Ranking data privacy techniques in cloud computing based on Tamir’s complex fuzzy Schweizer-Sklar aggregation approach

Abstract

In the era of cloud computing, it has become an important challenge to secure data privacy by storing and processing massive amounts of sensitive information in shared environments. Cloud platforms have become a necessary component for managing personal, commercial, and governmental data. Thus, the demand for effective data privacy techniques within cloud security frameworks has increased. Data privacy is no longer just an exercise in compliance but rather to reassure stakeholders and protect precious information from cyber-attacks. The decision-making (DM) landscape in the case of cloud providers, therefore, is extremely complex because they would need to select the optimal approach among the very wide gamut of privacy techniques, which range from encryption to anonymization. A novel complex fuzzy Schweizer-Sklar aggregation approach can rank and prioritize data privacy techniques and is particularly suitable for cloud settings. Our method can easily deal with uncertainties and multi-dimensional aspects of privacy evaluation. In this manuscript, first, we introduce the fundamental Schweizer-Sklar operational laws for a cartesian form of complex fuzzy framework. Then relying on these operational laws, we have initiated the notions of cartesian form of complex fuzzy Schweizer-Sklar power average and complex fuzzy Schweizer-Sklar power geometric AOs. We have developed the main properties related to these notions like Idempotency, Boundedness, and monotonicity. Also, we explored an algorithm for the utilization of the developed theory. Moreover, we provided an illustrative example and case study for the developed theory to show the ranking of data privacy techniques in cloud computing. At the end of the manuscript, we discuss the comparative analysis to show the supremacy of the introduced work.

Introduction

Cloud computing is now considered part of modern technology. Most organizations along with almost any individual can host, access, and process immense information all around the world. It has enormous potential benefits flexibility in access, cost-effectiveness, and the ability to scale. Therefore, it becomes the best option for businesses, governments, and individuals respectively. But along with this shift to cloud storage and processing comes the critical challenge of protecting sensitive information from security risks. Cloud environments store the data of several users in one place which poses a very realistic threat for security breaches. Cyber threats in the form of data breaches and unauthorized access evolve every day exposing confidential information and emphasizing the need for strong cloud security measures. Data privacy is the most important aspect of cloud security. However, it is the most important factor in establishing trust between users and cloud service providers since the privacy of personal and sensitive information about users is not only legal compliance. They expect that their data is safe from unauthorized access and misuse. Figure 1 shows how cloud computing connects with different services and also shows the type of cloud.

Fig. 1

This diagram shows how cloud computing connects with different services like storage, mobile, applications, and databases. It also shows the type of clouds.

For organizations, it means the prevention of costly breaches of data and damage to reputation while giving them a competitive edge over other firms. Protecting data privacy in the cloud is thus more than an operational requirement and it’s a strategic priority. Generally, a cloud environment would often represent shared infrastructures movement of data across international boundaries, and multiple user access some of which may lead to vulnerabilities. To combat these issues several techniques on data privacy are presently in use, each of which addresses particular aspects of privacy. Also, note that the choice of appropriate technique is a decision-making problem. For example, in the case of the health sector, healthcare organizations store patients on cloud platforms, to ensure the privacy of this information, the healthcare organization is evaluating several privacy techniques. The problem is that some techniques offer strong security of data but they are computationally expensive and some techniques provide good balance but may reduce accuracy. In these situations, healthcare organizations need to rank these techniques based on criteria such as security level, computation cost, ease of implementation, and impact on data utility.

Data privacy methods are essential in safeguarding sensitive data in the modern digital age. Encryption is one of the most common methods, whereby data is converted into an unreadable form using algorithms and keys so that only authorized individuals can view it. Tokenization substitutes sensitive data with non-sensitive equivalents known as tokens, which hold no exploitable value, making it very effective in safeguarding data such as credit card numbers. Data masking entails altering data so that it is still usable for testing or analysis but obscures the original values, typically utilized in non-production environments. Anonymization steps even further to erase personally identifiable information permanently, so data cannot be traced back to an individual. Finally, differential privacy adds mathematically generated noise into datasets so useful analysis is possible while keeping individual records private, generally applied in statistical and machine learning models. All three methods collectively present a solid framework for protecting data and facilitating compliance with privacy legislation. Figures 2 and 3 show the utilization of data privacy techniques by different sectors and the forecasted adoption of data privacy techniques respectively.

Fig. 2

Utilization of data privacy techniques by different sectors.

Fig. 3

Forecasted adoption of data privacy techniques (2024–2030).

Literature review

People storing data and processing information through cloud computing systems face fundamental doubts about their privacy and security protections. Multiple researchers have developed numerous technological approaches that aim to build trust for users while strengthening system resilience. According to Sun et al.¹ cloud environments face three main security issues involving data breaches with unauthorized access and unreliable availability of services. The authors reviewed encryption approaches together with access controls while identifying authentication as the foremost component for cloud security. The researcher Sun² executed an extensive examination of new security dangers including hacking from inside the organization and vulnerable APIs by developing hybrid cryptographic methods and implementing policy-based regulatory measures. Jakimoski³ reviewed both traditional encryption methods as well as contemporary strategies that protect cloud data with symmetric and asymmetric techniques and steganography and intrusion detection systems. He stressed how data protection needs multi-level security designs to work properly. Abrera⁴ composed a detailed overview that analyzed modern security approaches by categorizing them as proactive and reactive strategies and emphasized the combination of regulatory compliance with user transparency. Visual cryptography serves as a new approach for cloud platform data protection which Sultana and Srinivas⁵ developed. The research team divided data into separate visual components as their method to boost confidentiality while minimizing computational requirements. Gawande and Kapse⁶ conducted a comparative review of data confidentiality methods which demonstrated that homomorphic encryption together with obfuscation techniques yields positive outcomes for secure computation. Sood⁷ designed a collective framework that combines encryption functions with hash algorithms and digital signature processes to handle the protection of resting data as well as data transmission and system processing operations. This approach delivered better capabilities for managing secure data access efficiently. Singh et al.⁸ provided a privacy mechanism review that focused on identity protection and data-sharing security achieved through attribute-based encryption and anonymization techniques. Albugmi et al.⁹ analyzed how organizations can overcome practical deployment challenges related to cloud security while focusing on maintaining scalability and supporting interoperability. Different cloud service models require policy enforcement architectures that adapt according to service models proposed by them. The combination of cloud computing and big data with IoT requires scalable privacy solutions and efficient resource management standards for sustainable environments per Stergiou et al.¹⁰. Bruma¹¹ identified three security techniques for the cloud which include prevention detection and correction and recommended integrated access monitoring intrusion prevention systems and firewalls for complete security programs. The application of the Adaptive Neural Fuzzy Inference System (ANFIS) by Shahzadi et al.¹² led to real-time cyber threat detection and prevention which demonstrated enhanced accuracy when compared to ordinary systems. The research by Tariq et al.¹³ investigated how fuzzy logic serves cloud computing by analyzing security configuration decisions and threat evaluations. Intelligent systems demonstrate their ability to adjust their security responses to cloud infrastructure environments through the research of their team. Moreover, Feng et al.¹⁴ established the idea of privacy-preserving tensor composition over encrypted data in a federated cloud environment. Most recently, Feng et al.¹⁵ designed a Panther, a lightweight secure two-party neural network inference system safeguarding the privacy of server and client. Additionally, Zhang et al.¹⁶ discussed an effective truth discovery under local differential privacy by leveraging noise-aware probabilistic estimation and fusion. The researched publications showcase various sophisticated techniques for safeguarding cloud data which include both encryption methods and visual cryptography and artificial intelligence inference systems. Implementing reliable cloud platforms depends on how well intelligent algorithms work with security regulations alongside strong encryption systems.

MCDM background study

MCDM techniques in cloud computing have become essential to research because they solve complex competing factors when choosing optimal cloud services. AHP combined with TOPSIS used by Khan et al.¹⁷ exhibits an effective method to enable decision-makers to weigh privacy-related attributes so they can rank available alternatives by employing transparent consistent measures. Mostafa¹⁸ introduced an optimized MCDM framework that employs a “best-only” selection process to remove inferior alternatives early so it brings improved decision efficiency and reduced calculation time. Research conducted by Kumar et al.¹⁹ demonstrated how MCDM methods in cloud computing experience increased adoption of hybrid and fuzzy-enhanced models because user requirements and service descriptions remain vague. The implementation of intuitionistic fuzzy sets by Ghorui et al.²⁰ within MCDM enabled improved knowledge of provider capabilities through its managing abilities with conflicting evaluations and decision hesitations. Kumar et al.²¹ combined classic MCDM methods with fuzzy environments to develop human-oriented evaluation systems that tolerate uncertainties for service quality, cost, and scalability assessments. Aslam et al.²² proposed the combination of hesitant bipolar complex fuzzy sets and Dombi aggregation operators to make MCDM models more expressive and capable of handling intricate user preferences and conflicting attitudes. A hybrid MCDM model developed by Tomar et al.²³ enabled flexible processing of qualitative and quantitative factors for cloud service provider selection. Abdullah et al.²⁴ linked TOPSIS with artificial intelligence-based three-way decision-making to bridge classic MCDM with modern machine learning paradigms and enhance decision adaptability and precision. Abdullah et al.²⁵ built upon the previous model by introducing deep learning approaches and double hierarchy linguistic terms that enabled advanced decision management for evaluations with uncertain linguistic data. MCDM application growth within cloud computing becomes clearer through combined analysis of hybrid systems and fuzzy logic and artificial intelligence which renovates the assessment of cloud services with ranking decisions and privacy controls.

Motivation

In this section, we discuss the motivation and main contribution of the proposed work.

Motivation

The study of Ramot et al.²⁷ replies to the polar form of complex fuzzy set theory and in the case of Ramot et al.²⁷ approach, we can see that it uses the unit circle instead of the unit square. Moreover in the polar form of CFS the membership grade is given in the polar form that is $\:r{e}^{2\pi\:\theta\:}$and $\:r,\theta\:\in\:\left[0,\:1\right].\:$But notice that if the decision makers provide their assessment in the Cartesian form of complex fuzzy numbers like $\:0.9+\iota\:0.7$ then we can see that for this kind of data $\:r=\sqrt{0.{9}^{2}+0.{7}^{2}}=1.54\notin\:\left[0,\:1\right]$. Hence in this case the fundamental condition fails to hold as defined in Ramot et al.²⁷ approach. We can say that the Cartesian form of a complex fuzzy set is more reliable and advanced because it can handle such kind of information and uses the unit square instead of the unit circle. Many existing notions. The Cartesian form of CFS provides more space for decision-makers to discuss data with the second dimension. The main motivation of the developed theory is to discuss the Cartesian form of complex fuzzy set and avoid any kind of data loss because in the case of the Cartesian form of CFS, the chance of data loss is reduced and decision-makers can take theory information in decision making to make theory decision without any hesitation.

During the age of high-speed digital transformation, cloud computing is a pillar in data storage and processing across all sectors. Yet, with such convenience, there lies the essential challenge of maintaining strong data privacy. Data privacy methods in cloud computing are subjected to analyzing and ranking diverse, usually contradicting, criteria like encryption quality, computational overhead, scalability, and user trust. In addition, specialized opinions regarding these criteria are most often imprecise, doubtful, and uncertain. Classical crisp or even classical fuzzy methods tend to be unable to fully catch the entire variety of uncertainty and doubt involved in such decision processes. CFSs provide a more expressive format to represent such subtle and ambiguous information. CFSs provide a better representation of vagueness and hesitation, offering improved modeling of human logic used in expert estimation. Hence, using advanced fuzzy sets during the ranking stage gives a richer, more precise, and more reliable consideration of data protection methods within clouds, thereby driving more informed decisions.

Challenges and contribution

There are several challenges in ranking the data privacy techniques. These challenges with explanation are given by (1) Data privacy techniques must be evaluated across multi-dimensional criteria and proper weights to these criteria for fair ranking become difficult. (2) Multi-cloud infrastructures have different security requirements and constraints. A technique suitable for one environment may not be ideal for another, complicating ranking results. (3) Cyber threats are continuously evolving. Ranking needs to be continuously updated to remain relevant (4) Stronger privacy techniques are mostly computationally expensive and, in this case, high security may degrade the performance (5) Decision-making models mostly rely on expert input which can be biased, and ranking process may inherit this business which may reduce the trust in results. Ranking privacy methods in cloud computing is problematic because of uncertainty, subjective preference, and heterogeneity in criteria. Fuzzy set theory is a strong approach to handling vagueness with linguistic variables and fuzzy numbers. Fuzzy logic is also suitable to handle changing threats and heterogeneity in cloud infrastructures. Fuzzy-based models therefore support more flexible, realistic, and strong decision-making to rank privacy methods. Moreover, aggregation operators are the fundamental approach to converting the overall data into a single value that can help in decision-making problems. To cover these challenges and to provide a smooth way for the ranking process, the main contribution of the study is given as;

• (a) First, we develop several new aggregation operators under the conditions of CFSs. So, the list of newly developed operators is;

• CF Schweizer-Sklar power averaging (CFSSPA) operator.

• CF Schweizer-Sklar power weighted averaging (CFSSPWA) operator.

• CF Schweizer-Sklar power geometric (CFSSPG) operator.

• CF Schweizer-Sklar power weighted geometric (CFSSPWG) operator.

• (b) Development of MADM methodology with CF environment that helps to achieve effective alternatives in the ranking of data privacy techniques in cloud computing.

• (c) CF-MADM methodology gets applied to data privacy techniques in cloud computing for better classification and results through the mathematical and numerical implementation of CF data.

• (d) Moreover, our proposed research performs general comparative evaluations that test both the proposed CF Schweizer-Sklar power aggregation operators together with present approaches to demonstrate their ability to manage uncertainty in data privacy techniques.

Moreover, below Fig. 4 shows the graphical representation of the main contributions.

Fig. 4

The graphical representation of the main contributions.

Study layout

In Sect. 1 we have discussed the introduction of the application. Section 2 is about the literature study of the initiated work. Moreover, Sect. 3 is regarded as motivation of the developed approach, and in Sect. 4 we have discussed the fundamental notion that can help us to work on further theory. Section 5 is about aggregation theory and Sect. 6 is about the MADM algorithm to discuss the application point of view of the introduced approach. In Sect. 7, we have discussed the comparative analysis of the initiated theory to discuss the importance of the introduced work. In Sect. 8, we have discussed the conclusion remarks. The flow chart of the developed approach is given in Fig. 5.

Fig. 5

Graphical representation of the whole manuscript.

Preliminaries

In this section of the manuscript, we discuss the basic notions of FSs, and Polar form of Complex fuzzy sets, and the Cartesian form of compel fuzzy sets.

The idea of a fuzzy set is developed by Zadeh²⁶ to generalize the crisp set theory. The notion of FS was a great achievement and it opened the path for the researchers and a new area of research has been produced.

Definition 1

²⁶ Let $\:\mathfrak{U}$ be the universal set, then the idea of a fuzzy set is defined by

$$\:FS=\left\{\mathfrak{r},\mathfrak{\:}\vartheta\:\left(\mathfrak{r}\right)|\:\mathfrak{r}\in\:\mathfrak{U}\right\}$$

With information that $\:\vartheta\:;\mathfrak{U}\to\:\left[0,\:1\right]$ and $\:\vartheta\:\left(\mathfrak{r}\right)\in\:\left[0,\:1\right].$

The polar form of the complex fuzzy set has been developed by Ramot et al.²⁷ to generalize the fuzzy set theory and it uses the second dimension that shows the uniqueness of the polar form of complex fuzzy set.

Definition 2

²⁷ Let $\:\mathfrak{U}$ be the universal set, then the idea of the polar form of a complex fuzzy set is defined by

$$\:FS=\left\{\mathfrak{r},\mathfrak{\:}\vartheta\:\left(\mathfrak{r}\right)|\:\mathfrak{r}\in\:\mathfrak{U}\right\}$$

And in this case $\:\vartheta\:\left(\mathfrak{r}\right)=r{e}^{2\pi\:\theta\:}\:$where $\:r,\:\theta\:\in\:\left[0,\:1\right].$ In this case, see that the value of MD $\:\vartheta\:\left(\mathfrak{r}\right)$ is from the unit circle in the complex plan.

Definition 3

²⁸ Let $\:\mathfrak{U}$ be the universal set, then the idea of the Cartesian form of a complex fuzzy set is defined by

$$\:FS=\left\{\mathfrak{r},\mathfrak{\:}\vartheta\:\left(\mathfrak{r}\right)|\:\mathfrak{r}\in\:\mathfrak{U}\right\}$$

And in this case $\:\vartheta\:\left(\mathfrak{r}\right)=\left(\mathfrak{A}\left(\mathfrak{r}\right)+\iota\:\phi\:\left(\mathfrak{r}\right)\right).$ In this case, we can see that the value of MD $\:\mathfrak{A}\left(\mathfrak{r}\right)$ is in the unit square of the complex plan and $\:\mathfrak{A}\left(\mathfrak{r}\right)\:\text{a}\text{n}\text{d}\:\phi\:\left(\mathfrak{r}\right)\in\:\left[0,\:1\right]$.

The Cartesian form of a complex fuzzy set is proposed by Tamir et al.²⁸ and this notion uses the unit square as a range.

Cartesian form of complex fuzzy Schweizer-Sklar operational laws

In this section, we will discuss the basic operational laws for a Cartesian form of CFS. Moreover, we have discussed the notion of the score function and accuracy function that can help us manage the ranking of two CFNs. Moreover, we have discussed the aggregation theory based on developed laws.

Definition 4

Let $\:{CF}_{1}=\left({\mathfrak{A}}_{1}+\iota\:{\phi\:}_{1}\right)$ and $\:{CF}_{2}=\left({\mathfrak{A}}_{2}+\iota\:{\phi\:}_{2}\right)$, as two CFNs with $\:\varrho\:<0$ and $\:\varsigma\:>0$ then the CFSS operational laws originated as below

• 1) $\:{CF}_{1}\oplus\:{CF}_{2}=$ $\:\left(1-{\left({\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}+{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left({\left(1-{\phi\:}_{1}\right)}^{\varrho\:}+{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}\right)$

• 2)
  $$\:{CF}_{1}\otimes\:{CF}_{2}=\:\left({\left({\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}+{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}+\iota\:{\left({\left({\phi\:}_{1}\right)}^{\varrho\:}+{\left({\phi\:}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}\right)$$
• 3)
  $$\:\varsigma\:{CF}_{1}=\left(1-{\left(\varsigma\:{\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left(\varsigma\:-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left(\varsigma\:{\left(1-{\phi\:}_{1}\right)}^{\varrho\:}-\left(\varsigma\:-1\right)\right)}^{\frac{1}{\varrho\:}}\right)$$
• 4)
  $$\:{\left({CF}_{1}\right)}^{\varsigma\:}=\left({\left(\varsigma\:{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left(\varsigma\:-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:{\left(\varsigma\:{\left({\phi\:}_{1}\right)}^{\varrho\:}-\left(\varsigma\:-1\right)\right)}^{\frac{1}{\varrho\:}}\right)$$

Example 1

For two CFNs $\:{CF}_{1}=\left(0.3+\iota\:0.4\right)$ and $\:{CF}_{2}=\left(0.6+\iota\:0.7\right)$ and $\:\varsigma\:=2\:and\:\varrho\:=-1,$ the results of operational laws can be found out as;

• 1) $\:{CF}_{1}\oplus\:{CF}_{2}=$ $\:\left(1-{\left({\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}+{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left({\left(1-{\phi\:}_{1}\right)}^{\varrho\:}+{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}\right)$

  $$\:=\left(1-{\left({\left(1-0.3\right)}^{-1}+{\left(1-0.6\right)}^{-1}-1\right)}^{\frac{1}{-1}}+\iota\:1-{\left({\left(1-0.4\right)}^{-1}+{\left(1-0.7\right)}^{-1}-1\right)}^{\frac{1}{-1}}\right)$$
  $$\:=\left(0.6586+\iota\:0.7500\right)$$
• 2)
  $$\:{CF}_{1}\otimes\:{CF}_{2}=\:\left({\left({\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}+{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}+\iota\:{\left({\left({\phi\:}_{1}\right)}^{\varrho\:}+{\left({\phi\:}_{2}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}\right)$$
  $$\:=\left({\left({\left(0.3\right)}^{-1}+{\left(0.6\right)}^{-1}-1\right)}^{\frac{1}{-1}}+\iota\:{\left({\left(0.4\right)}^{\varrho\:}+{\left(0.7\right)}^{-1}-1\right)}^{\frac{1}{-1}}\right)$$
  $$\:=\left(0.2500+\iota\:0.3414\right)$$
• 3)
  $$\:2{CF}_{1}=\left(1-{\left(2\times\:{\left(1-0.3\right)}^{-1}-\left(2-1\right)\right)}^{\frac{1}{-1}}+\iota\:1-{\left(2{\times\:\left(1-0.4\right)}^{-1}-\left(2-1\right)\right)}^{\frac{1}{-1}}\right)$$
  $$\:=\left(0.4615+\iota\:0.58714\right)$$
• 4)
  $$\:{\left({CF}_{1}\right)}^{2}=\left({\left(2\times\:{\left(0.3\right)}^{-1}-\left(2-1\right)\right)}^{\frac{1}{-1}}+\iota\:{\left(2\times\:{\left(0.4\right)}^{-1}-\left(2-1\right)\right)}^{\frac{1}{-1}}\right)=\left(01764+\iota\:0.2500\right)$$

Theorem 1

Let $\:{CF}_{1}=\left({\mathfrak{A}}_{1}+\iota\:{\phi\:}_{1}\right)$ and $\:{CF}_{2}=\left({\mathfrak{A}}_{2}+\iota\:{\phi\:}_{2}\right)$ be two CFNs with $\:\varsigma\:,\:{\varsigma\:}_{1},\:{\varsigma\:}_{2}>0$ then the following results remain true

• 1)
  $$\:{CF}_{1}\oplus\:{CF}_{2}={CF}_{2}\oplus\:{CF}_{1}$$
• 2)
  $$\:{CF}_{1}\otimes\:{CF}_{2}={CF}_{2}\otimes\:{CF}_{1}$$
• 3)
  $$\:\varsigma\:\left({CF}_{1}\oplus\:{CF}_{2}\right)=\varsigma\:{CF}_{1}\oplus\:\varsigma\:{CF}_{2}$$
• 4)
  $$\:{{\varsigma\:}_{1}CF}_{1}\oplus\:{\varsigma\:}_{2}{CF}_{1}=\left({\varsigma\:}_{1}+{\varsigma\:}_{2}\right){CF}_{1}$$
• 5)
  $$\:{\left({CF}_{1}\right)}^{\varsigma\:}\otimes\:{\left({CF}_{2}\right)}^{\varsigma\:}={\left({CF}_{1}\otimes\:{CF}_{2}\right)}^{\varsigma\:}$$
• 6)
  $$\:{\left({CF}_{1}\right)}^{{\varsigma\:}_{1}}\otimes\:{\left({CF}_{1}\right)}^{{\varsigma\:}_{2}}={\left({CF}_{1}\right)}^{{\varsigma\:}_{1}+{\varsigma\:}_{1}}$$

Definition 5

For ranking the two CFS in Cartesian form, the notion of score function on Cartesian form of CFS $\:{CF}_{1}=\left({\mathfrak{A}}_{1}+\iota\:{\phi\:}_{1}\right)$ is defined by;

$$\:{S}_{cor.}\left({CF}_{1}\right)=\frac{1}{2}\left({1+\mathfrak{A}}_{1}+{\phi\:}_{1}\right)$$

$$\:{A}_{acur.}\:\left({CF}_{1}\right)=\frac{1}{2}\left({1+\mathfrak{A}}_{1}-{\phi\:}_{1}\right)$$

Notice that $\:scor.\left({CF}_{1}\right)\:\text{a}\text{n}\text{d}\:acur.\left({CF}_{1}\right)\in\:\left[0,\:1\right].$

Cartesian form of complex fuzzy Schweizer-Sklar aggregation operators

Since the Cartesian form of CFS is more advanced seen from the diagram, then based on this advanced structure and operational laws defined in def. (4), in this section, we will define the complex fuzzy Schweizer-Sklar power average (CFSSPA) aggregation operators.

Definition 6

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a family of CFNs. Then the CFSSPA AOs are given by

$$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\begin{array}{c}\mathcal{j}\mathcal{\:}\\\:\oplus\:\\\:\iota\:=1\end{array}\frac{\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{CF}_{CF\iota\:}\:$$

Where $\:\mathcal{T}\left({CF}_{CF\iota\:}\right)=\sum\:_{\begin{array}{c}\varrho\:=1\\\:\varrho\:\ne\:\iota\:\end{array}}^{\mathcal{j}}Sup\left({CF}_{CF\iota\:},\:{CF}_{CF\varrho\:}\right)\:$and $\:Sup\left({CF}_{CF\iota\:},\:{CF}_{CF\varrho\:}\right)$ represent the support between $\:{CF}_{CF\iota\:}$ and $\:{CF}_{CF\varrho\:}$ with constraints that

• 1)
  $$\:Sup\left({CF}_{CF\iota\:},\:{CF}_{CF\varrho\:}\right)\in\:\left[0,\:1\right]$$
• 2)
  $$\:Sup\left({CF}_{CF\iota\:},\:{CF}_{CF\varrho\:}\right)=Sup\left({CF}_{CF\varrho\:},\:{CF}_{CF\iota\:}\right)$$

• 3) $\begin{gathered} Sup\left( {CF_{{CF\iota }} ~,CF_{{CF\partial }} } \right) \ge Sup\left( {CF_{{CF\iota ^{*} }} ,~CF_{{CF\partial ^{*} }} } \right) \hfill \\ ~if\mathcal{Q}~\left( {CF_{{CF\iota }} ,~CF_{{CF\partial }} } \right) < \mathcal{Q}\left( {CF_{{CF\iota ^{*} }} ,~CF_{{CF\partial ^{*} }} } \right) \hfill \\ \end{gathered}$, where $\:\mathcal{Q}$ would be a distance measure.

Theorem 2

Take $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ as a family of CFNs. The outcome is again CFN obtained by using the CFSSPA operator given by

$$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\left(1-{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}\right)\:\:$$

(1)

In eq. $\:\left(1\right)$ $\:{\mathcal{w}}_{\iota\:}\:\left(\iota\:=1,\:2,\:3,\:\dots\:,\:\mathcal{j}\right)$ is a family of integrated weights$\:\:{\mathcal{w}}_{\iota\:}=\frac{\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}$.

Proof

For any value of$\:\mathcal{\:}\mathcal{w}={\left({\mathcal{w}}_{1},\:{\mathcal{w}}_{2},\:\dots\:,\:{\mathcal{w}}_{\mathcal{j}}\right)}^{T}$, the Eq. (1) will be reduced to the following

$$\:CFSSPA\left({CF}_{B{F}_{1}},\:{CF}_{B{F}_{2}},\:\dots\:,\:{CF}_{B{F}_{\mathcal{j}}\:}\right)=\left(\begin{array}{c}1-{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)\:\:$$

(2)

Here we use the mathematical induction method. When$\:\mathcal{\:}\mathcal{j}=2$, then

$$\:{\mathcal{w}}_{1}{CF}_{B{F}_{1}}=\left(1-{\left({\mathcal{w}}_{1}{\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left({\mathcal{w}}_{1}{\left(1-{\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)$$

and

$$\:{\mathcal{w}}_{2}{CF}_{CF2}=\left(1-{\left({\mathcal{w}}_{2}{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:1-{\left({\mathcal{w}}_{2}{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)$$

then,

$$\:{\mathcal{w}}_{1}{CF}_{B{F}_{1}}\oplus\:{\mathcal{w}}_{2}{CF}_{B{F}_{2}}=\left(\begin{array}{c}1-{\left({\mathcal{w}}_{1}{\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left({\mathcal{w}}_{1}{\left(1-{\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)\oplus\:\left(\begin{array}{c}1-{\left({\mathcal{w}}_{2}{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left({\mathcal{w}}_{2}{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}1-{\left(\begin{array}{c}{\left(1-\left(1-{\left({\mathcal{w}}_{1}{\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}\\\:+{\left(1-\left(1-{\left({\mathcal{w}}_{2}{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}-1\end{array}\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\begin{array}{c}{\left(1-\left(1-{\left({\mathcal{w}}_{1}{\left(1-{\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}\\\:+{\left(1-\left(1-{\left({\mathcal{w}}_{2}{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}-1\end{array}\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\:\left(\begin{array}{c}1-{\left({\mathcal{w}}_{1}{\left(1-{\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)+{\mathcal{w}}_{2}{\left(1-{\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)-1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left({\mathcal{w}}_{1}{\left(1-{\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)+{\mathcal{w}}_{2}{\left(1-{\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)-1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}1-{\left(\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

So, eq. $\:\left(2\right)$ is valid for$\:\mathcal{\:}\mathcal{j}=2$.

Suppose that Eq. (2) is true for$\:\mathcal{\:}\mathcal{j}=\mathcal{K}$, then$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}}\right)=\:\left(\begin{array}{c}1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$

Now for $\:\mathcal{j}=\mathcal{K}+1$, we get

$$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}},\:{CF}_{\mathcal{K}+1\:}\right)=CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}}\right)\oplus\:{CF}_{\mathcal{K}+1\:}$$

$$\:=\:\left(\begin{array}{c}1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:\oplus\:\left(\begin{array}{c}1-{\left({\mathcal{w}}_{\mathcal{K}+1}{\left(1-{\mathfrak{A}}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left({\mathcal{w}}_{\mathcal{K}+1}{\left(1-{\phi\:}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\:\left(\begin{array}{c}1-{\left(\begin{array}{c}{\left(1-\left(1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}\\\:+{\left(1-\left(1-{\left({\mathcal{w}}_{\mathcal{K}+1}{\left(1-{\mathfrak{A}}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}-1\end{array}\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\begin{array}{c}{\left(1-\left(1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}\\\:+{\left(1-\left(1-{\left({\mathcal{w}}_{\mathcal{K}+1}{\left(1-{\phi\:}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)\right)}^{\varrho\:}-1\end{array}\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}+1\:}{\mathcal{w}}_{\iota\:}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}+1}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:1-{\left(\sum\:_{\iota\:=1}^{\mathcal{K}+1\:}{\mathcal{w}}_{\iota\:}{\left(1-{\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}+1}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}},\:{CF}_{\mathcal{K}+1\:}\right)$$

.

Hence Eq. (2) is valid for $\:\mathcal{j}=\mathcal{K}+1.\:$So it is true for all$\:\mathcal{\:}\mathcal{j}$.

Theorem 3

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. The following properties hold.

• 1) (Idempotency): Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. Then if $\:{CF}_{\iota\:}=CF=\left(\mathfrak{A}+\iota\:\phi\:\right)\:\forall\:\:\iota\:$, we have

  $$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=CF$$

• 2) (Monotonicity): Let$\:\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)$, and $\:{CF}_{\iota\:}^{*}=\left({\mathfrak{A}}_{\iota\:}^{*}+\iota\:{\phi\:}_{\iota\:}^{*}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be two collections of CFNs. Then if $\:{\mathfrak{A}}_{\iota\:}\le\:{\mathfrak{A}}_{\iota\:}^{*}$and $\:{\phi\:}_{\iota\:}\le\:{\phi\:}_{\iota\:}^{*},$ we get

  $$\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:CFSSPA\left({CF}_{1}^{\star\:},\:{CF}_{2}^{\star\:},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}^{\star\:}\right)$$

• 3) (Boundedness): Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. Now if $\:{{CF}_{\iota\:}}^{-}=\left(\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\phi\:}_{\iota\:}\right\}\right)\:$and $\:{{CF}_{\iota\:}}^{+}=\left(\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\phi\:}_{\iota\:}\right\}\right)$, then

  $$\:{{CF}_{\iota\:}}^{-}\le\:CFSSPA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:{{CF}_{\iota\:}}^{+}$$

Complex fuzzy Schweizer-Sklar power weighted average aggregation operators

In this subsection, we will define the notion of complex fuzzy Schweizer-Sklar power weighted average (CFSSPWA) aggregation operators. Moreover, the characteristics analysis of the proposed theory has been developed to show the importance of these developed approaches.

Definition 7

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ as a family of CFNs. Then the CFSSPWA operator is as follows;

$$\:CFSSPWA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}}\right)=\begin{array}{c}\mathcal{j}\mathcal{\:}\\\:\oplus\:\\\:\iota\:=1\end{array}\frac{{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\sum\:_{\mathfrak{x}=1}^{\mathfrak{h}}{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{CF}_{\iota\:}\:$$

Where $\:\mathcal{T}\left({CF}_{\iota\:}\right)=\sum\:_{\begin{array}{c}\varrho\:=1\\\:\varrho\:\ne\:\iota\:\end{array}}^{\mathcal{j}}Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)\:$and $\:Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)$ represent the support between $\:{CF}_{\iota\:}$ and $\:{CF}_{\varrho\:}$ $\:\mathcal{V}=\left({\mathcal{V}}_{1},\:{\mathcal{V}}_{2},\:\dots\:,\:{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\right)$ is a WV with $\:\forall\:\mathcal{\:}\mathcal{j}\:0\le\:{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\le\:1$ and$\:\:\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}=1$.

Theorem 4

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. Then the outcome obtained by using CFSSPWA operators is again CFN.

$$\:CFSSPWA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}}\right)=\left(\begin{array}{c}1-{\left(\sum\:_{\mathfrak{x}=1}^{\mathfrak{h}\mathfrak{\:}}\frac{{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\sum\:_{\mathfrak{x}=1}^{\mathfrak{h}}{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\left(1-{\mathfrak{A}}_{\iota\:}\right)}^{h}\right)}^{\frac{1}{h}}+\iota\:\\\:1-{\left(\sum\:_{\mathfrak{x}=1}^{\mathfrak{h}\mathfrak{\:}}\frac{{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\sum\:_{\mathfrak{x}=1}^{\mathfrak{h}}{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\left(1+\mathcal{T}\left({CF}_{CF\iota\:}\right)\right)}{\left(1-{\phi\:}_{\iota\:}\right)}^{h}\right)}^{\frac{1}{h}}\end{array}\right)$$

Theorem 5

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. The following properties hold.

• 1) (Idempotency): Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. Then if $\:{CF}_{\iota\:}=CF=\left(\mathfrak{A}+\iota\:\phi\:\right)\:\forall\:\:\iota\:$, we have

  $$\:CFSSPWA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=CF$$

• 2) (Monotonicity): Let$\:\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)$, and $\:{CF}_{\iota\:}^{*}=\left({\mathfrak{A}}_{\iota\:}^{*}+\iota\:{\phi\:}_{\iota\:}^{*}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be two collections of CFNs. Then if $\:{\mathfrak{A}}_{\iota\:}\le\:{\mathfrak{A}}_{\iota\:}^{*}$and $\:{\phi\:}_{\iota\:}\le\:{\phi\:}_{\iota\:}^{*},$ we get

  $$\:CFSSPWA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:CFSSPWA\left({CF}_{1}^{\star\:},\:{CF}_{2}^{\star\:},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}^{\star\:}\right)$$

• 3) (Boundedness): Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be a collection of CFNs. Now if $\:{{CF}_{\iota\:}}^{-}=\left(\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\phi\:}_{\iota\:}\right\}\right)\:$and $\:{{CF}_{\iota\:}}^{+}=\left(\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\phi\:}_{\iota\:}\right\}\right)$, then

  $$\:{{CF}_{\iota\:}}^{-}\le\:CFSSPWA\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:{{CF}_{\iota\:}}^{+}.$$

Cartesian form of complex fuzzy Schweizer-Sklar power geometric aggregation operators

In this subsection, we will elaborate on the fundamental idea of CFSSPG AOs. Also, we will discuss the basic properties of these newly developed AOs.

Definition 8

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)=,\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. Then the CFSSPG AOs are given by

$$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\begin{array}{c}\mathcal{j}\mathcal{\:}\\\:\otimes\:\\\:\iota\:=1\end{array}{\left({CF}_{\iota\:}\right)}^{\frac{\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}}\:\:$$

.

Where $\:\mathcal{T}\left({CF}_{\iota\:}\right)=\sum\:_{\begin{array}{c}\varrho\:=1\\\:\varrho\:\ne\:\iota\:\end{array}}^{\mathcal{j}}Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)\:$and $\:Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)$ represent the support between $\:{CF}_{\iota\:}$ and $\:{CF}_{\varrho\:}$ with below axioms

• 1)
  $$\:Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)\in\:\left[0,\:1\right]$$
• 2)
  $$\:Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)=Sup\left({CF}_{\varrho\:},\:{CF}_{\iota\:}\right)$$

• 3) $\:Sup\left({CF}_{\iota\:}\:,{CF}_{\varrho\:}\right)\ge\:Sup\left({CF}_{{\iota\:}^{\star\:}},\:{CF}_{{\varrho\:}^{\star\:}}\right)\:$if $\:\mathcal{Q}\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)<\mathcal{Q}\left({CF}_{{\iota\:}^{\star\:}},\:{CF}_{{\varrho\:}^{\star\:}}\right)$ and $\:\mathcal{Q}$ is the distance measure.

Theorem 6

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)=,\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. Then the outcome obtained by using the CFSSPG operator is again CFN

$$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)==\left({\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}+\iota\:{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}\right)\:\:$$

(3)

Where $\:{\mathcal{w}}_{\iota\:}\:\left(\iota\:=1,\:2,\:3,\:\dots\:,\:\mathcal{j}\right)$ is a family of integrated weights$\:\:{\mathcal{w}}_{\iota\:}=\frac{\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}$.

Proof

For any value of $\:\mathcal{w}={\left({\mathcal{w}}_{1},\:{\mathcal{w}}_{2},\:\dots\:,\:{\mathcal{w}}_{\mathcal{j}}\right)}^{T}$, the eq. $\:\left(3\right)$ will degenerate into the form

$$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)\:\:$$

(4)

We will prove the result in Eq. (4) by using mathematical induction.

Now when we use$\:\mathcal{\:}\mathcal{j}=2$, then

$$\:{\left({CF}_{1}\right)}^{{\mathcal{w}}_{1}}=\left(\begin{array}{c}{\left({\mathcal{w}}_{1}{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{1}{\left({\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

And

$$\:{\left({CF}_{2}\right)}^{{\mathcal{w}}_{2}}=\left({\left({\mathcal{w}}_{2}{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:{\left({\mathcal{w}}_{2}{\left({\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)$$

Then,

$$\:{\left({CF}_{1}\right)}^{{\mathcal{w}}_{1}}{\otimes\:}_{\mathcal{S}\mathcal{S}}{\left({CF}_{2}\right)}^{{\mathcal{w}}_{2}}=\left(\begin{array}{c}{\left({\mathcal{w}}_{1}{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{1}{\left({\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:\otimes\:\left(\begin{array}{c}{\left({\mathcal{w}}_{2}{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{2}{\left({\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}{\left({\left({\left({\mathcal{w}}_{1}{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)}^{\varrho\:}+{\left({\left({\mathcal{w}}_{2}{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\left({\left({\mathcal{w}}_{1}{\left({\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)}^{\varrho\:}+{\left({\left({\mathcal{w}}_{2}{\left({\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)\right)}^{\frac{1}{\varrho\:}}\right)}^{\varrho\:}-1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}{\left({\mathcal{w}}_{1}{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)+{\mathcal{w}}_{2}{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)-1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{1}{\left({\phi\:}_{1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{1}-1\right)+{\mathcal{w}}_{2}{\left({\phi\:}_{2}\right)}^{\varrho\:}-\left({\mathcal{w}}_{2}-1\right)-1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}{\left({\mathcal{w}}_{1}{\left({\mathfrak{A}}_{1}\right)}^{\varrho\:}+{\mathcal{w}}_{2}{\left({\mathfrak{A}}_{2}\right)}^{\varrho\:}-{\mathcal{w}}_{1}-{\mathcal{w}}_{2}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{1}{\left({\phi\:}_{1}\right)}^{\varrho\:}+{\mathcal{w}}_{2}{\left({\phi\:}_{2}\right)}^{\varrho\:}-{\mathcal{w}}_{1}-{\mathcal{w}}_{2}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{2\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

So, eq.4 is valid when $\:\mathcal{j}=2.$

Now assume that Eq. (4) is true for$\:\mathcal{\:}\mathcal{j}=\mathcal{K}$, then$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}}\right)=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$

Now for$\:\mathcal{\:}\mathcal{j}=\mathcal{K}+1$, then

$$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}},\:{CF}_{\mathcal{K}+1\:}\right)=CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}}\right)\otimes\:{CF}_{\mathcal{K}+1\:}$$

$$\:=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}\mathcal{\:}}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:\otimes\:\left(\begin{array}{c}{\left({\mathcal{w}}_{\mathcal{K}+1}{\left({\mathfrak{A}}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left({\mathcal{w}}_{\mathcal{K}+1}{\left({\phi\:}_{\mathcal{K}+1}\right)}^{\varrho\:}-\left({\mathcal{w}}_{\mathcal{K}+1}-1\right)\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{\mathcal{K}+1}{\mathcal{w}}_{\iota\:}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}+1\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{\mathcal{K}+1}{\mathcal{w}}_{\iota\:}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}-\sum\:_{\iota\:=1}^{\mathcal{K}+1\:}{\mathcal{w}}_{\iota\:}+1\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

$$\:=CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{K}\mathcal{\:}},\:{CF}_{\mathcal{K}+1\:}\right)$$

Hence Eq. (4) is valid when $\:\mathcal{j}=\mathcal{K}+1.$ So, eq. $\:\left(4\right)$ is valid$\:\mathcal{\:}\forall\:\mathcal{\:}\mathcal{j}$

Theorem 7

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right)=,\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. The following properties hold true

• 1) (Idempotency): If $\:{CF}_{\iota\:}=CF=\left(\mathfrak{A}+\iota\:\phi\:\right)\:\forall\:\:\iota\:$, we have

  $$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=CF$$

• 2) (Monotonicity): If $\:{\mathfrak{A}}_{\iota\:}\le\:{\mathfrak{A}}_{\iota\:}^{*}$and $\:{\phi\:}_{\iota\:}\le\:{\phi\:}_{\iota\:}^{*},$ we get

  $$\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:CFSSPG\left({CF}_{1}^{\star\:},\:{CF}_{2}^{\star\:},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}^{\star\:}\right)$$

• 3) (Boundedness): If $\:{{CF}_{\iota\:}}^{-}=\left(\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\phi\:}_{\iota\:}\right\}\right)\:$and $\:{{CF}_{\iota\:}}^{+}=\left(\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\phi\:}_{\iota\:}\right\}\right)$, then

  $$\:{{CF}_{\iota\:}}^{-}\le\:CFSSPG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:{{CF}_{\iota\:}}^{+}.$$

Cartesian form of complex fuzzy Schweizer-Sklar power weighted geometric aggregation operators

In this sub-section, we will discuss the idea of the Cartesian form of CFSSPWG AO. Moreover, we will discuss the properties of these AOs.

Definition 9

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. Then the CFSSPWG operator is as follows.

$$\:CFSSPWG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\begin{array}{c}\mathcal{j}\mathcal{\:}\\\:\otimes\:\\\:\iota\:=1\end{array}{\left({CF}_{\iota\:}\right)}^{\frac{{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}}\:\:$$

Where $\:\mathcal{T}\left({CF}_{\iota\:}\right)=\sum\:_{\begin{array}{c}\varrho\:=1\\\:\varrho\:\ne\:\iota\:\end{array}}^{\mathcal{j}}Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)\:$and $\:Sup\left({CF}_{\iota\:},\:{CF}_{\varrho\:}\right)$ represent the support between $\:{CF}_{\iota\:}$ and $\:{CF}_{\varrho\:}$ and $\:\mathcal{V}=\left({\mathcal{V}}_{1},\:{\mathcal{V}}_{2},\:\dots\:,\:{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\right)$ is a WV $\:\forall\:\mathcal{\:}\mathcal{j}\:0\le\:{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}\le\:1$ and$\:\:\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\mathcal{j}\mathcal{\:}}=1$.

Theorem 8

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. Then the outcome obtained by CFSSPWG AOs is again CFN

$$\:CFSSPWG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=\left(\begin{array}{c}{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}\frac{{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\left({\mathfrak{A}}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}+\iota\:\\\:{\left(\sum\:_{\iota\:=1}^{\mathcal{j}\mathcal{\:}}\frac{{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\left({\phi\:}_{\iota\:}\right)}^{\varrho\:}\right)}^{\frac{1}{\varrho\:}}\end{array}\right)$$

.

Theorem 9

Let $\:{CF}_{\iota\:}=\left({\mathfrak{A}}_{\iota\:}+\iota\:{\phi\:}_{\iota\:}\right),\:\iota\:=1,\:2,\:..,\:\mathcal{j}$ be the collection of CFNs. The following properties hold for CFSSPWG AOs

• 1) (Idempotency): If $\:{CF}_{\iota\:}=CF=\left(\mathfrak{A}+\iota\:\phi\:\right)\:\forall\:\:\iota\:$, we have

  $$\:CFSSPWG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)=CF$$

• 2) (Monotonicity): If $\:{\mathfrak{A}}_{\iota\:}\le\:{\mathfrak{A}}_{\iota\:}^{*}$and $\:{\phi\:}_{\iota\:}\le\:{\phi\:}_{\iota\:}^{*},$ we get

  $$\:CFSSPWG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:CFSSPWG\left({CF}_{1}^{\star\:},\:{CF}_{2}^{\star\:},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}^{\star\:}\right)$$

• 3) (Boundedness): If $\:{{CF}_{\iota\:}}^{-}=\left(\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{i}\text{n}}\left\{{\phi\:}_{\iota\:}\right\}\right)\:$and $\:{{CF}_{\iota\:}}^{+}=\left(\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\mathfrak{A}}_{\iota\:}\right\}+\iota\:\underset{\iota\:}{\text{m}\text{a}\text{x}}\left\{{\phi\:}_{\iota\:}\right\}\right)$, then

  $$\:{{CF}_{\iota\:}}^{-}\le\:CFSSPWG\left({CF}_{1},\:{CF}_{2},\:\dots\:,\:{CF}_{\mathcal{j}\mathcal{\:}}\right)\le\:{{CF}_{\iota\:}}^{+}.$$

CF-MCDM and application

This section of the article is devoted to applying the introduced notion. For this purpose, we have developed an algorithm and utilized the notion of the Cartesian form of CFSSPWA and CFSSPWG AOs to discuss the applicability of the introduced notions. We have provided an illustrative example for this purpose the support the defined algorithm.

Assume that $\:{\mathcal{A}}_{1}$, $\:{\mathcal{A}}_{2}$, …, $\:{\mathcal{A}}_{\mathcal{j}\mathcal{\:}}\:$represent $\:\mathcal{j}$alternatives and $\:{atr.}_{1}$, $\:{atr.}_{2}$, …, $\:{atr.}_{\mathcal{p}}$ denote $\:\mathcal{p}$ attributes with WV denoted by $\:\mathcal{V}=\left({\mathcal{V}}_{1},\:{\mathcal{V}}_{2},\:\dots\:,\:{\mathcal{V}}_{\mathcal{p}}\right)$ by using the condition $\:\sum\:_{\varrho\:=1}^{\mathcal{p}}{\mathcal{V}}_{\mathcal{p}\mathcal{\:}}=1$. Then $\:M={\left({CF}_{\iota\:\varrho\:}\right)}_{\mathcal{j}\times\:\mathcal{p}}$ CF decision matrix can be obtained where each element would be in the form of CFN that is$\:\:{CF}_{\iota\:\varrho\:}=\left({\mathfrak{A}}_{\iota\:\varrho\:}+\iota\:{\phi\:}_{\iota\:\varrho\:}\right)$, where $\:{\mathfrak{A}}_{\iota\:\varrho\:}\:and\:{\phi\:}_{\iota\:\varrho\:}\in\:\left[0,\:1\right].$

Now we follow the steps-wise algorithm as follows;

Step-1

Collect the information in Cartesian form of CFNs given in the following matrix.

$$M = \left[ {\begin{array}{*{20}c} {{\mathfrak{A}}_{{11}} + \iota \varphi _{{11}} } & {{\mathfrak{A}}_{{12}} + \iota \varphi _{{12}} } & {{\mathfrak{A}}_{{2\partial }} + \iota \varphi _{{1\partial }} } \\ {{\mathfrak{A}}_{{21}} + \iota \varphi _{{21}} } & {{\mathfrak{A}}_{{22}} + \iota \varphi _{{22}} } & {{\mathfrak{A}}_{{2\partial }} + \iota \varphi _{{2\partial }} } \\ {\begin{array}{*{20}c} \ddots \\ {{\mathfrak{A}}_{{\iota 1}} + \iota \varphi _{{\iota 1}} } \\ \end{array} } & {\begin{array}{*{20}c} \ddots \\ {{\mathfrak{A}}_{{\iota 2}} + \iota \varphi _{{\iota 2}} } \\ \end{array} } & {\begin{array}{*{20}c} \ddots \\ {{\mathfrak{A}}_{{\iota \partial }} + \iota \varphi _{{\iota \partial }} } \\ \end{array} } \\ \end{array} } \right]$$

Step 1

To normalize the information given in Step 1, we will use the following formula.

$$\:N=\left\{\begin{array}{c}\left({\mathfrak{A}}_{\iota\:\varrho\:}+\iota\:{\phi\:}_{\iota\:\varrho\:}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:benefit\:type\\\:\left(1-{\mathfrak{A}}_{\iota\:\varrho\:})+\iota\:{(1-\phi\:}_{\iota\:\varrho\:})\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:cost\:type\:\:\:\:\:\:\:\end{array}\right.$$

Step 2

In this step, we can calculate the support by below eq.⁵

$$\:Sup\left({CF}_{\iota\:\varrho\:},{CF}_{\iota\:\varrho\:}\:\right)=1-\mathcal{Q}\left({CF}_{\iota\:\varrho\:},\:{CF}_{\iota\:\mathfrak{z}}\right)\:$$

(5)

$$\:\text{W}\text{h}\text{e}\text{r}\text{e}\:\iota\:=1,\:2,\:\dots\:.,\:\mathcal{j},\:\varrho\:,\:\mathfrak{z}=1,\:2,\:\dots\:,\:\mathcal{p}\:\text{a}\text{n}\text{d}\:\varrho\:\ne\:\mathfrak{z}.$$

Step 3

Find out $\:\mathcal{T}\left({CF}_{\iota\:\varrho\:}\right)$ by below Eq. (6)

$$\:\mathcal{T}\left({CF}_{\iota\:\varrho\:}\right)=\sum\:_{\begin{array}{c}z=1\\\:z\ne\:\varrho\:\end{array}}^{\mathcal{j}}Sup\left({CF}_{\iota\:\varrho\:},\:{CF}_{\iota\:\mathfrak{z}}\right)\:$$

(6)

$$\:\text{W}\text{h}\text{e}\text{r}\text{e}\:\iota\:=1,\:2,\:\dots\:.,\:\mathcal{j},\:\varrho\:,\:\mathfrak{z}=1,\:2,\:\dots\:,\:\mathcal{p}\:\text{a}\text{n}\text{d}\:\varrho\:\ne\:\mathfrak{z}.$$

Step 4

Find out the associated WV$\:\:{\mathcal{w}}_{\iota\:\varrho\:}$ of power operator for every CFN by employing the below Eq. (7)

$$\:{\mathcal{w}}_{\iota\:\varrho\:}=\frac{\left(1+\mathcal{T}\left({CF}_{\iota\:\varrho\:}\right)\right)}{\sum\:_{\mathfrak{z}=1}^{\mathcal{j}}\left(1+\mathcal{T}\left({CF}_{\iota\:\mathfrak{z}}\right)\right)}\:\:\:\:\:\:\:\text{o}\text{r}\:\frac{{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}{\sum\:_{\iota\:=1}^{\mathcal{j}}{\mathcal{V}}_{\iota\:}\left(1+\mathcal{T}\left({CF}_{\iota\:}\right)\right)}$$

(7)

Stage 5

Now utilize the idea of CFSSPA, CFSSPG, CFSSPWA and CFSSPWG operators to find out the aggregated results.

Step 6

Find out score values by using def. (5) and then get the ranking result to choose the best alternative.

Case study

XYZ Organization will implement a variety of data privacy techniques for its cloud-based infrastructure; such as data and business information that has sensitive elements from its valued customers. In the process of choosing the best technique, the organization will compare these techniques with the criteria below. The organization suppose four different alternatives of data privacy techniques. The alternatives are given in Table 1.

Table 1 Alternative table and their explanations.

Based on these alternatives the attributes are given in Table 2.

Table 2 Description of attributes.

Assume that WVs for these attributes are $\:\left(0.25,\:0.23,\:0.19,\:0.33\right).$ The experts provide their assessment in the form of CFN as given in Table 3.

Table 3 Cartesian form of CF information.

Now we apply the above algorithm to get the required result.

Step 1

No need to normalize the given data because all attributes are of benefit types.

Step 2

Utilize Eq. (5) to get support values as.

For $\:\iota\:=1$

$$\:Sup\left(11,\:12\right)=0.83,\:Sup\left(11,\:13\right)=0.765,\:Sup\left(11,\:14\right)=0.86,$$

$$\:Sup\left(12,\:11\right)=0.83,\:Sup\left(12,\:13\right)=0.765,\:Sup\left(12,\:14\right)=0.905,$$

$$\:Sup\left(13,\:11\right)=0.935,\:Sup\left(13,\:12\right)=0.765,\:Sup\left(13,\:14\right)=0.86,$$

$$\:Sup\left(14,\:11\right)=0.925,\:Sup\left(14,\:12\right)=0.905,\:Sup\left(14,\:13\right)=0.86,$$

For $\:\iota\:=2$

$$\:Sup\left(21,\:22\right)=0.95,\:Sup\left(21,\:23\right)=0.76,\:Sup\left(21,\:24\right)=0.87,$$

$$\:Sup\left(22,\:21\right)=0.95,\:Sup\left(22,\:23\right)=0.76,\:Sup\left(22,\:24\right)=0.89,$$

$$\:Sup\left(23,\:21\right)=0.71,\:Sup\left(23,\:22\right)=0.76,\:Sup\left(23,\:24\right)=0.87,$$

$$\:Sup\left(24,\:21\right)=0.84,\:Sup\left(24,\:22\right)=0.89,\:Sup\left(24,\:23\right)=0.87,$$

For $\:\iota\:=3$

$$\:Sup\left(31,\:32\right)=0.88,\:Sup\left(31,\:33\right)=0.76,\:Sup\left(31,\:34\right)=0.76,$$

$$\:Sup\left(32,\:31\right)=0.88,\:Sup\left(32,\:33\right)=0.76,\:Sup\left(32,\:34\right)=0.99,$$

$$\:Sup\left(33,\:31\right)=0.80,\:Sup\left(33,\:32\right)=0.76,\:Sup\left(33,\:34\right)=0.76,$$

$$\:Sup\left(34,\:31\right)=0.87,\:Sup\left(34,\:32\right)=0.99,\:Sup\left(34,\:33\right)=0.76,$$

For $\:\iota\:=4$

$$\:Sup\left(41,\:42\right)=0.98,\:Sup\left(41,\:43\right)=0.85,\:Sup\left(41,\:44\right)=0.80,$$

$$\:Sup\left(42,\:41\right)=0.98,\:Sup\left(42,\:43\right)=0.85,\:Sup\left(42,\:44\right)=0.95,$$

$$\:Sup\left(43,\:41\right)=0.83,\:Sup\left(43,\:42\right)=0.85,\:Sup\left(43,\:44\right)=0.80,$$

$$\:Sup\left(44,\:41\right)=0.93,\:Sup\left(44,\:42\right)=0.95,\:Sup\left(44,\:43\right)=0.80,$$

Here we utilized the formula

$$\:Sup\left({CF}_{\iota\:\varrho\:},{CF}_{\iota\:\mathfrak{z}}\:\right)=1-\mathcal{Q}\left({CF}_{\iota\:\varrho\:},\:{CF}_{\iota\:\mathfrak{z}}\right)=1-\left(\frac{1}{2}\times\:\left(\left|{\mathfrak{A}}_{\iota\:\varrho\:}-{\mathfrak{A}}_{\iota\:\mathfrak{z}}\right|+\left|{\phi\:}_{\iota\:\varrho\:}-{\phi\:}_{\iota\:\mathfrak{z}}\right|\right)\right)$$

.

Step 3

Through Eq. (6) we have.

$$\:\mathcal{T}\left({CF}_{11}\right)=2.455,\:\mathcal{T}\left({CF}_{12}\right)=2.5,\:\mathcal{T}\left({CF}_{13}\right)=2.56,\:\mathcal{T}\left({CF}_{14}\right)=2.69,$$

$$\:\mathcal{T}\left({CF}_{21}\right)=2.58,\:\mathcal{T}\left({CF}_{22}\right)=2.6,\:\mathcal{T}\left({CF}_{23}\right)=2.34,\:\mathcal{T}\left({CF}_{24}\right)=2.6,$$

$$\:\mathcal{T}\left({CF}_{31}\right)=2.4,\:\mathcal{T}\left({CF}_{32}\right)=2.63,\:\mathcal{T}\left({CF}_{33}\right)=2.32,\:\mathcal{T}\left({CF}_{34}\right)=2.62,$$

$$\:\mathcal{T}\left({CF}_{41}\right)=2.63,\:\mathcal{T}\left({CF}_{42}\right)=2.78,\:\mathcal{T}\left({CF}_{43}\right)=2.48,\:\mathcal{T}\left({CF}_{44}\right)=2.68,$$

Step 4

By Eq. (7) we have the associated weight vector$\:{\mathcal{w}}_{\iota\:\varrho\:}$

$$\:{\mathcal{w}}_{11}=0.243,\:{\mathcal{w}}_{12}=0.246,\:{\mathcal{w}}_{13}=0.251,\:{\mathcal{w}}_{14}=0.26,$$

$$\:{\mathcal{w}}_{21}=0.254,\:{\mathcal{w}}_{22}=0.255,\:{\mathcal{w}}_{23}=0.237,\:{\mathcal{w}}_{24}=0.255,$$

$$\:{\mathcal{w}}_{31}=0.243,\:{\mathcal{w}}_{32}=0.26,\:{\mathcal{w}}_{33}=0.238,\:{\mathcal{w}}_{34}=0.259,$$

$$\:{\mathcal{w}}_{41}=0.249,\:{\mathcal{w}}_{42}=0.259,\:{\mathcal{w}}_{43}=0.239,\:{\mathcal{w}}_{44}=0.253,$$

Assuming the weight vector $\:\left(0.25,\:0.23,\:0.19,\:0.33\right)$ and utilized Eq. (7) we have

$$\:{\mathcal{w}}_{11}^{\mathcal{w}}=0.242,\:{\mathcal{w}}_{12}^{\mathcal{w}}=0.226,\:{\mathcal{w}}_{13}^{\mathcal{w}}=0.19,\:{\mathcal{w}}_{14}^{\mathcal{w}}=0.342,$$

$$\:{\mathcal{w}}_{21}^{\mathcal{w}}=0.252,\:{\mathcal{w}}_{22}^{\mathcal{w}}=0.234,\:{\mathcal{w}}_{23}^{\mathcal{w}}=0.179,\:{\mathcal{w}}_{24}^{\mathcal{w}}=0.335,$$

$$\:{\mathcal{w}}_{31}^{\mathcal{w}}=0.242,\:{\mathcal{w}}_{32}^{\mathcal{w}}=0.238,\:{\mathcal{w}}_{33}^{\mathcal{w}}=0.18,\:{\mathcal{w}}_{34}^{\mathcal{w}}=0.34,$$

$$\:{\mathcal{w}}_{41}^{\mathcal{w}}=0.248,\:{\mathcal{w}}_{42}^{\mathcal{w}}=0.238,\:{\mathcal{w}}_{43}^{\mathcal{w}}=0.181,\:{\mathcal{w}}_{44}^{\mathcal{w}}=0.332,$$

.

Stage 5

The aggregated outcomes after utilizing deduced CFSSPA, CFSSPWA, CFSSPG, and CFSSPWG operators are discovered in Table 4.

Table 4 The outcomes by using CFSSPA, CFSSPWA, CFSSPG, and CFSSPWG operators.

Step 6

The required score values by using def. (5) are exhibited in Table 5.

Table 5 Score values of the alternatives.

Step 7

In Table 6, we have explored the ranking results.

Table 6 The ranking results.

Hence based on the result given in Table 6, we can conclude that $\:{CF}_{4}$ is the best result in all cases.

The graphical behavior of results is given in Figs. 6 and 7.

Fig. 6

Ranking data privacy techniques in cloud computing based on CFSSPWA operators.

Fig. 7

Ranking data privacy techniques in cloud computing based on CFSSPWG operators.

Sensitivity analysis

To strengthen the methodology of the delivered approach, in this section, we have proposed the sensitivity analysis of the introduced work by noticing the effects of different values of parameter $\varrho$ We have utilized different values of the parameter and observed the effects of the parameter on ranking results. The overall results are given in Tabe 7.

Table 7 Result of sensitivity analysis.

We can observe that in the case of CFSSPWA AOs when the values of parameter $\varrho \text{=-1,\:-2,\:-3,\:-4}$ then ranking results are $\:{\text{C}\text{F}}_{4}>{\text{C}\text{F}}_{1}>{\text{C}\text{F}}_{3}>{\text{C}\text{F}}_{2}$ and the best alternative in this case is $\:{\text{C}\text{F}}_{4}$ for all cases, however when the values of the parameter are $\varrho \text{=-5,\:-6,\:-7,\:-8}$ then ranking results $\:{\text{C}\text{F}}_{4}>{\text{C}\text{F}}_{3}>{\text{C}\text{F}}_{1}>{\text{C}\text{F}}_{2}\:$are slightly different from previous ranking results however the best alternative is $\:{\text{C}\text{F}}_{4}.$ Also notice that when parameter values are $\varrho\text{=-9,\:-10}$ the ranking results $\:{\text{C}\text{F}}_{4}>{\text{C}\text{F}}_{3}>{\text{C}\text{F}}_{2}>{\text{C}\text{F}}_{1}$ again differ from previous results however the best alternative is $\:{\text{C}\text{F}}_{4}.$ It means that the value of the parameter affects the ranking result however the best alternative remains the same in all cases. We can observe that in the case of CFSSPWG AOs, the values of the parameter do not affect the ranking results that show the stability of the proposed AOs.

Comparative analysis

This section of the article is devoted to discussing the comparative analysis of the prosed theory with some existing notions to discuss the superiority and advantages of the initiated work. Here we will compare our work with Ramot et al.²⁷ technique, Hu et al.²⁹ approach, Merigo and Casanovas³⁰ method Xia et al.³¹ approach, and Mardani et al.³² method. The overall discussion is given by.

• a) We can see in the case of Ramot et al.²⁷ approach that the idea is based on the polar form of the complex fuzzy set which uses the unit circle instead of the unit square which shows the limitation of Ramot et al.²⁷ theory. For example, when we see that if the decision makers provide theory information in the Cartesian form of a complex fuzzy set that is $\:\left(0.9+\iota\:0.8\right)$ then in this case observe that the basic condition on $\:"r"$ in the polar form of complex number fails to hold and so the weakness of the structure is obvious in this case. The defined notion is based on the Cartesian form of a complex fuzzy set and hence is more reliable than the existing theory of Ramot et al.²⁷.

• b) Case of Hu et al.²⁹ observe that this approach is based on the polar form of complex fuzzy numbers again and the polar form of complex fuzzy numbers is a limited approach because of the uses of the unit circle and its membership structure defined by $\:r{e}^{2\pi\:\theta\:}$ where $\:r,\:\theta\:\in\:\left[0,\:1\right].$ But in the case of data given in Cartesian form $\:0.7+\iota\:0.8$ we can see that we find out the value of $\:r=\sqrt{{0.7}^{2}+{0.8}^{2}}=1.34$ and in this case we can see that $\:r=1.34\notin\:\left[0,\:1\right].\:$So the condition for $\:"r"$ in the polar form of complex fuzzy number is violated and hence the limitation of the Hu et al.²⁹ approach is obvious. On the other hand, the developed approach can cope with this drawback and provide more space for decision-makers to take the information in a more generalized form, and in this case, the developed approach provides the opportunity to reduce the data loss option. Hence the initiated work is dominant to Hu et al.²⁹ approach.

• c) In the case of Merigo and Casanova’s³⁰ approach, it is based on pure fuzzy structure and it is free to discuss the second dimension. Hence Merigo and Casanovas³⁰ approach can never discuss the data whether it is in polar form or Cartesian form of a complex fuzzy set. While the developed approach is dominant to the Polar form of a complex fuzzy set and also dominant to Merigo and Casanovas³⁰ approach to discuss the second dimension.

• d) Xia et al.³¹ produced some hesitant fuzzy aggregation operators and discussed their applications in decision-making. However, this structure can never discuss the complex fuzzy data. The data discussed in this article is in a Cartesian form of a complex fuzzy set with property to discuss the second dimension. Although hesitant fuzzy structure is dominant to fuzzy set theory it can never discuss the complex fuzzy information that shows the limitation of the proposed theory.

• e) In the case of Mardani et al.³² method, they have produced fuzzy aggregation operators. Although this theory has its significance, in some decision-making situations, we have to utilize more advanced information to avoid any kind of data loss. The developed approach provides this opportunity for decision-makers to avoid this kind of hesitation and experts can take theory information in the Cartesian form of complex fuzzy numbers to solve the decision-making problems and to make decision-making more dominant and reliable. The overall results are given in Table 8.

Table 8 Comparative analysis and result.

Conclusion

Data privacy technique is one of the fundaments in cloud computing for protecting sensitive information from unauthorized access data breaches and non-compliance with pertinent regulations. Each has its advantages for which organizations can protect the data and match respective privacy as well as operational requirements. Advanced methods such as homomorphic encryption and differential privacy are advancing beyond the protection of data security that allows computations on sensitive data without having a breach on the privacy of such data. As cloud adoption continues to increase, the choice and deployment of a robust combination of these technologies will prove critical in forming a secure and compliant cloud environment to instill user confidence and satisfaction among stakeholders. Data utility alongside privacy will enable cloud computing to remain a viable and safe choice for organizations around the world. To discuss this application in the environment of complex fuzzy structure, we have developed first of all operational laws based on Schweizer-Sklar t-norm and t-conorm. Then based on these fundamental laws we have proposed a Cartesian form of CFSS power average and CFSS power geometric AOs. We have developed the main properties related to these notions like Idempotency, Boundedness, and monotonicity. To discuss the applicability of the proposed theory we have explored an algorithm along with an illustrative example to show that we can use the initiated work in cloud computing. Moreover, the comparative analysis of the developed theory shows the reliability and superiority of the introduced work.

Advantage of the proposed study

• The fuzzy environment establishes an efficient system for handling ambiguous inconsistent and imprecise details suitable for cloud data privacy assessment.

• The Schweizer–Sklar aggregation operators let decision-makers set flexible compensation methods between criteria to boost decision stability.

• The use of CFSs within this theory functions to process both attribute magnitude together with directional change which boosts the modeling accuracy of uncertain systems.

• The evaluation method generates better data privacy technique rankings through its effective multi-dimensional data combination process which minimizes assessment ties and score equivalencies.

• The framework incorporates several privacy-oriented attributes including security level performance and efficiency along with scalability and compliance measures as part of its complete rating system.

• Applicable across different types of cloud architectures (public, private, hybrid), making it versatile for real-world deployment.

• Through its Tamir extension, the model shows flexibility to modify its aggregation features for adjusting to evolving cloud privacy security threats.

• This alternative exhibits better precision in addition to higher dependability than traditional fuzzy systems intuitionistic fuzzy systems and real-valued MCDM systems.