A Novel Method for Determining Tourism Carrying Capacity in a Decision-Making Context Using q−Rung Orthopair Fuzzy Hypersoft Environment

1  Introduction

Making decisions can be daunting, particularly when we need more information and expertise in a specific area. However, we must not rely solely on our judgment but instead execute careful consideration and thoughtful analysis to make informed choices that result in favourable outcomes.

Different techniques have been used for decision-making problems, such as the application of RBF neural network optimal segmentation algorithm [1], stock intelligent investment strategy [2], smartphone app usage analysis [3], an algorithm for painting large objects [4], and multiscale feature extraction and multimodel fusion in visual question answering [5,6]. In 2022, Adak et al. [7] used a spherical distance measurement method for solving the MCDM problem under the Pythagorean fuzzy set. Debnath [8] used fuzzy hypersoft and developed a decision-making problem.

The idea of fuzzy logic was introduced by Zadeh [9,10], which involves the use of human experience to handle uncertain systems and assist decision-makers in making precise decisions [11,12]. Bellman and Zadeh proposed the concept of fuzzy sets [13]. Zadeh’s fuzzy set theory as a way to model uncertainty and imprecision in natural language, human reasoning, and complex systems. Since then, fuzzy set theory has been developed and extended in various directions, such as fuzzy logic, fuzzy relations, fuzzy measures, fuzzy analysis, possibility theory, type 2 fuzzy sets, etc. Fuzzy set theory has also found many applications in different disciplines, such as artificial intelligence, computer science, control engineering, decision theory, expert systems, logic, management sciences, operations research, robotics, and others [14].

However, the fuzzy set uses the degree of membership (MM) to describe the two states of support and opposition simultaneously. This may not be enough to grasp the uncertainty and imprecision in certain situations, where there may be a certain degree of hesitation or indeterminacy between support and opposition. To overcome this limitation, Atanassov [15] proposed an intuitionistic fuzzy set in 1983, an extension of a fuzzy set by adding a second index to measure the opposition state independently. In addition, a third index (i.e., degree of hesitation) can be derived from the degrees of MM and non-membership (N-MM) to quantify the state of indeterminacy. Since then, the intuitionistic fuzzy set theory has been developed and extended in various directions, such as interval values, type 2, neutrosophical, etc. Since then, the IFS idea has frequently been used to address real-world MCDM problems and challenges. IFS handles positive and negative membership grades only when the sum is less than or equal to 1. De et al. [16] created operations on intuitionistic fuzzy sets in 2002. Wang et al. [17] suggested several operations on IFS and created aggregation operators based on the fundamental operational laws. In addition, a multi-attribute decision-making (MADM) issue was created. Numerous studies [18,19] employed intuitionistic fuzzy sets in decision-making situations. The intuitionistic fuzzy set theory has also found many applications in different disciplines, such as artificial intelligence, computer science, control engineering, decision theory, expert systems, logic, management sciences, operations research, robotics, etc.

To overcome this limitation, Pythagorean fuzzy sets (PFS) were introduced by Yager et al. [20,21], which generalize IFS by ensuring that the square sum of MM and N-MM grades is equal to or less than 1. Several studies have developed different aggregation operators (AOs) in a Pythagorean fuzzy environment, distance measurement method and Pythagorean fuzzy power AOs [22,23].

In 2013, Cuong et al. [24] developed a picture fuzzy set. They assigned three degrees to each element of a universal set: a positive degree, a negative degree, and a neutral degree.

Hesitant fuzzy sets introduced by Torra [25] in 2010, hesitant fuzzy set assigns a set of possible degrees to each element of a universal set other than a single input.

Neutrosophic set initiated by Smarandache in 1998 [26], neutrosophic assigns three independent degrees to each element of a universal set, i.e., truth, indeterminacy, and falsity degree. The neutrosophic set can model the problem with a paradox or contradiction in the data, such as logic or philosophy.

Plithogenic set, also initiated by Smarandache in 2018 [27], are extensions of neutrosophic sets, where the truth, indeterminacy, and falsity degree are further refined into sub-degrees that shows different aspects of the data. For example, the truth degree can be divided into subjective, objective, and relative truth degrees.

However, fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets have some limitations, such as the inability to handle indeterminacy, lack of flexibility, and generality to represent different types of uncertainty. To overcome these problems, Yager further generalized the concept of Pythagorean fuzzy sets by defining q−Rung orthpair fuzzy sets in 2017 [28], which used a parameter q to control the degree of orthogonality between the degrees of MM and N-MM. q−ROF sets are a special case of orthopair fuzzy sets, defined by two functions that satisfy certain orthogonality conditions. The q−ROFS can be reduced to intuitionist and Pythagorean fuzzy sets when q equals one and two, respectively. q−ROF sets can also capture different types of uncertainty by varying the q value. Since their inception, q−ROF sets have attracted a lot of attention from researchers and practitioners and have been applied to various fields, such as decision-making, data mining, raw sets, topology, logic, etc.

The theory of soft sets, introduced by Molodtsov [29] in 1999, is a mathematical tool for managing uncertainty and inaccuracy in various fields. A software set can handle situations where the membership of an element to a set depends on the choice of attributes. However, the theory of soft sets has certain limitations, such as the inability to handle situations in which attributes must be divided into disjoint sets of attribute values or when more than one set of attributes is involved. Cagman et al. [30] developed a fuzzy soft set and its application. In 2010, Majumdar et al. [31] proposed the structure of a generalized fuzzy soft set. Many researchers used the theme of fuzzy soft sets and developed some operations on it [32,33].

To overcome these limitations, Smarandache [34] extended the concept of the soft set by defining the hypersoft set theory in 2018. A hypersoft set is a pair of a multi-argument function and a discourse universe, where the function maps several sets of attributes to subsets of the universe. A hypersoft set can handle situations where the MM of an element to a set depends on the choice of several attributes and their values. Since their inception, flexible and hypersoftset theories have attracted a lot of attention from researchers and practitioners and have been applied to various fields of decision-making.

Background: q−Rung orthopair fuzzy hypersoft set is a hybrid of q−Rung orthopair fuzzy soft and hypersoft set, which is used to express insufficient and indefinite information in decision-making problems [35]. This is a way for the q−Rung orthopair fuzzy hypersoft set that uses multiparameter approximation functions to handle the shortcomings of all other versions of the fuzzy set. It can deal not only with the MMD of NMMD but also with the degree of hesitation and indeterminacy. It has been applied to various fields, such as selecting construction companies, thermal energy storage techniques, and multi-attribute group decision-making [36].

Research gap: q−ROFHS is a hybrid concept of orthopair q−Rung fuzzy soft set and hypersoft set, used to express insufficient and indefinite information in decision-making problems. q−ROFHS is a generalization of intuitionist fuzzy sets, Pythagorean fuzzy sets, and q−ROFS, representing the DMM, the NMMD, and the hesitant degree of an element for a set. One of the reasons for using q−ROFHS is that it can capture more information and uncertainties than other fuzzy set extensions, such as intuitionist fuzzy sets, Pythagorean fuzzy sets, or q−ROFS. The q−ROFHS can represent more orthopairs that satisfy the limits of the orthopair fuzzy sets with qth power and can also incorporate the parameters of the soft sets and the membership functions of the hypersoft sets. Therefore, q−ROFHS can provide a more complete and realistic way to model complex and dynamic situations. The concept of q−Rung orthopair fuzzy hypersoft sets was introduced by Khan et al. [37,38], which utilizes specific operational rules and aggregation operators (AOs) to address various interactions among input arguments. Subsequently, Gurmani et al. [39] proposed different AOs-based basic operational laws. However, there has been no investigation on Einstein AOs in the existing literature despite efforts on fuzzy and hypersoft sets with q−Rung orthopairs. This paper addresses this gap by proposing using geometric and average Einstein AOs and developing software that integrates these operators into decision-making processes. The management and capacity planning of tourist attractions could be a potential area of application for the Einstein AOs.

The structure of this paper is as follows: In Section 2, we conduct a literature review of prior studies. Section 3 covers the fundamental materials. We introduce operational laws and aggregation operators (AOs) that can assist in decision-making problems in Section 4. Section 5 presents an algorithm for the Multi-Attribute Decision Making (MADM) technique that illustrates expected AOs in decision-making. To demonstrate the effectiveness of the proposed technique, we present a numerical example to analyze the technique and note that the final result resembles q−ROFHSN. In Section 6, we provide a comparative study of the proposed framework and existing structures. Finally, Section 7 concludes the paper by summarizing the results and highlighting future research directions.

1.1 Related Work

The tourist carrying capacity refers to the large number of visitors that a destination can accommodate sustainably without causing a negative impact on the environment, quality of residents, and culture. Some policymakers suggest balancing the development of tourism with the preservation of resources. Recently, Qiao et al. [40] studied embodiment theory and sensory compensation theory to examine the aspect of the tourism experience perspective of visually impaired tourists. Many researchers studied tourism with different perspectives, such as assessing quality tourism [41], tourism carrying capacity, a fuzzy approach [42], and economic and environmental impact of the tourism carrying capacity [43]. Many researchers worked on different decision-making approaches, such as deducting sudden rainstorm scenarios by decision-making [44] and an abstract syntax-based static fuzzing mutation for vulnerability evolution analysis [45]. Yuan et al. [46] in 2022 developed the system dynamic approach for evaluating the interconnection performance of cross-border transport. One of the applications of fuzzy sets and fuzzy logic in this estimation is the fuzzy linear programming model proposed by Fernández-Villarán et al. [47]. They developed a model to measure the TCC of an inhabited tourist destination (such as a country, region, or municipality), thanks to alerts that can help destination managers take action. The model considers the four dimensions of CBT: physical-ecological, social-demographic, economic, and perceptual. Each dimension has several indicators measured by vague numbers, representing the degree of satisfaction or dissatisfaction of tourists and residents with each indicator. Another example of using fuzzy sets and fuzzy logic to estimate CBT is the fuzzy set load capacity model (FTCC) proposed by Bertocchi et al. [48]. They focused on the case of Venice, one of the world’s most representative cases of over-tourism. Their objective is to determine a sustainable scenario for the number of tourists in Venice by looking for the best compromise between the local tourist sector’s monetary gains and the local population’s harmful effects on the destination. The model considers three types of tourists: tourists who sleep in hotels, tourists who sleep in other forms of accommodation, and day trips. Each type of tourist impacts the destination differently in terms of expenses, congestion, pollution, waste production, etc.

2  Materials and Methods

In this section, we gather some fundamental data that will be used to construct the outline of the article.

Definition 2.1. [35] The mathematical form of a q−Rung orthopair fuzzy hypersoft set (q−ROPHS set) over a universe of discourse X can be defined as:

Teij(mij)={⟨mij,φeij(mij),ψeij(mij)⟩:mij∈Ⓢ,eij∈® and q≥1}

where q−ROFH(Ⓢ) represents the family of all possible subsets of q−ROF X, where φij and ψij are the MM and N-MM with condition

0≤(φeij(mij))q+(ψeij(mij))q≤1,(q≥1)

For simplicity, Teij(mij)=⟨mij,φeij(mij),andψeij(mij)⟩ are denoted as Teij(mij)=⟨φeij,ψeij⟩ is represents a q−ROFHN.

Definition 2.2. [36] The q−Rung orthopair fuzzy hypersoft weighted average (q−ROFHWA) operator is a generalization of the traditional weighted average operator that considers the input information’s uncertainties and vagueness. The mathematical form of the q−ROFHWA operator can be expressed as follows:

q−ROFHWA(Ea11,Ea12,...,Eanm)=⊕j=1nvj(⊕i=1mwiEaij)(1)

where wi={1,2,...,n} and vj={1,2,...,m} are weight vectors with the condition wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1. And their mapping is defined as q−ROFHWA:Δn→Δ.

Definition 2.3. [36] The q−Rung orthopair fuzzy hypersoft weighted geometric (q−ROFHWG) operator is a generalization of the traditional weighted geometric operator that considers the uncertainties and vagueness of the input information. The mathematical form of the q−ROFHWG operator can be expressed as follows:

q−ROFHWG(Ea11,Ea12,...,Eanm)=⊗j=1nvj(⊗i=1mwiEaij)(2)

where wi={1,2,...,n} and vj={1,2,...,m} are weight vectors with the condition, wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1. And their mapping is defined as q−ROFHWG:Δn→Δ.

Definition 2.4. [37] The score function of the q−ROFHNs is defined as Seij=φeij(mij)q−ψeij(mij)q.

3  Use of Einstein Operations in the Context of q−ROFHNs

Einstein t-norms and t-conorms are constructed by using specific fixed values. Einstein operations refer to arithmetic operations employed to manipulate fuzzy sets. The Einstein operations for q−ROFHNs are presented below:

Definition 3.1. The Einstein product ⊗ and Einstein sum ⊕ are two fuzzy arithmetic operations commonly employed in fuzzy logic and fuzzy set theory. They represent t-norm and t-conorm, respectively, and are defined as follows:

T(a,♭)=a⊗♭=a♭1+(1−a)(1−♭);(3)

T∗(a,♭)=a⊕♭=a+♭1+a♭(4)

Our research now investigates Einstein’s operational laws, which can be described as follows:

Definition 3.2. For any two q−ROFHNs Ea11=(φa11,Ψa11) and Ea12=(φa12,Ψa12) with n > 2,

1.   Ea11⊕Ea12=((φ11q+φ12q1+φ11qφ12q)1q,(Ψ11Ψ12(1+(1−Ψ11q)(1−Ψ12q))1q))

2.   Ea11⊗Ea12=((φ11φ12(1+(1−φ11q)(1−φ12q))1q)1q,(Ψ11q+Ψ12q1+Ψ11qΨ12q)1q)

3.   λEa11=((1+φ11q)λ(1−φ11q)λ(1+φ11q)λ+(1−φ11q)λ)1q,(21qΨ11λ((2−Ψ11q)λ(Ψ11q))1q)1q

4.   Ea11λ=(21qφ11λ((2−φ11q)λ(φ11q))1q)1q,((1+Ψ11q)λ(1−Ψ11q)λ(1+Ψ11q)λ+(1−Ψ11q)λ)1q

Theorem 3.1. For any two q− ROFHNs Ea11=(φa11,Ψa11) and Ea12=(φa12,Ψa12) with any λ,λ1,λ2>2.

1.   Ea11⊗Ea12=Ea12⊗Ea11;

2.   Ea11⊕Ea12=Ea12⊕Ea11;

3.   Ea11λ⊗Ea12λ=(Ea11⊗Ea12)λ;

4.   λEa11⊕λEa12=λ(Ea11⊕Ea12);

5.   Ea11λ1⊗Ea11λ2=Ea11λ1+λ2;

6.   λ1Ea11⊕λ2Ea11=(λ1+λ2)Ea11;

3.1 Einstein Aggregation Operators

In this section, we will develop q−ROFHS Einstein weighted average (q−ROFHEWA), weighted geometric (q−ROFHEWG), ordered weighted average (q−ROFHEOWA), and ordered weighted geometric (q−ROFHEOWG) aggregation operators by using Einstein basic laws. These operators are then typically defined to overcome their q−ROFHN aggregation errors. The Einstein weighted average operator is a powerful tool for handling uncertainty and imprecision in data analysis and decision-making. It has many applications in areas such as finance, engineering, environmental science, and artificial intelligence, where decision-makers must combine multiple sources of uncertain information to reach a consensus or make a decision.

Definition 3.3. Let Eaij=(φij,Ψij) (i=1,2,...,m,t=1,2,...,n) be a q−ROFHNs, w, and v represents experts and attributes weights, respectively, with the condition Wj>0,∑j=1nWj=1,Vi>0,∑i=1mVi=1. We define the q−ROFHEWA operator as follows:

q−ROFHEWA(Ea11,Ea12,...,Eanm)=⨁j=1nWj(⨁i=1mViEaij)(5)

Theorem 3.2. Let Eaij=(φij,Ψij)\ (i=1,2,...,m,t=1,2,...,n)\ be q−\ROFHNs, with the weight of experts Wj\ and attributes Vi\ with Wj>0,∑j=1nWj=1,Vi>0,∑i=1mVi=1.\ Then, the aggregated result of the q−ROFHEWA operator is always q−\ROFHN obtained by the following equation:

q−ROFHEWA(Ea11,Ea12,...,Eanm)=⨁j=1nWj(⨁i=1mViEaij)=((∏j=1n(∏i=1m(1+φijq)Vi)Wj−∏j=1n(∏i=1m(1−φijq)Vi)Wj∏j=1n(∏i=1m(1+φijq)Vi)Wj+∏j=1n(∏i=1m(1−φijq)Vi)Wj)1q,21q∏j=1n(∏i=1m(Ψijq)Vi)Wj(∏j=1n(∏i=1m(2−Ψijq)Vi)Wj+∏j=1n(∏i=1m(Ψijq)Vi)Wj)1q)(6)

Proof. We will demonstrate the proof of this theorem using the mathematical induction method.

1.   When m=1,n=1 it follows that Vi=1,Wj=1 and thus the left-hand side of Eq. (5) can be expressed as follows:

q−ROFHEWA(Ea11,Ea12,...,Eanm)=Ea11=(φ11,Ψ11)

For the right side of Eq. (6) we have

=(((1+φ11q)−(1−φ11q)(1+φq)+(1−φq))1q,21q(Ψ11)(2−Ψ11q)+(Ψ11q)1q)=((2φ11q2)1q,21q(Ψ11)21q)=(φ11,Ψ11)

Consider the case where m=1,n=2. In this scenario, the following applies:

W1(V1Ea11)=((((1+φ11q)V1)W1−((1−φ11q)V1)W1((1+φ11q)V1)W1+((1−φ11q)V1)W1)1q,21q((Ψ11)V1)W1(((2−Ψ11q)V1)W1+((Ψ11q)V1)W1)1q)

W2(V1Ea12)=((((1+φ12q)V1)W2−((1−φ12q)V1)W2((1+φ12q)V1)W2+((1−φ12q)V1)W2)1q,21q((Ψ12)V1)W2(((2−Ψ12q)V1)W2+((Ψ12q)V1)W2)1q)q−ROFHEWA(Ea11,Ea12)=(W1(V1E11)⊕W2(V1E12))

((∏j=12((1+φ1wq)V1)Wj−∏j=12((1−φ1wq)V1)Wj∏j=12((1+φ1wq)V1)Wj+∏j=12((1−φ1wq)V1)Wj)1q,21q∏j=12((Ψ1tq)V1)Wj(∏j=12((2−Ψijq)V1)Wj+∏j=12((Ψijq)V1)Wj)1q)

Therefore, Eq. (6) holds for the values of m=1 and n=2.

3.   Assuming that Eq. (6) is valid for m=k,n=l, we obtain the following expression:

q−ROFHEWA(Ea11,Ea12,...,Eakl)=Ea11=(W1(V1Ea11))⊕(W2(V1Ea12))⊕...⊕(Wl(ViEaij))=⊕j=1lWj(⊕i=1kVkEakl)=((∏j=1l(∏i=1k(1+φijq)Vi)Wj−∏j=1l(∏i=1k(1−φijq)Vi)Wj∏j=1l(∏i=1k(1+φijq)Vi)Wj+∏j=1l(∏i=1k(1−φijq)Vi)Wj)1q,21q∏j=1l(∏i=1k(Ψijq)Vi)Wj(∏j=1l(∏i=1k(2−Ψijq)Vi)Wj+∏j=1l(∏i=1k(Ψijq)Vi)Wj)1q)

q−ROFHEWA(Ea11,Ea12,...,Ea(k+1)(l+1))=q−ROFHEWA(Ea11,Ea12,...,Eakl)⊕Ea(k+1)(l+1)=((∏j=1l(∏i=1k(1+φijq)Vi)Wj−∏j=1l(∏i=1k(1−φijq)Vi)Wj∏j=1l(∏i=1k(1+φijq)Vi)Wj+∏j=1l(∏i=1k(1−φijq)Vi)Wj)1q,21q∏j=1l(∏i=1k(Ψij)Vi)Wj(∏j=1l(∏i=1k(2−Ψijq)Vi)Wj+∏j=1l(∏i=1k(Ψijq)Vi)Wj)1q)⊕((((1+φ(k+1)(l+1)q)Vk+1)Wl+1−((1−φ(k+1)(l+1)q)Vk+1)Wl+1((1+φ(k+1)(l+1)q)Vk+1)Wl+1+((1−φ(k+1)(l+1)q)Vk+1)Wl+1)1q,21q((Ψ(k+1)(l+1))Vk+1)Wl+1(((2−Ψ(k+1)(l+1)q)Vk+1)Wl+1+((Ψ(k+1)(l+1)q)Vk+1)Wl+1)1q)=((∏j=1l+1(∏i=1k+1(1+φijq)Vi)Wj−∏j=1l+1(∏i=1k+1(1−φijq)Vi)Wj∏j=1l+1(∏i=1k+1(1+φijq)Vi)Wj+∏j=1l+1(∏i=1k+1(1−φijq)Vi)Wj)1q,21q∏j=1l+1(∏i=1k+1(Ψij)Vi)Wj(∏j=1l+1(∏i=1k+1(2−Ψijq)Vi)Wj+∏j=1l+1(∏i=1k+1(Ψijq)Vi)Wj)1q)

So the result is true for m=k+1 and n=l+1, and therefore true for all values of m,n.

The following theorem describes some fundamental properties of the proposed q−ROFHEWA operator.

Theorem 3.3. The q−ROFHEWA operator has the following properties:

1.   (Idempotency) If all Eaij(i=1,2,...,m,j=1,2,...,n) are equal, i.e., Eaij=Ea for all v and w, then q−ROFHEWA(Ea11,Ea12,...,Eaij)=Ea.

2.   (Boundedness) Let Eaij(i=1,2,...,m,j=1,2,...,n) be a family of q−ROFHNs, and let Ea−=(minφij,maxΨrt),Ea+=(maxφij,minΨrt), then Ea−⊆q−ROFHEWA(Ea11,Ea12,...,Eanm)⊆Ea+.

3.   (Monotonicity) Let Eaij(i=1,2,...,m,j=1,2,...,n) and E¨aij(i=1,2,...,m,j=1,2,...,n) be two sey = ts of q−ROFHNs, if Eaij⊆ E¨aij, for all v and w, then q−ROFHEWA(Ea11,Ea12,...,Eanm)⊆q−ROFHEWA(E¨a11,E¨a12,...,E¨aij).

Proof. (1) For Eaij=(φij,Ψrt)=Ea (i=1,2,...,m,j=1,2,...,n) by 3.5, the result is yielded below: q−ROFHEWA(Ea11,Ea12,...,Eanm)=⨁j=1nWj(⨁i=1mViEaij).

=((∏j=1n(∏i=1m(1+φijq)Vi)Wj−∏j=1n(∏i=1m(1−φijq)Vi)Wj∏j=1n(∏i=1m(1+φijq)Vi)Wj+∏j=1n(∏i=1m(1−φijq)Vi)Wj)1q,21q∏j=1n(∏i=1m(Ψij)Vi)Wj(∏j=1n(∏i=1m(2−Ψijq)Vi)Wj+∏j=1n(∏i=1m(Ψijq)Vi)Wj)1q)

=((((1+φijq)∑i=1mVi)∑j=1nWj−((1−φijq)∑i=1mVi)∑j=1nWj((1+φijq)∑i=1mVi)Wj+((1−φijq)∑i=1mVi)Wj)1q,21q((Ψij)∑i=1mVi)∑j=1nWj(((2−Ψijq)∑i=1mVi)∑j=1nWj+((Ψijq)∑i=1mVi)∑j=1nWj)1q)

=(((1+φijq)−(1−φijq)(1+φijq)+(1−φijq))1q,21qΨ((2−Ψijq)+Ψijq)1q)=((2φq2)1q,21qΨ(2)1q)=(φij,Ψrt)=Ea

(2) There exist the inequality Ea−≤Eaij≤Ea+ when Ea−=(minφij,maxΨrt) and Ea+=(maxφij,minΨrt). Thus, there also exists ⨁j=1lWj(⨁i=1kViEaij−)⊆⨁j=1lWj(⨁i=1kViEaij)⊆⨁j=1lWj(⨁i=1kViEaij+). Then the inequality Eaij−⊆⨁j=1lWj(⨁i=1kViEaij)⊆Eaij+ can be kept regarding the above properity (1), there is Eaij−⊆q−ROFHEWA(Ea11,Ea12,...,Eanm)⊆Eaij+.

(3) For Eaij⊆ E¨aij, there is the inequality ⨁j=1lWj(⨁i=1kViEaij)⊆⨁j=1lWj(⨁i=1kViE¨aij), i.e., q−ROFHEWA(Ea11,Ea12,...,Eanm)⊆q−ROFHEWA(E¨a11,E¨a12,...,E¨aij) exists. Therefore, all the above properties are true.

Further, we explain the q−ROFHEWG operator as follows:

Definition 3.4. Let Eaij=(φij,Ψij) (i=1,2,...,m,t=1,2,...,n) be a q−ROFHNs, w, and v represent the weight with the condition Wj>0,∑j=1nWj=1,Vi>0,∑i=1mVi=1. Then, the q−ROFHEWG operator is represented as:

q−ROFHEWG(Ea11,Ea12,...,Eanm)=⨂j=1nWj(⨂i=1mViEaij)

Theorem 3.4. Let Eaij=(φij,Ψij)\ (i=1,2,...,m,t=1,2,...,n)\ be q−\ROFHNs, with the weight of experts Wj\ and attributes Vi\ for Wj>0,∑j=1nWj=1,Vi>0,∑i=1mVi=1.\ Then the aggregated result of the q−ROFHEWG operator is again q−\ROFHN obtained by the following equation:

q−ROFHEWG(Ea11,Ea12,...,Eanm)=⨂j=1nWj(⨂i=1mViEaij)

=(21q∏j=1n(∏i=1m(φij)Vi)Wj(∏j=1n(∏i=1m(2−φijq)Vi)Wj+∏j=1n(∏i=1m(φijq)Vi)Wj)1q,(∏j=1n(∏i=1m(1+Ψijq)Vi)Wj−∏j=1n(∏i=1m(1−Ψijq)Vi)Wj∏j=1n(∏i=1m(1+Ψijq)Vi)Wj+∏j=1n(∏i=1m(1−Ψijq)Vi)Wj)1q,)(7)