Abstract
The selection of the most efficient actuator for biohybrid robots necessitates the implementation of precise and reliable decision-making (DM) methods. Dynamic aggregation operators (AOs) provide flexibility and consistency in DM by embracing time-dependent changes in data. The complex spherical fuzzy sets (CSFSs) adequately resolve multifaceted issue formulations characterized by spherical uncertainty and periodicity. This paper introduces two innovative AOs, namely, the complex spherical fuzzy dynamic Yager weighted averaging (CSFDYWA) operator and the complex spherical fuzzy dynamic Yager weighted geometric (CSFDYWG) operator. Notable characteristics of these operators are defined, and an enhanced score function is devised to rectify the deficiencies identified in the current score function in the CSF framework. In addition, the proposed operators are implemented to develop a methodical strategy for the multiple criteria decision-making (MCDM) situations to address the difficulties posed by inconsistent data during the selection procedure. These methodologies are also adeptly employed to address the MCDM problem, aiming to identify the most suitable actuator designed for precisely modelling human movement for biohybrid robots in CSF environment. Moreover, a comparative study is conducted to highlight the efficacy and legitimacy of the proposed methodologies in relation to the existing procedures.
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Introduction
This section is divided into four parts. “Background” section provides a detailed background of MCDM and its evolution through fuzzy set theories. “Literature review” section reviews the existing literature on fuzzy set extensions and their applications. “Research gaps and motivations of present study” section outlines the research gaps and motivations of the current study. “Objectives and contributions of current study” section highlights the objectives and major contributions of the present study and outlines the structure of the paper.
Background
MCDM is a methodical approach that aims to identify the optimal choice from a set of alternatives, enhancing the DM process by providing a formal framework for evaluating competing solutions based on their performance across several criteria. Nonetheless, this approach becomes incredibly challenging to implement when faced with ambiguous and unclear data. Decision-makers have traditionally perceived all information about access to alternatives as precise numerical data. Nevertheless, the growing uncertainty in data and the cognitive limitations faced by decision-makers frequently hinder their ability to articulate choices in precise numerical terms. Zadeh1 introduced the concept of FSs as an effective instrument for evaluating vague and ambiguous information, using fuzzy memberships that represent partial truth between the continuum of absolute truth and absolute falsehood. Research indicates that various domains consolidate traits and assess their disparities, thereby augmenting their reliance on FSs2. Atanassov3 came up with the creative idea of IFSs, which are an extension of FSs. An element’s membership grades in IFSs consist of two values, both within the closed unit interval [0,1]. One number represents positive membership support in these fuzzy sets, whereas the other negates this support, with their combined value residing between 0 and 1. Many scientists have proposed methodologies that use IFSs to tackle the challenges of incorporating and quantifying disparities across diverse features in multiple domains4,5,6. In scenarios where the aggregate of MD and NMD exceeds 1, more sophisticated approaches than IFSs are required. Yager7 developed the idea of PFSs as an extension to IFSs. In PFSs, the aggregate of the squares of MD and NMD lies within the closed unit interval. This enables decision-makers to represent fuzzy information with greater precision and flexibility compared to IFSs. Subsequently, Yager8 presented the theoretical foundation of q-ROFSs. In this framework, for any natural number q, (MD)q + (NMD)2 belongs to [0, 1], which gives a wider scope and improved capacity to address ambiguities relative to IFSs and PFSs.
Literature review
To enhance understanding of PFSs, readers are urged to consult references9,10,11,12, while the issues related to q-ROFSs are addressed in13,14,15,16,17. The concepts of IFS, PFS, and q-ROFS are relevant only in situations where FS theory is not applicable. These frameworks face challenges, especially in voting scenarios where opinions extend beyond simple ‘yes’ or ‘no’ classifications to include partial opposition and abstention. The neutral perspective is vital for representing the complexities of human cognition; however, it remains unaddressed in the aforementioned theories. Cuong18 introduced the concept of PicFSs to convey this information. These PicFSs include MD, NMD, and AD, and their cumulative total is limited to the closed unit interval. Qiyas et al.19 examined essential operational principles of PicFSs by utilizing Yager aggregation operations and incorporating them into the picture fuzzy context. However, PicFSs are unable to handle the situation where the summation of MD, NMD, and AD is greater than 1. Kahraman and Gundogdu20 proposed the concept of SFSs to address this issue, extending the concepts of PicFSs, where the square summation of MD, NMD, and AD remains within the closed unit interval [0, 1]. A multitude of successful studies grounded in SFS theory have been conducted across various scientific disciplines21,22,23,24,25.
Despite the emergence of many expansions of FS theory, the aforementioned models are insufficient for dealing with periodic information or two-dimensional phenomena. Romat et al.26,27 formulated the theoretical framework of CSFSs to address this issue. The authors proposed expanding the MD domain of CFSs from [0, 1] to a closed unit disk in the complex plane. Furthermore, they enhanced the concept of conventional FSs by including the phase term, which is crucial in the context of the DM process. This attribute differentiates CFSs as superior to other fuzzy theories. Using the idea of CFSs as a base, Alkouri and Salleh28 came up with CIFSs. These have complex-valued membership and non-membership functions inside a closed unit disk, and the sum of the real and imaginary parts of the MD and NMD can’t be greater than 1. This theory has been successfully implemented in various disciplines29,30,31. Nonetheless, the issue emerges when the decision-makers select the magnitudes of the real and imaginary components, resulting in a total that exceeds a closed unit disk. In acknowledgment of this constraint, Ullah et al.32 introduced a novel approach to CPFSs, wherein the square sum of the real and imaginary parts of these complex numbers is constrained within the closed unit disk. This concept’s effective implementation can be observed across multiple domains in33,34.
In addition to these scenarios, various issues and ambiguous situations arise in daily life, requiring the incorporation of options that convey neutral conduct within the datasets. During an election, we can classify voters into three distinct groups: supporters, opponents, and abstainers. Nonetheless, the previously described theories are inadequate for addressing two-dimensional scenarios. To address this deficiency, Akram et al.35 built upon the framework of CPFSs and introduced the novel concept of CPicFSs. This concept is characterized by the amplitude and phase components of MD, NMD, and AD, such that the sum of the real (imaginary) parts of MD, NMD, and AD remains inside the closed unit disk. The architecture of CPicFSs is crucial, as is its capacity to effectively manage human perception in a two-dimensional framework. However, challenges arise for CPicFSs when the sum of these three degrees exceeds 1. Ali et al.36 came up with the new framework of CSFSs to fix the problems with CPicFSs. CSFSs have membership, non-membership, and abstinence functions in a complex plane, alongside the more adaptable constraint that the squared sum of the real (imaginary) parts of MD, NMD, and AD remains inside the closed unit disk. The CSF system is superior to current two-dimensional methods. This remarkable topic has garnered considerable interest from researchers, as demonstrated by the multitude of studies focused on its examination. In order to highlight their significant attributes, Akram et al.37 proposed a hybrid DM technique that prioritizes critical success factor information using weighted averaging and geometric operators. Akram et al.38 devised the CSFOW AOs. They combined these with the CSFS VIKOR methodology to address challenges in group DM. The proposed method is based on the principle of prioritizing a conciliatory resolution by evaluating two critical factors: the collective advantage to the group and the personal remorse felt by the dissenting party. The study39 introduces complex spherical Dombi fuzzy aggregation operators for MCDM, showcasing their applicability in DM scenarios. Hussain et al.40 assessed the Aczel-Alsina operators inside the CSF environment and their application in the evaluation of power-battery vehicles. Akram et al.41 presented a strategic methodology for DM employing the ELECTRE method in the CSF context. Razzaque et al.42 suggested using Yager AOs in the realm of CSF systems to choose the best celestial object for space observation.
Research gaps and motivations of present study
The efficient processing of information and the ability to make decisions are indispensable components of an intelligent system that functions effectively. It is therefore essential to recognize that there are a variety of DM issues that cannot be resolved using various fuzzy environments, such as complex FSs, Cq-ROFSs, and CPFSs, with the sum of the MD, AD, and NMD lying outside the closed unit disk. The importance of these research gaps compels us to investigate the more robust theoretical framework of CSFSs, in which the square sum of these degrees lies within the region of a closed unit disk.
In addition, the application of AOs is a strategic approach for synthesizing and consolidating various data elements into manageable and meaningful resolutions. Dynamic AOs are required to handle ambiguous data and uncertainty evolving with time. Nevertheless, the current models utilizing specific t-norms (TNs) and t-conorms (TCNs) in the domain of CSF knowledge do not incorporate dynamic parameter adjustments according to the risk tolerance of decision-makers. Consequently, these current methodologies are unable to deliver precise estimations of the algebraic product and sum in a cohesive manner regarding their aggregation capabilities. This insensitivity may result in abrupt or less meaningful conclusions, rendering them less flexible and unable to account for individual preferences. Yager AOs (YAOs) are highly versatile, functioning effectively in decision-support systems across a wide range of scenarios, including ambiguous tasks. These operators have remarkable adaptability in managing operational contexts and show outstanding effectiveness in addressing DM problems. The Yager product and sum serve as efficient alternatives to the Einstein, Dombi, Hamacher, and other algebraic products in the context of intersection and union procedures. A notable feature of Yager aggregation operators is their capacity to dynamically modify parametric variables more efficiently, therefore yielding accurate estimations in a more seamless way than other existing approaches.
However, these current methodologies lack the ability to dynamically assess time-sensitive information, limiting their precision and adaptability. This rigidity could lead to a lack of real-time adaptability, an absence of prioritization over time, and suboptimal DM in dynamic systems. Since the accuracy of aggregated results is contingent upon recognizing the relevance of changing data points, there is a pressing need for more flexible solutions. In this regard, recently proposed dynamic Yager weighted AOs become particularly beneficial for facilitating prompt DM in complex and rapidly changing scenarios. Therefore, this study aims to develop and evaluate dynamic Yager AOs that are compatible with CSFSs, enabling the seamless adaptation of membership functions and aggregation methods in response to fluctuating conditions. Navigating varying degrees of uncertainty relies on adaptability; hence, traditional operators lacking the flexibility to consider time-related factors are less effective.
Objectives and contributions of current study
This study aims to achieve the following objectives:
-
i.
To propose an improved score function within the CSF framework to address limitations of existing score functions.
-
ii.
To formulate the two novel dynamic Yager weighted AOs, namely, the CSFDYWA and CSFDYWG operators, to streamline the aggregation of ambiguous data in CSF dynamic settings.
-
iii.
To investigate the important dynamic Yager operational laws for CSFNs and analyze the structural properties of the proposed operators.
-
iv.
To establish a step-by-step technique for the MCDM problem, utilizing the recently established approaches in the CSF environment.
-
v.
To demonstrate the efficacy of these strategies in identifying optimal bioactuators for biohybrid robots designed for human motion simulation and validate their utility through comparative analysis against existing techniques.
The primary contributions of this study are delineated as follows:
-
i.
We present an enhanced score function within the CSF framework to overcome the drawbacks of existing score function.
-
ii.
We explore the significant dynamic Yager operational laws for CSFNs.
-
iii.
We develop two innovative dynamic Yager weighted aggregation operators, referred to as the CSFDYWA and CSFDYWG operators, design to improve the integration of ambiguous data in CSF dynamic environments and examine the structural properties of these operators.
-
iv.
We design a systematic methodology addressing the MCDM problem, employing the newly developed strategies inside the CSF context.
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v.
By solving the MCDM problem of identifying the optimum bioactuators for biohybrid robots aimed at simulating human motion and confirming their intrinsic worth through a comparative analysis with existing methods.
We structure the upcoming sections of this research as follows: “Preliminaries” sectioncontains a brief exposition of the key definitions of CSFSs. “Formulization of enhanced score function for CSF settings” section develops an enhanced score function to effectively address the limitations of the existing score function for the DM challenges in the CSF framework. “Dynamic operations on CSFNs” section presents two novel dynamic Yager AOs, the CSFDYWA operator and the CSFDYWG operator, and investigates their basic features, including boundedness, monotonicity, and idempotency. “Developing and implementing mathematical algorithm for MCDM problems using the CSF dynamic knowledge” section provides a structured mathematical framework to address the intricate issues of MCDM that involve CSF information. In addition, we depict a numerical example of newly introduced methodologies used to handle the MCDM challenge of selecting an ideal bioactuator for biohybrid robots, enabling them to mimic human movement. Moreover, we conduct an in-depth comparison to demonstrate the feasibility and efficacy of these approaches compared to the established techniques. The conclusion of the research is provided in “Discussion” section.
Preliminaries
This section provides an overview of key terms related to CSFSs, including their vital characteristics, operations, and score function.
Definition 120
An SFS \(\Lambda\) over the universe \(E\) is defined as \(\Lambda =\left\{\left(x, \beth \left(x\right), \aleph \left(x\right),\Psi (x)\right) | x \in E\right\}\). Here \(\aleph , \beth\) and \(\Psi\) represent the functions which map the elements of \(E\) to \([0, 1]\) and are called abstinence, membership, and non-membership functions respectively. Note that 0 \(\le {\beth }^{2}(x)+{\aleph }^{2}(x)+{\Psi }^{2}(x) \le 1\). Moreover, the hesitance margin of \(\Lambda\) is expressed as:
Definition 226
A CFS \(\Lambda\) over the universe \(E\) is defined as \(\Lambda =\left\{\left(x, \beth \left(x\right)\right) \right| x \epsilon E\}\). Here \(\beth :E \to [0, 1]\) is a complex-valued membership function, defined by \(\beth \left(x\right)= \zeta \left(x\right). {e}^{i2\pi \varpi (x)}\). Also, \(\varpi \left(x\right)\) and \(\zeta \left(x\right)\) belong to \([0, 1]\) and are termed as phase and amplitude terms of \(\boldsymbol{\Lambda }\).
Definition 336
A CSFS \(\boldsymbol{\Lambda }\) over the universe \(E\) is defined as \(\boldsymbol{\Lambda }=\left\{\left(x, \beth \left(x\right), \aleph \left(x\right),\Psi \left(x\right)\right) \right| x \epsilon E\}\). Here, \(\beth , \aleph\) and \(\Psi\) are the functions which map the elements of \(E\) to \([0, 1]\) to the closed unit disk in a complex plane and are defined by \(\beth \left(x\right)= \zeta \left(x\right) \cdot {e}^{i2\pi \varpi (x)},\)\(\aleph \left(x\right)= \eta \left(x\right)\cdot {e}^{i2\pi \vartheta (x)}\), and \(\Psi \left(x\right)= \kappa \left(x\right)\cdot {e}^{i2\pi\Omega (x)}\). Also \(0\le \zeta \left(x\right), \eta \left(x\right), \kappa \left(x\right),\)\({\zeta }^{2}\left(x\right)+{\eta }^{2}\left(x\right) + {\kappa }^{2}\left(x\right)\le 1\) and 0 \(\le \varpi \left(x\right), \vartheta \left(x\right),\Omega (x),\)\({\varpi }^{2}(x)+{\vartheta }^{2}(x)+{\Omega }^{2}(x)\le 1\).
The term hesitancy margin of \(\boldsymbol{\Lambda }\) is represented by \(\gimel (x)=r\left(x\right){e}^{i2\pi \tau (x)}\) and is interpreted as:
For convenience, we express \(x=\left(\left(\zeta , \varpi \right), \left(\eta , \vartheta \right), \left(\kappa ,\Omega \right)\right)\) for \(x \epsilon E\) in the rest of the article. This specific depiction of \(x\) is designated as a CSFN, in which \(0\le \zeta , \eta , \kappa , {\zeta }^{2}+{\eta }^{2}+ {\kappa }^{2}\le 1\) and 0 \(\le \varpi , \vartheta ,\Omega , {\varpi }^{2}+{\vartheta }^{2}+{\Omega }^{2}\le 1\).
In addition, more relevant notions of the work presented in this article can be viewed in2,5,35,40.
Formulization of enhanced score function for CSF settings
The objective of this section is to clarify the constraints of the existing scoring function40 by proving an example where it does not work. So, we need to design a new scoring system to overcome this issue.
Example 1
Examine the CSFNs \(\tilde{\upsilon }_{1}\) = ((0.9, 0.1), (0.8, 0.7), (0.6, 0.5)) and \(\tilde{\upsilon }_{2}\) = ((0.5, 0.8), (0.6, \(0.9),\) (0.7, 0.39)). Applying scoring function40 to \(\tilde{\upsilon }_{1}\) and \(\tilde{\upsilon }_{2}\), it is evident that \({\mathcal{S}} (\tilde{\upsilon }_{1})\) = \({\mathcal{S}} (\tilde{\upsilon }_{2})\) = 0.51. Based on property 3 outlined in40, it is discernible that \(\tilde{\upsilon }_{1}\) and \(\tilde{\upsilon }_{2}\) are incomparable, i.e., \(\tilde{\upsilon }_{1} \sim _{2}\).
Consequently, the current scoring method demonstrates inadequacy in addressing specific conditions. This necessitates the enhancement of this function, resulting in the creation of a revised score function as delineated below.
Definition 4
For any CSFN, denoted as \(\tilde{\upsilon }=\left(\left(\zeta , \varpi \right), \left(\eta , \vartheta \right), \left(\kappa ,\Omega \right)\right)\). The modified scoring function for \(\tilde{\upsilon }\) is delineated as:
herein, \({\mathcal{S}} (\tilde{\upsilon })\) \(\in\) [0, 1].
Additionally, the comparison of two \({\tilde{\upsilon }}_{1}\) and \({\tilde{\upsilon }}_{2}\) within the context of the above score function is determined by the following principles, that is, if \({\mathcal{S}} ({\tilde{\upsilon }}_{1})\) \(>\) \({\mathcal{S}} ({\tilde{\upsilon }}_{2})\), then \({\tilde{\upsilon }}_{1}>{\tilde{\upsilon }}_{2}\), if \({\mathcal{S}}({\tilde{\upsilon }}_{1}\)) \(<\) \({\mathcal{S}}({\tilde{\upsilon }}_{2}\)), then \({\tilde{\upsilon }}_{1}<{\tilde{\upsilon }}_{2}\), and if \({\mathcal{S}}({\tilde{\upsilon }}_{1}\)) \(=\) \({\mathcal{S}}({\tilde{\upsilon }}_{2}\)), then \({\tilde{\upsilon }}_{1} \sim {\tilde{\upsilon }}_{2}\).
The succeeding example highlights the efficiency and veracity of the recently enacted score function designed specifically for CSFNs.
Example 2
Consider \(\tilde{\upsilon }_{1}\) = ((0.9, 0.1), (0.8, 0.7), (0.6, 0.5)) and \({\tilde{\upsilon }}_{2}\) = ((0.5, 0.8), (0.6, 0.9), (0.7, \(0.39)\)) are two CSFNs. Utilizing the structure as outlined in Definition 4 to these CSFNs \({\tilde{\upsilon }}_{1}\) and \({\tilde{\upsilon }}_{2}\), yields \({\mathcal{S}}(\tilde{\upsilon }_{1}\)) = 0.087 and \({\mathcal{S}}(\tilde{\upsilon }_{2}\)) = 0.074. In the light of 1st principle elucidated in Definition 4, it is apparent that \({\tilde{\upsilon }}_{1}\) is greater than \({\tilde{\upsilon }}_{2}\). The evidence at hand strongly signifies that \({\tilde{\upsilon }}_{1}\) is indeed preferable to \({\tilde{\upsilon }}_{2}\).
This indicates that the newly defined scoring function offers a better method for evaluating decisions.
Dynamic operations on CSFNs
This section elucidates the theory of CSF variables and delineates the fundamental dynamic Yager operational principles. Additionally, we design the CSFDYWA and CSFDYWG operators for integrating CSF data, particularly concerning time-independent parameters, and characterize their essential properties.
Dynamic Yager operational laws of CSFNs
This subsection introduces the notion of CSF variable and delves into several fundamental dynamic Yager operational principles that govern these variables.
Definition 5
Let us designate \(t\) as a time variable. The CSF variable \({\tilde{\upsilon }}_{t}\) is referred to as \(\left(\left({\zeta }_{t}, {\varpi }_{t}\right), \left({\eta }_{t}, {\vartheta }_{t}\right), \left({\kappa }_{t}, {\Omega }_{t}\right)\right)\), where \({\zeta }_{t}\), \({\varpi }_{t}\), \({\eta }_{t}\), \({\vartheta }_{t}\), \({\Omega }_{t}\in [\text{0,1}]\) and \(0\le {\left({\zeta }_{t}\right)}^{2}+{\left({\eta }_{t}\right)}^{2}+{\left({\kappa }_{t}\right)}^{2}, {\left({\varpi }_{t}\right)}^{2}+{\left({\vartheta }_{t}\right)}^{2}+{\left({\Omega }_{t}\right)}^{2}\le 1\).
While considering a CSF variable \({\tilde{\upsilon }}_{t}\), where \(t\) assumes values \({t}_{1},{t}_{2},\dots ,{t}_{p}\), it signifies the class of \(p\) CSF numbers collected at \(p\) distinct time periods, represented by \({\tilde{\upsilon }}_{{t}_{1}},{\tilde{\upsilon }}_{{t}_{2}},\dots ,{\tilde{\upsilon }}_{{t}_{p}}\).
Definition 6
Consider CSFNs \({\tilde{\upsilon }}_{{t}_{1}}=\left(\left({\zeta }_{{t}_{1}}, {\varpi }_{{t}_{1}}\right), \right.\)\(\left. \left({\eta }_{{t}_{1}}, {\vartheta }_{{t}_{1}}\right), \left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right)\right)\)\(\left. \left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right)) \right.\) and \({\tilde{\upsilon }}_{{t}_{2}}=\left(\left({\zeta }_{{t}_{2}}, {\varpi }_{{t}_{2}}\right), \right.\)\(\left. \left({\eta }_{{t}_{2}}, {\vartheta }_{{t}_{2}}\right), \left({\kappa }_{{t}_{2}}, {\Omega }_{{t}_{2}}\right)\right)\)\(\left. \left({\kappa }_{{t}_{2}}, {\Omega }_{{t}_{2}}\right)\right)\). The basic operations on \({\tilde{\upsilon }}_{{t}_{1}}\) and \({\tilde{\upsilon }}_{{t}_{2}}\) are delineated as follows:
\(({\text{i}})\qquad \tilde{\upsilon }_{{t_{1} }} \le \tilde{\upsilon }_{{t_{2} }}\) iff \({\zeta }_{{t}_{1}}\le {\zeta }_{{t}_{2}}, {\eta }_{{t}_{1}}\le {\eta }_{{t}_{2}}, {\kappa }_{{t}_{1}}\ge {\kappa }_{{t}_{2}}\) and \({\varpi }_{{t}_{1}}\le {\varpi }_{{t}_{2}}, {\vartheta }_{{t}_{1}}\le {\vartheta }_{{t}_{2}}, {\Omega }_{{t}_{1}}\ge {\Omega }_{{t}_{2}}\),
\((\text{ii})\qquad {\tilde{\upsilon }}_{{t}_{1}}^{c}\) = \(\left(\left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right), \left({\eta }_{{t}_{1}}, {\vartheta }_{{t}_{1}}\right), \left({\zeta }_{{t}_{1}}, {\varpi }_{{t}_{1}}\right)\right).\)
Definition 7
Consider CSFNs \({\tilde{\upsilon }}_{{t}_{1}}=\left(\left({\zeta }_{{t}_{1}}, {\varpi }_{{t}_{1}}\right), \right.\)\(\left. \left({\eta }_{{t}_{1}}, {\vartheta }_{{t}_{1}}\right), \right.\)\(\left. \left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{2}}=\left(\left({\zeta }_{{t}_{2}}, {\varpi }_{{t}_{2}}\right), \right.\)\(\left({\eta }_{{t}_{2}}, \right.\)\(\left. {\vartheta }_{{t}_{2}}\right),\)\(\left. \left({\kappa }_{{t}_{2}}, {\Omega }_{{t}_{2}}\right)\right)\), along with \({\mathfrak{U}}\) > 0 and \(\mathcal{X}>0\). Herein, we express the basic dynamic Yager operations for CSFNs in the subsequent way:
Structural analysis of the CSFDYWA operator
Here, we examine the CSFDYWA operator and delineate its core attributes.
Definition 8
Let there be a class \(\mathcal{L}\) of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{1}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\) with \(i\) spans 1 to \(m\). In addition, let \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\) be the weight vector related to the time periods \({t}_{i}\), with \({\beta }_{{t}_{i}}\in [0, 1]\) subject to the constraint \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). The CSF dynamic Yager weighted averaging operator is a mapping CSFDYWA: \({\mathcal{L}}^{m} \to \mathcal{L}\), given by:
Theorem 1
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\), along with their related weight vector \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), where each \({\beta }_{{t}_{i}}\in [0, 1]\) satisfies the constraint \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). In the context of CSFDYWA operator, the aggregation process preserves the CSFN structure, yielding the following formulation:
Proof
The theorem’s validity is substantiated through mathematical induction. Take into account the initial case, when \(m=2\). We come across two CSFNs represented as \({\tilde{\upsilon }}_{{t}_{1}}=\left(\left({\zeta }_{{t}_{1}}, {\varpi }_{{t}_{1}}\right), \left({\eta }_{{t}_{1}}, {\vartheta }_{{t}_{1}}\right), \left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{2}}=\left(\left({\zeta }_{{t}_{2}}, {\varpi }_{{t}_{2}}\right), \left({\eta }_{{t}_{2}}, {\vartheta }_{{t}_{2}}\right), \left({\kappa }_{{t}_{2}}, {\Omega }_{{t}_{2}}\right)\right).\) By utilizing the procedures designed in Definition 7 for the CSFNs, we obtain the subsequent expressions:
and
Based on Definition 8, the aggregated value of \({\tilde{\upsilon }}_{1}\) and \({\tilde{\upsilon }}_{2}\) is outlined below:
That being the case:
Thus, the theorem holds for \(m=2.\) Presuming its truth for m = k > 2, we derive:
Subsequently, we prove that the result holds true for \(m=k+1.\)
Evidently, this shows that
The assertion is applicable when \(m=k+1\). Hence, invoking mathematical induction, its validity is established for all \(m \in {\mathbb{Z}}^{+}\).
The following illustrated example substantiates the theoretical assertion of Theorem 1.
Example 3
Suppose \({\tilde{\upsilon }}_{{t}_{1}}=\left(\left(0.4, 0.8\right), \left(0.7, 0.27\right), \left(0.5, 0.6\right)\right),\)\(\tilde{\upsilon }_{{t_{2} }} = \left( {\left( {0.13,0.22} \right),\left( {0.36,0.1} \right),\left( {0.25,0.6} \right)} \right),\)\({\tilde{\upsilon }}_{{t}_{3}}=\left(\left(0.26, 0.74\right), \left(0.64, 0.35\right), \left(0.55, 0.3\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{4}}=\left(\left(0.63, 0.42\right), \left(0.72, 0.81\right), \left(0.16, 0.21\right)\right)\) are any four CSFNs. Take \(\beta ={\left(0.11, 0.17, 0.32, 0.4\right)}^{T}\) as weight vector of the given four time points and \(\mathcal{X}=3\). To amalgamate these values using CSFDYWA operator, we have.
Similarly,
It follows that
Therefore, it deduced that the previously stated discussion again results in a CSFN.
The forthcoming result proves the idempotency property of CSFNs under the dynamic Yager weighted averaging operator framework.
Theorem 2
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\) and the weight vector \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\) related to \({t}_{i}\), adheres to the constraint \({\beta }_{{t}_{i}}\in [0, 1]\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). If \({\tilde{\upsilon }}_{{t}_{i}}={\tilde{\upsilon }}_{{t}_{o}}\) ∀ \(i\), where \({\tilde{\upsilon }}_{{t}_{o}}=\left(\left({\zeta }_{{t}_{o}}, {\varpi }_{{t}_{o}}\right), \left({\eta }_{{t}_{o}}, {\vartheta }_{{t}_{o}}\right), \left({\kappa }_{{t}_{o}}, {\Omega }_{{t}_{o}}\right)\right)\). Then.
Proof
Upon observing the condition \({\tilde{\upsilon }}_{{t}_{i}}={\tilde{\upsilon }}_{{t}_{o}}\) ∀ \(i\). We have, \({{\zeta }_{{t}_{i}}={\zeta }_{{t}_{o}}, \varpi }_{{t}_{i}}={\varpi }_{{t}_{o}},\)\({\eta }_{{t}_{i}}={\eta }_{{t}_{o}}, {\vartheta }_{{t}_{i}}={\vartheta }_{{t}_{o}}, {\kappa }_{{t}_{i}}={\kappa }_{{t}_{o}},\) and \({\Omega }_{{t}_{i}}={\Omega }_{{t}_{o}}\). By substituting the above values in Eq. 4, we obtain.
Consequently,
The following result demonstrates that each class of CSFNs satisfies the monotonicity property within the context of the CSFDYWA operator.
Theorem 3
Let there be two classes of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{i}}^{*}=\left(\left({\zeta }_{{t}_{i}}^{*}, {\varpi }_{{t}_{i}}^{*}\right), \left({\eta }_{{t}_{i}}^{*}, {\vartheta }_{{t}_{i}}^{*}\right), \left({\kappa }_{{t}_{i}}^{*}, {\Omega }_{{t}_{i}}^{*}\right)\right)\), at various time instants \({t}_{i}\). The weight vector related to \({t}_{i}\) is \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), satisfying the condition \(0\le {\beta }_{{t}_{i}}\le 1\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). Assuming that the following requirements are satisfied for each \(i\): \({\zeta }_{{t}_{i}}\le {\zeta }_{{t}_{i}}^{*}, {\varpi }_{{t}_{i}}\le {\varpi }_{{t}_{i}}^{*}, {\eta }_{{t}_{i}}\le {\eta }_{{t}_{i}}^{*}, {\vartheta }_{{t}_{i}}\le {\vartheta }_{{t}_{i}}^{*}, {\kappa }_{{t}_{i}}\ge {\kappa }_{{t}_{i}}^{*}\), and \({\Omega }_{{t}_{i}}\ge {\Omega }_{{t}_{i}}^{*}\). Then.
Proof
Given that \({\zeta }_{{t}_{i}}\le {\zeta }_{{t}_{i} }^{*} \Rightarrow {\zeta }_{{t}_{i} }^{2\mathcal{X}}\le {\left({\zeta }_{{t}_{i} }^{*}\right)}^{2\mathcal{X}}\) \(\Rightarrow \sum_{i=1}^{m}{\beta }_{{t}_{i}}.{\zeta }_{{t}_{i} }^{2\mathcal{X}} \le \sum_{i=1}^{m}{\beta }_{{t}_{i}}.{\left({\zeta }_{{t}_{i} }^{*}\right)}^{2\mathcal{X}}\)
This implies the following
Analogously, we establish the following relations by applying the preceding mathematical procedure within the context of the inequalities \({\varpi }_{{t}_{i}}\le {\varpi }_{{t}_{i}}^{*}, {\eta }_{{t}_{i}}\le {\eta }_{{t}_{i}}^{*}, {\vartheta }_{{t}_{i}}\le {\vartheta }_{{t}_{i}}^{*}\):
In addition, considering \({\kappa }_{{t}_{i}}\ge {\kappa }_{{t}_{i}}^{*}\), we deduce that \(1-{\left({\kappa }_{{t}_{i}}\right)}^{2\mathcal{X}}\le {1-\left({\kappa }_{{t}_{i}}^{*}\right)}^{2\mathcal{X}}\)
As a result, it follows that
Similarly, the following expression is derived by employing the aforementioned mathematical procedure for \({\Omega }_{{t_{i} }} \ge {\Omega }_{{t_{i} }}^{*}\):
By comparing the relationships from 5 to 10 and employing Definition 6, we conclude the following:
\({\text{CSFDYWA}} \left( {\tilde{\upsilon }_{{t_{1} }} ,\tilde{\upsilon }_{{t_{2} }} , \ldots ,\tilde{\upsilon }_{{t_{m} }} } \right) \le {\text{CSFDYWA}} \left( {\tilde{\upsilon }_{{t_{1} }}^{*} ,\tilde{\upsilon }_{{t_{2} }}^{*} , \ldots , \tilde{\upsilon }_{{t_{m} }}^{*} } \right)\).
Within the framework of the CSFDYWA operator, any finite set of CSFNs adheres to the boundedness property, as illustrated by the subsequent result.
Theorem 4
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \right.\)\(\left({\eta }_{{t}_{i}}, \right.\)\(\left. {\vartheta }_{{t}_{i}}\right),\)\(\left({\kappa }_{{t}_{i}},\right.\)\(\left. \left.{\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\). The weight vector related to \({t}_{i}\) is \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), satisfying the condition \(0\le {\beta }_{{t}_{i}}\le 1\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). Additionally, let \({\tilde{\upsilon }}^{-}=\left\{\left(\underset{i}{\text{min}}\left\{{\zeta }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\varpi }_{{t}_{i}}\right\}\right), \left(\underset{i}{\text{max}}\left\{{\eta }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\vartheta }_{{t}_{i}}\right\}\right), \left(\underset{i}{\text{max}}\left\{{\kappa }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\Omega }_{{t}_{i}}\right\}\right)\right\}\) and \({\tilde{\upsilon }}^{+}=\left\{\left(\underset{i}{\text{max}}\left\{{\zeta }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\varpi }_{{t}_{i}}\right\}\right),\left(\underset{i}{\text{min}}\left\{{\eta }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\vartheta }_{{t}_{i}}\right\}\right),\left(\underset{i}{\text{min}}\left\{{\kappa }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\Omega }_{{t}_{i}}\right\}\right)\right\}\) be the lower and upper bonds of these CSFNs. Then.
Proof
By observing the result derived through the application of CSFDYWA operator to the set of CSFN, denoted as \(\text{CSFDYWA} \left({\tilde{\upsilon }}_{{t}_{1}},{\tilde{\upsilon }}_{{t}_{2}}, \dots ,{\tilde{\upsilon }}_{{t}_{m}}\right)\) = \(\left(\left({\zeta }_{t}, {\varpi }_{t}\right), \left({\eta }_{t}, {\vartheta }_{t}\right), \left({\kappa }_{t}, {\Omega }_{t}\right)\right)\).
With respect to each \({\tilde{\upsilon }}_{{t}_{i}}\), \(\underset{i}{\text{min}}\left\{{\zeta }_{{t}_{i}}\right\}\le {\zeta }_{{t}_{i}}\le \underset{i}{\text{max}}\left\{{\zeta }_{{t}_{i}}\right\}\)
Because \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\), we inferred that
Hence,
Similarly, the succeeding relation is produced by applying the above mathematical steps to the inequality:
Moreover, by considering \(\underset{i}{\text{max}}\left\{{\eta }_{{t}_{i}}\right\}\le {\eta }_{{t}_{i}}\le \underset{i}{\text{min}}\left\{{\eta }_{{t}_{i}}\right\}\) \(\Rightarrow {\left(\underset{i}{\text{max}}\left\{{\eta }_{{t}_{i}}\right\}\right)}^{2}\le {\left({\eta }_{{t}_{i}}\right)}^{2}\le {\left(\underset{i}{\text{min}}\left\{{\eta }_{{t}_{i}}\right\}\right)}^{2}\) \(\Rightarrow 1-{\left(\underset{i}{\text{min}}\left\{{\eta }_{{t}_{i}}\right\}\right)}^{2}\le 1-{\left({\eta }_{{t}_{i}}\right)}^{2}\le {1-\left(\underset{i}{\text{max}}\left\{{\eta }_{{t}_{i}}\right\}\right)}^{2}\)
Because \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\), we deduce that
Therefore,
Furthermore, the subsequent expressions are obtained by implementing the aforementioned mathematical procedures using the given conditions:
And
By comparing the expressions from 11 to 16, we have
Structural properties of CSFDYWG operator
In this subsection, we present the notion of complex spherical fuzzy dynamic Yager weighted geometric (CSFDYWG) operator and elucidate its fundamental properties.
Definition 9
Let there be a class \(\mathcal{L}\) of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{1}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\). Moreover, let \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\) represent the weight vector related to the time periods \({t}_{i}\), with \({\beta }_{{t}_{i}}\in [0, 1]\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). The CSF dynamic Yager weighted geometric operator is a function CSFDYWG: \({\mathcal{L}}^{m} \to \mathcal{L}\), given by:
The ensuing outcome indicates that the application of the CSFDYWG operator yields a CSFNs, when amalgamating a finite set of CSFNs.
Theorem 5
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\). Moreover, let \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\) represent the weight vector related to the time periods \({t}_{i}\), with \({\beta }_{{t}_{i}}\in [0, 1]\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\). In the structure of CSFDYWG operator, the aggregated value derived from these CSFNs represents the CSFN itself, as obtained through the following expression:
Proof
The application of mathematical induction allows us to ensure the validity of the theorem. Take into account the initial case, when \(m=2\). We come across two CSFNs represented as \({\tilde{\upsilon }}_{{t}_{1}}=\left(\left({\zeta }_{{t}_{1}}, {\varpi }_{{t}_{1}}\right), \left({\eta }_{{t}_{1}}, {\vartheta }_{{t}_{1}}\right), \left({\kappa }_{{t}_{1}}, {\Omega }_{{t}_{1}}\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{2}}=\left(\left({\zeta }_{{t}_{2}}, {\varpi }_{{t}_{2}}\right), \left({\eta }_{{t}_{2}}, {\vartheta }_{{t}_{2}}\right), \left({\kappa }_{{t}_{2}}, {\Omega }_{{t}_{2}}\right)\right).\) Utilizing the operations formulated in Definition 7 for CSFNs, we derive the following mathematical formulations:
and
By virtue of Definition 9, the combined value of \({\tilde{\upsilon }}_{{t}_{1}}\) and \({\tilde{\upsilon }}_{{t}_{2}}\) is calculated as follows:
It follows that.
Therefore, we have adequately established the theorem’s authenticity in the fundamental case involving \(m=2\).
Subsequently, we proceed to the induction phase, herein we assert the validity of the theorem for \(m=k>2\).
As a result, it follows that
Moreover, consider
Consequently,
The assertion is applicable when \(m=k+1\). Hence, invoking mathematical induction, its validity is established for all \(m \in {\mathbb{Z}}^{+}\).
The following illustrated example substantiates the theoretical assertion of Theorem 5.
Example 4
Consider any four CSFNs, denotes as \({\tilde{\upsilon }}_{{t}_{1}}\) = (0.73, 0.64), (0.45, 0.27), (0.3, 0.51), \({ \tilde{\upsilon }}_{{t}_{2}}\) = (0.34, 0.42), (0.62, 0.13), (0.57, 0.7),\({\tilde{\upsilon }}_{{t}_{3}}\) = (0.6, 0.44), (0.37, 0.8), (0.25, 0.34), and \({\tilde{\upsilon }}_{{t}_{4}}\) = (0.1, 0.4), (0.77, 0.31), (0.28, 0.5). Take \(\beta ={\left(0.11, 0.17, 0.32, 0.4\right)}^{T}\) as weight vector of the given four time points and \(\mathcal{X}=5\). To amalgamate these values using CSFDYWG operator, we get.
\(\begin{aligned} \left( {\mathop \sum \limits_{i = 1}^{4} \beta_{{t_{i} }} .\left( {1 - \zeta_{{t_{i} }}^{2} } \right)^{{\mathcal{X}}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 {\mathcal{X}}}} \right. \kern-0pt} {\mathcal{X}}}}} = \, & \left( {0.11\left( {1 - 0.73^{2} } \right)^{5} + 0.17\left( {1 - 0.34^{2} } \right)^{5} + 0.32\left( {1 - 0.6^{2} } \right)^{5} + 0.4\left( {1 - 0.1^{2} } \right)^{5} } \right)^{1/5} \\ { = } & { 0}{\text{.8737}}{.} \\ \end{aligned}\)
Similarly,
It follows that
Hence, it infers that the prior discussion yet again leads to a CSFN.
The following result substantiates the idempotent nature of CSFNs under the CSFDYWG operator.
Theorem 6
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\) and the weight vector associated with \({t}_{i}\) is \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), adhere to the condition \({\beta }_{{t}_{i}}\in [0, 1]\) with \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). If \({\tilde{\upsilon }}_{{t}_{i}}={\tilde{\upsilon }}_{{t}_{o}}\) for all \(i\), where \({\tilde{\upsilon }}_{{t}_{o}}=\left(\left({\zeta }_{{t}_{o}}, {\varpi }_{{t}_{o}}\right), \left({\eta }_{{t}_{o}}, {\vartheta }_{{t}_{o}}\right), \left({\kappa }_{{t}_{o}}, {\Omega }_{{t}_{o}}\right)\right)\) being a CSFN itself. Then we can establish that:
Proof
By replicating the method employed in Theorem 2, the desired result follows immediately.
The subsequent result verifies that each class of CSFNs adheres to the monotonicity property within the framework of the CSFDYWG operator.
Theorem 7
Let there be two classes of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \left({\eta }_{{t}_{i}}, {\vartheta }_{{t}_{i}}\right), \left({\kappa }_{{t}_{i}}, {\Omega }_{{t}_{i}}\right)\right)\) and \({\tilde{\upsilon }}_{{t}_{i}}^{*}=\left(\left({\zeta }_{{t}_{i}}^{*}, {\varpi }_{{t}_{i}}^{*}\right), \left({\eta }_{{t}_{i}}^{*}, {\vartheta }_{{t}_{i}}^{*}\right), \left({\kappa }_{{t}_{i}}^{*}, {\Omega }_{{t}_{i}}^{*}\right)\right)\), at various time instants \({t}_{i}\). The weight vector related to \({t}_{i}\) is \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), where \(0\le {\beta }_{{t}_{i}}\le 1\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). Assuming that the subsequent conditions adhere to each \(i\): \({\zeta }_{{t}_{i}}\le {\zeta }_{{t}_{i}}^{*}, {\varpi }_{{t}_{i}}\le {\varpi }_{{t}_{i}}^{*}, {\eta }_{{t}_{i}}\le {\eta }_{{t}_{i}}^{*}, {\vartheta }_{{t}_{i}}\le {\vartheta }_{{t}_{i}}^{*}, {\kappa }_{{t}_{i}}\ge {\kappa }_{{t}_{i}}^{*}\), and \({\Omega }_{{t}_{i}}\ge {\Omega }_{{t}_{i}}^{*}\). Then.
Proof
By replicating the method employed in Theorem 3, the desired result follows immediately.
The following result confirms boundedness for finite CSFN sets under CSFDYWG operator.
Theorem 8
Let there be a class of CSFNs, expressed as \({\tilde{\upsilon }}_{{t}_{i}}=\left(\left({\zeta }_{{t}_{i}}, {\varpi }_{{t}_{i}}\right), \right.\)\(\left({\eta }_{{t}_{i}}, \right.\)\(\left. {\vartheta }_{{t}_{i}}\right),\)\(\left({\kappa }_{{t}_{i}}, \right.\)\(\left. \left. {\Omega }_{{t}_{i}}\right)\right)\), at various time instants \({t}_{i}\). The weight vector related to \({t}_{i}\) is \(\beta ={\left({\beta }_{{t}_{1}}, {\beta }_{{t}_{2}}, \dots , {\beta }_{{t}_{m}}\right)}^{T}\), where \(0\le {\beta }_{{t}_{i}}\le 1\) such that \(\sum_{i=1}^{m}{\beta }_{{t}_{i}}=1\) and \(\mathcal{X}>0\). Moreover, let \({\tilde{\upsilon }}^{-}\)\(=\left\{\left(\underset{i}{\text{min}}\left\{{\zeta }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\varpi }_{{t}_{i}}\right\}\right), \right.\)\(\left. \left(\underset{i}{\text{min}}\left\{{\eta }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\vartheta }_{{t}_{i}}\right\}\right), \right.\)\(\left. \left(\underset{i}{\text{max}}\left\{{\kappa }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\Omega }_{{t}_{i}}\right\}\right)\right\}\) and \({\tilde{\upsilon }}^{+}\) = \(\left\{\left(\underset{i}{\text{max}}\left\{{\zeta }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\varpi }_{{t}_{i}}\right\}\right), \right.\)\(\left. \left(\underset{i}{\text{max}}\left\{{\eta }_{{t}_{i}}\right\}, \underset{i}{\text{max}}\left\{{\vartheta }_{{t}_{i}}\right\}\right),\left(\underset{i}{\text{min}}\left\{{\kappa }_{{t}_{i}}\right\}, \underset{i}{\text{min}}\left\{{\Omega }_{{t}_{i}}\right\}\right)\right\}\) be the lower and upper bonds of these CSFNs. Then.
Proof
By replicating the method employed in Theorem 4, the desired result follows immediately.
Developing and implementing mathematical algorithm for MCDM problems using the CSF dynamic knowledge
In this section, we present an algorithm to navigate MCDM problems related to the CSF information, utilizing the recently suggested powerful CSF dynamic Yager AOs.
For this, suppose \(\mathcal{Q}=\left({\mathcal{Q}}_{1}, {\mathcal{Q}}_{2},{\mathcal{Q}}_{3},\dots , {\mathcal{Q}}_{m}\right)\) is a discrete set of alternatives. Additionally, let us contemplate a class of attributes represented by \(\mathcal{C}=\left({\mathcal{C}}_{1}, {\mathcal{C}}_{2}, {\mathcal{C}}_{3}, \dots , {\mathcal{C}}_{n}\right)\), wherein each is accompanied by a corresponding weight vector, denoted as \(\delta ={\left[{\delta }_{1}, {\delta }_{2}, {\delta }_{3}, \dots , {\delta }_{n}\right]}^{T}\), where \(0\le {\delta }_{j}\le 1\) and \(\sum_{j=1}^{n}{\delta }_{j}=1\). Moreover, let \(\omega ={\left({\omega }_{{t}_{1}}, {\omega }_{{t}_{2}}, \dots , {\omega }_{{t}_{p}}\right)}^{T}\) signify a weight vector related to the various time periods \({t}_{k}\), where \(k\) is from \(1\) to \(p\), such that \({\omega }_{{t}_{k}}\in [0, 1]\), adhere to the condition \(\sum_{k=1}^{p}{\omega }_{{t}_{k}}=1\).
Consider the CSF dynamic decision matrices, denoted by \({\mathcal{D}}_{{t}_{k}}=\)\(\left( {{\mathcal{Q}}_{{ij\left( {t_{k} } \right)}} } \right)_{{m \times n}} = \left( {\left( {\zeta _{{ij\left( {t_{k} } \right)}} ,\varpi _{{ij\left( {t_{k} } \right)}} } \right),} \right.\)\(\left. \left({\eta }_{ij\left({t}_{k}\right)}, {\vartheta }_{ij\left({t}_{k}\right)}\right), \right.\)\(\left({\kappa }_{ij\left({t}_{k}\right)}, {\omega }_{ij\left({t}_{k}\right)}\right)_{{m \times n}}\), across \(p\) distinct time periods \({t}_{k}\). The matrices encode \({\zeta }_{ij\left({t}_{k}\right)}\) and \({\varpi }_{ij\left({t}_{k}\right)}\) as indicators of alternative \({\mathcal{Q}}_{i}\)’s adherence to criterion \({\mathcal{C}}_{j}\) at \({t}_{k}\), whereas \({\eta }_{ij\left({t}_{k}\right)}\) and \({\vartheta }_{ij\left({t}_{k}\right)}\) express its sustained conformity. In contrast, \({\kappa }_{ij\left({t}_{k}\right)}\) and \({\Omega }_{ij\left({t}_{k}\right)}\) signify the extent of its deviation. It is essential to acknowledge that these values are restricted to the closed interval [0,1] and comply to the constraints \(0\le {\left({\zeta }_{ij\left({t}_{k}\right)}\right)}^{2}+ {\left({\eta }_{ij\left({t}_{k}\right)}\right)}^{2}+{\left({\kappa }_{ij\left({t}_{k}\right)}\right)}^{2}, {\left({\varpi }_{ij\left({t}_{k}\right)}\right)}^{2}+{\left({\vartheta }_{ij\left({t}_{k}\right)}\right)}^{2} + {\left({\Omega }_{ij\left({t}_{k}\right)}\right)}^{2} \le 1\), where \(i=1, 2, \dots , m\) and \(j=1, 2, \dots , n\).
The essential steps to arrange and select the most advantageous alternative based on the subsequent description, within the framework of proposed techniques are outlined as follows:
Algorithm for MCDM problem using CSFDYWA and CSFDYWG operators
Input: Set of alternatives \(\left({\mathcal{Q}}_{1}, {\mathcal{Q}}_{2},{\mathcal{Q}}_{3},\dots , {\mathcal{Q}}_{m}\right)\) and criteria \(\left({\mathcal{C}}_{1}, {\mathcal{C}}_{2}, {\mathcal{C}}_{3}, \dots , {\mathcal{C}}_{n}\right)\), with decision matrices obtained under different time points \({t}_{k}\).
Output: Optimal alternative based on calculated preference and score values.
Step 1 Accumulation of the CSF dynamic decision matrices
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i.
Accumulation of the CSF Dynamic Decision Matrices \(\mathcal{D}={\left({\mathcal{Q}}_{ij}\right)}_{m\times n}={\left(\left({\zeta }_{ij}, {\varpi }_{ij}\right), \left({\eta }_{ij}, {\vartheta }_{ij}\right), \left({\kappa }_{ij}, {\Omega }_{ij}\right)\right)}_{m\times n}\) by combining CSF dynamic decision matrices \({\mathcal{D}}_{{t}_{k}}\) across time points.
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ii.
Use the CSFDYWA or CSFDYWG operators for the aggregation:
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For CSFDYWA:
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For CSFDYWG:
Step 2 Calculation of Preference Values
For each alternative \({\mathcal{Q}}_{i}\) , calculate the preference values \({\mathcal{G}}_{i}\) from the accumulated decision matrix \(\mathcal{D}\) using the CSFYWA/CSFYWG operator:
-
For CSFYWA:
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For CSFYWG:
Step 3 Assessment of Score Values
Compute the score values \({\mathcal{S}} \left({\mathcal{G}}_{i}\right)\) for each \({\mathcal{Q}}_{i}\), according to Definition 4 in the manuscript.
Step 4 Selection of the Optimal Alternative
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i.
Rank the alternatives \({\mathcal{Q}}_{i}\) based on their score values \({\mathcal{S}} \left({\mathcal{G}}_{i}\right)\).
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ii.
Select the alternative with the highest score value as the most advantageous option.
The flow chart in Fig. 1 illustrates the diagrammatic view of the above methodologies.
Diagrammatical view of the algorithm using proposed strategies.
Implementation
This subsection highlights the utilization of the proposed strategies through the provision of a numerical example.
Case study (advanced bio-actuator selection for efficient human motion simulation in biohybrid robots using CSF dynamic knowledge)
Biohybrid robot technology43,44,45 signifies a revolutionary success in the field of robotics, as it integrates biological components with conventional robotic structures to generate systems that exhibit the characteristics of both mechanical devices and living organisms. This fusion creates intelligent, dynamic systems capable of integrating the unique functionalities exhibited by living organisms, including but not limited to growth, regeneration, metamorphosis, biodegradation, and adaptation to dynamic settings. Besides, these bio living materials in biological systems exhibit properties such as biodegradability, intrinsic compliance, and the capability of enabling flexible actuation and control. These properties align with advancements in multitype chatter detection via multichannel internal and external signals, which enhance precision in robotic movement control46. A key feature is actuation, essential for movement, where biohybrid actuators use biological muscles or cells alongside artificial structures. They derive inspiration from the locomotion and interaction patterns observed in biological beings. Such inspiration also contributes to modern trajectory-tracking control techniques that allow optimal and accurate responses in biohybrid robots47. Yu et al.48 investigated adaptive gait training for lower limb rehabilitation robots by the assessment of human–robot interaction forces. In49, integrated actuation and sensor strategies that enhance intelligent soft robotics were evaluated. Wang et al.50 assessed the contractile and tensile characteristics of molecular artificial muscles, thereby advancing biohybrid robotics, while recent developments in closed-loop adaptive control for quadruped robots51, impedance-based stiffness control in medical and rehabilitation robots52,53, and worm-inspired creeping gait strategies in wheeled mobile robots54 further contribute to the sophistication and adaptability of actuation and control in biohybrid systems.
During the nineteenth and twentieth centuries, an inherent advantage of biohybrid robots, as opposed to traditional robots, was their ability to incorporate biological cells into an actuator.
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I.
First time-period (1994 to 2000).
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The year 1994 marked the initial emergence of biohybrid robots.
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The radical envelope-of-noise hypothesis and evolutionary robotics were proposed in 1997.
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In 2000, biological life was reproduced through complex, autocatalyzing chemical reactions, combined computational and experimental approach.
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II.
Second time period (2001 to 2008).
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The proposal for braided pneumatic actuators with nonlinear force-length properties came in 2001.
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In 2008, A rapid prototyping techniques emerged for tissue engineering applications. For further advancement in the realm of biohybrid robots, readers are encouraged to look up references.
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III.
Third time period (2009 to 2016).
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The technique of using balloons to aid the aerial movement of radio-controlled insect biobots was showcased in 2009.
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In 2010, the muscle-powered cantilever for micro-tweezers with rat myotube actuators was introduced.
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IV.
Fourth Time Period (2017 to 2024).
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In 2017, a hexapod robot using soft dielectric elastomer actuators was designed.
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In 2018, bio-mimetic approach was employed to design soft robots that imitate the intricate motion patterns of octopus tentacles.
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As of 2024, the team from Southeast University in Nanjing, China is leading modern research that aims to combine actuation, the capability to move and interact with the surroundings—with sensing, which entails gathering data about the environment. This integration is crucial for the advancement of soft robotics, enabling the development of autonomous robots that can independently respond to and adjust to their environment.
The subsequent discourse presents a numerical illustration of a solution to the MCDM problem, thereby demonstrating the effectiveness of the novel suggested methodologies.
A robotics engineer has commenced research on biohybrid soft robots at Techbio Laboratory, a leading and prestigious research organization. His objective is to develop the next generation of biohybrid robots that can perform complex industrial tasks precisely. One of the most important aspects of this quest is to choose the most appropriate actuator to power these advanced robots so that he can optimize their strength and effectiveness. He designs a DM dilemma where the actuators previously outlined are considered as potential options. Let \(\left\{{\mathcal{Q}}_{1}=\text{ Electroactive polymers }\left(\text{EAPs}\right), {\mathcal{Q}}_{2}=\text{Hydrogel actuators}, {\mathcal{Q}}_{3}=\text{Piezoelectric actuators}, {\mathcal{Q}}_{4}=\text{ Skeletal muscles actuators}\right\}\) be the class of four alternatives that need to be selected for enhancing the durability and efficacy of the biohybrid robots. The weight vector specifies a number of parameters that can be used to choose targets, such as; \({\mathcal{C}}_{1}\): Biocompatibility, \({\mathcal{C}}_{2}\): Durability, \({\mathcal{C}}_{3}\): Energy Efficiency, and \({\mathcal{C}}_{4}\): Control and Feedback Systems.
In order to construct a CSFN, these components can be further subdivided into two separate traits, which are detailed below:
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Biocompatibility consists of material interactions and chemical compatibility.
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Durability consists of lifespan and biodegradation.
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Energy efficiency consists of power consumption and heat generation.
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Control and feedback Systems consist of integration with sensors and ease of control.
The robotic engineer is assigned the responsibility of assessing the four potential alternatives \({\mathcal{Q}}_{1}\), \({\mathcal{Q}}_{2}\), \({\mathcal{Q}}_{3}\), and \({\mathcal{Q}}_{4}\). This assessment will be based on the CSF information for the four attributes \({\mathcal{C}}_{1}\), \({\mathcal{C}}_{2}\), \({\mathcal{C}}_{3}\), and \({\mathcal{C}}_{4}\) within the designated time periods \({t}_{1}\), \({t}_{2}\), and \({t}_{3}\), which correspond to the years 2001 to 2008, 2009 to 2016, and 2017 to 2024, respectively. According on the experts’ input, the weight vector for the specified time periods is \(\omega\) = \({\left[\text{0.19,0.25,0.56}\right]}^{T}\). Similarly, the associated weight vector for the attributes is given as \(\delta\) = \({\left[0.21, 0.27, 0.29, 0.23\right]}^{T}\). His evaluation of the dependability of each alternative \({\mathcal{Q}}_{i}\) for each attribute \({\mathcal{C}}_{j}\) throughout the defined time periods \({t}_{k}\) is succinctly displayed in the CSF dynamic decision matrices \({\mathcal{D}}_{{t}_{k}}\), found in Tables 1, 2, and 3, respectively, with entries represented as CSFNs.
Step 1: Consolidate all the CSF decision matrices \({\mathcal{D}}_{{t}_{k}}\) into a single collective CSF decision matrix \(\mathcal{D}\), as displayed in Table 4, by using the CSFDYWA operator.
Step 2: Acquire the cumulative preference values \({\mathcal{G}}_{i}\) for each \({\mathcal{Q}}_{{\varvec{i}}}\) by using the CSFYWA operator as displayed in Table 5.
Step 3: Calculate the scores \({\mathcal{S}} {(\mathcal{G}}_{{\varvec{i}}}\)) for each \({\mathcal{Q}}_{{\varvec{i}}}\) by applying Definition 4 in the following way:
\({\mathcal{S}} {(\mathcal{G}}_{1}\)) = 0.4755, \({\mathcal{S}} {(\mathcal{G}}_{2}\)) = 0.3978, \({\mathcal{S}} {(\mathcal{G}}_{3}\)) = 0.4928, and \({\mathcal{S}} {(\mathcal{G}}_{4}\)) = 0.5433.
Step 4: Since \({\mathcal{S}}{(\mathcal{G}}_{4})\)> \({\mathcal{S}} {(\mathcal{G}}_{3})\)> \({\mathcal{S}} {(\mathcal{G}}_{1})\)> \({\mathcal{S}} {(\mathcal{G}}_{2})\), therefore, the alternatives are arranged in the following order: \({\mathcal{Q}}_{4}> {\mathcal{Q}}_{3}>\) \({\mathcal{Q}}_{1}>{\mathcal{Q}}_{2}\).
In the same way, the solution to the previously stated MCDM problem, as it pertains to the CSFDYWG operator, is as follows:
Step 1: In order to acquire an assembled CSF decision matrix \(\mathcal{D}\), utilize the CSFDYWG operator is shown in the Table 6.
Step 2: Retrieve the aggregated preference values \({\mathcal{G}}_{i}\) for each alternative \({\mathcal{Q}}_{i}\) from step 1 by utilizing the CSFYWG operator, displayed in Table 7.
Step 3: Calculate the score values \({\mathcal{S}} \left(\mathcal{G}{}_{i}\right)\) for each alternative \({\mathcal{Q}}_{i}\) using Definition 4 as follows:
\({\mathcal{S}} {(\mathcal{G}}_{1}\)) = 0.4439, \({\mathcal{S}} {(\mathcal{G}}_{2}\)) = 0.3680, \({\mathcal{S}} {(\mathcal{G}}_{3}\)) = 0.4479, and \({\mathcal{S}} {(\mathcal{G}}_{4}\)) = 0.5231.
Step 4: Since \({\mathcal{S}} (\mathcal{G}_{4})>\) \({\mathcal{S}} ({\mathcal{G}}_{3} ) >\) \({\mathcal{S}} ({\mathcal{G}}_{1})>\) \({\mathcal{S}} ({\mathcal{G}}_{2})\), therefore, alternatives are organized in the subsequent sequence: \({\mathcal{Q}}_{4}> {\mathcal{Q}}_{3}>\) \({\mathcal{Q}}_{1}>{\mathcal{Q}}_{2}\).
Consequently, \(\text{Skeletal\, muscles\, actuators}\) are the most advanced bio-actuators for efficient human motion simulation in biohybrid robots.
Comparative analysis
In this segment, we examine multiple advanced operators in CIF, SF and CSF contexts to assess the strength and legitimacy of our suggested operators. The following discussion presents the aggregate results produced from various operators and the ranking of the alternatives, illustrating the contrast between our methodologies developed in the literature. The summarized outcomes derived from the CIFDWA and CIFDWG operators are displayed in Table 8, while Table 9 offers a comprehensive representation of the alternatives’ ranking.
Comparison 1. The frameworks introduced by Alghazzawi et al.31 lack real-time parameter adaptability to accommodate decision-makers’ risk tolerance, while also neglecting neutral conditions. These restrictions curtail adaptability and neglect personalized considerations. Therefore, to rectify these limitations, the recently introduced operators demonstrate enhanced flexibility by integrating the abstinence function and parametric values.
Comparison 2. The structural deficiencies of SFS-based Yager AOs developed by Chinaram et al.24 are the absence of a phase term and dynamic features, rendering them incapable of processing the CSF data illustrated in supplementary Tables 3–5. In addition, the techniques formulated in24 affect the special cases of proposed operators by keeping the 2nd dimension constant and ignoring the dynamic aspects. On the other hand, our proposed CSF dynamic Yager AOs are more efficient, as these operators adeptly mitigate this situation without losing sensitive data.
Comparison 3. AOs based on CSFSs presented by Akram et al.38 do not account for time periods, which lead to a significant loss of information, and hence these operators cannot process the data presented in supplementary Tables 3–5. The suggested AOs are dynamic and capable of collecting data from various different time frames. In addition, the methodologies developed in38 also lack the capability to dynamically adjust parameters to be consistent with the decision-makers’ risk acceptance criteria. However, the recently proposed operators are more adaptable and have the ability to consider individual preferences more efficiently.
Comparison 4. The CSF-based Yager AOs proposed by Razzaque et al.43 are not dynamic, meaning that these methods cannot be employed when initial data for DM is sourced from many distinct time periods, as is the situation in our research, whereas the proposed CSF-dynamic Yager AOs have the potential to manage such settings more precisely.
A graphical representation of the alternative rankings derived through various operators is illustrated in Fig. 2.
Diagrammatical view of ranking of the alternatives using different operators.
Comparison of the proposed strategy with the real biohybrid experimental data
Table 10 describes the comparison of the proposed techniques with the real biohybrid simulation.
From above Table, we conclude that the CSFDYW aggregation methods successfully handle uncertainty, dynamism, and complex interactions in biohybrid robots. Compared to experimental data, this method shows strong alignment, indicating it can be a powerful tool for:
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Predictive modeling
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Performance optimization
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Intelligent decision-making in bio-robotics
Sensitivity analysis
In this subsection, we elucidate the variations in alternative behavior corresponding to distinct values of the operational parameter \(\mathcal{X}\) within the framework of the CSFDYW aggregation operators. It is important to observe that, for the CSFDYWA operator and CSFDYWG operator, the score values of the alternatives consistently increase as the parametric value rises, despite the increase in \(\mathcal{X}\), the relative positions of the alternatives remain the same. The outcomes reinforce the reliability of the suggested operators in maintaining the ranking consistency as operational parameters evolve. The results of this mathematical procedure within the context of CSFDYWA and CSFDYWG are presented in Table 11 and Table 12, respectively.
Empirical validation of current study with other state-of-the-art algorithms
Table 13 describes the empirical analysis of the proposed methodologies across several key criteria with established MCDM techniques like classical TOPSIS, AHP, Fuzzy TOPSIS, Fuzzy AHP, Fuzzy VIKOR and EDAS.
Following the empirical analysis in Table 13, the results validate that suggested methodologies outperform the existing MCDM theories in addressing time-periodic complicated issues more effectively.
Advantages of the current study
The proposed operators in this article are designed to handle uncertainty and dynamic changes in data, making them suitable for complex DM scenarios. They aggregate data by applying fuzzy logic and dynamic weighted based on real-time inputs and changing environments. These methods offer significant benefits, including the ability to manage uncertainty and adapt to dynamic conditions in data aggregation than traditional methods. They enhance the DM process by considering multiple factors simultaneously, allowing for a more nuanced understanding of complex scenarios.
Discussion
Building on the previous analysis, it is crucial to evaluate the positive aspects of the suggested operators over the established methods. The newly developed operators offer enhanced flexibility through dynamic parameters and the abstinence function. Unlike the earlier frameworks developed by Alghazzawi et al., which neglected impartial conditions and failed to incorporate real-time adaptation, the recently introduced operators provide improved flexibility by integrating dynamic parameters and an abstinence function. In contrast to the research conducted by Chinaram et al., which highlights that the lack of phase terms and dynamic capabilities hinders the efficacy of their models in CSF situations, the proposed dynamic Yager operators effectively address these challenges while preserving sensitive information. Moreover, Akram et al.'s dynamic models are capable of effectively managing the temporal fluctuations and risk preference adaptation exhibited by their operators, resulting in information loss over various timeframes. these issues are well managed by the suggested CSF dynamic models. Furthermore, the techniques developed by Razzaque et al. are inadequate for multi-period decision scenarios, whereas the dynamic Yager operators presented in this article are more effective in managing such scenarios. According to a sensitivity analysis, the reliability of the categorization is not affected by the parametric values, which confirms the robustness of the proposed methods. The empirical validation conclusively shows that the proposed CSFDYWA model surpasses conventional MCDM approaches, such as classical TOPSIS, AHP, Fuzzy TOPSIS, Fuzzy VIKOR, and EDAS, by adeptly addressing uncertainty, neutrality, dynamic parameters, and time sensitivity. Therefore, it is exceptionally appropriate for practical DM contexts.
Limitations of the current study
Notwithstanding the numerous benefits of the methodologies presented in this article, they are nonetheless constrained by particular limitations:
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1.
These methods are inadequate for addressing issues where the square summation of MD, AD, NMD exceeds the closed unit interval.
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2.
Another potential limitation is the absence of permutation-based ordering in the CSFDYW aggregating operators, which limits their ability to prioritize criteria or options in contexts where the sequence or ranking of inputs is significant.
Conclusions
This work aims to develop innovative techniques for addressing DM issues in dynamic CSF situations. Although there have been several advantageous operators developed in previous research, none of them have specifically focused on time intervals in the context of CSF dynamic Yager weighted aggregation knowledge. Therefore, the utilization of the proposed dynamic CSF environment model is a more effective approach for modeling time-periodic challenges, as it can manage two-dimensional data in a single set. Considering these factors, we have initiated the study of novel dynamic Yager-weighted AOs within the CSF framework in this article. We have carried out an in-depth analysis of the characteristics of these operators. In addition, we have formulated a novel score function designed specifically to evaluate and pinpoint the most favorable alternative. Within the context of our research, we have introduced a new strategy for resolving complex decision-making problems that involve multiple factors in the CSF domain. In addition, we have effectively utilized these cutting-edge algorithms to select efficient cell-based actuators for biohybrid robots in a CSF dynamic environment. Finally, we have performed a comparative analysis to emphasize the significance and reliability of these novel strategies compared to existing techniques.
The proposed CSF dynamic Yager weighted AOs are extremely successful for real-time DM applications. Owing to their adaptability, they have the potential to rapidly respond to newly acquired information or changes in the environment, rendering them optimal for making decisions that require precision in time.
Potential future research directions of the current study
Our future research will concentrate on expanding the scope of dynamic AOs in several ways, particularly the CSF dynamic Yager ordered weighted AOs and the CSF dynamic Yager exponential AOs. With these breakthroughs, our aim will be to improve the adaptability and utility of our model in many DM situations, including decision support technologies in healthcare, information security, assessment of environmental effects, and evaluation of voyager’s data.
Data availability
All data generated or analyzed during this study are included in this article.
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Funding
This Project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (GPIP: 2023-665-2024). The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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M.N.A.: Conceptualization, Methodology, Writing—original draft, Data analysis; A.R.: Investigation, Formal analysis, Validation; A.W.B.: Writing—original draft, Review and editing, Project administration; D.A.: Visualization, Writing—review and editing, Resources; G.A.: Data curation, Validation, Review and editing. All authors have read and approved the final manuscript.
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Ansari, M.N., Razaq, A., Baidar, A.W. et al. Advanced bioactuator selection for efficient human motion simulation in biohybrid robots using CSF dynamic knowledge. Sci Rep 15, 24309 (2025). https://doi.org/10.1038/s41598-025-07500-w
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DOI: https://doi.org/10.1038/s41598-025-07500-w
