1 Introduction

The Fuzzy decision support systems can help the participants choose the kind of exercise that best meets their requirements and preferences and reach their physical fitness and health objectives in a suitable and safe way. The application of fuzziness can be configured to correspond with particular fitness objectives. Fuzzy decision support systems can direct training decisions to match these goals, regardless of whether the primary aim is on strength, fitness, or development of skills, optimizing the efficacy of the training programs.

The ability to make decisions has been frequently pointed out as an essential attribute for effective functioning in sports authorities; nevertheless, there has been a lack of knowledge of how this critical ability can be strengthened through training programs [1]. Decision support systems can enhance the knowledge, autonomy, and skills to provide accurate decisions in the profession of safety-critical operations in sports [2]. Previous approaches used the machine learning strategy [3] focus on using clever approaches, including fuzzy theory and neural network training, to forecast eventual results. The artificial neural network (ANN) approach provides precise training predictions based on learned load patterns of sports and thresholds without explicit rule-based approaches and does not handle uncertainties well [4]. Analyzing training sessions and uncertain activities helps to make strategic decisions about the sports athletes involved in the team and individual sports clubs involved in the lack of training quality [5]. The intelligent ML algorithms has been used for ensuring training distance and fitness [6] based on fuzzy inference logic by collecting training sessions. Sequential unloading intervals show modified responses when exposed to intense training workloads [7]. Over a recurrent period, planned training techniques and procedures result in further training impacts.

The quality of teaching physical education and personnel training has been evaluated using multi-criteria decision-making. The weight of each decision-maker can vary in team-based training programs, which is considered a shortcoming [8]. Skiers acquire a decision system [9] based on appropriate virtual training. They are better able to make decisions quickly in sports, given the circumstances, and taking into account their abilities and skill levels with an accuracy of 94.5%. The skill abilities aligned toward specific physical fitness goals are ensured via decision-making, and training procedures are gradually improved through continually enhanced programs [10]. Fuzzy sets are a generalization of classical set theory that handle uncertainty and vagueness. Fuzzy sets allow elements to have partial membership, represented by membership functions that assign a degree of membership between 0 and 1 [11]. Fuzzy systems, also known as fuzzy logic systems, use fuzzy sets to represent linguistic variables and rules to approximate human reasoning in uncertain and imprecision situations. Fuzzy decision-making uses fuzzy logic principles to incorporate subjective judgments and expert knowledge, making it useful in situations where traditional approaches are insufficient [12].

Easier access to health services, calorie intake from food, and a more effective everyday meal program can be achieved by developing and modifying the decision support system for individuals with specific needs. Using voting with weights to combine grouping rule information with decision-making data, researchers can help trainers develop alternatives for training for sports [13]. However, this weighting assessment needs expert ideas to finalize the decisions about the training outcome. The sports training sessions contributed to technical and physical fitness ways where the individual’s unique tactics in virtual sports information and decision-making processes in a stable form need enhancement regarding updating the training programs [14]. Fuzzy decision support systems assist Internet service through their combined potential to personalize sports training programs, handle complex data effectively, and improve the overall athlete experience in sports fitness services [15]. However, they are still limited regarding the time and space available for training functions.

The goal of this project is to enhance fitness training programs for sports by utilizing Mamdani fuzzy inference systems and fuzzy decision support systems. In doing so, it hopes to alleviate some difficulties in tailoring training programs and activities to individual athletes. With the use of intuitionistic fuzzy numbers and fuzzy logic, this system can make better decisions about sports fitness, which in turn leads to more effective and tailored training regimens. The field of sports science benefits from this study.

An evaluation and improvement decision support model for sports fitness training programs is developed by combining intuitionistic fuzzy logic with Mamdani fuzzy inference systems. This method creates tailored, adaptive training regimens by accounting for unknowns and fluctuations in real-time fitness metrics. Fuzzy logic evaluation rules for athletic fitness levels are also investigated in the study, which offers a formal framework for making decisions in unpredictable situations.

The decision-influencing factors considered for better efficiency were training conditions, residual effects, overtraining, delay, and rebound effects, which need to be evaluated in the distinct athlete’s sports training programs [16]. The main objectives or key contributions of this paper are:

  1. 1.

    To create a novel decision support model for improving the management of uncertainties and fluctuations in real-time sports fitness data.

  2. 2.

    To incorporate intuitionistic fuzzy logic and Mamdani fuzzy inference for thorough training program evaluation.

  3. 3.

    To assess the way a person’s training methods keep them fit for athletic competition.

1.1 Structure of the Paper

The research article is ordered as follows, where Sect. 1 covers the introduction of the paper. Section 2 reviews the existing works related to the decision support system in the sports field in terms of training and fitness aspects. Section 3 gives a detailed implementation of the data source taken for evaluation. Section 4 explains the proposed Fitness Mamdani Decision System (FMDS) for evaluating the effectiveness of training programs in sports. Section 5 gives a comparative study and discussion about the results being assessed with the corresponding performance metrics. Section 6 concludes the research work and provides research ideas for further enhancement in the future.

2 Literature Survey

Agustina and Pramana [17] designed the fuzzy analytic hierarchy process (FAHP) technique as a decision support system for selecting appropriate exercises based on their fitness features. These features let users choose from various parameters, including their fitness level, workout targets, individual preferences, and other deciding variables. The method evaluates the scores for each sport taken from the Indonesian sports survey that an individual has picked, giving ratings to each factor, such as football, badminton, gymnastics, swimming, and walking. The study collected knowledge regarding the factors taken into consideration while choosing physical movements, examined data, and then analyzed knowledge regarding the possible actions.

Qu [18] evaluated the intensity load of real-time physical training objectives using fuzzy combined with radial basis function (F-RBF) neural network. The study analyzed the training operations using a sensing model with the fuzzy logic (FL) inference system. The key benefit of time-based accurate regulatory tracking used in this research was that it helped make decisions about changing loads and training programs for better outcomes for a sports athlete to achieve their goal in the training setting. The result showed that an improved convergence impact throughout sports increased training speed and posture predictability during sports training, as well as a lack of information regarding inference rule and knowledge base.

Hu et al. [19] depicted the effectiveness of qualified training capacity of physical education among college students using multiple attribute decision-making by analyzing fitness through exercising and habits. Considerable importance is placed on engaging in efficient physical activity, forming good fitness routines, and advocating for changes to the way sport is presented in institutions. For efficient decision-making, fuzzy number assisted intuitionistic fuzzy weighted Heronian mean (IFWHM). The result showed a better assessment of the physical wellness conditions of university students and promoted their cognitive success.

Cao [20] applied the enhanced Apriori method to uncover the latent connections within the data, with the findings from the research being relevant to the physical fitness measurements of learners in higher education. The initial result is obtained by judging the tiredness of various orientations based on how often they occur. Free-paced activities and functional martial arts exercises can be combined into a single, extensive training program for participants, similarly to the way unique workouts and spinning sports can be combined into a single, thorough fitness assessment program. The evaluated results were minimal error convergence and mean square error value while deciding.

Song [21] implemented a frequent set of Apriori algorithms to analyze the efficiency of the sports fitness index. Gaussian analysis of time series was used to strengthen the fidelity of sports activities. The objective base interpolation research and the use of C spatiotemporal methods for reducing noise increased the effectiveness of decision-making. The study provides direction and decision-making for the advancement of students’ physical fitness condition and the training of sports education. The guidelines for the association can be explored using six rules for deciding on training activity with the minimum condition of support and confidence as 10–80%. There is no adaptive feedback mechanism for altering the effectiveness of the training program.

He [22] recommended using decision support systems for predicting the sports players’ actions to avoid unwanted injuries and weather nature to make decisions to carry out the games. Trainers are highly skilled in assessing athletes’ physical well-being and offering practical guidance based on their fitness level. The study utilized any colony optimization to recognize and monitor athletes’ best traits, thereby enhancing the performance of both individuals and teams in sporting events. There are neither time nor space constraints on the players’ ability to train using this probabilistic approach to solve computational problems and reduce with suitable decisions.

Hoelbling et al. [23] assessed the performance of JudgED sports via video scenes through appropriate training in expert-based decision-making (EDM) under time duration. The linguistic difficulty level in making decisions is defined as very low, low, medium, and high based on the training sequence. The metric analyzed for kickboxing referees was 43% accuracy in decision-making, the kappa coefficient range was 0.166, and the response time was 1.022 s. The investigation’s small sample size may limit how broadly the results are applied, and there is no proper training platform for making decisions with a lack of retention measure of on-field performance.

Sotoudeh-Anvari [24] presented multi-criteria decision-making (MCDM) has increased due to the COVID-19 pandemic’s multidimensionality and complexity of health and socio-economic systems. In this publication, 72 papers from 37 peer-reviewed journals discuss MCDM approaches in several pandemic areas. India leads this field, followed by Turkey and China. The most used MCDM method is AHP (including fuzzy AHP), followed by TOPSIS and VIKOR. To reduce uncertainty and ambiguity, most research uses fuzzy sets and MCDM. There is no consensus on whether fuzzy methods are better than crisp methods for solution quality.

Moslem [25] displayed that the study’s primary objective is to improve Dublin’s public bus system by lowering emissions of carbon dioxide, increasing ridership, decreasing traffic jams, and making commuters happier. To assess the complicated problem and save time and effort on survey estimation, the parsimonious analytic hierarchy process (P-AHP) is employed. Potential applications may be encouraged by the model’s consistent and dependable results, which eliminate ambiguity and uncertainty in decision-making.

Shakeel and Baskar [26] showed that analysis of facial expressions and other textual data is a part of textual information mining. A difficulty in document categorization arises from the exponential increase of Web 2.0 documents labeled with emotions like grief, surprise, happiness, empathy, anger, warmth, boredom, and amusement. With the explosion of emotion-based data, new features for document categorization are becoming possible.

Kumar et al. [27] stated that online application development must balance security and usability. Fuzzy AHP–TOPSIS is suggested for usable-security assessment and attribute prioritization. It was tested on institutional web applications to prove its efficacy. The findings will assist developers in constructing usable-security web apps that satisfy users and reduce security’s detrimental influence on usability.

Sahu et al. [28] presented that users seeking efficient and affordable software services need web application development. Developing these apps needs time and cost management. Low-cost web application maintenance requires durability. Identifying durable traits is complex, and experts disagree. Multi-criteria decision-based symmetrical selection is used to choose durability characteristics in this study. Web application durability numerical evaluation affects service life and low-cost management. The authors evaluated six University projects to help developers build robust and maintainable web apps.

This article thoroughly reviews sports decision support systems with an emphasis on fitness and training. The Apriori algorithm, decision support systems, fuzzy analytic hierarchy process, and expert-based decision-making are some important studies. Training programs, decision-making, and performance outcomes can all be improved with the help of the several ways and methodologies shown in these research. An innovative decision support system is necessary to maximize training efficiency and enhance sports fitness results; this is highlighted in the literature review, which establishes the foundation for the planned research.

3 Dataset Briefing

The PMData [29] is an extensive dataset that combines lifelogging and sports activity logging. Subsequently, it offers significant insights that may be utilized in the construction of a fuzzy decision support system that is specifically customized to training regimens. The data comprises information gathered from sixteen participants over 5 months. These participants’ characteristics include age, height, gender, session time, and metrics determined by Fitbit, such as stride length for walking and running. To provide a comprehensive view of the participants’ wellness and training routines, this dataset, illustrated in Fig. 1, involves the collection of data on both everyday activities and fitness connected to sports. Continuous activity tracking is accomplished through utilizing Fitbit Versa 2.0, Google Forms, and the PMSys sports app. The data that is gathered is then saved in wellness.csv. Detailed information regarding primary sports fitness criteria is presented in Table 1. These parameters include heart rate zones, degrees of weariness, the quality of sleep, soreness, stress, mood, readiness for exercise, perceived exertion, and nutrition habits. These are all essential elements that must be considered when developing effective training programs.

Fig. 1
figure 1

Sample sports activity logging information dataset

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Table 1 Sports fitness parameters for an efficient training program
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The PMData combines traditional lifelogging with sports activity logging, is relevant to the application of a fuzzy decision support system to deliver training programs that consider both everyday activities and sports-related fitness activities, and contains logging data for 5 months from 16 individual participants p1–p16 with attributes are age ranges between (23–60), height, gender (male/female), session in months, stride walk and run observed from Fitbit.

The dataset offers a rich source for modifying methods for training because it combines lifelog and sports data, as shown in Fig. 1. The dataset mentions everyday fitness activities like weight and sleep patterns. The data collection system involves Fitbit Versa 2.0 in Fig. 1a, Google Forms in Fig. 1b, and other systems like the PMSys sports app in Fig. 1c allows for continuous activity tracking. The information stored in wellness.csv contains fitness and well-being-related attributes like data and time, fatigue, mood, readiness, sleep duration, sleep quality, soreness, and mental state, which are considered significant indication parameter factors in the fitness field and are given in Table 1.

4 Decision Support System Implementation Using FDMS

Sports fitness comprises a number of uncertain factors, including a person’s response to the training programs involved, rate of recovery, and environmental influencing variables. Training programs can be made more resilient and dynamic conditional changes using fuzzy logic to model uncertainties and variation. Fuzzy decision support systems can dynamically control training volume and intensity in response to real-time input and shifting circumstances. The fuzzy process is significant for evaluating the effectiveness of training programs with various sports fitness attributes. Figure 2 illustrates the proposed FDMS schematic representation.

Fig. 2
figure 2

FDMS schematic representation

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Sports fitness requires new decision support systems since optimizing training programs and improving athletic performance is complex and dynamic. Sports fitness needs novel decision support systems for these reasons:

Personalization and individualization: Athletes have distinct physical traits, training requirements, and performance objectives. A unique decision support system may personalize training plans to athletes’ strengths, limitations, and performance needs.

Data-driven insights: Wearables, sensors, and performance measurements provide a lot of data thanks to technology. A powerful decision support system can analyze this data to help coaches and athletes make smart choices.

Training program complexity: Sports fitness programs involve various variables, including intensity, length, recuperation, nutrition, and psychological considerations. A unique decision support system can handle this complexity and recommend an optimal balance of factors to improve performance and prevent damage.

Ability to adapt in real time: Sports surroundings alter quickly throughout training and competition. To enhance athlete performance, a responsive decision support system can alter training regimens in real time depending on feedback, performance indicators, and environmental conditions.

Better decision-making: Training techniques, workload management, injury avoidance, and performance enhancement are difficult considerations for coaches and players. Novel decision support systems can give evidence-based suggestions and simulations to aid decision-making.

Continuous improvement: he sports sector evolves with new training methods, technologies, and research. A new decision support system may combine sports science advances and respond to changing trends to keep training regimens effective and current.

Competitive advantage: A cutting-edge decision support system can give individuals and teams an edge in competitive sports. By using advanced analytics and predictive modeling, athletes can improve their training and performance.

4.1 Input and Output Membership Functions

Fuzzy decision support systems applied in this sports fitness field using the proposed FDMS model depicted in Fig. 2 involve the crisp values of each input variable. Information is processed by fuzzy logic, also known as the fuzzification process. A fuzzy set created by assembling fuzzy rule sets is the input for the defuzzification process, which generates a decision represented by an integer in the fuzzy set’s range. Stride lengths can be stated as intuitionistic triangular fuzzy numbers, which indicate feasible values, corresponding membership degrees, and a non-member in place of crisp values. From the input source, the stride length during walking can be denoted as (70/80.9/90); here, 70 represents the lower bound,80.9 is the crisp value, and 90 represents the upper bound. The fitness attributes include.

Training schedules can be continuously adjusted depending on changing lifelogging and sporting activity information using fuzzy logic to model and forecast sequences. Fuzzy decision support systems are highly adept at managing intricate and varied datasets, combining input from several sources to provide accurate decision-making. In real time, information can be used by fuzzy decision support systems to dynamically alter fitness regimens in response to changes in a person’s daily routine and sports participation. Understanding the attributes of fitness helps to evaluate workout plans, intensity adjustment, and recovery strategies. The general workout will help individuals who are still in their early training stages become more fit in a variety of fitness-related areas. Determining both the quantity and quality of sleep can help to make more accurate decisions about when and how hard to work out, in addition to how long to get back during training sessions.

4.2 Apply Fuzzification with Intuitionistic Triangular Fuzzy Numbers

Apply intuitionistic triangular fuzzy numbers to fuzzify crisp input values into fuzzy sets. A triangular intuitionistic fuzzy number is an intuitionistic fuzzy set in S with the following membership function. \(\tilde{S} \in {\mathbb{R}}\) as a real number set, calculated for an attribute fatigue. The lower bound l is taken as the minimum value for fatigue, the upper bound u represents the maximum upper value, and the hesitation margin α represents the degree of uncertainty or hesitation in assigning a precise value. The membership and non-membership functions are expressed as \(\tilde{S} = (l,\alpha ,u)\) in Eq. (1) and \(\tilde{s}_1 = (l^{\prime} ,\alpha ,u^{\prime} )\) in Eq. (2).

$$\mu_{\tilde{S} } (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {x < l} \hfill \\ {\frac{x - l}{{\alpha - l}},} \hfill & { l \le x \le \alpha } \hfill \\ {1,} \hfill & {x = \alpha } \hfill \\ {\frac{u - x}{{u - \alpha }},} \hfill & { \alpha \le x \le u} \hfill \\ {0,} \hfill & {x > u} \hfill \\ \end{array} } \right.,$$
(1)
$$\mu_{\tilde{s}_1 } (x) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {x < l^{\prime} } \hfill \\ {\frac{\alpha - x}{{\alpha - l}},} \hfill & {l^{\prime} \le x \le \alpha } \hfill \\ {0,} \hfill & {x = \alpha } \hfill \\ {\frac{x - \alpha }{{u - \alpha }},} \hfill & {\alpha \le x \le u^{\prime} } \hfill \\ {1,} \hfill & {x > u^{\prime} } \hfill \\ \end{array} } \right..$$
(2)

The above calculation shows that \(\mu_{\tilde{S} } (x) + \mu_{\tilde{s}_1 } (x) \le 1\), which shows the membership function of the input fitness variable, and analyzing this using the 1–5 scale with the measure of fatigue, sleep quality, soreness, stress level, and mood measures offers a numerical representation of subjective experiences. The mathematical representation of the fatigue membership function is graphed in Fig. 3 for efficient decision outcomes.

Fig. 3
figure 3

Intuitionistic fuzzy environment for fatigue attribute

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Given an input value x, the Equation \(\mu_{\tilde{S} } (x) + \mu_{\tilde{s}_1 } (x) \le 1\) calculates the total membership values of two fuzzy sets, \(\tilde{S}\) and \(\tilde{s}_1\). Fuzzy logic uses a membership value range of 0–1, with 0 signifying complete and 1 full membership. With these membership values added together, a negative total is impossible. The sum of the non-negative values between 0 and 2 will be \(\mu_{\tilde{S} } (x) + \mu_{\tilde{s}_1 } (x) \le 1\) as they are both non-negative numbers. So, to keep the overall membership from going over full membership (1), these numbers will never be more than 1. The total of these values cannot be negative; hence, the Equation \(\mu_{\tilde{S}{ }} (x) + \mu_{\tilde{s}_1 { }} (x) \le 1\) is correct in fuzzy logic.

4.3 Mamdani Fuzzy Inference System for Optimal Decision Support

Mamdani’s fuzzy inference system is a fuzzy logic-based method for modeling and decision-making in uncertain sports environments. The Mamdani type is a rule-based system that uses intuitionistic fuzzy sets and linguistic variables to infer decisions from input variables. Formulate fuzzy rules based on the input variables using linguistic terms and intuitionistic fuzzy numbers in a triangular form. The rule base is evaluated in response to the input and output variables. Fuzzy logic evaluation rules for sports fitness levels with varying input attributes are framed as:

  • Rule 1: IF fatigue is High AND mood is Low, THEN readiness is Low.

  • Rule 2: IF readiness is High AND stress is Low, THEN the training load is High.

  • Rule 3: IF soreness is High AND mood is Low, THEN rrate is High.

  • Rule 4: IF readiness is Low AND fatigue is Moderate, THEN the training load is High.

  • Rule 5: IF sleep quality is Poor, THEN stress level is High.

Within the Fitness Mamdani Decision System (FMDS), the Mamdani fuzzy inference system is an essential instrument that plays a significant role in the optimization of sports fitness training regimens. It uses linguistic variables and intuitionistic fuzzy numbers to construct rules depending on input variables such as weariness, mood, preparedness, sleep quality, and stress levels. These variables are used to generate the rules. The rules are evaluated by the system through the utilization of linguistic concepts and triangular membership functions, which ultimately results in the determination of optimal output decisions. A process known as defuzzification is then carried out by the system in order to transform the fuzzy output into a particular decision or action. An adaptive decision framework is provided by the system, which takes into account all of the uncertainties and variances that occur in sports fitness parameters in real time. In addition to this, it enables performance evaluation, which requires the examination of a variety of indicators, including adaptability index, training load capacity, long-term training impacts, and participation ratio. The ability of this system to handle uncertainties, linguistic factors, and complicated interactions between input and output parameters contributes to an improvement in decision-making in the sports fitness area.

The Mamdani fuzzy inference system is a mathematical framework that combines input variables, fuzzy sets, and fuzzy rules to optimize training programs in sports fitness. The system uses input variables like fatigue level, mood, readiness, stress levels, and sleep quality, and defines fuzzy sets for each variable. Fuzzy rules are expressed as if–then statements, determining the degree of membership for each linguistic term. The system then applies fuzzy rules to determine the degree of membership for each term, combining them using fuzzy logic operations to generate a fuzzy output. Finally, the system defuzzifies the output into a crisp value for decision-making. This structured approach allows for the optimization of training programs in sports fitness based on input variables and expert knowledge, providing a valuable tool for decision-making in uncertain environments. The mathematical representations of the Mamdani fuzzy inference system are as follows.

The variables for input and fuzzification: The crisp input variables can be represented as \(x_1 ,x_2 \ldots x_n\). Fuzzy sets representing linguistic concepts should be defined for each input variable \(x_i\), where \(j = 1,2,3 \ldots m_i\), Fuzzification involves transforming pure input values into fuzzy sets by use of membership functions, where \(\mu A_i^j (x_i )\), stands for the degree of membership.

Rule base: The rule \(R_k {:}\,{\text{IF}}\left( {x_1 is\,A_i^{P_{1k} } } \right){\text{AND}}\left( {x_2 is A_i^{P_{2k} } } \right){\text{AND}} \ldots {\text{THEN}}(y \,is\,B^{q_k } )\) stands for the linguistic word indices in the input set \(P_{ik}\) and the output set \(B^{q_k }\).

Inference from fuzzy sets: Determine the level of activation \(\alpha_k\), for every rule according to the level of membership of input variables in their different fuzzy sets. Use fuzzy logic operators to combine the outputs of the active rules.

Aggregation: To find the total fuzzy output, \(O*(y)\), aggregate the outputs of the activated rules using methods such as max–min or max-product.

Defuzzification: Apply defuzzification techniques, like centroid or weighted average, to transform the fuzzy output \(O*(y)\), into a clear value \(\hat{y}\).

Defuzzification and output variables: The system’s choice or action is represented by the final output value \(\hat{y}\) according to the input variables and the rule base. In real-world applications, these mathematical procedures offer a systematic framework for resolving uncertainties, modeling complicated systems, and making judgments based on fuzzy logic principles.

4.4 Adaptation Index

Training programs can be adapted to each individual by analyzing these fitness scores to help discover variations and patterns over training in sessions up to 5 months. Sportspeople become more productive and effective through consistent training. Training-related stress may result in a brief decline in functional or performance; however, both will eventually improve as a result of \(A_{\text{I}}\) Outcome. Due to adaptation and a need for regular difficulties, trainers should think about rotating their exercise routines weekly or monthly to speed up their recovery rate rrate. Starting the training process with a general program may result in beneficial changes in adaptations. The general training schedule for sports fitness is likely unlikely to be as beneficial for advancement and professional growth as participants get aged. There is a connection between the process of adaptation and training load. The adaptation technique is best to increase the training load steadily TL in accordance with the fatigue level flevel, so that the body can gradually adjust to stronger challenges while performing training. flevel is a form of exhaustion occurring due to a lack of energy in the body, especially due to low sleep quality.

$$A_{\text{I}} (\% ) = \frac{{({\text{perf}}\_{\text{metrics}} + r_{{\text{rate}}} )}}{{(T_{\text{L}} + f_{{\text{level}}} )}}.$$
(3)

In Eq. (3), the formula represents the adaptation index AI. As a percentage, training program adaption is calculated based on several criteria. \({\text{perf}}\_{\text{metrics}}\) indicates the athlete’s overall performance. Recovery rate rrate measures athlete recovery from training-induced stress. The training load TL indicates the intensity and volume of training sessions. The fatigue level flevel indicates the athlete’s tiredness may be impacted by sleep quality. Performance metrics and recovery rate are divided by training load and exhaustion level. The ratio of training intensity to recovery capacity shows how well the training program facilitates adaptation and performance development.

Figure 4 shows the fuzzy set of heart rate variable adaption index. Equation (3) explains that the training load TL changes dependent on the physical (heart rate) and psychological (perceived exertion) components. Consistent training improves efficiency, stress management, endurance, preparedness, and sprint times through adaptation. The graph above indicates that athletes can tailor their training load depending on heart rate variability and perceived exertion to improve and avoid injury.

Fig. 4
figure 4

Fuzzy set of heart rate variable

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Adaptation index, as described in Eq. (3), explains the process by which the TL changes according to the operational structures of the body and mind, that is, both physical (heart rate), as shown in Fig. 4 and psychological (perceived exertion). \({\text{perf}}\_{\text{metrics}}\), efficiency potential is increased due to load adaptability in addition to stress, endurance, readiness, and sprint times. Consequently, an athlete can enhance their performance through adaptations, which necessitates consistency in training.

4.4.1 Training Load

Sportspeople need to keep increasing the effort they put into getting fit. The attribute readiness indicator provides a general subjective assessment of the person’s level of exercise readiness. Trainers and athletes can use it as an important factor to determine whether to increase intensity or when to add rest and downtime. A quick and easy method for figuring out each player’s specific load for practice and competition is to use the rate of perceived exertion (RPE) expressed in Eq. (4).

$$ts_{{\text{load}}} = {\text{RPE}} \times {\text{training}}\_{\text{duration}}{.}$$
(4)

Align the commencement of stress placed with the targeted training load of an individual to facilitate the physical readiness on a scale of 1–100 of the athletes for sporting activities. Each participant will conclude that certain exercises, activities, and any specific training program seem more difficult or simpler than others. The RPE assessment should include comparisons between individuals to help evaluate the extent of evolution, even though collecting information to measure training programs is essential. Program diversity can be achieved by varying the amount of training in terms of volumes and the number of repetition sets; hence, it is represented in arbitrary units. Training might be carried out in a series of sessions ts or on an irregular basis based on the participant’s response to the training inputs during the week. TS can be represented in linguistic terms as short, medium, and long. The corresponding intuitionistic fuzzy sets in terms of triangular representation can be calculated as \(ts({\text{short}})\), \(ts({\text{medium}})\), and \(ts({\text{long}})\). The evaluated rule from the rule base, in accordance with the knowledge acquired from the fuzzy sets, can be generated as

  • Rule 6: IF ts is short AND RPE is low, THEN tsload is low.

  • Rule 7: IF ts is medium AND RPE is moderate, THEN tsload is moderate.

  • Rule 8: IF ts is long AND RPE is high, THEN tsload is high

The generated rule. base can be implemented in the inference stage of every fuzzy calculation process.

The ultimate value of the fuzzy process will be calculated based on the rule basis and will eventually become a crisp result.

Fuzzy systems can modify the level of training intensities in response to an athlete’s persistent tracking of performance using real-time data. This flexibility takes into account variables such as fatigue levels of difficulty, speed of recovery times, and external factors to minimize excessive training or inadequate training.

4.4.2 Recovery and Injury Rate Analysis Through Real-Time Monitoring

By taking into account variables like the degree of fatigue, anxiety, and the quality of sleep, fuzzy decision support systems can aid in optimizing recovery plans. Through the use of a holistic approach, recovery schedules are customized to meet the demands of each person involved in sports activity, boosting performance and overall fitness. The progressively increasing nature of training workload or intensity is vital, as is the planning and scheduling of rest and recovery. A well-planned and carefully monitored rest and recovery period should be implemented between fitness activities, throughout exercises, and between repetition practices to prevent injury or fatigue. Depending on the objective and level of fitness of the individual being trained, limit the amount of time spent resting among sets and fitness activities within a time frame of roughly 30–60 s. It is recommended that individuals take a minimum of one working day off once a week to prevent stress; nevertheless, sports individuals need to pay attention to their bodies and, if required, take more time off.

With the use of real-time input data from Fitbit wearable devices mentioned in the input source and other reporting tools for monitoring, fuzzy decision support systems are able to deliver immediate feedback on the health and performance of athletes. The proposed model makes it possible to modify training plans on an as-needed basis and guarantees that athletes are constantly operating within their ideal parameters.

4.5 Long-Term Training Impact

The goal of training sessions is to prepare participants for long-term achievement rather than simply providing them with an enjoyable physical activity. The effectiveness of the training program impact is evaluated from the fuzzy rule written in the form of \(\mu_{f_{{\text{level}}} } ({\text{Low}}) = (0,0,0.2)\), \(\mu_{f_{{\text{level}}} } ({\text{Medium}}) = (0,0.6,1)\), \(\mu_{f_{{\text{level}}} } ({\text{High}}) = (0.7,0.8,1)\).

  • Rule 9: IF fatigue is Low AND readiness is High, THEN effectiveness is High.

4.5.1 Participation Ratio

A participant who engages in training lazily never gains assurance or enhances their sporting potential. Instead, they stay completely reliant on their trainer or physical education instructor and never acquire new skills. As a result, the instructor or coach has to make sure that the athletes engage in the exercise with intensity and passion. Activity names include [‘team,’ ‘soccer’], [‘individual,’ ‘running’], [‘individual’, ‘endurance’]. Individuals are encouraged to take part in a team sport, particularly soccer. It means playing organized games in which two groups take on one another. The adaptive program suggests participation in a specific exercise, in this case, while running and walking. It recommends exercises incorporating running, like sprints and running for long distances. Activities involving prolonged periods of steady exertion are common in endurance-related sports.

4.6 Defuzzification

Use triangular intuitionistic numbers that are fuzzy to represent the result of the effectiveness of a training program to convey uncertainties. The distribution of membership, in this instance, peaked at one and diminished as it reached the outer limits. Assume that the training effectiveness represented in the fuzzy set is triangular, having edges positioned at (0, 0), (0.5, 1), and (1, 0) of varying fitness attributes. Based on acquired input knowledge from the fitness input attributes and generated fuzzy rules, a program’s effectiveness is determined by its value, and it rises when it reaches higher values. The centroid method is applied for defuzzification using Eq. (5) as follows:

$$C = \frac{{\sum_i x_i \cdot \mu_{\tilde{S} } (x)}}{{\sum_i \mu_{\tilde{S} } (x)}}.$$
(5)

\(x_i\) denotes the discrete values of the input variables, and \(\mu_{\tilde{S} } (x)\) represents the intuitionistic fuzzy form in a triangular base with the overall aggregated values to \(x\) set of input physical and psychological attributes. Based on the fuzzy rules and the information entered, the bigger the value, the higher the level of efficiency toward the training program is deemed to be.

4.6.1 Evaluation and Interaction with Training Programs

The feedback is provided for evaluating the sports fitness related to the influencing factors from the defuzzification process performed in the output layer. Here, the fuzzy membership function is used to enhance the effectiveness of training programs. The decisions are made based on the evaluated fuzzy membership value. Feedback and knowledge about the rationale behind specific athletic decisions based on training programs are given to coaches and trainees, improving interaction and comprehension. The term food support for increasing nutritional support describes the application of diet practices to improve recovery, maximize effectiveness, and promote general fitness for a healthy sports environment.

5 Result Analysis

The proposed method’s performance is analyzed using a comparative analysis discussed below. In this analysis, the metrics of adaptability, training load, recovery, long-term impact, and student participation are used. The existing FAHP [17], F-RBF [18], and EDM [23] methods are contrasted with the proposed decision model. Calculations were performed on consideration of two variants by using data from 16 participants (p1–p16) and training session duration as 5 months. The implementation of the proposed idea generates the output results that validate the fuzzy logic data results in the MATLAB system. Calculations using the MATLAB toolbox are more intricate, and the membership function establishes how accurate the fuzzy outcome is for each of the parameters and set of rules.

Adaptation index analysis, as shown in Fig. 5, demonstrates the increased efficiency that may result from an athlete’s increased capacity to effectively adapt to changing training intensities or loads. The goal of the training regimen’s demands can be met with an increased dynamic and efficient response as a result of higher adaptation results. A lower chance of injury is frequently linked to increased adaptation. The body’s capacity to adjust to loads and stimuli decreases the chance of strain or damage and helps avoid overload. A trend toward higher adaptability suggests that the program related to training sports individuals is tailored to the needs of each participant being trained and encourages steady advancement throughout training sessions. Subjective assessment of an activity’s difficulty is known as perceived effort on a scale of 1–10. Increased exhaustion is the measure of fatigue that may be indicated by higher perceived exertion levels, which could have an impact on increased motivation and output towards efficient endurance training. Variations in heart rate could be a sign of how the body is handling stress, \(f_{{\text{level}}}\), or recovery \(r_{{\text{rate}}}\).

Fig. 5
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Adaptation analysis

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The training load’s unit of measurement is a variable that is given as arbitrary and is determined by the particular calculation methodology performed. Training load is sometimes stated in terms of arbitrary units (AU) that possess a specified physical measurement and are approximate. Assuming that perceived exertion stays uniform, the training intensity or load illustrated in Fig. 6 will increase appropriately with an increase in session durations. Regulating loads lessens the chance of injuries brought on by overstressing the body and lowers the possibility of excessive training. Recovery tactics that ensure proper rest and regeneration in between sessions are guided by effective load evaluation. Recognizing training loads facilitates efficiency improvements by modifying volume and intensity in accordance with individual reactions and objectives of the training program.

Fig. 6
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Training load analysis

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Helping athletes recover faster, refuel their storage facilities for glycogen, and feel less tired after exercise by suggesting when and what to eat after exercises through adaptive training programs had a long-term training impact as shown in Fig. 7. The efficiency of training programs can be improved by incorporating food habits with the attribute weight, fluid intake, and meal intake to meet the energy demands of workouts and enhance the speedy \(r_{{\text{rate}}}\). Training programs are intended to increase a person’s physical readiness and capacity for specific sports in a long-term training impact among sports individuals. The adaptive training program consists of exercise for cardiovascular conditioning, corrective and restorative actions, and physical activity. In addition, it offers guidance on nutritional values and psychological and psychological fitness.

Fig. 7
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Efficiency analysis of long-term training impact

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The participation ratio analysis is depicted in Fig. 8 Sessions of training are kept engaging and challenging for individuals through the inclusion of a wide range of workouts and sports. Higher levels of engagement and a better participation proportion may result from the proposed FDMS approach. The training program helps to increase the participation ratio and then do some adjustments in the fitness activities, either by changing meal plans, cardiovascular training, or stride adjustment training if the participant ratio is low. The training program can be modified with guidance based on an understanding of the input factors impacting participation. A significant percentage increase means that most participants take part in training sessions regularly. The result is encouraging since it shows that participants are dedicated and driven and think the training programs are worthwhile. When training programs successfully engage participants’ attention and inspire loyalty, it is seen through an increased participation outcome. Trainers and coordinators of programs can apply focused tactics that continually improve and adjust their services for the best possible involvement as well as efficiency in the discipline of sports fitness by considering and understanding this fitness information.

Fig. 8
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Participation ratio analysis

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The comparative analysis shown in Tables 2 and 3 suggests that the proposed FMDS seems to be a better choice among the compared existing works and has superior performance across various significant metrics. The ability to enhance adaptation, effective training load management, sustained long-term training impacts, and higher participation ratios demonstrated that the proposed idea is a promising algorithm for making decisions related to sports fitness. In both variations, FMDS surpasses other approaches, suggesting that it is beneficial for fostering athletes’ ability to adjust to varying training loads. The fact that FMDS has a minimal amount of training indicates that it can accomplish its goals while putting less strain on the bodies of the participants as a whole. FMDS has strong long-term conditioning for training impact, highlighting its capacity to have enduring, beneficial effects on athletes’ fitness. FMDS exhibits a significant participation amount, suggesting that athletes are more involved in the exercise program. Generating the best training results requires greater commitment levels toward training programs.

Table 2 Comparative analysis for variant 1 (no. of participants)
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Table 3 Comparative analysis for variant 2 (training session)
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6 Conclusion and Future Work

A study that optimized fitness training programs for athletes using the Mamdani fuzzy inference system found that it outperformed the competition. An effective foundation for decision-making in ever-changing contexts, the system performs well when faced with training program modifications and uncertainties. A more intuitive and human-like decision-making process is enabled by its effective handling of linguistic factors and expert knowledge. Sports fitness settings can benefit from the system’s capacity to adapt training regimens on the fly in response to feedback and changing conditions. The decision-making process can be better understood and interpreted by users because to the open and transparent character of fuzzy logic rules, which in turn allows for greater insights and feedback to be provided for enhancing training programs. Complexity, data reliance, and interpretation cost are some of the disadvantages of the approach. The efficiency of a fuzzy inference system is highly dependent on the correctness and appropriateness of the input data, and its implementation and tuning could necessitate computer resources and specialized knowledge. Users without a solid foundation in fuzzy logic may find it daunting to understand and manage a large number of fuzzy rules. The suggested method’s unique benefits in athletic fitness training programs can be better understood by comparing it to other decision-making algorithms such as TOPSIS, CODAS, and GRA. The findings and justification for the approach’s adoption can be strengthened by providing quantitative metrics or case studies that show how the proposed method is better than current methods.

The study on decision support systems for sports fitness, using Mamdani’s fuzzy inference system and intuitionistic fuzzy logic, highlights the importance of the adaptation index in tailoring training programs to individual athletes. The system’s effectiveness in evaluating training programs and fitness attributes was demonstrated through a comparative analysis against existing methods like FAHP, F-RBF, and EDM. The system demonstrated superiority in terms of adaptation index, training load levels, long-term training effects, and participation ratio.