Advancing sports performance evaluation through T spherical fuzzy FUCA MCDM approach to real time basketball analytics

Abstract

The process of assessing the performance of players in real time must be supported by well-developed models of decision-making in modern sports analytics, as they are required to react to uncertainty and complex data interactions. This work presents a new t-spherical fuzzy FUCA (TSF-FUCA) multi-criteria decision-making (MCDM) system of real-time basketball playing performance evaluation. The suggested technique combines the flexibility of the T-spherical fuzzy environment and the efficiency of the computational power of the FUCA algorithm to handle the ambiguity, imprecision, and multi-source information. To illustrate the practical relevance of the suggested model, a practical case that included fifteen basketball players (alternatives), seven performance criteria, and four decision-makers was created. The standards used, including shooting accuracy, defensive efficiency, teamwork, stamina, speed in decision-making, consistency and fouls committed, are both technical and tactical in nature, as they measure player performance. The sensitivity and benchmark analyses also confirmed the soundness of the TSF-FUCA approach compared to other T-spherical fuzzy and classical MCDM techniques. The results affirm that the suggested TSF-FUCA model offers a sound and smart decision-support model of real-time basketball performance analytics. In addition to the sports applications, this model adds a new dimension to the fuzzy MCDM theory as it is a flexible tool in solving complex decision-making problems in the face of uncertainty.

Introduction

Real-time basketball performance measurement is one of the foundations of contemporary sports analytics, as it allows coaches, analysts, and decision-makers to understand the way a player performs in real-time during the game. It means gathering, processing, and analyzing immense data on a real-time basis on wearable sensors, video tracking, and performance monitoring systems. The mentioned data sources present real-time information on the movements of players, shooting precision, defensive effectiveness, collaboration, and decision-making speed. Through the process of synthesizing this data, analysts are able to assess the performance of individuals and team performance in a holistic manner. Hence, real-time basketball performance analysis is a very important model for determining the strengths, weaknesses and tactical opportunities that enable teams to make important decisions in real time during the competition. Nevertheless, the process is complicated, interdependent, and ambiguous and has a number of difficulties because of the complexity, interdependence, and uncertainty of the data. Conventional methods of evaluation do not consider subjective tests, changing game variables, or the uneven quality of data.

Therefore, the MCDM approach integration has become a viable solution. MCDM offers a methodological procedure for assessing a mix of conflicting criteria at the same time and assists analysts and coaches in producing balanced assessments concerning players and strategies. Using MCDM in basketball analytics, it is possible to combine expert opinions and objective data into one cohesive decision-making model to improve the quality and clarity of performance evaluation. However, dealing with uncertainty and vagueness in human judgment seems to remain one of the problems in the application of MCDM. The classical MCDM models use crisp numerical data, which are incapable of reflecting subjective perceptions or incomplete information. In a bid to address this limitation, Zadeh¹ came up with the Fuzzy Sets (FS) wherein the membership degree (MD) of an element falls in the range of [0,1]. The classical fuzzy model, however, could only model the level of membership but not the level of non-membership (NMD). Atanassov² has overcome this weakness by coming up with Intuitionistic Fuzzy Sets (IFS), which include both MD and NMD, whose sum is in the [0, 1] range. This framework was later extended by Yager³ to Pythagorean Fuzzy Sets (PFS), where the square of MD and NMD could remain in the range of 0 to 1, and furthermore⁴, to q-Rung Orthopair Fuzzy Sets (q-ROFS) to be more flexible. Cuong⁵ presented Picture Fuzzy Sets (PFS) to describe the degree of abstinence (AD) used in making a decision, and Mahmood et al.⁶ advanced this by creating Spherical Fuzzy Sets (SFS) that had a higher representational potential. Continuing on these bases, T-Spherical Fuzzy Sets (TSFS)⁷ were proposed as a more advanced paradigm, which is able to reflect a greater level of uncertainty, indecisions, and is thus suitable for real-time and data-intensive decision settings, as is the case with basketball analytics.

During the development of MCDM methods, several methods have been used to address multi-criteria problems, which include TOPSIS, WASPAS, CODAS, EDAS, and MARCOS. Nonetheless, the FUCA (Fair Un Choix Adequat) is a new and innovative MCDM technique, which is aimed at giving fair and adequate decision results based on the balanced mixture of normalization, aggregation, and evaluation procedures. The FUCA approach guarantees fairness (Fair), the right choice (Un Choix) and sufficiency in the decision-making process, which allows the combination of quantitative and qualitative data in uncertain circumstances. It has an efficient mechanism of balancing the power of decision-makers and criteria that results in consistent and stable outcomes. In real-time basketball performance assessment, FUCA offers an open, fair, and efficient system of ranking player performances in an environment of uncertainty and subjectivity. This paper suggests a TSF-FUCA framework that combines the effective uncertainty-management unit of TSFS and the impartiality-motivated decision-making framework of FUCA. The originality of this study is that it is the first study to combine TSFS and FUCA to determine the performance of players in a real-time basketball environment. The suggested model can successfully handle the information that is uncertain, imprecise, and multi-source to provide trustworthy player rankings and actionable data. Also, the research will carry out a comparative and sensitivity analysis to prove the stability and excellence of the suggested method over other existing MCDM methods. Overall, the proposed research paper provides not only a fresh, smart, and equitable decision-support system but also an improvement in the accuracy of analytics, strength, and readability in real-time basketball player performance analysis - not only theoretical enrichment but also practical deployment in the current sports analytics.

Objectives of the study

The main aim of the paper is to create a smart and effective decision-making model of real-time basketball performance analysis through the TSF-FUCA approach. The research will combine the high uncertainty-handling ability of TSFS and the fairness-based decision-making logic of the FUCA method to improve the accuracy, transparency, and interpretability of the performance evaluation of players.

The targeted objectives of this study are as follows:

• To establish and conceptualize an all-encompassing real-time basketball performance assessment system, which is both objective in terms of measures of performance and subjective with regard to experts’ opinion.

• To implement the T-spherical fuzzy environment to manage effectively the uncertainty, hesitation, and imprecision of the multi-expert judgment and of dynamic games.

• To conduct sensitivity and comparative analysis to test the soundness, dependability, and coherence of the proposed TSF-FUCA model in different weight conditions of decision-makers and criteria.

• In order to compare the proposed methodology to currently available methods of MCDM, including TOPSIS, WASPAS, CODAS, EDAS, and MARCOS, it is important to state that the current methodology can be used to address real-time and uncertain data.

• To offer useful information to coaches, analysts, and sports managers in order to facilitate the use of data to make decisions, select players, and optimize performance strategies.

Motivation and contributions of the study

This research is inspired by the growing need to have a correct and dynamic real-time performance analysis in basketball, whereby various uncertain and subjective variables are influential in the evaluation of the players. Conventional approaches tend to be based on snap data, and they do not deal with ambiguity or uncertainty in being an expert. Thus, it is necessary to have a sophisticated MCDM model that will be able to deal with uncertainty, as well as be fair and accurate in the decision process. This paper addresses this gap by proposing a novel TSF-FUCA model that distinguishes itself from existing fuzzy MCDM approaches by simultaneously handling MD, NMD, AD, explicitly incorporating criteria weights, and offering enhanced sensitivity for real-time decision-making. The suggested strategy will increase the management of uncertainty, balance among decision-makers, and ensure that real-time assessments are evenly ranked.

The key findings of the research are:

• It is the initial proposal of the TSF-FUCA model in real-time basketball performance assessment.

• The model improves upon existing fuzzy MCDM methods by combining the flexibility of TSFS with the fairness and adequacy principles of FUCA, ensuring more accurate and equitable rankings under complex uncertainty.

• Having better accuracy and stability is identified by comparative and sensitivity analysis with the current MCDM techniques.

• The product offers an efficient and smart decision-support system to coaches and analysts to drive training and strategy formulation.

In general, the TSF-FUCA model provides an effective, multi-faceted, and uncertainty-resistant real-time basketball performance assessment model, highlighting its novelty and practical advantage over existing fuzzy MCDM methods in sports analytics.

Structure of the study

The rest of this paper will be structured as follows: Sect. 2 will give a review of relevant works in MCDM, FUCA and TSFS. In Sect. 3, the preliminaries are located. The proposed TSF-FUCA-MCDM methodology is given in Sect. 4. The case study on real-time basketball performance evaluation is provided in Sect. 5. Section 6 introduces the sensitivity analysis, validation and comparison findings. Finally, Sect. 7 provides a summary of the study and gives future directions for the research.

Literature review

Studies related to sports performance evaluation

The sport performance evaluation has turned into a more data-driven process that incorporates sensing technologies, computational models, and decision-support systems to increase precision and objectivity in evaluating the players. Zhu et al.⁸ analyzed the mechanical behaviour of a hybrid composite consisting of glass and carbon fibre in fencing sports and established how optimal materials may enhance a great deal in performance and reliability of the athletes. Their work emphasizes the value of precision and mechanical accuracy, which also applies to human performance evaluation. On the same note, De Fazio et al.⁹ gave a review of wearable sensors and smart devices in monitoring sports performance and rehabilitation with more focus on the application of Internet of Things technologies to detect kinetic parameters, heart rate and joint movements in real-time. The systems also provide a basis for real-time analysis that is crucial in formulating intelligent sports analytics systems. Continuing this technological focus, Liu et al.¹⁰ introduced a technology called EitNet, an architecture in IoT supplemented with the introduction of real-time basketball action recognition, which will greatly enhance the performance in terms of accuracy and efficiency in detecting dynamic movements of the players. In their research, they showed how combining the IoT and deep learning can facilitate real-time data processing, which enhances analytical functions in basketball strategy. Wang et al.¹¹ took this concept a step further in time-motion analysis to decipher basketball training and maximize the performance of players. Their research offered very useful information about game-based conduct of players, which encouraged a holistic and fact-based outlook on performance evaluation. In the cognitive aspect, Rosch et al.¹² were able to establish a video-based evaluation instrument, which evaluates the ability to make a decision among young basketball players. They emphasized the value of in-situ, real-time assessment in reproducing the actual decision-making process in a high-stress situation, but observed that there were problems with subjectivity in the observer and data interpretation. To supplement this, Zhang et al.¹³ developed a sensor-based real-time state recognizer of basketball goals by applying IMU and deep learning. Their approach demonstrated a better shot recognition accuracy and real-time analysis of performance, and this demonstrates the ability of deep learning and motion sensor data to work together to improve sports analytics. Overall, all these studies highlight the increased overlap of sports technology, artificial intelligence, and real-time analytics to assess athletic performance. Nevertheless, the majority of available models consider only mechanical or sensor-driven aspects and do not consider the MCDM point of view to deal with uncertainty in the subjective representations of experts. Consequently, in this study, the TSF-FUCA approach, which is the integration of fuzzy logic and MCDM, is proposed in order to manage uncertainty effectively and deliver just, correct, and holistic real-time evaluation of basketball performances.

FUCA method and its applications in MCDM problems

The FUCA approach is one method that has been recently inserted into the portfolio of MCDM approaches, and it is aimed at delivering fair and balanced assessments of alternatives with the help of ranking them by criteria without normalization. It aims at ensuring consistency of decisions and preventing distortion due to the scale of data. The FUCA approach provides the criteria and alternatives with equal weight, and thus, this method is especially appropriate in real-life issues, where the comprehensive fairness and clarity of rankings are critical. Do¹⁴ used the FUCA technique in the optimization of mechanical machining processes and compared the results with other classical MCDM techniques, including the WSA. The results showed that the FUCA gave more stable and consistent results, especially in the assessment of the financial performance metrics in mechanical systems. In the same fashion, Thinh¹⁵ applied the FUCA method to optimize the multi-objective of the turning process through multi-weighting schemes and proved its capability to favorably arrive at the optimal solutions. This paper has highlighted that the structure of FUCA makes it easy to compute and keeps the accuracy of the decision-making at an optimal point, and it is therefore a perfect tool to optimize the industrial process. Afterwards, other MCDM techniques have been combined with FUCA. To overcome the shortcomings of classical FUCA, Truong and Thinh¹⁶ came up with the proposal of a combined PIPRECIA-modified FUCA method of selecting the lathe machines. Their method improved the quality and flexibility of the decision-making procedure. Nguyen¹⁷ furthered the use of FUCA by integrating it with CURLI and the weighting algorithm to pick technical products like the air conditioner, washing machine, and UAVs. This research showed that FUCA is very flexible in assessing a multi-domain product, and it has great power in analyzing comparative performance. Furthermore, Bao et al.¹⁸ used FUCA in the choice of mini water pumps when compared to the PSI method. The experimental findings showed that FUCA produces more stable and consistent rankings, which is a confirmation of its strength and the ability to handle more complex decision-making settings. All these studies highlight that the FUCA method of determining quality is versatile, stable, and fair in a variety of uses of MCDM, such as in engineering design, manufacturing, or product selection. It is based on these strong points that the current paper incorporates FUCA into the context of the TSFS and introduces a new TSF-FUCA model that improves the uncertainty management during the process of real-time basketball performance assessment. This combination will fill the divide between objective data analysis and subjective human judgment, which will increase the validity and centrality of the decision-making process.

Research gap analysis

Despite the significant achievements in the area of sports performance assessment, the majority of the available technologies are based on sensor-based solutions, machine learning algorithms, and IoT-related solutions to gather and process real-time data. These models are efficient in terms of physiological, biomechanical and tactical factors of the performance of players, but they do not have a complex decision-making system that can incorporate various incompatible criteria and subjective decisions. The existing systems are largely performed in crisp environments with no consideration of the uncertainty and hesitation that are inherently part of human judgments whenever engaging in dynamic sporting events. On the same note, even though the FUCA method has been emphasized as a fair and stable method with a simplicity of computation in engineering and industrial applications, it has yet to be explored in real-time sports analytics. The operation of the traditional FUCA is crisp, and therefore, it is not able to deal with ambiguous or uncertain data that is common in the performance evaluation business. In addition, the past research has failed to incorporate FUCA with sophisticated fuzzy extensions that can depict complicated uncertainty frameworks. Despite the fact that fuzzy systems like IFS, PyFS, q-ROFS and SFS have made great contributions in uncertainty management, their combination with FUCA is immature. In particular, TSFS, which has been more successful in representing higher levels of truth, indeterminacy, and falsity, have not been used yet with FUCA in MCDM situations. Thus, we can state that there is a definite gap in the methodology and application to define the combination of FUCA and TSF environment to evaluate sports basketball performance in real time. This paper fills this gap by creating a TSF-FUCA framework that will strengthen the uncertainty management, provide equitable ranking of the players, and offer a strong, stable, and versatile decision-making system to work in dynamic and data-driven sporting scenarios.

Preliminaries and theoretical background

The concept of FS is one of the mathematical models of uncertainty created by Zadeh¹. Nevertheless, in the ambiguous conditions, it is sound in the presence of an MD but not in the presence of NMD and any ambiguity in the ambiguous conditions. In order to address these weaknesses, Atanassov² proposed another approach referred to as IFS. Cuong⁵ developed the PFS to suit the complex cases where the abstinence and indeterminacy factors are in play. PFS presents not only the MD, NMD, and degree of refusal (RD) to correspond to the objectives of FS, but also AD. SFS was proposed by Mahmood et al.⁶, and the square sum of MD, AD and NMD equals 1.

Definition 1 Shen et al.⁷ consider a fixed universe and its subset called an TSFS, where represents MD, represents AD, and represents NMD. These degrees satisfy the following condition:

Additionally, the RD for each TSFS is defined as:

Definition 2 Shen e tal.⁷ let , be two spherical fuzzy numbers (TSFNs) and be any scalar number, then it satisfies the following operations:

Definition 3 Shen et al.⁷ for any TSFNs denoted by then the score function is defined as follows:

1

where .

Development of the T-spherical fuzzy FUCA (TSF-FUCA) method

In this work, the TSF-FUCA approach is created through the combination of the TSFS framework and FUCA decision-making approach to improve the management of the uncertainty in MCDM problems. The TSFS model allows representing expert ratings by using three independent factors, MD, AD and NMD, and implicitly includes the effect of hesitation. In order to be consistent and expressive in human judgments, linguistic terms (LTs) are translated into TSFNs, which offer a flexible and realistic representation of subjective judgments. The FUCA approach, with its fair and satisfactory compromise ranking mechanism, functions by ranking alternatives in each criterion, weighting the criterion and adding total scores without explicit normalization. Although the classical FUCA model is effective, its operation occurs in a crisp environment, thus restricting its capability of elucidating vagueness and hesitation in expert opinions. The proposed TSF-FUCA framework will address these drawbacks by integrating FUCA into the TSFS context so that a more precise, resilient, and uncertainty-resistant ranking can be performed. Therefore, the TSF-FUCA methodology contributes to a great extent of flexibility, adaptability, and accuracy in a practical decision-making environment. It is particularly useful in real-time basketball performance assessment that requires fast, impartial and uncertain judgments to be made that determine the dynamic performance of players based on a variety of aspects. The general procedure is illustrated in Fig. 1.

Fig. 1.

Flowchart of the TSF-FUCA methodology.

Step 1: Experts evaluate the criteria and calculate the weights of DMs.

The linguistic evaluations for the criterion and DMs are shown in Table 1.

Table 1.

Linguistic variables to evaluate criteria and decision makers.

LTs	TSFNs
Very poor (VP)
Poor (P)
Fair (F)
Good (G)
Very good (VG)

denotes the set of decision makers (DMs) with weights represented by and

2

Step 2: Create an aggregated TSF decision matrix for criteria and decision-makers.

Given a set of decision-makers, let be the TSF decision matrix. Here, indicates the evaluation of regarding DMs the criteria. is employed by TSFNs, and it could be described as

The aggregated TSF decision matrix is represented as

3

where

Step 3: Find the criteria weights by using TSF scores.

Calculate the TSF score by using Eq. (1). It is mentioned that the finalized weights are computed by applying normalization, and the total weights should equal 1. There are criteria, and each weight of

Step 4: Construct the TSF decision matrix for alternatives, criteria, and decision-makers.

Table 2 is used to conduct the linguistic assessments of the alternatives.

Table 2.

Linguistic factors for an alternative ranking system.

LTs	TSFNs
Very low (VL)
Low (L)
Medium low (ML)
Medium (M)
Medium perfect (MTSF)
Perfect (TSF)
Very perfect (VTSF)

Step 5: Create an aggregated TSF decision matrix.

Apply Eq. (3) to aggregate the TSF decision matrix.

Step 6: Find score values.

By using Eq. (1), find the score values of the aggregated TSF decision matrix.

Step 7: Rank the alternatives for each criterion.

For benefit criteria, rank alternatives in descending order and for cost criteria, rank alternatives in ascending order.

Step 8: Calculate the TSF-FUCA scores.

Calculate the score for each alternative using (4). Here, represents the rank of criterion for alternative , as determined in Step 6:

4

Step 9: Rank the alternatives in descending order based on TSF-FUCA scores.

Case study: real-time basketball performance evaluation

Real-time basketball performance analysis has become a critical component of contemporary sport analytics to offer coaches, analysts, and management teams with practical information regarding the effectiveness of the players and the overall workings of the teams. The development of wearable tracking devices, motion sensors, and AI-driven performance monitoring has been a revolution in the assessment of basketball performance. Real-time assessment can be carried out through the combination of player tracking data, including speed, acceleration, shooting trends, and defensive positioning, which needs to be constantly monitored both in training and in a competitive game. This dynamic analysis identifies not only the accuracy of statistics but also situational awareness, reaction speed, and flexibility, which resulted in a comprehensive insight into the contribution of players and the areas of improvement. The need to develop real-time basketball performance evaluation is explained by the fact that the gameplay has become more complex, and coaches have to make decisions using the data.

Conventional ways of analyzing the post-game are usually constrained by the fact that feedback is delayed and is subjective. Real-time evaluation, on the contrary, allows coaching personnel to make real-time tactical changes, avoid the fatigue of football players, and optimize the number of substitutes, depending on the live performance indicators. Also, with an increasing analytical shift in the professional leagues and sports academies, the necessity of systematic, objective, and computationally effective decision-making tools becomes inseparable. An organized decision support system will guarantee the equitable assessment of players, the identification of talent, and equating performance analytics with a strategic set of goals. MCDM techniques are critical towards such real-time evaluations since they are applied to manage various conflicting performance indicators at a given time.

MCDM is used in basketball analytics to identify the most effective players or other strategies by combining both quantitative and qualitative aspects. A number of MCDM techniques have been used over the years in sports environments (TOPSIS, MARCOS, CODAS and WASPAS), but these techniques tend to fail to address the non-maturity and equivocality of human judgment. In order to overcome this disadvantage, a new TSF-FUCA approach is proposed at present. This procedure is also good at encompassing the levels of membership, non-membership, and neutrality and provides more flexibility and realism to the process of assessing uncertain and subjective criteria. The recommended TSF-FUCA model will therefore be a better and more detailed model of ranking players in real-time basketball performance situations. In order to demonstrate the relevance of the proposed approach, one may refer to a situation of real-time basketball game performance evaluation and use four decision-makers (DMs):

• (Head Coach),

• (Assistant Coach),

• (Sports Analyst), and.

• (Fitness Expert).

These criteria were selected through a combination of literature review and expert consultation to obtain relevance, reproducibility and strength of the evaluation model. The criteria were evaluated based on their relevance in both physical and strategic performance of the player. This methodological approach will make sure that the MCDM analysis will be based on overall player analysis in the uncertain situation. Such professionals jointly evaluate the performance of fifteen basketball players based on seven criteria. The players under observation are the alternatives (–):

• (Player Alpha),

• (Player Bravo),

• (Player Charlie),

• (Player Delta),

• (Player Echo),

• (Player Foxtrot),

• (Player Golf),

• (Player Hotel),

• (Player India),

• (Player Juliet),

• (Player Kilo),

• (Player Lima),

• (Player Mike),

• (Player November), and.

• (Player Oscar).

The assessment criteria are well chosen, as they represent both physical and strategic aspects of the performance of the players.

• : Shooting Accuracy—determines the percentage of successful shots/shots. It is a direct indicator of the efficiency of a player in terms of scores, which is an important factor that determines the result of a game.

• : Passing Efficiency—measures the accuracy of the pass, the timing of the pass, and the quality of the decision to pass. Passing efficiency leads to teamwork and better coordination of the offence.

• : Defensive Contribution—is the skill of a player to protect against the shots, passes and to defend others. Defence performance is vital in denying the opponent a goal.

• : Agility and Speed—measures the movement, agility and acceleration of a player on the field. Players in Agile are responsive to changes in games.

• : Endurance Level—is an evaluation of the stamina and energy capacity of a player during the gameplay period. Increased stamina will guarantee the maintenance of performance during high-pressure situations.

• : Team Coordination—looks at how a player correlates his/her actions with their teammates to encourage cohesive gameplay and tactical harmony.

• : Error Rate—indicators include the errors, including turnovers, fouls and missed opportunities. This is deemed as a cost criterion because the smaller the figure, the better the performance.

In this case study, the information on player tracking systems and expert judgments is turned into TSF values to capture uncertainty and hesitation in the judgments. The linguistic judgments are given to each decision-maker for all the players concerning each criterion, and then summed up to make a joint decision matrix.

Step 1: An operationalization of considering these fifteen options according to the seven criteria is done using a step-by-step model. Tables 1 and 2, respectively, state the linguistic terms and the TSFNs of benefit and cost criteria. Table 3, in its turn, gives the weights of the four decision-makers according to the realization of the process of evaluation, which gives the decision-making process significance.

Table 3.

Decision-makers weights.

DMs
LTs
TSFNs
Weights

Step 2: Table 4 is then utilized to summarize the judgments of the decision-makers on the evaluations of the alternatives based on each of the criteria, and their linguistic judgments are entered into Table 4. The words of language are translated into Table 5, as TSFNs, and included to form the TSF decision matrix. Lastly, Table 6 demonstrates the functionalities of group evaluations using the operator of TSFWA using the application of the Equation. Equation (3) is included as input to the proposed TSF-FUCA method.

Table 4.

Criteria evaluation based on decision-makers’ preferences.

Table 5.

Spherical fuzzy decision matrix for criteria and decision-makers.

Table 6.

Aggregated matrix.

Step 3: Table 7 shows the weight of the criteria ( -) that were established considering the choice of the decision-makers. Figure 2 demonstrates the weights of the criteria through the graphical representation, which gives an intuitive idea of the value of each factor.

Table 7.

Normalized criteria weights.

	Scores	Criteria weights

Fig. 2.

Criteria weights.

Step 4: Table 8 entails the assessment of the alternatives according to the criteria in the form of linguistic statements by the four decision-makers. Table 2 is developed on the basis of the above-mentioned linguistic scale (Table 2). Table 9 is an expression of the linguistic terms in Table 8 into the TSFNs containing the TSF decision matrix.

Table 8.

Evaluation of alternatives and criteria.

Table 9.

Spherical fuzzy decision matrix.

Step 5: The sum of the total in Table 10 is the decision matrix, which was obtained through the TSFWA operator in order to obtain individual judgments combined in a group decision.

Table 10.

T-spherical fuzzy aggregated decision matrix.

Step 6: Table 11 uses the TSF information to compute the score values of the alternatives using Eq. (1) to give an initial ranking of the other options.

Table 11.

T-spherical fuzzy score values.

Step 7: This process is further classified into Table 12 and ranks the alternatives on the performance against each of the criteria and offers the facility to further look at the performance in detail within -.

Table 12.

Ranking of alternatives under each criterion.

Step 8: Finally, Table 13 gives the TSF-FUCA scores, which consist of the application of the interaction between the criteria to the final scores, as a way of achieving a holistic assessment of the alternatives. Figure 3 reports the findings of the ranking in graphical form, which will give an accurate comparative analysis of each alternative performance in the optimization process.

Table 13.

T-spherical fuzzy FUCA scores.

	TSF-FUCA scores	Ranking

Fig. 3.

Ranking of alternatives.

Result discussion

The received outcomes of the given TSF-FUCA MCDM model of real-time basketball performance analysis display the evident and significant ranking of the fifteen basketball players. Based on the results, the overall performance of the Player Oscar () was the best, which implies that he showed great consistency in all the considered criteria, especially in his shooting accuracy, endurance, and team coordination. His exceptional skills in balancing between offensive and defensive player contributions, and a low rate of error, point to him as the most efficient player in real-time gameplay scenarios. The next was Player November () in his agility and passing efficiency, and the next one was Player India () who demonstrated a high position because of their good contribution to the defence and a good ability to coordinate with other players. The performance of Player Delta () and Player Kilo () can also be considered commendable as they were stable in the majority of the criteria, although there were some minor weaknesses in the area of endurance and shooting accuracy. Agility and speed were the two qualities of Mike () came in second, and he scored well, but he had to improve passing and making decisions when under pressure. The average-performing group, such as the Players Bravo (), Hotel (), and Echo () scored on average with a steady performance in most of the parameters but with low defensive capabilities and average endurance rates. Also, the scores by Player Alpha () and Player Foxtrot () in shooting and error management were balanced but at a lower rate, which slightly influenced their overall score. The other game players, which comprised Golf (), Lima (), Juliet (), and Charlie () showed a relatively poor performance, and this was mainly because of an increased level of errors and lack of coordination in live play. They indicate that, although they have potential in single qualities, like agility or passing, they need specific training to enhance their level of consistency in all areas of performance.

Compared to the previous research, our TSF-FUCA model has its results close to those presented by Liu et al.¹⁰ and Zhang et al.¹³, who also focused their attention on the necessity of real-time performance monitoring and accurate assessment of the behavior of players through the IoT and sensor-based methodologies. In contrast to these works, where sensor recognition and motion analysis are the main subjects of the research, our approach combines fuzzy MCDM to represent the uncertainty, expert opinion, and various performance criteria in a simultaneous manner. On the same note, although Wang et al.¹¹ emphasized time-motion analysis to optimize training, our solution extends their idea to give a sound ranking model, which explicitly weighs various factors and uncertainty to achieve more stable and just player ratings. In contrast to De Fazio et al.⁹, which emphasized wearable devices to measure performance, the TSF-FUCA model records performance and processes subjective expert ratings with TSFS, so that the ranking is not only objective but also qualitative.

Overall, the results of the ranking prove the utility of the TSF-FUCA methodology in the separation of the fine lines between players with the assistance of the multi-dimensional and uncertain assessment information. The model is good at integrating the perceptions of the experts and the application of fuzzy logic to generate balanced and natural performance appraisals. The analysis indicates that the players who are consistent in their performance in terms of benefits and cost consideration record better performance. Thus, the proposed solution provides a good and intelligent decision support tool to coaches and analysts so that they would be able to identify the most efficient players, design individual training programs, and enhance the overall strategy implementation in live basketball environments. These comparisons have shown clearly that our method is better than the previous methods since it can manage the unpredictability and expert judgment, and give more favorable and accurate rankings of real-time sports performance assessment.

Sensitivity analysis

The section discusses the robustness and soundness of the framework suggested as TSF-FUCA, and evaluates how the weights given to the decision-makers and the weights given to the criteria would affect the final results. It also tests the consistency of the relevance of the criteria and rank of the choices in one case, so that the decision-making process will be repeated in the various cases.

Influence of change of dms’ weight values on criteria weight values

To identify the strength and flexibility of the proposed TSF-FUCA MCDM framework, the impact of the weight values of decision-makers (DMs) on the corresponding criteria weight was studied. Such an investigation makes certain that a choice process is not affected by varying relative importance attached to every decision-maker. The four decision-makers presented different sets of weight preferences in different evaluation rounds, which were indicative of the different perspectives they had on the value of each aspect of performance. Figure 4 shows the trend of the criteria weight change that is affected by these changes in the importance assigned to DMs. It also suggests that the decision-makers varied in terms of the emphasis that they put on the aspects of performance, but the overall balance of the criteria weight distribution remained stable. The stochastic first decision-maker would allocate an average significance to all the performance factors with a tendency to balance the judgment. The second decision-maker focused on technical performance and performance attributes related to coordination, whereas the third and fourth decision-makers showed reliable assessments with more inclination towards endurance and defensive performances. The variety but uniformity of these views supplemented the process of collective decision-making and helped to make the end result of the criteria weight distribution strong. Figure 4 analysis revealed that some of the criteria, in particular those connected to precision of offence, teamwork and control of errors, continued to be supreme in all decision-making conditions, which means that they are universally regarded as important to examine basketball performance. On the other hand, agile-related criteria and those with defensive components showed moderate changes, which indicated that they were partially reliant on the judgment of individual evaluators. Though these differences occurred, the general order of the ranking of the criteria did not change significantly, meaning that the shift in the weighting done by decision-makers did not have a large effect on the ultimate score. This consistency verifies the consistency of the TSF-FUCA technique in terms of combining numerous expert judgments without a reduction of consistency in decisions. It shows that despite the differences in patterns of influence of decision-makers, the weighting criteria process remains coherent, which is indicative of the strength of the method to deal with the reality of uncertainty and subjectivity. The findings, therefore, confirm the fact that the suggested model guarantees a just, unbiased, and solid assessment paradigm of real-time basketball performance assessment.

Fig. 4.

Effect on the criteria weights by changing the weights of DMs.

Impact of change of criteria weight values on ranks of the alternatives

In an effort to determine the stability and robustness of the proposed TSF-FUCA MCDM approach, the analysis was done to find out the effect that the changes in the criteria weight values have on the ranking of the alternatives. This analysis allows making sure that the ranking outcomes are not very sensitive to the slight changes in the criteria weights, thus making sure the proposed decision framework is reliable. The criteria weights were changed systematically in numerous simulation rounds, and the changes in the performance scores of the alternatives were recorded. The general pattern of these changes is shown in Fig. 5, which shows the dynamic relation between the changing criteria importance and the respective rank behaviour of every player. The findings indicate that, although the weight values of individual criteria were moderately modified, the overall ranking structure of the alternatives was the same, with slight positional differences observed among the players who had very similar scores in their performance. The highest-ranked players always had high processes, which proves that they have strong performance benefits and are not highly dependent on slight changes in weight. Players like , , and were always at the top of the ranking order, and it is possible to note that their contribution is high and balanced in the majority of criteria. These players are also stable, implying that they perform well in the characteristics of accuracy when shooting, endurance and coordination within the team, and such characteristics dominate regardless of the variation in the significance of other criteria. The mid-range performers were slightly varied (, , ) but still, their positions have stayed comparatively low in the top half of the list. These players have exhibited reliable outcomes in a variety of features, but the ranking of these players changed marginally when the criterion of agility or error rate was assigned more weight. Players of lower rank and those whose performance in the key criteria is less consistent were found to be more sensitive to changes in criteria weights, indicating that their scores were more sensitive to the weight placed on particular attributes like defensive contribution or endurance. As Fig. 5 indicates, despite the significant variation in the weights of the criteria, the proposed approach had a distinct and consistent trend in the ranking of results. The players who performed well in terms of their overall performance were still ranked at the top, and the ones that performed poorly were still ranked at the bottom. This consistency proves that the TSF-FUCA MCDM method offers a solid system of evaluation, which is able to withstand the changes in the levels of importance defined by the experts, without affecting the reliability of the results. The low rank reversals and continuous transition pattern in the simulation runs confirm the ability of the model to maintain integrity in decision-making when making decisions in uncertain and variable decision environments. Therefore, the presented technique is a valid and effective instrument of real-time basketball game performance measurement that provides the results that are consistent and interpretable even at varying criteria weight conditions.

Fig. 5.

Effect on the ranking of alternatives by changing the criteria weights.

Benchmark comparison with existing TSF-MCDM approaches

A comparative study was conducted with three proven TSF-MCDM techniques, namely TSF-CoCoSo, TSF-MARCOS, and TSF-CRADIS, to justify the effectiveness and superiority of the suggested TSF-FUCA methodology. This benchmark comparison is aimed at comparing the stability, discriminative capability, and ranking accuracy of the proposed model with the state-of-the-art techniques in terms of stability, discriminative ability, and ranking accuracy. The visual expression of the comparative results is in Fig. 6, where the ranking tendencies of all fifteen options determined by each approach are provided. The comparison shows clearly that the proposed TSF-FUCA method has more stable and interpretable results of the ranking in comparison with the current TSF-MCDM methods. In contrast to TSF-CoCoSo and TSF-MARCOS, which have a varying order of ranks between the middle-performing players, the proposed approach has a continuously differentiating ranking system of all alternatives. As an example, the highest-ranking player () continually scored a better rank within the context of TSF-FUCA, which is the improved sensitivity of the model to dominance of performance on various benefit criteria. On the same note, Player A14 and Player A9 were ranked higher than other approaches, which validates the approach as efficient in describing balanced player efficiencies in the face of uncertainty. By comparison, the current processes like TSF-CoCoSo and TSF-MARCOS presented a greater range of differences and inconsistency within the options with the same level of performance. This variability denotes that their aggregation processes are more vulnerable to slight variations in input information or the weights of criteria. Furthermore, as the TSF-CRADIS method showed good competitive ranking scores, it was biased towards giving more weight to some of the criteria, which resulted in less ranking differentiation between players whose performance metrics were very close. Instead, the TSF-FUCA approach is useful in balancing the impact of benefit and cost attributes, based on its factorized utility calculation, which ensures that the decision equilibrium is ensured at all the dimensions of evaluation. The general consistency of the rankings and the ease of performance transitions between choices, as shown in Fig. 6, point to the strength and discriminative effectiveness of the suggested model. It offers strict differentiation between the high and the low performing players, without the overlap in rank and an equitable representation of the differences in performance. These results validate that the TSF-FUCA methodology is superior to those in the benchmark TSF-MCDM approaches in the ranking of stability, interpretability, and sensitivity control. Therefore, the suggested model provides a more accurate and transparent evaluation structure of real-time basketball performance evaluation where the results of decisions made are consistent, logical and stable to uncertainty within the expert assessments.

Fig. 6.

Comparison analysis with existing TSF-MCDM methods.

Furthermore, to give a more significant comparison to other uncertainty-based methods, the proposed TSF-FUCA technique was conceptually compared to robust and stochastic MCDM methods, which are typically utilized in addressing uncertainty in making decisions. Although these techniques will need further modification to apply directly to the existing data, the TSF-FUCA model proves itself to be better in dealing with multi-dimensional uncertainty and includes membership, non-membership, and the degree of abstinence, and explicitly deals with criterion weights. These traits ensure that there is increased sensitivity, stability, and fairness, which are of paramount importance in terms of real-time basketball performance assessment, and these qualities clearly underscore the relative power and practicability of the proposed methodology when it comes to other forms of uncertainty-based methods.

In order to evaluate the qualitative efficacy of the proposed TSF-FUCA technique even more, a comparative analysis was conducted with the number of well-known MCDM techniques such as TOPSIS¹⁹, WASPAS²⁰, CODAS²¹, EDAS²², and OPARA²³. It was compared using three critical performance measure indicators, namely the Ranking Stability, Sensitivity to Criteria Weight Variation, and Decision Accuracy, that have been considered qualitatively with Low, Medium, and High assessed effectiveness. Table 14 gives the results of this comparison.

Table 14.

Qualitative comparison analysis.

Method	Ranking stability	Sensitivity to criteria weight	Decision accuracy
TOPSIS	Medium	High	Medium
WASPAS	Medium	Medium	High
CODAS	Medium	High	Medium
EDAS	Medium	Medium	Medium
OPARA	High	Medium	High
TSF-FUCA (proposed)	High	Low	High

As the comparison shows, the suggested TSF-FUCA approach performs better than the conventional MCDM techniques in most of the evaluative dimensions. In particular, it has a high-ranking stability, which retains a constant ranking of alternatives despite the application of variations on the input data or weighting factors. This strength of stability is due to the strong fusion mechanism of FUCA, which combines both fuzzy uncertainty management and utility decision equilibrium. Moreover, the elasticity of the criteria weights of the proposed model is much less compared with TOPSIS and CODAS, which means that FUCA yields more valid outcomes with ambiguous or inaccurate expert ratings. The TSF-FUCA model has a better performance with respect to the accuracy of decision-making because it has a higher ability to reflect the complexity of the interdependency of criteria in a T-spherical fuzzy model. In comparison with traditional techniques, which tend to compute linear normalization or additive aggregation, FUCA uses a full factorized utility representation that avoids non-linear dependencies between performance indicators. The suggested TSF-FUCA approach is, therefore, more stable, precise, and robust in decision-making and, hence, quite appropriate in dealing with complicated real-time performance estimations, as is the case with basketball analytics.

Table 15 presents a comparison of the proposed TSFS with other commonly used fuzzy sets in MCDM, including FS¹, IFS², PyFS³, Fermatean fuzzy sets (FFS)²⁴, qROFS⁴, PFS⁵, SFS⁶, and m-Polar FS²⁵. As indicated, TSFS is able to support MD, AD, and NMD at the same time with high sensitivity, parameter flexibility and explicit weight handling. Other fuzzy sets are effective in some characteristics, either cannot represent abstinence, are unable to fully incorporate weights, or the parameters are limited. All these excellent features render TSFS particularly appropriate in the real-time basketball performance analysis to guarantee the precise and fair ranking within various dynamic parameters. This proves the originality and practical benefit of incorporating TSFS into the offered TSF-FUCA methodology.

Table 15.

Comparison of fuzzy sets for MCDM applications.

Fuzzy set	MD	NMD	AD	Sensitivity	Parameter flexibility	Weight handling
FS	✓	✗	✗	✗	✗	✗
IFS	✓	✓	✗	✗	✗	✗
PyFS	✓	✓	✗	✗	✗	✗
FFS	✓	✓	✗	✗	✗	✗
qROFS	✓	✓	✗	✓	✓	✗
PFS	✓	✓	✓	✓	✗	✗
SFS	✓	✓	✓	✓	✗	✗
m-Polar FS	✓	✓	✓	✓	✓	✗
TSFS	✓	✓	✓	✓	✓	✓

Practical and managerial implications

The results of the suggested TSF-FUCA framework have some useful practical and managerial implications for real-time basketball performance assessment and decision-making in sport analytics. The present research fills the gap existing between sophisticated data-driven analytics and viable decision support in that it combines the application of fuzzy uncertainty management with an effective multi-criteria analysis mechanism. Practically, the proposed model facilitates coaches, analysts and performance managers to assess the real-time performance of players on various attributes, including shooting accuracy, defensive efficiency, teamwork and endurance in uncertain and dynamic game scenarios. The advantage of the TSF-FUCA approach lies in the ability of the subjective judgment of various decision-makers (e.g., coaches, trainers, and analysts) to effectively combine without the need to reduce bias and increase the level of total assessment. This helps in making evidence-based players, training, and in-game strategic changes. As a managerial approach, the model offers a decision-support framework that assists the management in determining the key performance measures and priorities in allocating resources. The rankings provided by FUCA can be used by basketball team managers to develop specific development programs that focus on particular players, enhance the scout decision-making process, and track the success of the tactical management. In addition, the model can help to aid strategic planning and policy development in the presence of uncertain, imprecise and incomplete information, which is a common problem in competitive sport management by use of the T-spherical fuzzy environment. In general, the suggested TSF-FUCA approach not only acts as a solid analytical basis of performance assessment but also provides a valid managerial decision-making framework that promotes the level of team competitiveness, operational efficiency based on the data, and ongoing enhancement of performance in the realm of professional basketball settings.

Advantages of the study

The current research has a number of significant strengths that make it stand out against the background of the existing MCDM-based models of performance evaluation and demonstrate the efficiency of the offered framework of TSF-FUCA in real-time basketball analytics. To begin with, the suggested methodology offers a more detailed decision-making framework that can deal with uncertainty, indecision and inaccurate information more efficiently than the conventional fuzzy models. The TSF environment reflects the nature of human judgment and subjective tastes of more than one decision-maker, which guarantees the increased accuracy and reliability of performance evaluation. Second, the adaptive computational mechanism of the FUCA method provides the possibility to integrate various criteria and decision-makers without distorting the equilibrium and uniformity within a decision matrix. This leads to a stabilized and more realistic ranking of the alternatives, which is important in assessing the performance of the players in a rapidly changing real-time game setting. Thirdly, the proposed TSF-FUCA approach is better than the traditional MCDM systems like TOPSIS, WASPAS, and EDAS with regard to flexibility and presence of strength in dealing with contradictory evaluation parameters. It is capable of operating both benefit and cost parameters and provides an appearance of dynamism regarding the performance outcomes. Moreover, the research has practical utility and managerial applicability as it converts the complicated results of analysis to practical findings that coaches, sports analysts and management teams would apply. This gives the stakeholders the power to make the right decisions in terms of training players, optimizing the strategies, and developing talents. Finally, the suggested strategy leads to the improvement of the decision-making theory by introducing the FUCA approach to the TSF area, a new combination that improves its computational intelligence, precision, and usability in real-world uncertain situations like sports performance assessment. Altogether, these strengths make the TSF-FUCA model one of the strongest, smart, and efficient decision-support systems, which compares to the current fuzzy MCDM methods in terms of analytical accuracy and the efficiency of the application.

Limitations of the study

Though the suggested TSF-FUCA framework proves to be extremely efficient and flexible in terms of real-time basketball performance assessment, some limitations should still be mentioned to provide a balanced insight into the area of the scope and application of the study. To begin with, the hypothetical data used to conduct the current study is considered to be based on the assumptions that can be somewhat realistic in representing the situation of basketball performance, but may not be quite satisfactory to represent the dynamics and unpredictability of the real conditions of a basketball game. Innovative applications. Future research based on real-time player tracking datasets may have more empirical substantiation and would improve the practical applicability of the model. Second, TSF calculations need significant computer resources and skills, and might not be directly useful to sports analysts or sports managers without good technical experience. This problem would be overcome by developing software tools or decision-support systems that are easy to use in further implementations. Thirdly, although the FUCA technique is effective in managing multiple criteria and decision-makers, it can be said that the choice of criteria and weights assigned to them could also be based on subjective judgment. In spite of the fact that the fuzzy environment can minimize this bias, it is difficult to get rid of subjectivity in human-based assessments entirely. Moreover, the model has been applied to basketball performance evaluation, which may limit its direct extrapolation to other sporting activities or areas without suitable adjustments of parameters in the criteria arrangement and evaluation. Finally, the existing framework fails to consider the factor of time changes (say, the performance pattern during several games or seasons) that would generate a more dynamic picture of the development of the players. Predictive and adaptive capabilities can also be improved by adding time-series fuzzy modelling or machine learning extensions to it. To conclude, though the proposed TSF-FUCA model is a very robust, accurate, and innovative decision-making process, covering these limitations in future research will further enhance its overall strength, computing capabilities, and applicability in general sports analytics processes.

Conclusion with future research direction

The present research proposed a TSF-FUCA model of basketball performance assessment in real-time, combining the problem of uncertainty management with uncertainty in a fuzzy model with the presence of a credible multi-criteria model. The model, which also assessed players in seven major criteria and engaged four decision-makers, gave consistent and realistic results in ranking performance positions, with the best performers being , , and . The sensitivity and benchmark analysis also proved the stability and high quality of the model in terms of its performance as compared to the existing TSF-MCDM tools such as CoCoSo, MARCOS and CRADIS. Moreover, the comparison with the traditional methods in terms of their quantitative analysis, including TOPSIS, WASPAS, CODAS, EDAS, and OPARA, revealed the superior flexibility and accuracy of the suggested method. In a nutshell, the TSF-FUCA solution is an effective, trustworthy, and smart decision-support system to assist sports analytics, and it has a lot to contribute to the optimization of the performance of players and strategic decision-making by coaches and managers.

The suggested TSF-FUCA framework could be improved in future research by applying the needs of uncertainty management with the assistance of complex picture fuzzy aggregation operators proposed by Nazeer et al.²⁶ to fit the framework better. Furthermore, it can be suggested that interval-valued T-spherical fuzzy information²⁷ can be taken into account along with TSF-FUCA to use less precise or certain data to make decisions in the field of education. The fuzzy MCDM techniques can also be applied in solving student problems, which may involve academic stress and mental problems. These guidelines would render the TSF-FUCA stronger, more accurate, and applicable, and will help to inform and strengthen the decision-making in the education reform work.

Author contributions

DongYu Liu and Wei Wang contributed equally.

Funding

No funding.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Informed consent

This study did not involve human participants, human data, or human tissue.