Selecting an Optimal Approach to Reduce Drivers of Climate Change in a Complex Intuitionistic Fuzzy Environment

Abstract

The sustainability of the climate is a critical concern in the modern world. A variety of acts are included in sustainability that strive to lessen our carbon footprint and maintain the fragile balance of our world. To preserve a sustainable future for future generations, we must cooperate in adopting renewable energy sources, supporting green transportation, and implementing responsible land use. In this article, we propose the concepts of complex intuitionistic fuzzy Dombi hybrid averaging (CIFDHA) and complex intuitionistic fuzzy Dombi hybrid geometric (CIFDHG) operators within the framework of a complex intuitionistic fuzzy environment. Furthermore, we explore several additional important features of these operators. To overcome the limitations of the existing score function within the CIF knowledge context, we present a new and improved score function. Additionally, we apply the proposed score function and newly defined operators to select an optimal strategy for mitigating the drivers of climate change and saving the planet’s valuable resources for a more livable and resilient planet. In order to demonstrate the validity and practicality of the suggested strategies, we conducted a comparative study of these novel techniques with existing methods.

1. Introduction

Decisions involving several choices, actions, or candidates that must be prioritized according to a list of criteria are called multiple criteria decision-making (MCDM) problems. Since MCDM via aggregation operators (AOs) can easily tackle real-world problems in practically any field, including science, engineering, the environment, sociology, and many others, it is gaining in popularity. The purpose of AOs is to combine all the values in a set into a single value from that set, taking into account all the values in the original set. Before AOs existed, judgments were made using crisp sets. However, in practice, membership in a set is not often so precise, notably in the biological and social sciences, language and linguistics, psychology and economics, and more generally, the soft sciences, where traditional mathematical tools have limited efficacy. In 1965, Zadeh [1] developed the concept of partial belongingness of a set, which he called a fuzzy set (FS), to address this issue. Kahne [2] proposed a strategy for DM in 1975, applicable in cases where many criteria of varied weights must be used to evaluate potential options. A DM approach for computing the fuzzy optimal alternative was first given by Jain [3] in 1977. Fundamental operations on fuzzy sets were addressed by Dubois and Prade [4] in 1978. AOs for fuzzy sets were first presented by Yager [5].

Atanassov [6] developed an extension of fuzzy sets in 1986, termed intuitionistic fuzzy sets (IFS), which describes both membership degree (MD) and non-membership degree (NMD) under the constraint that the sum of MD and NMD should be less than or equal to 1. To deal with MCDM issues, Chen and Tan [7] introduced the concept of a score function inside the IFS theory in 1994. A technique for finding the answer to a group decision problem within the IFS settings was designed by Szmidt and Kaeprzyk [8] in 1996. Some basic linear programming methods and techniques were investigated for MCDM in IF contexts by Li [9] in 2005. Some geometric AOs on IFS were defined by Xu and Yager [10] in 2006. A few basic arithmetic aggregation procedures were created by Xu [11] in an IFS setting back in 2007. To address the MCDM issue, Zhao et al. [12] proposed the use of generalized aggregation operations on IFS in 2010. Group decision-making issues by applying AOs in IFS settings were addressed in [13]. Using Hamacher AOs on IFS, Huang [14] developed a DM strategy in 2014. Bonferroni mean operators for IFS were first described by Verma [15] in 2015. In order to address MCDM issues, Senapati et al. [16] presented the Aczel-Alsina AOs of IFS.

Researchers have found that Dombi AOs are among the most useful tools for tackling MCDM difficulties. In 2019, Pythagorean fuzzy Dombi AOs were designed by Akram et al. [17]. Afterward, QoF Dombi AOs [18], BF Dombi AOs [19], and PF Dombi AOs [20] were defined. In 2020, spherical fuzzy Dombi AOs were proposed by Ashraf et al. [21]. Liu et al. [22] investigated Dombi AOs for the hesitant fuzzy set for use in assessing risks to computer systems and networks in 2020. In addition, Dombi operators on IFS were first presented by Seikh and Mandal [23] in 2021. The tentative T-SF Dombi AOs and their implementation in MCDM were introduced by Karaaslan and Husseinawi [24]. In order to identify the most optimal solution to the MCDM problem, Alhamzi et al. [25] created interval-valued Pythagorean fuzzy Dombi AOs in 2023.

Both FS and IFS are mainly concerned with dealing with problems of a single dimension. The two-dimensional problems are what initially obscure the intriguing scenario. In 2002, Ramot et al. [26] established a stunning innovative ideology called complex fuzzy set (CFS) to tackle problems in two dimensions. Therefore, CFS theory, by including a second dimension in the declaration of MD, modifies the core concept of fuzzy membership. Numerous physical problems, such as risk assessment, data mining, wave function, pattern recognition, impedance in electrical engineering, and complex amplitude, have been solved with the help of CFS. Applications where several fuzzy variables are linked in a way that standard fuzzy operations cannot fully figure out, like advanced control and predicting recurring events, demonstrate the significance of this approach. In the current research of CFSs, MDs are employed to deal with the uncertainties in the data. However, there is a risk that this method may obscure relevant details and hinder decision-making. Considering the deficiency in CFS, Alkouri and Salleh [27] developed the complex intuitionistic fuzzy set (CIFS) by extending the concept of CFS with a complex NMD function in 2012. Therefore, a CIFS is an umbrella term for several different theories, including FS, IFS, and CFS.

Table 1. Comparison of CIFS model with other existing models.

Models	Uncertainty	Hesitation	Falsity	Periodicity	Handles 2D Data	Have Generalization
FS	$Y$	$N$	$N$	$N$	$N$	$N$
IFS	$Y$	$Y$	$Y$	$N$	$N$	$N$
CFS	$Y$	$N$	$N$	$Y$	$Y$	$N$
CIFS	$Y$	$Y$	$Y$	$Y$	$Y$	$Y$

provides a comparison of CIFS characteristics to those of other sets. In the table below, the symbol $Y$ denotes one’s ability to deal with the scenario in a certain fuzzy setting. Important factors like uncertainty, falsity, hesitation, periodicity, and lack of two-dimensional information about a physical phenomenon can all be counted on by the CIFS model, but the other environments listed in the table can only count on some of these characteristics.

In addition, CF AOs were created in [28,29]. In the CIF setting, arithmetic and geometric AOs were suggested by Garg and Rani [30]. Hamacher AOs on CIFS were discussed by Akram et al. [31]. CIF Aczel-Alsina AOs were first proposed by Mahmood et al. [32] in 2022. Masmali et al. [33] investigated CIF Dombi AOs in order to identify the most optimal solution to the MCDM problem in 2023. Additionally, the most recent advancements in these theories can be studied in [34,35,36,37,38,39,40,41,42].

Only Earth, out of all the incredible worlds in our solar system, is known to be habitable because of the unique combination of gases that surround the globe and serve to sustain life by providing air to breathe, shielding us from the sun’s harmful ultraviolet radiations, keeping the world warm, and reducing the extremes in temperature that occurs between the day and night. However, greenhouse gas emissions from expanding industries and the burning of fossil fuels act as a blanket over the Earth, preventing the sun’s heat from escaping and leading to climate change. Increasing temperatures are often cited as the most obvious consequence of climate change. But the rising temperatures are not what kicks off the story. Since Earth is a system in which everything is interconnected, changes in one region may have an effect on changes in all areas. Climate change affects many aspects of society, including health, food production, housing, safety, and work. We are all vulnerable to climate change, but those of us living in low-lying island nations and other developing countries are especially at risk. Some of the present impacts of climate change include intense droughts, a lack of water, severe fires, rising sea levels, flooding, the melting of polar ice, catastrophic storms, and a reduction in biodiversity. Chronic droughts have increased the risk of starvation while rising sea levels and saltwater intrusion have grown so dangerous that entire communities must be abandoned. Many more “climate refugees” will likely emerge in the years to come. In a nutshell, climate change poses a serious obstacle to progress. It could have a negative impact on economic growth and lead to increased poverty. At the same time, national developmental strategies and investments in meeting the energy, food, and water needs of a growing population can either exacerbate climate change and global risks or contribute to finding solutions. In order to mitigate climate change, governments around the world will need to ensure that their economies continue to expand [43,44,45,46,47]. In the present study, the foremost aim is to choose a reliable strategy to reduce drivers of climate change under the CIF Dombi knowledge.

The exceptional flexibility of the general purpose Dombi operators, along with their aggregation feature, operational qualities, and DM abilities, enables them to effectively handle imprecise information. By employing aggregation procedures, these operators condense the data into a single numerical representation. Their ability to swiftly adapt to changing operational conditions and tackle complex DM problems makes them highly efficient. Furthermore, the CIF Dombi AOs effectively address the issue of changing preferences caused by information loss in existing IF operators. This article proposes strategies that are more comprehensive and generalized compared to other existing techniques using CIF Dombi AOs. The addition of parametric values to these newly defined operators enhances their adaptability. When discussing practical MCDM challenges, the CIF Dombi AOs, in conjunction with other powerful mathematical tools, improve the precision and reliability of optimal results by taking into account all data throughout the aggregation process. Hence, they provide a groundbreaking approach for resolving MCDM issues, as discussed earlier.

The first and foremost aim of this research work is to create practical methods for dealing with many MADM issues in CIFS settings. The approach presented in this article is superior because it controls input dependencies. Because of this, our approach can be used in a wider variety of scenarios. Since the parameter cannot be dynamically adjusted to reflect the decision-makers’ risk aversion, it is difficult to put the MCDM approach into practice. However, the strategies discussed here are more than capable of making up for this deficiency. Since there is now no practical solution to the difficulty of resolving the issue of reducing climate-changing factors in a CIF context, it is important to emphasize that the method outlined in this article is rather novel in its own right. The goal of this publication is to introduce several aggregation methods for combining distinct types of CIFS. These techniques were developed to account for the correlation between MD pairs. The MD, a subset of the real numbers, is used in current research on fuzzy and its expansions to account for data uncertainty. Because of this, important data are lost, which may have an impact on subsequent choices. CIFS is a more general case of these that can store and process data in two dimensions simultaneously. Using degrees with ranges that go from the real subset to the complex subset with a unit disk makes it possible to deal with uncertainties, which is necessary for this purpose. This motivates us to formulate some unique aggregation methods and apply them to provide an answer to the research question of choosing a suitable strategy for reducing drivers of climate change using innovative approaches in the CIF Dombi environment.

The sections of this paper are categorized as follows: In Section 2, we take a look at a few common definitions. In Section 3, we discuss a shortcoming of the previous score function and present a new score function to overcome the deficiency inside the CIF setting. In Section 4, we present Dombi aggregation operations on CIFS and explore some of their fundamental characteristics. In Section 5, we apply the newly established operators to determine the optimal approach to reduce climate-changing factors and give a comparison of the proposed techniques with existing methodologies that illustrates the validity and feasibility of these strategies. Finally, we outline the conclusions from the present study.

2. Preliminaries

In order to comprehend the work presented in this article, it is beneficial to familiarize oneself with the fundamental definitions provided in this section:

Definition 1 

([6]). An IFS $S$ of $Y$ is defined as $S=\left\{\left(y,{\mu }_S\left(y\right),{\nu }_S\left(y\right)\right)|y\in Y\right\},$ where ${\mu }_S,{\nu }_S:Y\to [0,1]$ represents the membership and non-membership functions, respectively, satisfying $0\le {\mu }_S\left(y\right)+{\nu }_S\left(y\right)\le 1$. The degree of hesitation is defined as ${\pi }_S\left(y\right)=1-{\mu }_S\left(y\right)-{\nu }_S\left(y\right)$.

Example 1. 

An IFS $S$ of the universal set $Y=\{a,b,c,d\}$ is defined as follows:

$$S=\left\{\left(a,0.6,0.3\right),\left(b,0.4,0.3\right),\left(c,0.8,0.1\right),\left(d,0.3,0.6\right)\right\}.$$

Definition 2 

([26]). A CFS $S$  defined on a universe of discourse $Y$  is defined as $S=\left\{\left(y,{\mu }_S\left(y\right)\right)|y\in Y\right\},$  where ${\mu }_S$  maps each element of $Y$  to a closed unit circle in complex plane and is defined as ${\widehat{ᵲ}}_S\left(y\right)e^{i2\pi {\widehat{\text{ϣ}}}_S\left(y\right)}$ , where ${\widehat{ᵲ}}_S\left(y\right)$ denotes the real value from $Y$ to [0,1] and $e^{i2\pi {\widehat{\text{ϣ}}}_S\left(y\right)}$ is a value of the periodic membership function, and $0\le {\widehat{\text{ϣ}}}_S\left(y\right)\le 1$, respectively.

Example 2. 

A CFS $S$ of the universal set $Y=\{a,b,c\}$ is defined as follows:

$$S=\left\{\left(a,0.4e^{i2\pi \left(0.5\right)}\right),\left(b,0.6e^{i2\pi \left(0.7\right)}\right),\left(c,0.7e^{i2\pi \left(0.4\right)}\right)\right\}.$$

Definition 3 

([27]). A CIFS $S$ of $Y$ is defined as $S=\left\{\left(y,{\mu }_S\left(y\right),{\nu }_S\left(y\right)\right)|y\in Y\right\},$where ${\mu }_S\left(y\right)$ and ${\nu }_S\left(y\right)$are the complex-valued MD and NMD, respectively, defined from  $Y$ to the unit closed circle, with ${\mu }_S\left(y\right)={\widehat{ᵲ}}_S\left(y\right)e^{i2\pi {\widehat{\text{ϣ}}}_S\left(y\right)}$, ${\nu }_S\left(y\right)={\widehat{ᵵ}}_S\left(y\right)e^{i2\pi {\widehat{ϥ}}_S\left(y\right)}$ as their respective MD and NMD for each $y \in Y$ with $0\le {\widehat{ᵲ}}_S\left(y\right),{\widehat{ᵵ}}_S\left(y\right),{\widehat{\text{ϣ}}}_S\left(y\right),{\widehat{ϥ}}_S\left(y\right),{\widehat{ᵲ}}_S\left(y\right)+{\widehat{ᵵ}}_S\left(y\right),{\widehat{\text{ϣ}}}_S\left(y\right)+{\widehat{ϥ}}_S\left(y\right)\le 1.$

Example 3. 

A CIFS $S$ of the universal set $Y=\{a,b,c\}$ is defined as follows:

$$S=\left\{\begin{array}{c}\left(a,0.4e^{i2\pi \left(0.5\right)},0.5e^{i2\pi \left(0.3\right)}\right),\left(b,0.6e^{i2\pi \left(0.7\right)},0.3e^{i2\pi \left(0.1\right)}\right),\\ \left(c,0.7e^{i2\pi \left(0.4\right)},0.2e^{i2\pi \left(0.3\right)}\right)\end{array}\right\}.$$

It may be noted that in a CIFS environment, the presentation of an element $y=\left(\left(\widehat{ᵲ},\widehat{\text{ϣ}}\right),\left(\widehat{ᵵ},\widehat{ϥ}\right)\right)$ is called a complex intuitionistic fuzzy number (CIFN) that satisfies $0\le \widehat{ᵲ},\widehat{ᵵ},\widehat{ᵲ}+\widehat{ᵵ}\le 1$ and $0\le \widehat{\text{ϣ}},\widehat{ϥ},\widehat{\text{ϣ}}+\widehat{ϥ}\le 1$. We use this specific presentation of a CIFN in the subsequent study of this article.

Definition 4 

([27]). Any two CIFNs ${ß}_1=\left(\left({\widehat{ᵲ}}_1,{\widehat{\text{ϣ}}}_1\right),\left({\widehat{ᵵ}}_1,{\widehat{ϥ}}_1\right)\right)$ and ${ß}_2=\left(\left({\widehat{ᵲ}}_2,{\widehat{\text{ϣ}}}_2\right),\left({\widehat{ᵵ}}_2,{\widehat{ϥ}}_2\right)\right)$ admit the following properties:

1. 

${ß}_1\prec {ß}_2, if {\widehat{ᵲ}}_1<{\widehat{ᵲ}}_2,{\widehat{ᵵ}}_1>{\widehat{ᵵ}}_2,{\widehat{\text{ϣ}}}_1<{\widehat{\text{ϣ}}}_2 and {\widehat{ϥ}}_1>{\widehat{ϥ}}_2,$

2. 

${ß}_1={ß}_2, if {\widehat{ᵲ}}_1={\widehat{ᵲ}}_2,{\widehat{ᵵ}}_1={\widehat{ᵵ}}_2,{\widehat{\text{ϣ}}}_1={\widehat{\text{ϣ}}}_2 and {\widehat{ϥ}}_1={\widehat{ϥ}}_2,$

3. 

${ß}_1^c=\left(\left({\widehat{ᵵ}}_1,{\widehat{ϥ}}_1\right),\left({\widehat{ᵲ}}_1,{\widehat{\text{ϣ}}}_1\right)\right).$

Example 4. 

Let  ${ß}_1=\left(\left(0.6,0.5\right),\left(0.3,0.5\right)\right), {ß}_2=\left(\left(0.5,0.4\right),\left(0.1,0.45\right)\right)$ and ${ß}_3=\left(\left(0.6,0.5\right),\left(0.3,0.5\right)\right)$. Then, ${ß}_2\prec {ß}_1,$  ${ß}_1={ß}_3$ and ${ß}_1^c=\left(\left(0.3,0.5\right),\left(0.6,0.5\right)\right).$

Definition 5 

([30]). A CIF hybrid averaging (CIFHA) operator of dimension $\text{ք}$ is characterized by a function $CIFHA: {ß}^{\text{ք}}\to ß$ with corresponding weight vector $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{\text{ք}}\right)}^T$ such that

$$CIFHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={\oplus }_{\tau =1}^{\text{ք}}\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right)=\left(\begin{array}{c}\left(1-{\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left(1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }},1-\prod _{\tau =1}^{\text{ք}}{\left(1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }}\right),\\ \left({\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left({\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }},{\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left({\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }}\right)\end{array}\right)$$

where  $0\le {\text{ϣ}}_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք} and \sum _{\tau =1}^{\text{ք}}{\text{ϣ}}_{\tau }=1.$ Note that  ${\dot{ß}}_{\sigma \left(\tau \right)}$ is the  $\tau th$ largest of the weighted CIFNs  ${\dot{ß}}_{\tau }=\text{ք}{\text{ϣ}}_{\tau }{ß}_{\tau }$,  ${ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ and  $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ is the aggregation-associated weight vector such that  $0\le {\xi }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  $\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }=1.$

Definition 6 

([30]). A CIF hybrid geometric (CIFHG) operator of dimension $\text{ք}$ is characterized by a function $CIFHG: {ß}^{\text{ք}}\to ß$ with corresponding weight vector $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\text{ϣ}}_{\text{ք}}\right)}^T$ such that

$$CIFHG\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={\otimes }_{\tau =1}^{\text{ք}}{\left({\dot{ß}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }}=\left(\begin{array}{c}\left({\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left({\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }},{\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left({\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }}\right),\\ \left(1-{\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left({1-\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }},{\displaystyle \prod _{\tau =1}^{\text{ք}}}{\left(1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }}\right)\end{array}\right)$$

 where  $0\le {\omega }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք} and \sum _{\tau =1}^{\text{ք}}{\omega }_{\tau }=1.$ Note that  ${\dot{ ß}}_{\sigma \left(\tau \right)}$ is the  $\tau th$ largest of the weighted CIFNs  ${\dot{ ß}}_{\tau }={\left({ß}_{\tau }\right)}^{\text{ք}{\text{ϣ}}_{\tau }}$,  ${ ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ and  $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ is the aggregation-associated weight vector such that  $0\le {\xi }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  $\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }=1.$

Definition 7 

([48]). Let $a$ and $b$ be two real numbers. Then, Dombi $t$-norm and $t$-conorm are defined by:

1. 

$Dom\left(a,b\right)=\frac{1}{1+{\left\{{\left(\frac{1-a}{a}\right)}^{\zeta }+{\left(\frac{1-b}{b}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}$,

2. 

${Dom}^{\prime }\left(a,b\right)=\frac{1}{1+{\left\{{\left(\frac{a}{1-a}\right)}^{\zeta }+{\left(\frac{b}{1-b}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}$.

where  $\zeta \ge 1$ and  $(a,b) \in [0,1]\times [0,1].$ In the above discussion, (1) represents the Dombi product, and (2) represents the Dombi sum.

Example 5. 

 Let $a=0.5$ and $b=0.3$ be two real numbers. Then, for $\zeta =3$, we have $Dom\left(0.5,0.3\right)=0.2947$ and ${Dom}^{\prime }\left(0.5,0.3\right)=0.4937$.

Definition 8 

([27]). Let ${ß}_0=\left(\left({\widehat{ᵲ}}_0,{\widehat{\text{ϣ}}}_0\right),\left({\widehat{ᵵ}}_0,{\widehat{ϥ}}_0\right)\right)$ be a CIFN. The following expressions describe the score function and the accuracy function:

$$Ổ\left({ß}_0\right)={\widehat{ᵲ}}_0+{\widehat{\text{ϣ}}}_0-{\widehat{ᵵ}}_0-{\widehat{ϥ}}_0$$

 and

$$Ẩ\left({ß}_0\right)={\widehat{ᵲ}}_0+{\widehat{\text{ϣ}}}_0+{\widehat{ᵵ}}_0+{\widehat{ϥ}}_0$$

 respectively, where  $Ổ\left({ß}_0\right)\in [-\mathrm{2,2}]$ and  $Ẩ\left({ß}_0\right)\in \left[0,2\right].$ In addition, any two CIFNs  ${ß}_1$ and  ${ß}_2$ satisfy the following conditions:

1. 

If  $Ổ\left({ß}_1\right)>Ổ\left({ß}_2\right)$, then  ${ß}_1\succ {ß}_2$,

2. 

If  $Ổ\left({ß}_1\right)<Ổ\left({ß}_2\right)$  , then  ${ß}_1\prec {ß}_2$  ,

3. 

If  $Ổ\left({ß}_1\right)=Ổ\left({ß}_2\right)$  , then

a 

$Ẩ\left({ß}_1\right)>Ẩ\left({ß}_2\right)\Rightarrow {ß}_1\succ {ß}_2$  ,

b 

$Ẩ\left({ß}_1\right)<Ẩ\left({ß}_2\right)\Rightarrow {ß}_1\prec {ß}_2$  and

c. 

$Ẩ\left({ß}_1\right)=Ẩ\left({ß}_2\right)\Rightarrow {ß}_1{~ß}_2$

3. An Improvement to CIFS’s Existing Score Function

In this segment, we provide an illustration of the limitations of the CIFN score function formulated in [27] and explore how to address these issues.

Example 6. 

Suppose ${ß}_1=\left(\right(0.45,0.3),(0.4,0.5\left)\right)$ and ${ß}_2=\left(\right(0.6,0.15),(0.3,0.6\left)\right)$ be any two CIFNs. Applying Definition 8 on ${ß}_1$ and ${ß}_2$, we have $Ổ\left({ß}_1\right)=Ổ\left({ß}_2\right)=-0.15$ and $Ẩ\left({ß}_1\right)=Ẩ\left({ß}_2\right)=1.65$. In view of Definition 8 (property 3(c)), ${ß}_1~{ß}_2$.

This demonstrates that the score function being evaluated has some sort of shortcoming. In light of this discussion, we propose an improved definition of this score function.

Definition 9. 

Let  ${ß}_0=\left(\left({\widehat{ᵲ}}_0,{\widehat{\text{ϣ}}}_0\right),\left({\widehat{ᵵ}}_0,{\widehat{ϥ}}_0\right)\right)$ be a CIFN. The modified score function $\text{Ҧ}\left({ß}_0\right)$ of CIFN is defined as follows:

$$\text{Ҧ}\left({ß}_0\right)=\frac{{\widehat{ᵵ}}_0+{\widehat{ϥ}}_0-{\widehat{ᵲ}}_0-{\widehat{\text{ϣ}}}_0}{2}+\frac{{\widehat{ᵲ}}_0+{\widehat{\text{ϣ}}}_0+2({\widehat{ᵲ}}_0{\widehat{\text{ϣ}}}_0-{\widehat{ᵵ}}_0{\widehat{ϥ}}_0)}{{\widehat{ᵲ}}_0+{\widehat{\text{ϣ}}}_0+{\widehat{ᵵ}}_0+{\widehat{ϥ}}_0}$$

 where  $g\left({ß}_0\right)\in [0,2].$

In addition, any two CIFNs ${ß}_1$ and ${ß}_2$ admit comparison law by means of the above-proposed function:

1.

$\text{Ҧ}\left({ß}_1\right)>\text{Ҧ}\left({ß}_2\right)\Rightarrow {ß}_1\succ {ß}_2,$

2.

$\text{Ҧ}\left({ß}_1\right)<\text{Ҧ}\left({ß}_2\right)\Rightarrow {ß}_1\prec {ß}_2$ and

3.

$\text{Ҧ}\left({ß}_1\right)=\text{Ҧ}\left({ß}_2\right)\Rightarrow {ß}_1~{ß}_2.$

To demonstrate the accuracy of the suggested score function for CIFN, let us examine the subsequent example.

Example 7. 

Suppose ${ß}_1=\left(\right(0.45,0.3),(0.4,0.5\left)\right)$and ${ß}_2=\left(\right(0.6,0.15),(0.3,0.6\left)\right)$ are two CIFNs. By applying Definition 9 on ${ß}_1$  and ${ß}_2$  gives that $\text{Ҧ}\left({ß}_1\right)=0.4508$  and $\text{Ҧ}\left({ß}_2\right)=0.4205$. Thus, in light of property 2 of Definition 9, we have ${ß}_1\succ {ß}_2$. This demonstrates the superiority of ${ß}_{1 }$over ${ß}_2$.

Based on our discussion thus far, it is easy to see why the proposed score function is superior for decision analysis and delivers more reliable outcomes.

4. Dombi Operations on CIFNs

In this section, we define and establish Dombi operations within a CIF infrastructure and their various key properties. The general purpose Dombi operators’ extraordinary versatility, combined with their aggregation feature, operational characteristics, and DM skills, allows them to handle inaccurate information with ease. These operations combine the data into a single numerical representation by using aggregation processes. They are extremely effective because of their quick capacity to adjust to shifting operational situations and solve challenging DM issues. The CIF Dombi AOs also successfully address the problem of evolving preferences brought on by information loss among current IF operators.

Definition 10. 

The basic operations on any two CIFNs ${ß}_1=\left(\left({\widehat{ᵲ}}_1,{\widehat{\text{ϣ}}}_1\right),\left({\widehat{ᵵ}}_1,{\widehat{ϥ}}_1\right)\right)$ and ${ß}_2=\left(\left({\widehat{ᵲ}}_2,{\widehat{\text{ϣ}}}_2\right),\left({\widehat{ᵵ}}_2,{\widehat{ϥ}}_2\right)\right)$ for $\zeta \ge 1$ and $\psi >0$ are interpreted in the following way:

1. 

${ß}_1⨁{ß}_2=\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{{\left(\frac{{\widehat{ᵲ}}_1}{1-{\widehat{ᵲ}}_1}\right)}^{\zeta }+{\left(\frac{{\widehat{ᵲ}}_2}{1-{\widehat{ᵲ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{{\left(\frac{{\widehat{\text{ϣ}}}_1}{1-{\widehat{\text{ϣ}}}_1}\right)}^{\zeta }+{\left(\frac{{\widehat{\text{ϣ}}}_2}{1-{\widehat{\text{ϣ}}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{{\left(\frac{1-{\widehat{ᵵ}}_1}{{\widehat{ᵵ}}_1}\right)}^{\zeta }+{\left(\frac{1-{\widehat{ᵵ}}_2}{{\widehat{ᵵ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{{\left(\frac{1-{\widehat{ϥ}}_1}{{\widehat{ϥ}}_1}\right)}^{\zeta }+{\left(\frac{1-{\widehat{ϥ}}_2}{{\widehat{ϥ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right),$

2. 

${ß}_1⨂{ß}_2=\left(\begin{array}{c}\left(\frac{1}{1+{\left\{{\left(\frac{1-{\widehat{ᵲ}}_1}{{\widehat{ᵲ}}_1}\right)}^{\zeta }+{\left(\frac{1-{\widehat{ᵲ}}_2}{{\widehat{ᵲ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{{\left(\frac{1-{\widehat{\text{ϣ}}}_1}{{\widehat{\text{ϣ}}}_1}\right)}^{\zeta }+{\left(\frac{1-{\widehat{\text{ϣ}}}_2}{{\widehat{\text{ϣ}}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(1-\frac{1}{1+{\left\{{\left(\frac{{\widehat{ᵵ}}_1}{1-{\widehat{ᵵ}}_1}\right)}^{\zeta }+{\left(\frac{{\widehat{ᵵ}}_2}{1-{\widehat{ᵵ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{{\left(\frac{{\widehat{ϥ}}_1}{1-{\widehat{ϥ}}_1}\right)}^{\zeta }+{\left(\frac{{\widehat{ϥ}}_2}{1-{\widehat{ϥ}}_2}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right),$

3. 

$\psi {ß}_1=\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\psi {\left(\frac{{\widehat{ᵲ}}_1}{1-{\widehat{ᵲ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{{\psi \left(\frac{{\widehat{\text{ϣ}}}_1}{1-{\widehat{\text{ϣ}}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{{\psi \left(\frac{1-{\widehat{ᵵ}}_1}{{\widehat{ᵵ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\psi {\left(\frac{1-{\widehat{ϥ}}_1}{{\widehat{ϥ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right),$

4. 

${ß}_1^{\psi }=\left(\begin{array}{c}\left(\frac{1}{1+{\left\{{\psi \left(\frac{1-{\widehat{ᵲ}}_1}{{\widehat{ᵲ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{{\psi \left(\frac{1-{\widehat{\text{ϣ}}}_1}{{\widehat{\text{ϣ}}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(1-\frac{1}{1+{\left\{\psi {\left(\frac{{\widehat{ᵵ}}_1}{1-{\widehat{ᵵ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\psi {\left(\frac{{\widehat{ϥ}}_1}{1-{\widehat{ϥ}}_1}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right).$

The subsequent definition introduces the notion of the Dombi arithmetic operator in CIF settings, namely, the CIFDHA operator.

Definition 11. 

Consider the CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3\dots \text{ք}\right)$. The complex intuitionistic fuzzy Dombi hybrid averaging (CIFDHA) operator is a mapping $CIFDHA: {ß}^{\text{ք}}\to ß$ having an associated weight vector $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ with $0\le {\xi }_{\tau }\le 1$, $\tau =1,2,3,\dots ,\text{ք}$; ${\sum }_{\tau =1}^{\text{ք}}{\xi }_{\tau }=1$ defined by

$$CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={\oplus }_{\tau =1}^{\text{ք}}\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right),$$

 where  ${\dot{ß}}_{\sigma \left(\tau \right)}$ is the  $\tau th$ largest of the weighted CIFNs  ${\dot{ ß}}_{\tau }$ and  ${\dot{ß}}_{\tau }=\text{ք}{\text{ϣ}}_{\tau }{ß}_{\tau }$,  $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{\text{ք}}\right)}^T$ is the weighted vector of  ${ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ such that  $0\le {\omega }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  ${\sum }_{\tau =1}^{\text{ք}}{\text{ϣ}}_{\tau }=1$.

Theorem 1. 

Let  ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$ be the collection of CIFNs. Then, using the $CIFDHA$ operator, we can deduce that all of these CIFNs aggregate to make up another CIFN and are computed as follows:

$$\begin{gathered} CIFDHA\left({\mathit{ß}}_1,{\mathit{ß}}_2,\dots ,{\mathit{ß}}_{\text{ք}}\right)={\oplus }_{\mathit{\tau }=1}^{\text{ք}}\left({\mathit{\xi }}_{\mathit{\tau }}{\dot{\mathit{ß}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}\right) \\ =\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\mathit{\tau }=1}^{\text{ք}}{\mathit{\xi }}_{\mathit{\tau }}{\left(\frac{{\dot{\widehat{\mathit{ᵲ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}{1-{\dot{\widehat{\mathit{ᵲ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}\right)}^{\mathit{\zeta }}\right\}}^{\frac{1}{\mathit{\zeta }}}},1-\frac{1}{1+{\left\{\sum _{\mathit{\tau }=1}^{\text{ք}}{{\mathit{\xi }}_{\mathit{\tau }}\left(\frac{{\dot{\widehat{\mathit{ϣ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}{1-{\dot{\widehat{\mathit{ϣ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}\right)}^{\mathit{\zeta }}\right\}}^{\frac{1}{\mathit{\zeta }}}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\mathit{\tau }=1}^{\text{ք}}{{\mathit{\xi }}_{\mathit{\tau }}\left(\frac{1-{\dot{\widehat{\mathit{ᵵ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}{{\dot{\widehat{\mathit{ᵵ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}\right)}^{\mathit{\zeta }}\right\}}^{\frac{1}{\mathit{\zeta }}}},\frac{1}{1+{\left\{\sum _{\mathit{\tau }=1}^{\text{ք}}{{\mathit{\xi }}_{\mathit{\tau }}\left(\frac{1-{\dot{\widehat{\mathit{ϥ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}{{\dot{\widehat{\mathit{ϥ}}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}}\right)}^{\mathit{\zeta }}\right\}}^{\frac{1}{\mathit{\zeta }}}}\right)\end{array}\right) \end{gathered}$$

(1)

 where  $\zeta >0$,  ${\dot{ß}}_{\sigma \left(\tau \right)}$ is the  $\tau th$ largest of  ${\dot{ß}}_{\tau }=\text{ք}{\text{ϣ}}_{\tau }{ß}_{\tau }$,  $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{\text{ք}}\right)}^T$ is the weighted vector of  ${ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ with  $0\le {\omega }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}, {\sum }_{\tau =1}^{\text{ք}}{\omega }_{\tau }=1$ and  $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ is the aggregation-associated vector such that  $0\le {\xi }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  ${\sum }_{\tau =1}^{\text{ք}}{\xi }_{\tau }=1$.

Proof. 

We prove this theorem by mathematical induction on $\tau$.

When $\tau =1$, then clearly ${\xi }_1=1,{\text{ϣ}}_1=1$ and ${{\dot{\mathrm{ß}}}_{\sigma \left(1\right)}=\mathrm{ß}}_{\sigma \left(1\right)}={\mathrm{ß}}_1$.

$$\begin{gathered} CIFDHA\left({ß}_1\right)=\left(\left(1-\frac{1}{1+\frac{{\widehat{ᵲ}}_1}{1-{\widehat{ᵲ}}_1}},1-\frac{1}{1+\frac{{\widehat{\text{ϣ}}}_1}{1-{\widehat{\text{ϣ}}}_1}}\right),\left(\frac{1}{1+\frac{1-{\widehat{ᵵ}}_1}{{\widehat{ᵵ}}_1}},\frac{1}{1+\frac{1-{\widehat{ϥ}}_1}{{\widehat{ϥ}}_1}}\right)\right) \\ =\left(\left(1-\frac{1-{\widehat{ᵲ}}_1}{1-{\widehat{ᵲ}}_1+{\widehat{ᵲ}}_1},1-\frac{1-{\widehat{\text{ϣ}}}_1}{1-{\widehat{\text{ϣ}}}_1+{\widehat{\text{ϣ}}}_1}\right),\left(\frac{{\widehat{ᵵ}}_1}{{\widehat{ᵵ}}_1+1-{\widehat{ᵵ}}_1},\frac{{\widehat{ϥ}}_1}{{\widehat{ϥ}}_1+1-{\widehat{ϥ}}_1}\right)\right)=\left(\left({\widehat{ᵲ}}_1,{\widehat{\text{ϣ}}}_1\right),\left({\widehat{ᵵ}}_1,{\widehat{ϥ}}_1\right)\right) \end{gathered}$$

This means that

$$CIFDHA\left({ß}_1\right)={ß}_1$$

Therefore, Equation (1) holds for $\tau =1$.

In addition, the following result is obtained when Definition 11 is applied for $\tau =2$:

$$\begin{gathered} CIFDHA={\xi }_1{\dot{ß}}_{\sigma \left(1\right)}⨁{\xi }_2{\dot{ß}}_{\sigma \left(2\right)} \\ =\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{{\xi }_1{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(1\right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(1\right)}}\right)}^{\zeta }+{\xi }_2{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(2\right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(2\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{{\xi }_1{\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(1\right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(1\right)}}\right)}^{\zeta }+{\xi }_2{\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(2\right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(2\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{{\xi }_1{\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(1\right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(1\right)}}\right)}^{\zeta }+{\xi }_2{\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(2\right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(2\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{{{\xi }_1\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(1\right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(1\right)}}\right)}^{\zeta }+{{\xi }_2\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(2\right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(2\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \end{gathered}$$

This means that

$$CIFDHA\left({ß}_1,{ß}_2\right)=\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^2{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^2{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\tau =1}^2{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^2{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right)$$

Thus, the result is true for $\tau =2$.

Assume that the statement is true for $\tau =s$, we have

$$\begin{gathered} CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_s\right)=\left({\xi }_1{\dot{ß}}_{\sigma \left(1\right)}\right)⨁{(\xi }_2{\dot{ß}}_{\sigma \left(2\right)})⨁\dots ⨁({\xi }_s{\dot{ß}}_{\sigma \left(s\right)})={\oplus }_{\tau =1}^s\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right) \\ =\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^s{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \end{gathered}$$

Moreover, for $\tau =s+1$, we have

$$\begin{gathered} CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_s,{ß}_{s+1}\right)=\left({\xi }_1{\dot{ß}}_{\sigma \left(1\right)}\right)⨁{(\xi }_2{\dot{ß}}_{\sigma \left(2\right)})⨁\dots ⨁({\xi }_s{\dot{ß}}_{\sigma \left(s\right)})⨁{(\xi }_{s+1}{\dot{ß}}_{\sigma \left(s+1\right)}) \\ ={\oplus }_{\tau =1}^s\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right)\oplus \left({(\xi }_{s+1}{\dot{ß}}_{\sigma \left(s+1\right)}\right)) \\ =\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^s{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^s{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \\ ⨁\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{{\xi }_{s+1}{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(s+1\right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(s+1\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{{{\xi }_{s+1}\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(s+1\right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(s+1\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{{{\xi }_{s+1}\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(s+1\right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(s+1\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{{{\xi }_{s+1}\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(s+1\right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(s+1\right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \end{gathered}$$

This means that

$$CIFDHA{(ß}_1,{ß}_2,\dots ,{ß}_{s+1})=\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^{s+1}{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^{s+1}{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\tau =1}^{s+1}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^{s+1}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right)$$

For any positive integral values, we deduce that the statement is correct. □

Here is an illustration of the aforementioned fact:

Example 8. 

Consider the CIFNs ${ß}_1=\left(\left(0.55,0.4\right),\left(0.23,0.4\right)\right),{ ß}_2=\left(\left(0.45,0.6\right),\left(0.5,0.3\right)\right), {ß}_3=\left(\left(0.44,0.5\right),\left(0.3,0.4\right)\right), {ß}_4=\left(\left(0.4,0.5\right),\left(0.4,0.3\right)\right)$ and the importance factor of CIFNs $\omega ={\left(0.15,0.25,0.35,0.25\right)}^T$. We first compute the weighted CIFNs by using ${\dot{ß}}_{\tau }=\text{ք}{\omega }_{\tau }{ß}_{\tau }$ for $\zeta =5$ as

$$\begin{gathered} {\dot{ß}}_1=4(0.15)\left(\left(0.55,0.4\right),\left(0.23,0.4\right)\right)=0.6\left(\left(0.55,0.4\right),\left(0.23,0.4\right)\right) \\ =\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{0.6{\left(\frac{0.55}{1-0.55}\right)}^5\right\}}^{\frac{1}{5}}},1-\frac{1}{1+{\left\{{0.6\left(\frac{0.4}{1-0.4}\right)}^5\right\}}^{\frac{1}{5}}}\right),\\ \left(\frac{1}{1+{\left\{{0.6\left(\frac{1-0.23}{0.23}\right)}^5\right\}}^{\frac{1}{5}}},\frac{1}{1+{\left\{0.6{\left(\frac{1-0.4}{0.4}\right)}^5\right\}}^{\frac{1}{5}}}\right)\end{array}\right) \\ =\left(\left(0.525,0.376\right),\left(0.249,0.425\right)\right) \end{gathered}$$

Similarly, we can obtain ${\dot{\mathrm{ß}}}_2=\left(\left(0.450,0.600\right),\left(0.500,0.300\right)\right)$, ${\dot{\mathrm{ß}}}_3=\left(\left(0.457,0.517\right),\left(0.286,0.384\right)\right)$, and ${\dot{\mathrm{ß}}}_4=\left(\left(0.400,0.500\right),\left(0.400,0.300\right)\right)$.

Now, we permute these numbers by using Definition 9 and obtain the following information.

$$\text{Ҧ}\left({\dot{ß}}_1\right)=0.575 \text{Ҧ}\left({\dot{ß}}_2\right)=0.572, \text{Ҧ}\left({\dot{ß}}_3\right)=0.594, \text{Ҧ}\left({\dot{ß}}_4\right)=0.563$$

The permuted values of CIFNs are derived as follows:

${\dot{ß}}_{\sigma \left(1\right)}=\left(\left(0.457,0.517\right),\left(0.286,0.384\right)\right),{\dot{ß}}_{\sigma \left(2\right)}=\left(\left(0.525,0.376\right),\left(0.249,0.425\right)\right),$${\dot{ß}}_{\sigma \left(3\right)=}\left(\left(0.450,0.600\right),\left(0.500,0.300\right)\right)$, and ${\dot{ß}}_{\sigma \left(4\right)}=\left(\left(0.400,0.50\right),\left(0.400,0.30\right)\right)$.

Then, for the weight vector $\xi ={(0.35,0.2,0.3,0.15)}^T$ of $CIFDHA$ operator, we have

$${\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{{1-\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^5\right\}}^{\frac{1}{5}}=0.905,{\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{{1-\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^5\right\}}^{\frac{1}{5}}=1.240,$$

$${\left\{\sum _{\tau =1}^4{\xi }_{\tau }{\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^5\right\}}^{\frac{1}{5}}=2.433,{\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^5\right\}}^{\frac{1}{5}}=2.044.$$

This implies that

$$CIFDHA\left({ß}_1,{ß}_2,{ß}_3,{ß}_4\right)={⨁}_{\tau =1}^4\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right)=\left(\right(0.475,0.554),(0.291,0.329\left)\right)$$

It follows that the result of the preceding discussion is also CIFN.

Theorem 2. 

(Idempotency property) Consider the CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$such that ${ß}_{\tau }={ß}_0\forall \tau$, where ${ß}_0=\left(\left({\widehat{ᵲ}}_0,{\widehat{\text{ϣ}}}_0\right),\left({\widehat{ᵵ}}_0,{\widehat{ϥ}}_0\right)\right)$ is a CIFN. Then $CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={ß}_0$.

Proof. 

Since ${\mathrm{ß}}_{\tau }={\mathrm{ß}}_0\forall \tau$. Then, by Definition 4, ${\dot{\widehat{\mathrm{ᵲ}}}}_{\sigma \left(\tau \right)}={\widehat{\mathrm{ᵲ}}}_{\tau }={\widehat{\mathrm{ᵲ}}}_0,{\dot{\widehat{\mathrm{ϣ}}}}_{\sigma \left(\tau \right)}={\widehat{\mathrm{ϣ}}}_{\tau }={\widehat{\mathrm{ϣ}}}_0,{{\dot{\widehat{\mathrm{ᵵ}}}}_{\sigma \left(\tau \right)}=\widehat{\mathrm{ᵵ}}}_{\tau }={\widehat{\mathrm{ᵵ}}}_0$ and ${{\dot{\widehat{\mathrm{ϥ}}}}_{\sigma \left(\tau \right)}=\widehat{\mathrm{ϥ}}}_{\tau }={\widehat{\mathrm{ϥ}}}_0\forall \tau$. Substituting the aforementioned relations in Theorem 1, we obtain

$$\begin{gathered} CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)=\left(\begin{array}{c}\left(1-\frac{1}{1+\left(\frac{{\widehat{ᵲ}}_0}{1-{\widehat{ᵲ}}_0}\right){\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+\left(\frac{{\widehat{\text{ϣ}}}_0}{1-{\widehat{\text{ϣ}}}_0}\right){\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left(\frac{1-{\widehat{ᵵ}}_0}{{\widehat{ᵵ}}_0}\right)\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+\left(\frac{1-{\widehat{ϥ}}_0}{{\widehat{ϥ}}_0}\right){\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \\ =\left(\left(1-\frac{1}{1+\left(\frac{{\widehat{ᵲ}}_0}{1-{\widehat{ᵲ}}_0}\right)},1-\frac{1}{1+\left(\frac{{\widehat{\text{ϣ}}}_0}{1-{\widehat{\text{ϣ}}}_0}\right)}\right),\left(\frac{1}{1+\left(\frac{1-{\widehat{ᵵ}}_0}{{\widehat{ᵵ}}_0}\right)},\frac{1}{1+\left(\frac{1-{\widehat{ϥ}}_0}{{\widehat{ϥ}}_0}\right)}\right)\right) \\ =\left(\left({\widehat{ᵲ}}_0,{\widehat{\text{ϣ}}}_0\right),\left({\widehat{ᵵ}}_0,{\widehat{ϥ}}_0\right)\right) \end{gathered}$$

This shows that

$$CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={ß}_0$$

□

Theorem 3. 

(Boundedness property) Consider CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$. Let ${ ß}^{-}=\underset{\tau }{\mathit{min}}\left\{{ß}_{\tau }\right\}=\left(\left({\widehat{ᵲ}}^{-},{\widehat{\text{ϣ}}}^{-}\right),\left({\widehat{ᵵ}}^{-},{\widehat{ϥ}}^{-}\right)\right)$and ${ß}^{+}=\underset{\tau }{\mathit{max}}{\{ß}_{\tau }\}=\left(\left({\widehat{ᵲ}}^{+},{\widehat{\text{ϣ}}}^{+}\right), \left({\widehat{ᵵ}}^{+},{\widehat{ϥ}}^{+}\right)\right)$ where ${\widehat{ᵲ}}^{-}=\underset{\tau }{\mathit{min}}\left\{{\widehat{ᵲ}}_{\tau }\right\},{\widehat{\text{ϣ}}}^{-}=\underset{\tau }{\mathit{min}}\{{\widehat{\text{ϣ}}}_{\tau }\}, {\widehat{ᵵ}}^{-}=\underset{\tau }{\mathit{max}}\left\{{\widehat{ᵵ}}_{\tau }\right\}, {\widehat{ϥ}}^{-}=\underset{\tau }{\mathit{max}}{\{\widehat{ϥ}}_{\tau }\}, {\widehat{ᵲ}}^{+}=\underset{\tau }{\mathit{max}}\left\{{\widehat{ᵲ}}_{\tau }\right\},{\widehat{\text{ϣ}}}^{+}=\underset{\tau }{\mathit{max}}\{{\widehat{\text{ϣ}}}_{\tau }\},{\widehat{ᵵ}}^{+}=\underset{\tau }{\mathit{min}}\left\{{\widehat{ᵵ}}_{\tau }\right\},{\widehat{ϥ}}^{+}=\underset{\tau }{\mathit{min}}{\{\widehat{ϥ}}_{\tau }\}.$Then, ${ß}^{-} CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)\le {ß}^{+}$.

Proof. 

In the context of the conditions given, we have

$$1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\widehat{ᵲ}}^{-}}{1-{\widehat{ᵲ}}^{-}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\le 1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\le 1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\widehat{ᵲ}}^{+}}{1-{\widehat{ᵲ}}^{+}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},$$

$$1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\widehat{\text{ϣ}}}^{-}}{1-{\widehat{\text{ϣ}}}^{-}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\le 1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\le 1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\widehat{\text{ϣ}}}^{+}}{1-{\widehat{\text{ϣ}}}^{+}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},$$

$$\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\widehat{ᵵ}}^{-}}{{\widehat{ᵵ}}^{-}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\ge \frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\ge \frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\widehat{ᵵ}}^{+}}{{\widehat{ᵵ}}^{+}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},$$

$$\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\widehat{ϥ}}^{-}}{{\widehat{ϥ}}^{-}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\ge \frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\ge \frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\widehat{ϥ}}^{+}}{{\widehat{ϥ}}^{+}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}.$$

Based on Definition 4 and the above conditions, we obtain  ${ß}^{-}\le CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)\le {ß}^{+}.$□

Theorem 4. 

(Monotonicity property) Suppose ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)$ and ${ß}_{\tau }^{\prime }=\left(\left({\widehat{ᵲ}}_{\tau }^{\prime },{\widehat{\text{ϣ}}}_{\tau }^{\prime }\right),\left({\widehat{ᵵ}}_{\tau }^{\prime },{\widehat{ϥ}}_{\tau }^{\prime }\right)\right)$ for $\tau =1,2,3,\dots ,\text{ք}$ be two collections of CIFNs. If ${\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}\le {\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}^{\prime },{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}\ge {\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}^{\prime }, {\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}\le {\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}^{\prime }$ and ${\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}\ge {\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}^{\prime } \forall \tau .$ Then, $CIFDHA\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)\le CIFDHA\left({ß}_1^{\prime },{ß}_2^{\prime },\dots ,{ß}_{\text{ք}}^{\prime }\right).$

Proof. 

A straightforward application of Definition 4 provides the proof. □

The following definition presents the concept of the Dombi geometric operator in CIF settings, namely, the CIF Dombi hybrid geometric (CIFDHG) operator.

Definition 12. 

Consider the CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3\dots \text{ք}\right)$. The complex intuitionistic fuzzy Dombi hybrid geometric (CIFDHG) operator is a mapping $CIFDHG: {ß}^{\text{ք}}\to ß$ having an associated weight vector $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ with $0\le {\xi }_{\tau }\le 1$, $\tau =1,2,3,\dots ,\text{ք}$; $\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }=1$ defined by

$$CIFDHG\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={⨂}_{\tau =1}^{\text{ք}}{\left({\dot{ß}}_{\sigma \left(\tau \right)}\right)}^{{\xi }_{\tau }},$$

 where  ${\dot{ß}}_{\sigma \left(\tau \right)}$  is the  $\tau th$ largest of the weighted CIFNs  ${\dot{ß}}_{\tau }$ and  ${\dot{ ß}}_{\tau }={\left({ß}_{\tau }\right)}^{\text{ք}{\text{ϣ}}_{\tau }}$,  $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{\text{ք}}\right)}^T$ is the weighted vector of  ${ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ such that  $0\le {\omega }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  $\sum _{\tau =1}^{\text{ք}}{\omega }_{\tau }=1.$

Theorem 5. 

Let ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$ be the collection of CIFNs. Then, using the $CIFDHG$ operator, we can deduce that all of these CIFNs aggregate to make up another CIFN and are computed as follows:

$$\begin{gathered} \mathit{C}\mathit{I}\mathit{F}\mathit{D}\mathit{H}\mathit{G}\left({\mathit{ß}}_1,{\mathit{ß}}_2,\dots ,{\mathit{ß}}_{\text{ք}}\right)={⨂}_{\mathit{\tau }=1}^{\text{ք}}{\left({\dot{\mathit{ß}}}_{\mathit{\sigma }\left(\mathit{\tau }\right)}\right)}^{{\mathit{\xi }}_{\mathit{\tau }}} \\ =\left(\begin{array}{c}\left(\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right) \end{gathered}$$

(2)

 where  $\zeta >0$,  ${\dot{ß}}_{\sigma \left(\tau \right)}$ is the  $\tau th$ largest of  ${\dot{ß}}_{\tau }={\left({ß}_{\tau }\right)}^{\text{ք}{ϣ}_{\tau }}$,  $ϣ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_{\text{ք}}\right)}^T$ is the weighted vector of  ${ß}_{\tau }\left(\tau =1,2,3,\dots ,\text{ք}\right)$ with  $0\le {\omega }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}, \sum _{\tau =1}^{\text{ք}}{ϣ}_{\tau }=1$ and  $\xi ={({\xi }_1,{\xi }_2,\dots ,{\xi }_{\text{ք}})}^T$ is the aggregation-associated vector such that  $0\le {\xi }_{\tau }\le 1$,  $\tau =1,2,3,\dots ,\text{ք}$ and  $\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }=1.$

Proof. 

The theorem’s proof bears an analogy to Theorem 1. □

Here is an illustration of the aforementioned fact:

Example 9. 

Consider the CIFNs ${ß}_1=\left(\right(0.35,0.7),(0.3,0.2\left)\right),{ß}_2=\left(\right(0.2,0.5),(0.6,0.3\left)\right), {ß}_3=\left(\right(0.6,0.5),(0.3,0.2\left)\right), {ß}_4=\left(\right(0.55,0.7),(0.3,0.1\left)\right)$ and the importance factor of CIFNs $\omega ={\left(0.3,0.3,0.2,0.2\right)}^T$. We first compute the weighted CIFNs by using ${\dot{ß}}_{\tau }={\left({ß}_{\tau }\right)}^{\text{ք}{\text{ϣ}}_{\tau }}$ for $\zeta =4$ as

$$\begin{gathered} {\dot{ß}}_1=4(0.3)((0.35,0.7),(0.3,0.2))=1.2((0.35,0.7),(0.3,0.2)) \\ =\left(\begin{array}{c}\left(\frac{1}{1+{\left\{{1.2\left(\frac{1-0.35}{0.35}\right)}^4\right\}}^{\frac{1}{4}}},\frac{1}{1+{\left\{{1.2\left(\frac{1-0.7}{0.7}\right)}^4\right\}}^{\frac{1}{4}}}\right),\\ \left(1-\frac{1}{1+{\left\{1.2{\left(\frac{0.3}{1-0.3}\right)}^4\right\}}^{\frac{1}{4}}},1-\frac{1}{1+{\left\{1.2{\left(\frac{0.2}{1-0.2}\right)}^4\right\}}^{\frac{1}{4}}}\right)\end{array}\right) \\ =((0.340,0.690),(0.310,0.207) \end{gathered}$$

Similarly, we can obtain  ${\dot{ß}}_2=\left(\right(0.193,0.489),(0.611,0.310\left)\right)$,  ${\dot{ß}}_3=\left(\right(0.613,0.514),(0.288,0.191\left)\right)$ and  ${\dot{ß}}_4=\left(\right(0.594,0.712),(0.288,0.095\left)\right)$.

Now, we permute these numbers by using Definition 9  and obtain the following information.

$$\text{Ҧ}\left({\dot{ß}}_1\right)=0.630 \text{Ҧ}\left({\dot{ß}}_2\right)=0.426, \text{Ҧ}\left({\dot{ß}}_3\right)=0.702, \text{Ҧ}\left({\dot{ß}}_4\right)=0.774$$

The permuted values of CIFNs are derived as follow:

${\dot{ß}}_{\sigma \left(1\right)}=\left(\left(0.564,0.712\right),\left(0.288,0.095\right)\right),{\dot{ß}}_{\sigma \left(2\right)}=\left(\left(0.613,0.514\right),\left(0.288,0.191\right)\right),$  ${\dot{ß}}_{\sigma \left(3\right)=}\left(\left(0.340,0.690\right),\left(0.310,0.207\right)\right)$, and  ${\dot{ß}}_{\sigma \left(4\right)}=\left(\right(0.193,0.489),(0.611,0.31\left)\right)$.

Then, for the weight vector  $\xi ={(0.4,0.25,0.25,0.1)}^T$ of  $CIFDHG$ operator, we have

$${\left\{\sum _{\tau =1}^4{\xi }_{\tau }{\left(\frac{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}}\right)}^4\right\}}^{\frac{1}{4}}=2.420,{\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}}\right)}^4\right\}}^{\frac{1}{4}}=0.764.$$

$${\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}{{1-\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}}\right)}^4\right\}}^{\frac{1}{4}}=0.893,{\left\{\sum _{\tau =1}^4{{\xi }_{\tau }\left(\frac{{\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}{{1-\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}}\right)}^4\right\}}^{\frac{1}{4}}=0.279,$$

This implies that

$$\begin{gathered} CIFDHG\left({ß}_1,{ß}_2,{ß}_3,{ß}_4\right)={⨂}_{\tau =1}^{40}\left({\xi }_{\tau }{\dot{ß}}_{\sigma \left(\tau \right)}\right) \\ =\left(\left(0.2924,0.567\right),\left(0.4718,0.2181\right)\right) \end{gathered}$$

It follows that the result of the preceding discussion is also CIFN.

Theorem 6. 

(Idempotency property) Consider the CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$ such that ${ß}_{\tau }={ß}_0\forall \tau$,  where ${ß}_0=\left(\left({\widehat{ᵲ}}_0,{\widehat{\text{ϣ}}}_0\right),\left({\widehat{ᵵ}}_0,{\widehat{ϥ}}_0\right)\right)$ is a CIFN. Then, $CIFDHG\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)={ß}_0$.

Proof. 

The theorem’s proof bears an analogy to Theorem 2. □

Theorem 7. 

(Boundedness property) Consider CIFNs ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)\left(\tau =1,2,3,\dots ,\text{ք}\right)$. Let${ ß}^{-}=\underset{\tau }{\mathit{min}}\left\{{ß}_{\tau }\right\}=\left(\left({\widehat{ᵲ}}^{-},{\widehat{\text{ϣ}}}^{-}\right),\left({\widehat{ᵵ}}^{-},{\widehat{ϥ}}^{-}\right)\right)$ and ${ß}^{+}=\underset{\tau }{\mathit{max}}{\{ß}_{\tau }\}=\left(\left({\widehat{ᵲ}}^{+},{\widehat{\text{ϣ}}}^{+}\right), \left({\widehat{ᵵ}}^{+},{\widehat{ϥ}}^{+}\right)\right)$where ${\widehat{ᵲ}}^{-}=\underset{\tau }{\mathit{min}}\left\{{\widehat{ᵲ}}_{\tau }\right\},{\widehat{\text{ϣ}}}^{-}=\underset{\tau }{\mathit{min}}\{{\widehat{\text{ϣ}}}_{\tau }\}, {\widehat{ᵵ}}^{-}=\underset{\tau }{\mathit{max}}\left\{{\widehat{ᵵ}}_{\tau }\right\}, {\widehat{ϥ}}^{-}=\underset{\tau }{\mathit{max}}{\{\widehat{ϥ}}_{\tau }\}, {\widehat{ᵲ}}^{+}=\underset{\tau }{\mathit{max}}\left\{{\widehat{ᵲ}}_{\tau }\right\},{\widehat{\text{ϣ}}}^{+}=\underset{\tau }{\mathit{max}}\{{\widehat{\text{ϣ}}}_{\tau }\},{\widehat{ᵵ}}^{+}=\underset{\tau }{\mathit{min}}\left\{{\widehat{ᵵ}}_{\tau }\right\},{\widehat{ϥ}}^{+}=\underset{\tau }{\mathit{min}}{\{\widehat{ϥ}}_{\tau }\}.$Then, ${ß}^{-} CIFDHG\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)\le {ß}^{+}.$

Proof. 

The theorem’s proof bears an analogy to Theorem 3. □

Theorem 8. 

(Monotonicity property) Suppose  ${ß}_{\tau }=\left(\left({\widehat{ᵲ}}_{\tau },{\widehat{\text{ϣ}}}_{\tau }\right),\left({\widehat{ᵵ}}_{\tau },{\widehat{ϥ}}_{\tau }\right)\right)$ and ${ß}_{\tau }^{\prime }=\left(\left({\widehat{ᵲ}}_{\tau }^{\prime },{\widehat{\text{ϣ}}}_{\tau }^{\prime }\right),\left({\widehat{ᵵ}}_{\tau }^{\prime },{\widehat{ϥ}}_{\tau }^{\prime }\right)\right)$ for $\tau =1,2,3,\dots ,\text{ք}$  be two collections of CIFNs. If ${\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}\le {\dot{\widehat{ᵲ}}}_{\sigma \left(\tau \right)}^{\prime },{\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}\ge {\dot{\widehat{ᵵ}}}_{\sigma \left(\tau \right)}^{\prime }, {\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}\le {\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\tau \right)}^{\prime }$ and  ${\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}\ge {\dot{\widehat{ϥ}}}_{\sigma \left(\tau \right)}^{\prime } \forall \tau .$ Then, $CIFDHG\left({ß}_1,{ß}_2,\dots ,{ß}_{\text{ք}}\right)\le CIFDHG\left({ß}_1^{\prime },{ß}_2^{\prime },\dots ,{ß}_{\text{ք}}^{\prime }\right)$.

Proof. 

A straightforward application of Definition 4 provides the proof. □

5. Results and Discussions

5.1. Application of Proposed CIF Dombi AOs in MCDM Problem

Here, we introduce a method for dealing with MCDM issues using CIF data by making use of CIF Dombi AOs. Let $\left\{{Ả}_1,{Ả}_2,{Ả}_3,\dots ,{Ả}_{\mathrm{Ֆ}}\right\}$ represent the set of various alternatives, $\{{Ẻ}_1,{Ẻ}_2,{Ẻ}_3,\dots ,{Ẻ}_{\text{ք}}\}$ present the set of criteria, $\xi =\left({\xi }_1,{\xi }_2,{\xi }_3,\dots ,{\xi }_{\text{ք}}\right)$ represent the weight vector associated with those attributes, and $\mathsf{\omega }=({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,{\mathsf{\omega }}_3,\dots ,{\mathsf{\omega }}_{\text{ք}})$ symbolize the importance factor of the attributes with ${0<{\xi }_{\tau },\mathsf{\omega }}_{\tau }<1$ for all $\tau =1,2,3,\dots ,\text{ք}$ and ${\sum }_{\tau =1}^{\text{ք}}{\xi }_{\tau }=\sum _{\tau =1}^{\text{ք}}{\omega }_{\tau }=1$. Consider the CIF decision matrix $\mathcal{D}={\left({\phi }_{\delta \tau }\right)}_{\mathsf{Ֆ}\times \text{ք}}={\left(\left({\widehat{ᵲ}}_{\delta \tau },{\widehat{\text{ϣ}}}_{\delta \tau }\right),\left({\widehat{ᵵ}}_{\delta \tau },{\widehat{ϥ}}_{\delta \tau }\right)\right)}_{\mathsf{Ֆ}\times \text{ք}}$, where $\left({\widehat{ᵲ}}_{\delta \tau },{\widehat{\text{ϣ}}}_{\delta \tau }\right)$ and $\left({\widehat{ᵵ}}_{\delta \tau },{\widehat{ϥ}}_{\delta \tau }\right)$ represent the expert-assigned MD and NMD, respectively, for which each alternative ${Ả}_{\delta }$ meets the criteria ${Ẻ}_{\tau }$. Moreover, ${\widehat{ᵲ}}_{\delta \tau },{\widehat{ᵵ}}_{\delta \tau },{\widehat{\text{ϣ}}}_{\delta \tau },{\widehat{ϥ}}_{\delta \tau }\in \left[0,1\right]$ such that $0\le {\widehat{ᵲ}}_{\delta \tau }+{\widehat{ᵵ}}_{\delta \tau },{\widehat{\text{ϣ}}}_{\delta \tau }+{\widehat{ϥ}}_{\delta \tau }\le 1$. The algorithm intended for the solution of the MCDM problem within the CIF Dombi environment is as follows:

Step 1.

Summarize the preference values of decision maker in the CIF decision matrix as

$$\mathcal{D}=\left(\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}\left(\left({\widehat{ᵲ}}_{11},{\widehat{\text{ϣ}}}_{11}\right),\left({\widehat{ᵵ}}_{11},{\widehat{ϥ}}_{11}\right)\right)& \left(\left({\widehat{ᵲ}}_{12},{\widehat{\text{ϣ}}}_{12}\right),\left({\widehat{ᵵ}}_{12},{\widehat{ϥ}}_{12}\right)\right)\end{array}\\ \begin{array}{cc}\left(\left({\widehat{ᵲ}}_{21},{\widehat{\text{ϣ}}}_{21}\right),\left({\widehat{ᵵ}}_{21},{\widehat{ϥ}}_{21}\right)\right)& \left(\left({\widehat{ᵲ}}_{22},{\widehat{\text{ϣ}}}_{22}\right),\left({\widehat{ᵵ}}_{22},{\widehat{ϥ}}_{22}\right)\right)\end{array}\end{array}& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}{c}\left(\left({\widehat{ᵲ}}_{1\text{ք}},{\widehat{\text{ϣ}}}_{1\text{ք}}\right),\left({\widehat{ᵵ}}_{1\text{ք}},{\widehat{ϥ}}_{1\text{ք}}\right)\right)\\ \left(\left({\widehat{ᵲ}}_{2\text{ք}},{\widehat{\text{ϣ}}}_{2\text{ք}}\right),\left({\widehat{ᵵ}}_{2\text{ք}},{\widehat{ϥ}}_{2\text{ք}}\right)\right)\end{array}\\ \begin{array}{cc}⋮& ⋮\end{array}& \ddots & ⋮\\ \begin{array}{cc}\left(\left({\widehat{ᵲ}}_{\mathsf{Ֆ}1},{\widehat{\text{ϣ}}}_{\mathsf{Ֆ}1}\right),\left({\widehat{ᵵ}}_{\mathsf{Ֆ}1},{\widehat{ϥ}}_{\mathsf{Ֆ}1}\right)\right)& \left(\left({\widehat{ᵲ}}_{\mathsf{Ֆ}2},{\widehat{\text{ϣ}}}_{\mathsf{Ֆ}2}\right),\left({\widehat{ᵵ}}_{\mathsf{Ֆ}2},{\widehat{ϥ}}_{\mathsf{Ֆ}2}\right)\right)\end{array}& \cdots & \left(\left({\widehat{ᵲ}}_{\mathsf{Ֆ}\text{ք}},{\widehat{\text{ϣ}}}_{\mathsf{Ֆ}\text{ք}}\right),\left({\widehat{ᵵ}}_{\mathsf{Ֆ}\text{ք}},{\widehat{ϥ}}_{\mathsf{Ֆ}\text{ք}}\right)\right)\end{array}\right)$$

Step 2.

Compute the aggregated value ${\phi }_{\delta }=\left(\left({\widehat{ᵲ}}_{\delta },{\widehat{\text{ϣ}}}_{\delta }\right),\left({\widehat{ᵵ}}_{\delta },{\widehat{ϥ}}_{\delta }\right)\right)$ corresponding to each alternative ${Ả}_{\delta }$ by using $CIFDHA$ operator in the following way:

$$CIFDHA\left({\phi }_{\delta 1},{\phi }_{\delta 2},\dots ,{\phi }_{\delta \text{ք}}\right)={⨁}_{\tau =1}^{\text{ք}}\left({\xi }_{\tau }{\dot{\phi }}_{\delta \tau }\right)$$

The application of Theorem 1 in the above relation yields that

$$CIFDHA\left({\phi }_{\delta 1},{\phi }_{\delta 2},\dots ,{\phi }_{\delta \text{ք}}\right)=\left(\begin{array}{c}\left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵲ}}}_{\sigma \left(\delta \tau \right)}}{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\delta \tau \right)}}{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\delta \tau \right)}}{{\dot{\widehat{ᵵ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\delta \tau \right)}}{{\dot{\widehat{ϥ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right)$$

Similarly, compute the aggregated value ${\phi }_{\delta }=\left(\left({\widehat{ᵲ}}_{\delta },{\widehat{\text{ϣ}}}_{\delta }\right),\left({\widehat{ᵵ}}_{\delta },{\widehat{ϥ}}_{\delta }\right)\right)$ corresponding to each alternative ${Ả}_{\delta }$ by using $CIFDHG$ operator in the following way:

$$CIFDHG\left({\phi }_{\delta 1},{\phi }_{\delta 2},\dots ,{\phi }_{\delta \text{ք}}\right)={⨂}_{\tau =1}^{\text{ք}}{\left({\dot{\phi }}_{\delta \tau }\right)}^{{\xi }_{\tau }}$$

The application of Theorem 5 in the above relation yields that

$$CIFDHG\left({\phi }_{\delta 1},{\phi }_{\delta 2},\dots ,{\phi }_{\delta \text{ք}}\right)=\left(\begin{array}{c}\left(\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{ᵲ}}}_{\sigma \left(\delta \tau \right)}}{{\dot{\widehat{ᵲ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{1-{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\delta \tau \right)}}{{\dot{\widehat{\text{ϣ}}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right),\\ \left(1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{\xi }_{\tau }{\left(\frac{{\dot{\widehat{ᵵ}}}_{\sigma \left(\delta \tau \right)}}{1-{\dot{\widehat{ᵵ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}},1-\frac{1}{1+{\left\{\sum _{\tau =1}^{\text{ք}}{{\xi }_{\tau }\left(\frac{{\dot{\widehat{ϥ}}}_{\sigma \left(\delta \tau \right)}}{1-{\dot{\widehat{ϥ}}}_{\sigma \left(\delta \tau \right)}}\right)}^{\zeta }\right\}}^{\frac{1}{\zeta }}}\right)\end{array}\right)$$

Step 3.

Calculate the score value for each computed aggregated value ${\phi }_{\delta }$ by using Definition 9.

Step 4.

Choose the most suitable value on the basis of the information obtained from the previous step.

Now, we will illustrate a case study to illustrate how the suggested score function and aggregation operators might be useful.

Example 10 

([30]). An expert wants to select a machine on the basis of four criteria ${Ẻ}_1$: Reliability, ${Ẻ}_2$: Safety, ${Ẻ}_3$: Flexibility, and ${Ẻ}_4$ : Productivity of selecting machine. The importance of these factors is $\omega {=(0.35,0.3,0.1,0.25)}^T$ , and the weight vector assigned by the expert is $\xi ={\left(0.4,0.25,0.15,0.2\right)}^T.$He considered two machines, ${Ả}_1$ and ${Ả}_2$ and assigned values in the context of CIFNs as given in

Table 2. Decision matrix.

	${\mathbf{Ẻ}}_1$	${\mathbf{Ẻ}}_2$	${\mathbf{Ẻ}}_3$	${\mathbf{Ẻ}}_4$
${\mathrm{Ả}}_1$	$\left(\right(0.7,0.9),(0.1,0.1\left)\right)$	$\left(\right(0.8,0.5),(0.1,0.4\left)\right)$	$\left(\right(0.6,0.6),(0.3,0.2\left)\right)$	$\left(\right(0.7,0.7),(0.3,0.2\left)\right)$
${\mathrm{Ả}}_2$	$\left(\right(0.4,0.8),(0.5,0.1\left)\right)$	$\left(\right(0.7,0.3),(0.3,0.3\left)\right)$	$\left(\right(0.6,0.5),(0.1,0.3\left)\right)$	$\left(\right(0.5,0.5),(0.3,0.4\left)\right)$

.

Within the context of Definition 9, the permuted weighted CIF decision matrices by using ${\dot{ß}}_{\tau }=4{\mathsf{\omega }}_{\tau }{ß}_{\tau }$ and ${\left({ß}_{\tau }\right)}^{\text{ք}{\text{ϣ}}_{\tau }}$ with $\zeta =3$ and ${\dot{ß}}_{\sigma \left(\tau -1\right)}\ge {\dot{ß}}_{\sigma \left(\tau \right)}\forall \tau$ are given in

Table 3. Permuted weighted CIF decision matrix for CIFDHA.

	${\mathbf{Ẻ}}_1$	${\mathbf{Ẻ}}_2$	${\mathbf{Ẻ}}_3$	${\mathbf{Ẻ}}_4$
${\mathbf{Ả}}_1$	$\left(\right(0.73,0.91),(0.09,0.09\left)\right)$	$\left(\right(0.68,0.68),(0.32,0.21\left)\right)$	$\left(\right(0.80,0.50),(0.10,0.40\left)\right)$	$\left(\right(0.56,0.56),(0.34,0.23\left)\right)$
${\mathbf{Ả}}_2$	$\left(\right(0.56,0.46),(0.12,0.34\left)\right)$	$\left(\right(0.44,0.82),(0.46,0.09\left)\right)$	$\left(\right(0.70,0.30),(0.30,0.30\left)\right)$	$\left(\right(0.48,0.48),(0.32,0.42\left)\right)$

and

Table 4. Permuted weighted CIF decision matrix for CIFDHA.

	${\mathbf{Ẻ}}_1$	${\mathbf{Ẻ}}_2$	${\mathbf{Ẻ}}_3$	${\mathbf{Ẻ}}_4$
${\mathbf{Ả}}_1$	$\left(\right(0.67,0.88),(0.12,0.12\left)\right)$	$\left(\right(0.72,0.72),(0.28,0.19\left)\right)$	$\left(\right(0.64,0.64),(0.27,0.17\left)\right)$	$\left(\right(0.80,0.50),(0.10,0.40\left)\right)$
${\mathbf{Ả}}_2$	$\left(\right(0.64,0.54),(0.09,0.27\left)\right)$	$\left(\right(0.368,0.77),(0.54,0.12\left)\right)$	$\left(\right(0.52,0.52),(0.28,0.38\left)\right)$	$\left(\right(0.70,0.30),(0.30,0.30\left)\right)$

, respectively.

By applying the CIFDHA operator for $\zeta =3$ on the values of ${Ả}_1$ and ${Ả}_2$, we have

$${Ả}_1=\left(\right(0.7318,0.8819),(0.1102,0.1166\left)\right)$$

and

$${Ả}_2=\left(\right(0.5908,0.7427),(0.1550,0.1349\left)\right)$$

Calculating the score values by using Definition 9, we have $\text{Ҧ}\left({Ả}_1\right)=0.8707$ and $\text{Ҧ}\left({Ả}_2\right)=0.8144$. Clearly, we have $\text{Ҧ}\left({Ả}_1\right)>\text{Ҧ}\left({Ả}_2\right),$ which means that ${Ả}_1$ is the better alternative than ${Ả}_2.$

Similarly, within the context of the CIFDHG operator, we have

$${Ả}_1=\left(\right(0.6887,0.6159),(0.2229,0.2866\left)\right)$$

and

$${Ả}_2=\left(\right(0.4604,0.4188),(0.4300,0.2930\left)\right)$$

Calculating the score values by using Definition 9, we have $\text{Ҧ}\left({Ả}_1\right)=0.7188$ and $\text{Ҧ}\left({Ả}_2\right)=0.5519$. Clearly, we have $\text{Ҧ}\left({Ả}_1\right)>\text{Ҧ}\left({Ả}_2\right),$ which means that ${Ả}_1$ is the better alternative than ${Ả}_2.$

5.1.1. Implementation of the Newly Developed Strategies

Only Earth is known to be home to life among the astounding array of worlds in our solar system that has a concoction of gases that surrounds the globe and serves to support life by giving us air to breathe, protecting us from the sun’s harmful ultraviolet radiations, retaining heat to keep the world warm, and limiting the dramatic temperature variations between day and night. But increasing industries and fossil fuel combustion produce greenhouse gas emissions that serve as a blanket around the planet, trapping heat from the sun and increasing temperatures, causing climate change. There is a prevalent belief among individuals that the primary consequence of climate change is the escalation of temperatures. However, the narrative does not commence with the rise in temperature. Alterations in a particular location can potentially exert influence on modifications occurring in all other regions due to the Earth’s interconnectedness, as it operates as a unified system. Climate change has significant implications for various aspects of society, including health, food supply, housing, safety, and work. Certain segments of the population, namely inhabitants of tiny island nations and other underdeveloped nations, exhibit a heightened vulnerability to the impacts of climate change. The contemporary manifestations of climate change encompass a range of adverse consequences, including but not limited to severe droughts, water scarcity, devastating wildfires, escalating sea levels, inundation events, polar ice melting, catastrophic storms, and a reduction in biodiversity. Prolonged periods of drought are posing a significant threat to human populations, as they increase the risk of famine. Additionally, phenomena such as rising sea levels and the infiltration of saltwater have reached critical levels, necessitating the complete evacuation of entire towns. There will probably be more “climate refugees” in the future. In short, climate change is a major development challenge. It might make poverty worse and hinder economic development. While at the same time, how nations develop and the investments they make to fulfill the energy, food, and water needs of a growing population can either drive climate change and increase global dangers or help find answers. So, the government should find a way so that the economy must continue to grow; there is no going back on growth and reducing drivers of climate change. Using no plastic, taking public transportation, applying the 3R strategy, and many other measures can all help reduce the drivers of climate change, but these cannot be implemented at the government level; rather, they must be implemented on a person-to-person basis. World Bank Group President Jim Yong Kim [45] spoke to students at Georgetown University in Washington, D.C., on March 18. He discussed five major areas that governments must consider, where policies and growth choices can help lessen the causes of climate change by maintaining economic growth, and some factors influencing these strategies. We propose a step-by-step process for choosing an efficient strategy that can lessen the climate change drivers in the CIF Dombi environment as CIF Dombi operators are very important for solving real-world environmental science problems because they provide a strong mathematical framework for dealing with uncertainty, imprecision, conflicting information, and spatial-temporal complexities. Because they can deal with unclear and vague data, they are useful tools for modeling the environment, figuring out risks, making decisions, and performing many other important things in the field.

Let $\left\{{Ả}_1,{Ả}_2,{Ả}_3,{Ả}_4,{Ả}_5\right\}$ be the strategies (alternatives) to reduce the drivers of climate change;

1.

${Ả}_1$: Carbon pricing is a market-based strategy that assigns a monetary value to carbon emissions in order to encourage commercial and private consumers to cut back on their usage. Carbon pricing helps reduce greenhouse gas emissions by encouraging the switch to cleaner, more sustainable energy sources.

2.

${Ả}_2$: Fossil fuel subsidies: Efforts to switch to renewable energy sources are hampered by fossil fuel subsidies, which governments provide to encourage the production and consumption of fossil fuels. Subsidies for fossil fuels are hindering the transition to a low-carbon economy, sustainable development, and lowering greenhouse gas emissions.

3.

${Ả}_3$: Use of renewable energy sources: Alternatives to fossil fuels that make use of renewable energy sources like solar, wind, hydro, and geothermal power are more environmentally friendly and sustainable. Renewable energy generation has many benefits, including the diversification of energy sources, increased energy security, the creation of new jobs, the acceleration of technological advancement, and the acceleration of the transition to a more sustainable and resilient energy system.

4.

${Ả}_4$: Implement climate-smart agriculture and nurture forest landscapes: Farmers are better able to adapt to the changing climate and improve resilience, soil health, water conservation, and greenhouse gas emissions reductions when climate-smart agriculture and nurture forest landscape techniques are put into place. Promoting a more sustainable and resilient earth through caring for forest landscapes through responsible management, reforestation, and conservation activities helps reduce greenhouse gas emissions, keep biodiversity intact, shield water supplies, and supply vital ecosystem services.

5.

${Ả}_5$: Build low-carbon, resilient cities: Encourage compact and diversified land use, prioritize public transportation, walking, and bicycling, integrate renewable energy sources, and make buildings more energy efficient to construct a low-carbon, resilient strategy. A more sustainable and equitable future for cities is possible through the implementation of these policies, which reduce emissions of greenhouse gases, increase resilience to climate change, enhance air quality, and make communities more livable.

Let $\{{Ẻ}_1,{Ẻ}_2,{Ẻ}_3,{Ẻ}_4\}$ be the four factors that affect the above strategies;

1.

${Ẻ}_1$: Economic Factor

2.

${Ẻ}_2$: Technical Factor

3.

${Ẻ}_3$: Environmental Factor

4.

${Ẻ}_4$: Socio-political Factor

In order to generate a realistic CIF decision matrix, it is essential to classify these factors into specific categories, including the following:

➢

${Ẻ}_1$ consists of economic growth and investment costs.

• Economic growth: The key component is economic growth, which is defined as an increase in the output and consumption of goods and services within a given economy. Furthermore, a country’s ability to invest in measures to mitigate and adapt to climate change, increase resilience, and provide support to people most at risk can increase as its economy grows stronger.
• Investment cost: To successfully implement methods to reduce the drivers of climate change, it is important to evaluate the investment cost associated with doing so. Adequate funding promotes the uptake of new technology, finances research, and development, helps build infrastructure, incentivizes positive behavioral changes, and makes it easier to expand successful programs. Progress in adapting to climate change may be hampered without these investments.

➢

${Ẻ}_2$ is further categorized into the efficiency and maturity of the method.

• Efficiency of method: For optimal resource utilization, cost-effectiveness, large impact, rapid implementation, public support, scalability, and replication, it is essential to employ an efficient approach and plan to reduce the causes of climate change. Global efforts to reduce greenhouse gas emissions can be more successful and have a greater impact if efficiency is given top priority.
• Maturity of a method: Reliability, decreased hazards, cost savings, scalability, reproducibility, and policy backing all come into play as a method or plan for mitigating climate change reaches maturity. If we use tried-and-true techniques, we can make the greatest possible contribution to fighting climate change. It is essential, however, to keep looking for and creating novel approaches to tackling new problems and making the most of technological advances.

➢

${Ẻ}_3$ includes reducing the destructive impact on the biological and visual diversity of the region and reducing harmful gas emissions.

• Reducing the destructive impact on biological and visual diversity: Protecting ecosystems, preserving natural habitats, and increasing resilience are all ways in which reducing the negative impact on biological and visual diversity contributes to long-term sustainability and a healthy world, making it a crucial part of climate change mitigation methods.
• Reducing harmful gas emissions: Reducing harmful gas emissions is essential in climate change mitigation strategies because it reduces air pollution, encourages a shift to cleaner and more sustainable energy systems, and helps present and future generations live in a healthier environment.

➢

${Ẻ}_4$ consists of public and political acceptance.

• Public acceptance: In order to successfully execute and sustainably implement methods to mitigate climate change, public acceptance is essential. This acceptance creates support, motivates behavioral change, and mobilizes collective action.
• Political acceptance: Because it assures policy support, promotes effective governance, and facilitates international cooperation, political acceptance is crucial in implementing policies to reduce the drivers of climate change, creating a cohesive and coordinated strategy to meet the global challenge of climate change.

The expert opinion of a decision maker for each alternative corresponding to each criterion in the form of CIFN is summarized in

Table 5. CIF decision matrix.

	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
${\mathbf{Ả}}_1$	((0.20,0.30),(0.60,0.60))	((0.60,0.30),(0.20,0.50))	((0.30,0.40),(0.60,0.45))	((0.20,0.50),(0.60,0.20))
${\mathbf{Ả}}_2$	((0.30,0.50),(0.40,0.30))	((0.30,0.40),(0.50,0.40))	((0.40,0.30),(0.45,0.40))	((0.40,0.60),(0.40,0.30))
${\mathbf{Ả}}_3$	((0.35,0.70),(0.30,0.20))	((0.20,0.50),(0.60,0.30))	((0.60,0.50),(0.30,0.20))	((0.55,0.70),(0.30,0.10))
${\mathbf{Ả}}_4$	((0.70,0.80),(0.10,0.10))	((0.20,0.60),(0.60,0.20))	((0.85,0.70),(0.05,0.10))	((0.60,0.60),(0.20,0.15))
${\mathbf{Ả}}_5$	((0.30,0.40),(0.50,0.30))	((0.20,0.45),(0.55,0.40))	((0.40,0.30),(0.30,0.40))	((0.50,0.65),(0.30,0.30))

.

The importance of these factors is $\omega ={(0.2,0.3,0.3,0.2)}^T$ and the decision maker’s given weight vector is $\xi ={\left(0.25,0.4,0.25,0.1\right)}^T$, where $\sum _{\mathsf{\tau }=1}^4{\mathsf{\omega }}_{\mathsf{\tau }}=1=\sum _{\tau =1}^4{\xi }_{\tau }.$

Now, applying the $CIFDHA$ and $CIFDHG$ operators, we solve the decision matrix and select the optimal solution in the following discussion.

The solution to the aforementioned MCDM problem within the context of the $CIFDHA$ operator is as follows:

Step 1. The permuted weighted CIF decision matrix by using ${\dot{ß}}_{\tau }=\text{ք}{\mathsf{\omega }}_{\tau }{ß}_{\tau }$ with $\zeta =4$ and ${\dot{ß}}_{\sigma \left(\tau -1\right)}\ge {\dot{ß}}_{\sigma \left(\tau \right)}\forall \tau$ is given in

Table 6. Permuted weighted CIF decision matrix.

	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
${\mathbf{Ả}}_1$	((0.61,0.31),(0.19,0.49))	((0.19,0.49),(0.61,0.21))	((0.29,0.39),(0.61,0.46))	((0.21,0.31),(0.59, 0.59))
${\mathbf{Ả}}_2$	((0.39,0.59),(0.41,0.31))	((0.31,0.51),(0.39,0.29))	((0.31,0.41),(0.49,0.39))	((0.39,0.29),(0.46,0.41))
${\mathbf{Ả}}_3$	((0.54,0.69),(0.31,0.11))	((0.59,0.49),(0.31,0.21))	((0.36,0.71),(0.29,0.19))	((0.21,0.51),(0.59,0.29))
${\mathbf{Ả}}_4$	((0.84,0.69),(0.05,0.11))	((0.71,0.81),(0.10,0.10))	((0.59,0.59),(0.21,0.16))	((0.21,0.61),(0.59,0.19))
${\mathbf{Ả}}_5$	((0.49,0.64),(0.31,0.31))	((0.31,0.41),(0.49,0.29))	((0.39,0.29),(0.31,0.41))	((0.21,0.46),(0.54,0.39))

.

Step 2. Applying the $CIFDHA$ operator to the figures in

Table 7. Aggregated values of alternatives under CIFDHA operator.

Alternatives	${\mathsf{\phi }}_{\mathsf{\varsigma }}$
${\mathbf{Ả}}_1$	((0.5556,0.4166),(0.2312,0.2711))
${\mathbf{Ả}}_2$	((0.3598,0.5430),(0.4201,0.3205))
${\mathbf{Ả}}_3$	((0.5352,0.6749),(0.3108,0.1269))
${\mathbf{Ả}}_4$	((0.8114,0.7536),(0.0650,0.1098))
${\mathbf{Ả}}_5$	((0.4382,0.5844),(0.3343,0.3213))

for a given value of $\zeta =4$ yields:

Step 3. Using Definition 9, investigate the score outcomes for each ${\phi }_{\delta }$ derived in step 2.

$\mathrm{Ҧ}\left({\mathrm{Ả}}_1\right)=0.6533,\mathrm{Ҧ}\left({\mathrm{Ả}}_2\right)=0.5422,\mathrm{Ҧ}\left({\mathrm{Ả}}_3\right)=0.7387,\mathrm{Ҧ}\left({\mathrm{Ả}}_4\right)=0.8991$ and $\mathrm{Ҧ}\left({\mathrm{Ả}}_5\right)=0.6030$.

Step 4. Since $\text{Ҧ}\left({Ả}_4\right)>\text{Ҧ}\left({Ả}_3\right)>\text{Ҧ}\left({Ả}_1\right)>\text{Ҧ}\left({Ả}_5\right)>\text{Ҧ}\left({Ả}_2\right)$, therefore, the ranking order of alternatives is ${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$.

Step 5. The above discussion concludes that “implement climate smart agriculture and nurture forest landscapes” is the best strategy to counter the required challenge.

Similarly, the solution to the aforementioned MCDM problem within the context of the $CIFDHG$ operator is as follows:

Step 1. The permuted weighted CIF decision matrix by using ${\dot{ß}}_{\tau }={\left({ß}_{\tau }\right)}^{\text{ք}{\text{ϣ}}_{\tau }}$ for a specific value of $\zeta =4$ and ${\dot{ß}}_{\sigma \left(\tau -1\right)}\ge {\dot{ß}}_{\sigma \left(\tau \right)}\forall \tau$ is given in

Table 8. Permuted weighted CIF decision matrix.

	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
${\mathbf{Ả}}_1$	((0.59,0.29),(0.21,0.51))	((0.21,0.51),(0.59,0.19))	((0.31,0.41),(0.59,0.44))	((0.19,0.29),(0.61,0.61))
${\mathbf{Ả}}_2$	((0.41,0.61),(0.39,0.29))	((0.29,0.49),(0.41,0.31))	((0.41,0.31),(0.44,0.39))	((0.29,0.39),(0.51,0.41))
${\mathbf{Ả}}_3$	((0.56,0.71),(0.29,0.10))	((0.61,0.51),(0.29,0.19))	((0.34,0.69),(0.31,0.21))	((0.19,0.49),(0.61,0.31))
${\mathbf{Ả}}_4$	((0.86,0.71),(0.05,0.10))	((0.69,0.79),(0.10,0.10))	((0.61,0.61),(0.19,0.14))	((0.19,0.59),(0.61,0.21))
${\mathbf{Ả}}_5$	((0.51,0.66),(0.29,0.29))	((0.41,0.31),(0.29,0.39))	((0.29,0.39),(0.51,0.31))	((0.19,0.44),(0.56,0.41))

.

Step 2. Applying the $CIFDHG$ operator to the figures in

Table 9. Aggregated values of alternatives under CIFDHG operator.

Alternatives	${\mathsf{\phi }}_{\mathsf{\varsigma }}$
${\mathbf{Ả}}_1$	((0.2462,0.3243),(0.5606,0.5102))
${\mathbf{Ả}}_2$	((0.3370,0.3817),(0.4317,0.3502)))
${\mathbf{Ả}}_3$	((0.2924,0.5670),(0.4718,0.2181))
${\mathbf{Ả}}_4$	((0.2983,0.6562),(0.4691,0.1421))
${\mathbf{Ả}}_5$	((0.2836,0.3749),(0.4662,0.3502))

for a given value of $\zeta =4$ yields:

Step 3. Using Definition 9, investigate the score outcomes for each ${\phi }_{\delta }$ derived in step 2.

$\mathrm{Ҧ}\left({\mathrm{Ả}}_1\right)=0.3465,\mathrm{Ҧ}\left({\mathrm{Ả}}_2\right)=0.4805,\mathrm{Ҧ}\left({\mathrm{Ả}}_3\right)=0.5511,\mathrm{Ҧ}\left({\mathrm{Ả}}_4\right)=0.6029$ and $\mathrm{Ҧ}\left({\mathrm{Ả}}_5\right)=0.4482$.

Step 4. Since $\text{Ҧ}\left({Ả}_4\right)>\text{Ҧ}\left({Ả}_3\right)>\text{Ҧ}\left({Ả}_2\right)>\text{Ҧ}\left({Ả}_5\right)>\text{Ҧ}\left({Ả}_1\right)$, therefore, the ranking order of alternatives is ${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$.

Step 5. The above discussion concludes that “implement climate smart agriculture and nurture forest landscapes” is the best strategy to counter the required challenge.

In light of the newly specified CIF Dombi aggregation methods,

Table 10. Score values and ranking of alternatives.

Operators	${\mathcal{A}}_1$	${\mathcal{A}}_2$	${\mathcal{A}}_3$	${\mathcal{A}}_4$	${\mathcal{A}}_5$	Ranking
CIFDHA	0.6533	0.5422	0.7387	0.8991	0.6030	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
CIFDHG	0.3465	0.4805	0.5511	0.6029	0.4482	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$

summarizes all the data gathered above.

5.1.2. The Impact of the Operational Parameter $\mathsf{\zeta }$ within the Context of the CIFDHA and CIFDHG Framework

We explain how the alternatives’ behavior shifts for different values of the operational parameter $\zeta$ in the context of the CIF Dombi hybrid aggregation operators in the following discussion. Note that, in the case of the CIFDHA operator, the score values of the alternatives continue to grow progressively as the parametric value increases, whereas, in the case of the CIFDHG operator, the score values of the alternatives continue to decrease with the increase in $\zeta$, despite the fact that the relative positions of alternatives remain unchanged. Table 8 and Table 9 demonstrate the results of using these operators to rank the available alternatives. The steps involved in this process are summarized below in

Table 11. Score values and preference ranking by CIFDHA operator for various values of parameter $\zeta$.

$\mathbf{\zeta }$	$\mathbf{Ҧ}\left({\mathbf{Ả}}_1\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_2\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_3\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_4\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_5\right)$	Ranking
2	0.5950	0.5296	0.7146	0.8844	0.5638	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
4	0.6533	0.5422	0.7387	0.8991	0.6030	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
5	0.6717	0.5440	0.7378	0.9008	0.6000	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
10	0.7005	0.5574	0.7512	0.9085	0.6215	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
15	0.7101	0.5642	0.7563	0.9113	0.6295	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
20	0.7148	0.5681	0.7591	0.9127	0.6337	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
25	0.7177	0.5706	0.7610	0.9135	0.6363	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$

and

Table 12. Score values and preference ranking by CIFDHG operator for various values of parameter $\zeta$.

$\mathit{\zeta }$	$\mathbf{Ҧ}\left({\mathit{Ả}}_1\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_2\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_3\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_4\right)$	$\mathbf{Ҧ}\left({\mathit{Ả}}_5\right)$	Ranking
2	0.3783	0.4919	0.6105	0.6920	0.4728	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
4	0.3465	0.4805	0.5511	0.6029	0.4482	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
5	0.3365	0.4636	0.5253	0.5799	0.4209	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
10	0.3131	0.4387	0.4778	0.5351	0.3876	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
15	0.3039	0.4267	0.4606	0.5193	0.3742	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
20	0.2989	0.4201	0.4520	0.5115	0.3673	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
25	0.2956	0.4160	0.4468	0.5067	0.3628	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$

.

5.2. Comparative Analysis

In the subsequent section, we address the aforementioned MCDM issue using a number of pre-defined operators in IF and CIF settings, including IFDHA, IFDHG, CIFHA, and CIFHG.

Table 13. Aggregated values obtained from different existing operators.

	IFDHA [23]	IFDHG [23]	CIFHA [29]	CIFHG [29]
${\mathrm{Ả}}_1$	(0.5556,0.2312)	(0.2462,0.5606)	((0.4400,0.3672),(0.3404,0.4143))	((0.3224,0.4022),(0.4800,0.3988))
${\mathrm{Ả}}_1$	(0.3693,0.4186)	(0.3370,0.4317)	((0.3435,0.5025),(0.4085,0.3115))	((0.3744,0.4859),(0.4049,0.3190))
${\mathrm{Ả}}_1$	(0.5352,0.3108)	(0.2924,0.4718)	((0.4384,0.6466),(0.3251,0.1827))	((0.4489,0.6421),(0.3311,0.1714))
${\mathrm{Ả}}_1$	(0.7944,0.0714)	(0.2983,0.4691)	((0.6889,0.7292),(0.1234,0.1177))	((0.6350,0.7088),(0.1833,0.1173))
${\mathrm{Ả}}_1$	(0.4382,0.3343)	(0.2836,0.4662)	((0.3539,0.5018),(0.4195,0.3348))	((0.3830,0.4727),(0.3868,0.3190))

provides a summary of the results obtained by using these operators, and

Table 14. Score values and ranking of alternatives.

Operators	${\mathcal{A}}_1$	${\mathcal{A}}_2$	${\mathcal{A}}_3$	${\mathcal{A}}_4$	${\mathcal{A}}_5$	Ranking
IFDHA [23]	0.5440	0.4934	0.5204	0.5560	0.5153	${Ả}_4\succ {Ả}_1\succ {Ả}_3\succ {Ả}_5\succ {Ả}_2$
IFDHG [23]	0.4624	0.4858	0.4723	0.4741	0.4695	${Ả}_2\succ {Ả}_4\succ {Ả}_3\succ {Ả}_5\succ {Ả}_1$
CIFHA [29]	0.5169	0.5352	0.6739	0.8542	0.5269	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$
CIFHG [29]	0.4520	0.5415	0.6809	0.8169	0.5469	${Ả}_4\succ {Ả}_3\succ {Ả}_5\succ {Ả}_2\succ {Ả}_1$
CIFDHA	0.6533	0.5422	0.7387	0.8991	0.6030	${Ả}_4\succ {Ả}_3\succ {Ả}_1\succ {Ả}_5\succ {Ả}_2$
CIFDHG	0.3465	0.4805	0.5511	0.6029	0.4482	${Ả}_4\succ {Ả}_3\succ {Ả}_2\succ {Ả}_5\succ {Ả}_1$

provides a ranking of those findings.

The discussion above makes it clear that although each aggregation technique has its own benefits and drawbacks, and the ideal one to utilize will depend on the particular circumstances and objectives of the decision-making process, the strategies proposed in this article are more generalized than other existing techniques because the CIF Dombi aggregation operators successfully deal with the situation where the best preference changes as a result of the loss of information within the framework of existing IF operators. IF operators fail to show a similar ranking as they lack in the second dimension and can only handle the problems with one-dimensional information. Whereas, in comparison to average hybrid operators, Dombi hybrid operators have more versatility and can be used to make judgments with incremental acceptance/rejection of information, as well as handle asymmetric uncertainty, extreme circumstances, and ambiguous scenarios. Decision-makers have more control and precision in the aggregation process when using Dombi operators, which excel when dealing with subtle and complex uncertainty patterns than when using averaging operators. Although in the proposed case study, CIFHA and CIFHG operators show the same rankings, they may not be able to rank the alternatives that have a very large range of uncertain values. On the other hand, the proposed operators can best tackle two-dimensional problems and have greater versatility because of the parametric number they employ. It is also worth noting that the recently defined operators generalize the concept of aggregation operators developed by Masmali et al. [33] for certain values of the weight vector, whereas the operators defined by Sheik and Mandal [23], become the special cases of these newly defined operators by taking the second-dimension constant. It is important to note that although many hybridized aggregation operators are available in the literature in the context of IFS, our proposed methodologies have some advantages over existing operators:

1. Complex intuitionistic fuzzy sets work best in cases where uncertainty shows more complex behaviors, such as oscillations, dynamic changes, spatial complexity, multivariate interdependencies, and characteristics that do not add up. Even though Intuitionistic Fuzzy Sets are useful on their own, CIFS is a more expressive and flexible framework for dealing with complicated uncertainty and doubt in a wide range of real-world situations.

2. Dombi hybrid operators are better than averaging hybrid operators because they are more flexible, can handle asymmetric uncertainty, extreme cases, and unclear situations, and can be used to make decisions with gradual acceptance/rejection. Averaging operators are easier to use and can be used in many situations, while Dombi operators shine when dealing with complicated and nuanced uncertainty patterns, giving decision-makers more control and accuracy in the aggregation process in areas like fuzzy control systems, image and signal processing, environmental monitoring, and many others. In short, for solving two-dimensional, real-life problems with vague and asymmetric data, CIF Dombi hybrid aggregation operators do their best to select the optimal solution to these problems.

In short, for solving two-dimensional, real-life problems having vague and asymmetric data, CIF Dombi hybrid aggregation operators play their best to select the optimal solution to these problems.

6. Conclusions

In this study, inspired by the theories of Dombi aggregation operators and CIFS, the CIFDHA and CIFDHG operators for the CIF environment have been designed. The suggested aggregation operations are more flexible than others in the sense that a decision-maker can choose the most suitable value of the parameter $\zeta$ while making decisions. The score values of the alternatives continue to grow progressively as the parametric value increases within the context of the CIFDHA operator, whereas, in the case of the CIFDHG operator, the score values of the alternatives continue to decrease with the increase in $\zeta$, while the ranking of the alternatives remains unchanged within the context of these methods. Additionally, the operators defined in [33] become the special cases of these newly defined operators. The newly specified AOs have also been applied to formulate a mechanism to select an efficient approach to reduce the causes of climate change. Finally, a comparison has been made to the existing approaches to demonstrate the reliability and effectiveness of proposed strategies. It is crucial to remember that the strategy described in this article is better because it manages input dependencies. This makes our method applicable in a wider range of situations. It is challenging to implement the MCDM approach because the parameter cannot be dynamically changed to reflect the decision-makers’ risk aversion. The techniques covered here can more than makeup for this shortcoming, though. The primary focus of future efforts will be on creating a full-featured decision analysis tool using Dombi operators to increase its usefulness and applicability in the real world. Additionally, the hybrid Dombi operators will be used in upcoming studies to tackle numerous important MCDM issues effectively for complex picture, complex Pythagorean [39], complex bipolar [49], and interval-valued picture [50] fuzzy sets.