Document [original]

Hafiz Muhammad Athar Farid, Muhammad Riaz, Bandar Almohsin,

Dragan Marinkovic

Optimizing filtration technology for contamination

control in gas processing plants using hesitant

q-rung orthopair fuzzy information aggregation

Open Access via institutional repository of Technische Universität Berlin

Document type

Journal article | Accepted version

(i. e. final author-created version that incorporates referee comments and is the version accepted for

publication; also known as: Author’s Accepted Manuscript (AAM), Final Draft, Postprint)

This version is available at

https://doi.org/10.14279/depositonce-18498

Citation details

Farid, H. M. A., Riaz, M., Almohsin, B., & Marinkovic, D. (2023). Optimizing filtration technology for

contamination control in gas processing plants using hesitant q-rung orthopair fuzzy information aggregation.

In Soft Computing. Springer Science and Business Media LLC. https://doi.org/10.1007/s00500-023-08588-w.

This version of the article has been accepted for publication, after peer review and is subject to Springer

Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements,

or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s00500-023-08588-w

This work is protected by copyright and/or related rights. You are free to use this work in any way permitted by

the copyright and related rights legislation that applies to your usage. For other uses, you must obtain

permission from the rights-holder(s).

Optimizing filtration technology for contamination

control in gas processing plants using hesitant q-rung

orthopair fuzzy information aggregation

Hafiz Muhammad Athar Farid*

, Muhammad Riaz †

, Bandar Almohsin ‡Dragan Marinkovic §

Abstract

The natural gas industry faces significant challenges in implementing industrial filtration systems that

are both sustainable and maintainable. These are complex issues that necessitate a thorough and pragmatic

evaluation to ensure that the technology is both effective and efficient. Due to the increasing importance

of environmental sustainability and the need to reduce waste and emissions, the deployment of filtration

technology must be carefully evaluated to ensure that it meets these objectives. The goal of this study is to

develop a decision-making approach based on Aczel-Alsina (AA) operations, which provide a variety of ad-

vantages when dealing with real-world issues. To begin, we’ll go over some new hesitant q-rung orthopair

fuzzy set (Hq-ROPFS) operations such the Aczel-Alsina product, sum, exponent, and scalar multiplication.

The suggested AOs are based on AA operations and are used to prioritize industrial filtering technologies

while keeping sustainability and maintainability in mind. Several solutions are being studied for manag-

ing impurities produced by Pakistan’s natural gas sector. After careful evaluation, it has been determined

that the cyclo-filter technology is the most viable solution for the natural gas industry. This technology has

been chosen due to its adaptability with the regional and local conditions of the study area. By utilizing

this technology, policymakers can gain practical insight into process energy and environmental systems, and

effectively control contamination in the natural gas industry. This decision-making framework offers a new

and innovative approach to address sustainability and maintainability challenges in the industry.

Keywords: Aczel–Alsina operations, q-rung orthopair fuzzy numbers, filtration technology, Aggregation op-

erator, gas processing plants.

1 Introduction

The world is currently experiencing a period of rapid population growth. According to the United Nations

(U.N), the global population is expected to reach 9.7 billion by 2050, up from 7.8 billion in 2021. As the popula-

*Department of Mathematics, University of the Punjab, Lahore, Pakistan. Email: [email protected]

†Department of Mathematics, University of the Punjab, Lahore, Pakistan. Email: [email protected]

‡Mathematics Department, College of Science, King Saud University, Saudi Arabia. Email: [email protected]

§Faculty of Mechanical Engineering and Transport Systems, Technische Universitaet Berlin, 10623 Berlin, Germany. Email

[email protected]

tion grows, so too does the demand for energy. At present, fossil fuel resources still represent a main contrib-

utor to the energy supply, despite the increasing awareness of the environmental impact of these resources.

This essay will explore the reasons why the global population is increasing and why fossil fuels continue to be

a dominant source of energy, as well as the challenges and opportunities associated with these trends.

Despite the increase in the use of renewable energy sources in recent years, fossil fuels still represent a

main contributor to the energy supply. The primary reason for this is their availability and low cost. Fossil

fuels, such as coal, oil, and natural gas, are abundant and relatively easy to extract and process, making them a

cost-effective option for energy production. Moreover, they have a higher energy density than other renewable

sources, such as wind and solar power, which makes them more suitable for heavy industries and large-

scale energy production. However, the use of fossil fuels has significant environmental and social impacts.

Burning fossil fuels releases greenhouse gases, such as carbon dioxide and methane, into the atmosphere,

which contribute to climate change. In addition, fossil fuel extraction and processing can cause air and water

pollution, which can have adverse health effects on local communities. Moreover, the reliance on fossil fuels

has led to political and economic instability, as countries compete for access to these resources and their

prices fluctuate based on market conditions [1]. To address the challenges associated with increasing global

population and the continued use of fossil fuels, governments and international organizations have taken

steps to promote sustainable development and renewable energy sources. The U.N has set 17 "sustainable

development goals" (SDGs) to achieve by 2030, including affordable and clean energy, zero hunger, and good

health and well-being. To achieve these goals, countries are investing in renewable energy technologies and

implementing policies to reduce greenhouse gas emissions and promote sustainable practices. For instance,

several countries have set targets for net-zero emissions by 2050, and the global shift to electric vehicles

is accelerating, reducing the reliance on fossil fuels for transportation. Moreover, the rapid development

of renewable energy technologies is making them increasingly cost-effective and efficient. According to the

"international renewable energy agency", the cost of renewable energy sources, such as wind and solar power,

has declined significantly in recent years, making them competitive with fossil fuels in some regions.

The natural gas processing plant is an essential part of the energy industry, producing natural gas that is

used for heating, electricity generation, and transportation. However, this process also generates a significant

amount of waste and contaminants that can have adverse effects on the environment and human health.

As such, there is an urgent need for controlling and removing these contaminants to ensure the safety and

sustainability of the natural gas production process [2].

Methane is one of the main impurities created by the natural gas processing facility. Methane is a strong

greenhouse gas, having a 100-year warming potential that is 28 times larger than carbon dioxide. Natural

gas sector methane emissions are a substantial contribution to climate change, accounting for around 2% of

worldwide greenhouse gas emissions. To reduce these emissions, it is essential to implement effective control

measures, such as leak detection and repair programs, to identify and fix methane leaks in the natural gas

production process. In addition to methane, natural gas processing plants also generate other air pollutants,

such as "volatile organic compounds" (VOCs), "nitrogen oxides (NOx), and particulate matter. These pollutants

can have adverse effects on human health, including respiratory problems, cardiovascular disease, and cancer.

To protect human health, it is essential to implement effective control technologies, such as catalytic converters

and scrubbers, to reduce these emissions from natural gas processing plants. The natural gas processing plant

also generates wastewater, which can contain high levels of contaminants, such as salt, heavy metals, and

hydrocarbons. This wastewater is typically treated before being discharged into the environment, but the

treatment process can be challenging, as the contaminants are often difficult to remove. To ensure the safe

disposal of this wastewater, it is essential to develop and implement effective treatment technologies, such as

reverse osmosis and ion exchange, to remove these contaminants from the wastewater before it is discharged.

Another significant challenge associated with the natural gas processing plant is the disposal of solid waste.

The natural gas production process generates significant amounts of solid waste, such as drill cuttings, which

can contain high levels of contaminants, such as heavy metals and hydrocarbons. The safe disposal of this

waste is critical to avoid environmental contamination and human health risks. To address this challenge, it

is essential to develop and implement effective waste management practices, such as recycling and reuse, to

reduce the amount of waste generated and ensure the safe disposal of the remaining waste.

Moreover, the natural gas processing plant can have adverse impacts on local communities, particularly

those living near the plant. The noise, odors, and other environmental impacts associated with the plant can

have a significant impact on the quality of life of nearby residents. To address these concerns, it is essential

to develop and implement effective community engagement strategies, such as stakeholder consultation and

public participation, to ensure that local communities are informed and involved in the natural gas production

process. To address the urgent need for controlling and removing the contaminants produced by the natural

gas processing plant, governments, and the natural gas industry must work together to develop and implement

effective control measures and technologies. This includes implementing leak detection and repair programs

to reduce methane emissions, installing control technologies to reduce air pollutant emissions, developing and

implementing effective treatment technologies to remove contaminants from wastewater, and implementing

waste management practices to reduce the amount of solid waste generated. It is essential to engage with

local communities and other stakeholders to ensure that the natural gas production process is transparent

and accountable. This includes informing local communities about the potential environmental and health

impacts of the natural gas processing plant, providing opportunities for public participation, and implementing

mitigation measures to address community concerns [3].

Controlling and eliminating the impurities produced by the natural gas processing facility is a multi-

criteria decision-making challenge requiring careful evaluation of a variety of considerations. The decision to

implement specific control measures or technologies to address the contaminants produced by the plant must

be based on a thorough analysis of the potential environmental and health impacts, as well as the economic

and social implications of each option. One of the primary factors to consider when making this decision is the

effectiveness of the control measures or technologies in reducing or removing contaminants [4]. This includes

evaluating the performance of different technologies in removing specific contaminants, as well as the overall

impact of the control measures on reducing environmental and health risks. Another crucial factor is the

economic viability of the control measures or technologies. This includes evaluating the costs of implementing

and maintaining different technologies and comparing them to the potential economic benefits of reducing

environmental and health risks. Overall, controlling and removing the contaminants produced by the natural

gas processing plant is a complex issue that requires careful consideration of multiple factors. By taking

a multi-criteria decision-making approach, governments and the natural gas industry can make informed

decisions that promote the safety and sustainability of the natural gas production process [5].

When dealing with single-criterion situations, decision-making is a pretty simple procedure. In this sce-

nario, all we have to do is choose the option with the maximum preference rating. When dealing with alter-

natives with numerous criterion, however, several challenges, such as weighted criteria, preference reliance,

and attribute conflicts, complicate the issues. To solve these challenges, more refined and complex approaches

must be developed. The initial stage in dealing with MCDM problems is to estimate how many qualities or

criteria are present in the situations, as well as how to identify them. Uncertainty is an inescapable character-

istic of most real-world issues, which may be owing to the high number of mishandling and invested interests

that occur during the information collection process. However, due to the system’s complexity, it is impossible

to display the information precisely. Several theories have lately been established in order to deal with data

imprecision.

In his key study, Zadeh [6] established the concept of the fuzzy set, which is a useful tool for dealing with

unclear information. Fuzzy sets are incapable of dealing with the scenario of a neutral state, that is, a condi-

tion in which neither support nor opposition exists. Following that, Atanassov [7] presented the "intuitionistic

fuzzy set" (IFS) theory by expanding on Zadeh’s approach. IFS concepts are not very adaptable to dealing with

complex decision-making difficulties. For example, when the sum of “membership degree (MSD) and non-

membership degree (NMSD)” is more than one, the previously discussed theories are inapplicable. To address

such issues, Yager [8] developed the idea of "Pythagorean fuzzy set", in which the sum of squares of MSD and

NMSD is less than or equal to one. Yager [9] first proposed the idea of q-ROPFSs. His theory holds that the

total of the qth powers of both MSD and NMSD is less than or equal to one. AOs are mathematical tools that

play an important part in information fusion by reducing a set of fuzzy numbers to a single fuzzy number that

is a single representative of several fuzzy numbers. The importance of information fusion in MCDM cannot be

overstated. Due to the wide spectrum of uncertain real-life scenarios, a great number of AOs for data fusion

have been proposed.

Bonferroni mean AOs [10], Heronian mean AOs [11, 18], confidence level [12], Aczel–Alsina AOs [13], AOs

for soft sets [14], connection numbers based AOs [15], basic AOs [16] and Dombi AOs [17] are the AOs pro-

posed by the different authors. Lin et al. [19] proposed the linguistic q-ROPFSs and interactional partitioned

Heronian mean AOs for linguistic q-ROPFSs. Khan et al. [20] initiated the notion of knowledge base for

q-ROPFSs. Zeng et al. [21] proposed weighted induced logarithmic distance measures for q-ROPFSs with

MCDM. Sitara et al. [22] graph structures related to q-ROPFS with decision-making analysis. Farid and Riaz

[23] proposed generalized q-ROPF Einstein interactive geometric AOs. Saha et al. [24] proposed the idea of

hybrid hesitant fuzzy weighted AOs. Attaullah et al. [25] initiated the concept of q-rung orthopair hesitant

fuzzy rough AOs. In daily life, people are typically reluctant and indecisive while making decisions, which

makes it difficult for the decision maker to reach a conclusion. Torra [26] developed the idea of "hesitant

fuzzy set" (HFS) to address such situations. In this theory, the grade of membership for each element of the

reference set consists of a series of discrete values from the interval [0, 1] rather than a single number. In

uncertain fuzzy information, Hu et al. [27] developed the concepts of distance and similarity measure. In

addition, Zhu et al. [28] emphasized the absence of non-membership in HFSs and introduced the notion of

"dual hesitant fuzzy sets" (DHFSs), which include both membership and non-membership grades. Peng et al.

[29] proposed the hesitant IFS (HIFS), Khan et al. [30] gave the notion of hesitant PFS (HPFS) and Hussain et

al. [31] introduced hesitant q-ROPFS (Hq-ROPFS) as the hybrid structure of hesitant and IFS, PFS, q-ROPFS

respectively. There are many AOs on HFSs like Dombi AOs [32], Dombi–Archimedean AOs [33, 34], Einstein

AOs [35] and Hamacher AOs [36]. Moreover Choquet AOs [37], prioritized [38] and normalized geometric [39]

for single-valued neutrosophic hesitant fuzzy set.

Triangle norms were first introduced by Menger [40] in his hypothesis of probabilistic metric spaces. These

norms have since been found to be essential in fuzzy sets and structures, including the product t-norm and

probabilistic sum t-conorm [41], Einstein t-norm and t-conorm (TnTc) [42], and the Hamacher TnTc[43]. Re-

cently, Klement et al. [44] analyzed the properties and associated elements of triangular norms in depth. In

1982, Aczel and Alsina [45] introduced a new set of operations known as AA TnTcthat prioritize parameter

changeability. Wang et al. [46] proposed a score level fusion technique based on the AA TnTcthat simultane-

ously increases the distance between imposters. Senapati et al. [47, 48] extended the AA TnTcto IFSs and

interval-valued fuzzy sets respectively. Adak & Kumar [49] proposed some spherical distance measurement

and Smarandache [50] gave overview of plithogenic fuzzy set. Some extensive work can be seen in [51, 52].

Kwak et al. [53] proposed fuzzy modus ponens & tollens, Ramathilagam and Pitchipoo [54] introduced mod-

eling and development based on fuzzy logic. Mahmood et al. [55] proposed some machine learning techniques

with some engineering application. Shekhovtsov et al. [56], Fazlollahtabar & Kazemitash [57], Jagtap &

Karande [58] and Sivaprakasam & Angamuthu [59] introduced some decision-making methods.

Given the growing complexity of decision-making, it’s important to find effective ways of dealing with

uncertain information to identify the best alternative(s) in MCDM. One key challenge is managing the rela-

tionships between different inputs. To address these issues, this article aims to introduce several aggregation

operators for hesitant q-ROPF contexts, which we call Hq-ROPF AA AOs. While many innovative approaches

have been proposed in this area, we’ve conducted a thorough investigation to demonstrate that our strategy

outperforms all previous efforts in addressing the global concerns at hand.

The paper covers several key topics. The first section provides an overview of the fundamental con-

cepts related to AA TnTcand Hq-ROPFSs. The third section summarizes the AA operation laws for Hq-

ROPFNs. In Section 4, we discuss the the " hesitant q-rung orthopair fuzzy Aczel–Alsina weighted averaging

(Hq-ROPFAAWA) operator", " hesitant q-rung orthopair fuzzy Aczel–Alsina ordered weighted averaging (Hq-

ROPFAAOWA) operator", "hesitant q-rung orthopair fuzzy Aczel–Alsina hybrid averaging (Hq-ROPFAAHA)

operator", " hesitant q-rung orthopair fuzzy Aczel–Alsina weighted geometric (Hq-ROPFAAWG) operator," the

"hesitant q-rung orthopair fuzzy Aczel–Alsina ordered weighted geometric (Hq-ROPFAAOWG) operator", and

the " hesitant q-rung orthopair fuzzy Aczel–Alsina hybrid geometric (Hq-ROPFAAHG) operator" as well as

a few advantageous properties. In Section 5, we use the suggested operators to build a set of strategies for

addressing MCDM problems where the characteristic values are represented as Hq-ROPF data. Part 6 com-

prises of filtering technology for pollution control in petrol processing facilities and examples pertaining to

proposed AOs. Section 7 concludes by discussing the article’s major contributions.

2 Preliminaries

In this section, we will discuss several important ideas that are fundamental to the advancement of this work.

2.1 Basics about t-norm, t-conorm and Aczel-Alsina t-norm

Definition 2.1. [44] A function e

ħ: [0,1]2→[0,1] is a t-norm (Tn), if for all πγ,ϖγ,ϱγ∈[0,1], the consecutive

axioms are fulfilled:

1. e

ħ(πγ,ϖγ)=e

ħ(ϖγ,πγ);

2. e

ħ(πγ,ϖγ)≤e

ħ(πγ,ϱγ) if ϖγ≤u;

3. e

ħ(πγ,e

ħ(ϖγ,ϱγ)) =e

ħ(e

ħ(πγ,ϖγ),ϱγ);

4. e

ħ(πγ,1) =πγ.

These axioms are called, symmetry, monotonicity, associativity and "1" as identity respectively.

Definition 2.2. [44] A function e

k: [0,1]2→[0,1] is a t-conorm (Tc), if for all πγ,ϖγ,ϱγ∈[0,1], the consecutive

axioms are fulfilled:

1. e

k(πγ,ϖγ)=e

k(ϖγ,πγ);

2. e

k(πγ,ϖγ)≤e

k(πγ,ϱγ) if ϖγ≤ϱγ;

3. e

k(πγ,e

k(ϖγ,ϱγ)) =e

k(e

k(πγ,ϖγ),ϱγ);

4. e

k(πγ,0) =πγ.

These axioms are called, symmetry, monotonicity, associativity and "0" as identity respectively.

Example 2.3. Some famous t-norms are given as

•e

ħP(πγ,ϖγ)=πγ.ϖγ; (Product Tn)

•e

ħM(πγ,ϖγ)=min(πγ,ϖγ); (Minimum Tn)

•e

ħL(πγ,ϖγ)=max(πγ+ϖγ−1,0); (Lukasiewicz Tn)

•e

ħD(πγ,ϖγ)=









f, if ϖγ=1

ϖγ, if πγ=1

0, otherwise

for all πγ,ϖγ∈[0,1]; (Drastic Tn)

Example 2.4. Some famous t-conorms are given as

•e

kP(πγ,ϖγ)=πγ+ϖγ−πγ.ϖγ; (Probabilistic sum)

•e

kM(πγ,ϖγ)=max(πγ,ϖγ); (Maximum Tc)

•e

kL(πγ,ϖγ)=min(πγ+ϖγ,1); (Lukasiewicz Tc)

•e

kD(πγ,ϖγ)=









πγ, if ϖγ=0

ϖγ, if πγ=0

1, otherwise

for all πγ,ϖγ∈[0,1]; (Drastic Tc:)

Definition 2.5. [45] This class of Tnoriginally proposed by Aczel-Alsina in mid-1980s under the condition of

functional equations.

The category (e

ħג

A)ג∈[0,∞]of Aczel-Alsina Tns is stated by

ħג

A(πγ,ϖγ)=







e

ħD(πγ,ϖγ), if ג=0

min(πγ,ϖγ), if ג= ∞

e−((−logπγ)ג+(−logϖγ)ג)1/ג

, otherwise

The category (e

kג

A)ג∈[0,∞]of Aczel-Alsina Tcs is stated by

kג

A(πγ,ϖγ)=







e

kD(πγ,ϖγ), if ג=0

max(πγ,ϖγ), if ג= ∞

1−e−((−log(1−πγ))ג+(−log(1−ϖγ))ג)1/ג

, otherwise

Limiting Cases: e

ħ0

A=e

ħD,e

ħ1

A=e

ħP,e

ħ∞

A=min, e

A=e

kD,e

A=e

kP,e

k∞

A=max.

For every ג∈[0,∞] the Tne

ħג

Aand Tce

kג

Aare dual to each other. The class of Aczel-Alsina Tns is strictly

increasing and the class of Aczel-Alsina Tcs is strictly decreasing.

2.2 Hesitant q-rung orthopair fuzzy set

Definition 2.6. [9] A q-ROPFS Ron Xis defined as

R={〈x,µR(x),νR(x)〉:x∈X}

here µR,νR:X→[0,1] denotes the MSD and NMSD of the alternative x∈Xand ∀xwe have

0≤µq

R(x)+νq

R(x)≤1.

Furthermore, πR(x)=q

√1−µq

R(x)−νq

R(x) is called the “indeterminacy degree” of xto R.

Definition 2.7. [31] Let Xbe a universal set. Then a Hq-ROPFS Gdefined on Xis an object given by the

following:

G={<t,UG(t),VG(t))>q|t∈X,q≥1},

where UG(t) and VG(t) are the two subsets of [0,1], which represents the hesitant q-rung orthopair mem-

bership and hesitant q-rung orthopair non-membership grades of an object t∈Xto the set G. Moreover for

each element t∈X,∀µ(t)∈UG(t) there exist ν(t)∈VG(t), which holds the condition that 0 ≤(µ(t))q+(ν(t))q≤1

and similarly, ∀ν(t)∈VG(t), there exist µ(t)∈UG(t), which holds the condition that 0 ≤(µ(t))q+(ν(t))q≤1.

In perspective of the definition of Hq-ROPFS, for each tbelongs to Xthere are two sets, that is hesi-

tant q-rung orthopair membership grade UG(t) and hesitant q-rung orthopair non-membership grade VG(t).

The cardinality of UG(t) and VG(t) (number of the elements in UG(t) and VG(t) ) are represented by #UG(t)

and #VG(t). Actually Hq-ROPFSs provides a huge space for decision makers to select membership and non-

membership grades than the rest of the sets. For simplicity the pair G=< t,UG(t),VG(t))>represent a hesitant

-qrung fuzzy number (Hq-ROPFN), which is represented by αℓ=(Uαℓ,Vαℓ).

Definition 2.8. [31] For a Hq-ROPFN αℓ=(U,V), the score function of αℓis defined as

S(αℓ)=1

#U∑

µ∈U

µq−1

#V∑

ν∈V

νq,

where #Uand #Vrepresents the cardinality of Uand Vrespectively.

Let αℓi=(Ui,Vi)(i=1,2) be any two Hq-ROPFNs. Then

i: If S (αℓ1)>S(αℓ2), then αℓ1is superior than αℓ2, represented by αℓ1<αℓ2,

ii: If S(αℓ1)<S(αℓ2), then αℓ1is inferior than αℓ2, represented by αℓ14αℓ2,

Definition 2.9. [31] Consider αℓ=(U,V) is the Hq-ROPFN. Then the accuracy function of αℓis

A(αℓ)=1

#U∑

µ∈U

µq+1

#V∑

ν∈V

νq

Let αℓi=G(hi,Vi)(i=1,2) be any two Hq-ROPFNs. Then

If S(αℓ1)=ˆ

S(αℓ2), then

(a) if A(αℓ1)=A(αℓ2), then αℓ1is equivalent to αℓ2, represented by αℓ1≈αℓ2,

(b) if A(αℓ1)>A(αℓ2), then αℓ1is superior than αℓ2, represented by αℓ1<αℓ2.

3 AA operations for Hq-ROPFNs

This section introduces the AA operations for Hq-ROPFNs and examines some of its fundamental features.

Definition 3.1. Let αℓ=(Uαℓ,Vαℓ),αℓ1=(Uαℓ1,Vαℓ1), and αℓ2=(Uαℓ2,Vαℓ2)be three Hq-ROPFNs, ℵħ≥1 and

σ>0. Then, the AA Tnand Tcoperations of Hq-ROPFNs are defined as:

1.αℓ1⊕αℓ2=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







t1−e

−((−log(1−µq

αℓ1))ℵħ

+(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−((−logνq

αℓ1)ℵħ

+(−logνq

αℓ2)ℵħ)1

ℵħ







2.αℓ1⊗αℓ2=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−((−logµq

αℓ1)ℵħ

+(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−((−log(1−νq

αℓ1))ℵħ

+(−log(1−νq

αℓ2))ℵħ)1

ℵħ







3.σαℓ=∪

(µαℓ,ναℓ)∈(Uαℓ,Vαℓ)(q

√1−e−(σ(−log(1−µq

αℓ))ℵħ)1

ℵħ

√e−(σ(−logνq

αℓ)ℵħ)1

ℵħ)

4.αℓσ=∪

(µαℓ,ναℓ)∈(Uαℓ,Vαℓ)(q

√e−(σ(−logµq

αℓ)ℵħ)1

ℵħ

√1−e−(σ(−log(1−νq

αℓ))ℵħ)1

ℵħ)

Theorem 3.2. Let αℓ=(Uαℓ,Vαℓ),αℓ1=(Uαℓ1,Vαℓ1), and αℓ2=(Uαℓ2,Vαℓ2)be three q-ROPFNs, then we have

(i) αℓ1⊕αℓ2=αℓ2⊕αℓ1;

(ii) αℓ1⊗αℓ2=αℓ2⊗αℓ1;

(iii) ג(αℓ1⊕αℓ2)=גαℓ1⊕גαℓ2,ג>0;

(iv) (ג1+ג2)αℓ=ג1αℓ⊕ג2αℓ,ג1,ג2>0;

(v) (αℓ1⊗αℓ2)ג=αℓג

1⊗αℓג

2,ג>0;

(vi) αℓג1⊗αℓג2=αℓ(ג1+ג2),ג1,ג2>0.

Proof. For the three q-ROPFNs αℓ,αℓ1and αℓ2, and ג,ג1,ג2>0, we can get

(i)

αℓ1⊕αℓ2=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







t1−e

−((−log(1−µq

αℓ1))ℵħ

+(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−((−logνq

αℓ1)ℵħ

+(−logνq

αℓ2)ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







t1−e

−((−log(1−µq

αℓ2))ℵħ

+(−log(1−µq

αℓ1))ℵħ)1

ℵħ

−((−logνq

αℓ2)ℵħ

+(−logνq

αℓ1)ℵħ)1

ℵħ





=αℓ2⊕αℓ1.

(ii)

αℓ1⊗αℓ2=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−((−logµq

αℓ1)ℵħ

+(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−((−log(1−νq

αℓ1))ℵħ

+(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−((−logµq

αℓ2)ℵħ

+(−logµq

αℓ1)ℵħ)1

ℵħ

t1−e

−((−log(1−νq

αℓ2))ℵħ

+(−log(1−νq

αℓ1))ℵħ)1

ℵħ





=αℓ2⊗αℓ1.

(iii) Using this, we get

ג(αℓ1⊕αℓ2)=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)

ג





t1−e

−((−log(1−µq

αℓ1))ℵħ

+(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−((−logνq

αℓ1)ℵħ

+(−logνq

αℓ2)ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







t1−e

−(ג(−log(1−µq

αℓ1))ℵħ

+(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−(ג(−logνq

αℓ1)ℵħ

+(−logνq

αℓ2)ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







t1−e

−(ג(−log(1−µq

αℓ1))ℵħ)1

ℵħ

−(ג(−logνq

αℓ1)ℵħ)1

ℵħ





⊕





t1−e

−(ג(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−(ג(−logνq

αℓ2)ℵħ)1

ℵħ





=גαℓ1⊕גαℓ2.

(iv)

ג1αℓ⊕ג2αℓ=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







√1−e−(ג1(−log(1−µq

αℓ))ℵħ)1

ℵħ

√e−(ג1(−logνq

αℓ)ℵħ)1

ℵħ





⊕





√1−e−(ג2(−log(1−µq

αℓ))ℵħ)1

ℵħ

√e−(ג2(−logνq

αℓ)ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







√1−e−((ג1+ג2)(−log(1−µq

αℓ))ℵħ)1

ℵħ

√e−((ג1+ג2)(−logνq

αℓ)ℵħ)1

ℵħ





=(ג1+ג2)αℓ.

(v)

(αℓ1⊗αℓ2)ג

=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−((−logµq

αℓ1)ℵħ

+(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−((−log(1−νq

αℓ1))ℵħ

+(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−(ג(−logµq

αℓ1)ℵħ

+(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−(ג(−log(1−νq

αℓ1))ℵħ

+(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=∪

(µαℓw,ναℓw)∈(Uαℓw,Vαℓw)(w=1,2)







−(ג(−logµq

αℓ1)ℵħ)1

ℵħ

t1−e

−(ג(−log(1−νq

αℓ1))ℵħ)1

ℵħ





⊗







−(ג(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−(ג(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=αℓג

1⊗αℓג

(vi)

αℓג1⊗αℓג2=∪

(µαℓ,ναℓ)∈(Uαℓ,Vαℓ)







−(ג1(−logµq

αℓ1)ℵħ)1

ℵħ

t1−e

−(ג1(−log(1−νq

αℓ1))ℵħ)1

ℵħ





⊗∪

(µαℓ,ναℓ)∈(Uαℓ,Vαℓ)







−(ג2(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−(ג2(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=∪

(µαℓ,ναℓ)∈(Uαℓ,Vαℓ)







−((ג1+ג2)(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−((ג1+ג2)(−log(1−νq

αℓ2))ℵħ)1

ℵħ





=αℓ(ג1+ג2).

4 Hesitant q-rung orthopair fuzzy AA AOs

In this subsection, some Hq-ROPF aggregation operators based on AA operations are presented.

4.1 Hesitant q-rung orthopair fuzzy AA averaging AOs

Definition 4.1. Let αℓϕ=(Uαℓϕ,Vαℓϕ),(ϕ=1,2,...,i) be an accumulation of Hq-ROPFNs and ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)T

be the “weight vector” (WV) of αℓϕ, with ℵ℘ϕ>0 and ∑i

ϕ=1ℵ℘ϕ=1. Then Hq-ROPFAAWA operator is a map-

ping Hq-ROPFAAWA: (L∗)i→L∗, where

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)=(ℵ℘1αℓ1⊕ℵ℘2αℓ2⊕...,⊕ℵ℘iαℓi)(1)

Using AA operations on Hq-ROPFNs, we derive the following theorem.

Theorem 4.2. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be the collection of Hq-ROPFNs, then the consolidated output of them

using the Hq-ROPFAAWA operator is a Hq-ROPFNs, and

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)

⊕

ϕ=1(ℵ℘ϕαℓϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑i

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





, (2)

where ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)be the WV of αℓϕs.t ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1.

Proof. We would deduce Theorem 4.2 employing the mathematical induction approach as follows:

Based on the AA operations of Hq-ROPFNs with i=2, we derive

ℵ℘1αℓ1=∪

(µαℓ1,ναℓ1)∈(Uαℓ1,Vαℓ1)







t1−e

−(ℵ℘1(−log(1−µq

αℓ1))ℵħ)1

ℵħ

−(ℵ℘1(−logνq

αℓ1)ℵħ)1

ℵħ





ℵ℘2αℓ2=∪

(µαℓ2,ναℓ2)∈(Uαℓ2,Vαℓ2)







t1−e

−(ℵ℘2(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−(ℵ℘2(−logνq

αℓ2)ℵħ)1

ℵħ





Based on AA operations of Hq-ROPFNs, we obtain

Hq-ROPFAAWA(αℓ1,αℓ2)

=∪

(µαℓ1,ναℓ1)∈(Uαℓ1,Vαℓ1)







t1−e

−(ℵ℘1(−log(1−µq

αℓ1))ℵħ)1

ℵħ

−(ℵ℘1(−logνq

αℓ1)ℵħ)1

ℵħ





⊕

∪

(µαℓ2,ναℓ2)∈(Uαℓ2,Vαℓ2)







t1−e

−(ℵ℘2(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−(ℵ℘2(−logνq

αℓ2)ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2)







t1−e

−(ℵ℘1(−log(1−µq

αℓ1))ℵħ

+ℵ℘2(−log(1−µq

αℓ2))ℵħ)1

ℵħ

−(ℵ℘1(−log(νq

αℓ1))ℵħ

+ℵ℘2(−log(νq

αℓ2))ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2)







t1−e

−



∑q

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑q

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





Thus, it is true for i=2.

Consider Equation 4.2 is true for i=k, then we have

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓk)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k)







t1−e

−



∑k

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑k

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





we will prove that Equation 4.2 holds for i=k+1.

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓk,αℓk+1)

⊕(ℵ℘ϕαℓϕ)⊕(ℵ℘k+1αℓk+1)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k)







t1−e

−



∑k

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑k

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





⊕

∪

(µαℓk+1,ναℓk+1)∈(Uαℓk+1,Vαℓk+1)







t1−e

−(ℵ℘k+1(−log(1−µq

αℓk+1))ℵħ)1

ℵħ

−(ℵ℘k+1(−logνq

αℓk+1)ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k+1)







t1−e

−



∑k+1

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑k+1

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





Thus, we can establish that Equation 4.2 is valid for all values of i.

By utilizing the Hq-ROPFAAWA operator, we can effectively exhibit the following characteristics:

Theorem 4.3. If all αℓϕ=(Uαℓϕ,Vαℓϕ)are equal, that is, αℓϕ=αℓ∀ϕ, then

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)=αℓ.

Proof. Given that αℓϕ=(Uαℓϕ,Vαℓϕ), by Equation 4.2 we get

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ



,

−(∑i

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







t1−e

−

(−log(1−µq

αℓ)ℵħ)1

ℵħ

,

√e−((−logνq

αℓ)ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)(q

√1−elog(1−µq

αℓ),

√elogνq

αℓ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)(q

√µq

αℓ,q

√νq

αℓ)=αℓ.

Theorem 4.4. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let

αℓ−=min(αℓ1,αℓ2,...,αℓi)and αℓ+=max(αℓ1,αℓ2,...,αℓi). Then,

αℓ−≤Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)≤αℓ+.

Proof. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let αℓ−=min(αℓ1,αℓ2,...,αℓi)=(µ−

αℓ,ν−

αℓ)

and αℓ+=max(αℓ1,αℓ2,...,αℓi)=(µ+

αℓ,ν+

αℓ). We have, µ−

αℓ=minϕ{µαℓϕ},ν−

αℓ=maxϕ{ναℓϕ},µ+

αℓ=maxϕ{µαℓϕ},

and ν+

αℓ=minϕ{ναℓϕ}Hence, there have the subsequent inequalities,

∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−

∑i

ϕ=1ℵ℘ϕ(−log(1−µq

αℓ−)ℵħ)1

ℵħ



≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−µq

αℓϕ)ℵħ)1

ℵħ





≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−µq

αℓ+)ℵħ)1

ℵħ



,

and

∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

−(∑i

ϕ=1ℵ℘ϕ(−logνq

αℓ+)ℵħ)1

ℵħ

≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

−(∑i

ϕ=1ℵ℘ϕ(−logνq

αℓϕ)ℵħ)1

ℵħ

≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

√e−(∑i

ϕ=1ℵ℘ϕ(−logνq

αℓ−)ℵħ)1

ℵħ

Therefore, αℓ−≤Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)≤αℓ+

Theorem 4.5. Let αℓϕand αℓ′

ϕbe two sets of Hq-ROPFNs, if αℓϕ≤αℓ′

ϕ∀ϕ, then

Hq-ROPFAAWA(αℓ1,αℓ2,...,αℓi)≤Hq-ROPFAAWA(αℓ′

1,αℓ′

2,...,αℓ′

i).

Now, we present "hesitant q-rung orthopair fuzzy AA ordered weighted averaging (Hq-ROPFAAOWA) opera-

tor".

Definition 4.6. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Hq-ROPFAAOWA operator is

a mapping Hq-ROPFAAOWA: (L∗)i→L∗with the corresponding WV ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tsuch that

ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1, as

Hq-ROPFAAOWA(αℓ1,αℓ2,...,αℓi)=

⊕

ϕ=1(ℵ℘ϕαℓ˘

⨿(ϕ))

= ℵ℘1αℓ˘

⨿(1) ⊕ℵ℘2αℓ˘

⨿(2)⊕,...,⊕ℵ℘iαℓ˘

⨿(i),

where ( ˘

⨿(1), ˘

⨿(2),..., ˘

⨿(i)) are the permutation of (ϕ=1,2,...,i), including αℓ˘

⨿(ϕ−1) ≥αℓ˘

⨿(ϕ)∀ϕ=1,2,...,i.

Hence, the accompanying theorem is derived from AA operations on Hq-ROPFNs.

Theorem 4.7. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Hq-ROPFAAOWA operator is a

mapping Hq-ROPFAAOWA : (L∗)i→L∗with the corresponding vector ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tsuch that

ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1. Then,

Hq-ROPFAAOWA(αℓ1,αℓ2,...,αℓi)

⊕

ϕ=1(ℵ℘ϕαℓϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







t1−e

−



∑i

ϕ=1ℵ℘ϕ

−log(1−µq

αℓ˘

⨿(ϕ))ℵħ



ℵħ



,

−

∑i

ϕ=1ℵ℘ϕ(−logνq

αℓ˘

⨿(ϕ))ℵħ



ℵħ





,(3)

where ( ˘

⨿(1), ˘

⨿(2),..., ˘

⨿(i)) are the permutation of (ϕ=1,2,...,i), including αℓ˘

⨿(ϕ−1) ≥αℓ˘

⨿(ϕ)∀ϕ=1,2,...,i.

Proof. Same as Theorem 4.2.

By applying the Hq-ROPFAAOWA operator, we can illustrate the following features efficiently.

Theorem 4.8. If all αℓϕ=(Uαℓϕ,Vαℓϕ)are equal, that is, αℓϕ=αℓ∀ϕ, then

Hq-ROPFAAOWA(αℓ1,αℓ2,...,αℓi)=αℓ.

Proof. Same as Theorem 4.3.

Theorem 4.9. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let

αℓ−=min(αℓ1,αℓ2,...,αℓi)and αℓ+=max(αℓ1,αℓ2,...,αℓi). Then,

αℓ−≤Hq-ROPFAAOWA(αℓ1,αℓ2,...,αℓi)≤αℓ+.

Proof. Same as Theorem 4.4.

Theorem 4.10. Let αℓϕand αℓ′

ϕbe two sets of Hq-ROPFNs, if αℓϕ≤αℓ′

ϕ∀ϕ, then

Hq-ROPFAAOWA(αℓ1,αℓ2,...,αℓi)≤Hq-ROPFAAOWA(αℓ′

1,αℓ′

2,...,αℓ′

i).

The distinction between the Hq-ROPFAAWA and Hq-ROPFAAOWA operators is clear: the former weights

only the Hq-ROPFNs, while the latter weights only the ordered locations of the Hq-ROPFNs. However, these

two operators only consider one of these elements, and weights are used to indicate various elements of each.

To overcome this limitation, we propose a new operator called the "hesitant q-rung orthopair fuzzy AA hybrid

averaging (Hq-ROPFAAHA) operator," which takes into account both the Hq-ROPFNs and their respective

ordered positions.

Definition 4.11. Let αℓϕbe an accumulation of Hq-ROPFNs. Hq-ROPFAAHA operator is a mapping Hq-

ROPFAAHA: (L∗)i→L∗, s.t.

Hq-ROPFAAHA(αℓ1,αℓ2,...,αℓi)=

⊕

ϕ=1(ℵ℘ϕ¨

αℓ˘

⨿(ϕ))

= ℵ℘1¨

αℓ˘

⨿(1) ⊕ℵ℘2¨

αℓ˘

⨿(2)⊕,...,⊕ℵ℘i¨

αℓ˘

⨿(i),

where ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tis the weighting vector associated with the Hq-ROPFAAHA operator, with

ℵ℘ϕ∈[0,1] and ∑i

ϕ=1ℵ℘ϕ=1; ¨

αℓϕ=iδϕαℓϕ,ϕ=1,2,...,i,(¨

αℓ˘

⨿(1),¨

αℓ˘

⨿(2),..., ¨

αℓ˘

⨿(i))is any permutation of a

collection of the weighted Hq-ROPFNs (¨

αℓ1,¨

αℓ2,..., ¨

αℓi), s.t. ¨

αℓ˘

⨿(ϕ−1) ≥¨

αℓ˘

⨿(ϕ).δ=(δ1,δ2,...,δi)Tis the WV

of αℓϕ, with δϕ∈[0,1] and ∑i

ϕ=1δϕ=1, and iis the balancing coefficient, which plays a role of balance.

Theorem 4.12. Let αℓϕbe the collection of Hq-ROPFNs. Their aggregated value by Hq-ROPFAAHA operator

is still an Hq-ROPFN, and

Hq-ROPFAAHA(αℓ1,αℓ2,...,αℓi)

⊕

ϕ=1(ℵ℘ϕαℓϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







t1−e

−



∑i

ϕ=1ℵ℘ϕ

−log(1−µq

αℓ˘

⨿(ϕ))ℵħ



ℵħ



,

−

∑i

ϕ=1ℵ℘ϕ(−logνq

αℓ˘

⨿(ϕ))ℵħ



ℵħ





.(4)

Proof. Same as Theorem 4.2.

Theorem 4.13. The Hq-ROPFAAWA and Hq-ROPFAAOWA operators are special cases of the Hq-ROPFAAHA

operator.

Proof. (1) Let ℵ℘=(1/i,1/i,...,1/i)T. Then

Hq −ROPF AAH A (αℓ1,αℓ2,...,αℓi)= ℵ℘1˙

αℓi(1) ⊕ℵ℘2˙

αℓi(2) ⊕···⊕ℵ℘i˙

αℓi(i)

i(˙

αℓi(1) ⊕˙

αℓi(2) ⊕ ··· ⊕ ˙

αℓi(i))

= ℵ℘1αℓ1⊕ℵ℘2αℓ2⊕···⊕ ℵ℘iαℓi

=Hq −ROPF AAW A (αℓ1,αℓ2,...,αℓi)

(2) Let ℵ℘=(1/i,1/i,...,1/i)T. Then ˙

αℓϕ=αℓϕand

IF AAHAℵ℘,ℵ℘(αℓ1,αℓ2,...,αℓi)= ℵ℘1˙

αℓi(1) ⊕ℵ℘2˙

αℓi(2) ⊕···⊕ℵ℘i˙

αℓi(i)

= ℵ℘1αℓi(1) ⊕ℵ℘2αℓi(2) ⊕···⊕ℵ℘iαℓii(i)

=Hq −ROPF AAOW A (αℓ1,αℓ2,...,αℓi)

4.2 Hesitant q-rung orthopair fuzzy AA geometric AOs

Definition 4.14. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs and ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tbe

the WV of αℓϕ, with ℵ℘ϕ>0 and ∑i

ϕ=1ℵ℘ϕ=1. Then Hq-ROPFAAWG operator is a mapping Hq-ROPFAAWG:

(L∗)i→L∗, where

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)=(αℓℵ℘1

1⊗αℓℵ℘2

2⊗...,⊗αℓℵ℘i

i). (5)

Hence, the following theorem is derived from AA operations on Hq-ROPFNs.

Theorem 4.15. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs, then the consolidated output of

them using the Hq-ROPFAAWG operator is a Hq-ROPFNs, and

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)

⊗

ϕ=1(ℵ℘ϕαℓϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







−(∑i

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−ναℓϕ)ℵħ)1

ℵħ









, (6)

where ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)be the WV of αℓϕs.t ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1.

Proof. We can derive Theorem 4.15 in the following way using the mathematical induction technique:

For i=2, depend on AA operations of Hq-ROPFNs, we obtain

αℓℵ℘1

1=∪

(µαℓ1,ναℓ1)∈(Uαℓ1,Vαℓ1)







−(ℵ℘1(−logµq

αℓ1)ℵħ)1

ℵħ

t1−e

−



ℵ℘1(−log(1−νq

αℓ2)ℵħ)1

ℵħ









αℓℵ℘2

2=∪

(µαℓ2,ναℓ2)∈(Uαℓ2,Vαℓ2)







−(ℵ℘2(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−



ℵ℘2(−log(1−νq

αℓ2)ℵħ)2

ℵħ









Based on AA operations of Hq-ROPFNs, we obtain

Hq-ROPFAAWG(αℓ1,αℓ2)

=∪

(µαℓ1,ναℓ1)∈(Uαℓ1,Vαℓ1)







−(ℵ℘1(−logµq

αℓ1)ℵħ)1

ℵħ

t1−e

−



ℵ℘1(−log(1−νq

αℓ2)ℵħ)1

ℵħ









⊕

∪

(µαℓ2,ναℓ2)∈(Uαℓ2,Vαℓ2)







−(ℵ℘2(−logµq

αℓ2)ℵħ)1

ℵħ

t1−e

−



ℵ℘2(−log(1−νq

αℓ2)ℵħ)2

ℵħ









=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2)







−(∑q

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑q

ϕ=1ℵ℘ϕ(−log(1−ναℓϕ)ℵħ)1

ℵħ









Thus, it is true for i=2.

Consider Equation 4.15 is true for i=k, then we have

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓk)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k)







−(∑k

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑k

ϕ=1ℵ℘ϕ(−log(1−ναℓϕ)ℵħ)1

ℵħ









we will prove that Equation 4.15 holds for i=k+1.

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓk,αℓk+1)

⊕(ℵ℘ϕαℓϕ)⊕(ℵ℘k+1αℓk+1)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k)







−(∑k

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑k

ϕ=1ℵ℘ϕ(−log(1−ναℓϕ)ℵħ)1

ℵħ









⊕

∪

(µαℓk+1,ναℓk+1)∈(Uαℓk+1,Vαℓk+1)







−(ℵ℘k+1(−log(µq

αℓk+1))ℵħ)1

ℵħ

t1−e

−



ℵ℘k+1(−log(1−νq

αℓk+1)ℵħ)1

ℵħ









=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,k+1)







−(∑k+1

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑k+1

ϕ=1ℵ℘ϕ(−log(1−ναℓϕ)ℵħ)1

ℵħ









As a result, we can conclude that Equation 4.15 stands true for any i.

By applying the Hq-ROPFAAWG operator, we can illustrate the following features efficiently.

Theorem 4.16. If all αℓϕ=(Uαℓϕ,Vαℓϕ)are equal, that is, αℓϕ=αℓ∀ϕ, then

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)=αℓ.

Proof. Given that αℓϕ=(Uαℓϕ,Vαℓϕ), by Equation 4.15 we get

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







−(∑i

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−νq

αℓϕ)ℵħ)1

ℵħ









=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







−

(−log(µq

αℓ)ℵħ)1

ℵħ

,

√1−e−((−log(1−νq

αℓ))ℵħ)1

ℵħ





=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)(q

√elogµq

αℓ

√1−elog(1−νq

αℓ))

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)(q

√µq

αℓ,q

√νq

αℓ)=αℓ.

Theorem 4.17. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let

αℓ−=min(αℓ1,αℓ2,...,αℓi)and αℓ+=max(αℓ1,αℓ2,...,αℓi). Then,

αℓ−≤Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)≤αℓ+.

Proof. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let αℓ−=min(αℓ1,αℓ2,...,αℓi)=(µ−

αℓ,ν−

αℓ)

and αℓ+=max(αℓ1,αℓ2,...,αℓi)=(µ+

αℓ,ν+

αℓ). We have, µ−

αℓ=minϕ{µαℓϕ},ν−

αℓ=maxϕ{ναℓϕ},µ+

αℓ=maxϕ{µαℓϕ},

and ν+

αℓ=minϕ{ναℓϕ}Hence, there have the subsequent inequalities,

∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

−(∑i

ϕ=1ℵ℘ϕ(−logµq

αℓ+)ℵħ)1

ℵħ

≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

−(∑i

ϕ=1ℵ℘ϕ(−logµq

αℓϕ)ℵħ)1

ℵħ

≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

√e−(∑i

ϕ=1ℵ℘ϕ(−logµq

αℓ−)ℵħ)1

ℵħ

and

∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−

∑i

ϕ=1ℵ℘ϕ(−log(1−νq

αℓ−)ℵħ)1

ℵħ



≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−νq

αℓϕ)ℵħ)1

ℵħ





≤∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)

t1−e

−



∑i

ϕ=1ℵ℘ϕ(−log(1−νq

αℓ+)ℵħ)1

ℵħ



,

Therefore, αℓ−≤Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)≤αℓ+

Theorem 4.18. Let αℓϕand αℓ′

ϕbe two sets of Hq-ROPFNs, if αℓϕ≤αℓ′

ϕ∀ϕ, then

Hq-ROPFAAWG(αℓ1,αℓ2,...,αℓi)≤Hq-ROPFAAWG(αℓ′

1,αℓ′

2,...,αℓ′

i).

Now, we present "q-rung orthopair fuzzy AA ordered weighted geometric (Hq-ROPFAAOWG) operator".

Definition 4.19. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Hq-ROPFAAOWG operator is

a mapping Hq-ROPFAAOWG: (L∗)i→L∗with the corresponding WV ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tsuch that

ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1, as

Hq-ROPFAAOWG(αℓ1,αℓ2,...,αℓi)=

⊗

ϕ=1(αℓℵ℘ϕ

⨿(ϕ))

=αℓℵ℘1

⨿(1) ⊕αℓℵ℘2

⨿(2)⊕,...,⊕αℓℵ℘i

⨿(i),

where ( ˘

⨿(1), ˘

⨿(2),..., ˘

⨿(i)) are the permutation of (ϕ=1,2,...,i), including αℓ˘

⨿(ϕ−1) ≥αℓ˘

⨿(ϕ)∀ϕ=1,2,...,i.

Thus, the following theorem is obtained using AA operations on Hq-ROPFNs.

Theorem 4.20. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Hq-ROPFAAOWG operator is a

mapping Hq-ROPFAAOWG : (L∗)i→L∗with the corresponding vector ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tsuch that

ℵ℘ϕ>0, and ∑i

ϕ=1ℵ℘ϕ=1. Then,

Hq-ROPFAAOWG(αℓ1,αℓ2,...,αℓi)

⊗

ϕ=1(αℓℵ℘ϕ

ϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







−

∑i

ϕ=1ℵ℘ϕ(−logµq

αℓ˘

⨿(ϕ))ℵħ



ℵħ

t1−e

−



∑i

ϕ=1ℵ℘ϕ

−log(1−νq

αℓ˘

⨿(ϕ))ℵħ



ℵħ









,(7)

where (˘

⨿(1), ˘

⨿(2),..., ˘

⨿(i)) are the permutation of (ϕ=1,2,...,i), including αℓ˘

⨿(ϕ−1) ≥αℓ˘

⨿(ϕ)∀ϕ=1,2,...,i.

Proof. Same as Theorem 4.15.

By applying the Hq-ROPFAAOWG operator, we can illustrate the following features efficiently.

Theorem 4.21. If all αℓϕ=(Uαℓϕ,Vαℓϕ)are equal, that is, αℓϕ=αℓ∀ϕ, then

Hq-ROPFAAOWG(αℓ1,αℓ2,...,αℓi)=αℓ.

Proof. Same as Theorem 4.16.

Theorem 4.22. Let αℓϕ=(Uαℓϕ,Vαℓϕ)be an accumulation of Hq-ROPFNs. Let

αℓ−=min(αℓ1,αℓ2,...,αℓi)and αℓ+=max(αℓ1,αℓ2,...,αℓi). Then,

αℓ−≤Hq-ROPFAAOWG(αℓ1,αℓ2,...,αℓi)≤αℓ+.

Proof. Same as Theorem 4.17.

Theorem 4.23. Let αℓϕand αℓ′

ϕbe two sets of Hq-ROPFNs, if αℓϕ≤αℓ′

ϕ∀ϕ, then

Hq-ROPFAAOWG(αℓ1,αℓ2,...,αℓi)≤Hq-ROPFAAOWG(αℓ′

1,αℓ′

2,...,αℓ′

i).

It’s clear that the Hq-ROPFAAWG operator weights only the Hq-ROPFNs, while the Hq-ROPFAAOWG op-

erator weights only the ordered locations of the Hq-ROPFNs. Weights are used to indicate various elements

of these two operators. However, each operator considers only one of these elements. To overcome this limi-

tation, we propose a new operator called the "q-rung orthopair fuzzy AA hybrid geometric (Hq-ROPFAAHG)

operator," which takes into account both the Hq-ROPFNs and their respective ordered positions.

Definition 4.24. Let αℓϕbe an accumulation of Hq-ROPFNs. Hq-ROPFAAHG operator is a mapping Hq-

ROPFAAHG: (L∗)i→L∗, s.t.

Hq −ROPF AAHG (αℓ1,αℓ2,...,αℓi)=

⊕

ϕ=1(¨

αℓℵ℘ϕ

⨿(ϕ))

= ℵ℘1¨

αℓ˘

⨿(1) ⊕ℵ℘2¨

αℓ˘

⨿(2)⊕,...,⊕ℵ℘i¨

αℓ˘

⨿(i),

where ℵ℘=(ℵ℘1,ℵ℘2,...,ℵ℘i)Tis the weighting vector associated with the Hq-ROPFAAHG operator, with

ℵ℘ϕ∈[0,1] and ∑i

ϕ=1ℵ℘ϕ=1; ¨

αℓϕ=iδϕαℓϕ,ϕ=1,2,...,i,(¨

αℓ˘

⨿(1),¨

αℓ˘

⨿(2),..., ¨

αℓ˘

⨿(i))is any permutation of a

collection of the weighted Hq-ROPFNs (¨

αℓ1,¨

αℓ2,..., ¨

αℓi), s.t. ¨

αℓ˘

⨿(ϕ−1) ≥¨

αℓ˘

⨿(ϕ).δ=(δ1,δ2,...,δi)Tis the WV

of αℓϕ, with δϕ∈[0,1] and ∑i

ϕ=1δϕ=1, and iis the balancing coefficient, which plays a role of balance.

Theorem 4.25. Let αℓϕbe the collection of Hq-ROPFNs. Their aggregated value by Hq-ROPFAAHG operator

is still an Hq-ROPFN, and

Hq-ROPFAAHG(αℓ1,αℓ2,...,αℓi)

⊗

ϕ=1(αℓℵ℘ϕ

ϕ)

=∪

(µαℓϕ,ναℓϕ)∈(Uαℓϕ,Vαℓϕ)(ϕ=1,2...,i)







−

∑i

ϕ=1ℵ℘ϕ(−logµq

αℓ˘

⨿(ϕ))ℵħ



ℵħ

t1−e

−



∑i

ϕ=1ℵ℘ϕ

−log(1−νq

αℓ˘

⨿(ϕ))ℵħ



ℵħ









(8)

Proof. Same as Theorem 4.15.

Theorem 4.26. The Hq-ROPFAAWG and Hq-ROPFAAOWG operators are special cases of the Hq-ROPFAAHG

operator.

5 Decision making algorithm based on proposed AOs

We explore an MCDM issue by comparing each one of the ndistinct choices to a set of mdistinct qualities. It

is vital to supply a team of specialists.

Assume that the alternative Aגj(j=1,2,...,n) can be deduced from the DMs, with the criteria Ok

i(i=

1,2,...,m), and the evaluation outcome is expressed in terms of Hq-ROPFNs, ˘

⨿ji =(Uji,Vji). Moreover, ϖt

should be the WV for the parameter Ok

iunder the conditions, ϖt≥0 and ∑m

t=1ϖt=1. The proposed operator

is being used to develop an MCDM for the Hq-ROPF information, which includes the following steps:

Algorithm

Step 1:

Attain the decision matrix EG=(Yji)n×musing the Hq-ROPFNs from the DMs.

1Ok

2Ok













Aג1(µ11,ν11) (µ12,ν12)······ (µ1m,ν1m)

Aג2(µ21,ν21) (µ22,ν22)······ (µ2m,ν2m)

........

Aגn(µn1,νn1) (µn2,νn2)······ (µnm,νnm)

Step 2:

If necessary, normalise the Hq-ROPFNs by transforming all cost kind parameters (τc) to benefit kind at-

tributes (τb) using the given formula:

(ℵħN

ji )n×m=





(Yji)c;i∈τc

Yji;i∈τb.

(9)

where (Yji)cshow the compliment of (Yji). The normalised decision matrix will be ΓN=(ℵħN

ji )n×m=(˘

µji,˘

νji)n×m.

Step 3:

Evaluate the aggregated Hq-ROPF decision matrix by using normalized decision matrix ΓNand the WV ϖγ.

We utilized one of the proposed AOs.

Step 4:

Compute the score value of the aggregated value using SF.

Step 5:

All options are ordered in descending order according to their score values, and the option with the highest

score value is chosen as the best.

6 Explanation of the problem

Filtration technology plays a crucial role in the natural gas processing industry as it ensures the safe and

efficient processing of natural gas by removing impurities and contaminants. In natural gas processing plants,

filtration technology is used to maintain the quality of the processed gas to meet the required standards for

distribution to customers.

Contamination control is a critical aspect of natural gas processing due to the potential impacts on the

environment, public health, and the equipment and pipelines used in processing and distribution. The use

of effective filtration technology is therefore essential for ensuring the reliability and safety of natural gas

processing operations. There are various types of filtration technologies used in natural gas processing plants,

each with its own advantages and limitations. Some of the most common filtration technologies include me-

chanical filters, coalescing filters, and adsorption filters. The choice of filtration technology depends on the

specific requirements of the natural gas processing plant and the type of contaminants present in the natural

gas stream.

Mechanical filters, such as cyclo-filters and basket strainers, use physical processes to separate impurities

from the natural gas stream. These filters work by utilizing centrifugal force to separate the heavier impurities

from the lighter natural gas, which is then directed to the outlet for further processing. Mechanical filters are

highly efficient and easy to operate, making them a cost-effective solution for many natural gas processing

plants. Coalescing filters, such as gravity separators, are used to remove entrained liquids from the natural

gas stream. These filters are particularly effective at removing small droplets of liquids, which can cause

significant problems in the processing and distribution of natural gas. Coalescing filters are commonly used in

applications where the natural gas stream contains high levels of liquids, such as in offshore platforms or gas

production wells. Adsorption filters, such as activated carbon filters, work by absorbing impurities from the

natural gas stream. These filters use a porous material, such as activated carbon, which has a large surface

area for adsorbing impurities. Adsorption filters are commonly used in applications where the natural gas

stream contains volatile organic compounds (VOCs), which can have negative impacts on the environment

and public health.

One of the main advantages of filtration technology is its versatility. Filtration technology is capable of

removing a wide range of impurities, including liquids, solids, and entrained gases. This helps to ensure

that the processed gas meets the required quality standards, and reduces the risk of damage to pipelines and

equipment used in the processing and distribution of natural gas. Filtration technology is also environmentally

friendly, as it can be designed to minimize the impact on the environment. For example, mechanical filters

do not use any chemicals or generate any waste products, making them an environmentally safe solution for

natural gas processing. Similarly, adsorption filters can use environmentally safe materials, such as activated

carbon, to absorb impurities from the natural gas stream. Filtration technology is highly efficient, with many

filters achieving removal rates of 99% or higher for impurities in the natural gas stream. This high level of

efficiency minimizes the amount of contaminants that are carried over to the processed gas, and ensures that

the processed gas meets the required quality standards. Filtration technology is also relatively easy to operate

and requires minimal maintenance. This makes it a cost-effective solution for natural gas processing plants,

as it reduces the need for highly skilled personnel and minimizes the amount of time and resources required

for maintenance and repair.

6.1 Cyclo-filters

Cyclo-filters are commonly used in natural gas processing plants to remove impurities and contaminants from

the gas stream. They are a type of mechanical filter that work by utilizing centrifugal force to separate

impurities from the natural gas.

In a cyclo-filter, the natural gas enters the filter and is directed into a cylindrical chamber. The chamber

rapidly spins, causing the heavier impurities to be thrown to the outside of the chamber and collected at the

bottom, while the lighter natural gas rises to the top and exits the filter. Cyclo-filters are highly efficient at

removing impurities from the natural gas stream, ensuring that the processed gas meets the required quality

standards. Cyclo-filters are relatively simple in design and require minimal maintenance, making them a

cost-effective solution for natural gas processing plants. Cyclo-filters are easy to operate, requiring only min-

imal operator intervention. Cyclo-filters can be used to remove a wide range of impurities, including liquids,

solids, and entrained gases, making them a versatile solution for natural gas processing plants. Cyclo-filters

do not use any chemicals or generate any waste products, making them an environmentally friendly solution

for natural gas processing. Cyclo-filters are an important tool in natural gas processing plants, offering sev-

eral advantages including high efficiency, low maintenance, ease of operation, versatility, and environmental

sustainability. By incorporating cyclo-filters into their filtration systems, natural gas processing plants can

ensure the safe and efficient processing of natural gas, and maintain the highest quality standards. Schematic

diagram of the cyclo-filter technology in natural gas processing plant is indicated in Figure 1 [2].

6.2 Gravity separators

Gravity separators are widely used in natural gas processing plants to remove impurities and contaminants

from the gas stream. The principle behind gravity separation is simple: impurities with a higher density than

the natural gas settle at the bottom of the separator while the lighter gas rises to the top.

There are several advantages to using gravity separators in natural gas processing plants:

• Efficient Separation: Gravity separators are highly efficient at separating impurities from the natural

gas stream, ensuring that the processed gas meets the required quality standards.

• Cost-effective: Gravity separators are relatively simple in design and require minimal maintenance,

making them a cost-effective solution for natural gas processing plants.

• Easy to Operate: Gravity separators are easy to operate, requiring only minimal operator intervention.

• Versatility: Gravity separators can be used to remove a wide range of impurities, including liquids,

solids, and entrained gases, making them a versatile solution for natural gas processing plants.

• Environmental Benefits: Gravity separators do not use any chemicals or generate any waste products,

making them an environmentally friendly solution for natural gas processing.

Figure 1: Schematic diagram of the cyclo-filter technology

Gravity separators are a valuable tool in natural gas processing plants, offering several advantages including

high efficiency, cost-effectiveness, ease of operation, versatility, and environmental sustainability. By incorpo-

rating gravity separators into their filtration systems, natural gas processing plants can ensure the safe and

efficient processing of natural gas, and maintain the highest quality standards [2]. A schematic of a gravity

separator system is shown in Figure 2.

Figure 2: Working of gravity separator

6.3 Backwash

Backwash is a crucial process in natural gas processing plants as it helps maintain the efficiency and longevity

of filtration systems. In simple terms, backwash is the process of reversing the flow of the filtration system,

which cleans the filter media, removes accumulated debris, and restores the filtration system’s original perfor-

mance.

There are several advantages of using backwash in natural gas processing plants. Firstly, backwash helps

to prevent clogging of the filtration system, which can reduce its efficiency and lead to downtime. By regu-

larly cleaning the filter media, backwash ensures that the filtration system operates at optimal performance

levels. Secondly, backwash helps to reduce the cost of maintenance and replacement of filtration systems.

By removing accumulated debris and restoring the system’s performance, backwash extends the lifespan of

the filtration system and reduces the frequency of maintenance and replacement. Thirdly, backwash helps to

improve the quality of the processed natural gas. By removing contaminants and debris, backwash ensures

that the processed natural gas meets the required quality standards, reducing the risk of pipeline blockages

and ensuring safe and efficient transportation. Finally, backwash is an environmentally friendly process that

helps to reduce the amount of waste generated by natural gas processing plants. By removing debris and

contaminants, backwash reduces the need for disposal of contaminated waste, thus minimizing the impact on

the environment [4].

Backwash is a crucial process in natural gas processing plants that offers several advantages, including

improved filtration efficiency, reduced maintenance and replacement costs, improved natural gas quality, and

environmental sustainability. By incorporating backwash into their filtration systems, natural gas processing

plants can ensure the safe and efficient processing of natural gas, and maintain the highest quality standards

[2]. A schematic of the backwash system is illustrated in Figure 3.

Figure 3: Gas assisted backwash system

6.4 Cyclone separators

Cyclone separators are widely used in natural gas processing plants to remove impurities and contaminants

from the gas stream. A cyclone separator works by utilizing centrifugal force to separate heavy impurities

from the natural gas.

In a cyclone separator, the natural gas is directed into a cylindrical chamber where it is rapidly spun. The

spinning motion causes the heavier impurities to be thrown to the outside of the chamber and collected at the

bottom, while the lighter natural gas rises to the top and is then processed further. There are several advan-

tages to using cyclone separators in natural gas processing plants. Cyclone separators are highly efficient at

removing impurities from the natural gas stream, ensuring that the processed gas meets the required quality

standards. Cyclone separators are relatively simple in design and require minimal maintenance, making them

a cost-effective solution for natural gas processing plants. Cyclone separators are easy to operate, requiring

only minimal operator intervention. Cyclone separators can be used to remove a wide range of impurities,

including liquids, solids, and entrained gases, making them a versatile solution for natural gas processing

plants [60]. Cyclone separators do not use any chemicals or generate any waste products, making them an

environmentally friendly solution for natural gas processing.

Cyclone separators are an important tool in natural gas processing plants, offering several advantages

including high efficiency, low maintenance, ease of operation, versatility, and environmental sustainability. By

incorporating cyclone separators into their filtration systems, natural gas processing plants can ensure the

safe and efficient processing of natural gas, and maintain the highest quality standards. Figure 4 shows a

schematic of a system of cyclone separation [2].

6.5 Basket strainers

Basket strainers are commonly used in natural gas processing plants to remove impurities and contaminants

from the gas stream. They are a type of mechanical filter that work by trapping impurities within a mesh or

perforated basket, allowing only the filtered natural gas to pass through. In a basket strainer, the natural gas

enters the strainer through an inlet and flows through the basket, where any impurities or contaminants are

trapped within the mesh or perforations. The filtered natural gas then exits the strainer through an outlet.

There are several advantages to using basket strainers in natural gas processing plants. Basket strainers

are relatively simple in design and cost-effective to manufacture and maintain, making them an attractive

solution for natural gas processing plants. Basket strainers are easy to maintain, requiring only periodic

cleaning or replacement of the basket to ensure optimal performance. Basket strainers can be used to remove

a wide range of impurities, including liquids, solids, and entrained gases, making them a versatile solution

for natural gas processing plants. Basket strainers do not use any chemicals or generate any waste products,

making them an environmentally friendly solution for natural gas processing. Basket strainers are an im-

portant tool in natural gas processing plants, offering several advantages including cost-effectiveness, ease

of maintenance, versatility, and environmental sustainability. By incorporating basket strainers into their

filtration systems, natural gas processing plants can ensure the safe and efficient processing of natural gas,

and maintain the highest quality standards.

Filtration technology has advanced significantly over the years, and there are now several different criteria

for choosing the best filtration technology for contamination control in gas processing plants [61].

• Efficiency: efficiency is the primary criterion for filtration technology in gas processing plants. The

filtration system must effectively remove contaminants from the gas stream without affecting the gas

quality. The efficiency of a filtration system is measured in terms of its ability to capture and retain

particles of a certain size. The filtration system must be able to remove particles as small as possible,

without reducing the flow rate of the gas.

• Capacity: The filtration system’s capacity refers to the volume of gas that can pass through the system

Figure 4: Working of cyclone separation

without causing blockages or pressure drops. A filtration system with a high capacity can process a

larger volume of gas in a shorter amount of time, making it more efficient and cost-effective. However,

the capacity must be balanced with the efficiency of the system, as increasing the capacity may lead to a

reduction in filtration efficiency.

• Maintenance requirements: The filtration system must be easy to maintain, and the maintenance re-

quirements should be minimal. The filters should be easy to replace, and the system should not require

significant downtime to maintain. The filtration system should also be designed to withstand harsh

operating conditions, such as high temperatures and pressures, to reduce the need for frequent replace-

ments.

• Filtration medium: The filtration medium is the material used to capture and retain contaminants in

the gas stream. The choice of filtration medium depends on the type of contaminants present in the gas

stream, as well as the operating conditions of the filtration system. The most common filtration media

include pleated paper, wire mesh, ceramic, and activated carbon.

• Particle size distribution: The particle size distribution of the gas stream is also an important criterion

for filtration technology. The filtration system must be designed to remove particles of a specific size

range, depending on the gas product’s purity requirements. The particle size distribution can vary

depending on the source of the gas, and the filtration system must be able to adapt to these changes.

• Operating temperature and pressure: The filtration system must be able to operate under the gas pro-

cessing plant’s operating conditions, including temperature and pressure. The system must be able to

withstand high temperatures and pressures without degrading or malfunctioning. The operating condi-

tions may also impact the choice of filtration medium and the design of the filtration system.

• Flow rate: The flow rate of the gas stream is an essential criterion for filtration technology. The filtration

system must be able to process the gas stream at the required flow rate, without reducing the efficiency

of the system. The design of the filtration system, including the size and number of filters, must be

optimized to handle the required flow rate.

• Cost: The cost of the filtration system is an important consideration for gas processing plants. The

filtration system must provide efficient and effective contamination control at an affordable cost. The

initial cost of the system, as well as the ongoing maintenance and replacement costs, must be considered

when choosing the filtration technology.

Overall, choosing the right filtration technology for contamination control in gas processing plants involves

a careful evaluation of the efficiency, capacity, maintenance requirements, filtration medium, particle size

distribution, operating temperature and pressure, flow rate, and cost. Each of these criteria must be balanced

to provide the best filtration system for the specific gas product and processing plant’s operating conditions.

The use of advanced filtration technology can significantly improve the quality and purity of gas products,

reducing equipment damage, and maintenance costs, and ensuring the safety of the plant.

6.6 The problem statement

The Sui gas field is Pakistan’s most significant natural gas field at the moment. It may be found in the vicinity

of Sui in Balochistan. Late in the year 1952, the gas field was found, and the following year, in 1955, com-

mercial production of gas from the field started. Six percent of Pakistan’s total gas output comes from the

Sui gas field. The rise of oil and gas-related companies in the area has caused a number of environmental

issues, including an increase in the amount of pollution in the region’s air, water, and land as a result of the

establishment of new businesses such as refineries. The widespread pollution caused by the gas and petroleum

industries has contaminated areas and affected agricultural lands, as well as rural, nomadic, and urban areas.

This is the case despite the significance of the gas and petroleum industries to the region’s economic situation

and the creation of new jobs. It is important to note that there has been no study done on industrial filtration

technology in Sui city, Pakistan, for the purpose of regulating the pollutants generated by these enterprizes.

To solve this problem, the natural gas and oil industries need to select the most effective technology for indus-

trial filtering, one that adheres to the principles of sustainability and maintainability, so that pollution may

be kept under control. This has the potential to have a considerable effect on the region under consideration,

particularly in light of the growing pressure from environmental restrictions and people located nearby. The

environmental effect of the gas and petroleum sector can be mitigated by mandating the use of an appropriate

industrial filtration system in the Sui region, with the goal of improving pollutants management in a way that

is sustainable.

The specific statement about the filtration technology selection for contamination control in gas processing

plants problem is described as follows:

If we consider there are four alterenatives, namely Aג1= Gravity separator, Aג2= Backwash, Aג3= Cyclo-

filters, Aג5= Basket strainers and Aג5= Cyclone separator. The DMs are to be appointed for evalu-

ating the five filtration technology under the criterion given in Table 4. The WV ϖtfor the criterion is

(0.30,0.25,0.20,0.25).

Table 1: Criterion for filtration technology

Criterion

1Efficiency

2Cost (economic aspect )

3Maintenance requirements

4Operating temperature, pressure and flow rate

Here, we take q=4 and ℵħ=8.

6.7 Decision-making process

Step 1:

Obtain the decision matrix EG=(Yji)n×min the format of Hq-ROPFNs from DMs. The judgement values,

given by the DMs, is given in Table 2.

Table 2: Assessment matrix acquired from DMs

1Ok

2Ok

3Ok

Aג1{〈0.35,0.54〉

〈0.61〉} {〈0.65〉

〈0.43,0.32〉} {〈0.84,0.93〉

〈0.45〉} {〈0.95,〉

〈0.34,0.54,0.34〉}

Aג2{〈0.57,0.43,0.64〉

〈0.61,0.15〉} {〈0.65〉

〈0.43,〉} {〈0.43〉

〈0.32,0.56〉} {〈0.78,0.92〉

〈0.45,〉}

Aג3{〈0.76,0.56〉

〈0.43,0.51〉} {〈0.43,0.51〉

〈0.32〉} {〈0.93〉

〈0.45〉} {〈0.98〉

〈0.54,0.34〉}

Aג4{〈0.75,0.32〉

〈0.10〉} {〈0.87〉

〈0.65,0.33〉} {〈0.98〉

〈0.45〉} {〈0.93,0.42〉

〈0.56〉}

Aג5{〈0.87〉

〈0.15〉} {〈0.95〉

〈0.12〉} {〈0.63〉

〈0.45〉} {〈0.98,0.23,0.62〉

〈0.58,0.45〉}

Step 2:

Here Ok

2is cost type attribute so, the normalized decision matrix will be ΓN=(ℵħN

ji )n×m, given in Table 3.

Table 3: Normalized matrix acquired from DMs

1Ok

2Ok

3Ok

Aג1{〈0.35,0.54〉

〈0.61〉} {〈0.43,0.32〉

〈0.65〉} {〈0.84,0.93〉

〈0.45〉} {〈0.95,〉

〈0.34,0.54,0.34〉}

Aג2{〈0.57,0.43,0.64〉

〈0.61,0.15〉} {〈0.43〉

〈0.65〉} {〈0.43〉

〈0.32,0.56〉} {〈0.78,0.92〉

〈0.45,〉}

Aג3{〈0.76,0.56〉

〈0.43,0.51〉} {〈0.32〉

〈0.43,0.51〉} {〈0.93〉

〈0.45〉} {〈0.98〉

〈0.54,0.34〉}

Aג4{〈0.75,0.32〉

〈0.10〉} {〈0.65,0.33〉

〈0.87〉} {〈0.98〉

〈0.45〉} {〈0.93,0.42〉

〈0.56〉}

Aג5{〈0.87〉

〈0.15〉} {〈0.12〉

〈0.95〉} {〈0.63〉

〈0.45〉} {〈0.98,0.23,0.62〉

〈0.58,0.45〉}

Step 3:

Evaluate the aggregated Hq-ROPF matrix by using Hq-ROPFAAWA AOs.

Table 4: Aggregated Hq-ROPF matrix

Aג1{〈0.930,0.935,0.933,0.935,0.933,0.935,0.933,0.935〉,〈0.400,0.513,0.400〉}

Aג2{〈0.753,0.899,0.753,0.899,0.753,0.899〉,〈0.390,0.507,0.195,0.195〉}

Aג3{〈0.969,0.970〉,〈0.446,0.389,0.462,0.393,0.466,0.394,0.477,0.396〉}

Aג4{〈0.968,0.968,0.968,0.967,0.967,0.968,0.966,0.968〉,〈0.138〉}

Aג5{〈0.969,0.849,0.849〉,〈0.196,0.195〉}

Step 4:

Compute the score value of the aggregated value using SF.

S(Aג

1)=0.79995

S(Aג

2)=0.51037

S(Aג

3)=0.91841

S(Aג

4)=0.87657

S(Aג

5)=0.64171

Step 5:

At the end, the final ranking will be

Aג

3≻Aג

4≻Aג

1≻Aג

5≻Aג

As a result, alternative Aג5is the best filtration technology for contamination control in gas processing plants.

7 Comparative analysis

This section compares numerous planned AOs to current AOs. By solving the data with preexisting AOs, we

get an optimal solution equivalent to our findings. This displays the longevity and effectiveness of the AO.

Compared to a number of previously reported AOs, our method is more applicable and better. Our optimum

solution is validated by running it through a number of current operators. Our optimal decision is identical,

proving the validity of our suggested AOs.

Table 5: Comparison of proposed operators with some exiting operators

Authors AOs Ranking of alternatives The optimal alternative

Hussain et al. [31] Hq-ROPFWA Aג3≻Aג1≻Aג4≻Aג5≻Aג2Aג3

Hq-ROPFWG Aג3≻Aג2≻Aג1≻Aג5≻Aג4Aג3

Proposed Hq-ROPFAAWA Aג3≻Aג4≻Aג1≻Aג5≻Aג2Aג3

Hq-ROPFAAWG Aג3≻Aג2≻Aג1≻Aג5≻Aג4Aג3

8 Conclusion

The industrial operations in the oil and natural gas industries have resulted in a proliferation of pollutants

that pose essential global issues, such as climate change and global warming. To prevent environmental and

human damage, it is vital to use industrial filtration technology in these industries to regulate and manage

these pollutants. In natural gas production, failure to eradicate these impurities might result in a number of

process-related complications. Energy production systems, such as oil and natural gas, coal and metal mining,

and waste management, play a significant role in environmental contamination, a concern for both developed

and developing nations. The long-term effects of environmental contamination require fast deliberation and

action. Assessing and managing filtering technologies in the processing sector is an essential task to imple-

ment suitable industrial systems. To effectively select the most appropriate technology and mitigate contami-

nant emissions in energy sector projects, a robust decision-making method is necessary. Such a method should

be able to assess these technologies under real conditions and uncertainties and provide valuable insights to

DMs and policymakers. The primary goal of this research is to develop a novel approach called AA AOs for

Hq-ROPFNs to prioritize industrial filtration technologies. This approach takes into account various criteria,

allowing DMs to make informed decisions that are aligned with their preferences and real-world conditions.

For this we first extended the Aczel–Alsina t-norm and t-conorm to Hq-ROPF contexts, then established and

assessed a number of innovative operational principles for Hq-ROPFNs. The fundamental properties of the

proposed laws are discussed in detail. Based on the proposed laws, we define several AOs to aggregate the

Hq-ROPF information, namely the Hq-ROPFAAWA operator, Hq-ROPFAAOWA operator, Hq-ROPFAAHA op-

erator, Hq-ROPFAAWG operator, Hq-ROPFAAOWG operator, and Hq-ROPFAAHG operator. In addition, the

fundamental axioms of the operators are met by the suggested work. The suggested operators have been

applied to the MCDM method with Hq-ROPF data, and a numerical model exhibiting the decision-making

method has been produced.

Many applications, such as network analysis, risk assessment, cognitive science, reinforcement learning,

signal processing, and others with uncertain contexts, will eventually use the aforementioned operators and

approaches. Throughout the aggregation process of future initiatives, we want to evaluate the interrelation-

ships between attribute pairs. In addition, we intend to develop more broad information metrics to assist us

to comprehend the information we encounter on a regular basis.

Compliance with ethical standards

Conflict of interest:

The authors declare that they have no conflict of interest.

Ethical approval:

This article does not contain any studies with human participants or animals performed by any of the authors.

Authorship contributions:

All authors contributed to the design and implementation of the research, to the analysis of the results and to

the writing of the manuscript.

Acknowledgment:

Bandar Almohsen Supporting Project number (RSP2023R158), King Saud University, Riyadh, Saudi Arabia.

References

[1] F. Alhamad, M. Ali, M. Ali, H. Abid, H. Hoteit, S. Iglauer, A. Keshavarz, Effect of methyl orange on wettability of

sandstone formations: Implications for enhanced oil recovery, Energy Reports, 8(2022), 12357–12365.

[2] T. Wines, S. Mokhatab, Contamination Control in the Natural Gas Industry. Contamination Control in the Natural

Gas Industry, first ed. Elsevier. https://doi.org/10.1016/c2017-0-00554-2.

[3] A. Fetanat, M. Tayebi, H. Mofid, Combining life cycle sustainability assessment and fuzzy multicriteria decision

making method for prioritizing the flare technologies in the oil, gas, and chemical plants, Environmental Progress &

Sustainable Energy, (2022), Doi:10.1002/ep.13837.

[4] A. Manzardo, J. Ren, A. Mazzi, A. Scipioni, A grey-based group decision-making methodology for the selection of

hydrogen technologies in life cycle sustainability perspective, International Journal of Hydrogen Energy, 37(2012),

17663–17670.

[5] J. Ren, A. Manzardo, S. Toniolo, A. Scipioni, Sustainability of hydrogen supply chain. Part I: Identification of crit-

ical criteria and cause-effect analysis for enhancing the sustainability using DEMATEL, International Journal of

Hydrogen Energy, 38(2013), 14159–14171.

[6] L. A. Zadeh,Fuzzy sets, Information and Control, 8(1965), 338-353.

[7] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets ans Systems, 20(1)(1986), 87-96.

[8] R. R. Yager and A. M. Abbasov, Pythagorean membership grades, complex numbers, and decision making, Interna-

tional Journal of Intelligent Systems, 28(2013), 436-452,.

[9] R. R. Yager, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy Systems, 25(2017), 1222-1230.

[10] P. Liu, J. Liu, Some q-rung orthopair fuzzy bonferroni mean operators and their application to multi-attribute group

decision making, International Journal of Intelligent Systems, 33(2) 2018, 315-347.

[11] Z. Liu, S. Wang, P. Liu, Multiple attribute group decision making based on q-rung orthopair fuzzy Heronianmean

operators, International Journal of Intelligent Systems, 33(12)(2018), 2341-2364.

[12] B. P. Joshi, A. Gegov, Confidence levels q-rung orthopair fuzzy aggregation operators and its applications to MCDM

problems, International Journal of Intelligent Systems, (2020), 125-149.

[13] H. M. A. Farid, M. Riaz, q-rung orthopair fuzzy Aczel–Alsina aggregation operators with multi-criteria decision-

making, Engineering Applications of Artificial Intelligence, 122(2023), 106105.

[14] M. Riaz, M. T. Hamid, H.M. A. Farid, D. Afzal, TOPSIS, VIKOR and aggregation operators based on q-rung orthopair

fuzzy soft sets and their applications, Journal of Intelligent & Fuzzy Systems, 39(5) (2020), 6903-6917.

[15] H. Garg, CN-ROFS: Connection number-based q-rung orthopair fuzzy set and their application to decision-making

process , International Journal of Intelligent Systems, 36(7)(2021), 3106–3143.

[16] P. Liu, P. Wang, Some q-rung orthopair fuzzy aggregation operator and their application to multi-attribute decision

making, International Journal of Intelligent Systems, 33(2)(2018), 259-280.

[17] C. Jana, G. Muhiuddin, M. Pal, Some Dombi aggregation of q-rung orthopair fuzzy numbers in multiple-attribute

decision making, International Journal of Intelligent Systems, 34(12)(2019), 3220–3240.

[18] G. Wei, H. Gao, Y. Wei, Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making,

International Journal of Intelligent Systems, 33(7)(2018), 1426–1458.

[19] M. Lin, X. Li, L. Chen, Linguistic q-rung orthopair fuzzy sets and their interactional partitioned Heronian mean

aggregation operators, International Journal of Intelligent Systems, 35(2)(2020), 217–249.

[20] M. J. Khan, M. I. Ali, P. Kumam A new ranking technique for q-rung orthopair fuzzy values, International Journal of

Intelligent Systems, 36(1)(2021), 558–592.

[21] S. Z. Zeng, Y. J. Hu, X. Y. Xie, q-Rung orthopair fuzzy weighted induced logarithmic distance measures and their appli-

cation in multiple attribute decision making, Engineering Applications of Artificial Intelligence, 100(2021), 104167.

[22] M. Sitara, M. Akram, M. Riaz, Decision-making analysis based on q-rung picture fuzzy graph structures, Journal of

Applied Mathematics and Computing, (2021), https://doi.org/10.1007/s12190-020-01471-z.

[23] H. M. A. Farid, M. Riaz, Some generalized q-rung orthopair fuzzy Einstein interactive geometric aggregation operators

with improved operational laws, International Journal of Intelligent Systems, 36(2021), 7239-7273.

[24] A. Saha, D. Dutta, S. Kar, Some new hybrid hesitant fuzzy weighted aggregation operators based on Archimedean and

Dombi operations for multi-attribute decision making, Neural Computing & Application, 33(14)(2021), 8753-8776.

[25] Attaullah, S. Ashraf, N. Rehman, A. Khan, C. Park, A decision making algorithm for wind power plant based on

q-rung orthopair hesitant fuzzy rough aggregation information and TOPSIS, AIMS Mathematics, 7(4)(2022), 5241-

5274.

[26] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent System, 25(2010), 529-539.

[27] J. Hu, X. Zhang, X. Chen, Y. Liu, Hesitant fuzzy information measures and their applications in multi-criteria decision

making, International Journal of Systems Science, 47(2016), 62-76.

[28] B. Zhu, Z. S. Xu, M. M. Xia, Dual hesitant fuzzy sets, Journal of Applied Mathematics, 2012, Article ID 879629, 13

pages, 2012. https://doi.org/10.1155/2012/879629.

[29] J. J. Peng, J. Q. Wang, X. H. Wu, H. Y. Zhang, X. H. Chen, The fuzzy cross-entropy for intuitionistic hesitant fuzzy

sets and their application in multi-criteria decision-making, International Journal of Systems Science, 46(2015),

2335-2350.

[30] M. S. A. Khan, S. Abdullah, A. Ali, N. Siddiqui, F. Amin, Pythagorean hesitant fuzzy sets and their application to

group decision making with incomplete weight information, Journal Intelligent & Fuzzy Syst, 33(2017), 3971–3985.

[31] A. Hussain, M. I. Ali, T. Mahmood, Hesitant q-rung orthopair fuzzy aggregation operators with their applications in

multi-criteria decision making, Iranian Journal of Fuzzy Systems, 17(3)(2020), 117-134.

[32] He, X. Typhoon disaster assessment based on Dombi hesitant fuzzy information aggregation operators, Natural Haz-

ards, 90(2018), 1153-1175.

[33] P. Liu, A. Saha, D. Dutta, S. Kar, Multi-Attribute Decision-Making Using Hesitant Fuzzy Dombi–Archimedean

Weighted Aggregation Operators, International Journal of Computational Intelligence Systems, 14(2021), 386-411.

[34] A. Saha, D. Dutta, S. Kar, Some new hybrid hesitant fuzzy weighted aggregation operators based on Archimedean and

Dombi operations for multi-attribute decision making, Neural Computing and Applications, 33(2021), 8753–8776.

[35] Q. Yu, F. Hou, Y. Zhai, Y. Du, Some Hesitant Fuzzy Einstein Aggregation Operators and Their Application to Multiple

Attribute Group Decision Making, International Journal of Intelligent System, 31(2015), 722-746.

[36] L. Zhou, X. Zhao, G. Wei, Hesitant fuzzy Hamacher Aggregation operators and their application to multiple attribute

decision making, Journal of Intelligent & Fuzzy Systems, 26(2014), 2689-2699.

[37] X. Li, X. Zhang, Single-Valued Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators for Multi-Attribute Deci-

sion Making, Symmetry, 10( 2018), 50.

[38] R. Wang, Y. Li, Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators and Their

Applications to Multiple Criteria Decision-Making, Information, 9(2018), 10.

[39] L. Wang, Y. L. Bao, Multiple-Attribute Decision-Making Method Based on Normalized Geometric Aggregation Opera-

tors of Single-Valued Neutrosophic Hesitant Fuzzy Information, Complexity, 2021(2021), 5580761.

[40] K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences of the United States of America,

28(12)(1942), 535-537.

[41] M. Xia, Z. Xu, B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean tconorm and

t-norm, Knowledge Based Systems, 31(2012), 78-88.

[42] W. Wang, X. Liu, Intuitionistic fuzzy information aggregation using Einstein operations, IEEE Transactions on Fuzzy

Systems, 20(5)(2012), 923-938.

[43] P. Liu, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their

application to group decision making, IEEE Transactions on Fuzzy Systems, 22(2014), 83-97.

[44] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Dordrecht: Kluwer Academic Publishers, 2000.

[45] J. Aczel, C. Alsina, Characterization of some classes of quasilinear functions with applications to triangular norms

and to synthesizing judgements, Aequationes Math, 25(1)(1982), 313-315.

[46] N. Wang, Q. Li, A. A. A. El-Latif, X. Yan, X. Niu, A novel hybrid multibiometrics based on the fusion of dual Iris,

visible and thermal face images, In nnternational Symposium on Biometrics and Security Technologies, Chengdu,

(2013), 217-223.

[47] T. Senapati, G. Chen, R. R. Yager, Aczel–alsina aggregation operators and their application to intuitionistic fuzzy

multiple attribute decision making, International Journal of Intelligent Systems, 37(2022), 1529–1551.

[48] T. Senapati, G. Chen, R. Mesiar, R. R. Yager, Novel Aczel.Alsina operations-based interval-valued intuitionistic fuzzy

aggregation operatorsandtheir applications in multiple attribute decision-making process, International Journal of

Intelligent Systems, (2021), https://doi.org/10.1002/int.22751.

[49] A. K. Adak, G. Kumar, Spherical Distance Measurement Method for Solving MCDM Problems under Pythagorean

Fuzzy Environment, Journal of Fuzzy Extension & Applications, 4(1)(2023), 28-39.

[50] F. Smarandache, An Overview of Plithogenic Set and Symbolic Plithogenic Algebraic Structures, Journal of Fuzzy

Extension & Applications, 4(1)(2023), 52-59.

[51] D. Panchal, Reliability analysis of turbine unit using Intuitionistic Fuzzy Lambda-Tau approach, Reports in Mechan-

ical Engineering, 4(1)(2023), 47-61.

[52] A. Ashraf, K. Ullah, A. Hussain, M. Bari, Interval-Valued Picture Fuzzy Maclaurin Symmetric Mean Operator with

application in Multiple Attribute Decision-Making, Reports in Mechanical Engineering, 3(1)(2022), 210-226.

[53] C. J. Kwak, K. C. Ri, S. I. Kwak, K. J. Kim, U. S. Ryu, O. C. Kwon, N. H. Ki, Fuzzy modus ponens and tollens based

on moving distance in SISO fuzzy system, Romanian Journal of Information Science and Technology, 24(3)(2021),

257-283.

[54] A. Ramathilagam, P. Pitchipoo, Modeling and development of fuzzy logic-based intelligent decision support system,

Romanian Journal of Information Science and Technology, 25(1)(2022), 58-79.

[55] L. Mahmood, Z. Bahroun, M. Ghommem, H. Alshraideh, Assessment and performance analysis of Machine learning

techniques for gas sensing E-nose systems, Facta Universitatis, Series: Mechanical Engineering, 20(3)(2022), 479-501.

[56] A. Shekhovtsov, J. Wieckowski, B. Kizielewicz, W. Salabun, Towards Reliable Decision-Making in the green urban

transport domain, Facta Universitatis, Series: Mechanical Engineering, 20(2)(2022), 381-398.

[57] H. Fazlollahtabar, N. Kazemitash, Green supplier selection based on the information system performance evaluation

using the integrated Best-Worst Method, Facta Universitatis, Series: Mechanical Engineering, 19(3)(2021), 345-360.

[58] M. Jagtap, P. Karande, The m-polar fuzzy set ELECTRE-I with revised Simos’ and AHP weight calculation methods

for selection of non-traditional machining processes, Decision Making: Applications in Management and Engineering,

6(1)(2023), 240-281.

[59] P. Sivaprakasam, M. Angamuthu, Generalized Z-fuzzy soft ß-covering based rough matrices and its application

to MAGDM problem based on AHP method, Decision Making: Applications in Management and Engineering,

6(1)(2023), 134-152.

[60] S. Sangrola, A. Kumar, S. Nivedhitha, J. Chatterjee, S. Subbiah, S. Narayanasamy, Optimization of backwash param-

eters for hollow fiber membrane filters used for water purification, Journal of Water Supply: Research and Technology-

Aqua, 69(2020), 523–537.

[61] M. Tayebi, A. Bemani, A. Fetanat, M. Fehresti-Sani, A decision support system for sustainability prioritization of air

pollution control technologies in energy and carbon management: Oil & gas industry of Iran, Journal of Natural Gas

Science and Engineering, 99(2022), 104416.