{SV} (ℂ_{CP - 1}) = S_{SV} (ℂ_{CP - 2}) \Rightarrow$

When $ℍ_{AV} (ℂ_{CP - 1}) > ℍ_{AV} (ℂ_{CP - 2}) \Rightarrow ℂ_{CP - 1} > ℂ_{CP - 2}$ ;

When $ℍ_{AV} (ℂ_{CP - 1}) < ℍ_{AV} (ℂ_{CP - 2}) \Rightarrow ℂ_{CP - 1} < ℂ_{CP - 2}$ ;

When $ℍ_{AV} (ℂ_{CP - 1}) = ℍ_{AV} (ℂ_{CP - 2}) \Rightarrow ℂ_{CP - 1} = ℂ_{CP - 2}$ .

Definition 2 [32] Let $\hat{Ξ} = [Ξ_{α}, Ξ_{β}], Ξ_{α}, Ξ_{β} \in \bar{\bar{Ξ}}$ , where $\bar{\bar{Ξ}} = {Ξ_{α} : α \in R^{+}}$ is called ULS. By using any two ULSs ${\hat{Ξ}}_{1} = [Ξ_{α_{1}}, Ξ_{β_{1}}]$ and ${\hat{Ξ}}_{2} = [Ξ_{α_{2}}, Ξ_{β_{2}}]$ , then ${\hat{Ξ}}_{1} \oplus {\hat{Ξ}}_{2} = [Ξ_{α_{1}}, Ξ_{β_{1}}] \oplus [Ξ_{α_{2}}, Ξ_{β_{2}}] = [Ξ_{α_{1} + α_{2}}, Ξ_{β_{1} + β_{2}}]$ (8) ${\hat{Ξ}}_{1} \otimes {\hat{Ξ}}_{2} = [Ξ_{α_{1}}, Ξ_{β_{1}}] \otimes [Ξ_{α_{2}}, Ξ_{β_{2}}] = [Ξ_{α_{1} * α_{2}}, Ξ_{β_{1} * β_{2}}]$ (9) $δ_{SC} {\hat{Ξ}}_{1} = δ_{SC} [Ξ_{α_{1}}, Ξ_{β_{1}}] = [Ξ_{δ_{SC} * α_{1}}, Ξ_{δ_{SC} * β_{1}}]$ (10) ${\hat{Ξ}}_{1}^{δ_{SC}} = {[Ξ_{α_{1}}, Ξ_{β_{1}}]}^{δ_{SC}} = [Ξ_{α_{1}^{δ_{SC}}}, Ξ_{β_{1}^{δ_{SC}}}]$ (11)

Definition 3 [36] A MM operator is initiated by: ${MM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = {(\frac{1}{n_{NE}!} (\sum_{σ \in Ξ_{NE}} (\prod_{i = 1}^{n_{NE}} ℂ_{CF - σ (i)}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}$ (12) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector.

3 Complex Spherical fuzzy uncertain linguistic sets

In certain genuine life troubles, we go over numerous circumstances where we need to measure the vulnerability existing in the information to settle on ideal choices. Data measures are significant devices for taking care of unsure data present in our day-to-day life issues. Different sorts of operators, methods, and inclusion, process the inconsistent information and facilitate us to reach some conclusion. Recently, these measures have gained much attention from many authors due to their wide applications in various fields, such as pattern recognition, medical diagnosis, clustering analysis, and image segment. All the prevailing approaches of decision-making, based on operators, in IFS, PyFS, and PFSs theories, deal with only the grades of truth, abstinence, and falsity, which are real numbers. In CSFS theory, truth, abstinence, and falsity grades are complex-valued and are represented in polar coordinates. The amplitude term corresponding to truth and falsity degrees gives the extent of membership, abstinence, and non-membership of an object in a CSFS with a rule that is the sum of the squares of the real and unreal parts of triplet grades is limited to the unit interval. The phase terms are novel parameters of the truth, abstinence, and falsity degrees and these are the parameters that distinguish the CSFS and traditional SFS theories. SFS theory deals with only one dimension at a time, which results in information loss in some instances. However, in real life, we come across complex natural phenomena where it becomes essential to add the second dimension to the expression of truth and falsity grades. By introducing this second dimension, the complete information can be projected in one set, and hence, loss of information can be avoided. To cope with such sorts of issues, the principle of a complex spherical fuzzy set was developed by Ali et al. [29]. But there are still certain issues if an individual provides information’s in the form of truth, abstinence, falsity, and uncertain linguistic terms, then the principle of CSFS has been failed. For coping with sorts of issues, the principle of the complex spherical fuzzy uncertain linguistic set is proposed in this studied by combing the two distinct ideas such as CSFSs and ULSs to elaborate the CSFULSs and their basic operational laws.

Definition 4. A CSFULS $ℂ_{CPL}$ is initiated by: $ℂ_{CPL} = {(X_{EL}, ([Ξ_{α}, Ξ_{β}], (T_{C_{CPL}} (X_{EL}), A_{C_{CPL}} (X_{EL}), F_{C_{CPL}} (X_{EL})))) : X_{EL} \in X^{UNI}}$ (13) where $T_{C_{CPL}} (X_{EL}) = T_{C_{RP}} (X_{EL}) e^{i 2 π (T_{C_{IP}} (X_{EL}))}, A_{C_{CPL}} (X_{EL}) = A_{C_{RP}} (X_{EL}) e^{i 2 π (A_{C_{IP}} (X_{EL}))}$ and $F_{C_{CPL}} (X_{EL}) = F_{C_{RP}} (X_{EL}) e^{i 2 π (F_{C_{IP}} (X_{EL}))}$ with the certain rule that is $0 ⩽ T_{ℂ_{RP}}^{2} (X_{EL}) + A_{ℂ_{RP}}^{2} (X_{EL}) + F_{ℂ_{RP}}^{2} (X_{EL}) ⩽ 1$ and $0 ⩽ T_{ℂ_{IP}}^{2} (X_{EL}) + A_{ℂ_{IP}}^{2} (X_{EL}) + F_{ℂ_{IP}}^{2} (X_{EL}) ⩽ 1$ with $\hat{Ξ} = [Ξ_{α}, Ξ_{β}], Ξ_{α}, Ξ_{β} \in \bar{\bar{Ξ}}$ . The complex spherical fuzzy uncertain linguistic numbers (CSFULNs) are dented by $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ .

Definition 5. By using any two CSFULNs $ℂ_{CPL - 1} = ([Ξ_{α_{1}}, Ξ_{β_{1}}], (T_{C_{RP - 1}} e^{i 2 π (T_{C_{IP - 1}})}, A_{C_{RP - 1}} e^{i 2 π (A_{C_{IP - 1}})}, F_{C_{RP - 1}} e^{i 2 π (F_{C_{IP - 1}})}))$ and $ℂ_{CPL - 2} = ([Ξ_{α_{2}}, Ξ_{β_{2}}], (T_{C_{RP - 2}} e^{i 2 π (T_{C_{IP - 2}})}, A_{C_{RP - 2}} e^{i 2 π (A_{C_{IP - 2}})}, F_{C_{RP - 2}} e^{i 2 π (F_{C_{IP - 2}})}))$ , with constant δ_SC, then $ℂ_{CPL - 1} \oplus ℂ_{CPL - 2} = ([Ξ_{α_{1} + α_{2}}, Ξ_{β_{1} + β_{2}}], (\begin{matrix} (\begin{matrix} T_{ℂ RP - 1}^{2} + T_{ℂ RP - 2}^{2} \\ - T_{ℂ RP - 1}^{2} T_{ℂ RP - 2}^{2} \end{matrix}) \frac{1}{2} e^{i 2 π} {(T_{ℂ IP - 1}^{2} + T_{ℂ IP - 2}^{2} - T_{ℂ IP - 1}^{2} T_{ℂ IP - 2}^{2})}^{\frac{1}{2}}, \\ \begin{matrix} A_{ℂ RP - 1} A_{ℂ RP - 2} e^{i 2 π (A_{C}^{IP - 1} A_{C}^{IP - 2}),} \\ F_{ℂ_{RP - 1}} F_{ℂ RP - 2} e^{i 2 π (F_{C_{IP - 1}} F_{C}^{IP - 2})} \end{matrix} \end{matrix}))$ (14) $ℂ_{CPL - 1} \otimes ℂ_{CPL - 2} = ([Ξ_{α_{1} * α_{2}}, Ξ_{β_{1} * β_{2}}], (\begin{matrix} (T_{C_{RP - 1}} T_{C_{RP - 2}}) e^{i 2 π} (T_{C_{IP - 1}} T_{C_{IP - 2}}), \\ \begin{matrix} {(\begin{matrix} A_{ℂ RP - 1}^{2} + A_{ℂ_{RP - 2}}^{2} \\ - A_{ℂ_{RP - 1}}^{2} + A_{ℂ_{RP - 2}}^{2} \end{matrix})}^{\frac{1}{2}} e^{i 2 π (A_{ℂ_{IP - 1}}^{2} + A_{ℂ_{IP - 2}}^{2} - A_{ℂ_{IP - 1}}^{2} + A_{ℂ_{IP - 2}}^{2}) \frac{1}{2}}, \\ {(\begin{matrix} F_{ℂ_{RP - 1}}^{2} + F_{ℂ_{RP - 2}}^{2} \\ - F_{ℂ_{RP - 1}}^{2} + F_{ℂ_{RP - 2}}^{2} \end{matrix})}^{\frac{1}{2}} e^{i 2 π {(F_{ℂ_{IP - 1}}^{2} + F_{ℂ_{IP - 2}}^{2} - F_{ℂ_{IP - 1}}^{2} + F_{ℂ_{IP - 2}}^{2})}^{\frac{1}{2}}} \end{matrix} \end{matrix}))$ (15) $δ_{SC} ℂ_{CPL - 1} = ([Ξ_{δ_{SC} * α_{1}}, Ξ_{δ_{SC} * β_{1}}], (\begin{matrix} {(1 - {(1 - T_{ℂ_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - T_{ℂ_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}}, \\ A_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (A_{C_{IP - 1}}^{δ_{SC}})}, F_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (F_{C_{IP - 1}}^{δ_{SC}})} \end{matrix}))$ (16) $ℂ_{CPL - 1}^{δ_{SC}} = ([Ξ_{α_{1}^{δ_{SC}}}, Ξ_{β_{1}^{δ_{SC}}}], (\begin{matrix} T_{C_{RP - 1}}^{δ_{SC}} e^{i 2 π (T_{C_{IP - 1}}^{δ_{SC}})}, {(1 - {(1 - A_{C_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - A_{C_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}} \\ {(1 - {(1 - F_{ℂ_{RP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - F_{ℂ_{IP - 1}}^{2})}^{δ_{SC}})}^{\frac{1}{2}}} \end{matrix})) .$ (17)

Definition 6 By using any CSFULN $ℂ_{CPL - 1} = ([Ξ_{α_{1}}, Ξ_{β_{1}}], (T_{C_{RP - 1}} e^{i 2 π (T_{C_{IP - 1}})}, A_{C_{RP - 1}} e^{i 2 π (A_{C_{IP - 1}})}, F_{C_{RP - 1}} e^{i 2 π (F_{C_{IP - 1}})}))$ , we initiated the score value and accuracy value as the form: $S_{SV} (ℂ_{CPL - 1}) = (α_{1} + β_{1}) * \frac{1}{2} ((T_{C_{RP - 1}}^{2} - A_{C_{RP - 1}}^{2} - F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} - A_{C_{IP - 1}}^{2} - F_{C_{IP - 1}}^{2}))$ (18) $ℍ_{AV} (ℂ_{CPL - 1}) = (α_{1} + β_{1}) * \frac{1}{2} ((T_{C_{RP - 1}}^{2} + A_{C_{RP - 1}}^{2} + F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} + A_{C_{IP - 1}}^{2} + F_{C_{IP - 1}}^{2}))$ (19)

To determine the relations among any numbers of arrangements, we determine some rules such as:

When $S_{SV} (ℂ_{CPL - 1}) > S_{SV} (ℂ_{CPL - 2}) \Rightarrow ℂ_{CPL - 1} > ℂ_{CPL - 2}$ ;

When $S_{SV} (ℂ_{CPL - 1}) < S_{SV} (ℂ_{CPL - 2}) \Rightarrow ℂ_{CPL - 1} < ℂ_{CPL - 2}$ ;

When $S_{SV} (ℂ_{CPL - 1}) = S_{SV} (ℂ_{CPL - 2}) \Rightarrow$

When $ℍ_{AV} (ℂ_{CPL - 1}) > ℍ_{AV} (ℂ_{CPL - 2}) \Rightarrow ℂ_{CPL - 1} > ℂ_{CPL - 2}$ ;

When $ℍ_{AV} (ℂ_{CPL - 1}) < ℍ_{AV} (ℂ_{CPL - 2}) \Rightarrow ℂ_{CPL - 1} < ℂ_{CPL - 2}$ ;

When $ℍ_{AV} (ℂ_{CPL - 1}) = ℍ_{AV} (ℂ_{CPL - 2}) \Rightarrow ℂ_{CPL - 1} = ℂ_{CPL - 2}$ .

4 Complex Spherical Fuzzy Uncertain Linguistic MM operators

The MM operators offer a flexible arrangement with its modifiable factors because of Muirhead’s general structure. On the other hand, MM aggregation operators perform a significant role in conveying the magnitude level of options and characteristics. By using the MM aggregation operators based on CSFULS is to elaborate the CSFULMM, CSFULWMM, CSFULDMM, CSFULDWMM operators, and their important results are also elaborated with the help of some remarkable cases.

Definition 7. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , then the CSFULMM operator is initiated by: ${CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = {(\frac{1}{n_{NE}!} (\oplus_{σ \in Ξ_{n_{NE}}} (\otimes_{i = 1}^{n_{NE}} ℂ_{CF - σ (i)}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}$ (20) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector.

Theorem 1. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, then by using Equation (20), we get $\begin{matrix} {CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - nNE}) \\ = (\begin{matrix} [Ξ_{{(\frac{1}{n_{NE}!} (\sum_{σ \in Ξ n_{NE}} (\prod_{i = 1}^{n_{NE}} α_{σ (i)}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}, Ξ_{{(\frac{1}{n_{NE}!} (\sum_{σ \in Ξ n_{NE}} (\prod_{i = 1}^{n_{NE}} β_{σ (i)}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}], \\ (\begin{matrix} {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - {(\prod_{i = 1}^{n_{NE}} T_{C_{RP - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}} \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - {(\prod_{i = 1}^{n_{NE}} T_{C_{IP - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}, \\ {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{RP - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}} \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{IP - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}} \\ {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - F_{ℂ_{RP - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}} \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - F_{ℂ_{IP - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}) \end{matrix}$ (21) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector.

Proof. As shown above, we prove that Equation (21), we have $ℂ_{C F - σ (i)}^{℘_{i}} = (\begin{array}{l} [\begin{matrix} Ξ_{α_{σ (i)}^{℘_{i}}}, & Ξ_{β_{σ (i)}^{℘_{i}}} \end{matrix}], \\ (\begin{matrix} \prod_{i = 1}^{n_{N E}} T_{ℂ_{R P - σ (i)}}^{℘_{i}} e^{i 2 π} (\prod_{i = 1}^{n_{N E}} T_{ℂ_{I P - σ (i)}}^{℘_{i}}), {(1 - {(1 - A_{ℂ_{R P - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} e^{i 2 π} {(1 - {(1 - A_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} \\ {(1 - {(1 - ℱ_{ℂ_{R P - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} e^{i 2 π} {(1 - {(1 - ℱ_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} \end{matrix}) \end{array})$ $\otimes_{i = 1}^{n_{NE}} ℂ_{CF - σ (i)}^{℘_{i}} = (\begin{matrix} [Ξ_{\prod_{i = 1}^{n_{NE}} α_{σ (i)}^{℘_{i}}}, Ξ_{\prod_{i = 1}^{n_{NE}} β_{σ (i)}^{℘_{i}}}], \\ (\begin{matrix} \prod_{i = 1}^{n_{NE}} T_{C_{RP - σ (i)}}^{℘_{i}} e^{i 2 π} (\prod_{i = 1}^{n_{NE}} T_{C_{IP - σ (i)}}^{℘_{i}}), {(1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{RP - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} e^{i 2 π} {(1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{IP - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} \\ {(1 - \prod_{i = 1}^{n_{NE}} {(1 - F_{ℂ_{RP - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} e^{i 2 π} {(1 - \prod_{i = 1}^{n_{NE}} {(1 - F_{ℂ_{IP - σ (i)}}^{2})}^{℘_{i}})}^{\frac{1}{2}} \end{matrix}) \end{matrix})$ $\oplus_{σ \in Ξ n_{N E}} (\otimes_{i = 1}^{n_{N E}} ℂ_{C F - σ (i)}^{℘_{i}}) = (\begin{matrix} [Ξ_{\sum_{σ \in Ξ n_{N E}}} (\prod_{i = 1}^{n_{N E}} α_{σ (i)}^{℘_{i}}), Ξ_{\sum_{σ \in Ξ n_{N E}}} (\prod_{i = 1}^{n_{N E}} β_{σ (i)}^{℘_{i}})], \\ (\begin{array}{l} (1 - {(\prod_{σ \in Ξ n_{N E}} (1 - {(\prod_{i = 1}^{n_{N E}} T_{ℂ_{R P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{2}}) \\ e^{i 2 π (1 - {(\prod_{σ \in Ξ n_{N E}} (1 - {(\prod_{i = 1}^{n_{N E}} T_{ℂ_{I P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{2}})} \\ {(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - A_{ℂ_{R P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{2}} \\ e^{i 2 π {(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - A_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{2}}}, \\ {(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - ℱ_{ℂ_{R P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{2}} \\ e^{i 2 π} {(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - ℱ_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{2}} \end{array}) \end{matrix})$ $\frac{1}{n_{N E}!} (\oplus_{σ \in Ξ n_{N E}} (\otimes_{i = 1}^{n_{N E}} ℂ_{C F - σ (i)}^{℘_{i}})) = (\begin{matrix} [Ξ_{\frac{1}{n_{N E}!}} (\sum_{σ \in Ξ n_{N E}} (\prod_{i = 1}^{n_{N E}} α_{σ (i)}^{℘_{i}})), Ξ_{\frac{1}{n_{N E}!}} (\sum_{σ \in Ξ n_{N E}} (\prod_{i = 1}^{n_{N E}} β_{σ (i)}^{℘_{i}}))], \\ (\begin{array}{l} ({(1 - {(\prod_{σ \in Ξ n_{N E}} (1 - {(\prod_{i = 1}^{n_{N E}} T_{ℂ_{R P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}}) \\ e^{i 2 π ({(1 - {(\prod_{σ \in Ξ n_{N E}} (1 - {(\prod_{i = 1}^{n_{N E}} T_{ℂ_{I P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}})} \\ {({(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - A_{ℂ_{R P - σ (i)}^{}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}} \\ e^{i 2 π {({(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - A_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}}} \\ {({(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - ℱ_{ℂ_{R P - σ (i)}^{}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}} \\ e^{i 2 π {({(\prod_{σ \in Ξ n_{N E}} (1 - \prod_{i = 1}^{n_{N E}} {(1 - ℱ_{ℂ_{I P - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}}} \end{array}) \end{matrix})$

By using Equation (21), we discussed some results, and based on the above results, we concluded some properties such as Idempotency, Monotonicity, and boundedness.

Property 1. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} = ℂ_{CPL}$ , then by using Equation (20), we get ${CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = ℂ_{CPL}$ (22)

Proof. By using Equation (20), we get $\begin{array}{l} C S F U L M M^{℘_{i}} (ℂ_{C F - 1}, ℂ_{C F - 2}, \dots, ℂ_{C F - n_{N E}}) = {(\frac{1}{n_{N E}!} (\sum_{σ \in Ξ n_{N E}}^{} (\prod_{i = 1}^{n_{N E}} ℂ_{C F}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} \\ = {(\frac{1}{n_{N E}!} (\sum_{σ \in Ξ n_{N E}} (ℂ_{C F}^{\sum_{i = 1}^{n_{N E}} ℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} = {(\frac{1}{n_{N E}!} (n_{N E}! (ℂ_{C F}^{\sum_{i = 1}^{n_{N E}}}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} = {(ℂ_{C F}^{\sum_{i = 1}^{n_{N E}} ℘_{i}})}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} \\ = ℂ_{C P L} \end{array}$

Property 2. By using any two families of CSFULNs $ℂ_{CPL - i}^{*} = ([Ξ_{α_{i}^{*}}, Ξ_{β_{i}^{*}}], (T_{C_{RP - i}}^{*} e^{i 2 π (T_{C_{IP - i}}^{*})}, A_{C_{RP - i}}^{*} e^{i 2 π (A_{C_{IP - i}}^{*})}, F_{C_{RP - i}}^{*} e^{i 2 π (F_{C_{IP - i}}^{*})}))$ and $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} ⩽ ℂ_{CPL - i}^{*}$ , then by using Equation (20), we get ${CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ {CSFULMM}^{℘_{i}} (ℂ_{CF - 1}^{*}, ℂ_{CF - 2}^{*}, \dots, ℂ_{CF - n_{NE}}^{*})$ (23)

Property 3. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i}^{+} = ([Ξ_{\max α_{i}}, Ξ_{\max β_{i}}], (\max T_{ℂ_{RP - i}} e^{i 2 π (\max T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})}, \min F_{ℂ_{RP - i}} e^{i 2 π (\min F_{ℂ_{IP - i}})}))$ } and $ℂ_{CPL - i}^{-} = ([Ξ_{\min α_{i}}, Ξ_{\min β_{i}}], (\min T_{ℂ_{RP - i}} e^{i 2 π (\min T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})}, \max F_{ℂ_{RP - i}} e^{i 2 π (\max F_{ℂ_{IP - i}})}))$ , then by using Equation (20), we get} $ℂ_{CPL - i}^{-} ⩽ {CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ ℂ_{CPL - i}^{+}$ (24)

The proof of this result is the special case of property 1. Furthermore, by using the information in Equation (21), we get the following cases, such that

For ℘ = (1, 0, …, 0), the CSFULMM operator will convert to complex spherical fuzzy uncertain linguistic (CSFUL) arithmetic averaging (CSFULAA) operator is shown in the form of Equation (25), such that $\begin{matrix} {CSFULMM}^{(1, 0, \dots, 0)} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) \\ = ([\frac{Ξ_{\sum_{i = 1}^{n_{NE}} α_{i}},}{n_{NE}} \frac{Ξ_{\sum_{i = 1}^{n_{NE}} β_{i}}}{n_{NE}}], (\begin{matrix} {(1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{RP - i}}^{2})}^{\frac{1}{π}})}^{\frac{1}{2}} e^{i 2 π} {(1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{IP - i}}^{2})}^{\frac{1}{π}})}^{\frac{1}{2}} \\ \prod_{i = 1}^{n_{NE}} A_{C RP - i}^{\frac{1}{π}} e^{i 2 π} (\prod_{i = 1}^{n_{NE}} A_{C IP - i}^{\frac{1}{π}}), \prod_{i = 1}^{n_{NE}} F_{C_{RP - i}}^{\frac{1}{π}} e^{i 2 π} (\prod_{i = 1}^{n_{NE}} F_{C_{IP - i}}^{\frac{1}{π}}) \end{matrix})) \end{matrix}$ (25)

For ℘ = (γ, 0, …, 0), the CSFULMM operator will convert to generalized CSFUL arithmetic averaging (GCSFULAA) operator is shown in the form of Equation (26), such that $\begin{matrix} {CSFULMM}^{(γ, 0, \dots, 0)} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - nNE}) \\ = (\begin{matrix} [Ξ_{{(\frac{\sum_{i = 1}^{n_{NE}} α_{i}^{γ}}{n_{NE}})}^{\frac{1}{γ}}}, Ξ_{{(\frac{\sum_{i = 1}^{n_{NE}} β_{i}^{γ}}{n_{NE}})}^{\frac{1}{γ}}}], \\ (\begin{matrix} {({(1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{RP - i}}^{2 γ})}^{\frac{1}{π}})}^{\frac{1}{2}})}^{\frac{1}{γ}} e^{i 2 π {({(1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{IP - i}}^{2 γ})}^{\frac{1}{π}})}^{\frac{1}{2}})}^{\frac{1}{γ}}}, \\ {(1 - {(1 - \prod_{i = 1}^{n_{NE}} {(1 - {(1 - A_{ℂ_{RP - i}}^{2 γ_{i}})}^{γ})}^{\frac{1}{n_{NE}}})}^{\frac{1}{γ}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - \prod_{i = 1}^{n_{NE}} {(1 - {(1 - A_{ℂ_{IP - i}}^{2})}^{γ})}^{\frac{1}{n_{NE}}})}^{\frac{1}{γ}})}^{\frac{1}{2}}}, \\ {(1 - {(1 - \prod_{i = 1}^{n_{NE}} {(1 - {(1 - F_{ℂ_{RP - i}}^{2})}^{γ})}^{\frac{1}{n_{NE}}})}^{\frac{1}{γ}})}^{\frac{1}{2}} e^{i 2 π {(1 - {(1 - \prod_{i = 1}^{n_{NE}} {(1 - {(1 - F_{ℂ_{IP - i}}^{2})}^{γ})}^{\frac{1}{n_{NE}}})}^{\frac{1}{γ}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}) \end{matrix}$ (26)

For $℘ = (\begin{matrix} n_{NE} \\ 1, 1, \dots, 1 \end{matrix}, \begin{matrix} n_{NE} - k \\ 0, 0, \dots, 0 \end{matrix})$ , the CSFULMM operator will convert to CSFUL McLaurin symmetric mean (CSFULMSM) operator.

For ℘ = (1, 1, …, 1), the CSFULMM operator will convert to CSFUL geometric averaging (CSFULGA) operator is shown in the form of Equation (28), such that

For $℘ = (\frac{1}{n_{NE}}, \frac{1}{n_{NE}}, \dots, \frac{1}{n_{NE}})$ , the CSFULMM operator will convert to the CSFUL geometric averaging (CSFULGA) operator.

Definition 8. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , then the CSFULWMM operator is initiated by: ${CSFULWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = {(\frac{1}{n_{NE}!} (\oplus_{σ \in Ξ_{n_{NE}}} (\otimes_{i = 1}^{n_{NE}} {(ω_{i} ℂ_{CF - σ (i)})}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}$ (27) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector with a weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ .

Theorem 2. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, then by using Equation (30), we get $\begin{matrix} {CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - nNE}) \\ = (\begin{matrix} [Ξ_{{(\frac{1}{n_{NE}!} (\sum_{σ \in Ξ n_{NE}} (\prod_{i = 1}^{n_{NE}} {(ω_{i} α_{σ (i)})}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}, Ξ_{{(\frac{1}{n_{NE}!} (\sum_{σ \in Ξ n_{NE}} (\prod_{i = 1}^{n_{NE}} {(ω_{i} β_{σ (i)})}^{℘_{i}})))}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}], \\ (\begin{matrix} {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - {(1 - T_{ℂ_{RP - σ (i)}}^{2})}^{ω_{i}})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}} \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - (1 - T_{ℂ_{IP - σ (i)}}^{2}) ω_{i})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}}, \\ {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{RP - σ (i)}}^{2 ω_{i}})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}} \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - A_{ℂ_{IP - σ (i)}}^{2 ω_{i}})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}} \\ (1 - {(1 - {(\prod_{σ \in Ξ n_{N E}} {(1 - (\prod_{i = 1}^{n_{N E}} {(1 - ℱ_{ℂ_{R P - σ (i)}}^{2 ω_{i}})}^{℘_{i}}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}})}^{\frac{1}{2}}) \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} (1 - F_{ℂ_{IP - σ (i)}}^{2 ω_{i}})^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}) \end{matrix}$ (28) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector with a weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ .

By using Equation (28), we discussed some results, and based on the above results, we concluded some properties such as Idempotency, Monotonicity, and boundedness.

Property 4. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} = ℂ_{CPL}$ , then by using Equation (28), we get ${CSFULDWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = ℂ_{CPL}$ (29)

Property 5. By using any two families of CSFULNs $ℂ_{CPL - i}^{*} = ([Ξ_{α_{i}^{*}}, Ξ_{β_{i}^{*}}], (T_{C_{RP - i}}^{*} e^{i 2 π (T_{C_{IP - i}}^{*})}, A_{C_{RP - i}}^{*} e^{i 2 π (A_{C_{IP - i}}^{*})}, F_{C_{RP - i}}^{*} e^{i 2 π (F_{C_{IP - i}}^{*})}))$ and $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} ⩽ ℂ_{CPL - i}^{*}$ , then by using Equation. (28), we get ${CSFULWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ {CSFULWMM}^{℘_{i}} (ℂ_{CF - 1}^{*}, ℂ_{CF - 2}^{*}, \dots, ℂ_{CF - n_{NE}}^{*})$ (30)

Property 6. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i}^{+} = ([Ξ_{\max α_{i}}, Ξ_{\max β_{i}}], (\max T_{ℂ_{RP - i}} e^{i 2 π (\max T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})},, \min F_{ℂ_{RP - i}} e^{i 2 π (\min F_{ℂ_{IP - i}})}))$ and $ℂ_{CPL - i}^{-} = ([Ξ_{\min α_{i}}, Ξ_{\min β_{i}}], (\min T_{ℂ_{RP - i}} e^{i 2 π (\min T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})}, \max F_{ℂ_{RP - i}} e^{i 2 π (\max F_{ℂ_{IP - i}})}))$ , then by using Equation (28), we get $ℂ_{CPL - i}^{-} ⩽ {CSFULWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ ℂ_{CPL - i}^{+}$ (31)

Furthermore, by assigning the different values in the CSFULWMM operator in Equation (28), we get the variable cases, such that the complex spherical fuzzy uncertain linguistic (CSFUL) weighted arithmetic averaging (CSFULWAA) operator (℘ = (1, 0, …, 0)) and the CSFUL weighted McLaurin symmetric mean (CSFULWMSM) operator ( $℘ = (\begin{matrix} n_{NE} \\ 1, 1, \dots, 1 \end{matrix}, \begin{matrix} n_{NE} - k \\ 0, 0, \dots, 0 \end{matrix})$ ).

Definition 9. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , then the CSFULDMM operator is initiated by: ${CSFULDMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = {(\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} (\otimes_{σ \in Ξ_{NE}} (\oplus_{i = 1}^{n_{NE}} ℘_{i} ℂ_{CF - σ (i)})))}^{\frac{1}{n_{NE}!}}$ (32)

where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector.

Theorem 3. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, then by using Equation (32), we get $\begin{matrix} {CSFULMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - nNE}) \\ = (\begin{matrix} [Ξ_{{(\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} (\sum_{σ \in Ξ n_{NE}} (\sum_{i = 1}^{n_{NE}} ℘_{i} α_{σ (i)})))}^{\frac{1}{n_{NE}!}}}, Ξ_{{(\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} (\sum_{σ \in Ξ n_{NE}} (\sum_{i = 1}^{n_{NE}} ℘_{i} β_{σ (i)})))}^{\frac{1}{n_{NE}}}}], \\ (\begin{matrix} {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{RP - σ (i)}}^{2})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}} \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{IP - σ (i)}}^{2})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}}, \\ {({(1 - {(\prod_{σ \in Ξ n_{N E}} (1 - {(\prod_{i = 1}^{n_{N E}} A_{C_{R P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - {(\prod_{i = 1}^{n_{NE}} A_{C_{IP - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}} \\ ({(1 - {(\prod_{σ \in Ξ n_{N E}} {(1 - (\prod_{i = 1}^{n_{N E}} {(ℱ_{C_{R P - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}})}^{\frac{1}{2}}) \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - {(\prod_{i = 1}^{n_{NE}} F_{C_{IP - σ (i)}}^{℘_{i}})}^{2}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}} \end{matrix}) \end{matrix}) \end{matrix}$ (33) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector.

By using Equation (33), we discussed some results, and based on the above results, we concluded some properties such as Idempotency, Monotonicity, and boundedness.

Property 7. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} = ℂ_{CPL}$ , then by using Equation (33), we get ${CSFULDMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = ℂ_{CPL}$ (34)

Property 8. By using any two families of CSFULNs $ℂ_{CPL - i}^{*} = ([Ξ_{α_{i}^{*}}, Ξ_{β_{i}^{*}}], (T_{C_{RP - i}}^{*} e^{i 2 π (T_{C_{IP - i}}^{*})}, A_{C_{RP - i}}^{*} e^{i 2 π (A_{C_{IP - i}}^{*})}, F_{C_{RP - i}}^{*} e^{i 2 π (F_{C_{IP - i}}^{*})}))$ and $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i} ⩽ ℂ_{CPL - i}^{*}$ , then by using Equation. (33), we get ${CSFULDMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ {CSFULDMM}^{℘_{i}} (ℂ_{CF - 1}^{*}, ℂ_{CF - 2}^{*}, \dots, ℂ_{CF - n_{NE}}^{*})$ (35)

Property 9. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, if $ℂ_{CPL - i}^{+} = ([Ξ_{\max α_{i}}, Ξ_{\max β_{i}}], (\max T_{ℂ_{RP - i}} e^{i 2 π (\max T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})}, \min F_{ℂ_{RP - i}} e^{i 2 π (\min F_{ℂ_{IP - i}})}))$ } and $ℂ_{CPL - i}^{-} = ([Ξ_{\min α_{i}}, Ξ_{\min β_{i}}], (\min T_{ℂ_{RP - i}} e^{i 2 π (\min T_{ℂ_{IP - i}})}, \min A_{ℂ_{RP - i}} e^{i 2 π (\min A_{ℂ_{IP - i}})}, \max F_{ℂ_{RP - i}} e^{i 2 π (\max F_{ℂ_{IP - i}})}))$ , then by using Equation (33), we get} $ℂ_{CPL - i}^{-} ⩽ {CSFULDMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) ⩽ ℂ_{CPL - i}^{+}$ (36)

Furthermore, by using the information in Equation (33), we get the following cases, such that the complex spherical fuzzy uncertain linguistic (CSFUL) geometric averaging (CSFULGA) operator ( $P = (1, 0, \dots, 0)$ ), the generalized CSFUL geometric averaging (GCSFULGA) operator (℘ = (γ, 0, …, 0)), the CSFUL geometric McLaurin symmetric mean (CSFULGMSM) operator ( $℘ = (\underset{1, 1, \dots, 1}{n_{NE}}, \underset{0, 0, \dots, 0}{n_{NE} - k})$ ), the CSFUL arithmetic averaging (CSFULAA) operator (℘ = (1, 1, …, 1)), the CSFUL arithmetic averaging (CSFULAA) operator ( $℘ = (\frac{1}{n_{NE}}, \frac{1}{n_{NE}}, \dots, \frac{1}{n_{NE}})$ ).

Definition 10. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , then the CSFULDWMM operator is initiated by: ${CSFULDWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - n_{NE}}) = \frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} {((\otimes_{σ \in Ξ_{n_{NE}}} (\oplus_{i = 1}^{n_{NE}} {(℘_{i} ℂ_{CF - σ (i)})}^{ω_{i}})))}^{\frac{1}{n_{NE}!}}$ (37) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector with a weight vector $ω = (ω_{1}, ω_{2}, \dots, ω_{n_{N E}})$ based on the rule that is $\sum_{i = 1} ω_{i} = 1$ .

Theorem 4. By using any family of CSFULNs $ℂ_{CPL - i} = ([Ξ_{α_{i}}, Ξ_{β_{i}}], (T_{C_{RP - i}} e^{i 2 π (T_{C_{IP - i}})}, A_{C_{RP - i}} e^{i 2 π (A_{C_{IP - i}})}, F_{C_{RP - i}} e^{i 2 π (F_{C_{IP - i}})})), i = 1, 2, \dots, n_{NE}$ , with parameter vector ℘_i, then by using Equation (40), we get $\begin{matrix} {CSFULDWMM}^{℘_{i}} (ℂ_{CF - 1}, ℂ_{CF - 2}, \dots, ℂ_{CF - nNE}) \\ = (\begin{matrix} [Ξ_{{(\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} ((\prod_{σ \in Ξ n_{NE}} {(\sum_{i = 1}^{n_{NE}} ℘_{i} α_{σ (i)})}^{ω_{i}})))}^{\frac{1}{n_{NE}!}}}, Ξ_{{(\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}} ((\prod_{σ \in Ξ n_{NE}} {(\sum_{i = 1}^{n_{NE}} ℘_{i} β_{σ (i)})}^{ω_{i}})))}^{\frac{1}{n_{NE}!}}}], \\ (\begin{matrix} {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{RP - σ (i)}}^{2_{ω_{i}}})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}} \\ e^{i 2 π {(1 - {(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - \prod_{i = 1}^{n_{NE}} {(1 - T_{ℂ_{IP - σ (i)}}^{2 ω_{i}})}^{℘_{i}}))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}})}^{\frac{1}{2}}}, \\ {({(1 - (\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - {(1 - A_{ℂ_{RP - σ (i)}}^{2})}^{ω_{i}})}^{℘_{i}}))) \frac{1}{n_{NE}!})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}} \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - {(1 - A_{ℂ_{IP - σ (i)}}^{2})}^{ω_{i}})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}} \\ {({(1 - {(\prod_{σ \in Ξ n_{N E}} (1 - (\prod_{i = 1}^{n_{N E}} {(1 - {(1 - ℱ_{ℂ_{R P - σ (i)}}^{2})}^{ω_{i}})}^{℘_{i}})))}^{\frac{1}{n_{N E}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{N E}} ℘_{i}}} \\ e^{i 2 π {({(1 - {(\prod_{σ \in Ξ n_{NE}} (1 - (\prod_{i = 1}^{n_{NE}} {(1 - {(1 - F_{ℂ_{IP - σ (i)}}^{2})}^{ω_{i}})}^{℘_{i}})))}^{\frac{1}{n_{NE}!}})}^{\frac{1}{2}})}^{\frac{1}{\sum_{i = 1}^{n_{NE}} ℘_{i}}}} \end{matrix}) \end{matrix}) \end{matrix}$ (38) where σ (i) expressed the permutation of (i = 1, 2, …, n_NE) and the Ξ_{n
_NE} is the family of σ (i) and ℘_i is a parameters vector with a weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ .

5 Model for MADM with complex spherical fuzzy uncertain linguistic information

Certain scholars have utilized numerous operators in the circumstances of the MADM technique. To determine the consistency and legality of the elaborated operators with the help of MADM, in this study, we discussed the MADM troubles by using the elaborated operators based on CSFULSs. To resolve the above troubles, we choose the family of alternatives $ℂ_{AL} = {ℂ_{AL - 1}, ℂ_{AL - 2}, \dots, ℂ_{AL - m}}$ and the set $ℂ_{AT} = {ℂ_{AT - 1}, ℂ_{AT - 2}, \dots, ℂ_{AT - n}}$ expressed the family of attributes with their weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ . By using the above information, the MULTIMOORA method is discussed below.

5.1 MULTIMOORA-CSFUL based on muirhead mean operators

By using the investigated operators, we discussed the MULTIMOORA technique comprises various Moora techniques and multiplicative utility function (MUF). These techniques are preservative utility purpose, mention point methodology, and MUF. In the ratio approach and complete multiplicative structure elements of the MULTIMOORA technique, the intended MM operators are utilized. Thus, mutually the excellent elasticity of universal consideration of the MM operator and the DMM operator convey the characteristics improved. The MULTIMOORA MADM with CSFULNs can be clarified by the subsequent cases:

5.1.1 Method 1 based on CSFULWMM operator:

Step 1: By using the family of alternatives and their attributes, we develop the Metrix, whose every term in the form of CSFUL information’s such as: $ℂ_{CPL - ij} = ([Ξ_{α_{ij}}, Ξ_{β_{ij}}], (T_{C_{RP - ij}} e^{i 2 π (T_{C_{IP - ij}})}, A_{C_{RP - ij}} e^{i 2 π (A_{C_{IP - ij}})}, F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})})), i, j = 1, 2, \dots, n_{NE}, m_{NE}$ with a decision matrix such as $R = {[ℂ_{CPL - ij}]}_{m_{NE} \times n_{NE}}$ . By using the suggested matrix, we normalized the decision matrix based on the following rules, such that $\begin{matrix} ℂ_{CPL - ij} \\ = {\begin{matrix} ([Ξ_{α ij}, Ξ_{β ij}], (T_{C_{RP - ij}} e^{i 2 π (T_{C_{IP - ij}})}, A_{C_{RP - ij}} e^{i 2 π (A_{C_{IP - ij}}),} F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})})) for benefit types of sorts \\ ([Ξ_{α ij}, Ξ_{β ij}], (F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})}, A_{C_{RP - ij}} e^{i 2 π (A_{C_{IP - ij}}),} T_{C_{RP - ij}} e^{i 2 π (T_{C_{IP - ij}})})) for cost types of sorts \end{matrix} \end{matrix}$

If normalized is needed.

Step 2: By using the information of the developed matrix, we use the CSFULWMM operator defined in Equation (31) to aggregate the decision matrix with the weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ .

Step 3: By using the information in Case 2, we use the Equation (8) (score value formula) to determine the score values by using the aggregated values in Case 2, such that $S_{SV} (ℂ_{CPL - j}) = ({\overset{ˇ}{α}}_{j} + {\overset{ˇ}{β}}_{j}) * \frac{1}{2} ((T_{C_{RP - 1}}^{2} - A_{C_{RP - 1}}^{2} - F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} - A_{C_{IP - 1}}^{2} - F_{C_{IP - 1}}^{2}))$

Step 4: By using the information in case 3, we use the following formula to find the raking results to determine the best optimal, such that $ℂ_{CPL - j}^{RS} = \frac{S_{SV} (ℂ_{CPL - j})}{max (S_{SV} (ℂ_{CPL - j}))}$

From the above analysis, we obtained the ration systems which are expressed by “RS”.

5.1.2 Method 2 based on CSFULDWMM operator:

Step 5: By using the information of the developed matrix, we use the CSFULDWMM operator in Equation (32) to aggregate the decision matrix with the weight vector ω = (ω₁, ω₂, …, ω_{n
_NE}) based on the rule that is $\sum_{i = 1}^{n_{NE}} ω_{i} = 1$ .

Step 6: By using the information in Step 5, we use the Equation (8) (score value formula) to determine the score values by using the aggregated values in Case 5, such that $S_{SV} (ℂ_{CPL - j}) = ({\overset{ˇ}{α}}_{j} + {\overset{ˇ}{β}}_{j}) * \frac{1}{2} ((T_{C_{RP - 1}}^{2} - A_{C_{RP - 1}}^{2} - F_{C_{RP - 1}}^{2}) + (T_{C_{IP - 1}}^{2} - A_{C_{IP - 1}}^{2} - F_{C_{IP - 1}}^{2}))$

Step 7: By using the information in Step 6, we use the following formula to find the raking results to determine the best optimal, such that $ℂ_{CPL - j}^{RP} = \frac{S_{SV} (ℂ_{CPL - j})}{max (S_{SV} (ℂ_{CPL - j}))}$

From the above analysis, we obtained the reference point which is expressed by “RP”.

5.1.3 Method 3 based on similarity measures:

Step 8: By using the information of the developed matrix, we determine the positive idea solutions with the help of the following rules, such that $\begin{matrix} ℂ_{CF} \\ = {\begin{matrix} ([Ξ_{\max α_{i 1}}, Ξ_{\max β_{i 1}}], (\max T_{ℂ_{RP - i 1}} e^{i 2 π (\max T_{ℂ_{IP - i 1}})}, \min A_{ℂ_{RP - i 1}} e^{i 2 π (\min A_{ℂ_{IP - i 1}}),} \min F_{ℂ_{RP - i 1}} e^{i 2 π (\min F_{ℂ_{IP - i 1}})})), \\ ([Ξ_{\max α_{i 2}}, Ξ_{\max β_{i 2}}], (\max T_{ℂ_{RP - i 2}} e^{i 2 π (\max T_{ℂ_{IP - i 2}})}, \min A_{ℂ_{RP - i 2}} e^{i 2 π (\min A_{ℂ_{IP - i 2}}),} \min F_{ℂ_{RP - i 2}} e^{i 2 π (\min F_{ℂ_{IP - i 2}})})), \\ ([Ξ_{\max α_{im}}, Ξ_{\max β_{im}}], (\max T_{ℂ_{RP - im}} e^{i 2 π (\max T_{ℂ_{IP - im}})}, \min A_{ℂ_{RP - im}} e^{i 2 π (\min A_{ℂ_{IP - im}}),} \min F_{ℂ_{RP - im}} e^{i 2 π (\min F_{ℂ_{IP - im}})})) \end{matrix}} \end{matrix}$

Step 9: By using the information in Step 8, we use the following formulas to determine the similarity measures among positive idea and unknown parameters given in the decision matrix is discussed below: $\begin{matrix} S_{SV} (ℂ_{CPL - j}) = S_{SM} (ℂ_{CPL - i}, ℂ_{CP}) \\ = \frac{1}{3 n_{NE}} \sum_{i = 1}^{n_{NE}} ({| T_{C_{RP - i}} - T_{C_{RP}}^{PI} |}^{2} + {| T_{C_{IP - i}} - T_{C_{IP}}^{PI} |}^{2} + {| A_{C_{RP - i}} - A_{C_{RP}}^{PI} |}^{2} + {| A_{C_{IP - i}} - A_{C_{IP}}^{PI} |}^{2} \\ {+ {| F_{C_{RP - i}} - F_{C_{RP}}^{PI} |}^{2} + {| F_{C_{IP - i}} - F_{C_{IP}}^{PI} |}^{2})}^{\frac{1}{2}} \end{matrix}$

Step 10: By using the information in Step 9, we use the following formula to find the raking results to determine the best optimal, such that $ℂ_{CPL - j}^{MF} = \frac{S_{SV} (ℂ_{CPL - j})}{max (S_{SV} (ℂ_{CPL - j}))}$

From the above analysis, we obtained the multiplicative form which is expressed by “MF”. By using the above procedure, we will be utilized in the environment of numerical examples to determine the consistency and dominance of the elaborated approaches. Basically, by using one algorithm we obtained the three different types of ranking, which are more deeply and clearly explained with the help of the below example.

5.2 Numerical illustration

To improve educational teaching and encourage the construction of an education association, the university of supervision in an international Islamic university Islamabad, Pakistan institution of higher education intends to announce abroad exceptional professors. This initiation request has drawn good attention from the university. At this beginning, a committee of decision-makers was established, with the accountability of international Islamic university Islamabad, Pakistan president (Prof. Dr. Hathal Homoud Alotaibi) and the dean (Prof. Dr. Muhammad Arshad Zia), the university board, and individual reserves administrator. They received a stringent assessment for 5 applicants ${ℂ_{AL - 1}, ℂ_{AL - 2}, ℂ_{AL - 3}, ℂ_{AL - 4}, ℂ_{AL - 5}}$ . Corresponding to the subsequent four characteristics: $ℂ_{AT - 1}$ is the ethics; $ℂ_{AT - 2}$ is the research expertise; $ℂ_{AT - 3}$ is the teaching ability; $ℂ_{AT - 4}$ is the education experience. The president (Prof. Dr. Hathal Homoud Alotaibi) of the International Islamic University Islamabad, Pakistan is knowledgeable in the important significance of decision-making procedure, the dean (Prof. Dr. Muhammad Arshad Zia) of the administration university comes later. Moreover, this establishment will be in exacting agreement with the dean of amalgamate capability with constitutional honesty. For a decision, producers give their opinions in the form of the weight vector (0.4, 0.3, 0.2, 0.1), Thus, mutually the excellent elasticity of universal consideration of the MM operator and the DMM operator convey the characteristics improved. The MULTIMOORA MADM with CSFULNs can be clarified by the subsequent cases:

5.2.1 Method 1 based on CSFULWMM operator:

Table 1
Decision matrix with complex spherical fuzzy uncertain linguistic numbers

$ℂ_{AT - 1}$ $ℂ_{AT - 2}$ $ℂ_{AT - 3}$ $ℂ_{AT - 4}$

$ℂ_{AL - 1}$ $(\begin{matrix} [Ξ_{0}, Ξ_{1}], \\ (\begin{matrix} 0.6 e^{i 2 π (0.5)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.2 e^{i 2 π (0.2)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.61 e^{i 2 π (0.51)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.21 e^{i 2 π (0.21)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.62 e^{i 2 π (0.52)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.22 e^{i 2 π (0.22)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{3}], \\ (\begin{matrix} 0.63 e^{i 2 π (0.53)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.23 e^{i 2 π (0.23)} \end{matrix}) \end{matrix})$

$ℂ_{AL - 2}$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], \\ (\begin{matrix} 0.5 e^{i 2 π (0.4)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.1 e^{i 2 π (0.1)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{3}], \\ (\begin{matrix} 0.51 e^{i 2 π (0.41)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.11 e^{i 2 π (0.11)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{4}], \\ (\begin{matrix} 0.52 e^{i 2 π (0.42)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.12 e^{i 2 π (0.12)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{5}], \\ (\begin{matrix} 0.53 e^{i 2 π (0.43)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.13 e^{i 2 π (0.13)} \end{matrix}) \end{matrix})$

$ℂ_{AL - 3}$ $(\begin{matrix} [Ξ_{0}, Ξ_{3}], \\ (\begin{matrix} 0.4 e^{i 2 π (0.3)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.3 e^{i 2 π (0.4)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{2}, Ξ_{3}], \\ (\begin{matrix} 0.41 e^{i 2 π (0.31)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.31 e^{i 2 π (0.41)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{3}], \\ (\begin{matrix} 0.42 e^{i 2 π (0.32)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.32 e^{i 2 π (0.42)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.43 e^{i 2 π (0.33)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.33 e^{i 2 π (0.43)} \end{matrix}) \end{matrix})$

$ℂ_{AL - 4}$ $(\begin{matrix} [Ξ_{1}, Ξ_{5}], \\ (\begin{matrix} 0.7 e^{i 2 π (0.8)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.1 e^{i 2 π (0.1)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{4}], \\ (\begin{matrix} 0.71 e^{i 2 π (0.81)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.11 e^{i 2 π (0.11)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{3}], \\ (\begin{matrix} 0.72 e^{i 2 π (0.82)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.12 e^{i 2 π (0.12)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{1}, Ξ_{2}], \\ (\begin{matrix} 0.73 e^{i 2 π (0.83)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.13 e^{i 2 π (0.13)} \end{matrix}) \end{matrix})$

$ℂ_{AL - 5}$ $(\begin{matrix} [Ξ_{3}, Ξ_{4}], \\ (\begin{matrix} 0.2 e^{i 2 π (0.1)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.2 e^{i 2 π (0.2)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{5}], \\ (\begin{matrix} 0.21 e^{i 2 π (0.11)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.21 e^{i 2 π (0.21)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{2}, Ξ_{4}], \\ (\begin{matrix} 0.22 e^{i 2 π (0.12)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.22 e^{i 2 π (0.22)} \end{matrix}) \end{matrix})$ $(\begin{matrix} [Ξ_{3}, Ξ_{3}], \\ (\begin{matrix} 0.23 e^{i 2 π (0.13)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.23 e^{i 2 π (0.23)} \end{matrix}) \end{matrix})$

	$ℂ_{AT - 1}$	$ℂ_{AT - 2}$	$ℂ_{AT - 3}$	$ℂ_{AT - 4}$
$ℂ_{AL - 1}$	$(\begin{matrix} [Ξ_{0}, Ξ_{1}], \\ (\begin{matrix} 0.6 e^{i 2 π (0.5)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.2 e^{i 2 π (0.2)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.61 e^{i 2 π (0.51)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.21 e^{i 2 π (0.21)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.62 e^{i 2 π (0.52)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.22 e^{i 2 π (0.22)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{3}], \\ (\begin{matrix} 0.63 e^{i 2 π (0.53)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.23 e^{i 2 π (0.23)} \end{matrix}) \end{matrix})$
$ℂ_{AL - 2}$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], \\ (\begin{matrix} 0.5 e^{i 2 π (0.4)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.1 e^{i 2 π (0.1)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{3}], \\ (\begin{matrix} 0.51 e^{i 2 π (0.41)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.11 e^{i 2 π (0.11)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{4}], \\ (\begin{matrix} 0.52 e^{i 2 π (0.42)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.12 e^{i 2 π (0.12)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{5}], \\ (\begin{matrix} 0.53 e^{i 2 π (0.43)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.13 e^{i 2 π (0.13)} \end{matrix}) \end{matrix})$
$ℂ_{AL - 3}$	$(\begin{matrix} [Ξ_{0}, Ξ_{3}], \\ (\begin{matrix} 0.4 e^{i 2 π (0.3)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.3 e^{i 2 π (0.4)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{3}], \\ (\begin{matrix} 0.41 e^{i 2 π (0.31)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.31 e^{i 2 π (0.41)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{3}], \\ (\begin{matrix} 0.42 e^{i 2 π (0.32)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.32 e^{i 2 π (0.42)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{0}, Ξ_{2}], \\ (\begin{matrix} 0.43 e^{i 2 π (0.33)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.33 e^{i 2 π (0.43)} \end{matrix}) \end{matrix})$
$ℂ_{AL - 4}$	$(\begin{matrix} [Ξ_{1}, Ξ_{5}], \\ (\begin{matrix} 0.7 e^{i 2 π (0.8)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.1 e^{i 2 π (0.1)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{4}], \\ (\begin{matrix} 0.71 e^{i 2 π (0.81)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.11 e^{i 2 π (0.11)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{3}], \\ (\begin{matrix} 0.72 e^{i 2 π (0.82)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.12 e^{i 2 π (0.12)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{1}, Ξ_{2}], \\ (\begin{matrix} 0.73 e^{i 2 π (0.83)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.13 e^{i 2 π (0.13)} \end{matrix}) \end{matrix})$
$ℂ_{AL - 5}$	$(\begin{matrix} [Ξ_{3}, Ξ_{4}], \\ (\begin{matrix} 0.2 e^{i 2 π (0.1)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.2 e^{i 2 π (0.2)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{5}], \\ (\begin{matrix} 0.21 e^{i 2 π (0.11)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.21 e^{i 2 π (0.21)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{2}, Ξ_{4}], \\ (\begin{matrix} 0.22 e^{i 2 π (0.12)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.22 e^{i 2 π (0.22)} \end{matrix}) \end{matrix})$	$(\begin{matrix} [Ξ_{3}, Ξ_{3}], \\ (\begin{matrix} 0.23 e^{i 2 π (0.13)}, \\ 0.01 e^{i 2 π (0.01)}, \\ 0.23 e^{i 2 π (0.23)} \end{matrix}) \end{matrix})$

By using the suggested matrix, we normalized the decision matrix based on the following rules, such that $\begin{matrix} ℂ_{CPL - ij} \\ = {\begin{matrix} ([Ξ_{α_{ij}}, Ξ_{β_{ij}}], (T_{C_{RP - ij}} e^{i 2 π (T_{C_{IP - ij}})}, A_{C_{RP - ij}} e^{i 2 π (A_{C_{IP - ij}}),} F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})})) for benefit types of sorts \\ ([Ξ_{α_{ij}}, Ξ_{β_{ij}}], (F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})}, A_{C_{RP - ij}} e^{i 2 π (A_{C_{IP - ij}}),} F_{C_{RP - ij}} e^{i 2 π (F_{C_{IP - ij}})})) for cost types of sorts \end{matrix} \end{matrix}$

The information in Table 1 cannot need to be normalized. Because we will select all the benefit types of pieces of information.

Step 2: By using the information of the developed matrix in Table 1, we use the CSFULWMM operator to aggregate the decision matrix with the weight vector ω = (0.4, 0.3, 0.2, 0.1) based on the rule that is 0.4 + 0.3 + 0.2 + 0.1 = 1, are discussed in the form of Table 2.

Table 2

By using the information in Table 1, we aggregate the decision matrix

Δ _i	${CSFULDWMM}^{℘_{i}} (ℂ_{CF - i})$
$ℂ_{AL - 1}$	([Ξ₀, Ξ_0.0417] , (0.9915e^i2π(0.9885), 0.000121e^{i2π(0.000121)}, 0.3973e^i2π(0.3973)))
$ℂ_{AL - 2}$	([Ξ_0.0167, Ξ_0.0583] , (0.9885e^i2π(0.9848), 0.000121e^{i2π(0.000121)}, 0.4532e^i2π(0.4532)))
$ℂ_{AL - 3}$	([Ξ_0.0333, Ξ_0.075] , (0.9848e^i2π(0.9801), 0.000121e^{i2π(0.000121)}, 0.3582e^i2π(0.3262)))
$ℂ_{AL - 4}$	([Ξ_0.0167, Ξ_0.1083] , (0.9941e^i2π(0.9963), 0.000121e^{i2π(0.000121)}, 0.4532e^i2π(0.4532)))
$ℂ_{AL - 5}$	([Ξ_0.05, Ξ_0.0917] , (0.9735e^i2π(0.9624), 0.000121e^{i2π(0.000121)}, 0.3973e^i2π(0.3973)))

Step 3: By using the information in Case 2, we use the Equation (8) (score value formula) to determine the score values by using the aggregated values in Case 2, such that $\begin{matrix} S_{SV} (ℂ_{CPL - 1}) = - 0.03614, S_{SV} (ℂ_{CPL - 2}) = - 0.06216, S_{SV} (ℂ_{CPL - 3}) \\ = - 0.09186, S_{SV} (ℂ_{CPL - 4}) = - 0.1093, S_{SV} (ℂ_{CPL - 5}) = - 0.1085 \end{matrix}$

Step 4: By using the information in case 3, we use the following formula $ℂ_{CPL - j}^{RS}$ to find the raking results to determine the best optimal, such that $ℂ_{CPL - 1}^{RS} = 1, ℂ_{CPL - 2}^{RS} = 1.72017, ℂ_{CPL - 3}^{RS} = 2.5421, ℂ_{CPL - 4}^{RS} = 3.0257, ℂ_{CPL - 5}^{RS} = 3.0257$

From the above analysis, we obtained the ration systems which are expressed by “RS”.

5.2.2 Method 2 based on CSFULDWMM operator:

Step 5: By using the information of the developed matrix, we use the CSFULDWMM operator to aggregate the decision matrix with the weight vector ω = (0.4, 0.3, 0.2, 0.1) based on the rule that is 0.4 + 0.3 + 0.2 + 0.1 = 1, are discussed in the form of Table 3.

Table 3
By using the information in Table 1, we aggregate the decision matrix

Δ _i ${CSFULWMM}^{℘_{i}} (ℂ_{CF - i})$

$ℂ_{AL - 1}$ ([Ξ₀, Ξ_0.04167] , (0.2696e^i2π(0.2972), 0.000121e^{i2π(0.000121)}, 0.9735e^i2π(0.9735)))

$ℂ_{AL - 2}$ ([Ξ_0.01667, Ξ_0.05833] , (0.2972e^i2π(0.3261), 0.000121e^{i2π(0.000121)}, 0.9624e^i2π(0.9623)))

$ℂ_{AL - 3}$ ([Ξ_0.0333, Ξ_0.075] , (0.3261e^i2π(0.3582), 0.000121e^{i2π(0.000121)}, 0.98013e^i2π(0.9848)))

$ℂ_{AL - 4}$ ([Ξ_0.01667, Ξ_0.1083] , (0.2416e^i2π(0.2109), 0.000121e^{i2π(0.000121)}, 0.9623e^i2π(0.9623)))

$ℂ_{AL - 5}$ ([Ξ_0.05, Ξ_0.09167] , (0.3973e^i2π(0.4532), 0.000121e^{i2π(0.000121)}, 0.9735e^i2π(0.9735)))

Δ _i	${CSFULWMM}^{℘_{i}} (ℂ_{CF - i})$
$ℂ_{AL - 1}$	([Ξ₀, Ξ_0.04167] , (0.2696e^i2π(0.2972), 0.000121e^{i2π(0.000121)}, 0.9735e^i2π(0.9735)))
$ℂ_{AL - 2}$	([Ξ_0.01667, Ξ_0.05833] , (0.2972e^i2π(0.3261), 0.000121e^{i2π(0.000121)}, 0.9624e^i2π(0.9623)))
$ℂ_{AL - 3}$	([Ξ_0.0333, Ξ_0.075] , (0.3261e^i2π(0.3582), 0.000121e^{i2π(0.000121)}, 0.98013e^i2π(0.9848)))
$ℂ_{AL - 4}$	([Ξ_0.01667, Ξ_0.1083] , (0.2416e^i2π(0.2109), 0.000121e^{i2π(0.000121)}, 0.9623e^i2π(0.9623)))
$ℂ_{AL - 5}$	([Ξ_0.05, Ξ_0.09167] , (0.3973e^i2π(0.4532), 0.000121e^{i2π(0.000121)}, 0.9735e^i2π(0.9735)))

Step 6: By using the information in Case 5, we use the Equation (8) (score value formula) to determine the score values by using the aggregated values in Case 5, such that $\begin{matrix} S_{SV} (ℂ_{CPL - 1}) = 0.0343, S_{SV} (ℂ_{CPL - 2}) = 0.0576, S_{SV} (ℂ_{CPL - 3}) \\ = 0.0919, S_{SV} (ℂ_{CPL - 4}) = 0.0981, S_{SV} (ℂ_{CPL - 5}) = 0.1104 \end{matrix}$

Step 7: By using the information in case 6, we use the following formula $ℂ_{CPL - j}^{RP}$ to find the raking results to determine the best optimal, such that $ℂ_{CPL - 1}^{RP} = 0.3104, ℂ_{CPL - 2}^{RP} = 0.522, ℂ_{CPL - 3}^{RP} = 0.8323, ℂ_{CPL - 4}^{RP} = 0.889, ℂ_{CPL - 5}^{RP} = 1$

From the above analysis, we obtained the reference point which is expressed by “RP”.

5.2.3 Method 3 based on similarity measures:

Step 8: By using the information of the developed matrix, we determine the positive idea solutions with the help of the following rules, such that $ℂ_{CF} = {\begin{matrix} ([Ξ_{3}, Ξ_{5}], (0.7 e^{i 2 π (0.8)}, 0.01 e^{i 2 π (0.01)}, 0.1 e^{i 2 π (0.1)})), ([Ξ_{3}, Ξ_{5}], (0.71 e^{i 2 π (0.81)}, 0.01 e^{i 2 π (0.01)}, 0.11 e^{i 2 π (0.11)})), \\ ([Ξ_{3}, Ξ_{4}], (0.72 e^{i 2 π (0.82)}, 0.01 e^{i 2 π (0.01)}, 0.12 e^{i 2 π (0.12)})), ([Ξ_{3}, Ξ_{5}], (0.73 e^{i 2 π (0.83)}, 0.01 e^{i 2 π (0.01)}, 0.13 e^{i 2 π (0.13)})) \end{matrix}}$

Step 9: By using the information in Case 8, we use the following formulas $S_{SV} (ℂ_{CPL - j}) = S_{SM} (ℂ_{CPL - i}, ℂ_{CP})$ to determine the similarity measures among positive idea and unknown parameters given in the decision matrix is discussed below: $\begin{matrix} S_{SV} (ℂ_{CPL - 1}) = S_{SM} (ℂ_{CPL - 1}, ℂ_{CP}) = 1.6887, S_{SV} (ℂ_{CPL - 2}) = S_{SM} (ℂ_{CPL - 2}, ℂ_{CP}) \\ = 2.7392, S_{SV} (ℂ_{CPL - 3}) = S_{SM} (ℂ_{CPL - 3}, ℂ_{CP}) = 4.1991, S_{SV} (ℂ_{CPL - 4}) = S_{SM} (ℂ_{CPL - 4}, ℂ_{CP}) \\ = 0, S_{SV} (ℂ_{CPL - 5}) = S_{SM} (ℂ_{CPL - 5}, ℂ_{CP}) = 6.3204 \end{matrix}$

Step 10: By using the information in case 9, we use the following formula $ℂ_{CPL - j}^{MF}$ to find the raking results to determine the best optimal, such that $ℂ_{CPL - 1}^{MF} = 0.2672, ℂ_{CPL - 2}^{MF} = 0.4334, ℂ_{CPL - 3}^{MF} = 0.6644, ℂ_{CPL - 4}^{MF} = 0, ℂ_{CPL - 5}^{MF} = 0.2672$

From the above analysis, we obtained the multiplicative form which is expressed by “MF”. By using the above procedure, we discussed the ranking values in the form of Table 4.

Table 4
By using dominance theory expressions of the MULTIMOORA methods

Symbols Ration Systems References Points Multiplicative form MULTIMOORA Methods

$ℂ_{AL - 1}$ 5 5 3 3

$ℂ_{AL - 2}$ 4 4 2 2

$ℂ_{AL - 3}$ 3 3 1 1

$ℂ_{AL - 4}$ 2 2 5 5

$ℂ_{AL - 5}$ 1 1 4 4

Symbols	Ration Systems	References Points	Multiplicative form	MULTIMOORA Methods
$ℂ_{AL - 1}$	5	5	3	3
$ℂ_{AL - 2}$	4	4	2	2
$ℂ_{AL - 3}$	3	3	1	1
$ℂ_{AL - 4}$	2	2	5	5
$ℂ_{AL - 5}$	1	1	4	4

By using different sorts of rules, we get distinct results such as the best optimal are two by using three different rules, such are 5 and 3. Moreover, to determine the consistency and validity of the elaborated operators by using the information in Table 1, resolved it by using the elaborated and existing operators to improve the quality of the research works.

By using the information of Example 5.2, by choosing the imaginary parts will be zero, then the information of Table 1 is reduced into the information in Table 5. Then by using the procedure of the MULTIMOORA method, the result is discussed in the form of Table 5.

Table 5

By using dominance theory expressions of the MULTIMOORA methods

Symbols	Ration Systems	References Points	Multiplicative form	MULTIMOORA Methods
$ℂ_{AL - 1}$	5	5	3	3
$ℂ_{AL - 2}$	4	4	2	2
$ℂ_{AL - 3}$	3	3	1	1
$ℂ_{AL - 4}$	2	2	5	5
$ℂ_{AL - 5}$	1	1	4	4

Still the results are the same in Table 5, by choosing the values of imaginary parts will be zero. By using different sorts of rules, we get distinct results such as the best optimal are two by using three different rules, such are 5 and 3. Moreover, to determine the consistency and validity of the elaborated operators by using the information in Table 1 (with and without imaginary parts), resolved it by using the elaborated and existing operators to improve the quality of the research works.

5.3 Comparative analysis

Certain scholars have utilized different sorts of theories in the environment of MADM to determine the reliability and consistency of the elaborated approaches. In this study, we elaborated the principle of complex spherical fuzzy uncertain linguistic information and by using these ideas, we applied in the environment of Muirhead mean operators and extended the MULTIMOORA method. Further, to determine the consistency of the explored operators based on the MULTIMOORA method with some prevailing ideas by using the information in Table 1 (with and without imaginary parts). The information’s above prevailing operators are discussed as follows: Liu and Liu [37] developed the intuitionistic uncertain linguistic partitioned Bonferroni mean operators, Lu and Wei [38] proposed the Pythagorean uncertain linguistic aggregation operators, and Garg and Rani [24] developed the aggregation operators for CIFSs. By using the information of Table 1, the comparative analysis of the explored and prevailing operators is discussed in the form of Table 6. Their geomatical expression of the information in Table 6, is discussed in the form of Fig. 5.

Table 6
Comparative analysis of the explored and prevailing operators by using the information of Table 1 (with imaginary parts)

Methods Score Values Ranking Values

Liu and Liu [37] Cannot be Resolved Cannot be Resolved

Lu and Wei [38] Cannot be Resolved Cannot be Resolved

Garg and Rani [24] Cannot be Resolved Cannot be Resolved

CSFULWMM operator $ℂ_{CPL - 1}^{RS} = 1, ℂ_{CPL - 2}^{RS} = 1.72017,$ $ℂ_{CPL - 3}^{RS} = 2.5421, ℂ_{CPL - 4}^{RS} = 3.0257, ℂ_{CPL - 5}^{RS} = 3.0257$ $ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$

CSFULDWMM operator $ℂ_{CPL - 1}^{RP} = 0.3104, ℂ_{CPL - 2}^{RP} = 0.522,$ $ℂ_{CPL - 3}^{RP} = 0.8323, ℂ_{CPL - 4}^{RP} = 0.889, ℂ_{CPL - 5}^{RP} = 1$ $ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$

Multiplicative form $ℂ_{CPL - 1}^{MF} = 0.2672, ℂ_{CPL - 2}^{MF} = 0.4334,$ $ℂ_{CPL - 3}^{MF} = 0.6644, ℂ_{CPL - 4}^{MF} = 0, ℂ_{CPL - 5}^{MF} = 0.2672$ $ℂ_{CPL - 3}^{MF} ⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$ $⩾ ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF}$

Methods	Score Values	Ranking Values
Liu and Liu [37]	Cannot be Resolved	Cannot be Resolved
Lu and Wei [38]	Cannot be Resolved	Cannot be Resolved
Garg and Rani [24]	Cannot be Resolved	Cannot be Resolved
CSFULWMM operator	$ℂ_{CPL - 1}^{RS} = 1, ℂ_{CPL - 2}^{RS} = 1.72017,$ $ℂ_{CPL - 3}^{RS} = 2.5421, ℂ_{CPL - 4}^{RS} = 3.0257, ℂ_{CPL - 5}^{RS} = 3.0257$	$ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$
CSFULDWMM operator	$ℂ_{CPL - 1}^{RP} = 0.3104, ℂ_{CPL - 2}^{RP} = 0.522,$ $ℂ_{CPL - 3}^{RP} = 0.8323, ℂ_{CPL - 4}^{RP} = 0.889, ℂ_{CPL - 5}^{RP} = 1$	$ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$
Multiplicative form	$ℂ_{CPL - 1}^{MF} = 0.2672, ℂ_{CPL - 2}^{MF} = 0.4334,$ $ℂ_{CPL - 3}^{MF} = 0.6644, ℂ_{CPL - 4}^{MF} = 0, ℂ_{CPL - 5}^{MF} = 0.2672$	$ℂ_{CPL - 3}^{MF} ⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$ $⩾ ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF}$

Fig. 5

Geometrical expressions of the information of the Table 6.

Fig. 6

Geometrical expressions of the information of the Table 7.

As shown in Table 6, we know that the prevailing operators [37, 38] are not able to resolve the information in Table 1. Because Table 1, contains CSFUL information, and the prevailing operators are elaborated based on picture, spherical, and complex picture fuzzy uncertain linguistic sets.

Their geometrical expressions of the information in Table 7 are discussed in the shape of Fig. 6.

Table 7

Comparative analysis of the explored and prevailing operators by using the information of Table 1 (without imaginary parts)

Methods	Score Values	Ranking Values
Liu and Liu [37]	Cannot be Resolved	Cannot be Resolved
Lu and Wei [38]	$ℂ_{CPL - 1}^{RS} = 1.1241, ℂ_{CPL - 2}^{RS} = 1.83127,$ $ℂ_{CPL - 3}^{RS} = 2.6531, ℂ_{CPL - 4}^{RS} = 3.1368, ℂ_{CPL - 5}^{RS} = 3.1367$	$ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$
Garg and Rani [24]	Cannot be Resolved	Cannot be Resolved
CSFULWMM operator	$ℂ_{CPL - 1}^{RS} = 1.2351, ℂ_{CPL - 2}^{RS} = 1.94227,$ $ℂ_{CPL - 3}^{RS} = 2.7641, ℂ_{CPL - 4}^{RS} = 3.2479, ℂ_{CPL - 5}^{RS} = 3.2478$	$ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$
CSFULDWMM operator	$ℂ_{CPL - 1}^{RP} = 0.5326, ℂ_{CPL - 2}^{RP} = 0.7441,$ $ℂ_{CPL - 3}^{RP} = 0.9545, ℂ_{CPL - 4}^{RP} = 0.99400, ℂ_{CPL - 5}^{RP} = 1.2532$	$ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF} ⩾ ℂ_{CPL - 3}^{MF}$ $⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$
Multiplicative form	$ℂ_{CPL - 1}^{MF} = 0.4894, ℂ_{CPL - 2}^{MF} = 0.6556,$ $ℂ_{CPL - 3}^{MF} = 0.8866, ℂ_{CPL - 4}^{MF} = 0, ℂ_{CPL - 5}^{MF} = 0.4894$	$ℂ_{CPL - 3}^{MF} ⩾ ℂ_{CPL - 2}^{MF} ⩾ ℂ_{CPL - 1}^{MF}$ $⩾ ℂ_{CPL - 5}^{MF} ⩾ ℂ_{CPL - 4}^{MF}$

If we choose the CSFUL information, then the prevailing operators [37, 38] are not able to cope with it. But, if we choose prevailing types of information

such as pictures, spherical, and complex picture fuzzy uncertain linguistic sets, then the elaborated operators based on CSFULSs can resolve it. Further, to show the consistency of the proposed operators, we choose the information of Table 1 (without imaginary parts) resolved by using proposed and prevailing operators in the form of Table 7. The comparative analysis of the explored and prevailing operators by using the information of Table 1 (without imaginary parts) is discussed in the form of Table 7.

As shown above, by using the prevailing sorts of information, some prevailing types of operators [38] are also resolved. If we choose the CSFUL information, then the prevailing operators [38] are not able to cope with it. But, if we choose prevailing types of information such as pictures, spherical, and complex picture fuzzy uncertain linguistic sets, then the elaborated operators based on CSFULSs can resolve it. Therefore, the complex spherical fuzzy uncertain linguistic information is more useful and more generalized than prevailing operators [24 , 38] to determine the consistency and validity of the elaborated operators to cope with awkward information in genuine life troubles.

6 Conclusion

The goal of this study is to combine the principle of CSFS and ULS to elaborate the principle of CSFULS. CSFULS covers the grade of truth, abstinence, falsity, and their uncertain linguistic terms to accomplish with awkward and intricate data in actual life dilemmas. Some operational laws by using the elaborated CSFULSs are also settled. Furthermore, MM operators offer a flexible arrangement with its modifiable factors because of Muirhead’s general structure. On the other hand, MM aggregation operators perform a significant role in conveying the magnitude level of options and characteristics. By using the MM aggregation operators based on CSFULS to elaborate the CSFULMM, CSFULWMM, CSFULDMM, CSFULDWMM operators, and their important results are also elaborated with the help of some remarkable cases. Additionally, by using the elaborated operators based on CSFULS, a MADM dilemma. To determine the rationality and reliability of the elaborated operators, some numerical examples are illustrated. Finally, the supremacy and comparative analysis of the elaborated approaches with the help of graphical expressions are also developed.

In the future, we will extend these ideas into complex q-rung orthopair fuzzy sets [39 –41], T-spherical fuzzy sets [42, 43] and [44 –50] to improve the quality of the research works.

Footnotes

Acknowledgments

This work is supported by the Social Sciences Planning Projects of Zhejiang (21QNYC11ZD), Major Fundamental Research Funds for the Provincial Universities of Zhejiang (SJWZ2020002), Longyuan Construction Financial Research Project of Ningbo University (LYYB2002), Statistical Scientific Key Research Project of China (2021LZ33), Statistical Scientific Key Research Project of Zhejiang (21TJZZ25), Statistical Scientifififific Key Research Project of China and the First Class Discipline of Zhejiang-A (Zhejiang Gongshang University - Statistics).

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets And Systems 20(1) (1986), 87–96.

Atanassov

K.T.

, Interval valued intuitionistic fuzzy sets, In Intuitionistic Fuzzy Sets (pp. 139–177). Physica, Heidelberg. (1999).

Xia

, Xu

and Zhu

, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012), 78–88.

Garg

and Kumar

, A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory, Artificial Intelligence Review 53(1) (2020), 595–624.

Garg

, A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems, Applied Soft Computing 38 (2016), 988–999.

Yager

R.R.

, Pythagorean fuzzy subsets. In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (2013), (pp. 57–61). IEEE.

Luo

D.D.

, Zeng

S.Z.

, Zhang

C.H.

and Yu

G.S.

, Pythagorean fuzzy investment multiple attribute decision making method based on combined aggregation method, Journal of Intelligent & Fuzzy Systems 39(1) (2020), 949–959.

Garg

, Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process, International Journal of Intelligent Systems 33(6) (2018), 1234–1263.

10.

Z.M.

, Zeng

S.Z.

and Wang

P.Y.

, Novel approach to multi-attribute group decision-making based on interval-valued Pythagorean fuzzy power Maclaurin symmetric mean operator, Computers & Industrial Engineering 155 (2021), 107049.

11.

Garg

, Some methods for strategic decision-making problems with immediate probabilities in Pythagorean fuzzy environment, International Journal of Intelligent Systems 33(4) (2018), 687–712.

12.

Cuong

B.C.

, Picture fuzzy sets, Journal of Computer Science and Cybernetics 30(4) (2001), 409–420.

13.

, Das

and Kar

, An approach to rank picture fuzzy numbers for decision making problems, Decision Making: Applications in Management and Engineering 2(2) (2019), 54–64.

14.

Ullah

, Ali

, Jan

, Mahmood

and Maqsood

, Multi-attribute decision making based on averaging aggregation operators for picture hesitant fuzzy sets, Technical Journal 23(04) (2018), 84–95.

15.

Jan

, Ali

, Ullah

and Mahmood

, Some generalized distance and similarity measures for picture hesitant fuzzy sets and their applications in building material recognition and multi-attribute decision making, Punjab Univ J Math 51(7) (2019), 51–70.

16.

Mahmood

, Ullah

, Khan

and Jan

, An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets, Neural Computing and Applications 31(11) (2019), 7041–7053.

17.

Ullah

, Garg

, Mahmood

, Jan

and Ali

, Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making, Soft Computing 24(3) (2020), 1647–1659.

18.

Mahmood

, Ilyas

, Ali

and Gumaei

, Spherical Fuzzy Sets-Based Cosine Similarity and Information Measures for Pattern Recognition and Medical Diagnosis, IEEE Access 9 (2021), 25835–25842.

19.

Ramot

, Milo

, Friedman

and Kandel

, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems 10(2) (2002), 171–186.

20.

Alkouri

A.M.D.J.S.

and Salleh

A.R.

, Complex intuitionistic fuzzy sets. In AIP Conference Proceedings (Vol. 1482, No. 1, pp. 464–470). American Institute of Physics. (2012, September).

21.

Garg

and Rani

, Complex interval-valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae 164(1) (2019), 61–101.

22.

Rani

and Garg

, Distance measures between the complex intuitionistic fuzzy sets and their applications to the decision-making process, International Journal for Uncertainty Quantification 7(5) (2017).

23.

Garg

and Rani

, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Applied Intelligence 49(2) (2019), 496–512.

24.

Garg

and Rani

, Novel aggregation operators and ranking method for complex intuitionistic fuzzy sets and their applications to decision-making process, Artificial Intelligence Review (2019), 1–26.

25.

Garg

and Rani

, Complex interval-valued intuitionistic fuzzy sets and their aggregation operators, Fundamenta Informaticae 164(1) (2019), 61–101.

26.

Ullah

, Mahmood

, Ali

and Jan

, On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Complex & Intelligent Systems 6(1) (2020), 15–27.

27.

Akram

and Naz

, A novel decision-making approach under complex Pythagorean fuzzy environment, Mathematical and Computational Applications 24(3) (2019), 73.

28.

Akram

, Bashir

and Garg

, Decision-making model under complex picture fuzzy Hamacher aggregation operators, Computational and Applied Mathematics 39(3) (2020), 1–38.

29.

Ali

, Mahmood

and Yang

M.S.

, TOPSIS Method Based on Complex Spherical Fuzzy Sets with Bonferroni Mean Operators, Mathematics 8(10) (2020), 1739.

30.

Ali

, Mahmood

and Yang

M.S.

, Complex T-spherical fuzzy aggregation operators with application to multi-attribute decision making, Symmetry 12(8) (2020), 1311.

31.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning—I, Information Sciences 8(3) (1975), 199–249.

32.

, Uncertain linguistic aggregation operators-based approach to multiple attribute group decision making under uncertain linguistic environment, Information Sciences 168(1–4) (2004), 171–184.

33.

, Induced uncertain linguistic OWA operators applied to group decision making, Information Fusion 7(2) (2006), 231–238.

34.

Zavadskas

E.K.

, Antucheviciene

, Razavi Hajiagha

S.H.

and Hashemi

S.S.

, The interval-valued intuitionistic fuzzy MULTIMOORA method for group decision making in engineering, Mathematical Problems in Engineering, 2015. (2015).

35.

Z.H.

, An extension of the MULTIMOORA method for multiple criteria group decision making based upon hesitant fuzzy sets, Journal of Applied Mathematics 2014. (2014).

36.

Qin

, Liu

, Hong

and Jia

, Hesitant fuzzy muirhead mean operators and its application to multiple attribute decision making, Int J Innov Comput Inf Control 14(4) (2018), 1223–1238.

37.

Liu

and Liu

, Intuitionistic uncertain linguistic partitioned Bonferroni means and their application to multiple attribute decision-making, International Journal of Systems Science 48(5) (2017), 1092–1105.

38.

and Wei

G.W.

, Pythagorean uncertain linguistic aggregation operators for multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 21(3) (2017), 165–179.

39.

Ali

and Mahmood

, Maclaurin symmetric mean operators and their applications in the environment of complex q-rung orthopair fuzzy sets, Computational and Applied Mathematics 39 (2020), 1–27.

40.

Mahmood

and Ali

, Entropy measure and TOPSIS method based on correlation coefficient using complex q-rung orthopair fuzzy information and its application to multi-attribute decision making, Soft Computing 25(2) (2021), 1249–1275.

41.

Zeng

S.Z.

, Hu

Y.J.

and Xie

X.Y.

, Q-rung orthopair fuzzy weighted induced logarithmic distance measures and their application in multiple attribute decision making, Engineering Applications of Artificial Intelligence 100 (2021), 104167.

42.

Liu

, Zhu

and Wang

, A multi-attribute decision-making approach based on spherical fuzzy sets for Yunnan Baiyao’s R&D project selection problem, International Journal of Fuzzy Systems 21(7) (2019), 2168–2191.

43.

Khan

M.J.

, Kumam

, Deebani

, Kumam

and Shah

, Distance and similarity measures for spherical fuzzy sets and their applications in selecting mega projects, Mathematics 8(4) (2020), 519.

44.

Mahmood

, A Novel Approach toward Bipolar Soft Sets and Their Applications, Journal of Mathematics, (ISSN: 2314-4629) Volume 2020 (2020) Article ID 4690808. (2020).

45.

Gündoğdu

F.K.

and Kahraman

, A novel fuzzy TOPSIS method using emerging interval-valued spherical fuzzy sets, Engineering Applications of Artificial Intelligence 85 (2019), 307–323.

46.

Naeem

, Riaz

and Karaaslan

, Some novel features of Pythagorean m-polar fuzzy sets with applications, Complex & Intelligent Systems 7(1) (2021), 459–475.

47.

Pérez-Dominguez

, Durán

S.N.A.

, López

R.R.

, Pérez-Olguin

I.J.C.

, Luviano-Cruz

and Gómez

J.A. H.

, Assessment urban transport service and Pythagorean Fuzzy Sets CODAS method: Acase of study of Ciudad Juárez, Sustainability 13(3) (2021), 1281.

48.

Büyüközkan

, Göçer

and Uztürk

, A novel Pythagorean fuzzy set integrated Choquet integral approach for vertical farming technology assessment, Computers & Industrial Engineering 158 (2021), 107384.

49.

Dinakaran

, Analyzing Online Food Delivery Industries using Pythagorean Fuzzy Relation and Composition, International Journal of Hospitality & Tourism Systems 14(2) (2021).

50.

Tang

and Yang

, Sustainable e-bike sharing recycling supplier selection: An interval-valued Pythagorean fuzzy MAGDM method based on preference information technology, Journal of Cleaner Production 287 (2021), 125530.

A Multi-MOORA decision making method based on Muirhead mean operators and complex spherical fuzzy uncertain linguistic setting

Abstract

Keywords

1 Introduction

5.1 MULTIMOORA-CSFUL based on muirhead mean operators

5.1.1 Method 1 based on CSFULWMM operator:

5.1.2 Method 2 based on CSFULDWMM operator:

5.1.3 Method 3 based on similarity measures:

5.2 Numerical illustration

5.2.1 Method 1 based on CSFULWMM operator:

Table 4 By using dominance theory expressions of the MULTIMOORA methods Symbols Ration Systems References Points Multiplicative form MULTIMOORA Methods ℂ AL - 1 5 5 3 3 ℂ AL - 2 4 4 2 2 ℂ AL - 3 3 3 1 1 ℂ AL - 4 2 2 5 5 ℂ AL - 5 1 1 4 4

Footnotes

Acknowledgments

References

Table 4
By using dominance theory expressions of the MULTIMOORA methods

Symbols Ration Systems References Points Multiplicative form MULTIMOORA Methods

$ℂ_{AL - 1}$ 5 5 3 3

$ℂ_{AL - 2}$ 4 4 2 2

$ℂ_{AL - 3}$ 3 3 1 1

$ℂ_{AL - 4}$ 2 2 5 5

$ℂ_{AL - 5}$ 1 1 4 4