Evaluating the swimming technique analysis: a multi-criteria decision approach using hesitant bipolar fuzzy with a GRA approach

Abstract

Modern technology has played a significant role in improving the analysis and training of swimming, providing more accurate results. Also, finding the best swimming technique is still complicated because there are conflicting standards and uncertainty among experts. This study builds a decision framework using hesitant bipolar fuzzy sets (HBFS) and the grey relational analysis (GRA) method to deal with expert hesitations and bipolar opinions. Freestyle, breaststroke, butterfly, backstroke, and sidestroke are tested against what they accomplish in terms of speed, energy they use, how smooth they look during execution, and the efficiency of their breathing. Based on the HBFS-GRA approach, the experts considered the butterfly stroke the most suitable technique. The new process increases objectivity and makes decisions in sports science less vague. This study also recognizes some challenges, including using experts’ subjective opinions, missing data, and the need to regularly update the analysis due to the quick development of sports technologies. In the future, the system could be enhanced by including data from wearable devices and discussion with artificial intelligence, as well as by adapting it to cover different sports disciplines. This research promotes using data and well-structured models to assess athlete performance.

Introduction

Swimming is a highly technical sport that requires precise body movement, optimal energy efficiency, and well-coordinated limb action. Therefore, the analysis and enhancement of swimming techniques are paramount to athletes, coaches, and sports scientists. Traditionally, human observation has evaluated swimming techniques, often confirmed by manual video analysis. Unfortunately, such firsthand analysis is usually prone to inconsistencies and lacks accuracy.

With the advancement of technology, an extensive array of analytical tools, such as motion capture systems, biomechanical sensors¹, and artificial intelligence (AI)-based systems², has emerged, offering enhanced reliability and performance evaluation capabilities. Real-time feedback from these tools provides superior training insights and enables the detection of even slight inefficiencies in swimming techniques. This ultimately aids in improving performance. Technology is pivotal in modern sports training, especially refining stroke efficiency, body alignment, and reducing hydrodynamic resistance. With the integration of high-speed cameras, wearable sensors, and AI-driven motion analysis systems³, swimmers’ movements can now be quantified with high precision. These systems allow for the identification of technical flaws and the application of data-driven recommendations. Consequently, swimmers benefit from improved technique, enhanced propulsion, and minimized drag. However, selecting the most appropriate technology for swimming technique analysis remains a complex task, involving multiple considerations such as accuracy, cost-effectiveness, real-time feedback capabilities, ease of integration, and practical adaptability.

Multi-criteria decision-making (MCDM) methods provide a robust framework for evaluating and ranking different alternatives based on various criteria. In the realm of swimming technique analysis, MCDM can be applied to assess and compare a range of technological tools to determine the most suitable option. The complexity of such decisions is often heightened by uncertainty, conflicting opinions, and hesitation among experts, which necessitates a fuzzy decision-making approach. One advanced and flexible model to handle such complexities is the HBFS, which accounts for both positive and negative hesitation in expert opinions.

This study presents a structured decision-making approach based on integrating HBFS and the GRA method⁴ to evaluate and rank various swimming analysis technologies. The HBFS-GRA framework enables a more realistic and comprehensive assessment by incorporating expert uncertainty and bipolar hesitations, often present in performance evaluation scenarios.

This research aims to bridge the gap in current literature by applying the HBFS-based MCDM approach to the domain of swimming technique analysis. This application has remained largely unexplored. The proposed framework evaluates the technologies based on multiple criteria, including accuracy, real-time feedback, implementation cost, and practical adaptability. The outcome offers a reliable and systematic ranking mechanism for decision-makers in sports science.

The study also compares AI-based video analysis systems⁵, real-time motion tracking systems⁶, and other emerging tools, establishing a broader context for applying hesitant fuzzy decision models. The GRA method, known for its ability to perform under incomplete information, enables closeness-based ranking of alternatives and ensures that the selected solution aligns with the ideal decision scenario.

The proposed combination of HBFS and GRA provides a data-driven, uncertainty-resilient model for selecting the most appropriate swimming analysis technology. This structured framework will significantly assist coaches, athletes, and sports analysts in making informed decisions, thereby improving performance assessment strategies and fostering deeper investigations into applying hesitant fuzzy logic in sports science^7,8.

Literature review

Recent advancements in swimming performance analysis have been driven by progress in motion tracking systems⁹, wearable sensors¹⁰, and AI-enabled technologies¹¹. Traditional evaluation methods, such as manual video observation and biomechanical assessment, are increasingly criticized for subjectivity and susceptibility to error. Nowadays, technologies like motion capture systems¹², accelerometers¹³, gyroscopes¹⁴, and deep learning models have provided precise, real-time insights into swimmers’ strokes, body position, and propulsion.

However, selecting the optimum technology is still challenging because of competing factors such as cost, accuracy, ease, and adaptability. MCDM methods like TOPSIS¹⁵, AHP¹⁶, and ELECTRE¹⁶ are then employed to rank and compare such technologies using multiple performance criteria to arrive at a standard best solution. Where MCDM can provide a systematic evaluation, traditional methods cannot handle uncertainty, hesitation, and disagreements among experts. To address this problem, fuzzy logic should be included in decision-making.

Fuzzy set theory, introduced by Zadeh¹⁷, provides a mathematical means to address vagueness and subjectivity through membership degrees rather than binary logic. Fuzzy logic models have been widely used in sports performance evaluation for assessing techniques, training strategies, and technology selection. However, conventional fuzzy models are still limited in capturing conflicting sentiments or degrees of hesitation within expert evaluations.

To overcome these limitations, HBFS¹⁸ simultaneously allows several positive and negative membership values. This is very valuable for evaluating swimming analysis tools because these values include the advantages and disadvantages of each technology. Although HBFS has already been helpful in athlete training assessment, adjusting rehabilitation methods, and deciding on sports equipment, this study is unique in blending it with GRA for examining different technologies for swimming.

Recently, HBFSs have attracted attention as an effective method for facing uncertainty, conflicting opinions, and pausing in decision-making. Unlike traditional fuzzy sets, HBFSs showcase not one but multiple degrees of membership for both good and bad factors simultaneously, improving the idea of confusion and hesitation seen in reality. Xu and Wei¹⁹ outlined the use of hesitant bipolar fuzzy decision rules to improve the reliability of evaluations in situations involving multiple attributes. Mandal and Ranadive²⁰ strengthened the theory by outlining hesitant bipolar-valued fuzzy sets and explaining how they help capture bipolar-valued and hesitant information in group decision-making. Recently, Aslam et al.²¹ showed how HBFSs using hesitant bipolar complex fuzzy Dombi aggregation operators played a significant role in cloud service provider selection by supporting difficult evaluation decisions. Ali²² likewise developed probabilistic hesitant bipolar fuzzy methods for MCDM to make the model more reliable and effective.

Besides, recent investigations have used Fermatean fuzzy sets²³ and spherical fuzzy sets²⁴, both types of advanced fuzzy models, for sports and decision-making purposes. As an illustration, spherical fuzzy MCDM has assisted in reviewing training environments, while Fermatean fuzzy sets have improved the performance in settings with conflicting decisions. Until now, a complete MCDM approach to swimming analysis based on HBFS has not been available, and this research fills that gap. As a result, this paper offers the first MCDM system using HBFS-GRA, designed to analyze swimming technology. This work contributes to the current knowledge and provides valuable advice to athletes, coaches, and sports scientists.

Motivation

Swimming is just a field in the form, but a few modifications in terms of stroke efficiency, body positioning, and correct launch can do an impressive performance. Motion tracking, wearable sensors, and even artificial intelligence have advanced with technology and disease; however, choosing the right tool would still be difficult. Conventional approaches to decision-making cannot handle uncertainty, doubt, or conflicting expert opinions. Therefore, there is a need for a more robust approach. A method that has been proven to be fruitful is the procedure of HBFS combined with MCDM to give one a comprehensive structure that evaluates technologies based on merits and demerits. It also tries to fill the gap with a systematic decision model that considers uncertainty for the specific genre of swimming analysis tools. The output of the study would be useful for coaches, sports scientists, and athletes in terms of providing data-based information on training and assessment.

• Swimming technique enhancements can lead to better performance in the future.

• There are many forms of advanced technologies available, but it is difficult to choose the best.

• Traditional methods have problems such as uncertainty and conflicting expert opinions.

• HBFS are a better way to approach decision-making challenges.

• Developing a structured uncertainty-handling model for technology selection is the goal of this research.

• Findings will help coaches, sports scientists, and athletes optimize their performance.

Research objectives and question

Selecting the most appropriate swimming analysis technology can significantly improve stroke technique, body positioning, and overall performance. However, current methods often fail to adequately address factors such as uncertainty, hesitation, and conflicting expert opinions, which complicate decision-making processes. This study aims to integrate HBFS with MCDM techniques to develop a structured evaluation framework specifically designed for this purpose. By considering both the advantages and limitations of each technology, the proposed model will provide reliable rankings that reflect their relative value and usefulness. The findings will assist sports scientists, coaches, and training programs make effective, evidence-based decisions.

The specific objectives of this study are to:

• Review and analyze available swimming performance analysis technologies.

• Identify the most relevant criteria for evaluating these technologies.

• Integrate HBFS with MCDM methods to handle uncertainty and hesitation.

• Develop a structured decision-making model that addresses uncertainty and hesitation in expert assessments.

• Compare and rank technologies based on performance, accuracy, cost, adaptability, and ease of use.

• Provide a reliable, practical selection framework for coaches, athletes, and sports scientists.

For this study, the following research questions will serve as guidance.

• How can HBFS handle both favorable and unfavorable hesitations in evaluating tools for swimming analysis?

• Which technologies for evaluating swimming performance stand out the most when analyzed using factors such as precision, cost, simplicity, and ability to adapt using the HBFS-GRA approach?

• How is decision-making when faced with uncertainty improved when using HBFS and GRA, compared to the traditional fuzzy methods for MCDM?

• How can the proposed framework be helpful to coaches, athletes, and sports scientists to maximize performance in swimming analysis?

This research is expected to help stakeholders make data-based decisions and raise the level of training and sports results.

Layout

For the best technologies for swimming technique evaluation, this paper will be structured into the following sections: Section "Introduction" presents an introduction to swimming techniques and their decision-making; Section "Preliminaries" states some basic concepts while Section "Hesitant bipolar fuzzy averaging operators (HBFAO)" refers to the introduction of the HBFS and its methodology is presented in Section "Evaluating swimming technique analysis using HBFWAO-GRA approach" which is further employed by real-life decision-making problem. Section "Conclusion" summarizes the discussion, setting out the limitations and future research directions.

Preliminaries

This part explains the fundamental working principles of FS, BFS, and HFS for implementing these algorithms. Table 1 displays the list of symbols used in this paper.

Table 1 Table of symbols.

Definition 1

¹⁸: The fuzzy set (FS) $\text{F}$ is defined as:

$$\text{F}=\left\{\left({\varrho }_{j},{\xi }_{\text{F}}\left({\varrho }_{j}\right)\right):{\varrho }_{j}\in U\right\}$$

(1)

that defines a membership grade (MG)${\upxi }_{\text{F}}:{\upxi }_{\text{F}}\in \left[\text{0,1}\right]$.

Definition 2

¹⁸: The bipolar fuzzy set (BFS) $\text{F}$ is defined as:

$$\text{F}=\left\{<{\varrho }_{j}\left({\xi }_{\text{F}}^{+}\left({\varrho }_{j}\right),{\zeta }_{\text{F}}^{-}\left({\varrho }_{j}\right)>|{\varrho }_{j}\in U\right)\right\}$$

(2)

In a BFS, an element ${\varrho }_{j}$ is characterized by two membership grades: the positive MG ${\xi }_{\text{F}}^{+}\left({\varrho }_{j}\right)$ which indicates its level of agreement with a certain property, and the negative MG ${\zeta }_{\text{F}}^{-}\left({\varrho }_{j}\right)$ which reflects its degree of opposition to the same property. The BFS representation of any element ${\varrho }_{j}$ is given by $\text{F}{\varrho }_{j}=\left({\xi }_{\text{F}}^{+}\left({\varrho }_{j}\right),{\zeta }_{\text{F}}^{-}\left({\varrho }_{j}\right)\right)$ shows the constraints $0\le {\xi }^{+}\le 1$ and $-1\le {\zeta }^{-}\le 0$ to provide its balanced evaluation in terms of positive and negative attributes.

Definition 3

¹⁸: The fundamental operations of bipolar fuzzy numbers (BFNs) are as follows:

• ${F}_{1}\oplus {F}_{2}=\left({\upxi }_{1}^{+}+{\upxi }_{2}^{+}-{\upxi }_{1}^{+}{\upxi }_{2}^{+},-\left|{\zeta }_{1}^{-}\right|\left|{\zeta }_{2}^{-}\right|\right)$

• ${F}_{1}\otimes {F}_{2}=\left({\upxi }_{1}^{+}{\upxi }_{2}^{+},{\zeta }_{1}^{-}+{\zeta }_{2}^{-}-{\zeta }_{1}^{-}{\zeta }_{2}^{-}\right)$

• $\gamma F=\left(1-{\left(1-{\upxi }^{+}\right)}^{\gamma },-{\left|{\zeta }^{-}\right|}^{\gamma }\right),\gamma >0$

• ${\left(F\right)}^{\gamma }=\left({\left({\upxi }^{+}\right)}^{V},-1+{\left|{1+\zeta }^{-}\right|}^{\gamma }\right),\gamma >0$

• ${F}^{c}=\left(1-{\upxi }^{+},\left|{\zeta }^{-}\right|-1\right)$

• ${F}_{1}\subseteq {F}_{2},\Leftrightarrow {\upxi }_{1}^{+}\le {\upxi }_{2}^{+} and {\zeta }_{1}^{-}\ge {\zeta }_{2}^{-}$

• ${F}_{1}\cup {F}_{2}=\left(max\left\{{\upxi }_{1}^{+},{\upxi }_{2}^{+}\right\},min\left\{{\zeta }_{1}^{-},{\zeta }_{2}^{-}\right\}\right)$.

• ${F}_{1}\cap {F}_{2}=\left(min\left\{{\upxi }_{1}^{+},{\upxi }_{2}^{+}\right\},max\left\{{\zeta }_{1}^{-},{\zeta }_{2}^{-}\right\}\right)$;

Theorem 1

¹⁸: Consider the two BFNs ${F}_{1}=\left({\upxi }_{1}^{+},{\zeta }_{1}^{-}\right)$ and ${F}_{2}=\left({\upxi }_{2}^{+},{\zeta }_{2}^{-}\right)$, where $\gamma ,{\gamma }_{1},{\gamma }_{2}>0$. The following operations are as follows:

• ${F}_{1}\oplus {F}_{2}={F}_{2}\oplus {F}_{1}$

• ${F}_{1}\otimes {F}_{2}={F}_{2}\otimes {F}_{1}$

• $\gamma \left({F}_{1}\oplus {F}_{2}\right)=\gamma {F}_{1}\oplus \gamma {F}_{2}$

• ${\left({F}_{1}\otimes {F}_{2}\right)}^{\gamma }={\left({F}_{1}\right)}^{\gamma }\otimes {\left({F}_{2}\right)}^{\gamma }$

• ${\gamma }_{1}{F}_{1}\oplus {\gamma }_{2}{F}_{1}=\left({\gamma }_{1}+{\gamma }_{2}\right){F}_{1}$

• ${\left({F}_{1}\right)}^{{\gamma }_{1}}\otimes {\left({F}_{1}\right)}^{{\gamma }_{2}}={\left({F}_{1}\right)}^{\left({\gamma }_{1}+{\gamma }_{2}\right)}$

• ${\left({\left({F}_{1}\right)}^{{\gamma }_{1}}\right)}^{{\gamma }_{2}}={\left({F}_{1}\right)}^{{\gamma }_{1}{\gamma }_{2}}$

Hesitant bipolar fuzzy averaging operators (HBFAO)

This part presents some ingenious and specialized types of aggregation operator designed for HBFAO. This operator is efficiently gathering and evaluate the HBFAO, thus rendering them useful in a variety of decision-making and analytical contexts. Then the essential aspects of these operators are investigated by operating on them using the fundamental operations, thus allowing for a better understanding of their behavior and performance. This discussion could lead to some good methods of handling uncertain and bipolar data.

Definition 4

The HBFS ${F}^{*}$ is defined as:

$${F}^{*}=\left\{<{\varrho }_{j},{{B}}_{{F}^{*}\left({\varrho }_{j}\right)}>|{\varrho }_{j}\in U\right\}$$

(3)

where, ${{{B}}}_{{F}^{*}\left({\varrho }_{j}\right)}$ is a collection of BFNs. Specifically, $B_{{F^{*} \left( {{\varrho }_{j} } \right)}} = \cup _{{\left( {\upxi _{{F^{*} }}^{ + } \left( {{\varrho }_{j} } \right),\zeta _{{F^{*} }}^{ - } \left( {{\varrho }_{j} } \right)} \right) \in B_{F}^{*} }} \left( {{\varrho }_{j} } \right)\left( {\upxi _{{F^{*} }}^{ + } \left( {{\varrho }_{j} } \right),\zeta _{{F^{*} }}^{ - } \left( {{\varrho }_{j} } \right)} \right)$where ${\upxi }_{{F}^{*}}^{+}\left({\varrho }_{j}\right)$ be the positive MG indicating the extent to which a particular ${\varrho }_{j}$ satisfies an HBFS and ${\zeta }_{{F}^{*}}^{-}\left({\varrho }_{j}\right)$ indicates an opposing negative MG referring to the extent to which ${\varrho }_{j}$ satisfies any contrary or opposite property related to the HBFS ${F}^{*}$.

The following conditions limit these MG: $0\le {\upxi }_{{F}^{*}}^{+}\left({\varrho }_{j}\right)\le 1$ and $-1\le {\zeta }_{{F}^{*}}^{-}\left({\varrho }_{j}\right)\le 0$ for all ${\varrho }_{j}\in U$.

The pair $h\left({\varrho }_{j}\right)=\left\{\left({\upxi }^{+}\left({\varrho }_{j}\right),{\zeta }^{-}\left({\varrho }_{j}\right)\right)\right\}$ is called an HBFN and denoted as $h=\left({\upxi }^{+},{\zeta }^{-}\right)$, $0\le {\varphi }^{+}\le 1$ and $-1\le {\psi }^{-}\le 0$, $\left({\varphi }^{+},{\psi }^{-}\right)\in \left({\upxi }^{+},{\zeta }^{-}\right)$.

Using comparative laws to determine the HBFNs helps to have a systematic approach to differentiating these various HBFNs.

Definition 5

Let ${h}_{i}=\left({\upxi }_{i}^{+},{\zeta }_{i}^{-}\right)$ be any HBFN. Then,

$${\mathbbm{s}}\left({h}_{i}\right)=\frac{1}{{\tilde{\#}}{h}_{i}}\sum_{i=1}^{{\tilde{\#}}{h}_{i}}\frac{1+{\varphi }^{+}+{\psi }^{-}}{2}$$

(4)

${\mathbbm{s}}\left({h}_{i}\right)$ represents score function of ${h}_{i}=\left({\upxi }_{i}^{+},{\zeta }_{i}^{-}\right)$.

Definition 6

Let ${h}_{i}=\left({\upxi }_{i}^{+},{\zeta }_{i}^{-}\right)$ be any HBFN. The accuracy function of ${h}_{i}=\left({\upxi }_{i}^{+},{\zeta }_{i}^{-}\right)$ is;

$${\mathbbm{s}}^{*}\left({h}_{i}\right)=\frac{1}{{\tilde{\#}}{h}_{i}}\sum_{i=1}^{{\tilde{\#}}h}\frac{{\varphi }^{+}-{\psi }^{-}}{2}$$

(5)

where ${\tilde{\#}}{h}_{i}$ is number of elements in ${h}_{i}$.

The following operational laws would facilitate the integration of HBFNs in various contexts and allowing easier comparison and analysis.

(a) ${h}^{\gamma }={\cup }_{\left({\varphi }^{+},{\psi }^{-}\right)\in \left({\upxi }^{+},{\zeta }^{-}\right)}\left\{\begin{array}{c}{\left({\varphi }^{+}\right)}^{\gamma },\\ -1+{\left|1+{\psi }^{-}\right|}^{\gamma }\end{array}\right\},\gamma >0$;

(b) $\gamma h={\cup }_{\left({\varphi }^{+},{\psi }^{-}\right)\in \left({\upxi }^{+},{\zeta }^{-}\right)}\left\{\begin{array}{c}1-{\left(1-{\varphi }^{+}\right)}^{\gamma },\\ {\left|{\psi }^{-}\right|}^{\gamma }\end{array}\right\},\gamma >0$;

(c) ${h}_{1}\oplus {h}_{2}={\cup }_{\left({\varphi }_{1}^{+}{,\psi }_{1}^{-}\right)\in \left({\upxi }_{1}^{+}{,\zeta }_{1}^{-}\right),\left({\varphi }_{2}^{+}{,\psi }_{2}^{-}\right)\in \left({\upxi }_{2}^{+}{,\zeta }_{2}^{-}\right)}\left\{\begin{array}{c}{\varphi }_{1}^{+}+{\varphi }_{2}^{+}-{\varphi }_{1}^{+}{\varphi }_{2}^{+},\\ -\left|{\psi }_{1}^{-}\right|\left|{\psi }_{2}^{-}\right|\end{array}\right\}$

(d) ${h}_{1}\otimes {h}_{2}={\cup }_{\left({\varphi }_{1}^{+}{,\psi }_{1}^{-}\right)\in \left({\upxi }_{1}^{+}{,\zeta }_{1}^{-}\right),\left({\varphi }_{2}^{+}{,\psi }_{2}^{-}\right)\in \left({\upxi }_{2}^{+}{,\zeta }_{2}^{-}\right)}\left\{\begin{array}{c}{\varphi }_{1}^{+}{\varphi }_{2}^{+},\\ {\psi }_{1}^{-}+{\psi }_{2}^{-}-{\psi }_{1}^{-}{\psi }_{1}^{-}\end{array}\right\}$

Hesitant bipolar fuzzy weighted averaging operators (HBFWAO)

In this section, the HBFAO allows us to combine HBFN in a prepared way for further analysis and decision-making.

Definition 7

Let ${h}_{j}=\left({\upxi }_{j}^{+},{\zeta }_{j}^{-}\right)$ be the collection of HBFN. The HBFWAO is defined as:

$${HBFWAO}_{\mathcal{w}}\left({h}_{1},{h}_{2},\dots ,{h}_{n}\right)=\sum_{j=1}^{n}\left({\mathcal{w}}_{j}{h}_{j}\right)$$

(6)

where, $\mathcal{w}={\left({\mathcal{w}}_{1},{\mathcal{w}}_{1},\ldots ,{\mathcal{w}}_{1}\right)}^{\mathfrak{t}}$ be the weight vector of ${h}_{j}$ with ${\mathcal{w}}_{j}>0$ and $\sum_{j=1}^{n}\left({\mathcal{w}}_{j}\right)=1.$

Theorem 2

The HBFWAO provides an HBFN with

$${HBFWAO}_{\mathcal{w}}\left({h}_{1},{h}_{2},\dots ,{h}_{n}\right)=\sum_{j=1}^{n}\left({\mathcal{w}}_{j}{h}_{j}\right)$$

$${HBFWAO}_{\mathcal{w}}\left({h}_{1},{h}_{2},\dots ,{h}_{n}\right)={\cup }_{\left({\varphi }_{j}^{+}{,\psi }_{j}^{-}\right)\in \left({\upxi }_{j}^{+}{,\zeta }_{j}^{-}\right)}\left\{1-\prod_{j=1}^{n}{\left(1-{\varphi }_{j}^{+}\right)}^{{\mathcal{w}}_{j}},-\prod_{j=1}^{n}{\left|{\psi }_{j}^{-}\right|}^{{\mathcal{w}}_{j}}\right\}$$

(7)

Evaluating swimming technique analysis using HBFWAO-GRA approach

To determine the best technology for analyzing swimming techniques by combining the proposed HBFWAO and HBFWAO GRA approach, the range of alternatives as ${\acute{\text{{\AA}}}}=\left\{{{\acute{\text{{\AA}}}}}_{1},{{\acute{\text{{\AA}}}}}_{2},\dots ,{{\acute{\text{{\AA}}}}}_{m}\right\}$ and range of criteria is $\mathcal{C}=\left\{{\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{n}\right\}$. The weight vectors of these criteria are given by $\mathcal{w}=\left\{{\mathcal{w}}_{1},{\mathcal{w}}_{2},\dots ,{\mathcal{w}}_{n}\right\}$, where ${\mathcal{w}}_{j}\ge 0 ;\forall j$ and $\sum_{j=1}^{n}\left({\mathcal{w}}_{j}\right)=1.$ ${H=\left[{h}_{ij}\right]}_{m\times n}={\left[\left({\upxi }_{ij}^{+},{ \zeta }_{ij}^{-}\right)\right]}_{m\times n}$ represents the HBF decision matrix.

The methodology of solving a MCDM problem is as follows:

(1) For assessing an MCDM problem, the decision matrix should be formed based on a HBF framework.

$${{H}_{\mathfrak{i}\mathfrak{j}}}^{\left(h\right)}=\left[\begin{array}{cccccc}{h}_{11}& {h}_{12}& \cdots & {h}_{1\mathfrak{j}}& \cdots & {h}_{1m}\\ {h}_{21}& {h}_{22}& \cdots & {h}_{2\mathfrak{j}}& \cdots & {h}_{2m}\\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {h}_{\mathfrak{i}1}& {h}_{\mathfrak{i}2}& \cdots & {h}_{\mathfrak{i}\mathfrak{j}}& \cdots & {h}_{\mathfrak{i}m}\\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {h}_{\mathfrak{n}1}& {h}_{\mathfrak{n}2}& \cdots & {h}_{\mathfrak{n}\mathfrak{j}}& \cdots & {h}_{\mathfrak{n}m}\end{array}\right]={\left({H}_{\mathfrak{i}\mathfrak{j}}\right)}_{\mathfrak{n}\times m}$$

(8)

(2) Applying the HBFWA operator to process the information in the matrix $H$ as;

$${HBFWAO}_{\mathcal{w}}\left({h}_{i1},{h}_{i2},\dots ,{h}_{in}\right)=\sum_{j=1}^{n}\left({\mathcal{w}}_{j}{h}_{ij}\right)={\cup }_{\left({\varphi }_{ij}^{+}{,\psi }_{ij}^{-}\right)\in \left({\upxi }_{ij}^{+}{,\zeta }_{ij}^{-}\right)}\left\{\begin{array}{c}1-\prod_{j=1}^{n}{\left(1-{\varphi }_{ij}^{+}\right)}^{{\mathcal{w}}_{j}},\\ -\prod_{j=1}^{n}{\left|{\psi }_{ij}^{-}\right|}^{{\mathcal{w}}_{j}}\end{array}\right\}$$

(9)

such that $\mathcal{w}=\left({\mathcal{w}}_{1},{\mathcal{w}}_{2},\dots ,{\mathcal{w}}_{m}\right) , \sum_{\mathfrak{j}=1}^{m}{\mathcal{w}}_{\mathfrak{j}}=1$.

(3) Normalization ${H}^{N}$ of the first sequence about a particular target value to calculate the deviation sequences through the use of

$${H}^{N}=1-\sum_{j=1}^{n}\left({\mathcal{w}}_{j}{h}_{ij}\right)$$

(10)

(4) The grey relational coefficient ${{\upeta}}_{i}\left(H\right)$ calculates the association between the reference and comparison sequences. ${\upeta } =0.5$ is used as an optimistic estimation based on the normal distribution.

$${{\upeta}}_{i}\left(H\right)=\frac{{\Delta }_{min}+{\upeta}{\Delta }_{max}}{\Delta \left(H\right)+{\upeta}{\Delta }_{max}}$$

(11)

(5) Calculation of overall grey relational grading. The grey relational grades (GRG) are obtained from the average value of GRC corresponding to each of the process responses and are indicated as follows with the relevant formula:

$$H=\frac{1}{n}\sum_{j=1}^{n}\left({{\upeta}}_{i}\left(H\right)*{\mathcal{w}}_{j}\right)$$

(12)

(6) Calculate the score by;

$${\mathbbm{s}}\left({h}_{i}\right)=\frac{1}{{\tilde{\#}}{h}_{i}}\sum_{i=1}^{{\tilde{\#}}{h}_{i}}\frac{1+{\varphi }^{+}+{\psi }^{-}}{2}$$

(7) Rank the alternatives ${{\acute{\text{{\AA}}}}}_{i}$ based on their scores. If two scores are identical, then calculate accuracy function to rank alternatives.

(8) Choose the appropriate alternatives based on score values.

Strengths and limitations of the GRA method and the proposed improvements

In MCDM, the GRA technique is popular mainly because it is easy to use, simple, and works well with all criteria using only a limited number of values. The method is especially suitable for small data sets and highlights the closeness of each alternative to the perfect solution. Still, standard GRA can struggle to represent fairly complex uncertainties, vague terms, and how hesitation impacts decisions made by multiple individuals. This work joins the GRA approach with the HBFS setting to handle these issues. Using HBFS makes it possible to model an expert’s multiple favorable and unfavorable opinions by assigning different membership values. Also, the experts were consulted when constructing the decision matrix, and a sensitivity analysis was done to prove that the GRA rankings would remain reliable as the grey relational coefficient ($\eta$) varied. The additional details they include boost the classical GRA approach, so it can be used more effectively for complex real-life problems where subjectivity plays an important role.

Illustrative example

Swimming is an aquatic sport that embraces speed, endurance, and efficiency. It varies in terms of techniques as every swimmer has personal preferences in swimming styles, such as competition, training, and lifesaving. The best selection of techniques is often based on speed considerations but would include many factors: energy consumption, stroke efficiency, and breath control. Some strokes are designed for maximum speed, while others conserve energy or maintain balanced conditions in water. Thus, GRA objectively evaluates and compares various swimming techniques using its GRA, a multi-criteria decision-making model. This gives one up to that extent in analysis involving different factors and ranks the result based on overall performance.

This paper examines five swimming techniques and sets four major performance standards as evaluative criteria. Speed is essential in swimming as it covers the distance in as little time as possible, especially in competition. However, in most cases, this speed requires higher energy demands. Thus, energy consumption emerges as another crucial criterion. Efficient swimming would, therefore, be about having propulsion while minimizing resistance; hence, stroke efficiency becomes a consideration. Adding to this is the significant contribution of breathing to endurance and performance. Thus, breathing efficiency is examined to determine how applicable each technique is to optimal oxygen intake. This, put together, would tell how much speed, efficiency, or energy consumption one can balance with which technique. These results in the analysis would help swimmers, coaches, and common researchers organize the proper technique into their specific performance targets.

The aforementioned swimming alternatives comprise the five styles that are defined as follows:

• ${{\acute{\text{{\AA}}}}}_{1}$ is Freestyle: The fastest stroke as it offers low resistance.

• ${{\acute{\text{{\AA}}}}}_{2}$ is Breaststroke: One of the slower techniques yet most energy-efficient.

• ${{\acute{\text{{\AA}}}}}_{3}$ is Butterfly: A highly power-oriented stroke that needs exhaustive technique and endurance.

• ${{\acute{\text{{\AA}}}}}_{4}$ is Backstroke: A stroke specifying endurance and balance in the water.

• ${{\acute{\text{{\AA}}}}}_{5}$ is Sidestroke: Primarily for lifesaving because this consumes very low energy.

These models will be assessed by a panel of experts on four premises.

• ${\mathcal{C}}_{1}$ is Speed (Time per 50 m, sec): This will indicate the swiftness of how fast a swimmer completes a particular distance.

• ${\mathcal{C}}_{2}$ Energy Consumption (kcal/min): Measures the energy expended in sustaining the technique.

• ${\mathcal{C}}_{3}$ Stroke Efficiency (Power vs. Drag): Measures the capability of a stroke to convert power into movement with reduced maximum resistance.

• ${\mathcal{C}}_{4}$ is Breathing Efficiency (per stroke oxygen intake): Indicate how well the athlete controls oxygen absorption while performing concerning performance.

The decision-makers attribute to each of the criteria a weight by which they have reflected the hypothetical assumptions, that is, the weight vector $\mathcal{w}=(\text{0.32,0.05,0.43,0.20})$.

The evaluation of the swimming styles based on the criteria thus uses the above methodology:

• (1) Decision matrix development is shown in Table 2 as reflective of decision-makers’ perspectives in an HBF context to address the MCDM problem.

• (2) By after the directly above step apply the HBFWAO to procedure the information in a decision matrix $H$ is shown in Table 3.

• (3) Normalization of the sequence about a particular target value to calculate the deviation sequences using Eq. 8 and its result values is shown in Table 4.

• (4) Calculation of the association between the reference and comparison sequences ${{\upeta}}_{i}\left(H\right)$, where, ${\upeta } =0.5$ by the use of Eq. 9, shown in Table 5.

• (5) Calculate overall grey relational grading using Eq. 13 and its result is shown in Table 6.

• (6) Determine the score by using Eq. 4 and its result is shown in Table 7.

• (7) The table below, Table 8, shows the best alternatives based on their score values.

Table 2 Representation of the matrix.

Table 3 Overall values of alternatives by using HBFWAO.

Table 4 Normalized sequence deviations.

Table 5 Reference-comparison association.

Table 6 Overall GRA rational grading.

Table 7 Score values.

Table 8 Ranking of the best technology for analyzing swimming technique.

Figure 1 shows how the ranking of alternatives is portrayed graphically.

Fig. 1

Ranking of alternatives.

Result discussion

The GRA method evaluates different swimming styles by looking at speed, energy consumption, stroke efficiency, and breathing efficiency. The results reveal that the butterfly stroke (${{\acute{\text{{\AA}}}}}_{3}$) is placed first, the sidestroke comes in second (${{\acute{\text{{\AA}}}}}_{5}$), and freestyle (${{\acute{\text{{\AA}}}}}_{1}$) is in third. Its position arises because butterfly swimmers must focus more on speed and rhythm. Having excellent stroke and breathing rates helps them use their energy more wisely to achieve maximum results in competition. The special biomechanics of the butterfly enable athletes to move quickly and use oxygen more efficiently, making this stroke very effective. Still, since it takes a lot of energy, it may not be ideal for long-lasting activities. In comparison, the sidestroke saves energy, which is most useful in rescue situations or for people swimming a long distance. Switching to freestyle allows a swimmer to move faster, but it requires more effort than using the butterfly. The Backstroke and Breaststroke came in lower on the list since they are less efficient with energy use, which past studies on biomechanics have shown. Their findings highlight the importance of using specific strokes according to an athlete’s main aims, since improving performance relies on knowing both the body’s needs and the game conditions.

Practical applications

The study results present helpful directions for swimmers, coaches, and sports scientists to adjust their training routines according to their unique goals. Because butterflies are so fast and efficient, they work well for swimmers aiming for top results, though managing tiredness by using energy-saving exercises is still important. Teaching the sidestroke to athletes or rescue workers makes it easier for them to maintain energy and avoid feeling tired during long activities. The HBFS-GRA evaluation framework helps coaches follow a plan for comparing swimming styles, ensuring they make informed selections and plan workouts using performance data. The model makes it possible to find the pros and cons of different strokes so that swimmers train according to their needs and abilities. This way of thinking helps people swim faster in competitions and every day in safety-related situations.

Sensitivity analysis

The HBFS-GRA outcomes were checked for robustness by a sensitivity analysis, changing the distinguishing coefficient ${\varvec{\eta}}$ from $0.1$ to $0.9$. Coefficient η shows how closely the reference and comparison sequences are related, usually taking ${\varvec{\eta}}=0.5$, as it is seen as a positive interpretation for a typical distribution case. In Fig. 2, it can be seen that the position of each alternative does not change regardless of the parameter settings. It is important to note that ${{{\acute{\text{{\AA}}}}}}_{3}$ (butterfly stroke) performs the best and tops the rankings, which is closely followed by scores for ${{{\acute{\text{{\AA}}}}}}_{5}$ (sidestroke) and ${{{\acute{\text{{\AA}}}}}}_{1}$ (freestyle). This demonstrates that HBFS-GRA provides consistent results when parameters are varied. Because the model is so robust, it meets the expectations of use in real scenarios set by the existing MCDM literature^25,26.

Fig. 2

Sensitivity analysis by changing the parameter values.

Comparative analysis

Comparative analyses in Table 9 have been conducted, emphasizing the proposed strengths of the HBFS with the GRA Method over well-known MCDM techniques such as MERCE, TOPSIS, COCOSO, MULTIMOORA, EDAS, and VIKOR. Traditional MCDM techniques like TOPSIS and VIKOR have proved more stable but are somewhat sensitive to extreme values since they use distance-based measures. On the other hand, COCOSO and MULTIMOORA introduce various criteria and prefer integrated approaches toward decision-making but are less flexible about incorporating hesitant and bipolar information. MERCE and EDAS exhibit increased robustness; however, their deluxe version may run into trouble in multi-level uncertainty situations. Comparatively, the flexibility in the design and enhanced interaction offered by parametric and attribute modulation makes HBF-GRA highly adaptable to complex decision-making environments, rendering it much more compatible.

Table 9 Comparative analysis of HBF-GRA with existing MCDM Methods.

Table 10 underlines the strong and weak points of intuitionistic fuzzy sets (IFS)³², BFS¹⁸, HFS³³, and the proposed HBFS. While IFS has moderate or medium-level capabilities, it cannot process both hesitancy and bipolar opinions at the same time. Both BFS and HFS perform better when it comes to bipolarity or hesitancy, but still have problems with inconsistencies and in making detailed evaluations. HBFS offers a better method since it addresses hesitancy, different forms of disagreement, and highly detailed membership degrees. Thus, HBFS is especially useful for decisions requiring sorting through detailed, perhaps conflicting advice and facts.

Table 10 Comparative analysis with other fuzzy sets.

Conclusion

The current study was carried out to measure the correct assessment of the applied technology for swimming analysis through the use of the HBFS in combination with MCDM methods. By carefully considering the various uncertainties, hesitations, and conflicting expert opinions affecting technology selection, the research has thus successfully evolved a general decision-making framework. It was concluded that the technologies could be sequenced concerning the rest of the factors considered of accuracy, price, adaptability, and user-friendly nature-creating an easy path for coaches, sports scientists, and athletes to choose. HBFS went on to show its capability to portray the complicated and subtle nature of any technology evaluation where traditional decision-making models are challenged. Specifically, this model’s inclusion of positive and negative criteria adds to its reliability and practicality for real-life applications. The study thus corroborated the need to use fuzzy logic-based MCDM methods to confront further the difficulties in assessing advanced sports analysis technologies, leading to greater precision and customization of performance assessments. The approach will promote sports science by providing practical swimming training, technology application, and performance assessment. Clarifying rankings and evaluation criteria allows decision-makers to select technologies best suited for their needs, thus enhancing athlete performance. Moreover, the HBFS-MCDM framework can be extended for other sports, creating a solid model that can be applied to various athletic endeavors.

Limitations

Limitations of the study although this study offers a comprehensive framework for evaluating swimming analysis technologies, it is not without some limitations, which are discussed as follows:

• The study has considered a particular class of swimming analysis technologies. It may cover a range of possible tools available in the market. Further, the model needs to be updated as newer technologies keep emerging. Even though HBFS is used, the involved experts still have an inherent subjectivity regarding rating criteria. Experts’ knowledge and experience levels could introduce possible bias in the overall evaluation.

• Data availability was essential for ranking technologies, as some of these data could be sparse or incomplete. The accuracy of the evaluation will be highly improved with comprehensive datasets and empirical validation of the technologies in real-life conditions.

• The criteria that were taken into consideration for technology evaluation in this study were considered to be relatively static. A change in cost, a sudden user need, or a technological advancement may affect the criteria that need frequent updates in the model.

• The HBFS-MCDM framework is a compound one regarding calculations and uses fuzzy logic complemented with multiple decision-making methodologies. This will render its implementation difficult for all stakeholders without prior training and support.

• This avoids considering potential integration problems due to differences in technology, sensor compatibility, data analytics packages, and other equipment used in swimming performance analysis. Such real-life issues may limit the model’s actual use in some contexts.

Future direction

The presented study can be seen as an excellent basis, but it still allows for several directions for further study and improvement.

• The model proposed here may be fine-tuned and tested for other sports resources dependent on similar technology assessments of athletics, cycling, and gymnastics, by fine-tuning the framework for different disciplines. This would, in turn, enhance the versatility of this framework, therefore applying it to the broader spectrum of sports science.

• One of the promising directions for expansion is integrating real-time performance data obtained from wearables and sensors. It would permit dynamic on-the-fly decision-making, whereby coaches and athletes receive immediate feedback for adjusting during training. Also, including real-time data would help provide a better assessment and general effectiveness of the training sessions.

• AI developments will drive the future of analysis technology in sports. More sophisticated AI models and methods, such as machine learning algorithms and deep learning networks, may help address even more decision-making situations in an advanced manner.

• These techniques will deal with vast data and accurately predict individual training programs and performance optimizations for single athletes.

• Additionally, the confluence of AI and HBFS-MCDM would allow this system to learn and improve over time-based on the incoming data and changing training conditions, thus sufficiently qualifying the program as a good candidate for long-term athlete development.

• The principles used in this study would be extended beyond sports to other domains where the selection of technology is paramount, e.g., in healthcare (evaluation of medical devices), education (evaluation of e-learning platforms), or smart cities (promotion of technologies for urban planning). Cross-domain applications, where more development and acceptance will go toward bending the utility of HBFS-MCDM as a viable tool in complex decision-making.

An emerging frontier for future research could be to augment the precision and flexibility in the management of decision alternatives. The proposed operators can be extended by advanced fuzzy systems³⁴, neutrosophic framework³⁵, and a cubic framework³⁶ that account for hesitation and complex scenarios. Moreover, the research can further extend to incorporate aggregation operators^{37,38,39,40,41,42} in general, which are suitable for noisy or incomplete data. In the future, we aim to carry out a detailed comparison of our proposed HBFS-GRA approach with well-known MCDM methods such as TOPSIS²⁷, VIKOR²⁸, MULTIMOORA²⁹, COCOSO²⁸, MERCE³⁰, and DEMATEL³¹. We also aim to do more research with neutrosophic fuzzy sets⁴³ to confirm and compare the effectiveness of our proposed solution. Adding such operators with enhanced fuzzy methods will produce better and more flexible decision-making models.