Multi-criteria consensus-based group decision making with prospect-regret TOPSIS for human-AI tool selection under linguistic Z-numbers

Abstract

The rapid advances in artificial intelligence (AI) and machine learning have transformed computer technology. Now, computers can learn and adapt autonomously, make sound judgments, and create environments in which humans and AI work in harmony. Analytical frameworks that can process data with uncertainty, imprecision, and multiple perspectives are essential for the success of human-AI collaboration. In this connection, linguistic Z-number (LZN) evolution models play a crucial role in selecting tools for human-AI collaboration because they can better capture uncertainty and fuzziness. Given this, we extend multicriteria group decision-making (MCGDM) by incorporating LZNs. The structure of MCGDM under LZNs, the consensus-building process, estimation of experts’ weights, and the ranking of options are the main challenges. In the experts’ primary opinion-based MCGDM, we believe that expert participation and the opportunity to revise their opinions with original linguistic terms (LTs) would be more practical. However, in existing studies, experts have no scope to revise their opinions based on the original LTs. Most studies on consensus-building either do so automatically or advise experts to revise their opinions based on virtual LTs. But either automatically or virtually, LTs cannot be physically represented by words, which is a significant drawback. Given these facts, we propose an interactive, strategy-based consensus-building in which we advise experts to revise their opinions within the original LTs. To estimate experts’ weights, we developed an integrated subjective and objective method. By simultaneously assigning subjective weights and opinion-based relative-relevance indices to the experts, we derive the adjusted subjective expert weights. The objective weights of experts are derived from the criterion-based entropy method. The ranking of options, we proposed the prospect and regret theory-based TOPSIS ranking approach to rank the options. Finally, a case study on the optimal selection of an AI tool for human-AI collaboration validates the proposed method. Through a comparative analysis with alternative methodologies, we demonstrate the feasibility, stability, and superiority of the proposed model.

Introduction

The proliferation of artificial intelligence (AI) across industrial, scientific, and social domains has redefined the interface between human cognition and computational intelligence. Modern AI systems no longer operate as isolated tools but as adaptive partners capable of learning, reasoning, and making autonomous decisions in domains such as robotics, autonomous driving, face recognition, natural language processing, and virtual assistants^1–4. This rapid evolution has intensified the need for effective human–AI collaboration, where interpretability, trust, and joint decision coherence are paramount. To address these requirements, researchers have sought analytical frameworks that can accommodate uncertainty, linguistic ambiguity, and multiple expert perspectives. Within this context, several studies have incorporated fuzzy and neutrosophic theories into AI-driven decision models. Ashraf et al.⁵ applied AI-assisted fuzzy decision systems to the selection of hydrogen power plants using single-valued neutrosophic sine–trigonometric aggregation operators, while Abdullah and Ashraf⁶ extended such frameworks to agricultural decision models under STSV neutrosophic data. Wang⁷ further explored T-spherical fuzzy sets with Aczel–Alsina aggregation operators for evaluating AI tools in collaborative settings. Building on this foundation, the present study develops a multi-criteria consensus-based group decision-making (MCGDM) model with a prospect–regret theory–based selection mechanism under linguistic Z-numbers (LZNs), aimed at enhancing the precision and behavioral accuracy of AI tool management for human–AI collaboration.

Addressing the diverse judgments involved in human–AI collaboration, MCGDM provides a systematic framework for integrating the evaluations of multiple experts across competing criteria. As a central branch of decision science, MCGDM enables collective reasoning in situations where information is incomplete, uncertain, or subjective. In such contexts, linguistic information offers experts a flexible medium to articulate their assessments using qualitative terms that capture human perception more naturally than numerical scales. This linguistic representation helps overcome cognitive limitations and biases that often arise when experts must quantify imprecise judgments⁸. To enhance expressiveness and precision, numerous extensions of linguistic modeling have been proposed, including the 2-tuple linguistic (2TL) model⁹, the Pythagorean linguistic set (PLS)¹⁰, hesitant fuzzy linguistic term sets (HFLTSs)¹¹, and probabilistic linguistic term sets (PLTSs)¹², among others. These approaches have substantially expanded the applicability of MCGDM to complex, high-dimensional decision environments.

The aforementioned linguistic models, despite their effectiveness in accurately describing the qualitative content of expert judgments, neglect the reliability or confidence associated with those evaluations. In real decision environments, involving human–AI interaction, experts’ assessments are often influenced by psychological, contextual, or informational uncertainty, leading to varying degrees of dependability in their expressed opinions. Consequently, there is a need for a representation model that can express linguistic information with scientific accuracy and semantic completeness. In this regard, the linguistic Z-number (LZN), introduced by Wang et al.¹³, provides a powerful framework for expressing uncertain and reliability-weighted linguistic evaluations. As a result of this LZNs have been proved particularly suitable for complex MCGDM problems where experts’ opinions may differ not only in content but also in certainty^14–17.

The concept of LZNs originates from Zadeh’s theory of Z-numbers (ZNs)¹⁸, which capture both an expert’s evaluation and its associated reliability within a unified representation. Recognizing the need for more expressive forms, Ezadi et al.¹⁹ proposed a multi-layer framework for assessing and ranking ZNs, and Allahviranloo et al.²⁰ introduced advanced ZNs to enhance their computational flexibility. Building on these developments, Wang et al.¹³ extended the framework by coupling ZNs with linguistic terms (LTs), producing linguistic Z-numbers (LZNs) that align more closely with the structure of human reasoning. Recent research on LZNs can be broadly classified into three major domains: (i) foundational studies establishing theoretical formalisms, standardization, distance measures, aggregation operators, and data representation techniques^21,22; (ii) applied studies addressing decision-making problems such as MCGDM^23–27, large-scale GDM^28,29, and consensus-based evaluation^30,31; and (iii) hybrid models integrating LZNs with other methodologies, including the best–worst method³², the gained-and-lost dominance score method³³, and failure mode and effects analysis (FMEA)³⁴.

Effective consensus building in MCGDM requires two essential conditions, the individual assessments of experts must be both consistent and compatible³⁵. In this context, notable studies, such as the minimum-adjustment consensus under multi-granular hesitant linguistic information, establish an important optimisation-based consensus-reach process (CRP)³⁶, personalised individual semantics and consistency control in linguistic CRP³⁷, and social network group decision-making that enhances CRP-based MCGDM³⁸. However, most existing MCGDM models developed under the LZN framework fail to guarantee these properties^{13,17,22–28,33}. Consequently, there is a continuing need for procedures that promote consistency and agreement among experts during implementation. Several attempts have been made to address this issue. Mandal et al.³⁰ proposed a consensus-based MCGDM approach in which consensus was modelled as the minimum adjustment sum. Xu et al.³¹ designed the consensus based on a minimum-compensation-cost optimisation model. Du et al.²⁹ proposed a consensus-based MCGDM based on the minimum cost with Type constraints, where a large number of experts are involved in decision-making.

The above studies have made valuable contributions both to the development of MCGDM frameworks for selecting AI tools in human–AI collaboration and to the advancement of MCGDM methodologies under LZN environments. Nevertheless, several limitations remain, which can be summarized as follows:

1. In complex and uncertain decision environments, existing models^5–7,22 fail to incorporate procedures for determining subjective expert weights. Subjective weights derive from established facts, such as an expert’s renown, prestige, position, and knowledge of the problem at hand. Subjective weights assigned to experts may inherently reflect bias. It may not be appropriate to assign weights in advance. For example, if certain experts have high weights yet their judgment differs substantially from the group’s overall view, their preferences may be overly concentrated, which will make the final answers less accurate and precise.

2. The consensus-based MCGDM approaches proposed in^29–31 rely primarily on minimum-adjustment and optimization techniques. These optimization models tend to be automatic adjustments producing virtual terms, limited expert interpretability/acceptance, and inflexible adjustment constraints. However, expert feedback is essential to making the decision outcome more acceptable and persuasive. As noted in³⁹, achieving a high degree of group consensus requires consensus feedback, while it is recognized that feedback techniques help experts align their thoughts more closely with the group’s⁴⁰.

3. In existing studies on LZNs, it may be challenging to get experts to comprehend and adopt the updated LTs as their new preferences because the revised LTs are virtual LTs derived from consistency-improvement algorithms presented in^29–31.

4. The ranking of alternatives studied in^22,29–31 is score-based; none of them considers the psychological characteristics of loss aversion and regret aversion of experts.

Based on the above analysis, we outline below the motivation and significance of this study.

1. To enable effective and reliable cooperation between humans and AI systems, decision models must be capable of handling ambiguity, uncertainty, and diversified multi-element judgements. Traditional decision frameworks often struggle to process nebulous data or subjective human perspectives. Prior studies have shown the most effective way to address these challenges is to integrate MCGDM techniques with LZNs. To improve decision-making testing for the optimal implementation of collaborative system designs, MCGDM approaches utilise complex analytical systems to evaluate uncertain data with imprecise components. In this article, we’ll demonstrate how to utilise an MCGDM within the LZN framework to assess and improve human-AI collaboration in software engineering.

2. Traditionally, artificial intelligence systems have been built to process input that is crisp, numerical, or binary. They have great potential in controlled environments. But, as is common in the actual world of decision-making, they can struggle faced with opposing, realistic, or absent human inputs. The use of modelling LTs, such as very good, moderately good, partially good, etc., in linguistic situations introduces ambiguity. This adaptability enables the effective acquisition and processing of human ideas, which are often conveyed in subjective or fluid forms. Thus, the fuzzy system can be seen as both a bridge and a barrier between human reasoning and machine logic; it also aids in decision-making when data is incomplete or when views are contradictory or have varying values. The framework outperforms more conventional AI approaches for processing nuanced human-AI interactions due to the incorporation of LZNs, which take into account not only primary information but also capture relative information of primary information.

3. In this paper, we propose an adjustment-based process for determining expert subjective weights in the LZN environment. We assume that each group member has background knowledge, skills, and familiarity with all options and qualities, enabling them to adjust the assigned subjective weights. To obtain the modified subjective expert weights, we combine the upfront weight assignment with the experts’ relative relevance. After providing judgment information for each alternative, each expert must assess the relative importance of the other experts. To compute the expert objective weight, we employ the entropy method. Finally, we propose an approach to determining the integrated expert weight based on the expert’s subjective and objective weights.

4. An expert interactive consensus-based MCGDM under LZNs is proposed, in which experts can modify their opinions through the consensus-reaching process. With the proposed feedback technique, it is easy to identify the experts and their corresponding positions required for modifications.

5. In the proposed consensus-reaching process, advice generation with recommendations for the expert to revise their opinions in light of the original LTs, aligning their thoughts more closely with the group’s.

6. People who use AI tools are generally risk-averse because they believe that mistakes could incur substantial costs (operational, reputational, compliance, and safety), and they later regret their choice when a tool they selected doesn’t perform as well as others. Standard rational frameworks, such as traditional AHP/TOPSIS, presuppose stable value and fail to explicitly incorporate loss aversion and regret effects that influence expert assessments in uncertain contexts. Considering the psychological characteristics of loss aversion and regret aversion among experts, a prospect-regret theory (PRT)-based TOPSIS ranking approach is proposed. Our PRT-based TOPSIS stage is therefore not “added for novelty,” but to model a well-known managerial reality: experts’ evaluations deviate from purely rational scoring when downside risk is salient.

The text’s outline is as follows: The core concepts of LTSs, LZNs, Regret theory (RT), and Prospect theory (PT) are introduced in “Preliminaries” section. In “The proposed consensus-based MCGDM method under LZN” section, we give the consensus-based MCGDM under LZNs. In this section, we present a novel approach to determining expert weights, the weights assigned to the criteria, and the consensus process algorithm. A case study on the optimal selection of AI tools for human-AI collaboration, along with comparative and statistical analysis, is discussed in “Example with discussion” section. A comprehensive overview of the work is presented in “Conclusion” section, which also explores potential avenues for future research.

Preliminaries

Linguistic term set (LTS) and linguistic Z number (LZN) are the fundamental ideas we explore below, which will be helpful for our next discussion.

Linguistic term set with scale function

Let be a LTS of cardinality with the following properties³⁰: (1) if and only if . (2) (3) if . (4) if .

Following^41,42, the linguistic scale function (LSF) is a monotonically increasing function, which is defined below.

Definition 1

Let be a LTS and . Then the LSF is defined as: , , where if .

In general, the LSFs come in three varieties, each of which corresponds to distinct semantics.

1

2

3

In equation (2) and (3), and .

Linguistic Z number

Definition 2

^21,30 Let be a finite alternative set, and be two LTSs, where and are positive integers. Then, for any and , the LZN in is defined as , where denotes the fuzzy restriction applied to the domain of the uncertain variables, while serves as a measure of the reliability associated with . For simplicity, in this study we express LZN instated of , where and .

Example 1

For and , then let two LTSs Particularly poor, Very poor, Marginally poor, Slightly poor, Medium, Marginally good, Good, Very good, Particularly good , and Not at all sure, Not sure, Slightly not sure, Fair, Slightly sure, Sure, Very sure , are respectively. These two LTSs represent evaluation and credibility, respectively. The expert evaluates the project risk as “I’m not sure that the risk of the project is very good,” represented as an LZN Very good, Not sure .

For any two LZNs, and , , , the “max” and “min” operations are calculated in the following way:

1. ,

2. .

As introduced in¹⁴, the distance between two LZNs, and is computed in the following way:

4

where with and .

In accordance to⁴³, we use the following rule to compare two LZNs in this paper.

Definition 3

Let and be two LZNs. Then if and only if , where

5

In equation (5), , and and .

Definition 4

Let , be a set of objects with corresponding criteria set . Then with the basis of and , an information system (IS) is represented by , where , () is a domain of and is a function such that for each and . If be an LZN, then the IS is called LZN IS (LZNIS).

Regret and prospect theories

Both regret and prospect theories are valuable instruments for resolving decision-making issues. They offer us experts, practical ways to think, and analytical tools to better understand and anticipate potential behavioural deviations and psychological reactions during the decision-making process. They also provide expert guidance and scientific ways of thinking to reduce decision-making risk and increase decision-making impact.

Definition 5

⁴⁴ Let be the regret-rejoice function, which is expressed in the following way:

6

where a and represent the risk aversion coefficient and the two schemes’ utility difference, respectively. and indicate the joy and regret values, respectively.

Tversky et al. (1992)⁴⁵ introduced prospect theory, which extends the idea of limited rationality and accounts for experts’ individual risk preferences. The value function, decision weights, and the reference point for gains and losses make up the three main parts of the theoretical model.

Definition 6

The following is an expression for the value functions:

7

where denotes the the deviation of from a reference point . and indicates the gain and losses, are respectively. The coefficient of loss aversion is denoted by c, whereas b and d stand for the expert’s sensitivity to gains and losses, respectively.

Remark 1

Kahneman and Tversky⁴⁵ have shown that, when and , the choice outcomes closely resemble the actual experts’ perceived gains and losses, which are influenced by the results of prospect theory analysis.

The proposed consensus-based MCGDM method under LZN

This section introduces the principle and approach of the consensus-based MCGDM method. We also explain the key steps of the decision-making process of the proposed method.

The MCGDM problem

Let be the set of experts with their corresponding weight vector set is such that . Let , be a set of alternatives, be a set of criteria with corresponding criteria weight vector , where such that . Let be the k-th experts’ opinion judgements for the i-th alternative about j-th criteria, i.e., , where is an LZN information of the k-th experts’ opinion judgements for the i-th object about j-th attribute and , are two pre-defined distinct LTSs. Let be the evaluation matrix of k-th expert, where

8

The normalized decision matrix (NDM) of is denoted by , where

9

Computation of the weights of criteria

Based on studies^33,43, we use the correlation coefficient technique to determine the weights of the criteria. From the evaluation matrix (8) of k-th expert, we get the

10

where

11

Let B and C denote the sets of benefit and cost criteria, respectively. Then, in equation (11), and similarly calculated and .

Equation (10) computes the objective weight of criterion from the perspective of expert . Conceptually, it assigns a larger weight to a criterion if that criterion contributes more unique (non-redundant) information compared with the other criteria. (1) In our framework, each criterion induces a profile across the alternatives through the LZN evaluations . (2) Equation (11) defines , which measures the correlation (dependency / redundancy) between the profiles of criteria and . (3) Equation (10) then uses as an “independence degree” (or “non-redundancy degree”) between criteria and . (4) Finally, equation (10) aggregates over all v and normalizes the result so that . Thus, equation (10) is an objective, data-driven weighting rule: criteria with lower correlation (less overlap) with the remaining criteria receive higher weights, while highly redundant criteria receive lower weights.

We have the following justification for the correlation-based criteria structure suitable in our context:

1. Avoiding double-counting when criteria are interdependent In human–AI collaboration management, criteria such as interpretability/trust, transparency, and real-time availability can be partially overlapping. If two criteria are strongly correlated, assigning both high weights risks “counting the same information twice.” The correlation-based formulation reduces this redundancy by down-weighting highly correlated criteria.

2. Objective weighting without extra elicitation burden Many subjective weighting methods require additional pairwise comparisons or preference inputs (which increases expert workload and may introduce further inconsistency). Equation (10) uses only the already-provided evaluation matrices, making it operationally practical.

3. Compatibility with the LZN environment Because evaluations are linguistic Z-numbers, direct numeric correlation on raw values is not meaningful. Equation (11) constructs correlation using normalized distance-based quantities derived from the LZN distance function and ideal points (benefit/cost handling), which is consistent with the uncertainty-aware information structure of LZNs.

Computation of weight of experts and aggregation of expert opinion

According to studies in^33,46, when making real-world decisions, we can’t discount experts’ academic credentials and professional experience when assessing their weight. Inspired by these studies, we present an integrated process for determining expert weights. In making real-world decisions, any one group member evaluating and weighing other members based on their regard for their knowledge or opinions on the matter at hand is not possible in practice. Therefore, we determine the relative relevance indices of experts by analysing the intensity of the disparities in the judgments expressed by group members. One expert receives a higher score when their judgment is somewhat consistent with that of other experts. Conversely, when the judgment information is more diverse, they receive a lower value. In this connection, one must first modify the given subjective weights, then calculate the objective weights, and finally combine the modified subjective weights with the objective weights to obtain the integrated expert weights.

For the computation of the expert subjective weights, we follow the following steps.

Step 1: From the evaluation judgement matrix (8), we compute the relative relevance indices of from ’s point of view in the following way:

12

for all , .

Step 2: The overall relative relevance indices of k-th expert according to the evaluation judgement matrix of other experts computed compute in the following way:

13

Step 3: Based on the overall relative relevance indices of k-th expert , we compute the adjusted subjective weight of expert in the following way:

14

where , , , are the subjective weights of experts given in advanced.

For the computation of the expert objective weights, we follow the following steps.

Step 1: With the basis of the criteria weight determinant from the expert opinion by equation (10), we construct the weight NDM of each as:

15

where .

Step 2: Compute the arithmetic mean of in the following way:

16

where .

Step 3: Compute the entropy of in the following way:

17

where .

Step 4: Compute the entropy of in the following way:

18

where .

Step 5: Compute the relative proximity of from equations (17) and (18) in the following way:

19

Step 6: Compute the expert object weight of expert from equation (19) in the following way:

20

where .

Remark 2

The entropy technique is introduced for computing expert objective weights because it provides a principled way to objectively quantify information quality/dispersion in expert assessments, without requiring additional subjective inputs. Specifically:

1. Entropy captures the dispersion (information concentration vs. uniformity) of an expert’s assessments: In group decision-making, an expert whose weighted evaluations are nearly uniform across alternatives/criteria contributes less discriminative information than one who expresses clearer differentiation (while still being consistent). Entropy provides a well-established scalar measure of this dispersion pattern after normalization.

2. Entropy enables an objective benchmark against the group via : Equation (19) uses to measure deviation from the group’s overall information structure. This is important because our goal is not merely to reward “more extreme” or “more uniform” assessments, but to give higher objective weight to experts whose information contribution is more aligned with the collective structure.

3. It reduces the need for purely subjective expert weighting while remaining computationally efficient. In real management practice, asking participants to assign detailed subjective credibility scores for all experts can be sensitive and difficult. The entropy-based component provides a data-driven alternative that is easy to compute (linear in nm per expert after normalization), which also supports the practical controllability of time cost. Based on adjusted subjective weight defined in equation (14) and objective weight defined in equation (20), the integrated weight of expert is computed in the following way:

   21

   Based on expert weight of expert and inspire by Ref.²¹, the k number of experts’ opinions are aggregated and denoted by , and defined in the following way:

   22

Decision process

Phase 1: Consistency analysis with acceptable collective opinion of expert

Step 1: Three levels consensus index of expert opinion

The three levels consensus index (CI) of expert opinion decision matrix are:

• Elements: .

• Alternatives: .

• Evaluation matrix: .

In addition, the group consensus level (GCL) is as: .

Step 2: Identification of inconsistent positions

Let be a consensus threshold value. If , then the GCL is acceptable. Otherwise, in the following way we identify the inconsistent positions of opinions of experts.

Step 3: Generation of advice with recommendation

In the k-th experts’ opinion judgements for the i-th alternative about j-th criteria have two parts, first part is and second part is . Therefore, in the following we propose the generation of advice for two parts of expert :

1. If (), then the expert , decrees his/her first part (second part) in in the position (i, j).

2. If (), then the expert , increase his/her first part (second part) in in the position (i, j).

3. If (), then the expert , unchanged his/her first part (second part) in in the position (i, j).

The element of decision matrix of expert have two parts and , which are included into the two distinct predefined LTSs and . This indicates that and are original LTSs. Therefore, we believe that the recommendation of reconsider LTs are original and it is better than the virtual LTs. In this connection, we offer the set of two distinct original LTs for and of expert for choosing LTs of and of reconsidering their (i, j)-th position opinion. Therefore, the recommendation set of two distinct original LTs of experts for and are determined in the following way:

For :

23

where is the (i, j)-th position nearest original LT of corresponding (i, j)-th position LT of .

For :

24

where is the (i, j)-th position nearest original LT of corresponding (i, j)-th position LT of .

Based on the above discussion of the consensus process, we present Algorithm 1.

Algorithm 1.

The computation process of acceptable collective opinion

Theorem 1

The Algorithm 1is convergent.

Remark 3

The assignment operations are involved in lines 2 and 16 and therefore the time complexity of each step is O(l). In line 3, the weight of criteria experts weights computation complexity is O(nm). In lines from 5 to 8, the experts weights computation complexity is . The complexity of line 7 is calculated by the aggregation operator, therefore it is . The complexity of line 8 is determinate by the three level consensus measure of expert opinions, therefore it is . No loop in line 9, thus in this step 0 is the complexity. In line 11, the complexity is O(4lmn). line 12 is the modification of the object under corresponding criteria, therefore, in this step the complexity is . Finally, the total complexity of Algorithm 1 is .

Remark 4 If the experts are still unable to reach a consensus within the original term intervals after multiple iterations, we will follow the following mechanism: (I) We will set the maximum iteration, i.e., . (II) Advice generation with recommendation plans when consensus is not reached within original LTs in intervals: If GCL remains below the threshold after , we will include alternative managerial options: (1) reduce/adjust the consensus threshold to a practically acceptable value. (2) expand the recommendation interval by one adjacent LT step (still interpretable), rather than switching immediately to fully virtual terms. (3) allow an optional automatic adjustment step for only the remaining inconsistent entries, while keeping the majority of updates within original LTs.

Phase 2: PRT TOPSIS-based ranking method

With the basis of studies in^47,48, in this section, we integrated PT and RT into an acceptable collective opinion where for ranking of alternatives. The comprehensive methods are outlined below:

1. Computation of positive ideal object (PIO) and negative ideal object (NIO) for each criteria.

   • PIO: .
   • NIO: .
2. Computation of the distance from alternatives to the PIO and NIO We use the following computation technique for determination of the distance from alternatives to the PIO, denoted by , and the distance from alternatives to the NIO, denoted by .

   25

   26

3. Computation of PRT-based relative closeness index (RCI) of each alternative Based on their findings, the authors of⁴⁹ suggest that experts’ psychological utility of physical distances from current alternatives to the PIO and NIO can be reflected in RCI determination using PT. Then, to calculate RCI, we will include RT and PT. Compared to the PT, the RT improves the match between people’s actions and the utility function by incorporating regret, an additional psychological component. In the proposed RCI, we say that regret is the diminished pleasure that results from opting out of the PIO and that joy is the increased pleasure that results from opting out of the NIO. Therefore, we will now present the detailed process for calculating the PRT-based RCI for each alternative.

With the basis of PT, we use the following computation technique for determination of the prospect value of each alternative concerning PIO and NIO as follows:

• PPV: , if

• NPV: , if ,

where and called positive prospect value (PPV) and negative prospect value (NPV) of .

Using the principles of RT, we apply the following computation technique to determine the regret-rejoice value for each alternative in relation to NPV and PPV:

• PRgV: , if

• PRjV: , if ,

where and called prospect-regret value (PRgV) and prospect-rejoice value (PRjV) of .

Based on the PRgV and PRjV of , the PRT-based perceptive utility of is as follows:

• GRjPU: ,

• LRgPU: ,

where and called gain-rejoining perceptive utility (GRjPU) and loss-regret perceptive utility (LRgPU) of .

Based on GRjPU and LRgPU of , in the following we propose PT-based RCI computation technique.

27

The options are sorted from best to worst based on the descending order of the enhanced relative proximity coefficient.

In summary, Fig. 1 illustrates our proposed multi-criteria consensus-based group decision-making process under LZN with the prospect-regret TOPSIS ranking method.

Fig. 1.

Visualisation of our proposed multi-criteria consensus-based group decision making with prospect-regret TOPSIS-based ranking approach.

Example with discussion

To demonstrate the proposed approach, we consider a case study of the optimal selection of AI tools for human-AI collaboration in this section. This section also performs a sensitive analysis of related parameters using a comparative approach.

An example: a case study of the optimal selection of AI tools for human-AI collaboration

Unpredictable human preferences, including the use of AI tools and behavioural patterns, continue to pose obstacles to the advancement of computer technologies and AI, making successful human-AI collaboration an elusive goal. An appropriate evaluation strategy for this partnership would include creating a model to examine several qualitative and quantitative factors in an unpredictable environment. This paper’s overarching goal is to provide an MCGDM under a linguistic information framework for evaluating and improving human-AI cooperation strategies in IT development. The framework’s systematic assessment procedure uses AI technologies and systems as examples, assessing them against specified criteria.

1. The ability to be understood and to have faith in someone ().

2. How well the system works ().

3. Users’ capacity to adjust ).

4. Availability in near-real time and being open and honest ().

To generate firm conclusions from linguistic evaluations, a consensus-based MCGDM with a PRT TOPSIS-based ranking model will be employed. When dealing with the complex assessment process for human-AI collaboration in imprecise and unpredictable scenarios, it is vital to include LZNs within the proposed MCGDM framework. By providing greater flexibility, the LZNs’ information enables better representations of human choice and a clearer expression of expert judgements. The combination of human decision-making and AI operating behaviours might lead to confusing assessment outcomes, which requires special consideration in the development of computer technology. By effectively leveraging its expressive and flexible characteristics, the LZN information framework can process complex language inputs and inconsistent expert evaluations, leading to more easily understandable judgements. Incorporating LZN into assessment methods offers several benefits, including improved accuracy in defining how well the system works, greater clarity in understanding, trust in someone’s criteria, and the availability of near-real-time feedback, along with transparency as an AI tool.

We plan to analyse eight human-AI collaboration technologies , considering the above-specified criteria. The evaluation, which relies on decision analysis and artificial intelligence, was conducted by three recognised experts with their weight vector in the field using the predefined LTSs are Particularly poor, Very poor, Marginally poor, Slightly poor, Medium, Marginally good, Good, Very good, Particularly good , and Not at all sure, Not sure, Slightly not sure, Fair, Slightly sure, Sure, Very sure from various disciplines in relevant fields to assess the options based on the criteria .

The description of three experts is given in the following:

1. : An expert with over a decade of experience in academia and industry, he teaches courses on intelligent decision support systems and AI system architecture.

2. : An accomplished data scientist who focuses on developing AI with natural language understanding and flexible learning algorithms.

3. : A human-computer interaction research fellow with expertise on ethical alignment, interpretability, and trust in AI-based systems.

The description of four criterion is given in the following:

1. : An AI system’s ability to be understood and to have faith in someone is defined as its capacity to provide human-traceable, comprehensible rationale for its outputs and the level of specificity with which end-users evaluate the AI system’s performance and outcomes. When an AI system presents its reasoning in a way that humans can comprehend, whether through explanations or visual representations, it achieves “particularly good” and the relative information is “very sure”, meaning the system produces accurate and dependable results.

2. : When assessing the reliability of AI systems and tools, experts look at how fast and accurately the computers are. When basic AI systems demonstrate they can handle many inputs and outputs effectively, they are considered “particularly good”, and the relative information is “very sure”, indicating the AI system’s high performance and efficiency under dynamic conditions.

3. : The criteria evaluate a user’s capacity to adjust, usability, and learning potential in relation to an AI system. When the AI is flexible, the user experience is “particularly good” and the relative information is “very sure”, meaning the system is designed to adjust to high data utilisation and the level of decision-making logic.

4. : In terms of availability in near-real time, openness and honesty, privacy, and accountability, the evaluation assesses how well an AI system follows the rules and handles more users, more data, and more complex tasks. A “particularly good” suggests that the function is widely applicable in different contexts, and the relative information is “very sure”, meaning the system actively safeguards ethical considerations.

In Table 1, we show the opinion judgements for human-AI collaboration of AI tools.

Table 1.

The evaluation information expressed by the LZNs provided by three experts.

Expert	Alternative

Phase 1: Consistency analysis of expert opinion with aggregation

Set , , and . The expert subjective weights are , , and . Then by Algorithm 1, , , , , and the acceptable collective opinion is

28

Phase 2: PRT TOPSIS-based ranking method

Set . The PRT TOPSIS-based ranking of alternatives is shown in Table 2.

Table 2.

Ranking order (RO) of alternatives.

										RO
	0.5387	0.5729	0.5802	− 1.3781	0.2298	0.4621	0.8100	-0.9160	0.4693	6
	0.5152	0.5615	0.5579	− 1.3539	0.2220	0.4563	0.7799	-0.8977	0.4649	7
	0.6419	0.4353	0.6770	− 1.0823	0.2626	0.3856	0.9396	-0.6968	0.5742	1
	0.5647	0.4809	0.6047	− 1.1813	0.2383	0.4123	0.8430	-0.7690	0.5230	3
	0.5050	0.5389	0.5482	− 1.3058	0.2186	0.4444	0.7667	-0.8615	0.4709	5
	0.6826	0.6173	0.7146	− 1.4717	0.2750	0.4843	0.9896	-0.9874	0.5006	4
	0.4718	0.6267	0.5163	− 1.4913	0.2073	0.4889	0.7236	-1.0025	0.4192	8
	0.5780	0.4735	0.6173	− 1.1655	0.2425	0.4081	0.8599	-0.7573	0.5317	2

Therefore, from Table 2 the ranking of the object is . Then we suggest to build as a AI tool for human-AI collaboration.

Discussion

In this section, we discuss the effects of the parameters on decision outcomes and compare our proposed consensus-based MCGDM approach with existing studies.

Sensitivity analysis

Our proposed consensus-based MCGDM approach involved , , the consensus threshold value , and the parameter related to RT . Sensitive analysis involving the other three parameters related to PT cannot be performed here. This restriction is due to the extensive examination of experimental data conducted by Tversky and Kahneman⁴⁵ to determine these three parameters. The values of parameters are noted in Remark 1.

(1) Effect of and on decision result: Here, we will examine the impact of and on the CI of the experts’ evaluation matrix at the matrix level, as well as on GCL with ranking outcomes. Two parameters indicate the relative significance of evaluative information and its dependability. If the importance of the assessment information surpasses its dependability, then . If dependability supersedes evaluative information, then . If the significance of assessment information and dependability is identical, then . We conduct a sensitivity study by defining the range of values from 0 to 1 and the range of values from 1 to 0, using a step size of 0.1, along with the CI of the evaluation matrix level and GCL ranking outcomes. The results are displayed in Fig. 2. Figure 2 illustrates that the ranking results exhibit relatively significant variation with the changes in and . The findings demonstrate that considering the significance of dependability substantially influences the CI of the experts’ evaluation matrix at the matrix level and of the GCL in ranking outcomes.

Fig. 2.

The values of CI, GCL, and the number of iterations, along with the ranking of alternatives that change for different values of , are presented while keeping , , and .

(2) Effect of on decision result: The distance metric in equation (4) is the Minkowski distance. The main advantage of this distance metric is that it can be adjusted for different needs by changing its parameter : when , it becomes the Hamming distance; when , it changes to the Euclidean distance; and as tends to infinity, it turns into the Chebyshev distance. Subsequently, we will examine how changing affects the CI of the experts’ evaluation matrix level and GCL with ranking results across six cases: . Figure 3 displays the results. The results show that using different distance functions yielded distinct CIs for the experts’ evaluation matrix levels and GCL ranking results. Therefore, the set of is dependent on each particle’s needs.

Fig. 3.

The values of CI, GCL, and the number of iterations, along with the ranking of alternatives that change for different values of , are presented while keeping , , and .

(3) Effect of on decision result: To elucidate the precise impact of on decision outcomes, we initially conduct a sensitivity analysis by incrementing by 0.2 from 0.85 to 1. For each value of , we calculate the CI of expert opinion () as well as GCL and finally the ranking order of alternatives. The detailed results are shown in Fig. 4. In Fig. 4, we observe that maintains the top rank. Again, we see that after increased from 0.9 to 1, the number of adjustments increased, making the process more time-consuming, while the top-rank status of alternatives remained unchanged.  

Fig. 4.

The values of CI, GCL, and the number of iterations, along with the ranking of alternatives that change for different values of , are presented while keeping , , and .

(4) Effect of a on decision result: This section will involve sensitivity studies examining concurrent changes to the parameter a. We now set , , and , allowing a to incrementally rise from 0 to 1. The experimental findings are presented in Fig. 5. Figure  5 indicates that the influence of variations in a on the ranking outcomes is minimal, underscoring the robustness of the suggested model.

Fig. 5.

The values of CI, GCL, and the number of iterations, along with the ranking of alternatives that change for different values of a, are presented while keeping , , and .

(5) Quantitative analysis regarding the relationship between and : To quantify the managerial time cost of the consensus-reaching phase, we record the number of iterations required by Algorithm 1 to achieve different consensus thresholds (see Fig. 4). As increases from 0.85 to 1, increases monotonically but remains small shown in Table 3. Hence, the number of feedback/adjustment rounds is even under the strictest consensus requirement. Moreover, because the time complexity of Algorithm 1 is , the total computational complexity scales linearly with , i.e., . For the case study (), this yields a very small computational burden, indicating that the practical time cost is mainly driven by a limited number of feedback rounds, which is controllable. Consistent with Fig. 4, although higher increases the number of adjustments (especially when rises from 0.9 to 1.0), the top-ranked alternative remains unchanged.

Table 3.

Relationship between and in Algorithm 1 (fixed parameters: , , ).

	0.85	0.86	0.88	0.90	0.92	0.94	0.96	0.98	1
	1	1	1	2	2	3	3	3	4

Comparative analysis

In this section, we compare our proposed approach from two perspectives: one perspective compares it with related MCGDM approaches, while the other compares it with consensus-based approaches in the LZN environment.

Compared with related MCGDM approaches

The MCGDM model examined in^22,33,50 may be employed to resolve the issue of selecting AI tools for human-AI collaboration, which is the focus in this section. In this connection, we compare our proposed approach with four aspects: (I) determination of criteria weights, (II) determination of expert weights, (III) considering the opinions of experts without and with consensus, and (IV) ranking technique of alternatives.

(I) Determination of criteria weights:

In²², the weights of criteria and order positions are assumed. However, in³⁰, it was noted that the weight of criteria plays an essential role in MCGDM and in final decision outcomes. For the computation of criteria weights, Liu et al.’s³³ approach adopts a hybrid technique, which is a combination of subjective and objective weights. In their approach, they need additional information, such as subjective preference degrees between criteria, which are provided by experts based on comparisons of criteria with respect to Best-to-Others and Others-to-Worst. Zheng et al.’s⁵⁰ approach uses the maximum deviation method to calculate criterion weights, based on a linear optimisation technique. These approaches are well-established for computing criterion weights that align with practical needs. However, the criteria of AI tools for human-AI collaboration management are highly dependent on and correlated with one another. In this view, we proposed the correlation coefficient method for determining criterion weights, which is more effective, as indicated in the study^10,51.

(II) Determination of expert weights:

The determination of experts’ weights is avoided in^22,33,50; however,^30,47 points out that it cannot be avoided in the MCGDM because experts come from different fields, and their knowledge differs based on experience and expertise. In this paper, we propose a novel approach to computing experts’ weights based on the study^46,52.

(III) Considering the opinions of experts with and without consensus:

The approach employed in^22,33,50 does not account for expert agreement. However, the decision’s outcome is questionable due to the lack of a consensus process in the studies referenced in^31,47. In this context, we analysed the studies by Chen et al.²² approach¹, Liu et al.³³ approach², and Zhang et al.⁵⁰ approach³ along with our approach, which considers expert opinions with and without consensus. Tables 4 and 5 display the detailed results. From Tables 4 and 5, we observe that the optimal alternative remains unchanged when expert opinion consensus is considered, indicating that the consistency of expert opinions significantly affects decision outcomes. However, considering expert opinions without consensus significantly alters the ranking results.

Table 4.

Assessment of AI tools for human-AI collaboration by the studies in^22,33,50 vs. our approach with consensus case.

Alternatives	Chen et al.22 approach		Liu et al.33 approach		Zhang et al.50 approach		Our approach
	Ranking value	Ranking order	Ranking value	Ranking order	Ranking value	Ranking order	Ranking value	Ranking order
	0.5300	2	− 0.0327	6	0.4127	5	0.4693	6
	0.4916	7	− 0.0520	7	0.3677	6	0.4649	7
	0.5340	1	0.0536	1	0.5453	1	0.5742	1
	0.5220	4	0.0268	4	0.4757	3	0.5230	3
	0.4995	5	0.0198	5	0.3606	7	0.4709	5
	0.4936	6	0.0482	2	0.4725	4	0.5006	4
	0.5284	3	− 0.0698	8	0.3274	8	0.4192	8
	0.4897	8	0.0326	3	0.4868	2	0.5317	2

Table 5.

Assessment of AI tools for human-AI collaboration by the studies in^22,33,50 vs. our approach without consensus case.

Alternatives	Chen et al.22 approach		Liu et al.33 approach		Zhang et al.50 approach		Our approach
	Ranking value	Ranking order	Ranking value	Ranking order	Ranking value	Ranking order	Ranking value	Ranking order
	0.4748	2	− 0.0520	7	0.4092	4	0.4660	8
	0.4352	6	− 0.0297	6	0.4081	5	0.5446	3
	0.4655	4	0.0499	2	0.4732	2	0.5696	2
	0.4663	3	0.0007	5	0.3785	7	0.5148	4
	0.4576	5	0.0281	3	0.3511	8	0.5096	5
	0.4304	7	0.0268	4	0.4230	3	0.5038	6
	0.4772	1	− 0.0852	8	0.3807	6	0.4679	7
	0.4165	8	0.0711	1	0.4848	1	0.5880	1

(IV) The ranking technique of alternatives.

The ranking method in Chen et al.²² is based on LZN scores, which, in some cases, fails, as shown in⁴³. For ranking alternatives, Liu et al.’s³³ uses the original GLDS method, which is based on PT. They avoid another RT. The ranking method of Zheng et al.⁵⁰ is based on the ORESTE approach. However, in his study, Steinhagen⁴⁸ pointed out that the ORESTE approach has notable shortcomings, specifically the Besson ranking and the distance-based criteria. He claims that in his research, these shortcomings in decision-making frameworks may lead to unexpected or even undesirable results. In our approach, we used a PRT-based TOPSIS ranking technique to rank alternatives, thereby addressing the shortcomings of Chen et al.’s²², Liu et al.’s³³, and Zheng et al.’s⁵⁰ approaches.

Compared with consensus-based approach

For validity analysis, we compare our model with Xu et al.’s³¹ approach⁴. Thus, based on the Xu et al. approach, we are selecting AI tools for human-AI collaboration. The steps are as follows:

Step 1: Convert the expert opinion shown in In Table 1. The results are shown in Table 6.

Table 6.

The converted decision matrix of each expert.

	0.9778	0.9856	0.8439	0.9856	0.9771	0.2814	0.7186	0.6202	0.8886	0.4717	0.7711	0.9856
	0.7838	0.6443	0.9778	0.7704	0.9161	0.9946	0.9459	0.9856	0.8462	0.9778	0.8608	0.6202
	0.9856	0.9771	0.3720	0.7186	0.9351	0.8886	0.8886	0.8164	0.9856	0.9351	0.8886	0.9778
	0.4594	0.7186	0.9778	0.9351	0.9778	0.8608	0.9856	0.9161	0.9203	0.9552	0.8608	0.5946
	0.9778	0.4594	0.8439	0.9778	0.9351	0.7186	0.9778	0.7704	0.7711	0.8462	0.9778	0.8608
	0.7930	1.0000	0.9161	0.0222	0.9771	0.9771	0.8462	0.6280	0.9856	0.8886	0.9203	0.9203
	0.9856	0.9778	0.7225	0.9778	0.8164	0.1392	0.9203	0.9161	0.4717	0.8382	0.9161	0.9771
	0.9771	0.6202	0.9552	0.9015	0.9161	0.9771	0.6443	0.6568	0.9771	0.9778	0.9771	0.9856

Step 2: The collective decision matrix is shown in Table 7.

Table 7.

The initial collective decision matrix.

	0.9478	0.5796	0.7779	0.8638
	0.8487	0.8723	0.9282	0.7921
	0.9688	0.9336	0.7164	0.8376
	0.7858	0.8449	0.9414	0.8153
	0.8947	0.6747	0.9332	0.8697
	0.9186	0.9552	0.8942	0.5235
	0.7579	0.6517	0.8529	0.957
	0.9567	0.8584	0.8589	0.848

Step 3: We have now calculated the individual consensus levels, which are , , and . Thus, the group consensus level is 0.8837, which is greater than 0.85. That means in this example, no feedback loop is needed. Thus, the initial decision matrix shown in Table 6 is the final decision matrix of each expert.

Step 4: We have now computed the weights for each expert: 0.1484, 0.4289, and 0.4227. The optimal weights for the criteria are 0.2377, 0.3207, 0.1296 and 0.3120.

Step 5: Based on the quantum-like Bayesian network (BN), the first layer probabilities are the weights for each expert. In the quantum-like Bayesian network (BN), the first-layer probabilities correspond to the weights for each expert. The conditional probability in the BN is , , , , , , , ,

, , , , , , , ,

, , , , , , , .

Therefore, the probability of each alternative is , , , , , , , .

Based on the value of the probability of each alternative, the ranking is .

Step 6: We have now computed the standardized Deng entropy (SDE), shown in equation (29).

29

Step 7: With the basis of quantum interference effects, the quantum probability of each alternative is , , , , , , , .

Based on the value of the quantum probability of each alternative, the ranking is . The results indicate that our method is compatible with Xu et al.’s³¹, since the optimal alternative remains the same when comparing the two.

Statistical analysis

In this section, we conduct a statistical analysis to identify AI tools for human-AI collaboration. Specifically, we rank the eight AI tools for human-AI collaboration, based on the approaches outlined in^22,31,33,50 and our proposed approach. First, we analyse it without consensus-based approaches^22,33,50, and then we include a consensus-based approach³¹ in our analysis.

Statistical analysis of without consensus-based approaches

In this section, we apply two popular techniques: Spearman’s rank correlation (SRC) coefficient⁵ and Kendall’s Tau tests⁶. We aim to quantify differences between the compared methods and measure similarities among rank sets. Figures 6, 7, and 8 display the detailed results. From Figs. 6, and 7, we observe that our method is highly consistent with Zheng et al.’s⁵⁰ approach, strongly consistent with Liu et al.’s³³ approach, and significantly divergent from Chen et al.’s²² approach, considering the opinions of experts with and without consensus. However, from Fig. 8, we observe that our method is strongly consistent with Liu et al.’s³³ approach, moderately consistent with Zheng et al.’s⁵⁰ approach, and significantly divergent from Chen et al.’s²² approach.

Fig. 6.

The statistical ranking tests of our method are compared with those examined in^22,33,50, emphasizing the consensus among experts’ opinions.

Fig. 7.

The statistical ranking tests of our method, compared to those examined in^22,33,50, highlight the lack of consensus among experts’ opinions.

Fig. 8.

The statistical ranking tests of our method, compared to those examined in^22,33,50.

Statistical analysis of consensus-based approaches

In this section, we apply two popular techniques: SRC coefficient and Kendall’s Tau tests. We aim to quantify differences between the compared methods and measure similarities among rank sets. Figure 9 displays the detailed results. From Fig. 9, we observe that our method shows a moderate correlation with Xu et al.’s³¹ approach in the absence of an interface effect. Still, there is no meaningful correlation when the interface effect is present. However, we observe that the optimal alternative is unchanged in Xu et al.’s³¹ and our approaches in both cases.

Fig. 9.

The statistical ranking tests of our vs. Xu et al.’s³¹ approaches.

Implications of this study for a real-world scenario

Experts who guide and engage AI systems in human-computer dialogue can make better judgments in uncertain contexts using the proposed LZN-based MCGDM paradigm. Due to its adaptability and linguistic reasoning capabilities, it finds usage in several practical fields:

1. In the medical field, the paradigm can be used to evaluate diagnostic AI tools by considering key AI properties, including the ability to be understood and to have faith in someone, how well the system works, users’ capacity to adjust, availability in near-real time and being open and honest. These properties are crucial for the confidence and acceptance of these tools in clinical contexts.

2. The semi-autonomous driving systems rely heavily on human-AI interaction. Based on their ability to be understood, the system’s effectiveness, users’ capacity to adjust, and safety in dynamic circumstances, the model may rate the perception and navigation modules.

3. In the customer service, responsiveness, user-friendliness, and dependability should be the pillars around which chatbots and virtual agents are built. Customer experience managers can utilise the framework to evaluate and select tools based on expert consensus.

4. In the education systems, human educators can evaluate the usefulness and timeliness of real-time feedback and corrections, as well as the ability to personalise a system for each user, in both intelligent tutoring systems and adaptive learning environments. The provided examples demonstrate the exceptional adaptability of the recommended model across various domains that require expert work and human-AI collaboration.

A multi-expert evaluation of eight AI tools in relation to four critical features validates the usefulness of the proposed methodology. The model’s robustness is demonstrated by the findings, which consistently show that the ranking order stays constant independent of the values of parameters , and a. On the other hand, statistical analysis in “Statistical analysis” section reveals that the rankings of the AI tools in some existing studies are highly correlated with our research.

To achieve the study’s goal of discovering a technique to advance human-AI collaboration, this empirical investigation demonstrates that the model is adaptable and trustworthy for monitoring human-like decision-making behaviour under uncertainty.

Conclusion

In this article, we explore the assessment of AI tools for human-AI collaboration using the consensus-based MCGDM under LZNs. Some important things that this study has brought to light are:

1. Discussed interactive, strategy-based consensus-building in which we advise experts to revise their opinions within the original LTs, which resolves the deficiency of related studies.

2. Developed how to obtain subjective and objective-based integrated expert weights with LZN-based decision information. For derived subjective expert weights, we compare the judgments of any two experts and derive the overall relative-relevance indices for each expert, which are then used to adjust the subjective weights. To obtain objective expert weights, the criterion-weight-based entropy method is employed.

3. To measure the psychological characteristics of loss aversion and regret aversion of experts, the PRT-based TOPSIS ranking approach is developed.

4. An essential part of the decision-making process outlined in this essay was how to measure the efficacy of AI tools for human-AI collaboration. Similarly, a wide range of characteristics may be considered to provide greater precision.

The advantages of our proposed model are listed below:

1. Our proposed consensus model has a simple, meaningful feedback cum advice generation process. It is easy to identify which experts and which positions oppose the consensus goal. In the advice-generation process, experts revise their opinions using original LTs, which are easier for them to understand than virtual LTs.

2. This is because the expert’s reputation, standing, occupation, degree of expertise, and other observable qualities underlie the subjective weights. Subjective weights assigned to experts may be inherently skewed. Therefore, it might not be reasonable to assign weights in advance. We present a realistic and bias-free method for determining an expert’s adjusted subjective weights.

3. By merging PRT-based TOPSIS, we derive six relative utility functions, which we then use to explain the limited rationality of DMs. Decisions are less likely to be impacted by subjective considerations and biases when these relative utility functions are used.

4. To account for experts’ bounded rationality, we combine prospect theory, regret theory, and TOPSIS to derive the RCI for each alternative. This PRT-based RCI for each alternative effectively reduces the influence of subjective factors on decision-making and the risk of decision errors due to subjective biases.

We demonstrate that the technique is successful in real-world settings and is both feasible and stable by comparing it with existing decision-making methods and conducting a parametric analysis of the results. This demonstrates that our approach not only enhances MCGDM’s utility but also provides more rigorous, scientific support for complex decision-making. However, the proposed model has several limitations, which are listed below:

1. The current model assumes static expert or agent weights.

2. Unlike automatic consensus models, the suggested model requires user interaction to make decisions, which adds time to the process. In⁵³, the advantages and disadvantages of the automated and interactive approaches were compared.

3. The model’s parameters are not always easy to pin down. However, the rigorous experimental analysis in^54,55 shows that and .

Despite these limitations indicate clear avenues for future work. First, a promising extension is to introduce dynamic weighting, in which the influence of human experts and AI agents evolves based on consistency, reliability, and trust calibration across iterations or decision rounds. Second, investigate computational acceleration, that is, decomposition across expert blocks, parallel implementation of consensus updates, and incremental consensus updating when new experts/alternatives arrive. Third, focus on domain-specific parameter calibration through empirical elicitation and data-driven estimation to ensure validity in AI-management contexts.

Author contributions

Prasenjit Mandal: Conceptualization of this study, methodology, validation, writing–original draft. Leo Mrsic: Funding acquisition, validation and visualization. Antonios Kalampakas: Data curation, formal analysis, and visualization. Tofigh Allahviranloo: Formal analysis, visualization, and supervision. Sovan Samanta: Formal analysis, investigation, resources, writing—review and editing.

Funding

Project related funding: project “Innovative digital platform for assessing brain activity levels based on EEG technology and advanced machine learning (BRAINACTIV) NPOO.C3.2.R3-I1.06.0082".

Data availability

The data will be made available on request.

Declarations

Competing interests

The authors declare no competing interests.

Informed consent

The authors confirm that written informed consent has been obtained from the involved experts.