Novel Complex T-Spherical Fuzzy 2-Tuple Linguistic Muirhead Mean Aggregation Operators and Their Application to Multi-Attribute Decision-Making

Peide Liu; Zeeshan Ali; Tahir Mahmood

doi:10.2991/ijcis.d.201207.003

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 14, Issue 1, 2021, Pages 295 - 331

Novel Complex T-Spherical Fuzzy 2-Tuple Linguistic Muirhead Mean Aggregation Operators and Their Application to Multi-Attribute Decision-Making

Authors

Peide Liu¹^{, *}^,, Zeeshan Ali², Tahir Mahmood²^,

¹School of Management Science and Engineering, Shandong University of Finance and Economics, Shandong Province, China

²Department of Mathematics & Statistics, International Islamic University, Islamabad, Pakistan

^*Corresponding author. Email: peide.liu@gmail.com

Corresponding Author

Peide Liu

Received 14 October 2020, Accepted 30 November 2020, Available Online 17 December 2020.

DOI: 10.2991/ijcis.d.201207.003 How to use a DOI?
Keywords: Complex T-spherical fuzzy 2-tuple linguistic sets; Muirhead mean operators; Dual Muirhead mean operators; Multi-attribute decision making
Abstract: Complex T-spherical fuzzy 2-tuple linguistic set (CTSF2-TLS), which is a combination of complex fuzzy set (CFS), T-spherical fuzzy set (TSFS), and 2-tulpe linguistic variable set (2-TLVS), is a proficient technique to express uncertain and awkward information in real decision-making. CTSF2-TLS contains 2-tuple linguistic variable, truth, abstinence, and falsity grades, which gives an extensive freedom for decision-makers to express the uncertain information compared to other existing notions. In this article, we firstly propose a new concept of CTSF2-TLS by using the CFS, TSFS, and 2-TLVS, and the operational laws and comparison methods for CTSF2-TLSs are established, then the complex T-spherical fuzzy 2-tuple linguistic Muirhead mean (CTSF2-TLMM) operator and the complex T-spherical fuzzy 2-tuple linguistic dual Muirhead mean (CTSF2-TLDMM) operator are explored, and some special cases and the desirable properties of the explored operators are also studied. Moreover, we establish a method to solve the multi-attribute decision-making (MADM) problems, in which the evaluation information is described by CTSF2-TLSs. Finally, we use some numerical examples to explain the advantages of the explored method by comparing with other existing methods.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Decision-making is a significant activity for picking the ideal option from a set of schemes based on the data acquired from specialists. We are confronted with numerous decision-making issues in different life circumstances [1], vulnerability and ambiguity should be considered. For adapting such sorts of issues, the theory of fuzzy set (FS) was presented by Zadeh [2] which can express the fuzzy information by the truth grade. But main problem of the FS is that it deals only with truth grade. Further, Atanassov [3] propounded the intuitionistic FS (IFS), which contains the truth and falsity grades, whose sum is belonging to the unit interval. Under this theory, a lot of scholars has explored their theories [4]. However, when a decision-maker (DM) gives the information in the form of pair (0.6,0.5) for truth and falsity grades, the condition of IFS is exceeded from unit interval. For coping with this kind of information, Yager [5] modified the IFS, and proposed the Pythagorean FS (PFS) with a condition that the sum of squares of the truth grade and falsity grade is not exceeded from unit interval. PFS has received more attention [6]. But there was still a problem, when a DM gives the information in the form of pair (0.9, 0.8) for truth and falsity grades, then the condition of IFS and the condition of PFS are exceeded from unit interval. For coping this kind of information, Yager [7] again modified the PFS, and explored the q-rung orthopair FS (q-ROFS) with a condition that the sum of q-powers of the truth grade and falsity grade is not exceeded from unit interval. Because there is an extensive information expression, since it was set up, it has been used in some fields of aggregation operators, similarity measures (SM), hybrid aggregation operators, and so on [8,9].

In real decision-making, DMs may offer their feelings with more responses for a decision index, such as positive, negative, neutral, and abstinence. In order to express this information, Cuong [10] investigated the picture FS (PFS) which contains positive, abstinence, and negative grades, whose sum is bounded to [0, 1]. However, in some cases, for a DM, it is very difficult to face some limitations. So, the spherical FS (SFS) was established by Mahmood et al. [11], which is more powerful compared to IFS and PFS. However, the constraint of SFS is that the sum of squares of positive, abstinence, and negative grades belongs to [0, 1]. When a DM gives 0.9 for positive grade, 0.85 for abstinence grade, and 0.8 for negative grade, then the PFS and SFS are not able to cope it. So, the idea of T-spherical fuzzy set (TSFS) was established by Mahmood et al. [11], in which the sum of q-powers of positive, abstinence, and negative grades belongs to [0, 1]. The TSFS is more powerful compared to PFS and SFS, and since it was established, it has received the attention of many researchers and it is utilized in the environment of aggregation operators, SMs, hybrid aggregation operators, and so on. The various existing works based on PFS, SFS, and TSFS are elaborated as follow as:

Operators based Approaches. Many scholars have successfully discussed the aggregation operators in the environment of PFS, SFS, TSFS. For instance, Ullah et al. [12] evaluated investment policy based on interval-valued T-SFS. Ullah et al. [13] explored geometric aggregation operator based on T-SFS. Guleria and Bajaj [14] explored the novel of T-spherical fuzzy soft set with their aggregation operators. Quek et al. [15] presented the generalized T-spherical fuzzy weighted aggregation operators. Ullah et al. [16] proposed the notion of the averaging aggregation operators based on T-SFS. Ullah et al. [17] presented the evaluation of investment performance based on T-spherical fuzzy Hamacher aggregation operators.
Measures based Approaches. SM is a proficient technique to accurately examine the degree between any two objects. Many scholars have developed some SMs for the different FSs. For example, Ullah et al. [18] explored the correlation co-efficient based on T-SFS and their application in medical diagnosis. The SMsbased on T-SFS was presented by Ullah et al. [19].
Hybrid Operators based Approaches. To find the interrelationships between two objects, the hybrid aggregation operators play an essential role in the environment of realistic decision-making. Some scholars explored different hybrid aggregation operators using the T-SFSs. For example, Liu et al. [20] explored the power Muirhead mean (MM) operators, Munir et al. [21] explored the Einstein hybrid aggregation operators, and Zeng et al. [22] presented the Probabilistic interactive aggregation operators based on T-SFSs.

The above studies are based on FS, IFS, PFS, QROFS, and all parts of them are expressed by real numbers. In order to process the complex situations, Ramot et al. [23] proposed complex fuzzy set (CFS), which contains complex-valued truth grade in the form of polar coordinates belonging to unit disc in a complex plane. But in some situations, the CFS cannot process the complicated and awkward information effectively. To resolve this kind of problems, Alkouri and Salleh [24] developed the complex IFS (CIFS), characterized by the complex-valued truth grade and complex-valued falsity grade with a condition that the sum of the real parts (also for the imaginary part) of the truth and falsity grades is bounded to the unit interval. Now, CIFS has received extensively attractions. For instance, Garg and Rani [25] established a new generalized Bonferroni mean aggregation operators for CIFS based on Archimedean t-norm and t-conorm. Rani and Garg [26] developed the distance measures for CIFSs and applied them to multi-attribute decision making (MADM) Process. When a DM gives the information in the form of pair 0.6ei2π0.6,0.5ei2π0.5 for complex-valued truth and complex-valued falsity grades, the conditions of CIFS is exceeded from unit interval. For coping these kinds of problems, Ullah et al. [27] modified the CIFS to explore the complex PFS (CPFS) with conditions that the sum of squares of the real part (also for the imaginary part) of the truth grade and the real part (also for the imaginary part) of the falsity grade is not exceeded from unit interval. CPFS has received more attention [28]. But there was still a problem, when a DM gives the information in the form of pair 0.9ei2π0.9,0.8ei2π0.8 for truth and falsity grades, the conditions of CIFS and the conditions of CPFS are exceeded from unit interval. For coping these kinds of problems, Liu et al. [29,30] modified the CPFS to explore the complex q-ROFS (Cq-ROFS) with a condition that the sum of q-powers of the real part (also for imaginary part) of the truth grade and the real part (also for imaginary part) of the falsity grade are not exceeded from unit interval, and now Cq-ROFS has received more attention [31].

Linguistic variable (LV) proposed by Zadeh [32] is an important tool to express the qualitative information, and many specialists have investigated the linguistic MADM issues, including linguistic approximation [33], uncertain LV [34], and linguistic 2-tuple information [35], and so on. In some practical problems, the single linguistic term set cannot be involved in those cases which contains two terms like truth and falsity grades. For dealing with such kinds of problems, Wang and Li [36] established the intuitionistic linguistic number (ILN), which contains a linguistic term, truth grade, and falsity grade. The ILN is more powerful idea to cope with uncertainty and vagueness. Further, Liu and Chen [37] explored the intuitionistic 2-tuple linguistic terms set, Peng and Yang [38] established Pythagorean fuzzy linguistic set, Wei et al. [39] explored the Pythagorean 2-tuple linguistic aggregation operators, Li et al. [40] proposed q-rung orthopair linguistic Heronian mean operators, Ju et al. [41] explored the q-rung orthopair fuzzy 2-tuple linguistic MM operators.

The interrelationship among the different attributes in real decision-making is ever-present. Muirhead [42] explored the MM operator, as an effective method to evaluate perfectly the interrelationship among the attributes, then its extensions for the different FSs were made. For instance, Liu and Li [43] explored the intuitionistic fuzzy MM operators. Zhu and Li [44] presented the Pythagorean fuzzy MM operators, Wang et al. [45] established q-rung orthopair fuzzy MM operators. On addition, there are some general aggregation operators for q-ROFS, such as the averaging aggregation operator based on q-ROFS [8], Partitioned Bonferroni mean operator (BMO) based on q-ROFS [46], and Maclaurin symmetric mean operator (MSMO) based on q-ROFS [47]. These operators are useful in genuine choice hypothesis. But, the two-dimensional information in a single set cannot be discussed in IFS, PyFS, q-ROFS, PFS, and TSFS, and only is discussed in the environment of CIFS, CPyFS, and Cq-ROFS, but these notions cannot contain the neutral grade, which is used in many real-life scenarios. For example, when a DM provides sS(LT−3),0.03,0.5ei2π0.3,0.4ei2π0.5,0.3ei2π0.4, for linguistic term, truth, abstinence, and falsity grades, the existing notions cannot deal with such kind of problems. So, we need to propose a new concept, that is, complex spherical fuzzy 2-tuple linguistic set. We can see that 0.52+0.42+0.32=0.25+0.16+0.09=0.49≤1, and 0.32+0.52+0.42=0.09+0.25+0.16=0.49≤1. But, there is one other complication, when a DM gives sS(LT−3),0.03,0.9ei2π0.8,0.8ei2π0.7,0.7ei2π0.6, for linguistic term, truth, abstinence, and falsity grades, the existing notions cannot deal with such kind of problems. For addressing with sun kind of issues, we explore the idea of complex T-spherical fuzzy 2-tuple linguistic sets (CTSF2-TLSs) with a condition that the sum of q-powers of the real parts of the truth, abstinence, and falsity grades is not exceeded form unit interval. So, for q=7, the above problem is solved effectively. Considering the intricacy in the real circumstances and maintaining the benefits of the MM operators and CTSF2-TLSs, the goals of this research are shown as follows.

To investigate the novel concept of CTSF2-TLSs and furthermore depict their operational laws.
To develop the MM operator and dual MM (DMM) operator based on the CTSF2-TLSs, and discuss some properties and special cases.
To investigate the MADM method utilizing the complex T-spherical fuzzy 2-tuple linguistic Muirhead mean (CTSF2-TLMM) operator and complex T-spherical fuzzy 2-tuple linguistic dual Muirhead mean (CTSF2-TLDMM) operator.
To show the advantages of the proposed method by some examples.

So we give the following structure. In Section 2, we review some concepts of FSs, CFSs, q-ROFSs, Tuple linguistic function (2-TLFs), inverse 2-TLFs, q-ROF2-TLSs and their operational laws. Further, the MM operator and DMM operator are also discussed. In Section 3, the notion of CTSF2-TLS using CFS, TSFS, and 2-tulpe linguistic variable set (2-TLVS) is defined. In Section 4, based on the established operational laws and comparison methods for CTSF2-TLSs, the CTSF2-TLMM aggregation operator and CTSF2-TLDMM aggregation operator are explored. Some special cases and the desirable properties are also studied. In Section 5, we establish a method to solve the multi-attribute group decision-making (MAGDM) problems, in which the evaluation information is expressed as CTSF2-TLSs. Finally, some numerical examples are given to explain the effectiveness and superiority of the explored method by comparing with other methods. The conclusion of this paper is discussed in Section 6.

2. PRELIMINARIES

In this part, we concisely review some useful notions of 2-TLF [35], inverse 2-TLF [35], TSFS [11] and their operational laws. Further, the MM operator and DMM operator are also discussed. The symbols UUNI,μ,ξ and η are represented the universal, the grade of truth, the grade of abstinence, and the grade of falsity. Where aSC,δSC,qSC≥1.

Definition 1.

[35] For a linguistic term set SLT=sSLT−0,sSLT−1,…,sSLT−g with βSC∈0,1, the 2-tuple linguistic function ΔLT is given by:

ΔLT:0,1→SSLT−j×−12g,12g(1)

ΔLTβSC=sSLT−j,aSC with sSLT−jj=roundβSC,gaSC=βSC−jgaSC∈−12g,12g(2)

The 2-tuple linguistic inverse function ΔLT−1 is given by:

ΔLT−1:SSLT−j×−12g,12g→0,1(3)

ΔLT−1sSLT−j,aSC=jg+aSC=βSC(4)

Definition 2.

[11] A TSFS is given by

ℑTS=(μℑTS(u),ξℑTS(u),ηℑTS(u)):u∈UUNI(5)

where μℑTS:UUNI→[0,1],ξℑTS:UUNI→[0,1] and ηℑTS:UUNI→[0,1], with a condition: 0≤μℑTSqSC(u)+ξℑTSqSC(u)+ηℑTSqSC(u)≤1. Moreover, ζℑTS(u)=1−μℑTSqSC(u)+ξℑTSqSC(u)+ηℑTSqSC(u)1qSC is called refusal grade, the T-spherical fuzzy number (TSFN) is represented by ℑTS=μℑTS(u),ξℑTS(u),ηℑTS(u).

Definition 3.

[11] For any two TSFNs ℑTS−1=μℑTS−1(u),ξℑTS−1(u),ηℑTS−1(u) and ℑTS−2=μℑTS−2(u),ξℑTS−2(u),ηℑTS−2(u), then

ℑTS−1⊕TSℑTS−2=μℑTS−1qSC(u)+μℑTS−2qSC(u)−μℑTS−1qSC(u)μℑTS−2qSC(u)1qSC,ξℑTS−1(u)ξℑTS−2(u),ηℑTS−1(u)ηℑTS−2(u);
ℑTS−1⊗TSℑTS−2=μℑTS−1(u)μℑTS−2(u),ξℑTS−1qSC(u)+ξℑTS−2qSC(u)−ξℑTS−1qSC(u)ξℑTS−2qSC(u)1qSC,ηℑTS−1qSC(u)+ηℑTS−2qSC(u)−ηℑTS−1qSC(u)ηℑTS−2qSC(u)1qSC;
ℑTS−1δSC=μℑTS−1δSC(u),1−1−ξℑTS−1qSC(u)δSC1qSC,1−1−ηℑTS−1qSC(u)δSC1qSC;
δSCℑTS−1=1−1−μℑTS−1qSC(u)δSC1qSC,ξℑTS−1δSC(u),ηℑTS−1δSC(u).

Definition 4.

[11] For any two TSFNs ℑTS−1=μℑTS−1(u),ξℑTS−1(u),ηℑTS−1(u) and ℑTS−2=μℑTS−2(u),ξℑTS−2(u),ηℑTS−2(u), the score and accuracy function are given by:

SℑTS−1=μℑTSqSC(u)−ξℑTSqSC(u)−ηℑTSqSC(u)(6)

HℑTS−1=μℑTSqSC(u)+ξℑTSqSC(u)+ηℑTSqSC(u)(7)

Based on the above two notions, the compassion between two TSFNs is given by:

If SℑTS−1>SℑTS−2, then ℑTS−1≻ℑTS−2;
If SℑTS−1=SℑTS−2, then:
1. If HℑTS−1>HℑTS−2, then ℑTS−1≻ℑTS−2;
2. If HℑTS−1=HℑTS−2, then ℑTS−1=ℑTS−2.

Definition 5.

[42] Choose the family of positive numbers ℑTS−j(j=1,2,3,…,n) with the vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn, the MM operator is given by:

MMtVPℑTS−1,ℑTS−2,…,ℑTS−n=1n!∑ϑ(j)∈sSC−n∏j=1nℑTS−ϑ(j)tVP−j1∑j=1ntVP−j(8)

The DMM is given by:

DMMtVPℑTS−1,ℑTS−2,…,ℑTS−n=1∑j=1ntVP−j∏ϑ(j)∈sSC−n∑j=1ntVP−jℑTS−ϑ(j)1n!(9)

where ϑ(j),(j=1,2,3,…,n) is an n permutation and the set of all permutation of 1 to n is denoted by SSC−n.

2.1. Complex T-Spherical Fuzzy 2-Tuple Linguistic Sets

In this part, we propose the novel concept of CTSF2-TLS, which is the mixture of CFS, TSFS, and 2-TLS. Some important fundamental operational laws of the CTSF2-TLS are also established.

Definition 6.

A CTSF2-TLS is given by:

ℑCTS=sSLT(u),aSC,μℑCTS(u),ξℑCTS(u),ηℑCTS(u):u∈UUNI(10)

where μℑCTS=μℑRPTLei2πWμℑIPTL,ξℑCTS=ξℑRPTLei2πWξℑIPTL and ηℑCTS=ηℑRPTLei2πWηℑIPTL, with a condition: 0≤μℑRPTLqSC(u)+ξℑRPTLqSC(u)+ηℑRPTLqSC(u)≤1,0≤WμℑIPTLqSC(u)+WξℑIPTLqSC(u)+WηℑIPTLqSC(u)≤1 and the pair sSLT(u),aSC is called 2-tulpe linguistic variable with aSC∈−12g,12g and sSLT(u)∈SLT. Moreover, ζℑTS(u)=ζℑRPTLei2πWζℑIPTL=1−μℑRPTLqSC(u)+ξℑRPTLqSC(u)+ηℑRPTLqSC(u)1qSCei2π1−WμℑIPTLqSC(u)+WξℑIPTLqSC(u)+WηℑIPTLqSC(u)1qSC is called refusal grade, the complex T-spherical fuzzy 2-tuple linguistic number (CTSF2-TLN) is represented by:

ℑCTS=sSLT(u),aSC,μℑCTS(u),ξℑCTS(u),ηℑCTS(u) or

ℑCTS=sSLT,aSC,μℑRPTLei2πWμℑIPTL,ξℑRPTLei2πWξℑIPTL,ηℑRPTLei2πWηℑIPTL.

An example of CTSF2-TLS is given as follows:

ℑCTS=sSLT−2,0.01,0.8ei2π(0.8),0.1ei2π(0.1),0.3ei2π(0.3),sSLT−2,0.02,0.18ei2π(0.81),0.11ei2π(0.11),0.31ei2π(0.31),sSLT−3,0.03,0.81ei2π(0.18),0.01ei2π(0.1),0.03ei2π(0.3),sSLT−3,0.04,0.018ei2π(0.081),0.011ei2π(0.11),0.031ei2π(0.31)

Moreover, we can give some special cases. If we ignore the terms of 2-tuple linguistic sets, complex-valued abstinence, and complex-valued nonmembership, then the CTSF2-TLS (Eq. (10)) will be converted to CFS, similarly, if we ignore the terms of 2-tuple linguistic sets and the imaginary parts of the complex-valued membership, complex-valued abstinence, and complex-valued nonmembership, then the CTSF2-TLS (Eq. (10)) will be converted to TSFSs. Finally, if we ignore the complex-valued membership, complex-valued abstinence, and complex-valued nonmembership, then the idea of CTSF2-TLS (Eq. (10)) will be converted to 2-tuple linguistic sets.

Definition 7.

For any two CTSF2-TLNs ℑCTS−1=sSLT−1,aSC−1,μℑRPTL−1ei2πWμℑIPTL−1,ξℑRPTL−1ei2πWξℑIPTL−1,ηℑRPTL−1ei2πWηℑIPTL−1 and ℑCTS−2=sSLT−2,aSC−2,μℑRPTL−2ei2πWμℑIPTL−2,ξℑRPTL−2ei2πWξℑIPTL−2,ηℑRPTL−2ei2πWηℑIPTL−2, then

ℑCTS−1⊕CTSℑCTS−2=△LTΔLT−1sSLT−1,aSC−1+ΔLT−1sSLT−2,aSC−2,μℑRPTL−1qSC+μℑRPTL−2qSC−μℑRPTL−1qSCμℑRPTL−2qSC1qSCei2πWμℑIPTL−1qSC+WμℑIPTL−2qSC−WμℑIPTL−1qSCWμℑIPTL−2qSC1qSC,ξℑRPTL−1ξℑRPTL−2ei2πWξℑIPTL−1WξℑIPTL−2,ηℑRPTL−1ηℑRPTL−2ei2πWηℑIPTL−1WηℑIPTL−2;
ℑCTS−1⊗CTSℑCTS−2=△LTΔLT−1sSLT−1,aSC−1×ΔLT−1sSLT−2,aSC−2,μℑRPTL−1μℑRPTL−2ei2πWμℑIPTL−1WμℑIPTL−2,ξℑRPTL−1qSC+ξℑRPTL−2qSC−ξℑRPTL−1qSCξℑRPTL−2qSC1qSCei2πWξℑIPTL−1qSC+WξℑIPTL−2qSC−WξℑIPTL−1qSCWξℑIPTL−2qSC1qSC,ηℑRPTL−1qSC+ηℑRPTL−2qSC−ηℑRPTL−1qSCηℑRPTL−2qSC1qSCei2πWηℑIPTL−1qSC+WηℑIPTL−2qSC−WηℑIPTL−1qSCWηℑIPTL−2qSC1qSC;
ℑCTS−1δSC=△LTΔLT−1sSLT−1,aSC−1δSC,μℑRPTL−1δSCei2πWμℑIPTL−1δSC,1−1−ξℑRPTL−1qSCδSC1qSCei2π1−1−WξℑIPTL−1qSCδSC1qSC,1−1−ηℑRPTL−1qSCδSC1qSCei2π1−1−WηℑIPTL−1qSCδSC1qSC;
δSCℑCTS−1=△LTδSC×ΔLT−1sSLT−1,aSC−1,1−1−μℑRPTL−1qSCδSC1qSCei2π1−1−WμℑIPTL−1qSCδSC1qSC,ξℑRPTL−1δSCei2πWξℑIPTL−1δSC,ηℑRPTL−1δSCei2πWηℑIPTL−1δSC.

Definition 8.

For any two CTSF2-TLNs ℑCTS−1=sSLT−1,aSC−1,μℑRPTL−1ei2πWμℑIPTL−1,ξℑRPTL−1ei2πWξℑIPTL−1,ηℑRPTL−1ei2πWηℑIPTL−1 and ℑCTS−2=sSLT−2,aSC−2,μℑRPTL−2ei2πWμℑIPTL−2,ξℑRPTL−2ei2πWξℑIPTL−2,ηℑRPTL−2ei2πWηℑIPTL−2, the score and accuracy function are given by:

SℑCTS−1=ΔLT−1sSLT−1,aSC−1×1+μℑTS−1qSC+WμℑIPTL−1qSC−ξℑTS−1qSC−WξℑIPTL−1qSC−ηℑTS−1qSC−WηℑIPTL−1qSC4(11)

HℑCTS−1=ΔLT−1sSLT−1,aSC−1×μℑTS−1qSC+WμℑIPTL−1qSC+ξℑTS−1qSC+WξℑIPTL−1qSC+ηℑTS−1qSC+WηℑIPTL−1qSC4(12)

Based on the above two notions, the compassion between two CTSF2-TLNs is given by:

If SℑCTS−1>SℑCTS−2, then ℑCTS−1≻ℑCTS−2;
If SℑCTS−1=SℑCTS−2, then:
1. If HℑCTS−1>HℑCTS−2, then ℑCTS−1≻ℑCTS−2;
2. If HℑCTS−1=HℑCTS−2, then ℑCTS−1=ℑCTS−2.

Example 1.

For any two CTSF2-TLNs ℑCTS−1=sSLT−2,0.01,0.8ei2π(0.8),0.1ei2π(0.1),0.3ei2π(0.3) and ℑCTS−2=sSLT−4,−0.02,0.9ei2π(0.9),0.1ei2π(0.1),0.2ei2π(0.2), and for qSC=δSC=2. Then

ℑCTS−1⊕CTSℑCTS−2=ΔLTΔLT−1sSLT−2,0.01+ΔLT−1sSLT−4,−0.02,0.82+0.92−0.82×0.92(1/2)ei2π0.82+0.92−0.82×0.92(1/2),(0.1×0.1)ei2π(0.1×0.1),(0.3×0.2)ei2π(0.3×0.2)=ΔLT24+0.01+44−0.02,0.9652ei2π(0.9652),0.01ei2π(0.01),0.06ei2π(0.06)=ΔLT(1.49),0.9652ei2π(0.9652),0.01ei2π(0.01),0.06ei2π(0.06)=(sSLT−4,0.49),0.9652ei2π(0.9652),0.01ei2π(0.01),0.06ei2π(0.06).
ℑCTS−1⊗CTSℑCTS−2=ΔLTΔLT−1sSLT−2,0.01×ΔLT−1sSLT−4,−0.02,(0.8×0.9)ei2π(0.8×0.9),0.12+0.12−0.12×0.12(1/2)ei2π0.12+0.12−0.12×0.12(1/2)0.32+0.22−0.32×0.22(1/2)ei2π0.32+0.22−0.32×0.22(1/2)=ΔLT24+0.01×44−0.02,0.72ei2π(0.72),0.1ei2π(0.1),0.36ei2π(0.36)=ΔLT(0.4998),0.72ei2π(0.72),0.1ei2π(0.1),0.36ei2π(0.36)=(sSLT−3,0.1665),0.72ei2π(0.72),0.1ei2π(0.1),0.36ei2π(0.36).
ℑCTS−12=ΔLTΔLT−1sSLT−2,0.012,0.82ei2π(0.82),1−1−0.122(1/2)ei2π1−1−0.122(1/2),1−1−0.322(1/2)ei2π1−1−0.322(1/2)=ΔLT0.512,0.64ei2π(0.64),1−(1−0.01)2(1/2)ei2π1−(1−0.01)2(1/2),1−(1−0.09)2(1/2)ei2π1−(1−0.09)2(1/2)=ΔLT(0.2601),0.64ei2π(0.64),0.14ei2π(0.14),0.42ei2π(0.42)=sSLT−1,0.0101,0.64ei2π(0.64),0.14ei2π(0.14),0.42ei2π(0.42).
2×ℑCTS−1=ΔLT2×ΔLT−1sSLT−2,0.01,1−1−0.822(1/2)ei2π1−1−0.822(1/2),0.12ei2π(0.1)2,0.32ei2π(0.3)2=ΔLT(2×0.51),1−(1−0.64)2(1/2)ei2π1−(1−0.64)2(1/2),0.01ei2π(0.01),0.09ei2π(0.09)=ΔLT(1.02),0.9330ei2π(0.9330),0.01ei2π(0.01),0.09ei2π(0.09)=sSLT−4,0.02,0.9330ei2π(0.9330),0.01ei2π(0.01),0.09ei2π(0.09).

Further, we examine the interrelationship between two CTSF2-TLNs based on the score functions, such that

SℑCTS−1=ΔLT−1sSLT−2,0.01×1+0.82+0.82−0.12−0.12−0.32−0.32/4=((0.51)×(2.08))/4=0.2652, SℑCTS−2=ΔLT−1sSLT−4,−0.02×1+0.92+0.92−0.12−0.12−0.22−0.22/4=((0.98)×(2.52))/4=0.6174,

So, SℑCTS−2≥SℑCTS−1.

3. MM AGGREGATION OPERATORS FOR CTSF2-TLSs

In this part, we investigate the MM and DMM operators based on a CTSF2-TLS, and they are called CTSF2-TLMM operator and CTSF2-TLDMM operator. Their advantages are that they are more generalized than averaging operator (AO), geometric operator (GO), BMO, and MSMO which are the special cases of the explored operators.

Definition 9.

Choose the family of CTSF2-TLNs ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn, the CTSF2-TLMM operator is given by:

CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=1n!⊕CTSϑ(j)∈sSC−n⊗CTSj=1nℑTS−ϑ(j)tVP−j1∑j=1ntVP−j(13)

where ϑ(j),(j=1,2,3,…,n) is an n permutation and the set of all permutation of 1 to n is denoted by SSC−n.

Based on the operational laws, we establish the following result.

Theorem 1.

Suppose the family of CTSF2-TLNs is ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn. Then the aggregated value of CTSF2-TLNs is again a CTSF2-TLN, and

CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC

Proof:

Using the Definition 7, we have

ℑCTS−ϑ(j)tVP−j=△LTΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j,μℑRPTL−ϑ(j)tVP−jei2πWμℑIPTL−ϑ(j)tVP−j,1−1−ξℑRPTL−ϑ(j)qSCtVP−j1qSCei2π1−1−WξℑIPTL−ϑ(j)qSCtVP−j1qSC,1−1−ηℑRPTL−ϑ(j)qSCtVP−j1qSCei2π1−1−WηℑIPTL−ϑ(j)qSCtVP−j1qSC

⊗CTSj=1nℑCTS−ϑ(j)tVP−j=△LT∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j,∏j=1nμℑRPTL−ϑ(j)tVP−jei2π∏j=1nWμℑIPTL−ϑ(j)tVP−j,1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1qSCei2π1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1qSC,1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1qSCei2π1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1qSC

⊕CTSϑ(j)∈sSC−n⊗CTSj=1nℑCTS−ϑ(j)tVP−j=ΔLT∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1qSCei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1qSC,∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1qSCei2π∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1qSC,∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1qSCei2π∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1qSC

1n!⊕CTSϑ(j)∈sSC−n⊗CTSj=1nℑTS−ϑ(j)tVP−j=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSCei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC,∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1qSC ei2π∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1n!1qSC ei2π∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j

1n!⊕CTSϑ(j)∈sSC−n⊗CTSj=1nℑTS−ϑ(j)tVP−j1∑j=1ntVP−j=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC

Hence, the result is proved.

Example 2.

For any three CTSF2-TLNs ℑCTS−1=sSLT−2,0.01,0.8ei2π(0.8),0.0ei2π(0.0),0.6ei2π(0.6), ℑCTS−2=sSLT−3,0.05,0.7ei2π(0.7),0.1ei2π(0.1),0.5ei2π(0.5), and ℑCTS−3=sSLT−4,−0.02,0.9ei2π(0.9),0.01ei2π(0.01),0.4ei2π(0.4), and their linguistic term set SLT=sSLT−0,sSLT−1,sSLT−2,sSLT−3,sSLT−4 with the value of parameter tVP=(1,2,1) for qSC=2. Then by CTSF2-TLMM operator, we get

ΔLT13!∑ϑ(j)∈SSC−3∏j=13ΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j11+2+1=ΔLT16ΔLT−1sSLT−2,0.011×ΔLT−1sSLT−3,0.052×ΔLT−1sSLT−4,−0.021+ΔLT−1sSLT−2,0.011×ΔLT−1sSLT−4,−0.022×ΔLT−1sSLT−3,0.051+ΔLT−1sSLT−3,0.051×ΔLT−1sSLT−2,0.012×ΔLT−1sSLT−4,−0.021+ΔLT−1sSLT−3,0.051×ΔLT−1sSLT−4,−0.022×ΔLT−1sSLT−2,0.011+ΔLT−1sSLT−4,−0.021×ΔLT−1sSLT−2,0.012×ΔLT−1sSLT−3,0.051+ΔLT−1sSLT−4,−0.021×ΔLT−1sSLT−3,0.052×ΔLT−1sSLT−2,0.01114=ΔLT0.7649=sSLT−3,0.0149;

Further, we calculate the value of the complex-valued truth grade of the CTSF2-TLMM operator, as follows:

1−∏ϑ(j)∈sSC−n1−∏j=13μℑRPTL−ϑ(j)qSCtVP−j161214ei2π1−∏ϑ(j)∈sSC−n1−∏j=13WμℑIPTL−ϑ(j)qSCtVP−j161214=1−1−0.82×1×0.72×2×0.92×1×1−0.82×1×0.92×2×0.72×1×1−0.72×1×0.82×2×0.92×1×1−0.72×1×0.92×2×0.82×1×1−0.92×1×0.72×2×0.82×1×1−0.92×1×0.82×2×0.72×1161214ei2π1−1−0.82×1×0.72×2×0.92×1×1−0.82×1×0.92×2×0.72×1×1−0.72×1×0.82×2×0.92×1×1−0.72×1×0.92×2×0.82×1×1−0.92×1×0.72×2×0.82×1×1−0.92×1×0.82×2×0.72×1161214=0.7983ei2π(0.7983);

Next, we get the value of the complex-valued abstinence grade of the CTSF2-TLMM operator, such that

1−1−∏ϑ(j)∈sSC−n1−∏j=131−ξℑRPTL−ϑ(j)qSCtVP−j13!1412ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j13!1412=1−1−1−1−0.021×1−0.122×1−0.0121×1−1−0.021×1−0.0122×1−0.121×1−1−0.121×1−0.022×1−0.0121×1−1−0.121×1−0.0122×1−0.021×1−1−0.0121×1−0.022×1−0.121×1−1−0.0121×1−0.122×1−0.02113!1412 ei2π1−1−1−1−0.021×1−0.122×1−0.0121×1−1−0.021×1−0.0122×1−0.121×1−1−0.121×1−0.022×1−0.0121×1−1−0.121×1−0.0122×1−0.021×1−1−0.0121×1−0.022×1−0.121×1−1−0.0121×1−0.122×1−0.02113!1412=0.0ei2π(0.0)

In last, we obtain the value of the complex-valued falsity grade of the CTSF2-TLMM operator, such that

1−1−∏ϑ(j)∈sSC−n1−∏j=131−ηℑRPTL−ϑ(j)qSCtVP−j13!1412ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j13!1412

=1−1−1−1−0.621×1−0.522×1−0.421×1−1−0.621×1−0.422×1−0.521×1−1−0.521×1−0.622×1−0.421×1−1−0.521×1−0.422×1−0.621×1−1−0.421×1−0.622×1−0.521×1−1−0.421×1−0.522×1−0.62113!1412ei2π1−1−1−1−0.621×1−0.522×1−0.421×1−1−0.621×1−0.422×1−0.521×1−1−0.521×1−0.622×1−0.421×1−1−0.521×1−0.422×1−0.621×1−1−0.421×1−0.622×1−0.521×1−1−0.421×1−0.522×1−0.62113!1412=0.5095ei2π(0.5095);

then 13!⊕CTSϑ(j)∈sSC−n⊗CTSj=13ℑTS−ϑ(j)tVP−j1(1+2+1)=sSLT−3,0.0149,0.7983ei2π(0.7983),0.0ei2π(0.0),0.5095ei2π(0.5095).

If we set zero to the imaginary parts of the complex-valued truth and falsity grades, then the Example 2 is converted for T-spherical fuzzy 2-tuple linguistic variables, which is a special case of the established operator.

Theorem 2.

Suppose the families of two CTSF2-TLNs are ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) and ℑCTS−∗j=sSLT−∗j,aSC−∗j,μℑRPTL−∗jei2πWμℑIPTL−∗j,ξℑRPTL−∗jei2πWξℑIPTL−∗j,ηℑRPTL−∗jei2πWηℑIPTL−∗j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn. Then the idempotency, monotonicity, and boundedness are shown as follows:

If ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) all are equal if and only if ℑCTS−j=ℑCTS, then

CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ℑCTS
If sSLT−j,aSC−j≥s^SLT−j,aSC−j,μℑRPTL−j≥μℑRPTL−j,WμℑIPTL−j≥WμℑIPTL−j,ξℑRPTL−j≤ξℑRPTL−j,WξℑIPTL−j≤WξℑIPTL−j, and

ηℑRPTL−j≤η^ℑRPTL−j,WηℑIPTL−j≤WηℑIPTL−j, then

CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≥CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n.
If sSLT−j−,aSC−j−=min1≤j≤nsSLT−j,aSC−j,sSLT−j+,aSC−j+=max1≤j≤nsSLT−j,aSC−j,

μℑRPTL−j+ei2πWμℑIPTL−j+=max1≤j≤nμℑRPTL−jei2πmax1≤j≤nWμℑIPTL−j,μℑRPTL−j−ei2πWμℑIPTL−j−=min1≤j≤nμℑRPTL−jei2πmin1≤j≤nWμℑIPTL−j,

ξℑRPTL−j+ei2πWξℑIPTL−j+=max1≤j≤nξℑRPTL−jei2πmax1≤j≤nWξℑIPTL−j,ξℑRPTL−j−ei2πWξℑIPTL−j−=min1≤j≤nξℑRPTL−jei2πmin1≤j≤nWξℑIPTL−j,

ηℑRPTL−j+ei2πWηℑIPTL−j+=max1≤j≤nηℑRPTL−jei2πmax1≤j≤nWηℑIPTL−j,ηℑRPTL−j−ei2πWηℑIPTL−j−=min1≤j≤nηℑRPTL−jei2πmin1≤j≤nWηℑIPTL−j, then

sSLT−j−,aSC−j−,μℑRPTL−j−ei2πWμℑIPTL−j−,ξℑRPTL−j−ei2πWξℑIPTL−j−,ηℑRPTL−j−ei2πWηℑIPTL−j−≤CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≤sSLT−j+,aSC−j+,μℑRPTL−j+ei2πWμℑIPTL−j+,ξℑRPTL−j+ei2πWξℑIPTL−j+,ηℑRPTL−j+ei2πWηℑIPTL−j+

Proof:

The proofs of the above three properties are shown as follows:

CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=

ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC

=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT,aSCtVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTLqSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTLqSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTLqSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTLqSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTLqSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTLqSCtVP−j1n!1∑j=1ntVP−j1qSC=sSLT,aSC,μℑRPTLei2πWμℑIPTL,ξℑRPTLei2πWξℑIPTL,ηℑRPTLei2πWηℑIPTL=ℑCTS

Hence, CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ℑCTS.
We know that
CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ℑCTS−j=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ηℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC
and
CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ℑ^CTS−j=ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1s^SLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWμℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−ξℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WξℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−η^ℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WηℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC
where sSLT−j,aSC−j≥s^SLT−j,aSC−j

⇒ΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j≥ΔLT−1s^SLT−ϑ(j),aSC−ϑ(j)tVP−j

⇒ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j≥ΔLT1n!∑ϑ(j)∈sSC−n∏j=1nΔLT−1s^SLT−ϑ(j),aSC−ϑ(j)tVP−j1∑j=1ntVP−j

Further, we check the real part of the complex-valued truth grade, such that μℑRPTL−j≥μℑRPTL−j
⇒μℑRPTL−ϑ(j)qSCtVP−j≥μ^ℑRPTL−ϑ(j)qSCtVP−j⇒1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j≤1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j⇒1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j≥1−∏ϑ(j)∈sSC−n1−∏j=1nμℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j

Similarly we can prove the imaginary part of the complex-valued truth grade, real and imaginary parts of the complex-valued abstinence and falsity grades, such that WμℑIPTL−j≥WμℑIPTL−j,ξℑRPTL−j≤ξℑRPTL−j,WξℑIPTL−j≤WξℑIPTL−j, and ηℑRPTL−j≤η^ℑRPTL−j,WηℑIPTL−j≤WηℑIPTL−j, then it is clear that If SℑCTS−j≻SℑCTS−j⇒ℑCTS−j≻ℑCTS−j;

If SℑCTS−j=SℑCTS−j, then we also use the accuracy function, such that if HℑCTS−j≻HℑCTS−j⇒ℑCTS−j≻ℑCTS−j; if HℑCTS−j=HℑCTS−j⇒ℑCTS−j=ℑCTS−j. Hence the inequality is holds true.
CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≥CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n.

The results has been completed.
By using the result 1 and result 2, we get the following result, such that
sSLT−j−,aSC−j−,μℑRPTL−j−ei2πWμℑIPTL−j−,ξ−ℑRPTL−jei2πWξℑIPTL−j−,η−ℑRPTL−jei2πWηℑIPTL−j−≤CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n
and
CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≤sSLT−j+,aSC−j+,μℑRPTL−j+ei2πWμℑIPTL−j+,ξℑRPTL−j+ei2πWξℑIPTL−j+,ηℑRPTL−j+ei2πWηℑIPTL−j+
then
sSLT−j−,aSC−j−,μℑRPTL−j−ei2πWμℑIPTL−j−,ξℑRPTL−j−ei2πWξℑIPTL−j−,ηℑRPTL−j−ei2πWηℑIPTL−j−≤CTSF2−TLMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≤sSLT−j+,aSC−j+,μℑRPTL−j+ei2πWμℑIPTL−j+,ξℑRPTL−j+ei2πWξℑIPTL−j+,ηℑRPTL−j+ei2πWηℑIPTL−j+.

Hence the result is proved.

Further, the special cases of the explored operator is discussed based on tVP.

Case 1: If we choose tVP=(1,0,0,‥‥,0), then the CTSF2-TLMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic arithmetic averaging operator shown as

CTSF2−TLAAtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=1−∏j=1n1−μℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WμℑIPTL−ϑ(j)qSC1n1qSC,ΔLT∑j=1n1n×ΔLT−1sSLT−ϑ(j),aSC−ϑ(j),1−∏j=1n1−μℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WμℑIPTL−ϑ(j)qSC1n1qSC,∏j=1nξℑRPTL−ϑ(j)1nei2π∏j=1nWξℑIPTL−ϑ(j)1n,∏j=1nηℑRPTL−ϑ(j)1nei2π∏j=1nWηℑIPTL−ϑ(j)1n1−∏j=1n1−μℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WμℑIPTL−ϑ(j)qSC1n1qSC,

Case 2: If we choose tVP=(1,1,0,‥‥,0), then the CTSF2-TLMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic BMO given by:

CTSF2−TLBMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT1n(n−1)∑j≠k=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)×ΔLT−1sSLT−ϑ(k),aSC−ϑ(k)12,1−∏j≠k=1n1−μℑRPTL−ϑ(j)qSC×μℑRPTL−ϑ(k)qSC1n(n−1)1qSC12ei2π1−∏j≠k=1n1−WμℑIPTL−ϑ(j)qSC×WμℑIPTL−ϑ(k)qSC1n(n−1)1qSC12,1−1−∏j=1n1−1−ξℑRPTL−ϑ(j)qSC×1−ξℑRPTL−ϑ(k)qSC1n(n−1)121qSC ei2π1−1−∏j=1n1−1−WξℑIPTL−ϑ(j)qSC×1−WξℑIPTL−ϑ(k)qSC1n(n−1)121qSC,1−1−∏j=1n1−1−ηℑRPTL−ϑ(j)qSC×1−ηℑRPTL−ϑ(k)qSC1n(n−1)121qSC ei2π1−1−∏j=1n1−1−WηℑIPTL−ϑ(j)qSC×1−WξηIPTL−ϑ(k)qSC1n(n−1)121qSC

Case 3: If we choose tVP=1,11,1,1︷n,….,0,0,0,0︷n−k, then the CTSF2-TLMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic MSMO given by:

CTSF2−TLMSMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT1Cnk∑1≤j1≤j2≤…≤jk≤n∏k=1nΔLT−1sSLT−ϑ(k),aSC−ϑ(k)1k,1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nμℑRPTL−ϑ(k)qSC1Cnk1qSC1kei2π1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nWμℑIPTL−ϑ(k)qSC1Cnk1qSC1k,1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−ξℑRPTL−ϑ(k)qSC1Cnk1k1qSC ei2π1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−WξℑIPTL−ϑ(k)qSC1Cnk1∑j=1ntVP−j1qSC,1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−ηℑRPTL−ϑ(k)qSC1Cnk1k1qSC ei2π1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−WηℑIPTL−ϑ(k)qSC1Cnk1k1qSC

Case 4: If we choose tVP=(1,1,1,‥‥,1), then the CTSF2-TLMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic geometric AO is given by:

CTSF2−TLGAtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)1n,1−∏j=1n1−ηℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WηℑIPTL−ϑ(j)qSC1n1qSC∏j=1nμℑRPTL−ϑ(j)1nei2π∏j=1nWμℑIPTL−ϑ(j)1n,1−∏j=1n1−ξℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WξℑIPTL−ϑ(j)qSC1n1qSC,1−∏j=1n1−ηℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WηℑIPTL−ϑ(j)qSC1n1qSC

Further, we investigate the DMM operator based on a CTSF2-TLS which is called CTSF2-TLDMM operator.

Definition 10.

Choose the family of CTSF2-TLNs ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn, the CTSF2-TLDMM operator is given by:

CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=1∑j=1ntVP−j⊗CTSϑ(j)∈sSC−n⊕CTSj=1ntVP−jℑTS−ϑ(j)1n!(14)

where ϑ(j),(j=1,2,3,…,n) is n permutation, and the set of all permutation of 1 to n is denoted by SSC−n.

Based on the above analysis related to operational laws and Definition 10, we establish the following results.

Theorem 3.

Suppose the family of CTSF2-TLNs ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn. Then the aggregated value of CTSF2-TLNs is a CTSF2-TLN, and

CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT1∑j=1ntVP−j∏ϑ(j)∈sSC−n∑j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)tVP−j1n!,1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−μℑRPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC ei2π1−1−∏ϑ(j)∈sSC−n1−∏j=1n1−WμℑIPTL−ϑ(j)qSCtVP−j1n!1∑j=1ntVP−j1qSC,1−∏ϑ(j)∈sSC−n1−∏j=1nξℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWξℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j,1−∏ϑ(j)∈sSC−n1−∏j=1nηℑRPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−jei2π1−∏ϑ(j)∈sSC−n1−∏j=1nWηℑIPTL−ϑ(j)qSCtVP−j1n!1qSC1∑j=1ntVP−j

Proof:

Straightforward. (The proof of this theorem is similar to Theorem 1).

Example 3.

Based on the information in Example 2, we can get the aggregated value by the CTSF2-TLDMM operator, such that

1∑j=1ntVP−j⊗CTSϑ(j)∈sSC−n⊕CTSj=1ntVP−jℑTS−ϑ(j)1n!=sSLT−3,0.0118,0.8165ei2π(0.8165),0.0ei2π(0.0),0.4966ei2π(0.4966).

If we set to zero to the imaginary parts of the complex-valued truth and falsity grades, then the Example 3 is converted to T-spherical fuzzy 2-tuple linguistic variables, which is the special cases of the established operator.

Theorem 4.

Suppose the families of two CTSF2-TLNs are ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) and ℑCTS−∗j=sSLT−∗j,aSC−∗j,μℑRPTL−∗jei2πWμℑIPTL−∗j,ξℑRPTL−∗jei2πWξℑIPTL−∗j,ηℑRPTL−∗jei2πWηℑIPTL−∗j,(j=1,2,3,‥,n) with vector of parameters tVP=tVP−1,tVP−2,…,tVP−n∈Rn. Then the idempotency, monotonicity, and boundedness are shown as follows:

If ℑCTS−j=sSLT−j,aSC−j,μℑRPTL−jei2πWμℑIPTL−j,ξℑRPTL−jei2πWξℑIPTL−j,ηℑRPTL−jei2πWηℑIPTL−j,(j=1,2,3,‥,n) are all equal if and only if ℑCTS−j=ℑCTS, then CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ℑCTS
If sSLT−j,aSC−j≥s^SLT−j,aSC−j,μℑRPTL−j≥μℑRPTL−j,WμℑIPTL−j≥WμℑIPTL−j,ξℑRPTL−j≤ξℑRPTL−j,WξℑIPTL−j≤WξℑIPTL−j, and ηℑRPTL−j≤η^ℑRPTL−j,WηℑIPTL−j≤WηℑIPTL−j, then

CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≥CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n.
If sSLT−j−,aSC−j−=min1≤j≤nsSLT−j,aSC−j,s+SLT−j,a+SC−j=max1≤j≤nsSLT−j,aSC−j,

μℑRPTL−j+ei2πWμℑIPTL−j+=max1≤j≤nμℑRPTL−jei2πmax1≤j≤nWμℑIPTL−j,μℑRPTL−j−ei2πWμℑIPTL−j−=min1≤j≤nμℑRPTL−jei2πmin1≤j≤nWμℑIPTL−j,

ξℑRPTL−j+ei2πWξℑIPTL−j+=max1≤j≤nξℑRPTL−jei2πmax1≤j≤nWξℑIPTL−j,ξ−ℑRPTL−jei2πWξℑIPTL−j−=min1≤j≤nξℑRPTL−jei2πmin1≤j≤nWξℑIPTL−j,

ηℑRPTL−j+ei2πWηℑIPTL−j+=max1≤j≤nηℑRPTL−jei2πmax1≤j≤nWηℑIPTL−j,η−ℑRPTL−jei2πWηℑIPTL−j−=min1≤j≤nηℑRPTL−jei2πmin1≤j≤nWηℑIPTL−j, then

sSLT−j−,aSC−j−,μℑRPTL−j−ei2πWμℑIPTL−j−,ξℑRPTL−j−ei2πWξℑIPTL−j−,ηℑRPTL−j−ei2πWηℑIPTL−j−≤CTSF2−TLDMMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n≤sSLT−j+,aSC−j+,μℑRPTL−j+ei2πWμℑIPTL−j+,ξℑRPTL−j+ei2πWξℑIPTL−j+,ηℑRPTL−j+ei2πWηℑIPTL−j+

Proof:

Straightforward. (The proof of this theorem is similar to the Theorem 2).

Further, the special cases of the explored operator is discussed based on the value of tVP.

Case 1: If we choose tVP=(0,0,0,‥‥,0), then the CTSF2-TLDMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic geometric AO given by:

CTSF2−TLGAtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT ∏j=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j) 1n, 1−∏j=1n1−ηℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WηℑIPTL−ϑ(j)qSC1n1qSC.∏j=1nμℑRPTL−ϑ(j)1nei2π∏j=1nWμℑIPTL−ϑ(j)1n, 1−∏j=1n1−ξℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WξℑIPTL−ϑ(j)qSC1n1qSC,1−∏j=1n1−ηℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WηℑIPTL−ϑ(j)qSC1n1qSC.

Case 2: If we choose tVP=(1,1,0,‥‥,0), then the CTSF2-TLDMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic geometric BMO given by:

CTSF2−TLGBMtVP(ℑCTS−1,ℑCTS−2,…,ℑCTS−n)=ΔLT12∏j≠k=1nΔLT−1sSLT−ϑ(j),aSC−ϑ(j)+ΔLT−1sSLT−ϑ(k),aSC−ϑ(k)1n(n−1),1−1−∏j=1n1−1−μℑRPTL−ϑ(j)qSC×1−μℑRPTL−ϑ(k)qSC1n(n−1)121qSC ei2π1−1−∏j=1n1−1−WμℑIPTL−ϑ(j)qSC×1−WμℑIPTL−ϑ(k)qSC1n(n−1)121qSC,1−∏j≠k=1n1−ξℑRPTL−ϑ(j)qSC×ξℑRPTL−ϑ(k)qSC1n(n−1)1qSC12ei2π1−∏j≠k=1n1−WξℑIPTL−ϑ(j)qSC×WξℑIPTL−ϑ(k)qSC1n(n−1)1qSC12,1−∏j≠k=1n1−ηℑRPTL−ϑ(j)qSC×ηℑRPTL−ϑ(k)qSC1n(n−1)1qSC12ei2π1−∏j≠k=1n1−WηℑIPTL−ϑ(j)qSC×WηℑIPTL−ϑ(k)qSC1n(n−1)1qSC12

Case 3: If we choose tVP=1,1,1,1,1,1︷k,….,0,0,0,0,0,0︷n−k, then the CTSF2-TLDMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic geometric MSMO given by:

CTSF2−TLGMSMtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT1k∏1≤j1≤j2≤…≤jk≤n∑k=1nΔLT−1sSLT−ϑ(k),aSC−ϑ(k)1Cnk,1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−μℑRPTL−ϑ(k)qSC1Cnk1k1qSC ei2π1−1−∏≤j1≤j2≤…≤jk≤n1−∏k=1n1−WμℑIPTL−ϑ(k)qSC1Cnk1∑j=1ntVP−j1qSC,1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nξℑRPTL−ϑ(k)qSC1Cnk1qSC1kei2π1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nWξℑIPTL−ϑ(k)qSC1Cnk1qSC1k,1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nηℑRPTL−ϑ(k)qSC1Cnk1qSC1kei2π1−∏≤j1≤j2≤…≤jk≤n1−∏k=1nWηℑIPTL−ϑ(k)qSC1Cnk1qSC1k

Case 4: If we choose tVP=(1,1,1,‥‥,1), then the CTSF2-TLDMM operator is reduced to the complex T-spherical fuzzy 2-tuple linguistic arithmetic AO given by:

CTSF2−TLAAtVPℑCTS−1,ℑCTS−2,…,ℑCTS−n=ΔLT∑j=1n1n×ΔLT−1sSLT−ϑ(j),aSC−ϑ(j),1−∏j=1n1−μℑRPTL−ϑ(j)qSC1n1qSCei2π1−∏j=1n1−WμℑIPTL−ϑ(j)qSC1n1qSC,∏j=1nξℑRPTL−ϑ(j)1nei2π∏j=1nWξℑIPTL−ϑ(j)1n,∏j=1nηℑRPTL−ϑ(j)1nei2π∏j=1nWηℑIPTL−ϑ(j)1n

4. MADM METHOD BASED ON COMPLEX T-SPHERICAL FUZZY 2-TUPLE LINGUISTIC INFORMATION

For a MADM problem based on complex T-spherical fuzzy 2-tuple linguistic information, we consider the families of the alternatives and attributes, which are stated as: AAl=AAl−1,AAl−2,…,AAl−m, CAT=CAT−1,CAT−2,…,CAT−n, and then construct the decision-making matrix RDM=rjkm×n, where rjk=sSLT−jk,aSC−jk,μℑRPTL−jkei2πWμℑIPTL−jk,ξℑRPTL−jkei2πWξℑIPTL−jk,ηℑRPTL−jkei2πWηℑIPTL−jk is in the form of CTSF2-TLNs for alternative AAL−j(j=1,2,3,…,m) under the attribute CAT−k(k=1,2,3,…,n), then the steps of this MADM problem based on CTSF2-TLNs are as follow:

By CTSF2-TLMM operator or CTSF2-TLDMM operator to get the aggregated result.
By Definition 4, we get the score values of the aggregated values.
By Definition 4, we get the ranking results, and then obtain the best one alternative.
The end.

Example 4.

The purpose of this example is to select the emergency alternative. Suppose there are four alternatives shown as: AAl=AAl−1,AAl−2,AAl−3,AAl−4, and there are four attributes which are explained as CAT−1= Preparing Capacity, CAT−2= Rescuing Capacity, CAT−3= Recovering Capacity, and CAT−4= Responding Time. Further, the linguistic term set SLT=sSLT−0=very−poor,sSLT−1=poor,sSLT−2=fair,sSLT−3=good,sSLT−4=very−good is adopted, and the decision matrix RDM=rjk4×4 is build up which is in the form CTSF2-TLNs shown in the Table 1. The steps are shown as follows:

By the CTSF2-TLMM operator, we can get the aggregated values for four alternatives, where, we select the parameter tVP=1,1,1,1 and qSC=3, then

AAl−1=CTSF2−TLMMtVPCAl−1,AT−1,CAl−1,AT−2,CAl−1,AT−3,CAl−1,AT−4=(sSLT−2,0.097669,0.0005125ei2π(0.0005125),0.32ei2π(0.32),0.23ei2π(0.23);

AAl−2=CTSF2−TLMMtVPCAl−2,AT−1,CAl−2,AT−2,CAl−2,AT−3,CAl−2,AT−4=(sSLT−1,0.22658,0.0ei2π(0.0),0.16ei2π(0.16),0.36ei2π(0.36);

AAl−3=CTSF2−TLMMtVPCAl−3,AT−1,CAl−3,AT−2,CAl−3,AT−3,CAl−3,AT−4=(sSLT−2,0.94927,0.0ei2π(0.0),0.22ei2π(0.22),0.18ei2π(0.18);

AAl−4=CTSF2−TLMMtVPCAl−4,AT−1,CAl−4,AT−2,CAl−4,AT−3,CAl−4,AT−4=(sSLT−1,0.24592,0.0ei2π(0.0),0.184ei2π(0.184),0.144ei2π(0.144).
By Definition 4, we get the score values of the aggregated values as follows.

SAAl−1=−0.22135,SAAl−2=−0.167,SAAl−3=−0.160,SAAl−4=−0.4385.
By Definition 4, we obtain the ranking result and the best one alternative.

AAl−3>AAl−2>AAl−1>AAl−4

and then AAl−3 is the best alternative for emergency preplan.
The end.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAl−1	((sSLT−3,0.03),(0.5ei2π(0.5),0.1ei2π(0.1),0.2ei2π(0.2)))	((sSLT−4,0.13),(0.9ei2π(0.9),0.01ei2π(0.01),0.01ei2π(0.01)))	((sSLT−3,0.11),(0.8ei2π(0.8),0.1ei2π(0.1),0.1ei2π(0.1)))	((sSLT−4,0.01),(0.19ei2π(0.19),0.2ei2π(0.2),0.23ei2π(0.23)))
AAl−2	((sSLT−3,0.01),(0.51ei2π(0.51),0.11ei2π(0.11),0.12ei2π(0.12)))	((sSLT−3,0.14),(0.6ei2π(0.6),0.11ei2π(0.11),0.15ei2π(0.15)))	((sSLT−3,0.13),(0.4ei2π(0.4),0.2ei2π(0.2),0.2ei2π(0.2)))	((sSLT−2,0.02),(0.36ei2π(0.36),0.3ei2π(0.3),0.13ei2π(0.13)))
AAl−3	((sSLT−4,0.0101),(0.7ei2π(0.7),0.1ei2π(0.1),0.1ei2π(0.1)))	((sSLT−3,0.15),(0.19ei2π(0.19),0.2ei2π(0.2),0.2ei2π(0.2)))	((sSLT−4,0.06),(0.3ei2π(0.3),0.2ei2π(0.2),0.12ei2π(0.12)))	((sSLT−3,0.15),(0.19ei2π(0.19),0.2ei2π(0.2),0.2ei2π(0.2)))
AAl−4	((sSLT−3,0.12),(0.7ei2π(0.7),0.2ei2π(0.2),0.01ei2π(0.01)))	((sSLT−2,0.16),(0.38ei2π(0.38),0.13ei2π(0.13),0.3ei2π(0.3)))	((sSLT−3,0.05),(0.28ei2π(0.28),0.2ei2π(0.2),0.22ei2π(0.22)))	((sSLT−2,0.16),(0.38ei2π(0.38),0.13ei2π(0.13),0.3ei2π(0.3)))

Table 1

Decision matrix in the form of complex picture fuzzy 2-tuple linguistic numbers.

4.1. Further Discussion

To evaluate the influence of the parameter tVP, based on the Example 4, we set the different values to the parameter tVP in the established approaches, that is, based on CTSF2-TLMM operator and CTSF2-TLDMM operator, then the ranking results are shown in Table 2.

Parameter Vector	Operators	Score Values	Ranking Result
tVP=1,1,1,1	MM	S(AAL−1)=−0.22135,S(AAL−2)=−0.167,S(AAL−3)=−0.160,S(AAL−4)=−0.4385	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,1,1,1	DMM	S(AAL−1)=−0.135,S(AAL−2)=−0.127,S(AAL−3)=−0.110,S(AAL−4)=−0.246	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,0,0,0	MM	S(AAL−1)=−0.1395,S(AAL−2)=−0.137,S(AAL−3)=−0.130,S(AAL−4)=−0.285	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,0,0,0	DMM	S(AAL−1)=−0.235,S(AAL−2)=−0.178,S(AAL−3)=−0.170,S(AAL−4)=−0.453	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,1,0,0	MM	S(AAL−1)=−0.254,S(AAL−2)=−0.247,S(AAL−3)=−0.240,S(AAL−4)=−0.558	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,1,0,0	DMM	S(AAL−1)=−0.227,S(AAL−2)=−0.169,S(AAL−3)=−0.164,S(AAL−4)=−0.445	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,1,1,0	MM	S(AAL−1)=−0.223,S(AAL−2)=−0.168,S(AAL−3)=−0.161,S(AAL−4)=−0.4402	AAl−3>AAl−2>AAl−1>AAl−4
tVP=1,1,1,0	DMM	S(AAL−1)=−0.221,S(AAL−2)=−0.173,S(AAL−3)=−0.163,S(AAL−4)=−0.438	AAl−3>AAl−2>AAl−1>AAl−4
tVP=2,2,2,2	MM	S(AAL−1)=−0.203,S(AAL−2)=−0.147,S(AAL−3)=−0.135,S(AAL−4)=−0.425	AAl−3>AAl−2>AAl−1>AAl−4
tVP=2,2,2,2	DMM	S(AAL−1)=−0.214,S(AAL−2)=−0.154,S(AAL−3)=−0.134,S(AAL−4)=−0.430	AAl−3>AAl−2>AAl−1>AAl−4
tVP=3,3,3,3	MM	S(AAL−1)=−0.189,S(AAL−2)=−0.1137,S(AAL−3)=−0.113,S(AAL−4)=−0.378	AAl−3>AAl−2>AAl−1>AAl−4
tVP=3,3,3,3	DMM	S(AAL−1)=−0.188,S(AAL−2)=−0.127,S(AAL−3)=−0.117,S(AAL−4)=−0.376	AAl−3>AAl−2>AAl−1>AAl−4

DMM, dual Muirhead mean; MM, Muirhead mean.

Table 2

Ranking results for different value of parameter tVP.

From the Table 1, we get the ranking results from the both methods with different values of the parameters, obviously, there is the same ranking result, and the best alternative is AAl−3. When we change the value of parameter tVP, the result is still remain same result.

4.2. Comparative Analysis

The purpose of this sub-section is to prove that the established operators in this manuscript are effective, based on Example 4, we make a comparison between explored operators with existing operators, such as averaging aggregation operator, geometric aggregation operator, geometric BMO, geometric MSMO based on the complex T-spherical fuzzy 2-tuple linguistic information, complex spherical fuzzy 2-tuple linguistic information, and complex picture fuzzy 2-tuple linguistic information. The comparisons between established operators with some existing operators are discussed in Table 3.

Methods	Operators	Score Values	Ranking Values
Wei et al. [48]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Ju et al. [49]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
T-spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 1)	MM	S(AAL−1)=−0.235,S(AAL−2)=−0.157,S(AAL−3)=−0.150,S(AAL−4)=−0.385	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.271,S(AAL−2)=−0.172,S(AAL−3)=−0.156,S(AAL−4)=−0.431	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.205,S(AAL−2)=−0.137,S(AAL−3)=−0.118,S(AAL−4)=−0.296	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.143,S(AAL−2)=−0.132,S(AAL−3)=−0.107,S(AAL−4)=−0.253	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 2)	MM	S(AAL−1)=−0.243,S(AAL−2)=−0.127,S(AAL−3)=−0.107,S(AAL−4)=−0.345	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.271,S(AAL−2)=−0.122,S(AAL−3)=−0.102,S(AAL−4)=−0.337	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.144,S(AAL−2)=−0.113,S(AAL−3)=−0.102,S(AAL−4)=−0.346	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.143,S(AAL−2)=−0.114,S(AAL−3)=−0.104,S(AAL−4)=−0.363	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 3)	MM	S(AAL−1)=−0.22135,S(AAL−2)=−0.167,S(AAL−3)=−0.160,S(AAL−4)=−0.438	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.2371,S(AAL−2)=−0.1662,S(AAL−3)=−0.152,S(AAL−4)=−0.437	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.135,S(AAL−2)=−0.127,S(AAL−3)=−0.110,S(AAL−4)=−0.246	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.133,S(AAL−2)=−0.128,S(AAL−3)=−0.104,S(AAL−4)=−0.263	AAl−3>AAl−2>AAl−1>AAl−4

DMM, dual Muirhead mean; MM, Muirhead mean; WDMM, weighted dual Muirhead mean; WMM, weighted Muirhead mean.

Table 3

Comparative analysis between established operators with existing operators by using Example 4.

From the Table 3, we obtain that some existing methods [48,49] cannot deal with this decision-making problem and the proposed method get the same ranking result AAl−3>AAl−2>AAl−1>AAl−4 and the best alternative is AAL−3. Obviously, the proposed method is more general than the methods [48,49]. The graphical interpretation based on Table 3 is shown in Figure 1.

In the Figure 1, we have discussed four different series, which denote the alternatives AAL−1 to AAL−4. From the Figure 1, it is clear that the series 3 gives greater values compared to other values in different series.

Example 5.

In this example, the meanings of alternatives and attributes are the same as Example 4, we only consider the complex spherical fuzzy 2-tuple linguistic information, which is shown in Table 4. The explored operators are compared with some existing operators to examine the reliability and proficiency of the established operators.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAl−1	((sSLT−3,0.03),(0.5ei2π(0.5),0.4ei2π(0.4),0.2ei2π(0.2)))	((sSLT−4,0.13),(0.9ei2π(0.9),0.1ei2π(0.1),0.01ei2π(0.01)))	((sSLT−3,0.11),(0.8ei2π(0.8),0.1ei2π(0.1),0.1ei2π(0.1)))	((sSLT−4,0.01),(0.19ei2π(0.19),0.2ei2π(0.2),0.23ei2π(0.23)))
AAl−2	((sSLT−3,0.01),(0.51ei2π(0.51),0.41ei2π(0.41),0.12ei2π(0.12)))	((sSLT−3,0.14),(0.6ei2π(0.6),0.31ei2π(0.31),0.15ei2π(0.15)))	((sSLT−3,0.13),(0.7ei2π(0.7),0.2ei2π(0.2),0.2ei2π(0.2)))	((sSLT−2,0.02),(0.36ei2π(0.36),0.3ei2π(0.3),0.13ei2π(0.13)))
AAl−3	((sSLT−4,0.0101),(0.7ei2π(0.7),0.3ei2π(0.3),0.1ei2π(0.1)))	((sSLT−3,0.15),(0.7ei2π(0.7),0.2ei2π(0.2),0.2ei2π(0.2)))	((sSLT−4,0.06),(0.8ei2π(0.8),0.2ei2π(0.2),0.12ei2π(0.12)))	((sSLT−3,0.15),(0.19ei2π(0.19),0.2ei2π(0.2),0.2ei2π(0.2)))
AAl−4	((sSLT−3,0.12),(0.7ei2π(0.7),0.4ei2π(0.4),0.01ei2π(0.01)))	((sSLT−2,0.16),(0.38ei2π(0.38),0.13ei2π(0.13),0.3ei2π(0.3)))	((sSLT−3,0.05),(0.8ei2π(0.8),0.1ei2π(0.1),0.22ei2π(0.22)))	((sSLT−2,0.16),(0.38ei2π(0.38),0.13ei2π(0.13),0.3ei2π(0.3)))

Table 4

Decision matrix in the form of complex spherical fuzzy 2-tuple linguistic numbers.

By using the above steps of the algorithm, the comparison results between the established approach and some existing operators are shown in Table 5 and Figure 2.

Methods	Operators	Score Values	Ranking Values
Wei et al. [48]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Ju et al. [49]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
T-spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 1)	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 2)	MM	S(AAL−1)=−0.354,S(AAL−2)=−0.204,S(AAL−3)=−0.172,S(AAL−4)=−0.554	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.371,S(AAL−2)=−0.222,S(AAL−3)=−0.187,S(AAL−4)=−0.537	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.251,S(AAL−2)=−0.178,S(AAL−3)=−0.142,S(AAL−4)=−0.343	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.255,S(AAL−2)=−0.174,S(AAL−3)=−0.144,S(AAL−4)=−0.353	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 3)	MM	S(AAL−1)=−0.322,S(AAL−2)=−0.184,S(AAL−3)=−0.163,S(AAL−4)=−0.424	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.321,S(AAL−2)=−0.192,S(AAL−3)=−0.167,S(AAL−4)=−0.437	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.351,S(AAL−2)=−0.188,S(AAL−3)=−0.148,S(AAL−4)=−0.356	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.349,S(AAL−2)=−0.164,S(AAL−3)=−0.143,S(AAL−4)=−0.361	AAl−3>AAl−2>AAl−1>AAl−4

DMM, dual Muirhead mean; MM, Muirhead mean.

Table 5

Comparative analysis between established operators with existing operators by using Example 5.

From the Table 5, we can see that some existing methods [48,49] cannot deal with this decision-making problem and the proposed method get the same ranking result AAl−3>AAl−2>AAl−1>AAl−4 and the best alternative is AAL−3. Obviously, the proposed method is more general than the methods [48,49]. The graphical interpretation based on Table 5 is shown in Figure 2.

Example 6.

In this example, the meanings of alternatives and attributes are the same as Example 4, we only consider the complex T-spherical fuzzy 2-tuple linguistic information which is shown in Table 6, and then the explored operators are compared with some existing operators to show the reliability and proficiency of the established operators.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAl−1	((sSLT−3,0.03),(0.8ei2π(0.5),0.7ei2π(0.1),0.2ei2π(0.2)))	((sSLT−4,0.13),(0.9ei2π(0.9),0.9ei2π(0.01),0.9ei2π(0.01)))	((sSLT−3,0.11),(0.8ei2π(0.8),0.79ei2π(0.1),0.71ei2π(0.1)))	((sSLT−4,0.01),(0.9ei2π(0.19),0.2ei2π(0.2),0.63ei2π(0.23)))
AAl−2	((sSLT−3,0.01),(0.59ei2π(0.51),0.91ei2π(0.11),0.12ei2π(0.12)))	((sSLT−3,0.14),(0.6ei2π(0.6),0.81ei2π(0.11),0.15ei2π(0.15)))	((sSLT−3,0.13),(0.84ei2π(0.4),0.82ei2π(0.2),0.82ei2π(0.2)))	((sSLT−2,0.02),(0.36ei2π(0.36),0.3ei2π(0.3),0.13ei2π(0.13)))
AAl−3	((sSLT−4,0.0101),(0.7ei2π(0.7),0.6ei2π(0.1),0.5ei2π(0.1)))	((sSLT−3,0.15),(0.89ei2π(0.19),0.72ei2π(0.2),0.2ei2π(0.2)))	((sSLT−4,0.06),(0.3ei2π(0.3),0.2ei2π(0.2),0.12ei2π(0.12)))	((sSLT−3,0.15),(0.19ei2π(0.19),0.2ei2π(0.2),0.2ei2π(0.2)))
AAl−4	((sSLT−3,0.12),(0.7ei2π(0.7),0.62ei2π(0.2),0.71ei2π(0.01)))	((sSLT−2,0.16),(0.88ei2π(0.38),0.83ei2π(0.13),0.39ei2π(0.3)))	((sSLT−3,0.05),(0.28ei2π(0.28),0.2ei2π(0.2),0.22ei2π(0.22)))	((sSLT−2,0.16),(0.38ei2π(0.38),0.13ei2π(0.13),0.3ei2π(0.3)))

Table 6

Decision matrix in the form of complex T-spherical fuzzy 2-tuple linguistic numbers.

By using the above steps of the algorithm, the comparison results between the established operators and some existing operators are shown in Table 7 and Figure 3.

Methods	Operators	Score Values	Ranking Values
Wei et al. [48]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Ju et al. [49]	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
T-spherical fuzzy 2-tuple linguistic variables	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 1)	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 2)	MM	Cannot be classified	−
	WMM	Cannot be classified	−
	DMM	Cannot be classified	−
	WDMM	Cannot be classified	−
Proposed method (q = 12)	MM	S(AAL−1)=−0.955,S(AAL−2)=−0.857,S(AAL−3)=−0.781,S(AAL−4)=−0.985	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.975,S(AAL−2)=−0.853,S(AAL−3)=−0.762,S(AAL−4)=−0.982	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.948,S(AAL−2)=−0.849,S(AAL−3)=−0.734,S(AAL−4)=−0.978	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.955,S(AAL−2)=−0.851,S(AAL−3)=−0.799,S(AAL−4)=−0.984	AAl−3>AAl−2>AAl−1>AAl−4

DMM, dual Muirhead mean; MM, Muirhead mean.

Table 7

Comparative analysis between established operators with existing operators by using Example 6.

From the Table 7, we can see that some existing methods [48,49] cannot deal with this decision-making problem and the proposed method get the same ranking result AAl−3>AAl−2>AAl−1>AAl−4 and the best alternative is AAL−3. Obviously, the proposed method is more general than the methods [48,49]. The graphical interpretation based on Table 7 is shown in Figure 3.

Example 7.

In this example, the meanings of alternatives and attributes are the same as Example 4, we only consider the picture fuzzy 2-tuple linguistic information which is shown in Table 8, and then the explored operators are compared with some existing operators to show the validity of the established operators.

Symbols	CAT−1	CAT−2	CAT−3	CAT−4
AAl−1	((sSLT−3,0.03),(0.5ei2π(0.0),0.1ei2π(0.0),0.2ei2π(0.0)))	((sSLT−4,0.13),(0.9ei2π(0.0),0.01ei2π(0.0),0.01ei2π(0.0)))	((sSLT−3,0.11),(0.8ei2π(0.0),0.1ei2π(0.0),0.1ei2π(0.0)))	((sSLT−4,0.01),(0.19ei2π(0.0),0.2ei2π(0.0),0.23ei2π(0.0)))
AAl−2	((sSLT−3,0.01),(0.51ei2π(0.0),0.11ei2π(0.0),0.12ei2π(0.0)))	((sSLT−3,0.14),(0.6ei2π(0.0),0.11ei2π(0.0),0.15ei2π(0.0)))	((sSLT−3,0.13),(0.4ei2π(0.0),0.2ei2π(0.0),0.2ei2π(0.0)))	((sSLT−2,0.02),(0.36ei2π(0.0),0.3ei2π(0.0),0.13ei2π(0.0)))
AAl−3	((sSLT−4,0.0101),(0.7ei2π(0.0),0.1ei2π(0.0),0.1ei2π(0.0)))	((sSLT−3,0.15),(0.19ei2π(0.0),0.2ei2π(0.0),0.2ei2π(0.0)))	((sSLT−4,0.06),(0.3ei2π(0.0),0.2ei2π(0.0),0.12ei2π(0.0)))	((sSLT−3,0.15),(0.19ei2π(0.0),0.2ei2π(0.0),0.2ei2π(0.0)))
AAl−4	((sSLT−3,0.12),(0.7ei2π(0.0),0.2ei2π(0.0),0.01ei2π(0.0)))	((sSLT−2,0.16),(0.38ei2π(0.0),0.13ei2π(0.0),0.3ei2π(0.0)))	((sSLT−3,0.05),(0.28ei2π(0.0),0.2ei2π(0.0),0.22ei2π(0.0)))	((sSLT−2,0.16),(0.38ei2π(0.0),0.13ei2π(0.0),0.3ei2π(0.0)))

Table 8

Decision matrix in the form of picture fuzzy 2-tuple linguistic numbers.

By using the above steps of the algorithm, the comparison results between the established operators and some existing operators are shown in Table 9 and Figure 4.

Methods	Operators	Score Values	Ranking Values
Wei et al. [48]	MM	S(AAL−1)=−0.438,S(AAL−2)=−0.342,S(AAL−3)=−0.316,S(AAL−4)=−0.640	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.433,S(AAL−2)=−0.338,S(AAL−3)=−0.311,S(AAL−4)=−0.653	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.237,S(AAL−2)=−0.127,S(AAL−3)=−0.113,S(AAL−4)=−0.336	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.235,S(AAL−2)=−0.128,S(AAL−3)=−0.120,S(AAL−4)=−0.353	AAl−3>AAl−2>AAl−1>AAl−4
Ju et al. [49]	MM	S(AAL−1)=−0.439,S(AAL−2)=−0.340,S(AAL−3)=−0.312,S(AAL−4)=−0.634	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.431,S(AAL−2)=−0.335,S(AAL−3)=−0.311,S(AAL−4)=−0.650	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.239,S(AAL−2)=−0.130,S(AAL−3)=−0.118,S(AAL−4)=−0.340	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.239,S(AAL−2)=−0.127,S(AAL−3)=−0.123,S(AAL−4)=−0.349	AAl−3>AAl−2>AAl−1>AAl−4
Spherical fuzzy 2-tuple linguistic variables	MM	S(AAL−1)=−0.395,S(AAL−2)=−0.278,S(AAL−3)=−0.247,S(AAL−4)=−0.534	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.389,S(AAL−2)=−0.272,S(AAL−3)=−0.240,S(AAL−4)=−0.527	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.391,S(AAL−2)=−0.279,S(AAL−3)=−0.249,S(AAL−4)=−0.530	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.390,S(AAL−2)=−0.279,S(AAL−3)=−0.245,S(AAL−4)=−0.531	AAl−3>AAl−2>AAl−1>AAl−4
T-spherical fuzzy 2-tuple linguistic variables	MM	S(AAL−1)=−0.274,S(AAL−2)=−0.245,S(AAL−3)=−0.225,S(AAL−4)=−0.316	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.271,S(AAL−2)=−0.243,S(AAL−3)=−0.227,S(AAL−4)=−0.307	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.281,S(AAL−2)=−0.254,S(AAL−3)=−0.221,S(AAL−4)=−0.310	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.276,S(AAL−2)=−0.249,S(AAL−3)=−0.217,S(AAL−4)=−0.198	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 1)	MM	S(AAL−1)=−0.438,S(AAL−2)=−0.342,S(AAL−3)=−0.316,S(AAL−4)=−0.640	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.433,S(AAL−2)=−0.338,S(AAL−3)=−0.311,S(AAL−4)=−0.653	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.237,S(AAL−2)=−0.127,S(AAL−3)=−0.113,S(AAL−4)=−0.336	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.235,S(AAL−2)=−0.128,S(AAL−3)=−0.120,S(AAL−4)=−0.353	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 2)	MM	S(AAL−1)=−0.395,S(AAL−2)=−0.278,S(AAL−3)=−0.247,S(AAL−4)=−0.534	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.389,S(AAL−2)=−0.272,S(AAL−3)=−0.240,S(AAL−4)=−0.527	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.391,S(AAL−2)=−0.279,S(AAL−3)=−0.249,S(AAL−4)=−0.530	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.390,S(AAL−2)=−0.279,S(AAL−3)=−0.245,S(AAL−4)=−0.531	AAl−3>AAl−2>AAl−1>AAl−4
Proposed method (q = 3)	MM	S(AAL−1)=−0.274,S(AAL−2)=−0.245,S(AAL−3)=−0.225,S(AAL−4)=−0.316	AAl−3>AAl−2>AAl−1>AAl−4
	WMM	S(AAL−1)=−0.271,S(AAL−2)=−0.243,S(AAL−3)=−0.227,S(AAL−4)=−0.307	AAl−3>AAl−2>AAl−1>AAl−4
	DMM	S(AAL−1)=−0.281,S(AAL−2)=−0.254,S(AAL−3)=−0.221,S(AAL−4)=−0.310	AAl−3>AAl−2>AAl−1>AAl−4
	WDMM	S(AAL−1)=−0.276,S(AAL−2)=−0.249,S(AAL−3)=−0.217,S(AAL−4)=−0.198	AAl−3>AAl−2>AAl−1>AAl−4

DMM, dual Muirhead mean; MM, Muirhead mean.

Table 9

Comparative analysis between established operators with existing operators by using Example 7.

From the Table 9 and Figure 4, we can see that all methods get the same ranking result AAl−3>AAl−2>AAl−1>AAl−4 and the best alternative is AAL−3. Obviously, this can explain the validity of the proposed method.

4.3. Advantages

The explored MM operator and DMM operator using the novel concept of CTSF2-TLSs are more powerful and more superior than existing operators which are discussed in Tables 3, 5, and 7, based on the complex picture fuzzy 2-tuple linguistic information and complex spherical fuzzy 2-tuple linguistic information. So the established approaches in this manuscript is more reliable and more efficient than existing methods [48,49].

5. CONCLUSIONS

CTSF2-TLS combined from CFS, TSFS, and 2-TLVS is a proficient technique to express uncertain and awkward information in real decision-making, which contains 2-tuple linguistic variable, truth, abstinence, and falsity grades, and gives more extensive freedom than some existing information expressions due to its constraint that the sum of q-powers of the real parts of the truth, abstinence, and falsity grades is not exceeded form unit interval. Based on the established operational laws and comparison methods for CTSF2-TLSs, the CTSF2-TLMM aggregation operator and CTSF2-TLDMM aggregation operator are explored. Some special cases and the desirable properties of the explored operators are also established and studied. Moreover, we establish a method to solve the MADM problems, in which the evaluation information is expressed by CTSF2-TLNs. Finally, we solve some numerical examples to explain the validity and advantaged of the explored method by comparing with some other methods. In a word, the proposed operators are a generalization of some existing operators such as averaging aggregation operator, geometric aggregation operator, geometric BMO, geometric MSMO based on the complex picture fuzzy 2-tuple linguistic information. In the future researches, we will explore some real applications based on the proposed operators, or some new operators based on CTSF2-TLNs.

CONFLICTS OF INTERESTS

The authors declare that there are no conflicts of interests regarding the publication of this article.

AUTHORS' CONTRIBUTIONS

All authors contributed equally.

DATA AVAILABILITY STATEMENT

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

ACKNOWLEDGMENTS

This paper is supported by the National Natural Science Foundation of China (No. 71771140), Project of cultural masters and “the four kinds of a batch” talents, the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

REFERENCES

1.V.I. Yukalov and D. Sornette, Physics of risk and uncertainty in quantum decision making, Eur. Phys. J. B, Vol. 71, 2009, pp. 533-548.

2.L.A. Zadeh, Fuzzy sets, Inf. Control, Vol. 8, 1965, pp. 338-353.

3.K.T. Atanassov, Intuitionistic fuzzy sets, K.T. Atanassov (editor), Intuitionistic fuzzy sets, Physica, Heidelberg, Germany, 1999, pp. 1-137.

4.R. Krishankumar, K.S. Ravichandran, and S.K. Tyagi, Solving cloud vendor selection problem using intuitionistic fuzzy decision framework, Neural Comput. Appl., Vol. 32, 2020, pp. 589-602.

5.R.R. Yager, Pythagorean fuzzy subsets, IEEE, in 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) (Edmonton, Canada), 2013, pp. 57-61.

6.B. Sarkar and A. Biswas, A unified method for Pythagorean fuzzy multicriteria group decision-making using entropy measure, linear programming and extended technique for ordering preference by similarity to ideal solution, Soft Comput., Vol. 24, 2020, pp. 5333-5344.

7.R.R. Yager, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst., Vol. 25, 2016, pp. 1222-1230.

8.P. Liu and P. Wang, Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making, Int. J. Intell. Syst., Vol. 33, 2018, pp. 259-280.

9.T. Senapati and R.R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods, Eng. Appl. Artif. Intell., Vol. 85, 2019, pp. 112-121.

10.B.C. Cường, Picture fuzzy sets, J. Comput. Sci. Cybern., Vol. 30, 2014, pp. 409.

11.T. Mahmood, K. Ullah, Q. Khan, and N. Jan, An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets, Neural Comput. Appl., Vol. 31, 2019, pp. 7041-7053.

12.K. Ullah, N. Hassan, T. Mahmood, N. Jan, and M. Hassan, Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators, Symmetry, Vol. 11, 2019, pp. 357.

13.K. Ullah, T. Mahmood, N. Jan, and Z. Ali, A note on geometric aggregation operators in T-spherical fuzzy environment and their applications in multi-attribute decision making, J. Eng. Appl. Sci., Vol. 37, 2018, pp. 75-86.

14.A. Guleria and R.K. Bajaj, T-spherical fuzzy soft sets and its aggregation operators with application in decision making, Scientia Iranica, 2019. In Press

15.S.G. Quek, G. Selvachandran, M. Munir, T. Mahmood, K. Ullah, L.H. Son, et al., Multi-attribute multi-perception decision-making based on generalized T-spherical fuzzy weighted aggregation operators on neutrosophic sets, Mathematics, Vol. 7, 2019, pp. 780.

16.K. Ullah, T. Mahmood, and N. Jan, Some averaging aggregation operators for t-spherical fuzzy sets and their applications in multi-attribute decision making, in Proceedings of the International Conference on Soft Computing & Machine Learning (ICSCML) (Wuhan, China), 2019, pp. 26-28.

17.K. Ullah, T. Mahmood, and H. Garg, Evaluation of the performance of search and rescue robots using T-spherical fuzzy hamacher aggregation operators, Int. J. Fuzzy Syst., Vol. 22, 2020, pp. 570-582.

18.K. Ullah, H. Garg, T. Mahmood, N. Jan, and Z. Ali, Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making, Soft Comput., Vol. 24, 2020, pp. 1647-1659.

19.K. Ullah, T. Mahmood, and N. Jan, Similarity measures for T-spherical fuzzy sets with applications in pattern recognition, Symmetry, Vol. 10, 2018, pp. 193.

20.P. Liu, Q. Khan, T. Mahmood, and N. Hassan, T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their application in multi-attribute group decision making, IEEE Access, Vol. 7, 2019, pp. 22613-22632.

21.M. Munir, H. Kalsoom, K. Ullah, T. Mahmood, and Y.M. Chu, T-spherical fuzzy einstein hybrid aggregation operators and their applications in multi-attribute decision making problems, Symmetry, Vol. 12, 2020, pp. 365.

22.S. Zeng, H. Garg, M. Munir, T. Mahmood, and A. Hussain, A multi-attribute decision making process with immediate probabilistic interactive averaging aggregation operators of T-spherical fuzzy sets and its application in the selection of solar cells, Energies, Vol. 12, 2019, pp. 4436.

23.D. Ramot, R. Milo, M. Friedman, and A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst., Vol. 10, 2002, pp. 171-186.

24.A.M.D.J.S. Alkouri and A.R. Salleh, Complex intuitionistic fuzzy sets, AIP Conf. Proc., Vol. 1482, 2012, pp. 464-470.

25.H. Garg and D. Rani, A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making, Appl. Intell., Vol. 49, 2019, pp. 496-512.

26.D. Rani and H. Garg, Distance measures between the complex intuitionistic fuzzy sets and their applications to the decision-making process, Int. J. Uncertain. Quan., Vol. 7, 2017, pp. 423-439.

27.K. Ullah, T. Mahmood, Z. Ali, and N. Jan, On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Complex Intell. Syst., Vol. 6, 2020, pp. 15-27.

28.M. Akram and S. Naz, A novel decision-making approach under complex Pythagorean fuzzy environment, Math. Comput. Appl., Vol. 24, 2019, pp. 73.

29.P. Liu, Z. Ali, and T. Mahmood, A method to multi-attribute group decision-making problem with complex q-rung orthopair linguistic information based on Heronian mean operators, Int. J. Comput. Intell. Syst., Vol. 12, 2019, pp. 1465-1496.

30.P. Liu, T. Mahmood, and Z. Ali, Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision making, Information, Vol. 11, 2020, pp. 5.

31.H. Garg, J. Gwak, T. Mahmood, and Z. Ali, Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications, Mathematics, Vol. 8, 2020, pp. 538.

32.L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-III, Inf. Sci., Vol. 9, 1975, pp. 43-80.

33.R. Degani and G. Bortolan, The problem of linguistic approximation in clinical decision making, Int. J. Approx. Reason., Vol. 2, 1988, pp. 143-162.

34.M. Delgado, J.L. Verdegay, and M.A. Vila, On aggregation operations of linguistic labels, Int. J. Intell. Syst., Vol. 8, 1993, pp. 351-370.

35.F. Herrera and L. Martinez, The 2-tuple linguistic computational model: Advantages of its linguistic description, accuracy and consistency, Int. J. Uncertain. Fuzziness Knowl. Based Syst., Vol. 9, 2001, pp. 33-48.

36.J.Q. Wang and H.B. Li, Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers, Control Decis., Vol. 25, 2010, pp. 1571-1574. In Chinese

37.P. Liu and S.M. Chen, Multiattribute group decision making based on intuitionistic 2-tuple linguistic information, Inf. Sci., Vol. 430, 2018, pp. 599-619.

38.X.D. Peng and Y. Yang, Multiple attribute group decision making methods based on Pythagorean fuzzy linguistic set, Comput. Eng. Appl., Vol. 52, 2016, pp. 50-54. In Chinese

39.G. Wei, M. Lu, F.E. Alsaadi, T. Hayat, and A. Alsaedi, Pythagorean 2-tuple linguistic aggregation operators in multiple attribute decision making, J. Intell. Fuzzy Syst., Vol. 33, 2017, pp. 1129-1142.

40.L. Li, R. Zhang, and X. Shang, Some q-rung orthopair linguistic Heronian meanoperators with their application to multi-attribute groupdecision making, Arch. Control Sci., Vol. 28, 2018, pp. 551-583.

41.Y. Ju, A. Wang, J. Ma, H. Gao, and E.D. Santibanez Gonzalez, Some q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making, Int. J. Intell. Syst., Vol. 35, 2020, pp. 184-213.

42.R.F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinburgh Math. Soc., Vol. 21, 1902, pp. 144-162.

43.P. Liu and D. Li, Some Muirhead mean operators for intuitionistic fuzzy numbers and their applications to group decision making, PLoS One, Vol. 12, 2017. e0168767

44.J. Zhu and Y. Li, Pythagorean fuzzy Muirhead mean operators and their application in multiple-criteria group decision-making, Information, Vol. 9, 2018, pp. 142.

45.J. Wang, R. Zhang, X. Zhu, Z. Zhou, X. Shang, and W. Li, Some q-rung orthopair fuzzy Muirhead means with their application to multi-attribute group decision making, J. Intell. Fuzzy Syst., Vol. 36, 2019, pp. 1599-1614.

46.W. Yang and Y. Pang, New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making, Int. J. Intell. Syst., Vol. 34, 2019, pp. 439-476.

47.G. Wei, C. Wei, J. Wang, H. Gao, and Y. Wei, Some q-rung orthopair fuzzy maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization, Int. J. Intell. Syst., Vol. 34, 2019, pp. 50-81.

48.G. Wei, F.E. Alsaadi, T. Hayat, and A. Alsaedi, Picture 2-tuple linguistic aggregation operators in multiple attribute decision making, Soft Comput., Vol. 22, 2018, pp. 989-1002.

49.Y. Ju, D. Ju, and A. Wang, A note on “Picture 2-tuple linguistic aggregation operators in multiple attribute decision making”, Soft Comput., Vol. 24, 2020, pp. 3937-3941.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 295 - 331
Publication Date: 2020/12/17
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.201207.003 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Peide Liu
AU  - Zeeshan Ali
AU  - Tahir Mahmood
PY  - 2020
DA  - 2020/12/17
TI  - Novel Complex T-Spherical Fuzzy 2-Tuple Linguistic Muirhead Mean Aggregation Operators and Their Application to Multi-Attribute Decision-Making
JO  - International Journal of Computational Intelligence Systems
SP  - 295
EP  - 331
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.201207.003
DO  - 10.2991/ijcis.d.201207.003
ID  - Liu2020
ER  -

download .riscopy to clipboard