Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.016727

ARTICLE

Novel Complex T-Spherical Dual Hesitant Uncertain Linguistic Muirhead Mean Operators and Their Application in Decision-Making

Shouzhen Zeng1,2,*, Zeeshan Ali3 and Tahir Mahmood3

1School of Business, Ningbo University, Ningbo, 315211, China
2College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China
3Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan
*Corresponding Author: Shouzhen Zeng. Email: zszzxl@163.com
Received: 20 March 2021; Accepted: 26 May 2021

1  Introduction

Owing to the extensive existence of uncertain information, some practical decision-making issues are often intractable and complicated, which are very difficult for a decision-maker to cope with. For managing such kinds of issues, the theory of intuitionistic fuzzy set (IFS) was discovered by Atanassove [1]. IFS is a modified version of the fuzzy set (FS) [2], composing the grade of truth and the grade of falsity with a condition that the sum of both grades’ cannot exceed the unit interval. Numerous scholars have widely explored the application of the IFS theory in different fields [3–5]. In general, the IFS must hold the limitation that the sum of both grades cannot exceed the unit interval. However, in awkward realistic decision issues, this limitation cannot always be held. For example, if a decision-maker provides the pair A=(0.7,0.6) for the grade of truth and the grade of falsity. As 0.7+0.6=1.3>1, then A cannot be handled by the IFS. To extend the information space that IFS cannot describe, Yager [6] discovered the Pythagorean fuzzy set (PyFS) with a constraint that the sum of the squares of both grades is limited in the interval [0, 1]. Compared to the IFS, the PyFS is more extensively proficient to handle awkward and vague information in realistic decision issues. However, there is still a problem in PyFS, i.e., when a decision-maker provides the pair A=(0.9,0.8) for the grade of truth and the grade of falsity, then A cannot be handled by PyFS and IFS as 0.92+0.82=0.81+0.64=1.45>1. To extend the PyFS’s information space, Yager [7] then discovered the q-rung orthopair fuzzy set (QROFS) with a constraint that the sum of the q-powers of both grades cannot exceed from [0, 1]. The QROFS is an useful extension of the PFS and IFS to solve the awkward and uncertain information in realistic decision issues. At present, Numerous applications of the QROFS have extensively been utilized in different fields [8–10].

There are no complications that the theory of IFS has an extensive technique to manage awkward and difficult information in real-life issues, but it still cannot precisely deal with some voting problems in reality. This category of voting divides into four parts, i.e., the vote in favor, abstinence, vote against, and refusal. For managing such kinds of problems, the theory of picture fuzzy set (PFS) was discovered by Cuong et al. [11]. The PFS composes the grade of truth, abstinence, and falsity with the condition that the sum of all grades shall limit in the unit interval. Until now, it has received numerous extensions and applications in different fields [12–14]. But, when a decision-maker provides the pair A=(0.6,0.5,0.3) for the grade of truth, the grade of abstinence, and the grade of falsity, the PFS cannot handle this evaluation as 0.6+0.5+0.3=1.4>1. To overcome this limitation, Mahmood et al. [15] then discovered the spherical fuzzy set (SFS) with a condition that the sum of the squares of the truth, abstinence, and falsity grades cannot exceed the unit interval. However, the SFS is useless to deal with the pair A=(0.9,0.8,0.7), wherein 0.9+0.8+0.7=2.4>1 and 0.92+0.82+0.72=0.81+0.64+0.49=1.94>1. To achieve a broader information space, Mahmood et al. [15] discovered the T-spherical fuzzy set (TSFS) with a situation that the sum of the q-powers of the truth, abstinence, and falsity grades shall list in the unit interval. Now the theory of TSFS has been extensively utilized in different areas [16–18].

The theory of complex IFS (CIFS) was discovered by Alkouri et al. [19]. CIFS is a basic modified version of complex FS (CFS) [20], composed by the grade of truth and falsity in the form of a complex number with the condition that the sum of the real part (also for the imaginary part) of both grades shall not exceed the unit interval. The graphical representation of the unit circle in a complex plane is discussed in the form of Fig. 1. It has attracted the attention of many researchers and has been widely utilized in various fields [21–23]. The CIFS must hold the limitation that the sum of the real part (also for the imaginary part) of both grades cannot exceed the unit interval. However, in awkward realistic decision issues, this limitation cannot be always held. Afterward, Ullah et al. [24] presented the complex PFS (CPFS) with a constraint that the sum of the real part (also for the imaginary part) of the squares of both grades’ cannot exceed from [0, 1]. Compared with the CIFS, the CPFS is more effective to cope with awkward and vague information in realistic decision issues. Based on the CPFS, Liu et al. [25,26] developed the complex QROFS (CQROFS) with the constraint that the sum of the real part (also for the imaginary part) of the q-powers of both grades cannot exceed from [0, 1] [27–33]. The geometrical interpretation of the existing notions is discussed with the help of Fig. 2.

Figure 1: Geometrical representation of the unit disc

Figure 2: Graphical representation of the IFS, PFS, and QROFS

The FS utilizes just one value to express the truth grade, nonetheless, decision-makers are often hesitant among a few qualities while building up the truth grade in extensive practical MADM issues. To successfully manage decision-makers high hesitancy, Torra [34] presented the idea of a hesitant FS (HFS), which permits the truth grade to be signified by a few single values rather than just one. Inferred from its great capability of finding fuzzy data and decision-makers hesitancy, HFS has been viewed as one of the most impressive and adaptable techniques in MADM methodology [35–37]. However, the HFS is coping only with the truth level, whereas it ignores the grade of falsity. Zhu et al. [38] then presented the dual HFS (DHFS), which contains the truth grade as well as the falsity grade in the form of subsets of [0, 1]. If compared with the HFS, the DHFS is more proficient and reliable to cope with complicated and awkward information in realistic decision issues. Numerous scholars have utilized the theory of DHFS in different fields [39–41].

In genuine decision-making, there are numerous complexities and difficulties which are hard to give attribute values by quantitative assessment. Nonetheless, they are anything but difficult to give the linguistic assessment. Since Zadeh [42] discovered the linguistic variable set (LVS), the exploration of multi-attribute decision-making issues based on linguistic assessment data has gotten rich accomplishments. Now and again, decision-makers do not communicate his/her feelings by choosing linguistic labels, but they can show the data by interval linguistic label, in other words, by uncertain linguistic variables (ULVs) [43,44]. 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 measures of information, such as aggregation operators, hybrid operators, 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 decision making, pattern recognition, and multi-attribute group decision making. All the prevailing approaches of decision-making, based on information measures, in picture fuzzy set (PFS), spherical fuzzy set (SFS) and T-spherical fuzzy sets (T-SFS) theories, deal with only the grades of truth, abstinence and falsity, which are real-valued. In CTSDHULS theory, truth, abstinence, and falsity grades are complex-valued and are represented in polar coordinates, with uncertain linguistic terms. The amplitude term corresponding to truth, abstinence and falsity degrees gives the extent of membership, abstinence, and non-membership of an object, whose real and unreal parts in the form of the finite subset of the unit interval. The phase terms are novel parameters of the truth, abstinence, and falsity degrees and these are the parameters that distinguish the CTSDHULS and traditional T-spherical dual hesitant uncertain linguistic set (TSDHULS) theories. The TSDHULS 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, abstinence, 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 illustrate the significance of the phase term, we give an example. Assume XYZ organization chooses to set up biometric-based participation gadgets (BBPGs) in the entirety of its workplaces spread everywhere in the country. For this, the organization counsels a specialist who gives the data concerning (i) demonstrates of BBPGs and (ii) creation dates of BBPGs. The organization needs to choose the most ideal model of BBPGs with its creation date all the while. Here, the issue is two-dimensional, to be specific, the model of BBPGs and the creation date of BBPGs. This kind of issue cannot be displayed precisely utilizing the conventional TSDHULS hypothesis as the TSDHULS hypothesis cannot handle both the measurements at the same time. The most ideal approach to address the entirety of the data given by the master is by utilizing the CTSDHULS hypothesis. The sufficiency terms in CTSDHULS might be utilized to give the organization’s choice regarding the model of BBPGs and the stage terms might be utilized to address the organization’s judgment concerning the creation date of BBPGs.

Presently, aggregation operator is one of the most important technique which will help not only in comparing one data entity with other but also show the extent of association between them and their direction. Also, CTSDHULSs have a powerful ability to model the imprecise and ambiguous information in real-world applications than the existing theories such as TSDHULS, spherical dual hesitant uncertain linguistic sets (SDHULS). Based on the ULVs, the theory of complex picture dual hesitant uncertain linguistic set (CPDHULS) is proposed, which is a very proficient technique with a condition that the sum of the supremum of the real part (also for the imaginary part) of the truth, abstinence, and falsity grades cannot exceed from the unit interval, i.e., max{0.2,0.3,0.4}+max{0.02,0.03,0.04}+max{0.1,0.2,0.3}=0.4+0.04+0.3=0.74≤1 and max{0.1,0.2,0.3}+max{0.01,0.02,0.03}+max{0.2,0.3}=0.3+0.03+0.3=0.63≤1. However, there is still a problem, given the pair {{0.2,0.3,0.7}ei2π{0.1,0.8,0.3},{0.02,0.03,0.07}ei2π{0.01,0.08,0.03},{0.1,0.2,0.4}ei2π{0.2,0.3}}, for the grade of truth, abstinence, and falsity, then the CPDHULS fails to cope with it. For coping with such types of issues, the theory of complex spherical dual hesitant uncertain linguistic set (CSDHULS) is a very proficient technique with the condition that the sum of the square of the supremum of the real part (also for the imaginary part) of the truth, abstinence, and falsity grades cannot exceed from the unit interval, i.e., (max{0.2,0.3,0.7})2+(max{0.02,0.03,0.07})2+(max{0.1,0.2,0.4})2=0.72+0.072+0.42=0.49+0.0049+0.16=0.6549≤1 and (max{0.1,0.8,0.3})2+(max{0.01,0.08,0.03})2+(max{0.2,0.3})2=0.82+0.082+0.32=0.64+0.0064+0.09=0.7364≤1. However, there is still a problem, when a decision-maker gives the pair {{0.2,0.3,0.9}ei2π{0.1,0.8,0.3}+{0.02,0.03,0.09}ei2π{0.01,0.08,0.03},{0.1,0.2,0.8}ei2π{0.2,0.7}}, for the grade of truth, abstinence, and falsity, then the CPDHULS and CSDHULS are not able to cope with it.

For coping with such types of issues, based on the work of complex TSFS (CTSFS) [45], in this paper, we shall develop a new fuzzy tool, called the theory of complex T-spherical dual hesitant uncertain linguistic set (CTSDHULS), a very proficient technique that can effectively solve the deficiency of existing methods in describing uncertain information. The summary of the discovered theory of this manuscript is followed as:

(1) To explore the theory of CTSDHULS and their operational laws.

(2) To develop some aggregation methods for the CTSDHULS, including the CTSDHULMM operator, CTSDHULWMM operator, CTSDHULDMM operator, and CTSDHULWDMM operator, and discuss some of their cases.

(3) A MADM technique is utilized on basis of the explored operators. The enterprise informatization level evaluation issue is presented to verify the proficiency and capability of the discovered approaches.

(4) Finally, the comparative analysis and graphical expressions of the explored works are further developed to express more extensive and flexibility than the existing methods.

The purpose of this study is organized in the following ways: In Section 2, we recall the notion of ULSs, DHFSs, CTSFSs, Muirhead mean (MM) operator, Dual Muirhead mean (DMM) operator and their operational laws. In Section 3, the notion of CTSDHULS and their operational laws are discovered. In Section 4, we explore the CTSDHULMM operator, CTSDHULWMM operator, CTSDHULDMM operator, and CTSDHULWDMM operator. Additionally, the special cases of the presented work are also discussed. In Section 5, a MADM technique is constructed and applied to solve the enterprise informatization level evaluation. Finally, a deep comparative analysis is presented to verify the proficiency and capability of the discovered approaches. The conclusion of this manuscript is discussed in Section 6.

2  Preliminaries

In this study, we recall the notion of ULSs, DHFSs, CTSFSs, Muirhead mean (MM) operator, Dual Muirhead means (DMM) operator, and their operational laws which will be used fully in the next section.

Definition 1: [42] A LTS is initiated by:

L={L0,L1,L2,…,Lk−1} (1)

where k should be odd, which holds the following conditions:

(1)   If k>k′, then Lk>Lk′;

(2)   The negative operator neg(Lk)=Lk′ with a condition k+k′=k+1;

(3)   If k≥k′, max(Lk,Lk′)=Lk, and if k≤k′, max(Lk,Lk′)=Lk.

Additionally, L^={Li:i∈R} denotes the LTSs. A set L=[Li,Lj],Li,Lj∈L^(i≤j) is called ULVs, where Li,Lj represent the upper and lower limits of L, respectively [45].

Definition 2: [38] A DHFS QDH is initiated by:

QDH={(MQDH(x),NQDH(x)):x∈XUNI} (2)

where the symbols MQDH={MQR−1,MQR−2,…,MQR−n} and NQR−m}NQDH={NQR−1,NQR−2,…, express the grade of truth and falsity in the form of a subset of [0, 1], with the condition: 0≤max(MQDH)+max(NQDH)≤1. The pair QDH=(MQDH,NQDH) is called dual hesitant fuzzy numbers (DHFNs).

Definition 3: [45] A CTSFS QCQ is initiated by:

QCQ={(MQCQ(x),AQCQ(x),NQCQ(x)):x∈XUNI} (3)

where the symbols MQCQ=MQRPei2π(MQIP),AQCQ=AQRPei2π(AQIP) and NQCQ=NQRPei2π(NQIP) represent the grade of truth, abstinence, and falsity with the conditions: 0≤MQRPqSC+AQRPqSC+NQRPqSC≤1 and 0≤MQIPqSC+AQIPqSC+NQIPqSC≤1, qSC≥1. Additionally, πQCQ=πQRPei2π(πQIP)=(1−(MQRPqSC+AQRPqSC+NQRPqSC))1qSCei2π(1−(MQIPqSC+AQIPqSC+NQIPqSC))1qSC is called the refusal grade. The pair QCQ=(MQRPei2π(MQIP),AQRPei2π(AQIP),NQRPei2π(NQIP)) is called complex T-spherical fuzzy numbers (CTSFNs). For any two CTSFNs

QCQ−1=(MQRP−1ei2π(MQIP−1),AQRP−1ei2π(AQIP−1),NQRP−1ei2π(NQIP−1)) and

QCQ−2=(MQRP−2ei2π(MQIP−2),AQRP−2ei2π(AQIP−2),NQRP−2ei2π(NQIP−2)),

then we define some operational laws, such that:

QCQ−1⊕QCQ−2=((MQRP−1qSC+MQRP−2qSC−MQRP−1qSCMQRP−2qSC)1qSCei2π(MQIP−1qSC+MQIP−2qSC−MQIP−1qSCMQIP−2qSC)1qSC,AQRP−1AQRP−2ei2π(AQIP−1AQIP−2),NQRP−1NQRP−2ei2π(NQIP−1NQIP−2)) (4)

QCQ−1⊗QCQ−2=(MQRP−1MQRP−2ei2π(MQIP−1MQIP−2),(AQRP−1qSC+AQRP−2qSC−AQRP−1qSCAQRP−2qSC)1qSCei2π(AQIP−1qSC+AQIP−2qSC−AQIP−1qSCAQIP−2qSC)1qSC,(NQRP−1qSC+NQRP−2qSC−NQRP−1qSCNQRP−2qSC)1qSCei2π(NQIP−1qSC+NQIP−2qSC−NQIP−1qSCNQIP−2qSC)1qSC) (5)

ΥSCQCQ−1=((1−(1−MQRP−1qSC)ΥSC)1qSCei2π(1−(1−MQIP−1qSC)ΥSC)1qSC,AQRP−1ΥSCei2π(AQIP−1ΥSC),NQRP−1ΥSCei2π(NQIP−1ΥSC)) (6)

QCQ−1ΥSC=(MQRP−1ΥSCei2π(MQIP−1ΥSC),(1−(1−AQRP−1qSC)ΥSC)1qSCei2π(1−(1−AQIP−1qSC)ΥSC)1qSC,(1−(1−NQRP−1qSC)ΥSC)1qSCei2π(1−(1−NQIP−1qSC)ΥSC)1qSC) (7)

Additionally, we introduce the score function (SF) and accuracy function (AF) of the CTSFN as below:

SSF(QCQ−1)=(MQRP−1qSC+MQIP−1qSC−AQRP−1qSC−AQIP−1qSC−NQRP−1qSC−NQIP−1qSC3) (8)

HAF(QCQ−1)=(MQRP−1qSC+MQIP−1qSC+AQRP−1qSC+AQIP−1qSC+NQRP−1qSC+NQIP−1qSC3) (9)

The relationship between any two CTSFNs can be cleared with the help of the following laws:

(1)   If SSF(QCQ−1)>SSF(QCQ−2), then QCQ−1>QCQ−2;

(2)   If SSF(QCQ−1)=SSF(QCQ−2); then

(1) If HAF(QCQ−1)>HAF(QCQ−2), then QCQ−1>QCQ−2;

(2) If HAF(QCQ−1)=HAF(QCQ−2), then QCQ−1=QCQ−2.

Definition 4: [46] For any QPI−i(i=1,2,3,…,n), then the MM operator is initiated by:

MMΦ(QPI−1,QPI−2,…,QPI−n)=(1n!∑O∈Rn∏i=1nQPI−O(i)λSC−i)1∑i=1nλSC−i (10)

where Φ=(λSC−1,λSC−2,…,λSC−n)∈Rn denotes all possible permutations and O(i),i=1,2,3,…,n is any one of Rn.

Definition 5: [47] For any QPI−i(i=1,2,3,…,n), then the DMM operator is initiated by:

DMMΦ(QPI−1,QPI−2,…,QPI−n)=1∑i=1nλSC−i(∏O∈Rn∑i=1nλSC−iQPI−O(i))1n! (11)

where Φ=(λSC−1,λSC−2,…,λSC−n)∈Rn expresses all possible permutations and O(i),i=1,2,3,…,n is any one of Rn.

3  Complex T-Spherical Dual Hesitant Uncertain Linguistic Sets

In this study, we discover the idea of CTSDHULSs and their basic laws which are very helpful in the next section.

Definition 6: A CTSDHULS QCD is initiated by:

QCD={([LΔ(i),L∇(j)],(MQCD(x),AQCD(x),NQCD(x))):x∈XUNI} (12)

where the symbols

MQCQ={MQRP−1,MQRP−2,…,MQRP−n}ei2π{MQIP−1,MQIP−2,…,MQIP−n},

AQCQ={AQRP−1,AQRP−2,…,AQRP−n}ei2π{AQIP−1,AQIP−2,…,AQIP−n}and

NQCQ={NQRP−1,NQRP−2,…,NQRP−m}ei2π{NQIP−1,NQIP−2,…,NQIP−m}

express the grade of truth, abstinence, and falsity, whose real and imaginary parts are in the form of a subset of [0, 1], with a condition: 0≤(max(MQRP−i))qSC+(max(AQRP−i))qSC+(max(NQRP−1))qSC≤1 and 0≤(max(MQIP−i))qSC+(max(AQIP−i))qSC+(max(NQIP−1))qSC≤1,qSC≥1, with [LΔ(i),L∇(j)]∈L^. Additionally, πQCD=πQRP−iei2π(πQIP−i)=(1−(MQRP−iqSC+AQRP−iqSC+NQRP−iqSC))1qSCei2π(1−(MQIP−iqSC+AQIP−iqSC+NQIP−iqSC))1qSC is called the refusal grade. The pair QCD=([LΔi,L∇j].(MQCD,AQCD(x),NQCD)) is called the complex T-spherical dual hesitant uncertain linguistic numbers (CTSDHULNs).

Definition 7: For any two CTSDHULNs QCD1=([LΔ1,L∇1].(MQCD1,AQCD1,NQCD1)) and QCD2=([LΔ2,L∇2].(MQCD2,AQCD2,NQCD2)), then some operational laws are defined as follows:

QCD1⊕QCD2=([LΔ1+Δ2,L∇1+∇2],(∐(MQRP−1∈MQCD1,MQRP−2∈MQCD2)(MQRP−1qSC+MQRP−2qSC−MQRP−1qSCMQRP−2qSC)1qSCei2π.∐(MQIP−1∈MQCD1,MQIP−2∈MQCD2)(MQIP−1qSC+MQIP−2qSC−MQIP−1qSCMQIP−2qSC)1qSC,∐(AQRP−1∈AQCD1,AQRP−2∈AQCD2)(AQRP−1AQRP−2)ei2π∐(AQIP−1∈AQCD1,AQIP−2∈AQCD2)(AQIP−1AQIP−2),∐(NQRP−1∈NQCD1,NQRP−2∈NQCD2)(NQRP−1NQRP−2)ei2π∐(NQIP−1∈NQCD1,NQIP−2∈NQCD2)(NQIP−1NQIP−2))) (13)

QCD1⊗QCD2=([LΔ1×Δ2,L∇1×∇2],(∐(MQRP−1∈MQCD1,MQRP−2∈MQCD2)(MQRP−1MQRP−2)ei2π∐(MQIP−1∈MQCD1,MQIP−2∈MQCD2)(MQIP−1MQIP−2),∐(AQRP−1∈AQCD1,AQRP−2∈AQCD2)(AQRP−1qSC+AQRP−2qSC−AQRP−1qSCAQRP−2qSC)1qSCei2π.∐(AQRP−1∈AQCD1,AQRP−2∈AQCD2)(AQIP−1qSC+AQIP−2qSC−AQIP−1qSCAQIP−2qSC)1qSC,∐(NQRP−1∈NQCD1,NQRP−2∈NQCD2)(NQRP−1qSC+NQRP−2qSC−NQRP−1qSCNQRP−2qSC)1qSCei2π.∐(NQRP−1∈NQCD1,NQRP−2∈NQCD2)(NQIP−1qSC+NQIP−2qSC−NQIP−1qSCNQIP−2qSC)1qSC)) (14)

ΥSCQCD1=([LΥSC×Δ1,LΥSC×∇1],(∐MQRP−1∈MQCD1(1−(1−MQRP−1qSC)ΥSC)1qSCei2π(∐MQIP−1∈MQCD1(1−(1−MQIP−1qSC)ΥSC)1qSC),∐AQRP−1∈AQCD1AQRP−1ΥSCei2π(∐AQIP−1∈AQCD1AQIP−1ΥSC),∐NQRP−1∈NQCD1NQRP−1ΥSCei2π(∐NQIP−1∈NQCD1NQIP−1ΥSC))) (15)

QCD1ΥSC=([LΔ1ΥSC,L∇1ΥSC],(∐MQRP−1∈MQCD1MQRP−1ΥSCei2π(∐MQIP−1∈MQCD1MQIP−1ΥSC),∐AQRP−1∈AQCD1(1−(1−AQRP−1qSC)ΥSC)1qSCei2π(∐AQIP−1∈AQCD1(1−(1−AQIP−1qSC)ΥSC)1qSC),∐NQRP−1∈NQCD1(1−(1−NQRP−1qSC)ΥSC)1qSCei2π(∐NQIP−1∈NQCD1(1−(1−NQIP−1qSC)ΥSC)1qSC))) (16)

By using any two CTSDHULNs QCD1=([L1,L2],({0.3,0.9}ei2π{0.2,0.3,0.8},{0.03,0.09}ei2π{0.02,0.03,0.08},{0.2,0.8}ei2π{0.2,0.3})) and QCD2=([L2,L3],({0.31,0.91}ei2π{0.21,0.31,0.81},{0.031,0.091}ei2π{0.021,0.031,0.081},{0.21,0.81}ei2π{0.2,0.31})), for qSC=6 and ΥSC=2, then by using the Eqs. (13)–(16), we get the following results: