Prioritizing disability support systems by using Tamir’s complex fuzzy Dombi aggregation operators

Abstract

Fuzzy sets can model the inherent ambiguity and subjectivity in disability assessment by allowing for flexible classification and decision-making. This contributes to the development of flexible clinical support systems that are effective in meeting individual needs. Subjective assessments of an individual’s needs and talents are common in disability. Conventional clear-cut methodologies classify people as “disabled” or "not disabled," which may not fully reflect the wide range of disabilities. Fuzzy sets allow for degrees of membership, such as (a) Person A has 70% mobility impairment. (b) Person B needs 50% personal care support. This method takes into account the subjectivity and variability of disability assessments. Designing adaptive systems for people with disabilities requires the use of fuzzy sets for clinical support. For this purpose, we have proposed the theory of aggregation operators based on a complex fuzzy set. Aggregation operators help us to convert overall information to a single value that can help our clinical support system. Moreover, for the application of the delivered approach, we have proposed an algorithm and utilized this approach for the selection of the best disability support system. We have provided a comparative analysis of the defined theory to discuss the advantages of the initiated work.

Introduction

Disability support systems (DSS)

The integrated clinical support system comes in support systems with a human disability system that is all-around technology enhancing the quality of life for people living with disabilities. It offers the much-needed support that improves upon the quality of life using both clinical support and personal care that is through technological tools that are integrated and incorporate the use of data-driven insights. This enables interventions, monitoring the status of their health, and communicating with the patients, caregivers, and professionals in health care by applying clinical data. It supports autonomy in technology-assisted care by way of ambulation aids, communication aid devices, and rehabilitation instruments. The system also satisfies psychosocial requirements concerning the impairment including counseling, skill building, and community-based support. Predictive diagnostics is possible and might enable early intervention and a patient-specific plan to prevent hospitalization and, thereby, the outcome. This all-inclusive approach fills the gap that exists between health providers and disabled persons while promoting accessibility, inclusiveness, and sustainable well-being.

Key features of disability support systems

The key features of disability support systems are given in Table 1.

Table 1 Key features of disability support systems.

Why disability support system is the need of the day

Because it allows persons with disabilities to live with equality, independence, and dignity, the disability assistance system is crucial in today’s society. These systems aid in bridging the gap between capabilities and barriers by offering specialized resources and assistive technologies. They enable access to healthcare, education, and employment, fostering self-reliance and reducing societal dependence. In a society that strives for inclusivity, these support networks promote equity by combating discrimination and removing physical and social barriers. They also promote a culture of mutual respect and understanding and inspire innovative, flexible solutions. By encouraging a range of contributions and helping individuals with disabilities reach their full potential, these strategies improve society.

Problem statement

Disability support systems continue to face significant challenges in providing tailored care and support to individuals with varying needs. Traditional approaches to categorizing and assessing disability can occasionally ignore the underlying ambiguity and complexity of individual requirements, which may differ in scope and character across a diverse population. When designing disability support programs, binary classifications or rigid categories are frequently employed, which may not fully take into consideration the nuances of an individual’s condition.

Fuzzy set theory offers a potential solution to this issue by enabling a more flexible and detailed description of individual desires. It enables the development of membership features that take into consideration varying degrees of impairment, which enhances decision-making on resource allocation, service customization, and intervention design. However, the application of fuzzy set theory to real disability assistance systems is limited by challenges such as choosing appropriate membership functions, handling contradictory data, and ensuring that the finished systems are effective and beneficial for users. This project aims to explore the use of fuzzy set theory for disability assistance programmers, with a focus on developing fuzzy models that better represent the diverse requirements of individuals.

Literature review

To solve the difficulties faced by people with disabilities, the study of human disability integrates medical, social, and psychological aspects from a variety of viewpoints. The social model, which emphasizes societal barriers and the need for inclusivity, has replaced the old medical paradigm, which only focuses on diagnosis and therapy, according to literature. Scholars have investigated how disabilities affect social engagement, work prospects, education, and quality of life. Promoting autonomy and equity for people with disabilities has been made possible in large part by developments in assistive technology and legislative initiatives. Furthermore, the way that age, gender, and socioeconomic status intersect with disability emphasizes the significance of customized strategies to meet a range of needs. Disability research is an important field for further investigation and action since, despite advancements, there are still gaps in tackling stigma, enhancing accessibility, and promoting an inclusive culture. Literature is rich in the study of human disability and researchers are working in this field to make human life easy. The creation of a decision support system to offer occupational therapy services to students with disabilities during transitional times is covered by Lerslip et al.¹. Mitra and Gao² studied social poly and disability. An integrated support system for individuals with intellectual disabilities was provided by Papadogiorgaki et al.³. A systematic review of independent living, personal help, and individuals with impairments was covered by Rioboo-Lois et al.⁴. Additionally, the framework for disability in the new ways of working was proposed by Klinksiek et al.⁵. Chen⁶ discussed the rights of a person with disabilities and explored the current status and future directions.

The idea of the fuzzy set (FS) was first introduced by Zadeh⁷ in 1965 which can handle fuzzy and ambiguous information easily. FS is the generalization of crisp set theory that deals with the situation in which we can decide in the form of yes or no and there is no other possibility. In many situations, we have to decide beyond these limitations where the information of ambiguous. In the case of crisp set theory, we can say that an element either belongs to a set or does not belong to a set. But in FS theory it is not the case. Terms like “tall”, “rich” and healthy, etc. are ambiguous and we cannot decide in terms of yes or no whether a person is tall, rich and healthy, etc. In this case, FS theory helps to decide that a person is how much tall, rich and healthy, etc. through its membership grade and its value belong to [0, 1]. In crisp set theory if an element fully belongs to a set, then we will provide its membership value equal to 1 and if it does not belong to a set then we provide the value of membership grade equal to 0. In the case of FS, the value of membership grade belongs to [0, 1] which is evident that FS is the generalization of crisp set theory. Moreover, in the situation where the crisp set theory fails to hold then FS works and provides assistance to decide in many decision-making scenarios. Many developments have been made by the researchers and they have proved the utilization of FS in human disability. A multidimensional assessment method should be developed to measure and interpret the residual capabilities of qualified disabled individuals about their strengths, weaknesses, and compatibility with job requirements and work environments. This will help to place these individuals in appropriate job positions. Therefore, Chen and He⁸ introduced the AHP process under the notion of FS to rate and rank the disability. Their main goal is to talk about how to organize a hierarchy that is connected to the issue of determining an Overall Disability Index (ODI) to gauge a person’s disability. Fuzzy set theory, entropy theory, and the analytic hierarchy process are used in this study. Navin and Krishnan⁹ delivered a fuzzy rule-based classifier model for evidence-based clinical decision-making. Castro et al.¹⁰ introduced a fuzzy multicriteria decision-making algorithm. This research offers a fresh viewpoint on evaluating educational technologies for kids with intellectual disabilities in terms of design elements, which are also made up of technical requirements and ergonomic design principles. Elfakki et al.¹¹ explored an intelligent tool based on fuzzy logic and a 3D virtual learning environment for disabled student academic performance assessment. This study examined various fuzzy logic-based methods for assessing students’ academic performance in a three-dimensional virtual learning environment (VLE). The study also discussed the creation and design of assessment systems that consider the fact that certain of the methods were particularly beneficial for students with disabilities. Fuzzy logic was employed in the study to examine the academic performance of impaired students over a year. Suib et al.¹² delivered an approach of the fuzzy Delphi method to discuss the risk management index in special education mathematics. Therefore, the purpose of this study was to create a risk management index in special education mathematics by identifying and weighting each main risk aspect using the Fuzzy Delphi Method (FDM). When school managers are making decisions on special education mathematics, they might use the risk management index. Al-qaysi et al.¹³ delivered a dynamic decision-making framework for the benchmarking brain-computer interface application. They have developed a fuzzy weighted inconsistency method for consistent weights and VIKOR for stable rank. Zulkifli et al.¹⁴ shared their research on designing the content of religious education learning to create sustainability among children with learning disabilities. Thus, the goal of this study was to use fuzzy Delphi to create religious education content for kids with learning difficulties. The design method and developmental research approach were employed in this study. Alharbi et al.¹⁵ delivered the user perception-based optimal route selection for vehicles of disabled persons in urban centers of Saudi Arabia. According to this study, users of powered wheelchairs highlighted the lack of sidewalks, users of manual wheelchairs preferred the length criterion, and users of artificial limbs were worried about slope. Only two of the ten routes had medium accessibility, according to the results, with the rest routes showing low accessibility. Additionally, Alamoodi et al.¹⁶ proposed a novel evaluation framework for medical by combining fuzzy logic and MCDM for medical relation and clinical concept extraction. Alzanin and Gumaei¹⁷ established a MCDM approach to creating an accessible environment to empower mobility-impaired individuals. Hamid et al.¹⁸ delivered a fuzzy decision-making framework for evaluating hybrid detection models of Trauma patients.

The idea of the polar form of compel fuzzy set (CFS) was developed by Ramot et al.¹⁹ and it has its significance. In the structure of Ramot et al.¹⁹ idea the value of membership grade belongs to the unit circle of the complex plane. To show the significance of the polar form of CFS, many developments have been made by the researchers working on FS theory. Yazdanbakhsh and Dick²⁰ provided a systematic review of CFS. Ramot et al.²¹ discuss and explore the CF logic in their study. Chen et al.²² discussed a neuro-fuzzy architecture by employing the notion of CFS. Ma et al.²³ explored CFS and provided its application in signals. Ma et al.²⁴ discussed a method for multiple periodic factor prediction using the idea of CFS. Hu et al.²⁵ delivered and explored the notions of distance of CFSs and discussed the continuity of C operators. The notion of the Cartesian form of CFS has been delivered by Tamir et al.²⁶ in which the value of memberships grade belongs to the unit square in the complex plane instead of the unit square.

Geometrical motivation of utilized approach

The geometric representation of both ideas shows they have their significance. But in the case of the Cartesian form of CFS, we can see that some points cannot be handled by the polar form CFS. For example, if we take the Cartesian form of CFN as $0.9+\iota 0.89$, then in this case we can see that the polar form fails because $\text{r}=1.2657\notin \left[0, 1\right].$ Hence in this case the condition on “r” in the polar form CFS fails and that kind of information cannot be handled by this structure. While the Cartesian form of CFS can discuss such kind of information easily.

The motivation of this article is this discuss the Cartesian form of CFS theory in more detail and discuss the significance of this study as compared to the polar form of CFS. Although both concepts are independent, we can observe geometrically that the notion of the Cartesian form of CFS is more advanced than the polar form of CFS. Because the decision makers take the value as $0.91+\iota 0.8$ then we can see that in the polar form $r{e}^{2\pi \theta }$ where $\text{r}\; \text{and}\; \theta \in \left[0, 1\right].$ But in this case, we can see that $\text{r}=\sqrt{0.9{1}^{2}+0.{8}^{2}}=1.2116\notin \left[0, 1\right]$ and the theory of the polar form of CFS fails to hold. Hence, in this case, only the Cartesian form of CFS works and can be useful. Based on these observations, in this article, we have developed aggregation theories such as complex fuzzy Dombi aggregation operators Moreover to discuss the utilization of these approaches we have developed an algorithm to discuss the application of these established works. The comparative analysis of the established work shows the significance of the introduced work.

Research gaps

This section of the article is about the research gaps of existing notions to discuss the need for a developed approach. In this section, we have, first of all, given Table 1 in which we have discussed the research gaps in the existing notions.

The research gaps of the existing studies are given in Table 2.

Table 2 Research gaps in the existing notions.

From Table 2, we can see that

• FS⁷ developed by Zadeh can generalize the crisp set theory but it can never discuss the extra fuzzy information and it is free to discuss the two-dimensional information. This characteristic makes the FS approach limited and there exists a research gap to discuss the two-dimensional information.

• Ramot et al.¹⁹ approach is based on the polar form of CFS and it is based on phase term and amplitude terms both belonging to [0, 1. We can notice from Fig. 1 that Ramot et al.¹⁹ approach uses the unit disc in the complex plan and in this case some fuzzy information that belongs to the unit square in the complex plan can never be discussed. It means there is a research gap in Ramot et al.¹⁹ approach. This issue is handled in Fig. 2.

• In the case of Ma et al.^{23, 24} approaches, it is based on Ramot et al.¹⁹ idea for CFS. In this case the chance of data loss increases. To make the decision-making process effective and valuable, there is a need to define such a structure in which the chance of data loss is minimum, and in the case of existing ideas the chance of data loss increases which shows the research gaps of the existing ideas.

• We may observe that the Bi et al.^27,28 techniques have created geometric AOs and CF averages. The polar form of CFS is the basis for these established methods, and the Bi et al.^27,28 technique has a higher risk of data loss. These are the research gaps of Bi et al.^27,28 approaches.

Fig. 1

Polar form of CFS with MG = re^2πθ and r, $\theta \in \left[0, 1\right]$ .

Fig. 2

Cartesian form of CFS.

Contribution of study

Based on the research gaps given in Table 2 and their explanation, we have observed that Tamir’s et al.²⁶ approach is reliable and beneficial, and more fruitful results can be obtained by using this notion. So in this article, we have contributed the notions of

• Basic Dombi operational laws for Tmair’s CF notions.

• The aggregation theory of complex fuzzy Dombi weighted average and complex fuzzy Dombi ordered weighted average and complex fuzzy Dombi hybrid average AOs.

• The aggregation theory of complex fuzzy Dombi weighted geometric and complex fuzzy Dombi ordered weighted geometric and complex fuzzy Dombi hybrid geometric AOs.

• The application of the introduced work to disability support systems.

• Comparative analysis of the developed approach to deal with the reliability and superiority of the initiated work.

The rest of the article is arranged as follows, in Section "Introduction", we have delivered the introduction of the proposed work. Section "Literature review" is about the literature review of the initiated work. In Section "Preliminaries" we have discussed the fundamental existing notions that can help to develop the further theory. Section "Fundamental operational rules for the Cartesian form of CFNs" is based on Dombi operational laws based on the Cartesian form of CFS. In Section "Cartesian form of complex fuzzy Dombi weighted average (CFDWA) AOs" we have discussed aggregation theory. In Section "The multi-attribute decision-making (MADM) for a Cartesian form of complex fuzzy sets", we have delivered an algorithm to discuss the application of the proposed study along with the application for prioritization of disability support system. In Section "Comparative analysis", we have discussed a comparative analysis of the initiated theory and Section "Conclusion" is about the conclusion remarks.

Moreover, the geometrical presentation of the paperwork is given in Fig. 3.

Fig. 3

Graphical representation of introduced work.

Preliminaries

In this section, we have discussed some fundamental notions that can further help us to define the notion initiated in this article. Here we have reviewed the definition of FS, the polar form of CFS, a Cartesian form of CFS, and some fundamental rules based on these notions. We have also reviewed the basic definition of Dombi t-norm and t-conorm for real numbers.

Definition 1

Zadeh⁷ Let $\text{U}$ be the universal set. The idea of FS was delivered by Zadeh in 1965 and its presentation was given by

$${\text{FS}} = \left\{ {{\text{x}},{\text{u}}\left( {\text{x}} \right)|{\text{x}} \in {\text{U}}} \right\}$$

Here ${\text{u}}\left({\text{x}}\right)$ is called the membership grade and ${\text{u}}\left({\text{x}}\right)\in \left[0, 1\right].$

Definition 2

Alamoodi et al.¹⁶ For the universal set $\text{U}$ the notion of the polar form of CFS is given by

$${\text{CFS}} = \left\{ {{\text{x}},\;{\text{r}}\left( {\text{x}} \right){\text{e}}^{{\iota {\text{w}}\left( {\text{x}} \right)}} } \right\}$$

Here ${\text{r}}\left({\text{x}}\right)$ is called amplitude term and ${\text{r}}\left({\text{x}}\right)\in \left[0, 1\right]$ and ${\text{w}}\left({\text{x}}\right)$ is called phase term.

Definition 3

Tamir et al.²⁶ The notion of the Cartesian form of CFS was delivered by Tamir et al.²⁶ and its representation is given by

$$\text{CFS}=\left\{{\text{x}},{\text{M}}\left({\text{x}}\right)|{\text{M}}\left({\text{x}}\right)={\text{u}}\left({\text{x}}\right)+{\text{i}}{\text{v}}\left({\text{x}}\right)\right\}$$

Here ${\text{M}}\left({\text{x}}\right)={\text{u}}\left({\text{x}}\right)+{\text{i}}{\text{v}}\left({\text{x}}\right)$ is called membership grade and ${\text{u}}\left({\text{x}}\right),{\text{v}}\left({\text{x}}\right)\in \left[0, 1\right]$

Definition 4

Tamir et al.²⁶ Assume that ${\text{I}}_{1}=\left({{\text{u}}}_{1}+{\text{i}}{{\text{v}}}_{1}\right)$ and ${\text{I}}_{2}=\left({{\text{u}}}_{2}+{\text{i}}{{\text{v}}}_{2}\right)$ represents two CFNs. The fundamental operational rules are given by

• (1) ${\text{I}}_{1}\oplus {\text{I}}_{2}=\left({{\text{u}}}_{1}+{{\text{u}}}_{2}-{{\text{u}}}_{1}{{\text{u}}}_{2}+{\text{i}}\left({{\text{v}}}_{1}+{{\text{v}}}_{2}-{{\text{v}}}_{1}{{\text{v}}}_{2}\right)\right)$

• (2) ${\text{I}}_{1}\otimes{\text{I}}_{2}=\left(\left({{\text{u}}}_{1}{{\text{u}}}_{2}+\iota {{\text{v}}}_{1}{{\text{v}}}_{2}\right)\right)$

• (3) ${\text{q}}{\text{I}}_{1}=\left(1-{\left(1-{{\text{u}}}_{1}\right)}^{{\text{q}}}+\iota \left(1- {\left(1-{{\text{v}}}_{1}\right)}^{{\text{q}}}\right)\right)$ for ${\text{q}} > 0$

• (4) ${{\text{I}}_{1}}^{{\text{q}}}=\left(\left({\left({{\text{u}}}_{1}\right)}^{{\text{q}}}+\iota {\left({{\text{v}}}_{1}\right)}^{{\text{q}}}\right) \right)$ for ${\text{q}} > 0$

Definition 5

Tamir et al.²⁶ Assume that ${\text{I}}_{1}=\left({{\text{u}}}_{1}+{\text{i}}{{\text{v}}}_{1}\right)$ and ${\text{I}}_{2}=\left({{\text{u}}}_{2}+{\text{i}}{{\text{v}}}_{2}\right)$ represents two CFNs, their union, intersection, and complement are defined as

• (1) ${\text{I}}_{1}\cup {\text{I}}_{2}=\left\{\left(\text{max}\left({{\text{u}}}_{1}, {{\text{u}}}_{2}\right)+\iota \text{max}\left({{\text{v}}}_{1}, {{\text{v}}}_{2}\right)\right)\right\}$

• (2) ${\text{I}}_{1}\cap {\text{I}}_{2}=\left\{\left(\text{min}\left({{\text{u}}}_{1}, {{\text{u}}}_{2}\right)+\iota \text{min}\left({{\text{v}}}_{1}, {{\text{v}}}_{2}\right)\right)\right\}$

• (3) ${\left({\text{I}}_{1}\right)}^{\text{c}}=\left(\left(\left(1-{{\text{u}}}_{1}\right)+\iota \left(1-{{\text{v}}}_{1}\right)\right)\right)$

Definition 6

Merigo et al.²⁹ For any two real numbers ${\propto }_{1}, {\propto }_{2}\; \text{and}\; \varrho \ge 1,$ the Dombi t-norms and t-conorm are given by

$${\text{T}}_{{{\text{norm}}}} \left( { \propto _{1} , \propto _{2} } \right) = \frac{1}{{1 + \left\{ {\left( {\frac{{1 - \propto _{1} }}{{ \propto _{1} }}} \right)^{{\varrho }} + \left( {\frac{{1 - \propto _{2} }}{{ \propto _{2} }}} \right)^{{\varrho }} } \right\}^{{\frac{1}{{\varrho }}}} }}$$

$${\text{T}}_{\text{conorm}}\left({\propto }_{1}, {\propto }_{2}\right)=1-\frac{1}{1+{\left\{{\left(\frac{{\propto }_{1}}{1-{\propto }_{1}}\right)}^{\varrho }+{\left(\frac{{\propto }_{2}}{1-{\propto }_{2}}\right)}^{\varrho }\right\}}^{\frac{1}{\varrho }}}$$

Fundamental operational rules for the Cartesian form of CFNs

This section of the article is about fundamental operational laws for a Cartesian form of CFS. Based on these operational laws we have developed the further aggregation theory. Moreover, the notion of score function and accuracy function is introduced.

Definition 7

Assume that ${\text{I}}_{1}=\left({{\text{u}}}_{1}+{\text{i}}{{\text{v}}}_{1}\right)$ and ${\text{I}}_{2}=\left({{\text{u}}}_{2}+{\text{i}}{{\text{v}}}_{2}\right)$ represents two CFNs and ${\text{s}}\ge 1$ and ${\text{q}}> 0.$ Then

Operational laws are given by

• (1) ${\text{I}}_{1} \oplus {\text{I}}_{2} = \left( {1 - \frac{1}{{1 + \left\{ {\left( {\frac{{{{\text{u}}}_{1} }}{{1 - {{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{{{\text{u}}}_{2} }}{{1 - {{\text{u}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {\left( {\frac{{{{\text{v}}}_{1} }}{{1 - {{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{{{\text{v}}}_{2} }}{{1 - {{\text{v}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right)$

• (2) ${\text{I}}_{1} \otimes {\text{I}}_{2} = \left( {\frac{1}{{1 + \left\{ {\left( {\frac{{1 - {{\text{u}}}_{1} }}{{{{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{1 - {{\text{u}}}_{2} }}{{{{\text{u}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {\left( {\frac{{1 - {{\text{v}}}_{1} }}{{{{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{1 - {{\text{v}}}_{2} }}{{{{\text{v}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right)$

• (3) ${{\text{q}}{\text{I}}}_{1} = \left( {1 - \frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{{{\text{u}}}_{1} }}{{1 - {{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{{{\text{v}}}_{1} }}{{1 - {{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right)$

• (4) ${\text{I}}_{1} ^{{{\text{q}}}} = \left( {\frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{1 - {{\text{u}}}_{1} }}{{{{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{1 - {{\text{v}}}_{1} }}{{{{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right)$

For the calculation perspective used in the application section, we have provided an example of these Dombi operational laws. Based on these laws the ideas of CFDWA and CFDWG AOs are developed and application is given. To understand the calculation process, we have provided an example given by

Example 1

Let ${\text{I}}_{1}=\left(0.7+{\text{i}}0.8\right)$ and ${\text{I}}_{2}=\left(0.3+{\text{i}}0.5\right)$ be two CFNs and ${\text{s}}=2$ and ${\text{q}}=3$, then

• (1) $\begin{aligned} {\text{I}}_{1} \oplus {\text{I}}_{2} & = \left( {1 - \frac{1}{{1 + \left\{ {\left( {\frac{{{{\text{u}}}_{1} }}{{1 - {{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{{{\text{u}}}_{2} }}{{1 - {{\text{u}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {\left( {\frac{{{{\text{v}}}_{1} }}{{1 - {{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{{{\text{v}}}_{2} }}{{1 - {{\text{v}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\left( {\frac{{0.7}}{{1 - 0.7}}} \right)^{2} + \left( {\frac{{0.3}}{{1 - 0.3}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {\left( {\frac{{0.8}}{{1 - 0.8}}} \right)^{2} + \left( {\frac{{0.5}}{{1 - 0.5}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }}} \right) = \left( {0.7034 + \iota 0.8048} \right) \\ \end{aligned}$

Similarly, if we take three CFNS then we add the first two CFNs to get another CFNS and add it to the third CFN and this process continues.

• (2) $\begin{aligned} {\text{I}}_{1} \otimes {\text{I}}_{2} & = \left( {\frac{1}{{1 + \left\{ {\left( {\frac{{1 - {{\text{u}}}_{1} }}{{{{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{1 - {{\text{u}}}_{2} }}{{{{\text{u}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {\left( {\frac{{1 - {{\text{v}}}_{1} }}{{{{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} + \left( {\frac{{1 - {{\text{v}}}_{2} }}{{{{\text{v}}}_{2} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right) \\ & = \left( {\frac{1}{{1 + \left\{ {\left( {\frac{{1 - 0.7}}{{0.7}}} \right)^{{{\text{s}}}} + \left( {\frac{{1 - 0.3}}{{0.3}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {\left( {\frac{{1 - 0.8}}{{0.8}}} \right)^{2} + \left( {\frac{{1 - 0.5}}{{0.5}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }}} \right) = \left( {0.2965 + \iota 0.4924} \right) \\ \end{aligned}$

• (3) $\begin{aligned} {{\text{q}}{\text{I}}}_{1} & = \left( {1 - \frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{{{\text{u}}}_{1} }}{{1 - {{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{{{\text{v}}}_{1} }}{{1 - {{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {3\left( {\frac{{0.7}}{{1 - 0.7}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {3\left( {\frac{{0.8}}{{1 - 0.8}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }}} \right) = \left( {0.8016 + \iota 0.8738} \right) \\ \end{aligned}$

• (4) $\begin{aligned} {\text{I}}_{1} ^{{{\text{q}}}} & = \left( {\frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{1 - {{\text{u}}}_{1} }}{{{{\text{u}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{1 - {{\text{v}}}_{1} }}{{{{\text{v}}}_{1} }}} \right)^{{{\text{s}}}} } \right\}^{{\frac{1}{{{\text{s}}}}}} }}} \right) \\ & = \left( {\frac{1}{{1 + \left\{ {3\left( {\frac{{1 - 0.7}}{{0.7}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }} + {\text{i}}\frac{1}{{1 + \left\{ {{{\text{q}}}\left( {\frac{{1 - 0.8}}{{0.8}}} \right)^{2} } \right\}^{{\frac{1}{2}}} }}} \right) = \left( {0.5739 + \iota 0.6978} \right) \\ \end{aligned}$

Definition 8

Assume that ${\text{I}}_{1}=\left({{\text{u}}}_{1}+{\text{i}}{{\text{v}}}_{1}\right)$ represents a CFN, the notion of score function and accuracy function is given by

$$\text{Scr}.\left({\text{I}}_{1}\right)=\frac{1}{4}\left({{\text{u}}}_{1}+{{\text{v}}}_{1}\right);\text{Scr}.\in \left[0, 1\right]$$

and

$$\text{Ac}.\left({\text{I}}_{1}\right)=\frac{1}{4}\left({{\text{u}}}_{1}-{{\text{v}}}_{1}\right);\text{Ac}.\in \left[0, 1\right]$$

Definition 9

Assume that ${\text{I}}_{1}=\left({{\text{u}}}_{1}+{\text{i}}{{\text{v}}}_{1}\right)$ and ${\text{I}}_{2}=\left({{\text{u}}}_{2}+{\text{i}}{{\text{v}}}_{2}\right)$ represents two CFNs. For ordering the CFNs, we have the following criteria

• (a) If $\text{Scr}.\left({\text{I}}_{1}\right)<\text{Scr}.\left({\text{I}}_{2}\right)$ then ${\text{I}}_{1}<{\text{I}}_{2}$

• (b) If $\text{Scr}.\left({\text{I}}_{1}\right)>\text{Scr}.\left({\text{I}}_{2}\right)$ then ${\text{I}}_{1}>{\text{I}}_{2}$

• (c) If $\text{Scr}.\left({\text{I}}_{1}\right)=\text{Scr}.\left({\text{I}}_{2}\right)$ then

  • I. If $\text{Ac}.\left({\text{I}}_{1}\right)>\text{Ac}.\left({\text{I}}_{2}\right)$ then ${\text{I}}_{1}>{\text{I}}_{2}$

  • II. If $\text{Ac}.\left({\text{I}}_{1}\right)<\text{Ac}.\left({\text{I}}_{2}\right)$ then ${\text{I}}_{1}<{\text{I}}_{2}$

  • III. $\text{Ac}.\left({\text{I}}_{1}\right)=\text{Ac}.\left({\text{I}}_{2}\right)$then ${\text{I}}_{1}={\text{I}}_{2}$

Cartesian form of complex fuzzy Dombi weighted average (CFDWA) AOs

In this section of the article, we have developed the AOs for a Cartesian form of CFSs. The AOs are (1) complex fuzzy Dombi weighted average (CFDWA) AOs (2) Complex fuzzy Dombi ordered weighted average (CFDOWA) AOs (3) complex fuzzy Dombi weighted geometric (CFDWG) AOs (4) Complex fuzzy Dombi ordered weighted geometric (CFDOWG) AOs. Here in this section, we have also explained the properties of these introduced notions.

Because of their special qualities and adaptability, Dombi AOs are frequently utilized in data aggregation and decision-making. A thorough justification for their application is given in Table 3.

Table 3 Rational of Dombi AOs.

Moreover, we can observe that Dombi AOs have demonstrated remarkable efficacy in tackling intricate research goals, especially in multi-criteria optimization and decision-making scenarios. Their adaptability comes from their programmable parameters, which enable the aggregation process to be fine-tuned to represent differing levels of significance or ambiguity related to various criteria. Because of their versatility, they are particularly useful in fuzzy logic settings where the data may be ambiguous or inaccurate. Dombi AOs, taken together, offer a flexible framework for obtaining precise, effective, and context-sensitive solutions in a variety of applications.

Definition 10

Let ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ be the collection of CFNs then the idea of CFDWA AOs is defined by the function ${\text{f}}:{\text{I}}^{\text{n}}\to {\text{I}}$

$$\text{CFDWA}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}{\text{n}}\\ \oplus \\ {\text{j}}=1\end{array}\left({{\text{w}}}_{{\text{j}}}{\text{I}}_{{\text{j}}}\right)$$

(1)

Here ${\text{w}}={\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)}^{\text{T}}$ represents the weight vector (WV) of ${\text{I}}_{{\text{j}}}\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ and ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right],\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1$.

Theorem 1

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs. The outcome by utilizing Eq. (1) is again CFN provided by

$$\text{CFDWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}{\text{n}}\\ \oplus \\ {\text{j}}=1\end{array}\left({{\text{w}}}_{{\text{j}}}{\text{I}}_{{\text{j}}}\right)=\left(1-\frac{1}{1+{\left\{\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}{\left(\frac{{{\text{u}}}_{{\text{j}}}}{1-{{\text{u}}}_{{\text{j}}}}\right)}^{{\text{s}}}\right\}}^{\frac{1}{{\text{s}}}}}+\text{i} 1-\frac{1}{1+{\left\{\sum_{{\text{j}}=1}^{\text{n}}{{{\text{w}}}_{{\text{j}}}\left(\frac{{{\text{v}}}_{{\text{j}}}}{1-{{\text{v}}}_{{\text{j}}}}\right)}^{{\text{s}}}\right\}}^{\frac{1}{{\text{s}}}}}\right)$$

(2)

Here ${\text{w}}={\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)}^{\text{T}}$ represents the weight vector (WV) of ${\text{I}}_{{\text{j}}}\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ and ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right],\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1$.

Proof

We utilize the method of mathematical induction to prove this result.

Note that

For ${\text{n}}=2$,

We get

$$\begin{aligned} \text{CFDWA}_{{\text{w}}} \left( {{\text{I}}_{1} ,\;{\text{I}}_{2} } \right) = {\text{I}}_{1} \oplus {\text{I}}_{2} & = \left( {1 - \frac{1}{{1 + \left\{ {{\text{w}}_{1} \left( {\frac{{{\text{u}}_{1} }}{{1 - {\text{u}}_{1} }}} \right)^{{\text{s}}} + {\text{w}}_{2} \left( {\frac{{{\text{u}}_{2} }}{{1 - {\text{u}}_{2} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {{\text{w}}_{1} \left( {\frac{{{\text{v}}_{1} }}{{1 - {\text{v}}_{1} }}} \right)^{{\text{s}}} + {\text{w}}_{2} \left( {\frac{{{\text{v}}_{2} }}{{1 - {\text{v}}_{2} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{2} {{\text{w}}_{{\text{j}}} } \left( {\frac{{{\text{u}}_{{\text{j}}} }}{{1 - {\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{2} {{\text{w}}_{{\text{j}}} \left( {\frac{{{\text{v}}_{{\text{j}}} }}{{1 - {\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Hence Eq. (2) is valid for ${\text{n}}=2$.

Assume that Eq. (2) is true for ${\text{n}} = {\text{k}}$

$$\begin{aligned} {\text{CFDWA}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{\text{k}}} } \right) & = \begin{array}{*{20}c} {\text{k}} \\ \oplus \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{\text{j}}} } \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{{\text{u}}_{{\text{j}}} }}{{1 - {\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{i}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} \left( {\frac{{{\text{v}}_{{\text{j}}} }}{{1 - {\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

To prove that Eq. (2) is valid for ${\text{n}}={\text{k}}+1$

$$\begin{aligned} {\text{CFDWA}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{{\text{k}} + 1}} } \right) & = \begin{array}{*{20}c} {\text{k}} \\ \oplus \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{\text{j}}} } \right) \oplus \left( {{\text{w}}_{{{\text{k}} + 1}} {\text{I}}_{{{\text{k}} + 1}} } \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{{\text{u}}_{{\text{j}}} }}{{1 - {\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} \left( {\frac{{{\text{v}}_{{\text{j}}} }}{{1 - {\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & \oplus \left( {1 - \frac{1}{{1 + \left\{ {{\text{w}}_{{{\text{k}} + 1}} \left( {\frac{{{\text{u}}_{{{\text{k}} + 1}} }}{{1 - {\text{u}}_{{{\text{k}} + 1}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {{\text{w}}_{{{\text{k}} + 1}} \left( {\frac{{{\text{v}}_{{{\text{k}} + 1}} }}{{1 - {\text{v}}_{{{\text{k}} + 1}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{{\text{k}} + 1}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{{\text{u}}_{{\text{j}}} }}{{1 - {\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{{\text{k}} + 1}} {{\text{w}}_{{\text{j}}} \left( {\frac{{{\text{v}}_{{\text{j}}} }}{{1 - {\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Hence the result is true for ${\text{n}}={\text{k}}+1$. So Eq. (2) is valid for all n.

Properties of CFDWA AOs

In this section we have to explore the basic properties (1) Idempotency (2) Boundedness (3) Monotonicity. The overall discussion is given by

• (a) (Idempotency) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs and ${\text{I}}_{{\text{j}}}={\text{I}}$ for all ${\text{j}}$ where ${\text{I}}=\left({\text{u}}+{\text{i}}{\text{v}}\right)$ then

$$\text{CFDWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)={\text{I}}$$

• (b) (Boundedness) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs and ${{\text{I}}_{{\text{j}}}}^{-}=\min\left({\text{I}}_{{\text{j}}}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, ${{\text{I}}_{{\text{j}}}}^{+}=\max\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, then

$${{\text{I}}_{{\text{j}}}}^{-}\le \text{CFDWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le {{\text{I}}_{{\text{j}}}}^{+}$$

• (c) (Monotonicity) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)$ and ${\text{I}}_{{\text{j}}}^{\star }=\left({{\text{u}}}_{{\text{j}}}^{\star }+{\text{i}}{{\text{v}}}_{{\text{j}}}^{\star }\right)$ are two families of CFNs. Now if ${\text{I}}_{{\text{j}}}\le {\text{I}}_{{\text{j}}}^{\star }\; \text{for}\; \text{all}\; {\text{j}}$, then

  $$\text{CFDWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le \text{CFDWA}_{{\text{w}}}\left({\text{I}}_{1}^{\star }, {\text{I}}_{2}^{\star }, {\text{I}}_{3}^{\star }, \dots , {\text{I}}_{\text{n}}^{\star }\right).$$

Cartesian form of complex fuzzy Dombi ordered weighted average (CFDOWA) AOs

The idea of CFDWA AO can only weigh the different CFNs but the notion of CFDOWA can order the preference. The main differences between these two aggregation operators are given in the following Table 4.

Table 4 Features of CFDWA and CFDOWA AOs.

Definition 11

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs. The idea of CFDOWA AOs is defined by a function ${\text{f}}:{\text{I}}^{\text{n}}\to {\text{I}}$ such that

$$\text{CFDOWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}{\text{n}}\\ \oplus \\ {\text{j}}=1\end{array}\left({\text{w}}_{{\text{j}}}{\text{I}}_{{\text{o}}({\text{j}})}\right)$$

(3)

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ is the linked WV and ${\text{w}}_{{\text{j}}}\in \left[0, 1\right], \sum_{{\text{j}}=1}^{\text{n}}{\text{w}}_{{\text{j}}}=1.$ Also,${\text{o}}\left(1\right),{\text{o}}\left(2\right),{\text{o}}\left(3\right),\dots,{\text{o}} ({{\text{n}}})$ are the permutation of ${\text{o}}\left({\text{j}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$, for which ${\text{I}}_{{\text{o}}({\text{j}}-1)}\ge {\text{I}}_{{\text{o}}({\text{j}})}\forall {\text{j}}=1, \text{2,3},\dots,\text{n}$.

Theorem 2

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs. The outcome obtained by utilizing the (3) is again CFN.

$$\begin{aligned} {\text{CFDOWA}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{\text{n}}} } \right) & = \begin{array}{*{20}c} {\text{n}} \\ \oplus \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{{\text{o}}({\text{j}})}} } \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{n}}} {{\text{w}}_{{{\text{o}}({\text{j}})}} } \left( {\frac{{{\text{u}}_{{{\text{o}}({\text{j}})}} }}{{1 - {\text{u}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{n}}} {{\text{w}}_{{{\text{o}}({\text{j}})}} \left( {\frac{{{\text{v}}_{{{\text{o}}({\text{j}})}} }}{{1 - {\text{v}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ is the linked WV and ${\text{w}}_{{\text{j}}}\in \left[0, 1\right].$ Also, ${\text{o}}\left(1\right),{\text{o}}\left(2\right),{\text{o}}\left(3\right),\dots,{\text{o}}\left(\text{n}\right)$ are the permutation of ${\text{o}}\left({\text{j}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$, for which ${\text{I}}_{{\text{o}}({\text{j}}-1)}\ge {\text{I}}_{{\text{o}}({\text{j}})}\forall {\text{j}}=1, \text{2,3},\dots,\text{n}$.

The notion of CFDOWA AO operator satisfies the same properties as CFDWA AOs do. The properties are given by

Properties of CFDOWA AOs

• (a) (Idempotency) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$ represent the family of CFNs and ${\text{I}}_{{\text{j}}}={\text{I}}$ for all ${\text{j}}$ where ${\text{I}}=\left({\text{u}}+{\text{i}}{\text{v}}\right)$ then

  $$\text{CFDOWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)={\text{I}}$$

• (b) (Boundedness) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$ represent the family of CFNs and ${{\text{I}}_{{\text{j}}}}^{-}=\text{min}\left({\text{I}}_{{\text{j}}}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, ${{\text{I}}_{{\text{j}}}}^{+}=\text{max}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, then

  $${{\text{I}}_{{\text{j}}}}^{-}\le \text{CFDOWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le {{\text{I}}_{{\text{j}}}}^{+}$$

• (c) (Monotonicity) Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)$ and ${\text{I}}_{{\text{j}}}^{\star }=\left({{\text{u}}}_{{\text{j}}}^{\star }+{\text{i}}{{\text{v}}}_{{\text{j}}}^{\star }\right)$ are two families of CFNs. Now if ${\text{I}}_{{\text{j}}}\le {\text{I}}_{{\text{j}}}^{\star } \text{for}\; \text{all} {\text{j}}$, then

  $$\text{CFDOWA}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le \text{CFDOWA}_{{\text{w}}}\left({\text{I}}_{1}^{\star }, {\text{I}}_{2}^{\star }, {\text{I}}_{3}^{\star }, \dots , {\text{I}}_{\text{n}}^{\star }\right).$$

The notion of CFDWA AOs targets only the CF values, while the CFDOWA AOs discuss the ordered positions of the CF values rather than the weights of the CF values themselves. By combining both properties in one structure, we define the CFDHA AOs.

Definition 12

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$ represent the family of CFNs. The notion of CFDHA AOs is a mapping given by ${\text{f}}:{{\text{I}}_{{\text{j}}}}^{\text{n}}\to {\text{I}}$

$${\text{CFDHA}}_{{\text{w}}} ,\left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{\text{n}}} } \right) = \begin{array}{*{20}c} {\text{n}} \\ \oplus \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{{\text{o}}({\text{j}})}} } \right) = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{{\text{o}}({\text{j}})}} } \left( {\frac{{{\text{u}}_{{{\text{o}}({\text{j}})}} }}{{1 - {\text{u}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{{\text{o}}({\text{j}})}} \left( {\frac{{{\text{v}}_{{{\text{o}}({\text{j}})}} }}{{1 - {\text{v}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right)$$

(4)

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ is the linked WV and ${\text{w}}_{{\text{j}}}\in \left[0, 1\right], \sum_{{\text{j}}=1}^{\text{n}}{\text{w}}_{{\text{j}}}=1.$ Also ${\text{I}}_{{\text{o}}({\text{j}}}^{*}$ is the largest weighted CF values ${\text{I}}_{{\text{o}}({\text{j}}}^{*}\left( {\text{I}}_{{\text{o}}({\text{j}})}^{*}=\left(n{\text{w}}\right){\text{I}}_{{\text{j}}},{\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ and ${\text{w}}=\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)$ is the WV of ${\text{I}}_{{\text{j}}},{{\text{w}}}_{{\text{j}}}\in \left[0, 1\right]$ and $\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1.$ Here n is called the balancing coefficient.

Cartesian form of complex fuzzy Dombi weighted geometric (CFDWG) AOs

In this section, we have to deliver the ideas of (1) complex fuzzy Dombi weighted geometric AO (2) complex fuzzy Dombi ordered weighted geometric AO (3) complex fuzzy Dombi hybrid geometric AO.

Definition 13

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , \text{n}\right)$ represent the family of CFNs. The idea of CFDWG AOs is defined by a function ${\text{f}}:{\text{I}}^{\text{n}}\to {\text{I}}$ such that

$$\text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}{\text{n}}\\ \otimes \\ {\text{j}}=1\end{array}{\left({\text{I}}_{{\text{j}}}\right)}^{{{\text{w}}}_{{\text{j}}}}$$

(5)

Here ${\text{w}}={\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)}^{\text{T}}$ represents the WV of ${\text{I}}_{{\text{j}}}\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ and ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right],\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1$.

Theorem 3

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent the family of CFNs. The outcome obtained by utilizing Eq. (5) is again CFN and

$$\text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}{\text{n}}\\ \otimes\\ {\text{j}}=1\end{array}{\left({\text{I}}_{{\text{j}}}\right)}^{{{\text{w}}}_{{\text{j}}}}=\left(\frac{1}{1+{\left\{\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}{\left(\frac{1-{{\text{u}}}_{{\text{j}}}}{{{\text{u}}}_{{\text{j}}}}\right)}^{{\text{s}}}\right\}}^{\frac{1}{{\text{s}}}}}+{\text{I}}\frac{1}{1+{\left\{\sum_{{\text{j}}=1}^{\text{n}}{{{\text{w}}}_{{\text{j}}}\left(\frac{1-{{\text{v}}}_{{\text{j}}}}{{{\text{v}}}_{{\text{j}}}}\right)}^{{\text{s}}}\right\}}^{\frac{1}{{\text{s}}}}}\right)$$

(6)

Here, ${\text{w}}={\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)}^{\text{T}}$ represents the WV of ${\text{I}}_{{\text{j}}}\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ with condition that ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right]$ and $\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1$.

Proof

The result of Eq. (6) can be proved by utilizing the method of mathematical induction. Now note that

For ${\text{n}}=2$,

We get

$$\begin{aligned} {\text{CFDWG}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} } \right) & = {\text{I}}_{1} \otimes {\text{I}}_{2} \\ & = \left( {\frac{1}{{1 + \left\{ {{\text{w}}_{1} \left( {\frac{{1 - {\text{u}}_{1} }}{{{\text{u}}_{1} }}} \right)^{{\text{s}}} + {\text{w}}_{2} \left( {\frac{{1 - {\text{u}}_{2} }}{{{\text{u}}_{2} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {{\text{w}}_{1} \left( {\frac{{1 - {\text{v}}_{1} }}{{{\text{v}}_{1} }}} \right)^{{\text{s}}} + {\text{w}}_{2} \left( {\frac{{1 - {\text{v}}_{2} }}{{{\text{v}}_{2} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{2} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{\text{j}}} }}{{{\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{2} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{\text{j}}} }}{{{\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Hence Eq. (6) is true for ${\text{n}}=2$.

Assume Eq. (6) is true for ${\text{n}}={\text{k}}$

$$\begin{aligned} {\text{CFDWG}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{\text{k}}} } \right) & = \begin{array}{*{20}c} {\text{k}} \\ \otimes \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{\text{j}}} } \right) \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{\text{j}}} }}{{{\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{\text{j}}} }}{{{\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

To prove Eq. (6) is true for ${\text{n}}={\text{k}}+1$

$$\begin{aligned} {\text{CFDWG}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{{{\text{k}} + 1}} } \right) & = \begin{array}{*{20}c} {\text{k}} \\ \otimes \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{I}}_{{\text{j}}} } \right) \otimes \left( {{\text{w}}_{{{\text{k}} + 1}} {\text{I}}_{{{\text{k}} + 1}} } \right) \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{\text{j}}} }}{{{\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{\text{k}}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{\text{j}}} }}{{{\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & \otimes \left( {\frac{1}{{1 + \left\{ {{\text{w}}_{{{\text{k}} + 1}} \left( {\frac{{1 - {\text{u}}_{{{\text{k}} + 1}} }}{{{\text{u}}_{{{\text{k}} + 1}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {{\text{w}}_{{{\text{k}} + 1}} \left( {\frac{{1 - {\text{v}}_{{{\text{k}} + 1}} }}{{{\text{v}}_{{{\text{k}} + 1}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{{\text{k}} + 1}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{\text{j}}} }}{{{\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{{{\text{k}} + 1}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{\text{j}}} }}{{{\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Hence the Eq. (6) is true for ${\text{n}}={\text{k}}+1$. Hence, it is valid for all n.

Properties of CFDWG AOs

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\; \text{and}\; {\text{I}}_{{\text{j}}}^{\star }=\left({{\text{u}}}_{{\text{j}}}^{\star }+{\text{i}}{{\text{v}}}_{{\text{j}}}^{\star }\right) \left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent two families of CFNs. The notion of CFDWG AOs satisfies the following properties.

• (a) (Idempotency) If ${\text{I}}_{{\text{j}}}={\text{I}}$ for all ${\text{j}}$ where ${\text{I}}=\left({\text{u}}+{\text{i}}{\text{v}}\right)$ then

$$\text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)={\text{I}}$$

• (b) (Boundedness) If ${{\text{I}}_{{\text{j}}}}^{-}=\text{min}\left({\text{I}}_{{\text{j}}}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$ and ${{\text{I}}_{{\text{j}}}}^{+}=\text{max}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, then

$${{\text{I}}_{{\text{j}}}}^{-}\le \text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le {{\text{I}}_{{\text{j}}}}^{+}$$

• (c) (Monotonicity) If ${\text{I}}_{{\text{j}}}\le {\text{I}}_{{\text{j}}}^{\star } \text{for all}{\text{j}}$, then

  $$\text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le \text{CFDWG}_{{\text{w}}}\left({\text{I}}_{1}^{\star }, {\text{I}}_{2}^{\star }, {\text{I}}_{3}^{\star }, \dots , {\text{I}}_{\text{n}}^{\star }\right).$$

Cartesian form of complex fuzzy Dombi ordered weighted geometric (CFDOWG) AOs

The notion of CFDWG AOs can only weight CF values. If we have to discuss the ordered position of the CF values rather than the weights of the CF values then the notion of CFDOWG AO can help us in this regard. Now in this subsection, we have to develop the idea of CFDOWG AOs. Moreover, the properties of these AOs have been developed.

Definition 14

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right) \left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represents the family of CFNs. The idea of CFDOWG AOs is a function ${\text{f}}:{\text{I}}^{\text{n}}\to {\text{I}}$

$$\text{CFDOWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)=\begin{array}{c}n\\ \otimes\\ {\text{j}}=1\end{array}{\left({\text{I}}_{{\text{o}}\left({\text{j}}\right)}\right)}^{{\text{w}}_{{\text{j}}}}$$

(7)

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ are the linked WVs with ${\text{w}}_{{\text{j}}}\in \left[0, 1\right].$ Also, ${\text{o}}\left(1\right),{\text{o}}\left(2\right),{\text{o}}\left(3\right),\dots,{\text{o}}\left(n\right)$ are the permutation of ${\text{o}}\left({\text{j}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$, for which ${\text{I}}_{{\text{o}}({\text{j}}-1)}\ge {\text{I}}_{{\text{o}}({\text{j}})}\forall {\text{j}}=1, \text{2,3},\dots,{\text{n}}$.

Theorem 4

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)$ represent the family of CFNs. The outcome obtained by using Eq. (7) is again CFN.

$$\begin{aligned} {\text{CFDOWG}}_{{\text{w}}} \left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{\text{n}} } \right) & = \begin{array}{*{20}c} n \\ \otimes \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{I}}_{{{\text{o}}\left( {\text{j}} \right)}} } \right)^{{{\text{w}}_{{\text{j}}} }} \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{{\text{o}}({\text{j}})}} }}{{{\text{u}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{{\text{o}}({\text{j}})}} }}{{{\text{v}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ are the linked WVs with ${\text{w}}_{{\text{j}}}\in \left[0, 1\right].$ Also, ${\text{o}}\left(1\right),{\text{o}}\left(2\right),{\text{o}}\left(3\right),\dots,{\text{o}}\left({\text{n}}\right)$ are called the permutation of ${\text{o}}\left({\text{j}}\right)\left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$, for which ${\text{I}}_{{\text{o}}({\text{j}}-1)}\ge {\text{I}}_{{\text{o}}({\text{j}})}\forall {\text{j}}=1, \text{2,3},\dots,{\text{n}}$.

Properties of CFDOWG AOs

In this subsection, we have to discuss the basic properties of CFDOWG AOs. The notion of CFDOWG AOs satisfies the same properties as CFDWG AOs do. These properties are given by.

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)\; \text{and}\; {\text{I}}_{{\text{j}}}^{\star }=\left({{\text{u}}}_{{\text{j}}}^{\star }+{\text{i}}{{\text{v}}}_{{\text{j}}}^{\star }\right) \left({\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ represent two families of CFNs. The

• (a) (Idempotency) If ${\text{I}}_{{\text{j}}}={\text{I}}$ for all ${\text{j}}$ where ${\text{I}}=\left({\text{u}}+{\text{i}}{\text{v}}\right)$ then

$$\text{CFDOWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)={\text{I}}$$

• (b) (Boundedness) If ${{\text{I}}_{{\text{j}}}}^{-}=\min\left({\text{I}}_{{\text{j}}}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$ and ${{\text{I}}_{{\text{j}}}}^{+}=\max\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)$, then

$${{\text{I}}_{{\text{j}}}}^{-}\le \text{CFDOWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le {{\text{I}}_{{\text{j}}}}^{+}$$

• (c) (Monotonicity) If ${\text{I}}_{{\text{j}}}\le {\text{I}}_{{\text{j}}}^{\star } \text{for} \text{all}{\text{j}}$, then

  $$\text{CFDOWG}_{{\text{w}}}\left({\text{I}}_{1}, {\text{I}}_{2}, {\text{I}}_{3}, \dots , {\text{I}}_{\text{n}}\right)\le \text{CFDOWG}_{{\text{w}}}\left({\text{I}}_{1}^{\star }, {\text{I}}_{2}^{\star }, {\text{I}}_{3}^{\star }, \dots , {\text{I}}_{\text{n}}^{\star }\right).$$

The notion of CFDWG AOs can only weigh the CF values and the notion of CFDOWG AOs can discuss the ordered positions of CF values rather than the weights of the CF values. To discuss both of these properties in one structure, the idea of CFDHG AO can help us in this regard. In the next definition, we have explored the idea of CFDHG AOs.

Definition 15

Assume ${\text{I}}_{{\text{j}}}=\left({{\text{u}}}_{{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{j}}}\right)$ represent the family of CFNs. The idea of CFDHG AOs is given by a function ${\text{f}}:{\text{I}}^{\text{n}}\to {\text{I}}$ and

$${\text{CFDHG}}_{{\text{w}}} ,\left( {{\text{I}}_{1} ,{\text{I}}_{2} ,{\text{I}}_{3} , \ldots ,{\text{I}}_{\text{n}} } \right) = \begin{array}{*{20}c} {\text{n}} \\ \otimes \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{I}}_{{{\text{o}}\left( {\text{j}} \right)}} } \right)^{{{\text{w}}_{{\text{j}}} }} = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{u}}_{{{\text{o}}({\text{j}})}} }}{{{\text{u}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{{\text{o}}({\text{j}})}} }}{{{\text{v}}_{{{\text{o}}({\text{j}})}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right)$$

(8)

Here ${\text{w}}=\left({\text{w}}_{1}, {\text{w}}_{2}, \dots , {\text{w}}_{\text{n}}\right)$ represents the linked WVs in Eq. (8) with conditions ${\text{w}}_{{\text{j}}}\in \left[0, 1\right]$, $\sum_{{\text{j}}=1}^{\text{n}}{\text{w}}_{{\text{j}}}=1.$ Also ${\text{I}}_{{\text{o}}({\text{j}}}^{*}$ is the largest weighted CF values ${\text{I}}_{{\text{o}}({\text{j}}}^{*}\left( {\text{I}}_{{\text{o}}({\text{j}}}^{*}=\left({\text{nw}}\right){\text{I}}_{{\text{j}}},{\text{j}}=1, 2, 3, \dots , {\text{n}}\right)$ and ${\text{w}}=\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)$ is the WV of ${\text{I}}_{{\text{j}}}$ and ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right]$ with $\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1.$ Here n represents the balancing coefficient.

The multi-attribute decision-making (MADM) for a Cartesian form of complex fuzzy sets

Consider the set of $"\text{m}"$ alternatives given by ${\text{A}}_{{{\text{alt}}}} = \left\{ {{\text{A}}_{{{\text{alt}} - 1}} ,\;{\text{A}}_{{{\text{alt}} - 2}} ,\;{\text{A}}_{{{\text{alt}} - 3}} ,\; \ldots ,\;{\text{A}}_{{{\text{alt}} - {\text{m}}}} } \right\}$. Also ${\text{A}}_{{{\text{atr}}}} = \left\{ {{\text{A}}_{{{\text{atr}} - 1}} ,\;{\text{A}}_{{{\text{atr}} - 2}} ,\;{\text{A}}_{{{\text{atr}} - 3}} , \ldots ,\;{\text{A}}_{{{\text{atr}} - {\text{n}}}} } \right\}$ represents the set of n attributes and ${\text{w}}=\left({{\text{w}}}_{1}, {{\text{w}}}_{2}, {{\text{w}}}_{3}, \dots , {{\text{w}}}_{\text{n}}\right)$ are the WVs of attributes with conditions that ${{\text{w}}}_{{\text{j}}}\in \left[0, 1\right]$, $\sum_{{\text{j}}=1}^{\text{n}}{{\text{w}}}_{{\text{j}}}=1.$ Assume that the decision analyst proposes their assessment in the Cartesian form of CFSs and the decision matrix in this way is represented as ${\text{m}} = \left( {{\text{L}}_{{{\text{ij}}}} } \right)_{{\text{m}}} \times {\text{n}} = \left( {\left( {{\text{u}}_{{{\text{ij}}}} + {\text{iv}}_{{{\text{ij}}}} } \right)} \right)_{{\text{m}}} \times {\text{n}}$ where $\left( {{\text{i}}:1,2, \ldots ,{\text{m}}} \right)\;{\text{and}}\;\left( {{\text{j}}:1,2, \ldots ,{\text{n}}} \right).$ Now the stepwise algorithm is given by

Decision algorithm

In this subsection, we have to explore the decision-making algorithm that can help us to decide on any decision-making problem.

Step 1: Assume that the information proposed by the decision analyst is in the form of Cartesian form of CFS and it is given in the following matrix

$${\text{m}} = \left[ {\begin{array}{*{20}l} {\left( {{\text{u}}_{{11}} + {\text{iv}}_{{11}} } \right)} \hfill & {\left( {{\text{u}}_{{21}} + {\text{iv}}_{{21}} } \right)} \hfill & \ldots \hfill & {\left( {{\text{u}}_{{{\text{m}} \times 1}} + {\text{iv}}_{{{\text{m}} \times 1}} } \right)} \hfill \\ {\left( {{\text{u}}_{{12}} + {\text{iv}}_{{12}} } \right)} \hfill & {\left( {{\text{u}}_{{22}} + {\text{iv}}_{{22}} } \right)} \hfill & \ldots \hfill & {\left( {{\text{ u}}_{{{\text{m}} \times 2}} + {\text{iv}}_{{{\text{m}} \times 2}} } \right)} \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {\left( {{\text{u}}_{{1{\text{n}}}} + {\text{iv}}_{{1{\text{n}}}} } \right)} \hfill & {\left( {{\text{u}}_{{2{\text{n}}}} + {\text{iv}}_{{2{\text{n}}}} } \right)} \hfill & \ldots \hfill & {\left( {{\text{u}}_{{{\text{m}} \times {\text{n}}}} + {\text{iv}}_{{{\text{m}} \times {\text{n}}}} } \right)} \hfill \\ \end{array} } \right]$$

Step 2: Normalize the decision matrix by using the following formulas

$${\text{N}} = \left\{ {\begin{array}{*{20}c} {\left( {{\text{u}}_{{{\text{ij}}}} + {\text{iv}}_{{{\text{ij}}}} } \right){\text{for}}\;{\text{benefit}}\;{\text{type}}\;{\text{attributes}}} \\ {\left( {1 - {\text{u}}_{{{\text{ij}}}} + {\text{i}}1 - {\text{v}}_{{{\text{ij}}}} } \right){\text{for}}\;{\text{cost}}\;{\text{type}}\;{\text{attributes}}} \\ \end{array} } \right.$$

where $\left(1-{{\text{u}}}_{{\text{i}}{\text{j}}}+{\text{i}}{1-{\text{v}}}_{{\text{i}}{\text{j}}}\right)$ is the complement of $\left({{\text{u}}}_{{\text{i}}{\text{j}}}+{\text{i}}{{\text{v}}}_{{\text{i}}{\text{j}}}\right)$.

Step 3: Utilize the explored idea of CFDWA AO or CFDWG AO to aggregate the data obtained from Step 2

$${\text{m}} = \left( {{\text{L}}_{{{\text{ij}}}} } \right)_{{{\text{m}} \times {\text{n}}}}$$

$$\begin{aligned} {\text{L}}_{{\text{i}}} = {\text{CFDWA}}\left( {{\text{L}}_{{{\text{i}}1}} ,{\text{L}}_{{{\text{i}}2}} ,{\text{L}}_{{{\text{i}}n}} , \ldots ,{\text{L}}_{{{\text{ij}}}} } \right) & = \begin{array}{*{20}c} {\text{n}} \\ \oplus \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{w}}_{{\text{j}}} {\text{L}}_{{{\text{ij}}}} } \right) \\ & = \left( {1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{{\text{u}}_{{\text{j}}} }}{{1 - {\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}1 - \frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} \left( {\frac{{{\text{v}}_{{\text{j}}} }}{{1 - {\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

$$\begin{aligned} {\text{L}}_{{\text{i}}} = {\text{CFDWG}}\left( {{\text{L}}_{{{\text{i}}1}} ,{\text{L}}_{{{\text{i}}2}} ,{\text{L}}_{{{\text{i}}n}} , \ldots ,{\text{L}}_{{{\text{ij}}}} } \right) & = \begin{array}{*{20}c} {\text{n}} \\ \otimes \\ {{\text{j}} = 1} \\ \end{array} \left( {{\text{L}}_{{{\text{ij}}}} } \right)^{{{\text{w}}_{{\text{j}}} }} \\ & = \left( {\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} } \left( {\frac{{1 - {\text{u}}_{{\text{j}}} }}{{{\text{u}}_{{\text{j}}} }}} \right)^{{\text{s}}} } \right\}^{{\frac{1}{{\text{s}}}}} }} + {\text{I}}\frac{1}{{1 + \left\{ {\sum\limits_{{{\text{j}} = 1}}^{\text{n}} {{\text{w}}_{{\text{j}}} \left( {\frac{{1 - {\text{v}}_{{\text{j}}} }}{{{\text{v}}_{{\text{j}}} }}} \right)^{{\text{s}}} } } \right\}^{{\frac{1}{{\text{s}}}}} }}} \right) \\ \end{aligned}$$

Step 4. To obtain the score values utilize the def. for ${{\text{L}}}_{{\text{i}}}$ to rank the alternatives.

Step 5. Ordering the alternatives ${\text{A}}_{{{\text{alt}} - 1}} ,\;{\text{A}}_{{{\text{alt}} - 2}} ,\;{\text{A}}_{{{\text{alt}} - 3}} ,\; \ldots ,\;{\text{A}}_{{{\text{alt}} - {\text{m}}}}$ to choose the best alternative.

Step 6. End.

Case study

Different disability support systems

The objective of the support system for the disabled is to provide people with disabilities access to tools and services to improve their lives, which helps them to actively participate in society. These systems help in clinical support systems. The common constituents are accessible accommodation, medical care, learning facilities, vocational training, and other financial support measures. All these measures to ensure proper accommodations for people with disability are coordinated through inter-linkages between government agencies, NGOs, and business groups. All of these will depend on the degree of the handicap, the needs of the individual, and the laws of the nation or area. Services can vary in different systems where it is supposed to help people with disabilities to protect rights, encourage independence, and break barriers.

These also include help with housing and transportation, making accessibility possible through house adaptations or access to accessible modes of transit. This reduces the physical barriers that would disqualify people from living their day-to-day lives. Other important aspects of the system of support for people with disabilities are protection under the law from discrimination and social inclusion. Under anti-discrimination legislation, people with disabilities should have equal opportunities in all spheres of society, like employment, public services, and accommodations. The support system for people with disabilities is dynamic and changes according to advancements in technology, societal views on disability, and the rights of people with disabilities. In bringing out such change, advocacy groups as well as the persons with a disability themselves are also vital to create a society where persons with disabilities can live freely and are supported to act in their communities actively. Four disability support systems are as an alternative follow. Discussion given by

• (1) Assistive technology

Assistive technology refers to tools and gadgets designed to assist people with disabilities in performing things that would otherwise be quite challenging for them due to their condition. For example, some people who have a problem with hearing wear cochlear implants or hearing aids, while others with mobility or dexterity issues use speech-to-text software, and blind people have screen readers. These technologies make better communication, mobility, and information access possible and help bridge the gap between freedom and impairment. Assisting devices advance with technology and offer increasingly specialized solutions for a spectrum of needs. Integration of AI has led to even more individualized experiences-including adaptive learning tools. Other important roles are performed by smartphones and home systems, allowing voice control over the thermostat, light bulbs, and other appliances. Wearable technology may follow the indicators of health that would be translated into increased well-being and safety. Assistive technology aims to encourage more freedom and inclusion. Several digital firms have focused on developing accessibility features to help people overcome disabilities in their lives, workplaces, and educational facilities.

• (2) Personalized support plans

Individualized methods aimed at providing the highest quality care and support for a person with disabilities are termed PSPs, or personalized support plans. Typically, these approaches entail an extensive assessment of the individual’s needs, preferences, and strengths. The purpose is to ensure that everything from physical care to social inclusion is provided so that the individual has the support and resources he or she requires in their life. These plans are often developed with input from a variety of experts, such as family members, educators, social workers, and physicians. PSPs can include support with communication, self-care, mobility, or employment. These approaches can improve quality of life by promoting independence and reducing dependence on others by focusing on the person. Moreover, the plans also consider the prospect of changing needs over time. The flexibility to change with changing conditions is allowed in these plans. Often, these plans include technology; thereby, gadgets or applications can be used for monitoring progress and ensuring consistency. As psychological care helps control disabilities, personalized support includes mental health and emotional well-being.

• (3) Inclusive education programs

In the inclusive education model, children with disabilities are put in mainstream classes with their non-disabled peers. The two key purposes of this model of inclusive education are to support equality and foster an inclusive culture of respect and diversity. This model of special education for children with physical, cognitive, or emotional impairments is composed of special education teachers and support personnel who have been trained to make adjustments to the lessons, provide additional resources, and make learning accessible. These programs often combine assistive technologies to facilitate improved learning. Some examples include the use of audiobooks, digital textbooks, as well as real-time captioning for students with hearing or other learning challenges. The bottom line of inclusive education would be to encourage cooperation, as well as empathy and social skills among students with vastly different skill levels. Teachers also receive special training to grasp and implement accommodations such as extra time for tests, or even different testing formats. By fostering understanding among classmates, this method helps out the impaired students as well as their peers. Also, equal opportunities in learning facilitate children with impairment to get ready for after-school life.

• (4) Employment support systems

The employment support programs are there to help people with disabilities find and keep fulfilling employment. These systems include workplace accommodations, employment coaching, and mentorship programs tailored to the special needs of disabled employees. It is aimed at setting an environment in the workplace that enables the sharing of knowledge and capabilities among disabled persons frequently through adapted technology or change of duties. The employment services for people with disabilities may liaise with businesses to eliminate any communication or physical barriers and train them on the need for accessibility. Various programs help a person with a disability prepare for work by providing career counseling, interview training, and skill development. Job matching services also help in placing applicants in jobs that match their skills. Vocational rehabilitation agencies also offer training programs to learn new skills or retrain for employment in fields with a promising job market. Legal safeguards, such as those covered in the Americans with Impairments Act (ADA), assure equal employment opportunities for the impaired. These support programs focus on enhancing financial independence and decreasing the unemployment rate among those who are impaired.

The graphical representation of these four alternative is given in Fig. 4.

Fig. 4

Graphical representation of disability support systems.

Based on this information of alternatives the attributes information is given in Table 5.

Table 5 Description of attributes.

Also, the graphical representation of attributes is given in Fig. 5.

Fig. 5

Graphical representation of attributes.

Step 1: The data provided by the decision analyst in the Cartesian form of CFS is given in Table 6.

Table 6 The Cartesian form of complex fuzzy information.

Step 2: No need to normalize the information because all the attributes are benefit types.

Step 3: To aggregate the information in Table 6 we have to utilize the notion of CFDWA AO and CFDWG AO. The overall results are given by

For CFDWA AOs

For the execution of CFDWA AO assume that ${\text{s}}=2$ and WVs for attributes are $\left(0.23, 0.29, 0.21, 0.27\right)$.

$${{\text{L}}}_{1}=\left(0.446+{\text{i}} 0.234\right), {{\text{L}}}_{2}=\left(0.569+{\text{i}} 0.436\right),$$

$${{\text{L}}}_{3}=\left(0.746+{\text{i}} 0.689\right), {{\text{L}}}_{4}=\left(0.917+{\text{i}} 0.503\right)$$

Step 4: Compute the score values of $\text{Scr}.\left({{\text{L}}}_{{\text{i}}}\right)\left({\text{i}}=1, \text{2,3}, 4\right)$ of the overall CFNs ${{\text{L}}}_{{\text{i}}} \left({\text{i}}=1, \text{2,3}, 4\right)$

$$\text{Scr}.\left({{\text{L}}}_{1}\right)=0.3400, \text{Scr}.\left({{\text{L}}}_{2}\right)=0.5030,$$

$$\text{Scr}.\left({{\text{L}}}_{3}\right)=0.7170, \text{Scr}.\left({{\text{L}}}_{4}\right)=0.7100$$

Step 5. Rank all the alternatives and find out the best alternative according to the result.

Hence the overall ranking result is as follows.

$${\text{A}}_{\text{alt-3}}> {\text{A}}_{\text{alt-4}}>{\text{A}}_{\text{alt-2}}>{\text{A}}_{\text{alt-1}}$$

According to the results, we can say ${\text{A}}_{\text{alt-3}}$ is the best choice.

Step 6. End.

The geometrical representation of the ranking result is given in Fig. 6.

Fig. 6

Geometrical representation of ranking results by using CFDWA AOs.

Figure 6 describes the score value and ranking results of the initiated work. We can observe that when utilizing the notion of CFDWA AOs then the best result or best alternative, in this case, is alternative 3 which shows the effective use of these developed approaches in decision-making.

For CFDWG AOs

Step 3: We use the idea of CFDWG AOs by taking ${\text{s}}=2$ to calculate values ${{\text{L}}}_{{\text{i}}}.$ So

$${{\text{L}}}_{1}=\left(0.0730+{\text{i}} 0.0360\right), {{\text{L}}}_{2}=\left(0.1820+{\text{i}} 0.0670\right)$$

$${{\text{L}}}_{3}=\left(0.6520+{\text{i}} 0.0810\right), {{\text{L}}}_{4}=\left(0.3070+{\text{i}} 0.1070\right)$$

Step 4: The score values $\text{Scr}.\left({{\text{L}}}_{{\text{i}}}\right)\left({\text{i}}=1, \text{2,3}, 4\right)$ of the CFNs ${{\text{L}}}_{{\text{i}}} \left({\text{i}}=1, \text{2,3}, 4\right)$ are given by $\text{Scr}.\left({{\text{L}}}_{1}\right)=0.0542,$ $\text{Scr}.\left({{\text{L}}}_{2}\right)=0.1245$, $\text{Scr}.\left({{\text{L}}}_{3}\right)=0.3663$, $\text{Scr}.\left({{\text{L}}}_{4}\right)=0.2067$

Step 5: Rank all the alternatives and find out the best alternative according to the result.

Hence the overall ranking result is as follows.

$${\text{A}}_{\text{alt-3}}> {\text{A}}_{\text{alt-4}}>{\text{A}}_{\text{alt-2}}>{\text{A}}_{\text{alt-1}}$$

According to the results, we can say ${\text{A}}_{\text{alt-3}}$ is the best choice.

Step 6: End.

Moreover, the graphical representation of ranking results using the notion of CFDWG AOs is given in Fig. 7.

Fig. 7

Graphical representation of ranking results by using CFDWG AOs.

Figure 7 describes the score value and ranking results of the initiated work. We can observe that when utilizing the notion of CFDWG AOs then the best result or best alternative, in this case, is alternative 3 which shows the effective use of these developed approaches in decision-making.

Comparative analysis

In this section, we have compared the proposed work with existing notions to discover the dominance and superiority of the established work. Here we have compared our work with the FS⁷ approach, Chen and He⁸ approach, Navin and Krishnan⁹ method, the polar form of complex fuzzy set¹⁹, Ma et al.^23,24 approach, Hu et al.²⁵ approach, Bi et al.^27,28 approach, Merigo et al.29 method and Hossain et al.³⁰. The overall discussion is given in the following section.

Result and discussion

This section of the article is devoted to discussing the results obtained through the developed approach and the advantages obtained through initiated work.

• When we compare our work with FS⁷ then we can see that the developed approaches are more advanced and it can discuss the more advanced data as compared to FS. The introduced notions are based on the Cartesian form of the CFS which provides the opportunity to discuss the two-dimensional data. The idea of FS can never discuss two-dimensional data. In disability support system there is a chance that decision makers provide their assessment in the Cartesian form of CFS the notions FS fails to hold.

• In the case of the polar form of CFS¹⁸, the developed approaches are more advanced because when decision-makers provide their assessment as $0.9+\iota 0.8$ then the main condition in the polar form of CFS fails to hold. In this case, we can observe that the amplitude terms $\text{r}\notin \left[0, 1\right]$ and the limitation of the polar form of CFS is obvious. Hence the developed approach provides more space for decision-makers to take their information in the Cartesian form of CFS.

• In the case of Bi et al.^24,25 approaches, we can see that they have developed CF average and geometric AOs. These developed approaches are based on the polar form of CFS and in the case of Bi et al.^{24, 25} approach, the chance of data loss increases. While the initiated theory can cover these limitations. Hence in disability support systems, these kinds of limitations can be covered and more reliable results can be obtained.

• Merigo et al.²⁶ approach is also based on the polar form of CFS and this approach also fails to handle the data of Table 6. The introduced approach discusses this information effectively and this behavior makes the delivered approach more dominant.

• Navin and Krishnan’s⁹ approach is based on fuzzy logic and they have utilized this approach in disability sectors. If the decision makers provide theory assessment in the form of $\text{a}+\iota \text{b}$ then the existing can never discuss such kind of data. The delivery approach can handle this data and assist in the disability sector.

• Chen and He’s⁸ approach and Hossain et al.²⁷ approach, have used the AHP process to rate and rank the disability but these approaches are based on FS. These existing notions can never discuss the two-dimensional information given in Table 6. The delivered approach can provide this opportunity to discover the data in Table 6. Hence the dominancy of the deliver approach is clear. To rate and rate the disability the delivered approaches are dominant because they can discuss the two-dimensional information in disability.

• The Ramot et al.¹⁸ concept for CFS serves as the foundation for the Ma et al.^20,21 methods. The likelihood of data loss rises in this situation. To make the decision-making process worthwhile and successful, it is necessary to create a structure where the likelihood of data loss is minimal and, in the event of an existing idea, the likelihood of data loss increases, revealing the gaps in the current ideas.

• Hu et al.²² developed the distance of a complex fuzzy set but this approach is less applicable due to the limitation of data loss. This approach utilizes Ramot’s CF structure and in the case of Ramot’s CFS, the value of membership grade belongs to the unit disc in a complex plan. The developed approach is based on Tamir’s CFS and in the case of Tamir’s CFSs the value of membership grade belongs to a a unit square in the complex plan. This case provides more space for decision-makers to handle more complex information. Moreover, in this case, extra fuzzy data can be handled to make decision-making effective and reliable.

The overall results are given in Table 7.

Table 7 Overall result.

Moreover, the geometrical representation of the results given in Table 7 is given in Fig. 8.

Fig. 8

Geometrical representation of data given in Table 7.

From the analysis of Fig. 8, we can say that all existing notions fail to handle the data in Table 6 and proposed theories can handle it easily making the defined approaches dominant to existing notions.

Sensitivity analysis

In this section, we will deliver the sensitivity analysis of the proposed work to establish the importance of the delivered approach. For this purpose, we will take different values of parameters used in initiated AOs and see the behavior of the delivered AOs. In this section, we will analyze the results for CFDWA AOs, and the overall results are given in Table 8.

Table 8 Results of sensitivity analysis.

Conclusion

A new and practical method for managing ambiguous and imprecise information in decision-making processes is provided by the suggested Cartesian form of complicated fuzzy Dombi aggregation operators. We have proposed the notion of CFDWA, CFDOWA, CFDHA, CFDWG, CFDOWG and CFDHG AOs. Moreover, this study shows how sophisticated fuzzy logic may be used in the prioritization of disability assistance systems by incorporating these operators, leading to more accurate and dependable results. In addition to boosting decision models’ adaptability in clinical systems, the presented methodology offers insightful information for better resource allocation in disability assistance systems. This work offers great promise for improving public services and meeting the requirements of people with disabilities by laying the groundwork for future developments in fuzzy decision-making techniques and their practical applications.

Since the introduced ideas are based on a complex fuzzy set in which the membership grade is regarded as a complex fuzzy number consisting of real and imaginary parts with the condition that these values must belong to [0, 1]. However, these developed ideas are limited due to the restriction of using only membership grades. Whenever the decision analysis wants to discuss non-membership grade in the form of a complex fuzzy number, then, in this case, the proposed idea fails to handle such kind of data. More if the data is available in the form of complex Pythagorean fuzzy numbers using the conditions that the sum of the square of the real part of (MG, NMG), the sum of the square of the imaginary part of (MG, NMG) must belong to [0, 1]. Then in this case the developed approach fails to handle.

In the future, we can extend these notions to intuitionistic fuzzy set theory as discussed in³¹ by adding the non-membership grade. Moreover, we can offer advanced structure in the form of a complex q-rung orthopair fuzzy set as given in³². The idea of a q-rung orthopair fuzzy set is a more generalized form of an intuitionistic fuzzy set. Additionally, to discuss the third membership grade which is called the abstinence grade in the structure, we can offer the notion of complex T-spherical fuzzy set theory as delivered in³³.

Furthermore, some useful abbreviation used throughout the article is given in Table 9.

Table 9 Some useful abbreviations.