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

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

Abstract: In this manuscript, the theory of complex T-spherical dual hesitant uncertain linguistic set is discovered, which is the mixture of three different ideas like the complex T-spherical fuzzy set, dual hesitant fuzzy set, and uncertain linguistic set. The complex T-spherical dual hesitant uncertain linguistic set composes the uncertain linguistic set, truth grade, abstinence grade, and falsity grade. Whose real and imaginary parts are the subset of a unit interval, and some of their operational laws are also presented. The theory of complex T-spherical dual hesitant uncertain linguistic Muirhead mean operator, complex T-spherical dual hesitant uncertain linguistic weighted Muirhead mean operator, complex T-spherical dual hesitant uncertain linguistic dual Muirhead mean operator and complex T-spherical dual hesitant uncertain linguistic weighted dual Muirhead mean operator are discovered. Some exceptional cases of the proposed operators are also examined. A multi-attribute decision making technique is further utilized based on explored operators. Moreover, an enterprise informatization level evaluation issue is resolved by using the presented operators to verify the proficiency and capability of the discovered approaches. Finally, some comparative analysis and advantages of the explored works are further developed to express that it is more flexible and effective than the existing methods.

Keywords: CTSDHULSs; Muirhead mean operators; MADM; enterprise informatization level evaluation

Abbreviations | |

CTSFS: | Complex T-spherical fuzzy sets. |

DHFS: | Dual hesitant fuzzy sets. |

ULS: | Uncertain linguistic sets. |

CTSDHULS: | Complex T-spherical dual hesitant uncertain linguistic sets. |

CTSDHULMM | Complex T-spherical dual hesitant uncertain linguistic Muirhead mean operator. |

CTSDHULWMM | Complex T-spherical dual hesitant uncertain linguistic weighted Muirhead mean operator. |

CTSDHULDMM | Complex T-spherical dual hesitant uncertain linguistic dual Muirhead mean operator. |

CTSDHULWDMM | Complex T-spherical dual hesitant uncertain linguistic weighted dual Muirhead mean operator. |

MADM: | Multi-attribute decision making. |

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

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

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.

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.,

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.

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:

where

(1) If

(2) The negative operator

(3) If

Additionally,

Definition 2: [38] A DHFS

where the symbols

Definition 3: [45] A CTSFS

where the symbols

then we define some operational laws, such that:

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

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

(1) If

(2) If

(1) If

(2) If

Definition 4: [46] For any

where

Definition 5: [47] For any

where

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

where the symbols

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:

Definition 7: For any two CTSDHULNs

By using any two CTSDHULNs

Additionally, we can examine the SF and AF of the CTSDHULN below:

For any two CTSDHULNs, we can examine their relationships with the help of the following laws:

(1) If

(2) If

1) If

2) If