ROLE OF HUMAN COLLABORATION IN ARTIFICIAL INTELLIGENCE WITH FUZZY BASED MATHEMATICAL MODELS AND DECISION MAKING PROBLEMS

AI has progressed so rapidly that it has transformed computer technology by creating advanced automation, intelligent decision systems, and adaptive systems. Advances in AI systems that connect with humans in complex operational domains make the method by which humans and AI systems work together a vital aspect. Using classical set operations to express human interaction with computers does not adequately cope with uncertain situations. Even the classical set operations fail to deal with various real-life terms. Zadeh proposed expressing uncertain terms known as fuzzy sets (FS)¹. The FS expresses the ambiguous terms using membership grade (MG), which indicates the degree of trust in human judgment, and is assigned values in the range [0,1]. This creative method established the groundwork for various FS applications in addressing practical problems. In some situations, the MG was insufficient to portray the situation using only the MG. Atanassov² established the idea of intuitionistic FS (IFS), which takes into account both MG and non-MG (NMG) to express the ambiguous situations with constraint MG + NMG ∈ [0,1]. By considering what belongs and what does not belong to a set, the IFS expansion improves the ability to describe uncertainty more thoroughly. It offers a more sophisticated framework for handling ambiguous situations. According to Zadeh, the MG assigned an object a value since it belonged to a specific situation. The FS has become very popular among academics and applies to many facets of life. Sometimes, the experts assign the values to MG and NMG such that their sum exceeds 1. To handle situations where the sum of the MG and NMG exceeded, Yager³ relaxed the requirement that the squares of the MG and NMG be from the unit interval to create the Pythagorean FS (PyFS). PyFS theory is a generalized version of FS theory, but it has some limitations. He added IFS with an extra grade known as NMG because the IFS could not handle those situations. For example, professionals have encountered four issues during an election: yes, abstention, no, neutral or refusal. The idea of PFSs has been disregarded when addressing these issues. To address the problems mentioned earlier, Cuong5 developed the theory of picture FSs (PFSs), which is a positive concept that describes the MG and abstinence value (AV) and a negative concept that describes the NMG of information whose range is in a unit interval with a prominent feature. In real-world situations, a decision-maker may assign values to an object's MG, AG, and NMG within the unit interval [0,1], but their sum exceeds 1. Standard PFSs were unable to handle these situations. Mahmood et al.7 developed a new idea for a T-spherical FS (TSFS) to deal with this situation. The main difference between PFSs and SFSs is that in the latter case, the square sums of MG, AG, and NMG lie inside the unit interval [0,1], even if the sum of these variables is greater than 1. The TSFS allows the experts to express their opinion to describe the ambiguous situations independently with the help of four degrees. Thus, the TSFS is the most suitable framework that can deal with human interaction more precisely.

Several models help in information fusion. Information aggregation is crucial in addressing human interaction, particularly in multi-attribute decision-making (MADM) problems, where diverse data must be synthesized into actionable insights. Various aggregation operators (AOs) have been developed, utilizing different t-norms and t-conorms to suit specific decision-making contexts. Arithmetic AOs have been explained by several frameworks, including IF arithmetic, geometric AOs, and fuzzy number AOs¹⁰. Weill proposed some arithmetic AOs with intuitionistic trapezoidal fuzzy numbers that apply to group decision-making. Ye¹² introduced IF hybrid arithmetic and geometric AOs to select mechanical design schemes. Wang and Liu¹³ introduced IF geometric AOs based on Einstein's operations, and in the context of IF, Zhang and Yu¹⁴ used Einstein's operations to introduce several geometric Choquet AOs. Ullah et al.¹⁵ suggested TSF Hamacher AOs for assessing the efficacy of search and rescue robots. Jana and Pal¹⁶ examined the application of PF Hamacher AOs in enterprise performance assessment. Wang et al.¹⁷ introduced PyF interactive Hamacher power AOs for evaluating express service quality using entropy weight. Ashraf et al.¹⁸ introduced using fuzzy decision support model AI tooling to select based on single-valued neutrosophic sine trigonometric AOs for hydrogen power plants. Trigonometric SFAOs were introduced by Qiyas and Abdullah¹⁹ and used in decision support systems. Garg introduced a special q-ROFAO based on TOs and their key characteristics²⁰. Ajay et al.²¹ introduced sine trigonometry operational principles for complicated neutrosophic sets and associated AOs in Material Selection. Abdullah and Ashraf²² proposed decision support model AI tooling for selecting agriculture, which depends on STSV neutrosophic data. The application of Pythagorean probabilistic hesitant fuzzy AOs to decision-making was suggested by Batool et al. ²³. Lindahl and Ramon²⁴ introduced DM's generalized hybrid averaging operator. The PyF TOPSIS technique was proposed by Ye and Chen²⁵ for selecting cotton fabrics. A unique DM method that depends on SHF logic was suggested by Naeem et al.²⁶. Sun et al.²⁷ proposed a sliding mode control approach for discrete-time interval type-2 fuzzy Markov jump systems with a preview target signal. Zhang et al.²⁸ suggested observer-based sliding mode control for fuzzy stochastic switching systems with deception attacks. Xia et al.²⁹ presented further results on the fuzzy sampled-data stabilization of chaotic nonlinear systems. Gao et al.³⁰ suggested SMC for semi-Markov jump TS fuzzy systems with time delay. Ge and Zhang³¹ suggested adaptive inventory control based on a fuzzy neural network under an uncertain environment. Sarwar and Li³² proposed fuzzy fixed point results and their applications to ordinary fuzzy differential equations in complex-valued metric spaces. Although many models have been proposed to find solutions in the field of fuzzy logic as far as making decisions is concerned, several limitations have not been resolved:

• Standard fuzzy and intuitionistic fuzzy methods do not adequately take into consideration such refusals or abstentions as often happen in real-life expert judgments.
• Most of the earlier models have used hard aggregation operators, such as arithmetic means, which are not sensitive to conflicting or uncertain values.
• Available models (e.g., those on IFS or PyFS) struggle to accommodate nonlinear human preferences and multi-expert disagreement in potentially ambiguous reality.

These constraints led to the incorporation of T-Spherical fuzzy sets, which can express four grades of expert wrinkled judgment (MG, AG, NMG, and RG), and the use of Aczel-Alsina-based operators, which provide flexible information aggregation as needed. The suggested model, therefore, provides a more comprehensive and flexible foundation for assessing the collaboration between humans and AI.

Some motivations of this study are listed as follows:

1. We need decision models to handle unclear conditions alongside uncertain scenarios and diverse multi-element decisions, enabling the most effective and benign collaboration between humans and AI systems. Traditional systems often fail to effectively process vague information or human opinions when humans collaborate with AI systems. The research report recommends using MADM decision-making procedures in conjunction with advanced fuzzy set theory to address existing problems. Fuzzy MADM methods are sophisticated analytical systems that process uncertain data with imprecise components to enhance decision-making tests for optimal collaborative system design implementation. The paper will demonstrate a fuzzy MCDM framework for assessing and improving human-AI collaboration within computer technology development processes.
2. Human-AI collaboration enables users and intelligent systems to establish two-way communication by pursuing unified objectives through dynamic adaptation of their operations. The quality of human-AI collaboration develops based on trust, explainability, performance, flexibility, and system transparency. Human-AI collaboration enhancement through the development of better algorithms, improved interfaces, and enhanced models has become a recent research focus.
3. Fuzzy set theory and its extensions have successfully integrated with MCDM methods to address these limitations. TSFSs extend the modelling capabilities to account for hesitation, uncertainty, and subjective aspects. The evaluation of complicated AI-assisted technologies benefits significantly from these evaluation methods.
4. Classic AI systems are typically designed to handle precise, binary, or numerically crisp data. In regularized settings, they can be very successful. Still, they can be terrible at processing perpendicular, realistic, or missing human inputs, as often arises in the real world of decision-making. Fuzzy environments, in turn, embody uncertainty in modelling values that take on a scale between 0 and 1, which are used as linguistic terms such as "highly reliable,” “moderately adaptable,” and “partially trustworthy" etc. Human opinions, usually expressed in subjective or unfixed terms, can be well captured and processed in light of this flexibility. Therefore, the fuzzy system can be viewed as a barrier and connector between machine logic and human thinking, and as an aid for decision-making when data are partial or inconsistent opinions have different values. By incorporating T-spherical fuzzy sets, the framework not only considers agreement or disagreement but also hesitation or refusal, thereby improving its performance over traditional AI methods in processing subtle human-AI interactions.

To help readers better comprehend this material, we define a few key terms in this section. This section outlines the ideas for the Aczel-Alsina TN, TCN, and TSFS. Definition 1 The set Π = {(ρ,δ (ρ),ι (ρ),ν (ρ))|ρ∈ Y} is called a TSFS defined over a universal set Y. The terms δ,ι, and v denote the MG, AG, and NMG such that δ,ι,ν : Υ → [0,1] With conditions 0 < δº + ıq + vº < 1Forq∈ Z⁺. The refusal grade (RG) is defined as π = 1 − (δ 9 +19 +vq) In addition, the triplet (δ, ι,ν) demonstrations of the T-spherical fuzzy value (TSFV). The TSFS is a framework for handling ambiguous data gleaned from real-world situations. However, the TSFVs display the information that was gathered. It is preferable to specify a scoring function for TSFV defuzzification. The following is the definition of the defuzzification score function. Definition 2 Consider Scr (p) be the score of a TSFV. Then sc(p) = δº + ιº + vº (1) The TNM and TCNM typically handle data in the form of unit intervals. In logic and its applications, various TNMs and TCNMs have proved crucial. When handling ambiguous information, the AATNM and AATCNM are highly adaptable. With the aid of existing parameters, the decision-maker may regulate the decision outputs for various scenarios by using AATNM and AATCNM. Definition 3 The functions T, S : [0,1]² → [0,1] are AATNM and AATCNM, respectively, and defined as Τι (γ,3) = { e-((-lny)M+(-1n3)M) 7 if M = 0 if M→∞ Otherwise. Where M∈ (0,0). S(,3) = { Τα (γ, 3) min (y, 3) if M = 0 if M→∞ Otherwise. 1 - e-((-ln(1-x))+(-ln(1-3))M) Particularly when human judgment is involved, TSFS can effectively represent ambiguous data. The AOs for TSFS, incorporating AATNM and AATCNM, were established by Hussain et al.³³ to aggregate the data and provide valuable outcomes. The following defines the TSFAAWA and TSFAAWG operators. Definition 4 A mapping TSFAAWA : П™ → П defined for TS- FVs Π₁ = (δj, j, v j) (j = 1,2, 3, . . ., n) is known as a TSFAAWA operator and is defined as TSFAAWA (Π,Π, 1' 2' =AA j=1 ( M1/M 1 - e-(Σ; (-1n(1-859)))1/ V1-e 1/M 1/M (((3)))/((()))/ n mapping TSFAAWG : Π" → П e defined for FVs Πj = (δزا,ز, v j) (j = 1,2, 3, ..., n) is known as a TSFAAWG operator and is defined as n e-((-1(8)))1/M (Π.Π.Π.) = (Π) = -((-(1-3))*)* TSFAAWG 1' 2' =AA j=1 j ) M1/M e

Successful human-AI collaboration remains an elusive goal because computer technologies and AI progress continue to face challenges due to unpredictable human preferences, such as the use of AI tools and AI behavioural patterns. A proper assessment method for this collaboration must develop a model to evaluate numerous quantitative and qualitative criteria while operating under uncertain conditions. The paper aims to develop a fuzzy MCDM framework that facilitates the assessment and improvement of human-AI collaboration methods in computer technology development. The framework consists of a systematic assessment process that uses examples such as AI tools and systems to evaluate them against specific criteria:

• Interpretability.
• Trustworthiness.
• System performance.
• User adaptability.
• Real-time responsiveness.
• Transparency.

The fuzzy model will utilize advanced aggregation operators to create robust decisions through linguistic evaluation using fuzzy logic methods. The integration of TSFS within the proposed MCDM framework becomes essential for handling the complex evaluation process of human-AI collaboration in uncertain and imprecise situations. The TSFS extends basic fuzzy and spherical fuzzy approaches by introducing extended freedom, which promotes the improved expression of expert judgments and representations of human choice. Computer technology development requires specific attention to unclear evaluation results that stem from the interaction between human decision-making and AI operational behaviour. The TSFS framework effectively leverages its flexible and expressive capabilities to handle inconsistent expert evaluations and ambiguous linguistic inputs, thereby generating more interpretable decisions. Including TSFS in assessment procedures enhances the evaluation process by providing greater accuracy in determining criteria related to interpretability, as well as real-time responsiveness and trustworthiness as an AI tool.

The proposed methodology consists of the following steps.

1. Criteria and alternative identification: The proposed fuzzy MADM framework evaluates human-AI collaboration in computer technology development using TSFS, which effectively handles the intricacies of expert judgments and doubtful conditions. TSFS offers better functionalities than conventional fuzzy models when handling vague elements, such as doubt and hesitation, in uncertain contexts, making them best suited for this field. TSFS models enable the framework to assess five AI-based solutions alongside six crucial performance factors: Interpretability, Trustworthiness, System Performance, User Adaptability, Real-Time Responsiveness, and Transparency. The evaluation framework consists of fundamental technical elements and human qualities essential for productive team collaboration. The aggregation process achieves greater accuracy and reliability through TSFS-based linguistic assessments, which allow decision-makers to use membership degrees and express opinions based on hesitancy and non-membership. The chosen alternatives fulfil both technical requirements and human expectations for uncertain situations through this approach.
2. Experts Inputs. TSFS serves within the proposed fuzzy MCDM framework to handle the uncertainty and intricacy of expert inputs for human-AI collaboration evaluation in computer technology development. The input collection process requires experts to make subjective assessments that contain uncertain elements, especially during evaluations of abstract criteria that focus on human elements. TSFS serves as an advanced evaluation structure because it delivers simultaneous membership and non-membership information along with values of indeterminacy. Five significant AI capabilities – AI recommendation systems, AI conversational agents, autonomous systems, intelligent diagnostic tools, and adaptive learning systems are assessed for their six essential characteristics: interpretation functionality, trustworthiness, functional performance, user adaptability, real-time response, and system transparency. Expert assessments expressed through natural language terms linked to TSFS scales provide a richer framework for combining specialist knowledge more realistically. Combining expert knowledge with these tools generates trustworthy decisions that connect software performance to human-operated expectations and joint work objectives.
3. Aggregation of information (Individually). The experts assessed the AI tools using TSFVs throughout all measurement criteria. The individual expert assessments for AI tools and criteria exist in tabular format as the first step. The individual responses from Expert-1, Expert-2, and Expert-3 get separate attention, which helps to trace their evaluation methods.
4. Aggregation of information (collectively). The experts provide private evaluations, which are summed up using the TSFAAWA and TSFAAWG methods. Expert opinions are integrated through these operators to produce a unified T-spherical fuzzy evaluation that applies to both AI tools and criteria. Expert weights determine the aggregation method through arithmetic averaging by the TSFAAWA operator and geometric means by the TSFAAWG operator.
5. Finding scores. After obtaining the T-spherical fuzzy collective evaluation results, the score function converts each TSFN into a crisp numerical value. Each AI tool receives its score evaluation according to its performance metrics under each criterion. The final score of each AI tool is calculated by summing the weighted scores from all evaluation criteria.
6. Ranking of AI-tools. The AI instruments determine their final rankings based on the score results they have achieved previously. The ranking methodology sorts AI tools by starting with the top-scoring option and then moving to the bottom-scoring option. The ranking system generates data-driven output through expert consensus, which emerges from T-spherical fuzzy environments. Figure 1 shows the steps of the evaluation process as follows. The following figure, i.e., Fig. 2, shows the conceptual framework of this study.

In this section, we intend to examine five AI tools based on several linguistic features. Details on the following attributes are presented. Characteristics 1: Interpretability (I). The interpretability of an AI system determines its ability to display clear, understandable reasoning behind its outputs that humans can understand and trace. A high MG rating exists when an AI system delivers its reasoning through human-understandable explanations or visual displays. In contrast, high NMG indicates systems that function without explainable outputs, and high AG describes interpretability, which varies according to context. The criterion reflects varying assessments of how an AI tool or system fits user cognitive requirements. Characteristics 2: Trustworthiness (T). This factor determines how specific end-users are about the performance and results produced by the AI system. The system generates reliable and accurate results when it shows high MG, but unreliable results occur when it shows NMG, while a mixed level of trust is observed with AG. Characteristics 3: System performance (SP). Experts evaluate computing speed and precision as key factors in the dependability of AI tools and systems. Basic AI systems receive MG when they demonstrate exceptional operational inputs and output performance. In contrast, those showing recurring operational faults receive NMG, and AG indicates systems whose measured output depends on environmental factors, such as network specifics or data sources. Characteristics 4: User adaptivity (UA). The criterion assesses a user's ability to adapt and utilise an Al system, as well as their ability to learn it. High MG values reflect the user experience flexibility, while poor usability results in NMG scores, and system adaptability leads to AG evaluations. Characteristics 5: Real-time responsiveness (RR). High-performance speed and effectiveness become evident when the AI system operates in dynamic situations. Low latency, along with context-aware behaviour, indicates a high MG rating. In contrast, the system receives NMG status when delays or static responses deteriorate performance and achieves Good ratings when response times fluctuate depending on specific workloads or interaction types. Characteristics 6: Transparency (TP). Transparency is the degree of openness in system design, data usage, and decision-making logic. The clarity of documentation and determination of the decision path result in a rating of MG. However, NMG appears when critical processes remain unclear or hidden, and AG represents situations with partial transparency or component-dependent clarity. Characteristics 7: Ethical Alignment (EA). The assessment determines the extent to which an AI system adheres to ethical principles regarding fairness, privacy requirements, and responsibility standards. The system incorporates active ethical protections in high MG ratings, whereas NMG points to possible risks arising from biased behaviour or unclear ethical implications. Systems with AG ratings have uncertain ethical elements. Characteristics 8: Scalability (S). System scalability refers to the ability of a system to function effectively with an increasing number of users, larger amounts of data, and evolving complexity. A high positive value of MG indicates widespread functionality across various situations. Still, NMG implies operational degradation when systems expand in scale, whereas AG signals uncertain results due to minimal real-world application testing efforts. Note that the evaluation (based on decision analysis and artificial intelligence) was carried out by three professionals who had defined credentials in the field:

• Expert 1: University professor in AI system architecture and intelligent decision support systems, with more than 10 years of educational and industrial experience.
• Expert 2: A highly experienced data scientist who operates in the creation of conversational artificial intelligence as well as adjustable learning frameworks.
• Expert 3: A research fellow of human-computer interaction specialising in trust, interpretability, and ethical alignment of AI-based technologies.

To evaluate the consistency of judgments and variability of expert judgment, we used descriptive statistics, including the mean and standard deviation (SD) and variance of the score values for each participant's AI tool (using the TSFAAWA operator). The findings have been tabulated below in Table 14. The findings support reliable ratings, with a minor variance observed in every tool. The variability is slightly higher in AI Tool 4, indicating more divergent expert opinions.

The suggested fuzzy MCDM model facilitates decisions in uncertain settings by experts who lead and involve Al systems in communication with human beings. It is applicable in several practical disciplines because of its flexibility and the ability to reason in languages:

• Healthcare Diagnostics: The evaluation of AI-based tools used in diagnostics can be implemented within the framework, taking into account key AI characteristics such as interpretability, ethical alignment, and real-time responsiveness, which are critical to the trust and adoption of such tools in clinical settings.
• Autonomous Vehicles Human-AI interaction is crucial to semi-autonomous driving systems. The model can rank the modules of navigation and perception in terms of their adaptability, transparency, and safety in dynamic states.
• Customer Service AI: Chatbots and virtual agents should be evaluated based on their responsiveness, ease of use for customers, and reliability. The framework enables customer experience managers to compare and select tools based on professional opinion.
• Education Technology (EdTech): It is possible in both intelligent tutoring systems and adaptive learning environments to allow human teachers to assess the quality and responsiveness of feedback, corrections made in real-time, and customizing a system to a user.

The given examples demonstrate the high versatility of the suggested model in application to various areas where human-AI collaboration and expert work are called upon. The practicality of the advocated framework is confirmed through a multi-expert assessment of five AI tools relative to eight key attributes. It is consistently indicated in the results that the ranking order remains stable, regardless of the values of parameters q, q, q and MMM, demonstrating the model's robustness. Additionally, box plots were added to visually depict the distribution of changes in the way experts evaluated it. In contrast, the metrics of statistical analysis (mean, standard deviation, and variance) would show the degree to which the evaluations by the experts were similar. This empirical analysis not only shows that the model is applicable but also that it is reliable for tracking human-like decision-making behaviour under uncertainties, thereby achieving the study's objective of finding a way to further human-AI collaboration.

In this paper, TSFS and the Aczel-Alsina aggregation are used because they allow for modelling highly complex and ambiguous human input. In this part, we contrast the proposed method with other popular fuzzy models of decision-making, which are adopted in situations of human-AI interaction: In Table 15, it can be observed that TSFS offers the most comprehensive framework for carrying out multidimensional human judgments. Along with Aczel-Alsina operators, this approach provides competent adjustability, which is central to reflecting diverse human ideas in team collaboration between humans and artificial intelligence agents.

The proposed model in this study can be utilized in various real-life problems. For example, the developed model can address slot management problems at airports³⁴, pivot path problems in self-reconfigurable satellites³⁵, pose control³⁶ using various factors, and analysis of grain boundary problems³⁶. The integration of AI and TSFS makes the model flexible in dealing with uncertain situations. Thus, the developed model can be beneficial to utilize in failure analysis in K-TIG welding³⁷, analysis of impacts of car body lightening in pedestrian³⁸, Prediction model on maximum potential pollution range of debris flows generated in tailings dam break³⁹, analysis of the friction coefficient estimation of the clutch in automatic transmission based on improved persistent excitation condition⁴⁰, analysis of the effects of serum containing natural cerebrolysin on glucose-regulated protein 78 and CCAAT enhancer-binding protein homologous protein expression in neuronal PC12 cells following tunicamycin-induced endoplasmic reticulum stress⁴¹. Similarly, the developed model can be utilized to analyze evidence-based traditional Chinese medicine research, including its two decades of development, impact, and breakthroughs ⁴², as well as the effects of Siegesbeckiae Herba in treating chronic pain⁴³, among other applications. Additionally, we can also fix complexities in real world problems and decision-making problems^44–46.

This article presents a sophisticated methodology utilizing fuzzy logic to evaluate AI tools for potential application in physical education. Decision-makers can modify the model's parameters by integrating the parameters q and M, following various evaluation scenarios and settings. This quantitative evaluation system enhances the accuracy and objectivity of language assessments by utilizing TSFAAWA and TSFAAWG operators. Among this study's significant contributions are:

• Experts can independently provide nuanced assessments of learners' proficiency through TSFS, increasing the accuracy of capturing a range of performance levels.
• Adding customizable parameters such as M and q allows decision-makers to adjust the evaluation model to varying degrees of uncertainty, making it suitable for various educational scenarios. Teachers can adapt tests to achieve good student results, which helps make language training more individualized.
• This article's decision methodology included critical elements for evaluating AI TOOL performance. Likewise, a large variety of attributes can be considered for more accuracy.
• This study developed a flexible and responsive framework that enhances instructors' understanding of each learner's progress in language and enables them to make more informed judgments.

In the future, we aim to apply the decision algorithm to enhance accuracy for various uncertain issues across multiple areas. Fuzzy logic has several practical advantages when it comes to teaching English. Understanding the variety and variability of human judgment makes it feasible to administer ever-more complex tests and offer tailored feedback that caters to the requirements of every learner and promotes targeted development. Using a structured, rule-based framework, fuzzy logic minimizes grading bias, ensures uniform assessment standards, and maintains flexibility for contextual deviations. Richer, multifaceted data helps teachers make well-informed judgments about curriculum modifications and instructional practices. Due to the methodology's excellent scalability and adaptability, it can be applied in various instructional scenarios and used with other languages or subjects.