key: cord-0778352-9yd2hdpd
authors: Tchier, Fairouz; Ali, Ghous; Gulzar, Muhammad; Pamučar, Dragan; Ghorai, Ganesh
title: A New Group Decision-Making Technique under Picture Fuzzy Soft Expert Information
date: 2021-09-07
journal: Entropy (Basel)
DOI: 10.3390/e23091176
sha: b4f271d8152c6adebbfaa21791ff4abba72198e8
doc_id: 778352
cord_uid: 9yd2hdpd

As an extension of intuitionistic fuzzy sets, the theory of picture fuzzy sets not only deals with the degrees of rejection and acceptance but also considers the degree of refusal during a decision-making process; therefore, by incorporating this competency of picture fuzzy sets, the goal of this study is to propose a novel hybrid model called picture fuzzy soft expert sets by combining picture fuzzy sets with soft expert sets for dealing with uncertainties in different real-world group decision-making problems. The proposed hybrid model is a more generalized form of intuitionistic fuzzy soft expert sets. Some novel desirable properties of the proposed model, namely, subset, equality, complement, union and intersection, are investigated together with their corresponding examples. Two well-known operations AND and OR are also studied for the developed model. Further, a decision-making method supporting by an algorithmic format under the proposed approach is presented. Moreover, an illustrative application is provided for its better demonstration, which is subjected to the selection of a suitable company of virtual reality devices. Finally, a comparison of the initiated method is explored with some existing models, including intuitionistic fuzzy soft expert sets.

Multi-attribute group decision making (MAGDM) is an efficient procedure that has the ability to provide the rankings for an available finite family of objects based on multiple parameters associated with these objects. A significant problem in practical decisionmaking processes is how to describe a numeric value to a given alternative more accurately and efficiently. Due to the existence of fuzziness in various complex decision-making realworld problems, it was not possible to describe objects with exact values. To overcome this issue, Zadeh [1] was the first who initiated the notion of a fuzzy set (FS), which is a superset of the classical set. FS actually deals with the conception of partial truth between "absolute true" and "absolute false". The membership function that delivers the membership values to objects from closed unit interval is very important.

The theory of FSs cannot work properly in some practical situations. For instance, when an expert gives judgment on a piece of information involving non-membership degree of an object that is obtained by considering the standard negation of the membership degree.

To deal with such types of difficulties, Atanassov [2] proposed the theory of intuitionistic fuzzy sets (IFSs) as a generalization of FSs. Atanassov [2] modified the definition of FSs the concepts of SESs by combining them with fuzzy theory, thus introducing fuzzy SESs. The powerful concept of SESs inspired many researchers to solve various group decisionmaking problems using the SESs, as in [35] [36] [37] . For instance, Broumi and Smarandache [38] introduced intuitionistic fuzzy SESs (IFSESs) and discussed their applications. Qudah and Hassan [39] introduced the bipolar fuzzy SES model and provided its applications. Ali and Akram [40] introduced N-SESs and fuzzy N-SESs with their applications in MAGDM situations under multinary information. Moreover, Akram et al. [41] introduced the m-polar fuzzy SES model by the combination of m-polar FSs with SESs and discussed its applications to solve MAGDM problems. Very recently, Ali et al. [42] proposed a novel hybrid model called fuzzy bipolar SESs and studied its application in group decision making.

Some of the prominent existing methods on the theory of PFSs are:

• The PFSS presented by Yang et al. [30] . •

The interval-valued PFSs developed by Cuong et al. [43] . •

The multi-valued PFSs proposed by Jan et al. [44] .

From the above studies, it has been observed that several models, including PFSs or PFSSs or interval-valued PFSs have been proposed in the last decade to compile effectively picture fuzzy information; however, an efficient hybrid model by combining the PFSs with SESs is still unattended. The main reasons behind this construction are outlined as below:

1.

The hybrid model, namely, IFSESs [38] is actually deal with two-dimensional information evaluated by multiple experts with respect to multiple parameters. This model fails to deal with the important idea of neutrality degree, which can be observed in various real-life situations when we face the experts' opinions in different types such as yes, no, abstain, refusal. For instance, in medical diagnosis, neutrality degree can be considered, that is, specific illnesses (heart or chest problems) may not have symptoms such as headache and temperature. In a similar manner, the symptoms chest pain and stomach pain have a neutral effect on different diseases, including typhoid, malaria and viral fever.

The concepts of PFS and SS are combined by Yang et al. [30] to form a novel hybrid model called PFSSs but this model cannot properly deal with multiple experts. We establish a novel hybrid model called picture fuzzy SESs by combining the PFSs with SESs in order to adequately deal with multiple experts.

The major contributions of the developed picture fuzzy SES (PFSES, henceforth) model are:

Inspired by the strength of PFSs to deal with uncertain and vague information in real-world problems, this paper focuses on initiating a new hybrid model, namely, PFSESs, as a combination of PFSs with SESs.

Some of its desirable properties, namely, subset, complement, union, intersection, OR operation and AND operation are investigated via corresponding examples.

A decision-making algorithm is developed based on PFSESs.

An illustrative application is provided for the better demonstration of the proposed approach.

Further, to prove the efficiency and reliability, the benefits and comparison of proposed hybrid model with some existing models, including intuitionistic fuzzy SESs are explored.

For more fruitful basic notions, the readers are referred to [45] [46] [47] [48] [49] [50] [51] .

The remaining sections of this paper are arranged as: Section 2 provides a detailed review of some fundamental notions, including SES, PFS, score function and accuracy function for PFSs and PFSSs. Section 2 presents a novel hybrid model called PFSESs as an efficient extension of PFSSs or IFSESs. Further, some basic properties and operations such as subset, complement, union, intersection, OR operation and AND operation are also investigated for PFSESs through illustrative numerical examples. Section 4 provides a daily-life application for the better demonstration of the proposed approach. Section 5 studies the benefits and comparison of presented model with existing ones, including

. Let X be a universal set, then for any

, its score and accuracy functions are given below:

∀ x ∈ X . Notice that for any two PFNs P 1 and P 2 , we say that P 1 is less than P 2 if S(P 1 ) < S(P 2 ), In case if S(P 1 ) = S(P 2 ), then we use accuracy function to compute whether given PFNs are equal or not. Now if H(P 1 ) < H(P 2 ), we say that P 1 is less than P 2 and if H(P 1 ) = H(P 2 ) then P 1 = P 2 .

Definition 5 ([30] ). Let X be a universe and Q be a set of parameters. For each A ⊆ Q, A pair ( f , A) is said to be a picture fuzzy soft set or PFSS over X where f is a function given as below:

f : A → P F (X ) .

This section provides the main notion of this study, namely, PFSESs together with some fundamental properties of the model that are explained by illustrative examples. Definition 6. Let X be a universe, Q a universe of parameters, E a set of experts and OP = {1 = agree, 0 = disagree} be a set of their opinions. For each A ⊆ S with S = Q × E × OP, a pair (Υ, A) is called a picture fuzzy soft expert set or PFSES where Υ is a function given as:

In set notation: the PFSES (Υ, A) over the universal set X is given below:

The tabular representation is a more precise and compact way to represent a PFSES (Υ, A). Assume that X = {x 1 , x 2 , . . . , x n } is a universal set, and Q = {q 1 , q 2 , . . . , q m } is a universe of parameters about the elements of X . Let E = {e 1 , e 2 , . . . , e t } be a collection of experts and OP = {0 = disagree, 1 = agree} be their opinions. Then, a PFSES (Υ, A) can also be presented by tabular arrangement as displayed in Table 1 . (q 1 , e 1 , 1) ζ A (q 1 , e 1 , 1), A (q 1 , e 1 , 1), γ A (q 1 , e 1 , 1) . . . ζ A (q 1 , e 1 , 1), A (q 1 , e 1 , 1), γ A (q 1 , e 1 , 1) (q 1 , e 2 , 1) ζ A (q 1 , e 2 , 1), A (q 1 , e 2 , 1), γ A (q 1 , e 2 , 1) . . . ζ A (q 1 , e 2 , 1), A (q 1 , e 2 , 1), γ A (q 1 , e 2 , 1) . . . . . . . . . . . .

(q m , e t , 1) ζ A (q m , e t , 1), A (q m , e t , 1), γ A (q m , e t , 1) . . . ζ A (q m , e t , 1), A (q m , e t , 1), γ A (q m , e t , 1) (q 1 , e 1 , 0) ζ A (q 1 , e 1 , 0), A (q 1 , e 1 , 0), γ A (q 1 , e 1 , 0) . . . ζ A (q 1 , e 1 , 0), A (q 1 , e 1 , 0), γ A (q 1 , e 1 , 0) (q 1 , e 2 , 0) ζ A (q 1 , e 2 , 0), A (q 1 , e 2 , 0), γ A (q 1 , e 2 , 0) . . . ζ A (q 1 , e 2 , 0), A (q 1 , e 2 , 0), γ A (q 1 , e 2 , 0) . . . . . . . . . . . .

(q m , e t , 0) ζ A (q m , e t , 0), A (q m , e t , 0), γ A (q m , e t , 0) . . . ζ A (q m , e t , 0), A (q m , e t , 0), γ A (q m , e t , 0) Example 1. Consider a company that wants to buy electric cars with the rapid development of the global electric commercial vehicle market in the world from six substitutes X = {x 1 = Mini Electric, x 2 = Tesla Model S, x 3 = BMW i3, x 4 = Hyundai Kona EV, x 5 = Polestar 2, x 6 = Jaguar I − PACE}. To choose the best electric car, the organization needs to take the opinions of three specialists E = {e 1 , e 2 , e 3 }. Let Q = {q 1 = touch screens, q 2 = f unky design, q 3 = rapid charging, q 4 = millage capacity} be a favorable set of parameters provided by the company to help experts in the evaluation process to fulfill their needs accordingly. The experts provide their judgments in the form of a PFSES (Υ, A), which is given in Table 2 where A ⊆ S = Q × E × OP. From Table 2 , it can be readily seen that the information closed in the first cell (0.4, 0.2, 0.2) is explained well as: The expert e 1 agrees if the belongingness, neutrality and non-belongingness degrees of the electric car 'x 1 ' are 0.4, 0.2 and 0.2, respectively, with respect to parameter q 1 , and so on for other cells of Table 2 . 

We now introduce some basic operations for PFSESs, including subset, equality, complement, union, intersection, AND and OR with corresponding examples. We start with the subset relation between PFSESs. A ⊆ B, 2.

Υ(a) is picture fuzzy soft expert subset of Ω(a) (symbolically, Υ(a) ⊆ Ω(a)) where a ∈ A and Υ(a) and Ω(a) are PFSs; therefore,

, for all x ∈ X . This subset relation is represented by (Υ, A)⊆(Ω, B). We can say, (Ω, B) is a picture fuzzy soft expert superset of (Υ, A).

Let us see an illustrative numerical example of picture fuzzy soft expert subset relation: Example 2. Consider Example 1 again and let (Υ, A) and (Ω, B) be two PFSESs over X given by Tables 3 and 4 , respectively, where A and B are given as: A = {(q 1 , e 2 , 1), (q 2 , e 1 , 1), (q 3 , e 1 , 1), (q 4 , e 3 , 1), (q 1 , e 3 , 0), (q 2 , e 3 , 0), (q 3 , e 2 , 0), (q 4 , e 1 , 0)} B = {(q 1 , e 2 , 1), (q 2 , e 1 , 1), (q 3 , e 1 , 1), (q 4 , e 3 , 1), (q 1 , e 3 , 0), (q 2 , e 3 , 0), (q 3 , e 2 , 0), (q 4 , e 1 , 0)} Clearly A ⊆ B and Υ(a) ⊆ Ω(a) for all a ∈ A. Thus (Υ, A)⊆(Ω, B). Table 3 . Tabular representation of the PFSES (Υ, A). Table 4 . Tabular representation of the PFSES (Ω, B).

(Ω, B) Table 5 below: Table 5 . Tabular representation of an agree-PFSES (Υ, A) 1 .

Example 4. Reconsider the PFSES (Υ, A) in Example 1. Then, its disagree-PFSES (Υ, A) 0 is provided by Table 6 . 

Definition 11. Let (Υ, A) be a PFSES on the universe X . Then, complement of PFSES (Υ, A) is represented by (Υ, A) c and is given as

Reconsider the PFSES (Υ, A) as defined in Example 1. Then, its complement (Υ, A) c is computed in Table 7 . Table 7 . Tabular form of the complement of PFSESs (Υ, A).

Definition 12. For any two PFSESs (Υ, A) and (Ω, B) on X , their union, denoted by (Υ, A) Table 9 below. (Ω, B) . Table 9 . Tabular form of the union of PFSESs (Υ, A) and (Ω, B) . 

It follows directly with similar arguments used in part (1). 

Reconsider the PFSES (Υ, A) in Example 1 and let (Ω, B) be another PFSES over X , which is displayed in Table 8 . Then, the intersection (Υ, A) (Ω, B) of PFSESs (Υ, A) and (Ω, B) is provided in Table 10 below. 

Proposition 3. Let (Υ 1 , A), (Υ 2 , B) and (Υ 3 , C) be three PFSESs on X . Then 1.

Proof.

From Definition 13, (Υ 1 , A) (Υ 2 , B) = (I 1 , J 1 ) with J 1 = A ∪ B and for all ϕ ∈ J 1 ,

Similarly, by Definition 13, (Υ 2 , B) (Υ 1 , A) = (I 2 , J 2 ) with J 2 = B ∪ A and for all ϕ ∈ J 2 ,

It follows easily from part (1).

, which is defined as: Table 13 . Table 11 . Tabular form of the PFSES (Υ, A). (Ω, B) 

Proposition 5. Let (Υ 1 , A), (Υ 2 , B) and (Υ 3 , C) be PFSESs over X . Then 1.

(Υ 1 , A)∧((Υ 2 , B)∧(Υ 3 , C)) = ((Υ 1 , A)∧(Υ 2 , B))∧(Υ 3 , C), 2.

(Υ 1 , A)∨((Υ 2 , B)∨(Υ 3 , C)) = ((Υ 1 , A)∨(Υ 2 , B))∨(Υ 3 , C), 3. (Υ 1 , A)∧((Υ 2 , B)∨(Υ 3 , C)) = ((Υ 1 , A)∧(Υ 2 , B))∨((Υ 1 , A)∧(Υ 3 , C)), 4. (Υ 1 , A)∨((Υ 2 , B)∧(Υ 3 , C)) = ((Υ 1 , A)∨(Υ 2 , B))∧((Υ 1 , A)∨(Υ 3 , C)).

Using Definition 14,

Using the Proposition 3, we obtain

The remaining parts followed similarly as part (1).

Virtual reality (VR) originated in 1957 by Morton Heilig. His interactive media gadget called the Sensorama is regarded one of the foremost VR systems. Actually, the term 'virtual reality' or VR was introduced in 1987 by analyzer Jaron Lanier, whose analysis and work promoted several areas of the VR industry. The technology of VR basically depends on the usage of computer engineering to generate a simulated circumstance. As a replacement for seeing a screen in front of the users, they are immersed and able to interconnect with a three-dimensional (3D) world. By simulating as many senses as achievable, namely, hearing, touch, sight and even smell, the computer is converted into a doorkeeper to this unnatural world. VR's most instantly identifiable element is the head-mounted display. There are many important scientific fields where VR is playing an outstanding role such as VR in healthcare, VR in education and VR in military. VR is utilized in multiple sectors of healthcare. Any type of medical circumstance can be simulated using VR, to permit the students to manage with it as in actual life. VR can be employed to improve student learning abilities. VR education can efficiently modify the process by which educational content is provided; it operates on the basis of producing a virtual real world or allows users to not only watch it but also interact with it. In the late 1920s and 1930s, flight trainers from the Link Company was the earliest utilization of simulators in a military environment. For training purposes, VR has also been used by military forces, including army, navy and air force. Further, the entertainment industry is one of the most enthusiastic applications of VR. At the beginning of 2017, the British Museum presented a VR experience of the British Museum, offering users an unparalleled digital experience by mobile device or using computer, and the opportunity to be completely engaged with a VR headset. The above discussion reveals an important fact that the selection of the best VR system manufacturing company is an uncertain problem due to multiple characteristics (parameters) of VR systems. So, it is a critical task for the buyers to choose the best option. The selection of an appropriate company manufacturing VR systems is possible with the help of different experts' evaluations about VR systems according to the favorable parameters of buyers (wholesale dealers). Suppose X = {x 1 = Applied VR, x 2 = Phaser Lock Interactive, x 3 = Lucid Sight, x 4 = Owlchemy Labs, x 5 = WEVR, x 6 = Unity Technologies} is the set of well-known companies that are producing VR systems. In order to obtain the best VR system producing company, consider Q = {q 1 , q 2 , . . . , q 12 } is the set of parameters used by a dealer for the selection of best company regarding the production of VR systems where • q 1 serves as interaction, • q 2 serves as video games, • q 3 serves as education, • q 4 serves as sensory feedback,

• q 5 serves as training, • q 6 serves as effective communication, • q 7 serves as convenience, • q 8 serves as comfort, • q 9 serves as building student skills, • q 10 serves as detail views, • q 11 serves as connect with people, • q 12 serves as realistic.

Assume that E = {e 1 , e 2 , e 3 } is a collection of three experts invited by the dealer to determine the most suitable company regarding manufacturing of VR systems and OP = {1, 0} is the set of opinions where 1 = agree and 0 = disagree. Suppose that experts provide their judgments in the form of a PFSES (Υ, A) where A ⊆ S = Q × E × OP. For simplicity, PFSES (Υ, A) is divided into agree and disagree components, respectively. In the following, Table 15 displays the agree-PFSES while Table 16 represents the disagree-PFSES. Table 17 . Score values of the agree-PFSES (Υ, A) 1 . 

We now provide an algorithm that will be helpful in the selection process for the best option under the picture fuzzy soft expert framework (see Algorithm 1).

By applying the above algorithm, the final score values are computed in the following Table 19 . 

From Table 19 , it can easily seen that the best choice for the dealer to select a virtual reality system producing company is x 4 ; therefore, the dealer will select company x 4 for his consignment order of VR systems.

To better understand the initiated algorithmic approach, its flowchart diagram is provided in Figure 1 . 

With the analysis of the last few decades, from the invention of FS theory to present day, we can easily observe that a number of research scholars from almost every domain of science put themselves into a race of producing different natural generalizations of FSs, such as IFSs, bipolar FSs, interval-valued FSs, PFSs, etc., or hybrid models of these extensions with other existing uncertainty theories, including soft sets, rough sets and SESs. One can easily notice that hybrid models of PFS with SES are still unable to handle picture fuzzy soft information more efficiently with 't' experts, t > 1. Motivated by this fact, in this study, a novel hybrid model called PFSESs is proposed by the combination of above-mentioned existing theories. Our developed model has the ability to deal with the evaluations of more than one expert regarding each alternative with respect to each parameter under consideration. The proposed approach is very reliable and feasible for dealing with imprecise and vague picture fuzzy soft expert information. Particularly, when the problem under consideration is based on picture fuzzy soft information collected with the judgments of different experts.

Many fruitful results have been produced as an extension of PFSs, such as PFSSs, to handle different problems of several scientific fields, including artificial intelligence. Since PFSS is a soft extension of the PFS model but fails to deal with the individual evaluation of more than one expert in a group decision-making situation, we created our proposed model as an efficient generalization of SESs, FSESs, IFSESs or PFSSs. We have checked the effectiveness of our proposed model by solving the application in Section 4 with the proposed approach and existing IFSES model. Clearly, we obtain optimal results; however, there is a minor change in their ranking order of sub-optimal decision objects. We provide the comparative analysis of the developed PFSES model in both qualitative and quantitative modes, which are displayed in Tables 20 and 21 and Figure 2 . We conclude that the initiated approach is more cogent and feasible to solve different real-world problems in the picture fuzzy soft expert environment. Table 21 . Comparison with existing hybrid models.

Soft sets [25] t = 1 Discrete data in soft form SESs [33] t ≥ 1 Discrete data in SES environment FSESs [34] t ≥ 1 Data in FSES form IFSESs [38] t ≥ 1 Data in IFSES environment Proposed PFSESs t ≥ 1 Data in PFSES form

In the past two decades, the impact of uncertainty theories in mathematical modelingbased real-world systems has increased substantially. A hybrid model is a combination of two or more theories that experts can use to make decisions. A good hybrid model has the ability to overcome some issue in existing theories and provide more accurate results than existing ones; however, it does not matter how good it is because models will almost always have limitations. In the following, we discuss the limitations of the initiated approach that we observed during its construction process.

The initiated model fails to address a situation involving three-dimensional information where membership value is 0.4, non-membership value is 0.7 and neutral value is 0.1. Clearly, 0.4 + 0.7 + 0.1 = 1.2 ≮ 1.

Since mathematical modeling is mainly dependent on the input data and evaluations. The speed of the proposed hybrid model regarding computation may be slow in the case of a large data-set. This deficiency is present in almost every existing model that can be overcome via an appropriate coding method with the help of software, including MATLAB.

Another difficulty of our initiated model is the change in rank of alternatives if existing parameters (or alternatives) are deleted or new parameters (or alternatives) are inserted in a group decision-making problem. The main reason behind these problems is the independent behavior of objects and parameters.

Group decision-making issues are of great significance in various fields ranging from medicine to engineering. To cope with such issues, PFSs are becoming a generic mathematical tool for handling imprecision and vagueness in different group decisionmaking situations. A noticeable growth is found regarding the use of PFSs and PFSSs in real-world decision-making issues. The major goal of this study is to present a new hybrid model, called PFSES, which is an extension of PFSS or FSES or IFSES. This novel concept can be employed to describe picture fuzzy knowledge in a more effective and general manner. Specifically, some fundamental operations, such as subset, equality, union, intersection, complement, OR operation, AND operation and agree-and disagree-PFSESs, are developed and are investigated with respective numerical examples. Further, certain De Morgan's laws for PFSESs are verified with respect to the AND and OR operations. Moreover, a method is presented to solve the MAGDM problems based upon the PFSES framework. Our proposed methodology is tested on a practical application to describe the validity and cogency of the developed hybrid model, that is, a selection of best VR system manufacturing company. Finally, a comparison of the presented approach with some existing models, including IFSESs [38] is provided. From Figure 2 and Table 20 , we have observed that the optimal decision object is the same (x 4 ) by solving the proposed application with both IFSESs and the initiated PFSES model. Thus, as an extension of PFSSs regarding experts, the presented approach for MAGDM is very feasible and more reliable than existing SES models. In future work, we are trying to extend our study with the following models:

Let (Υ, A) be a PFSES over the universal set X . Then, a disagree-PFSES (Υ, A) 0

on X is a picture fuzzy soft expert subset of (Υ, A) which is given below: (Υ, A) 0 = {Υ(a) : a ∈ Q × E × {0}}

Definition 15. Let (Υ, A) and (Ω, B) be two PFSESs over X . Then the operation 'OR' between PFSESs

Compute an agree-PFSES (Υ, A) 1 and disagree-PFSES (Υ, A) 0

By Definition 4, determine score values for both agree (Υ, A) 1 and disagree (Υ, A) 0 components of PFSES

Calculate accumulative scores a j = ∑ i e ij in the score table of agree-PFSES (Υ, A) 1

Calculate accumulative scores b j = ∑ i e ij in the score table of disagree-PFSES (Υ, A) 0

Determine z j = a j − b j , j = 1, 2, . . . , n. 7. Find r, for which z r = max z j . Output: The object having maximum final score value in step '7' will be the decision. In case if two or more values have similar final score then any one of them can be chosen as decision object

Rung orthopair picture fuzzy soft expert sets, 2. Interval-valued picture fuzzy soft expert sets, 3. Fuzzy parameterized picture fuzzy soft expert sets

Fuzzy sets

Intuitionistic fuzzy sets. Fuzzy Sets Syst

D numbers-FUCOM-fuzzy RAFSI model for selecting the group of construction machines for enabling mobility

A novel multi-criteria decision-making model for building material supplier selection based on entropy-AHP weighted TOPSIS

A fuzzy multiple criteria decision making approach with a complete user friendly computer implementation

Multi-criteria decision analysis towards robust service quality measurement

Picture Fuzzy Sets-First Results. Part 1 and Part 2, Seminar Neuro-Fuzzy Systems with Applications

Correlation coefficients for picture fuzzy sets

Generalized picture distance measure and applications to picture fuzzy clustering

Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arab

TODIM method for picture fuzzy multiple attribute decision making

Different approaches to multi-criteria group decision making problems for picture fuzzy environment

Career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory

Some similarity measures for picture fuzzy sets and their applications

CPT-TODIM method for picture fuzzy multiple attribute group decision making and its application to food enterprise quality credit evaluation

An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers

New operations on interval-valued picture fuzzy set, interval-valued picture fuzzy soft set and their applications

An adjustable weighted soft discernibility matrix based on generalized picture fuzzy soft set and its applications in decision making

MULTIMOORA based MCDM model for site selection of car sharing station under picture fuzzy environment

Picture fuzzy weighted distance measures and their application to investment selection

The linguistic picture fuzzy set and its application in multi-criteria decision-making: An illustration to the TOPSIS and TODIM methods based on entropy weight

Picture fuzzy extension of the CODAS method for multi-criteria vehicle shredding facility location

A dynamic distance measure of picture fuzzy sets and its application

Rough sets

Soft set theory-First results

On some new operations in soft set theory

An application of soft sets in a decision-making problem

A novel type of soft rough covering and its application to multicriteria group decision making

Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making

Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making

Group decision-making methods based on hesitant N-soft sets

Necessary and possible hesitant fuzzy sets: A novel model for group decision making

Soft expert sets

Fuzzy soft expert set and its application

Multi criteria decision making problem with soft expert set

Possibility fuzzy soft expert set

Cubic soft expert sets and their application in decision making

Intuitionistic fuzzy soft expert sets and its application in decision making

Bipolar fuzzy soft expert set and its application in decision making

Decision-making method based on fuzzy N-soft expert sets. Arab

Novel MCGDM analysis under m-polar fuzzy soft expert sets

Zain Ul Abidin, M. Ranking effectiveness of COVID-19 tests Using fuzzy bipolar soft expert sets

Some Operators on Interval-Valued Picture Fuzzy Sets and a Picture Clustering Algorithm on Picture Fuzzy Sets

Multi-valued picture fuzzy soft sets and their applications in group decision-making problems

Parameter reduction analysis under interval-valued m-polar fuzzy soft information

Einstein geometric aggregation operators using a novel complex interval-valued Pythagorean fuzzy setting with application in green supplier chain management

Resolving a location selection problem by means of an integrated AHP-RAFSI approach

Prioritizing the weights of the evaluation criteria under fuzziness: The fuzzy full consistency method-FUCOM-F

Picture fuzzy sets

Virtual reality technology: A tutorial

for all (α, β) ∈ A × B and x ∈ X . Example 9. Reconsider PFSESs (Υ, A) and (Ω, B), which are provided by Tables 11 and 12 , respectively. Then, the 'OR operation' between them is given by Table 14 . Informed Consent Statement: Not applicable.

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

The authors declare no conflict of interest.