A Novel Multi-Criteria Quantum Group Decision-Making Model Considering Decision Makers’ Risk Perception Based on Type-2 Fuzzy Numbers

Abstract

In multi-criteria group decision making, decision makers are commonly regarded as independent. However, in practice, heterogeneous backgrounds and complex cognitive processes lead to mutual interference among their judgments. To address this gap, a novel multi-criteria quantum group decision-making model is proposed that explicitly incorporates opinion interference effects. First, type-2 fuzzy numbers are employed to represent evaluation information, and a specialized Euclidean distance measure for them is introduced. Second, an extended distance-based criteria importance through an inter-criteria correlation method incorporating Deng entropy is developed to derive robust criteria weights under uncertainty. Third, the TODIM method integrates cumulative prospect theory to capture decision makers’ risk perceptions and computes prospect-based dominance degrees. Fourth, a quantum-inspired aggregation mechanism models the mutual interference in group opinions. Finally, a case study on FinTech startup investment demonstrates the model’s practical applicability, while sensitivity analysis and comparisons to established methods confirm its robustness and effectiveness.

1. Introduction

Multi-criteria group decision making (MCGDM) is a key domain in decision science. In standard MCGDM scenarios, decision makers (DMs) provide evaluations for various alternatives based on established criteria. Information fusion techniques are employed to aggregate individual opinions, resulting in a collective evaluation that forms the basis for ranking alternatives [1,2,3]. In practice, DMs often prefer linguistic information to express opinions due to the inherent uncertainty and subjectivity in human judgment, as well as the difficulty in precisely quantifying qualitative aspects. Consequently, MCGDM models utilizing linguistic information serve as valuable tools in managerial decision-making contexts. To meet diverse and complex decision-making needs, linguistic information has undergone further expansion into various representations, such as type-1 fuzzy sets (T1FSs) [4], type-2 fuzzy sets (T2FSs) [5], and interval type-2 fuzzy sets (IT2FSs) [6]. However, T1FSs and T2FSs often face challenges such as high computational complexity. In contrast, type-2 fuzzy numbers (T2FNs) offer a practical approach to representing DMs’ judgments with enhanced computational efficiency [7]. T2FNs model uncertainty using two variables: the primary membership, which represents the degree of truthfulness of a judgment, and the secondary membership, which reflects the reliability of that truthfulness. This dual-membership framework enables T2FNs to effectively address the complexities of decision-making under uncertainty, rendering them particularly suitable for practical managerial applications.

In MCGDM problems, the rational determination of criteria weights is a pivotal factor in ensuring the accuracy of decision outcomes. The distance-based criteria importance through the inter-criteria correlation (D-CRITIC) method, by integrating standard deviation and distance correlation, objectively captures nonlinear interactions among criteria, which significantly enhances the scientific rigor and fairness of weight allocation compared to traditional methods [8]. Leveraging these advantages, the D-CRITIC method has found extensive applications in fields such as intelligent transportation [9], enterprise management [10], and path planning [11]. However, its reliance on standard deviation makes it highly sensitive to outliers and limits its efficacy with highly uncertain data [12,13]. Deng entropy, which extends Shannon entropy and serves as a flexible uncertainty-measurement instrument, presents a promising means for quantifying uncertainty and handling highly fuzzy information [14]. Through its direct operation on the basic probability assignment within evidence theory, Deng entropy can more comprehensively capture conflict and complementarity relationships among evidence. This direct measurement of the evidence structure enhances its robustness in addressing nonlinear relationships and extreme data distributions, enabling a more precise quantification of complex uncertainty stemming from conflicting evidence and fuzzy assessments [15]. Considering these complementary strengths and limitations, integrating Deng entropy into the D-CRITIC framework constitutes a promising direction for improving objective weighting. Such a unified approach can mitigate sensitivity to outliers while leveraging Deng entropy’s nuanced handling of evidential complexity, ultimately leading to more reliable criteria weights and a more resilient decision-support tool in MCGDM.

Following the rational determination of criteria weights in MCGDM, the accurate ranking of alternatives becomes a central objective. However, this process relies not only on objective data but also on the psychological factors of DMs that exert a significant influence on their evaluation outcomes, particularly their risk perception and attitudes toward gains and losses. The TODIM method, proposed by Gomes and Lima [16,17], captures DMs’ behavioral characteristics, such as reference dependence and loss aversion, through a value function analogous to that in prospect theory [18]. To more comprehensively describe DMs’ risk preferences under uncertainty, cumulative prospect theory (CPT) [19] was integrated into the TODIM framework, leading to the CPT-TODIM method [20]. This method combines CPT’s value function and probability weighting function to more accurately reflect DMs’ psychological evaluation processes under conditions of risk and uncertainty. In recent years, CPT-TODIM has been successfully applied to group decision-making problems in fuzzy environments [21,22], demonstrating its effectiveness in handling complex risk attitudes. This study employs the CPT-TODIM method to derive individual DMs’ evaluation results for alternatives, thereby reflecting their psychological characteristics. However, despite CPT-TODIM’s excellent performance in modeling individual risk perception, it does not directly account for the mutual interference effects among DMs in group decision making, which provides the primary research motivation for the subsequent introduction of quantum probability theory (QPT).

The resolution of MCGDM problems requires combining separate assessments from individuals using a particular method for aggregation. Extensive research has been dedicated to aggregation techniques in MCGDM. However, a common limitation is that most current methods are predicated on the assumption of mutual independence among the involved DMs, meaning that their opinions are not influenced by one another. Contrary to this idealized view, real-world instances frequently demonstrate that social and environmental factors easily influence individuals’ opinions. Conjunctive or disjunctive attitudes might be adopted by individuals regarding the opinions of different DMs in actual decision making. This inherently introduces the fact that the assessments provided by various DMs become susceptible to mutual influence, potentially impacting the final decision. This interplay brings forth the concept of “interference”, a fundamental element within QPT. As a recent theoretical progression [23,24], QPT advances upon classical probability theory. It features robust probability descriptive power coupled with a rigorous axiomatic framework, facilitating its widespread study and application in various fields. Among these are psychology [25,26], cognitive science [27], and decision analysis [28,29]. Significantly, QPT also resolves many long-standing paradoxes unresolved by classical probability theory, including the disjunction fallacy [30], the Ellsberg paradox [31], and the order effect [32]. It is noteworthy that QPT offers a natural mechanism for reflecting interference terms. This inherent capability makes QPT a highly suitable and theoretically grounded means to explore the interference effects that manifest between DMs’ opinions in MCGDM applications. Consequently, leveraging QPT to model these interference effects within the decision-making process is considered an objective and rational approach.

Although the existing MCGDM methods have demonstrated strong performance in solving various decision problems, several critical issues need to be resolved:

(1)

A prevalent assumption in existing research on T2FNs is that DMs perceive linguistic variables linearly [33,34]. This simplification, however, often inadequately reflects actual human decision-making processes, wherein DMs typically exhibit greater sensitivity to extreme evaluations rather than intermediate ones. Consequently, current linear linguistic transformation methods may lack sufficient discriminability for these intermediate terms and can diminish the perceived impact of extreme values, thereby limiting their suitability and accuracy in practical applications that require a nuanced understanding of DMs’ judgments.

(2)

Since its inception, the D-CRITIC method has been widely applied and has demonstrated significant effectiveness across various decision-making contexts. However, the traditional D-CRITIC method calculates criteria weights based on standard deviation and distance correlation, which entails certain limitations. In particular, its reliance on standard deviation leads to the method being highly sensitive to outliers, thereby compromising the robustness and reliability of the weight determination process. Moreover, this approach fails to adequately address the inherent uncertainty present in MCGDM problems, which may result in distorted or biased weight assignments [8,11].

(3)

The CPT-TODIM method, when compared to the traditional TODIM method, maintains the strengths inherent in comparative analysis while also integrating DMs’ psychological dimensions. However, current studies on the CPT-TODIM framework mainly focus on individual DMs’ psychological aspects [35,36], neglecting the interference effects among them. This oversight results in the insufficient consideration of group psychological dynamics in decision-making processes.

Addressing the identified problems, a multi-criteria quantum group decision-making model based on T2FNs and the CPT-TODIM method is developed in this paper by incorporating DMs’ behavioral risk perception as well as the interference effects among their opinions. The model involves several steps: First, T2FNs are employed to process linguistic evaluation information, thus better formalizing DMs’ judgments. Subsequently, the D-CRITIC method combined with Deng entropy is utilized for the calculation of criteria weights. Next, individual evaluation results are pooled via a quantum aggregation framework, which enables the exploration of interference effects among DMs grounded in a rigorous quantum theoretical perspective. The main contributions of this paper are as follows:

(1)

We employ a normal distribution mapping function to convert linguistic variables into T2FNs, thereby capturing DMs’ nonlinear sensitivity to extreme evaluations. In contrast to conventional linear mappings, this approach more accurately reflects DMs’ heightened response to boundary terms over midpoints, a characteristic often overlooked by traditional linear transformations. By generating T2FNs that align more closely with actual human perception, this transformation significantly enhances the model’s psychological realism and its applicability in complex real-world decision-making scenarios.

(2)

We propose an enhanced D-CRITIC framework by integrating Deng entropy to address the limitations of traditional methods in quantifying complex uncertainties. This approach leverages Deng entropy’s unique capability to measure the uncertainty arising from both conflicting evidence and inherent fuzziness within criteria evaluations. Unlike simpler metrics, this provides a more nuanced uncertainty assessment, which proves particularly advantageous in complex decision environments. Consequently, when dealing with the decision problems marked by high evidential conflict or contradictory indicators, our method offers a more accurate quantification of uncertainty, significantly improving the robustness and validity of the determined criteria weights.

(3)

We propose modeling the MCGDM process by integrating the CPT-TODIM method for individual evaluations with a QPT-based aggregation framework. This dual approach allows us to first capture individual DMs’ complex psychological behaviors and risk perceptions via the CPT-TODIM method and then explicitly model the mutual interference effects among these viewpoints during aggregation using QPT. This integration provides a more comprehensive and psychologically grounded understanding of group decision dynamics compared to methods that address only individual biases or assume simple independence in aggregation.

The structure of the remainder of the paper is outlined as follows: Section 2 presents a literature review. Section 3 introduces the prerequisite knowledge for the study, primarily encompassing the fundamental concepts of T2FNs, Deng entropy, and quantum decision theory. Section 4 develops and elaborates on a multi-criteria quantum group decision-making model based on T2FNs and CPT-TODIM. Section 5 demonstrates the practical application and effectiveness of the proposed model through a case study on the selection of a financial technology startup. Section 6 conducts sensitivity and comparative analyses to evaluate the model’s robustness and highlight the advantages of the proposed approach. Finally, Section 7 provides a summary of the study and outlines future research directions.

2. Literature Review

This section reviews research on type-2 fuzzy set theory, recent advances in QPT, and the investigation and application of the TODIM method in decision-making contexts.

2.1. Type-2 Fuzzy Set Theory

Given the unavoidable presence of uncertainty, seeking methods capable of accurately depicting the complexity associated with real-world phenomena becomes vital. Uncertainty is a fundamental trait within decision-making processes, often stemming from incomplete or erroneous datasets and conflicting knowledge. However, human cognitive abilities allow for negotiating and reasoning with such fuzziness [5]. To address the nuances associated with modeling uncertainty, Zadeh introduced T2FSs [5] as a means to handle higher levels of fuzziness where the membership itself is uncertain. Unlike T1FSs [4], T2FSs incorporate secondary memberships, providing enhanced modeling capabilities. Owing to the practical difficulties in defining arbitrary secondary membership functions, Mendel [6] proposed IT2FSs. For IT2FSs, the secondary membership is fixed at 1, which allows them to be effectively characterized by two T1FSs. This significantly simplifies the structure and computation associated with T2FSs, enhancing their practical applicability.

Due to their advantage in reducing computational complexity compared to T2FSs, IT2FSs have been increasingly applied, for instance, in modeling linguistic perceptions for online service satisfaction [37] and selecting industrial maintenance strategies using integrated TOPSIS and AHP methods [38]. Nevertheless, IT2FSs have limitations, as their simplified secondary membership may omit finer uncertainty aspects, and their interval-based computations remain relatively complex compared to simpler fuzzy representations. Aiming to address these shortcomings, Zhu et al. [7] introduced T2FNs along with their fundamental computational principles. T2FNs, primarily consisting of a primary membership degree and a secondary membership degree, offer a more compact and computationally efficient structure. Li et al. [33] further established a distance measure and defined a score function for T2FNs, enhancing their distinguishability. By providing advantages in describing both uncertainty and reliability with potentially simpler computations than IT2FSs, T2FNs are particularly well-adapted to decision-making contexts. However, existing applications of T2FNs predominantly assume linear linguistic transformations, which may not adequately capture the nonlinear nature of human perception, particularly regarding extreme evaluations—a gap this study addresses.

2.2. Quantum Probability Theory

As an emerging theory, QPT is founded upon quantum mechanics and mathematics [39]. Relative to classical probability theory, QPT furnishes an alternative probabilistic framework for decision modeling and has demonstrated success in addressing behaviors deemed contradictory or irrational from a classical viewpoint [27]. It offers a unified general decision theory, enabling a superior depiction of the uncertainty, fuzziness, and risk intrinsic to human decision-making processes. Trueblood et al. [40] introduced a quantum reasoning framework derived from QPT. This framework aimed to achieve different information orderings and provided enhanced coherence regarding interference effects. He et al. [23] proposed a quantum decision model to address MCGDM problems, extending the research on Bayesian networks (BNs) to accommodate belief interference among non-independent individuals. Wu et al. [41] integrated the generalized TODIM method with QPT to develop a decision framework that incorporates considerations of bounded rationality in DMs. Wang et al. [42] designed a trust aggregation method guided by quantum cognition, further integrating quantum theory with social network perspectives.

In MCGDM, QPT addresses the limitations of classical probabilistic models in capturing complex cognitive phenomena. Quantum-like Bayesian networks (QLBNs) use complex probability amplitudes to model intricate dependencies and interference effects among DMs’ preferences [43]. These interference effects highlight non-additive belief interactions, a key feature of QLBNs. Han et al. [44] enhanced explanations of subjective and behavioral factors using a QLBN-based MCGDM approach. Jiang et al. [28] analyzed the interference in consensus processes using a quantum cognition model. These developments underscore QPT’s strength in modeling DMs’ interactions, rendering it particularly well-suited for capturing interference effects within complex decision-making processes. Despite these advances, the integration of QPT with behavioral decision-making methods like CPT-TODIM remains underexplored, particularly in contexts involving fuzzy linguistic information—a research gap that motivates the present study.

2.3. The TODIM Method

The TODIM method, a characteristic behavioral multi-criteria decision-making (MCDM) approach first described by Gomes and Lima [16,17], provides a framework for capturing key subjective psychological behaviors of DMs, namely reference dependence, loss aversion, and diminishing sensitivity [18]. The method has received substantial attention in recent decades and has been widely adopted across diverse application areas. For instance, Cao et al. [45] proposed a complex q-rung orthopair fuzzy generalized TODIM method that incorporates weighted power geometric operators for the appraisal of food waste treatment technologies. Wei et al. [46] presented a TODIM method that incorporates hesitant linguistic term sets for assessing telecommunication service providers. Hong et al. [47] introduced a TODIM method that utilizes fuzzy evaluation principles coupled with the Shapley index to facilitate the selection of product recycling channels. The utility of the TODIM method extends widely to numerous other decision scenarios beyond these illustrative examples. These include occupational health and safety risk assessment [48], commercial insurance selection [49], debtor enforcement likelihood determination [50], and the appraisal of self-driving vehicles relevant to smart airport operations [51].

The CPT-TODIM method, developed by Tian et al. [20], integrates CPT into the traditional TODIM framework to enhance the MCGDM process by accounting for DMs’ psychological behaviors. This method offers significant advantages over classical decision models by incorporating cognitive biases such as reference dependence, loss aversion, and diminishing sensitivity, allowing for a more accurate representation of how DMs evaluate risks and alternatives under uncertainty [18,35]. This refined modeling of individual DMs’ psychological factors within MCGDM yields a nuanced understanding of their risk perception and decision-making behavior. However, existing CPT-TODIM applications primarily focus on individual psychological behaviors while neglecting group dynamics and interference effects among DMs. Furthermore, integrating this approach with QPT enables a more comprehensive consideration of psychological behaviors throughout the decision-making process, though such integration has not been systematically explored in conjunction with advanced fuzzy representations like T2FNs and enhanced weighting methods, presenting a significant opportunity for methodological advancement.

3. Preliminaries

This section outlines the fundamental concepts related to T2FNs, Deng entropy, and quantum decision theory. These concepts are discussed in greater detail in the subsections below.

3.1. Type-2 Fuzzy Numbers

Based on the research by Mendel and John [52], T2FSs are defined as follows:

Definition 1

([52]). For a T2FS X, its type-2 membership function is ${\mu }_X(a,u)$, where $a\in \mathcal{A}$ represents an element from the universe of discourse $\mathcal{A}$, and $u\in [0,1]$, as follows:

$$X=\left\{\left((a,u),{\mu }_X(a,u)\right)\mid \forall a\in \mathcal{A},\forall u\in [0,1]\right\},$$

(1)

where ${\mu }_X(a,u)\in [0,1]$. In this definition, u serves as a primary membership degree for a. Following this, ${\mu }_X(a,u)$ represents the secondary membership degree.

Building on T2FSs, Zhu et al. [7] introduced the foundational concept of T2FNs. The formal description for a T2FN pair is provided:

Definition 2

([7]). For a given T2FS X, every variable $a\in \mathcal{A}$ has an associated primary membership degree u and a secondary membership degree $\mu \left(u\right)$, which are typically represented as the pair $t\left(a\right)=(u,\mu (u\left)\right)$, or in a more concise form t. The first part, u, represents the degree of truth, and the second part, $\mu \left(u\right)$, represents the degree of reliability. Such a t is denoted as a T2FN.

To quantify the uncertainty in T2FNs, Zhu et al. [7] defined an uncertainty function that modeled judgment behavior using fast and slow thinking. Although such prior uncertainty measures provided a basis for understanding T2FNs, they often overlooked the interaction between these cognitive processes [7]. To better formalize this interplay and address this limitation, Yu et al. [34] introduced a new uncertainty measure, known as the d score, for T2FNs:

Definition 3

([34]). Given a T2FN $t=(u,\mu (u\left)\right)$, where $u\in [0,1]$ and $\mu \left(u\right)\in [0,1)$, the uncertain degree is computed as

$$s\left(t\right)=\left\{\begin{array}{cc}\pi {(1-u)}^{{\displaystyle \frac{\tau +\zeta }{1+\tau }}}{(1-\mu \left(u\right))}^{{\displaystyle \frac{\tau \zeta +1}{1+\tau }}},\hfill & if\mu \left(u\right)\in [0,1),\hfill \\ \pi (1-u),\hfill & if\mu \left(u\right)=1,\hfill \end{array}\right.$$

(2)

where τ denotes the relative importance of a primary membership with respect to secondary membership. ζ denotes the interaction coefficient between fast thinking, which is captured by the primary membership degree and slow thinking, which is captured by the secondary membership degree. According to the research by [34], in this paper, τ is set to 0 and ζ is set to 1.

To facilitate the subsequent computations, $s\left(t\right)$ is normalized:

$$s^{\prime }\left(t\right)={\displaystyle \frac{s\left(t\right)}{maxs\left(t\right)}}.$$

(3)

Figure 1 depicts embedded uncertainties where the horizontal dimension corresponds to $1-u$ and the vertical dimension corresponds to $1-\mu \left(u\right)$. These complementary dimensions enclose an elliptical region that depicts the uncertain degree.

T2FN is expressed in combination with linguistic terms:

Definition 4

([7]). A linguistic type-2 fuzzy number (L-T2FN), denoted as $t^L$, is a T2FN where both the primary membership and the secondary membership are characterized by linguistic terms.

Let $S_1=\left\{s_l^1\right|l=1,2,\dots ,L\}$ and $S_2=\left\{s_p^2\right|p=1,2,\dots ,L\}$ represent two finite and ordered discrete linguistic term sets. Therefore, an L-T2FN is denoted by $t^L=(s_l^1,s_p^2)$.

Let f be the primary mapping function such that $f\left(s_l^1\right)\to u$, and g be the secondary mapping function such that $g\left(s_p^2\right)\to \mu \left(u\right)$. Since the secondary mapping function operates on the same principles, for brevity, only the primary mapping function is outlined here.

Primary mapping function characterized by normal distribution is as follows [7]:

$$f\left(s_l^1\right)=\left\{\begin{array}{cc}0.5+Y\left(0\right)-Y\left(\left(l-{\displaystyle \frac{L+1}{2}}\right)\sigma \right),\hfill & \mathrm{if}l>{\displaystyle \frac{L+1}{2}},\hfill \\ 0.5-\left(Y\left(0\right)-Y\left(\left(l-{\displaystyle \frac{L+1}{2}}\right)\sigma \right)\right),\hfill & \mathrm{if}l<{\displaystyle \frac{L+1}{2}},\hfill \\ 0.5,\hfill & \mathrm{if}l={\displaystyle \frac{L+1}{2}},\hfill \end{array}\right.l=1,2,\dots ,L,$$

(4)

where $Y(·)$, acting as the probability density function, describes a random variable distributed normally $N(0,\sigma )$, when $f\left(s_l^1\right)\in (0,1)$. If $f\left(s_l^1\right)\in (0,1)$ for all $l=1,2,\dots ,L$, the value of $\sigma$ is determined to be greater than $0.69$ for $L=5$, and greater than $0.79$ for $L=7$.

Zhu et al. [7] proposed two mapping functions for T2FNs: uniform and normal distribution. Compared to the uniform distribution, the normal distribution mapping function more effectively characterizes the complex and asymmetric phenomena prevalent in the real world, owing to its nonlinear nature. Accordingly, this study adopts the normal distribution mapping function for its modeling.

The relationship between language terms and primary membership degrees is represented by Figure 2.

Zhu et al. [7] defined arithmetic rules for T2FNs using the t-norm $Z\left(xy\right)=ab$ and t-conorm $R(a,b)=a+b-ab$. For T2FNs $t=(u,\mu \left(u\right)),t_1=(u_1,{\mu }_1\left(u\right))$, and $t_2=(u_2,{\mu }_2\left(u\right))$, the operational rules are as follows:

(1)

Addition: $t_1\oplus t_2=(R(u_1,u_2),R({\mu }_1\left(u\right),{\mu }_2\left(u\right))$.

(2)

Multiplication: $t_1\otimes t_2=(Z(u_1,u_2),Z({\mu }_1\left(u\right),{\mu }_2\left(u\right)).$

(3)

Scalar-multiplication: $\gamma t=(1-{(1-u)}^{\gamma },1-{(1-\mu \left(u\right))}^{\gamma }),\gamma >0$.

(4)

Power-multiplication: $t^{\gamma }=(u^{\gamma },\mu {\left(u\right)}^{\gamma }),\gamma >0$.

Similarly, the following properties hold under the defined operational rules:

(1)

$t_1\oplus t_2=t_2\oplus t_1$;

(2)

$t_1\otimes t_2=t_2\otimes t_1$;

(3)

$(t_1\oplus t_2)\oplus t=t_1\oplus (t_2\oplus t)$;

(4)

$(t_1\otimes t_2)\otimes t=t_1\otimes (t_2\otimes t)$.

Let $t_i=(u_i,{\mu }_i\left(u\right))$ denotes a set of T2FNs. Their weights are ${\omega }_i$, where ${\omega }_i\in [0,1]$ for $i=1,2,\dots ,m$.

Definition 5

([7]). Let Ω be the set of all T2FNs. The type-2 fuzzy weighted averaging (T2WA) operator is defined as follows:

$$\mathrm{T}2{\mathrm{WA}}_{\omega }(t_1,t_2,\dots ,t_m)={\omega }_1{t}_1\oplus {\omega }_2{t}_2\oplus \dots \oplus {\omega }_m{t}_m.$$

(5)

The aggregation result theorem for the operator is

Theorem 1

([7]). The calculation for aggregating T2FNs using the T2WA operator proceeds as follows:

$$\mathrm{T}2{\mathrm{WA}}_{\omega }\left(t_1,t_2,\dots ,t_m\right)=\left(1-\prod _{i=1}^m{\left(1-u_i\right)}^{{\omega }_i},1-\prod _{i=1}^m{\left(1-{\mu }_i\left(u\right)\right)}^{{\omega }_i}\right).$$

(6)

3.2. Deng Entropy

Quantifying uncertainty is essential for decision making, particularly in evidence theory where information is often conflicting and fuzzy. Deng entropy offers a robust approach by extending Shannon entropy to measure uncertainty directly from the belief structure within a frame of discernment [14], effectively handling such imprecise or conflicting evidence.

Definition 6

([15]). For the frame of discernment $\Xi =\{{\xi }_1,{\xi }_2,\dots ,{\xi }_N\}$ equipped with a mass function n, Deng entropy is defined as follows:

$$E_D\left(n\right)=-\sum _{\begin{array}{c}Q\in 2^{\Xi }\end{array}}n\left(Q\right){log}_2\left({\displaystyle \frac{n\left(Q\right)}{2^{\left|Q\right|}-1}}\right),$$

(7)

where Q is the focus element.

3.3. Quantum Decision Theory

3.3.1. Quantum Probability Theory

The framework of QPT, differing from classical probability theory, considers all possible events to be subspaces of a Hilbert space. The Hilbert space itself is an extension of Euclidean space, characterized as a complex vector space that has an orthogonal basis $D=\left\{\right|D_1\rangle ,\dots ,|D_n\rangle \}$ spanning it. Furthermore, in this Hilbert space, quantum superposition states $|\psi \rangle$ define all events, and these states can be formulated as linear combinations of the orthogonal basis vectors [53].

$$\begin{array}{cc}\hfill |\psi \rangle & =\langle D_1|\psi \rangle |D_1\rangle +\langle D_2|\psi \rangle |D_2\rangle +\dots +\langle D_n|\psi \rangle |D_n\rangle \hfill \\ \hfill & ={\alpha }_1|D_1\rangle +{\alpha }_2|D_2\rangle +\dots +{\alpha }_n|D_n\rangle \hfill \\ \hfill & ={\chi }_1{e}^{i{\theta }_1}|D_1\rangle +{\chi }_2{e}^{i{\theta }_2}|D_2\rangle +\dots +{\chi }_n{e}^{i{\theta }_n}|D_n\rangle .\hfill \end{array}$$

(8)

Describing the probability amplitude of state $D_k$ ($k=1,2,\dots ,n$) utilizes the complex number ${\alpha }_k$, ${\chi }_k{e}^{i{\theta }_k}$ ($k=1,2,\dots ,n$), given that ${\left({\chi }_k\right)}^2\in [0,1]$ and ${\sum }_{k=1}^n{\left({\chi }_k\right)}^2=1$. The phase of this complex wave function is denoted by $e^{i{\theta }_k}$. In line with Born’s rule [24], ${\left({\chi }_k\right)}^2$ directly corresponds to the observed probability of that state:

$$\begin{array}{cc}\hfill P\left(D_k\right)& =|{\alpha }_k{|}^2={\left|{\chi }_k{e}^{i{\theta }_k}\right|}^2\hfill \\ \hfill & =\left({\chi }_k{e}^{i{\theta }_k}\right)\times \left({\chi }_k{e}^{-i{\theta }_k}\right)\hfill \\ \hfill & ={\chi }_k{e}^{i{\theta }_k}\times {\chi }_k{e}^{-i{\theta }_k}={\left({\chi }_k\right)}^2(k=1,2,\dots ,n).\hfill \end{array}$$

(9)

3.3.2. Decision Making in Quantum Probability Theory

According to classical Bayesian decision theory [54], decision making may be understood as a sequential decision process.

As shown in Figure 3, a single path initiates at state A, continues to B, and finishes at C. Using conditional probabilities, it is computed as follows:

$$P(A\to C)=P\left(A\right)·P\left(B\right|A)·P(C\left|B\right).$$

(10)

For the single path described above, the probability within the QPT framework is given by

$$P(A\to C)=|{\psi }_A{|}^2·|{\psi }_{B|A}{|}^2·{\left|{\psi }_{C|A}\right|}^2=P\left(A\right)·P\left(B\right|A)·P\left(C\right|B).$$

(11)

This is shown in Figure 4a as a two-path graph. Beginning at state A, proceeding to either B or C, and ultimately reaching D. The probability is calculated using the law of full probability as

$$P(A\to D)=P\left(A\right)·P\left(B\right|A)·P(D\left|B\right)+P\left(A\right)·P\left(C\right|A)·P(D\left|C\right).$$

(12)

Because intermediate processes are unobservable in the QPT framework as depicted in Figure 4b, interference phenomena can occur among multiple possible paths. Hence, the final state can be viewed as a superposition of these distinct paths. The computation of the transition probability from an initial state A to a final state D requires summing the probability amplitudes associated with all potential paths from A to D to obtain the total probability amplitude. The probability is then derived by squaring the magnitude of this total amplitude:

$$\begin{array}{cc}\hfill P(A\to D)=& |{\psi }_A·{\psi }_{B|A}·{\psi }_{D|B}+{\psi }_A·{\psi }_{C|A}·{\psi }_{D|C}{|}^2\hfill \\ \hfill =& |{\psi }_A·{\psi }_{B|A}·{\psi }_{D|B}{|}^2+{|{\psi }_A·{\psi }_{C|A}·{\psi }_{D|C}|}^2\hfill \\ \hfill & +2·|{\psi }_A·{\psi }_{B|A}·{\psi }_{D|B}|·|{\psi }_A·{\psi }_{C|A}·{\psi }_{D|C}|·cos\left(\theta \right).\hfill \end{array}$$

(13)

The final term in Equation (13) represents the interference term, which results from the interaction between two independent paths. When these two paths are considered individually rather than collectively, QPT becomes equivalent to the classical full probability.

$$\begin{array}{cc}\hfill P(A\to D)=& |{\psi }_A·{\psi }_{B|A}·{\psi }_{D|B}{|}^2+{|{\psi }_A·{\psi }_{C|A}·{\psi }_{D|C}|}^2\hfill \\ \hfill =& P\left(A\right)·P\left(B\right|A)·P(D\left|B\right)+P\left(A\right)·P\left(C\right|A)·P(D\left|C\right).\hfill \end{array}$$

(14)

QPT provides a more effective framework for capturing individuals’ subjective psychological biases and subconscious factors. By inherently accounting for the cognitive interference that inevitably occurs during decision-making processes, QPT is particularly well-suited for the modeling and analysis conducted herein.

4. Multi-Criteria Quantum Group Decision Model Integrating T2FNs and the CPT-TODIM Method

A typical MCGDM problem is formulated with m alternatives $x_i(i=1,2,\dots ,m)$, n criteria $c_j(j=1,2,\dots ,n)$, and z DMs $d_s(s=1,2,\dots ,z)$. The criteria weights and DMs weights are denoted by $w={(w_1,w_2,\dots ,w_n)}^T$ and $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_z)}^T$, respectively.

4.1. Euclidean Distance Measure and Score Function for T2FNs

Euclidean distance is a fundamental and widely recognized metric, highly valued for its intuitive geometric interpretation and effectiveness in various domains. However, despite its merits, research specifically addressing Euclidean distance measures for T2FNs remains notably limited [33]. To bridge this gap and be inspired by the innovative distance metrics proposed in [55], a novel Euclidean distance measure tailored for T2FNs is introduced in this paper. Building upon this newly defined distance, a score function is subsequently proposed. This function, derived from the proposed distance measure, is designed to facilitate robust ranking and decision-making applications, thereby significantly enhancing the model’s practical applicability.

Step 1:

The DMs’ evaluation information, expressed using L-T2FNs, is collected. The evaluation matrix for DM $d_s$ is given by

$$d_s=\left(\begin{array}{cccc}{t}_{11}^s& t_{12}^s& \dots & t_{1n}^s\\ t_{21}^s& t_{22}^s& \dots & t_{2n}^s\\ ⋮& ⋮& \ddots & ⋮\\ t_{m1}^s& t_{m2}^s& \dots & t_{mn}^s\end{array}\right).$$

(15)

Step 2:

Convert the L-T2FNs matrices into T2FNs matrices. The evaluation matrix for DM $d_s$ after transformation is as follows

$$T^s=\left(\begin{array}{cccc}{\overline{t}}_{11}^s& {\overline{t}}_{12}^s& \dots & {\overline{t}}_{1n}^s\\ {\overline{t}}_{21}^s& {\overline{t}}_{22}^s& \dots & {\overline{t}}_{2n}^s\\ ⋮& ⋮& \ddots & ⋮\\ {\overline{t}}_{m1}^s& {\overline{t}}_{m2}^s& \dots & {\overline{t}}_{mn}^s\end{array}\right).$$

(16)

Based on the new uncertain degree, the Euclidean distance measure for T2FNs is defined:

Definition 7.

For any two T2FNs, $t_1=(u_1,{\mu }_1)$ and $t_2=(u_2,{\mu }_2)$, the new Euclidean distance measure between them is formulated as

$$D(t_1,t_2)=\sqrt{\varrho [{\displaystyle \frac{1}{2}}{(u_1-u_2)}^2+{({\mu }_1-{\mu }_2)}^2]+(1-\varrho ){(s^{\prime }\left(t_1\right)-s^{\prime }\left(t_2\right))}^2},$$

(17)

where $s^{\prime }\left(t_1\right)$ and $s^{\prime }\left(t_2\right)$ represent the new uncertain degrees of $t_1$ and $t_2$, respectively. For the ease of handling in this scenario, the parameter ϱ is designated as $0.5$ [33]. The distance measure $D(t_1,t_2)$ satisfies the following properties:

(1)

$0\le D(t_1,t_2)\le 1;$

(2)

$D(t_1,t_2)=0$ if and only if $t_1=t_2;$

(3)

$D(t_1,t_2)=D(t_2,t_1).$

Definition 8.

Combining Equations (2) and (17), the new score function can be expressed as

$$E\left(t_{j_1}\right)=\sum _{i_1=1}^m\sum _{i_2=1}^m(\phi /m)\times |s^{\prime }\left(t_{j_1{i}_1}\right)-s^{\prime }\left(t_{i_2{j}_1}\right)|+(1-\phi )/m\times D\left(t_{i_1{i}_2}^{j_1}\right),$$

(18)

where the variable φ corresponds to the score function threshold.

Step 3:

The comprehensive score matrix for DM $d_s$ can be obtained using Equation (18) as follows:

$$A^s=\left(\begin{array}{cccc}{a}_{11}^s& a_{12}^s& \dots & a_{1n}^s\\ a_{21}^s& a_{22}^s& \dots & a_{2n}^s\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}^s& a_{m2}^s& \dots & a_{mn}^s\end{array}\right).$$

(19)

4.2. Determining the Criteria Weights

The high precision with which Deng entropy measures uncertainty provides significant advantages when applied to multi-criteria decision analysis. Compared to Shannon entropy, which is limited to relying on a single event, it can more finely characterize the multi-level, nonlinear, uncertain relationships in complex systems [56]. The correction term of Deng entropy enhances its robustness against extreme data distributions, effectively suppressing weight polarization [14]. Therefore, motivated by these advantages, Deng entropy is utilized to develop a new CRITIC method.

The following are the main steps for calculating criteria weights using the D-CRITIC method integrated with Deng entropy.

Step 1:

Normalize the decision matrix, which includes benefit and cost criteria, using Equations (20) and (21):

$${\tilde{a}}_{ij}={\displaystyle \frac{a_{ij}-a_i^{min}}{a_i^{max}-a_i^{min}}},i=1,2,\dots ,m;j=1,2,\dots ,n,$$

(20)

$${\tilde{a}}_{ij}={\displaystyle \frac{a_{ij}-a_i^{max}}{a_i^{min}-a_i^{max}}},i=1,2,\dots ,m;j=1,2,\dots ,n,$$

(21)

where ${\tilde{a}}_{ij}$ refers to the normalized result of the i-th alternative’s evaluation relative to the j-th criterion in the decision matrix. $a_j^{max}$ is the maximum value in $\{a_{1j},a_{2j},\dots ,a_{mj}\}$, and $a_j^{min}$ is the minimum value in this set.

Step 2:

Calculate the Deng entropy $E_j$ for the j-th criterion:

$$E_j=-\left[\sum _{i=1}^m{v}_j{y}_{ij}{log}_2\left(v_j{y}_{ij}\right)\right]-(1-v_j){log}_2\left({\displaystyle \frac{1-v_j}{2^m-1}}\right),$$

(22)

where $y_{ij}$ is calculated as $y_{ij}={\displaystyle \frac{{\tilde{a}}_{ij}}{{\sum }_{i=1}^m{\tilde{a}}_{ij}}}$ and represents the standardized value for criterion $c_j$. Furthermore, $v_j$ is defined as $v_j={\displaystyle \frac{{\kappa }_j}{0.5}}$, where ${\kappa }_j$ stands for the standard deviation of the evaluation values related to criterion $c_j$. This coefficient $v_j$ indicates the degree of dispersion among the evaluations for different alternatives under the specific criterion $c_j$.

Step 3:

Normalize the Deng entropy $E_j$ of criterion $c_j$:

$$E_j^{\prime }={\displaystyle \frac{E_j}{{log}_2(2^m-1)}}.$$

(23)

Step 4:

Determine the average absolute deviation $A_{A{D}_j}$ for each criterion:

$$A_{A{D}_j}={\displaystyle \frac{1}{m}}\sum _{i=1}^m|{\tilde{a}}_{ij}-{\overline{a}}_j|,$$

(24)

where ${\overline{a}}_j$ is the standardized mean value of the criterion.

Step 5:

Calculate the distance correlation $dCor(c_j,c_{j^{\prime }})$ between criteria $c_j$ and $c_{j^{\prime }}$:

$$dCor(c_j,c_{j^{\prime }})={\displaystyle \frac{dCov(c_j,c_{j^{\prime }})}{\sqrt{dVar\left(c_j\right)·dVar\left(c_{j^{\prime }}\right)}}},$$

(25)

where the distance covariance for $c_j$ is $dVar\left(c_j\right)=dCov(c_j,c_j)$, and similarly, the distance variance for $c_{j^{\prime }}$ is $dVar\left(c_{j^{\prime }}\right)=dCov(c_{j^{\prime }},c_{j^{\prime }})$. One can consult [8] for the details concerning the distance correlation computation involving criteria $c_j$ and $c_{j^{\prime }}$.

Step 6:

Determine the information content $I_j$ associated with criterion $c_j$ using Equation (26):

$$I_j=\left((1-E_j^{\prime })+A_{A{D}_j}\right)\sum _{j^{\prime }=1}^n\left(1-d\mathrm{Cor}(c_j,c_{j^{\prime }})\right),j=1,2,\dots ,n.$$

(26)

Integrating Deng entropy into the D-CRITIC framework enhances criteria weights determination in MCGDM by managing uncertainties from conflicting criteria evaluations. Unlike Shannon entropy, Deng entropy’s cardinality-based correction reduces weight polarization. It mitigates D-CRITIC’s sensitivity to outliers, a limitation of standard deviation-based methods, ensuring stable and robust weights.

Step 7:

The criteria weights are determined by employing Equation (27):

$$w_j={\displaystyle \frac{I_j}{{\sum }_{j=1}^n{I}_j}},$$

(27)

where $w_j$ is the weight of criterion $c_j$.

4.3. The T2FNs-CPT-TODIM Method

After the evaluation information is gathered, the subsequent derivation of individual evaluation results relies on the CPT-TODIM method. This method is specifically designed to handle the MCGDM process described in the previous section. The implementation steps are as follows:

Step 1:

Normalize the decision matrix $A^s={\left(a_{ij}\right)}_{m\times n}$ to yield the normalized matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$:

$${\overline{a}}_{ij}^s=\left\{\begin{array}{cc}{a}_{ij}^s,\hfill & \mathrm{if}{c}_j\mathrm{is}\mathrm{benefit}\mathrm{criterion},\hfill \\ -a_{ij}^s,\hfill & \mathrm{if}{c}_j\mathrm{is}\cos \mathrm{t}\mathrm{criterion}.\hfill \end{array}\right.$$

(28)

Step 2:

Compute the transformed probability weight relating $x_i$ to $x_k$ for DM $d_s$ using Equation (29):

$${\vartheta }^s\left(w_j\right)=\left\{\begin{array}{cc}{\displaystyle \frac{w_j^{\gamma }}{{(w_j^{\gamma }+{(1-w_j)}^{\gamma })}^{\frac{1}{\gamma }}}},\hfill & \mathrm{if}{a}_{ij}^s\ge a_{kj}^s,\hfill \\ {\displaystyle \frac{w_j^{\delta }}{{(w_j^{\delta }+{(1-w_j)}^{\delta })}^{\frac{1}{\delta }}}},\hfill & \mathrm{if}{a}_{ij}^s<a_{kj}^s,\hfill \end{array}\right.$$

(29)

where the parameters $\delta$ and $\gamma$ denote both the modifications in declining sensitivity across profit/loss scenarios and the precise curvature attributed to the weighting function.

Step 3:

Compute the relative weight ${\overline{\vartheta }}_{ikj}^s\left(w_j\right)$ associated with $x_i$ relative to $x_k$ using Equation (30):

$${\overline{\vartheta }}_{ikj}^s\left(w_j\right)={\displaystyle \frac{{\vartheta }_{ikj}^s\left(w_j\right)}{max{\vartheta }_{ikj}^s\left(w_j\right)}},j\in y,\forall (i,k),$$

(30)

where ${\vartheta }_{ikj}^s\left(w_j\right)$ indicates the transformed probability weight; this weight is for the j-th criterion of alternative $x_i$ according to DM $d_s$.

Step 4:

Determine the relative prospect dominance degree of each alternative $x_i$ with respect to $d_s$ using Equation (31).

$${\varphi }_j^s(x_i,x_k)=\left\{\begin{array}{cc}{\displaystyle \frac{{\overline{\vartheta }}_{ikj}^s{(a_{ij}^s-a_{kj}^s)}^{\alpha }}{{\sum }_{j=1}^m{\overline{\vartheta }}_{ikj}^s}},\hfill & \mathrm{if}{a}_{ij}^s>a_{kj}^s,\hfill \\ 0,\hfill & \mathrm{if}{a}_{ij}^s=a_{kj}^s,\hfill \\ {\displaystyle \frac{-\lambda \left({\sum }_{j=1}^m{\overline{\vartheta }}_{ikj}^s{(a_{kj}^s-a_{ij}^s)}^{\eta }\right)}{{\overline{\vartheta }}_{ikj}^s}},\hfill & \mathrm{if}{a}_{ij}^s<a_{kj}^s,\hfill \end{array}\right.$$

(31)

where $\alpha$, $\lambda$, and $\eta$ denote the parameters.

Step 5:

For each alternative $x_i$, its relative prospect dominance degree values relative to every other alternative $x_k$, considering all criteria from $d_s$, are presented in an $m\times n$ dominance matrix, as follows:

$${\Phi }^s\left(x_i\right)={\left(\begin{array}{cccc}{\varphi }_1^s(x_i,x_1)& {\varphi }_2^s(x_i,x_1)& \dots & {\varphi }_n^s(x_i,x_1)\\ {\varphi }_1^s(x_i,x_2)& {\varphi }_2^s(x_i,x_2)& \dots & {\varphi }_n^s(x_i,x_2)\\ ⋮& ⋮& \ddots & ⋮\\ {\varphi }_1^s(x_i,x_m)& {\varphi }_2^s(x_i,x_m)& \dots & {\varphi }_n^s(x_i,x_m)\end{array}\right)}_{m\times n}.$$

(32)

By summing all elements within ${\Phi }^s\left(x_i\right)$, the global prospect dominance degree ${\varphi }^s\left(x_i\right)$ for each alternative $x_i$ from $d_s$ can be determined, where

$${\varphi }^s\left(x_i\right)=\sum _{k=1}^m\sum _{j=1}^n{\varphi }_j^s(x_i,x_k).$$

(33)

4.4. Aggregating the Individual Evaluation Result in Quantum Framework

Following the acquisition of individual assessment outcomes, reasonable information fusion techniques are essential for consolidating these evaluations. Consequently, this subsection details a quantum-based aggregation model specifically designed to account for the interference effects observed across DMs’ perspectives, as shown in Figure 5. The steps involved are outlined below:

Step 1:

The weights of the DMs are set as the first layer probabilities of the BN, where

$$P\left(d_s\right)={\omega }_s,s=1,2,\dots ,z.$$

(34)

Step 2:

The global prospect dominance degree, $\varphi \left(x_i\right)$, for each alternative $x_i$ associated with $d_s$, is determined using the CPT-TODIM method. Subsequently, $\varphi \left(x_i\right)$ is converted into exponent-based conditional probabilities:

$$P\left(x_i\right|d_s)={\displaystyle \frac{e^{\varphi \left(x_i\right)}}{{\sum }_i{e}^{\varphi \left(x_i\right)}}}.$$

(35)

Step 3:

In QPT, the probability of $x_i$ is determined by squaring the amplitude that corresponds to the cumulative summation of amplitude probabilities for each available path, calculated as follows:

$$P\left(x_i\right)={\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^z{\psi }_{d_s}{\psi }_{x_i|d_s}{e}^{i{\theta }_s}\right|}^2={\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^z\sqrt{P\left(d_s\right)}\sqrt{P\left(x_i\right|d_s)}{e}^{i{\theta }_s}\right|}^2,i=1,2,\dots ,m,$$

(36)

where

$$\aleph =\sum _{i=1}^n{\left|\sum _{s=1}^z{\psi }_{d_s}{\psi }_{x_i|d_s}{e}^{i{\theta }_s}\right|}^2=\sum _{i=1}^n{\left|\sum _{s=1}^z\sqrt{P\left(d_s\right)}\sqrt{P\left(x_i\right|d_s)}{e}^{i{\theta }_s}\right|}^2.$$

(37)

When Equation (36) undergoes expansion, we can obtain

$$\begin{array}{cc}\hfill P\left(x_i\right)=& {\displaystyle \frac{1}{\aleph }}[\sum _{s=1}^{z}P\left(d_s\right)P\left(x_i\right|d_s)\hfill \\ \hfill & +2\sum _{s=1}^{z-1}\sum _{s^{\prime }=s+1}^z\sqrt{P\left(d_s\right)P\left(x_i\right|d_s)P\left(d_{s^{\prime }}\right)P\left(x_i\right|d_{s^{\prime }})}·cos({\theta }_s-{\theta }_{s^{\prime }})].\hfill \end{array}$$

(38)

Step 4:

Finally, the quantum interference terms in Equation (38) are determined using method from Section 4.5. Subsequently, the final probability for each alternative is computed, which then allows for the establishment of a final ranking under the quantum framework.

4.5. Determination of the Quantum Interference Term

The interference effects occurring among DMs are indicated by the final term in Equation (38). If the opinion information supplied by DMs $d_s$ and $d_{s^{\prime }}$ is known, then ${\theta }_s$ and ${\theta }_{s^{\prime }}$ correspond to the respective two phases. We define ${\beta }_{s{s}^{\prime }}=|{\theta }_s-{\theta }_{s^{\prime }}|$ as the absolute difference between ${\theta }_s$ and ${\theta }_{s^{\prime }}$, ensuring ${\beta }_{s{s}^{\prime }}\in [0,2\pi ]$. Within Figure 6, three representative phases are shown below:

(1)

When $\beta ={\displaystyle \frac{\pi }{2}}$ or $\beta ={\displaystyle \frac{3\pi }{2}}$, the DMs are considered independent, meaning interference is null (as $cos\beta =0$, the interference term is 0).

(2)

When $\beta =0$ or $\beta =2\pi$, the DMs’ opinions are subject to purely positive influence, resulting in a disjunctive aggregation.

(3)

When $\beta =\pi$, a purely negative influence is exerted on opinions, leading to a conjunctive aggregation.

Considering the similarity heuristic method [23,53,57], the procedure for establishing $\beta$ is outlined below:

Step 1:

Based on the opinion information supplied by DMs $d_s$ and $d_{s^{\prime }}$, vectors $\overrightarrow{d_s}$ and $\overrightarrow{d_{s^{\prime }}}$ are constructed, followed by the calculation of the difference $\overrightarrow{d_{s{s}^{\prime }}}$.

$$\overrightarrow{d_{s{s}^{\prime }}}=\overrightarrow{d_s}-\overrightarrow{d_{s^{\prime }}},$$

(39)

where

$$\overrightarrow{d_s}={({\psi }_{d_s}{\psi }_{x_1|d_s},{\psi }_{d_s}{\psi }_{x_2|d_s},\dots ,{\psi }_{d_s}{\psi }_{x_m|d_s})}^T,$$

(40)

$$\overrightarrow{d_{s^{\prime }}}={({\psi }_{d_{s^{\prime }}}{\psi }_{x_1|d_{s^{\prime }}},{\psi }_{d_{s^{\prime }}}{\psi }_{x_2|d_{s^{\prime }}},\dots ,{\psi }_{d_{s^{\prime }}}{\psi }_{x_m|d_{s^{\prime }}})}^T.$$

(41)

Step 2:

The cosine theorem is utilized to calculate the angle between vectors $\overrightarrow{d_s}$, $\overrightarrow{d_{s^{\prime }}}$ and $\overrightarrow{d_{s{s}^{\prime }}}$, according to the following formulation:

$$\begin{array}{cc}\hfill {\theta }_{\overrightarrow{d_s}}& ={cos}^{-1}\left({\displaystyle \frac{\Vert \overrightarrow{d_{s^{\prime }}}{\Vert }^2+\Vert \overrightarrow{d_{s{s}^{\prime }}}{\Vert }^2-{\Vert \overrightarrow{d_s}\Vert }^2}{2\Vert \overrightarrow{d_{s^{\prime }}}\Vert \Vert \overrightarrow{d_{s{s}^{\prime }}}\Vert }}\right),\hfill \\ \hfill {\theta }_{\overrightarrow{d_{s^{\prime }}}}& ={cos}^{-1}\left({\displaystyle \frac{\Vert \overrightarrow{d_s}{\Vert }^2+\Vert \overrightarrow{d_{s{s}^{\prime }}}{\Vert }^2-{\Vert \overrightarrow{d_{s^{\prime }}}\Vert }^2}{2\Vert \overrightarrow{d_s}\Vert \Vert \overrightarrow{d_{s{s}^{\prime }}}\Vert }}\right),\hfill \\ \hfill {\theta }_{\overrightarrow{\beta }}& ={cos}^{-1}\left({\displaystyle \frac{\Vert \overrightarrow{d_s}{\Vert }^2+\Vert \overrightarrow{d_{s^{\prime }}}{\Vert }^2-{\Vert \overrightarrow{d_{s{s}^{\prime }}}\Vert }^2}{2\Vert \overrightarrow{d_s}\Vert \Vert \overrightarrow{d_{s^{\prime }}}\Vert }}\right),\hfill \end{array}$$

(42)

where ${\theta }_{\overrightarrow{d_s}}$, ${\theta }_{\overrightarrow{d_{s^{\prime }}}}$ and ${\theta }_{\overrightarrow{\beta }}$ represent the angles corresponding to the vectors $\overrightarrow{d_s}$, $\overrightarrow{d_{s^{\prime }}}$ and $\overrightarrow{d_{s{s}^{\prime }}}$.

Step 3:

Let the similarity measure be denoted by $ϵ$. Consequently, the interference angle ${\beta }_{s{s}^{\prime }}$ can be calculated as

$${\beta }_{s{s}^{\prime }}=\left\{\begin{array}{cc}0,\hfill & ϵ\le -2,\hfill \\ {\displaystyle \frac{\pi (1+0.5ϵ)}{2}},\hfill & -2\le ϵ\le 2,\hfill \\ \pi ,\hfill & ϵ\ge 2,\hfill \end{array}\right.$$

(43)

where $ϵ$ is computed as

$$ϵ={\displaystyle \frac{{\theta }_{\overrightarrow{\beta }}}{{\theta }_{\overrightarrow{d_s}}}}-{\displaystyle \frac{{\theta }_{\overrightarrow{d_{s^{\prime }}}}}{{\theta }_{\overrightarrow{d_s}}}}.$$

(44)

4.6. The Procedure for Multi-Criteria Quantum Group Decision Method

The proposed multi-criteria quantum decision-making model, developed using T2FNs and the CPT-TODIM method, is detailed in the following sequential steps, and its overall framework is illustrated in the flowchart in Figure 7.

Step 1:

Language assessment information from DMs is collected and transformed into T2FNs using Equation (4).

Step 2:

The decision matrices of DMs are transformed into comprehensive score matrices by means of Equation (18).

Step 3:

Criteria weights are determined using the D-CRITIC method combined with Deng entropy (Equation (27)).

Step 4:

The weights of DMs are set in the first layer of the BN.

Step 5:

The overall dominance value for alternative $x_i$ from $d_s$ is calculated using Equations (28)–(33).

Step 6:

Using Equation (35), the overall prospect dominance degree of $d^s$ with respect to $x_i$ is converted into a conditional probability and assigned to the second layer of the BN.

Step 7:

The probability of the alternative $x_i$ is calculated according to Equation (36).

Step 8:

The ranking of alternatives is determined by their calculated probabilities.

5. Case Study

This section describes the use of the proposed model for investment decision making in the domain of FinTech startups, thereby showcasing the proposed model’s effectiveness.

5.1. Problem Description

According to a 2025 investment plan, a venture capital fund plans to allocate USD 10 million to the FinTech sector. The objective is to select the most investable enterprise from four candidate startups. The four alternatives are ➀ $x_1$: Intelligent credit risk control platform, ➁ $x_2$: Quantitative investment robo-advisor, ➂ $x_3$: Intelligent insurance pricing system, and ➃ $x_4$: RegTech compliance platform. Four DMs, denoted as $d_1,d_2,d_3$, and $d_4$, participate in the decision process. The backgrounds of these DMs are shown in

Table 1. The composition and backgrounds of the DMs group for FinTech startup evaluation.

DM ID	Field of Expertise	Qualifications	Experience (Years)
$d_1$	Venture capital, FinTech investment	MBA (Finance), CFA Charterholder	15+
$d_2$	Software engineering, system architecture	Ph.D. in computer science	10+
$d_3$	FinTech entrepreneurship, product development	B.Sc. in business, serial entrepreneur	12+
$d_4$	Corporate law, FinTech regulation	LL.M. (Master of Laws), Bar Admission	8+

. Given their diverse yet complementary expertise, and to ensure a balanced contribution from each unique perspective without prior bias towards any specific viewpoint, the DMs are assigned equal weights. The vector $\omega ={(0.25,0.25,0.25,0.25)}^T$ specifies their respective weights. Through a collaborative discussion, the DMs established six criteria based on which they evaluate the four alternatives. These criteria are ➀ $c_1$: Technological innovation, ➁ $c_2$: Business model feasibility, ➂ $c_3$: Market potential, ➃ $c_4$: Team capability, ➄ $c_5$: Financial status, and ➅ $c_6$: Risk factors.

5.2. Decision-Making Process

For a more objective identification of the optimal FinTech startup, the principal steps in the decision-making process are outlined below:

Step 1:

Each of the four alternatives is evaluated by four DMs according to the linguistic term sets $S_1$ = {VL, L, M, H, VH} and $S_2$ = {unknown, maybe, usually, likely, very likely}. The specific evaluation language chosen is documented in

Table 2. The linguistic evaluation matrix.

DMs	Alternatives	Criteria
		${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$	${\mathit{c}}_{\mathbf{4}}$	${\mathit{c}}_{\mathbf{5}}$	${\mathit{c}}_{\mathbf{6}}$
$d_1$	$x_1$	(H, likely)	(VH, very likely)	(M, very likely)	(M, likely)	(H, likely)	(H, likely)
	$x_2$	(VL, very likely)	(L, very likely)	(H, very likely)	(M, usually)	(VH, maybe)	(M, usually)
	$x_3$	(M, maybe)	(VH, usually)	(M, maybe)	(M, usually)	(M, usually)	(H, likely)
	$x_4$	(L, very likely)	(H, maybe)	(H, very likely)	(H, maybe)	(L, maybe)	(VH, very likely)
$d_2$	$x_1$	(L, likely)	(L, usually)	(H, likely)	(M, unknown)	(L, likely)	(M, very likely)
	$x_2$	(VL, likely)	(H, likely)	(L, likely)	(M, maybe)	(M, likely)	(M, likely)
	$x_3$	(H, likely)	(VL, usually)	(VH, maybe)	(H, usually)	(L, maybe)	(H, usually)
	$x_4$	(VH, maybe)	(M, likely)	(VH, usually)	(M, unknown)	(VL, usually)	(M, maybe)
$d_3$	$x_1$	(H, usually)	(H, usually)	(M, likely)	(M, likely)	(VH, unknown)	(H, usually)
	$x_2$	(M, maybe)	(L, likely)	(H, likely)	(VH, very likely)	(H, very likely)	(VH, unknown)
	$x_3$	(L, usually)	(H, unknown)	(L, usually)	(M, likely)	(H, maybe)	(M, very likely)
	$x_4$	(H, usually)	(M, usually)	(H, maybe)	(VH, maybe)	(L, usually)	(H, likely)
$d_4$	$x_1$	(VL, usually)	(L, likely)	(L, very likely)	(H, very likely)	(M, maybe)	(VH, likely)
	$x_2$	(L, usually)	(VL, likely)	(L, likely)	(L, likely)	(L, maybe)	(M, usually)
	$x_3$	(VL, likely)	(L, likely)	(M, very likely)	(H, very likely)	(VH, maybe)	(H, maybe)
	$x_4$	(M, likely)	(M, maybe)	(L, usually)	(VH, usually)	(L, usually)	(M, usually)

.

Step 2:

Set $\sigma =0.9$ [7], and convert L-T2FNs to T2FNs using Equation (4). DMs’ evaluation matrices represented by T2FNs are displayed in

Table 3. The decision matrix represented by T2FNs.

DMs	Alternatives	Criteria
		${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$	${\mathit{c}}_{\mathbf{4}}$	${\mathit{c}}_{\mathbf{5}}$	${\mathit{c}}_{\mathbf{6}}$
$d_1$	$x_1$	(0.6744, 0.6744)	(0.8833, 0.8833)	(0.5000, 0.8833)	(0.5000, 0.6744)	(0.6744, 0.6744)	(0.6744, 0.6744)
	$x_2$	(0.1167, 0.8833)	(0.3256, 0.8833)	(0.6744, 0.8833)	(0.6744, 0.5000)	(0.8833, 0.3256)	(0.5000, 0.5000)
	$x_3$	(0.5000, 0.3256)	(0.8833, 0.5000)	(0.5000, 0.3256)	(0.5000, 0.5000)	(0.5000, 0.5000)	(0.6744, 0.6744)
	$x_4$	(0.3256, 0.8833)	(0.6744, 0.3256)	(0.6744, 0.8833)	(0.6744, 0.3256)	(0.3256, 0.3256)	(0.8833, 0.8833)
$d_2$	$x_1$	(0.3256, 0.6744)	(0.3256, 0.5000)	(0.6744, 0.6744)	(0.5000, 0.1167)	(0.3256, 0.6744)	(0.5000, 0.8833)
	$x_2$	(0.1167, 0.6744)	(0.6744, 0.6744)	(0.3256, 0.6744)	(0.5000, 0.3256)	(0.5000, 0.6744)	(0.5000, 0.6744)
	$x_3$	(0.6744, 0.6744)	(0.1167, 0.5000)	(0.8833, 0.3256)	(0.6744, 0.5000)	(0.3256, 0.3256)	(0.6744, 0.5000)
	$x_4$	(0.6744, 0.6744)	(0.1167, 0.5000)	(0.8833, 0.3256)	(0.6744, 0.5000)	(0.3256, 0.3256)	(0.6744, 0.5000)
$d_3$	$x_1$	(0.6744, 0.5000)	(0.6744, 0.5000)	(0.5000, 0.6744)	(0.5000, 0.6744)	(0.8833, 0.1167)	(0.6744, 0.5000)
	$x_2$	(0.5000, 0.3256)	(0.3256, 0.6744)	(0.6744, 0.6744)	(0.8833, 0.8833)	(0.6744, 0.8833)	(0.8833, 0.1167)
	$x_3$	(0.3256, 0.5000)	(0.6744, 0.1167)	(0.3256, 0.5000)	(0.5000, 0.6744)	(0.6744, 0.3256)	(0.5000, 0.8833)
	$x_4$	(0.6744, 0.5000)	(0.5000, 0.5000)	(0.6744, 0.3256)	(0.8833, 0.3256)	(0.3256, 0.5000)	(0.6744, 0.6744)
$d_4$	$x_1$	(0.1167, 0.5000)	(0.3256, 0.6744)	(0.3256, 0.8833)	(0.6744, 0.8833)	(0.5000, 0.3256)	(0.8833, 0.6744)
	$x_2$	(0.3256, 0.5000)	(0.1167, 0.6744)	(0.3256, 0.6744)	(0.3256, 0.6744)	(0.3256, 0.3256)	(0.5000, 0.5000)
	$x_3$	(0.1167, 0.6744)	(0.3256, 0.6744)	(0.5000, 0.8833)	(0.6744, 0.8833)	(0.8833, 0.3256)	(0.6744, 0.3256)
	$x_4$	(0.5000, 0.6744)	(0.5000, 0.3256)	(0.3256, 0.5000)	(0.8833, 0.5000)	(0.3256, 0.5000)	(0.5000, 0.5000)

.

Step 3:

Let $\phi =0.5$ [33]. Through Equation (18), the comprehensive score matrix for four DMs is obtained.

Table 4. The comprehensive score matrix.

DMs	Alternatives	Criteria
		${\mathit{c}}_{\mathbf{1}}$	${\mathit{c}}_{\mathbf{2}}$	${\mathit{c}}_{\mathbf{3}}$	${\mathit{c}}_{\mathbf{4}}$	${\mathit{c}}_{\mathbf{5}}$	${\mathit{c}}_{\mathbf{6}}$
$d_1$	$x_1$	0.1272	0.1405	0.1047	0.0717	0.1541	0.0697
	$x_2$	0.1302	0.1489	0.0961	0.0588	0.1759	0.1453
	$x_3$	0.2124	0.1171	0.2425	0.0680	0.1391	0.0697
	$x_4$	0.1206	0.1701	0.0961	0.0705	0.2318	0.1328
$d_2$	$x_1$	0.1216	0.1334	0.0906	0.1000	0.1324	0.1475
	$x_2$	0.1658	0.1688	0.1581	0.1017	0.1620	0.0930
	$x_3$	0.1282	0.1890	0.1031	0.1854	0.1502	0.1000
	$x_4$	0.1714	0.1336	0.0964	0.1000	0.1433	0.1697
$d_3$	$x_1$	0.0921	0.0982	0.0947	0.0927	0.1702	0.0953
	$x_2$	0.1026	0.1015	0.1180	0.1414	0.1984	0.1321
	$x_3$	0.1124	0.1232	0.1455	0.0927	0.1380	0.1237
	$x_4$	0.0921	0.0749	0.1114	0.1309	0.1968	0.0802
$d_4$	$x_1$	0.1367	0.0641	0.1147	0.0826	0.1106	0.1704
	$x_2$	0.0964	0.0846	0.1078	0.1649	0.1547	0.0783
	$x_3$	0.1024	0.0641	0.1268	0.0826	0.2360	0.0917
	$x_4$	0.1565	0.1213	0.1661	0.1191	0.1206	0.0783

displays the resultant values.

Step 4:

Using the D-CRITIC method combined with Deng entropy, weights corresponding to individual criteria are computed. These results are detailed in

Table 5. The weights for each criterion.

${\mathit{w}}_1$	${\mathit{w}}_2$	${\mathit{w}}_3$	${\mathit{w}}_4$	${\mathit{w}}_5$	${\mathit{w}}_6$
0.2519	0.1566	0.1447	0.1233	0.1620	0.1615

.

Step 5:

The weights of the four DMs are set to the first layer of the BN.

Step 6:

Based on Equation (31) and the parameter settings $\gamma =0.61$, $\delta =0.69$, $\alpha =0.88$, $\eta =0.88$, $\lambda =2.25$ [20], the prospect dominance degree measuring the relationship for each pair of alternatives $x_i$ and $x_k$ for criterion $c_j$ from $d_s$ is obtained.

The overall prospect dominance degree of alternative $x_i$ from $d_s$ are derived as

$$\begin{array}{c}\hfill {\varphi }^{\left(1\right)}\left(x_1\right)=-9.3456,{\varphi }^{\left(1\right)}\left(x_2\right)=-6.5967,\\ \hfill {\varphi }^{\left(1\right)}\left(x_3\right)=-7.7277,{\varphi }^{\left(1\right)}\left(x_4\right)=-4.8580,\\ \hfill {\varphi }^{\left(2\right)}\left(x_1\right)=-8.9033,{\varphi }^{\left(2\right)}\left(x_2\right)=-5.0310,\\ \hfill {\varphi }^{\left(2\right)}\left(x_3\right)=-4.9410,{\varphi }^{\left(2\right)}\left(x_4\right)=-5.6897,\\ \hfill {\varphi }^{\left(3\right)}\left(x_1\right)=-7.7257,{\varphi }^{\left(3\right)}\left(x_2\right)=-1.2132,\\ \hfill {\varphi }^{\left(3\right)}\left(x_3\right)=-5.1835,{\varphi }^{\left(3\right)}\left(x_4\right)=-6.0849,\\ \hfill {\varphi }^{\left(4\right)}\left(x_1\right)=-9.2711,{\varphi }^{\left(4\right)}\left(x_2\right)=-7.7770,\\ \hfill {\varphi }^{\left(4\right)}\left(x_3\right)=-7.9383,{\varphi }^{\left(4\right)}\left(x_4\right)=-5.7905.\end{array}$$

The conversion of the overall prospect dominance degree ${\psi }^s\left(x_i\right)$ into exponential-based conditional probabilities is then performed, giving the corresponding results:

$$\begin{array}{c}\hfill P\left(x_1\right|d_1)=0.0090,P\left(x_2\right|d_1)=0.1413,P\left(x_3\right|d_1)=0.0458,P\left(x_4\right|d_1)=0.8039,\\ \hfill P\left(x_1\right|d_2)=0.0079,P\left(x_2\right|d_2)=0.3799,P\left(x_3\right|d_2)=0.4156,P\left(x_4\right|d_2)=0.1966,\\ \hfill P\left(x_1\right|d_3)=0.0014,P\left(x_2\right|d_3)=0.9727,P\left(x_3\right|d_3)=0.0184,P\left(x_4\right|d_3)=0.0075,\\ \hfill P\left(x_1\right|d_4)=0.0240,P\left(x_2\right|d_4)=0.1068,P\left(x_3\right|d_4)=0.0909,P\left(x_4\right|d_4)=0.7784.\end{array}$$

Step 7:

Following QPT principles, where the probability of $x_i$ corresponds to the magnitude squared of the total amplitude probabilities from the complete set of paths, Equations (36)–(38) are applied to determine these probabilities as detailed below:

$$\begin{array}{cc}\hfill P\left(x_1\right)=& {\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^4\sqrt{P\left(d_s\right)}\sqrt{P\left(x_1\right|d_s)}{e}^{i{\theta }_{d_s}}\right|}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{\aleph }}(0.0106+2\times (0.0021cos{\beta }_{12}+0.0009cos{\beta }_{13}\hfill \\ \hfill & +0.0037cos{\beta }_{14}+0.0008cos{\beta }_{23}\hfill \\ \hfill & +0.0034cos{\beta }_{24}+0.0014cos{\beta }_{34})),\hfill \end{array}$$

$$\begin{array}{cc}\hfill P\left(x_2\right)=& {\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^4\sqrt{P\left(d_s\right)}\sqrt{P\left(x_2\right|d_s)}{e}^{i{\theta }_{d_s}}\right|}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{\aleph }}(0.4002+2\times (0.0579cos{\beta }_{12}+0.0927cos{\beta }_{13}\hfill \\ \hfill & +0.0307cos{\beta }_{14}+0.1520cos{\beta }_{23}\hfill \\ \hfill & +0.0504cos{\beta }_{24}+0.0806cos{\beta }_{34})),\hfill \end{array}$$

$$\begin{array}{cc}\hfill P\left(x_3\right)=& {\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^4\sqrt{P\left(d_s\right)}\sqrt{P\left(x_3\right|d_s)}{e}^{i{\theta }_{d_s}}\right|}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{\aleph }}(0.1427+2\times (0.0345cos{\beta }_{12}+0.0073cos{\beta }_{13}\hfill \\ \hfill & +0.0161cos{\beta }_{14}+0.0219cos{\beta }_{23}\hfill \\ \hfill & +0.0486cos{\beta }_{24}+0.0102cos{\beta }_{34})),\hfill \end{array}$$

$$\begin{array}{cc}\hfill P\left(x_4\right)=& {\displaystyle \frac{1}{\aleph }}{\left|\sum _{s=1}^4\sqrt{P\left(d_s\right)}\sqrt{P\left(x_4\right|d_s)}{e}^{i{\theta }_{d_s}}\right|}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{\aleph }}(0.4466+2\times (0.0994cos{\beta }_{12}+0.0194cos{\beta }_{13}\hfill \\ \hfill & +0.1978cos{\beta }_{14}+0.0096cos{\beta }_{23}\hfill \\ \hfill & +0.0978cos{\beta }_{24}+0.0191cos{\beta }_{34})),\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill \aleph =& \sum _{i=1}^4{\left|\sum _{s=1}^4\sqrt{P\left(d_s\right)}\sqrt{P\left(x_i\right|d_s)}{e}^{i{\theta }_s}\right|}^2\hfill \\ \hfill =& 1+0.3878cos{\beta }_{12}+0.2406cos{\beta }_{13}\hfill \\ \hfill & +0.4966cos{\beta }_{14}+0.3686cos{\beta }_{23}\hfill \\ \hfill & +0.4004cos\beta 24+0.2226cos{\beta }_{34}.\hfill \end{array}$$

Using Equations (39)–(44) to determine the interference angle between DMs.

$$\begin{array}{cccc}\hfill cos\left({\beta }_{12}\right)=0.3419,& cos\left({\beta }_{13}\right)=-0.0252,\hfill & \hfill cos\left({\beta }_{14}\right)& =0.6626,\hfill \\ \hfill cos\left({\beta }_{23}\right)=0.2949,& cos\left({\beta }_{24}\right)=0.3725,\hfill & \hfill cos\left({\beta }_{34}\right)& =-0.0721.\hfill \end{array}$$

Based on Equation (36), calculate the final probability for each alternative.

$$P\left(x_1\right)=0.0116,P\left(x_2\right)=0.3484,P\left(x_3\right)=0.1384,P\left(x_4\right)=0.5016.$$

Step 8:

According to the probability values of the alternatives, their final ranking is $x_4\succ x_2\succ x_3\succ x_1$. Based on the ranking results, the startup with the highest investment value is identified as $x_4$: RegTech compliance platform.

6. Sensitivity and Comparative Analysis

This section provides sensitivity and comparative analyses for the purpose of assessing the proposed method’s robustness and effectiveness.

6.1. Sensitivity Analysis

A robust justification for the default parameter values is a crucial prerequisite for a meaningful sensitivity analysis. Accordingly, the parameters for the CPT-TODIM method are directly derived from the foundational work of Tversky and Kahneman [19], whose pioneering study established these values through rigorous experimentation to characterize typical patterns of decision making under uncertainty.

Building upon this, we analyze the effect of three distinct CPT parameters $\alpha$, $\eta$, and $\lambda$ on both computational outcomes and the final ordering. These parameters capture the DMs’ risk attitudes and loss aversion features, such that diminished $\alpha$ and $\eta$ values denote higher risk aversion, and a decreased $\lambda$ points to an elevated inclination for risk-taking.

(1) To assess the influence of parameter $\alpha$ on the ranking outcomes, its value is systematically varied. We incrementally adjust $\alpha$ from 0 to 1 with a step size of 0.1, keeping other parameters fixed, and observe its impact on the ranking of alternatives $x_1,x_2,x_3$, and $x_4$. As depicted in Figure 8, the ranking results demonstrate consistent stability across the entire range of $\alpha$ values tested. The resulting ranking remains $x_4\succ x_2\succ x_3\succ x_1$ for all $\alpha \in [0,1]$.

(2) Similarly, to evaluate the influence of parameter $\eta$ on the ranking outcomes, $\eta$ is varied from 0 to 1 in increments of $0.1$, while other parameters are held constant. The corresponding results are presented in Figure 9. These results indicate that, for $\eta =0$, alternative $x_3$ outperforms $x_2$. However, for all other tested values of $\eta$, $x_2$ outperforms $x_3$.

The issue of rank reversal between $x_2$ and $x_3$ at $\eta =0$ arises primarily because $\eta =0$ represents a critical boundary point, where the CPT-TODIM method completely disregards the actual magnitude of losses in its loss calculation. However, when $\eta >0$, the model begins to account for the magnitude of losses, and since alternative $x_3$ exhibits a greater loss magnitude than $x_2$, it is ranked lower than $x_2$.

(3) To assess how the ranking of alternatives responds to the parameter $\lambda$, we systematically vary its value from 1.5 to 6.5, in steps of 0.5, holding other parameters fixed. The ranking outcomes for alternatives $x_1,x_2,x_3,$ and $x_4$, as determined across the entire range of $\lambda$, are clearly depicted in Figure 10. It can be observed that the ranking order exhibits complete robustness to changes in the value of $\lambda$. Specifically, the ranking order consistently remains $x_4\succ x_2\succ x_3\succ x_1$.

In conclusion, the sensitivity analysis evaluates how the primary CPT parameters affect the ultimate rankings of the alternatives, and the findings reveal that the proposed model is highly robust. The final ranking order remains entirely unchanged across the full range tested for parameter $\alpha \in [0,1]$ and a wide range for parameter $\lambda \in [1.5,6.5]$. While parameter $\eta$ shows a rank reversal between $x_2$ and $x_3$ specifically at the theoretical boundary of $\eta =0$, the ranking is consistently stable across the more behaviorally relevant range of $\eta \in (0,1]$. This specific sensitivity at $\eta =0$ is well-explained by the CPT mechanism, where setting $\eta$ to 0 effectively removes the influence of loss magnitude from the evaluation, thus demonstrating the model’s correct incorporation of behavioral nuances. Overall, the pronounced stability across wide variations in $\alpha$ and $\lambda$, coupled with the explainable and contained sensitivity of $\eta$, underscores the robustness and reliability of the proposed method. It suggests that the model’s outcomes are primarily driven by the inherent preference structure derived from the data and method, rather than being overly sensitive to the precise calibration of these psychological parameters within typical ranges, highlighting its practical applicability.

6.2. Comparative Analysis

This subsection presents a comparative analysis contrasting the method we designed with alternative ranking approaches, thereby validating the efficacy of our model. Detailed rankings derived from these distinct methods can be found in

Table 6. The comparison with different methods.

Methods	Ranking Indices	Final Rankings
TOPSIS [58]	$T_1$ = 0.1411, $T_2$ = 0.6341,	$x_3\succ x_4\succ x_2\succ x_1$
	$T_3$ = 0.7603, $T_4$ = 0.6870.
MABAC [59]	$M_1$ = −0.0115, $M_2$ = −0.0023,	$x_3\succ x_4\succ x_2\succ x_1$
	$M_3$ = 0.0058, $M_4$ = 0.00437.
CPT-TODIM [20]	$\varphi \left(x_1\right)$ = −7.2737, $\varphi \left(x_2\right)$ = −1.5273,	$x_4\succ x_2\succ x_3\succ x_1$
	$\varphi \left(x_3\right)$ = −2.0091, $\varphi \left(x_4\right)$ = −1.4969.
Quantum framework of [41]	$P\left(x_1\right)$ = 0.2262, $P\left(x_2\right)$ = 0.2580,	$x_4\succ x_2\succ x_3\succ x_1$
	$P\left(x_3\right)$ = 0.2570, $P\left(x_4\right)$ = 0.2589.
The proposed method	$P\left(x_1\right)$ = 0.0116, $P\left(x_2\right)$ = 0.3484,	$x_4\succ x_2\succ x_3\succ x_1$
	$P\left(x_3\right)$ = 0.1384, $P\left(x_4\right)$ = 0.5016.

.

According to Table 6, the ranking generated by our proposed method aligns with those from the CPT-TODIM method and the quantum model developed by Wu et al. [41]. Specifically, all three approaches rank the alternatives as $x_4\succ x_2\succ x_3\succ x_1$. In contrast, both the TOPSIS and MABAC methods yield the ranking $x_3\succ x_4\succ x_2\succ x_1$. The interpretation and further analysis of these comparative results are presented below:

(1)

The rankings generated by the proposed model show alignment with those from the quantum model developed by Wu et al. [41], underscoring the stability and effectiveness of our approach. As shown in Table 6, for alternatives $x_1$ and $x_4$, our proposed method yields probabilities of $P\left(x_1\right)=0.0116$ and $P\left(x_4\right)=0.5016$, whereas the model by Wu et al. [41] produces results of $P\left(x_1\right)=0.2262$ and $P\left(x_4\right)=0.2589$. This difference stems primarily from the CPT-TODIM method’s foundation in CPT, which places greater emphasis on loss aversion and sensitivity to gains. This leads to a substantial amplification for alternatives exhibiting pronounced advantages and shows higher sensitivity to both extremely low and extremely high probabilities, causing certain well-performing alternatives to acquire exceptionally high probabilities, while those performing slightly less well are assigned extremely low probabilities. The comparison results indicate that our proposed method, relative to the quantum model by Wu et al. [41], more closely aligns with the psychological decision-making processes of actual humans, is capable of simulating complex risk attitudes and probability distortions in greater detail, and offers a more comprehensive description of the behavioral psychology of DMs.

(2)

The optimal alternative selected by both the TOPSIS [58] and MABAC [59] methods is $x_3$, whereas the proposed method, along with the CPT-TODIM method and the quantum model, identifies $x_4$ as the optimal alternative. The primary reason for this inconsistency is that our proposed model, the CPT-TODIM method, and the quantum model all originate from the TODIM framework. The TODIM-based methods and their quantum probability extensions emphasize the cumulative advantages and penalty for disadvantages across all criteria between alternatives, prioritizing overall balanced stability in ranking. Consequently, $x_4$, which demonstrates balanced performance with minimal weaknesses, is ranked first. In contrast, TOPSIS and MABAC are grounded in calculating aggregate distances connecting alternatives to the “positive ideal solution”, potentially allowing superior results pertaining to specific criteria to greatly enhance an alternative’s rank. Thus, $x_3$ ranks above $x_4$ via these methods due to specific criteria benefits, yielding contrasting orderings for these alternatives across different methods.

(3)

The rankings derived from the classical CPT-TODIM method [20] exhibit consistency relative to our developed approach’s outcomes. It is because the CPT-TODIM method itself sets forth a clear and significant basis for the ranking that this consistency is observed. The relative increment introduced by the interference term from quantum probability is insufficient to overcome the original large ranking gaps. Although the resulting ranking is consistent, our method, compared to the traditional CPT-TODIM method, incorporates a consideration of the interference effects among DMs. This enables our model to theoretically better handle conflict and fuzziness, makes the model more flexible, and provides stronger explanatory power for deviations in complex behavior.

Overall, the comparative analysis in the preceding subsections systematically contrasts the ranking outcomes of the proposed model with those of several established decision-making methods. The observed alignments and divergences underscore the distinctive features and contributions of our approach. In particular, its consistent performance against methods sharing similar theoretical foundations, coupled with divergent results compared to models based on different assumptions, highlights the nuanced capabilities of our framework. This comprehensive evaluation confirms the robustness and effectiveness of our integrated quantum group decision model, particularly its ability to incorporate psychological factors and interference effects in group dynamics within complex decision environments.

6.3. Further Discussions on Symmetry

Beyond the direct performance evaluations presented in the sensitivity and comparative analyses, it is pertinent to delve into a foundational aspect that underpins the robustness and coherence of the proposed model: the role of symmetry. Symmetry is central to the theoretical formulation and the practical implementation of the model developed in this study. At the theoretical level, the membership functions for T2FNs in this study are constructed based on Gaussian functions, which inherently possess symmetric characteristics. This design ensures that the fuzzy number’s representation of uncertainty exhibits a symmetric decay around its core, facilitating a balanced characterization of vague concepts, particularly in the absence of prior preference information, thereby potentially mitigating biases from inappropriate membership function selection and positively contributing to the stability of decision evaluations. Furthermore, the aggregation process for group preferences within our model draws upon the mathematical framework of quantum theory, a framework deeply intertwined with symmetry principles in physics. Its unique mechanisms of probability amplitudes and interference offer a novel perspective for addressing complex inter-DM interactions. This quantum-inspired methodology enables the capture of intricate potential associations within the group, which may encompass both symmetric and asymmetric influence patterns. Consequently, it can lead to more stable and robust aggregated group preferences, providing a solid mathematical foundation for MCGDM and enhancing the model’s reliability and consistency.

In practical application, symmetry is further manifested in the design of evaluation criteria and weight allocation for the FinTech startup selection case study. By constructing a symmetric evaluation framework, the model balances risk and return, ensuring the fairness and interpretability of decision outcomes. For instance, within the value function of CPT-TODIM, symmetric characteristics enable a more balanced perception of positive and negative returns, thereby avoiding the extreme risk preferences that may arise in traditional asymmetric models. Sensitivity analysis indicates that the symmetric design significantly enhances the model’s adaptability to parameter variations, demonstrating higher stability and accuracy compared to asymmetric approaches. This synergy between theory and application underscores that symmetry is not only a core design principle of the model but also offers substantial practical value in real-world decision-making scenarios.

6.4. Managerial Implications

The novel decision-making framework proposed in this study offers a robust tool for navigating investment selection in complex, uncertain environments. Initially, the framework mitigates subjective bias by transforming DMs’ linguistic evaluations into T2FNs and subsequently applying a Deng entropy-enhanced D-CRITIC method to derive objective, nonlinear criteria weights. Furthermore, the integration of the CPT-TODIM model captures the psychological risk attitudes of DMs—such as loss aversion, reference dependence, and risk sensitivity–leveraging empirically validated parameters to help managers establish more precise decision thresholds. Crucially, the incorporation of QPT elucidates the interference effects among DMs’ opinions, enhancing the model’s fidelity to real-world scenarios. This dual focus on both individual psychology and group dynamics significantly strengthens the reliability of the final outcomes, thereby providing superior guidance for managerial practice.

7. Conclusions

Interference effects among DMs are a critical factor frequently impacting decision outcomes. To effectively capture and address these complex dynamics, this paper develops a novel multi-criteria quantum group decision-making framework employing T2FNs. The framework integrates an enhanced D-CRITIC weighting method incorporating Deng entropy, a CPT-TODIM method for capturing DMs’ risk perceptions, and a quantum-based aggregation mechanism to account for interference among opinions. The proposed model is applied to a case study of FinTech startup investment. Compared against classical MCDM methods and existing quantum decision models, our approach demonstrates superior robustness and stability: the rankings remain invariant under wide variations of CPT parameters, while sensitivity analysis reveals only explainable reversals at theoretical parameter boundaries. These results confirm that the model not only preserves the behavioral insights of CPT but also leverages QPT to capture latent inter-DM influences, thereby yielding a more comprehensive and reliable decision-support tool under uncertainty.

Notwithstanding its contributions, some limitations should be noted. First, the integration of multiple advanced techniques inevitably elevates the model’s computational complexity. Consequently, the scalability and computational efficiency of our proposed methodology may be constrained when applied to large-scale, high-dimensional problems. Second, our model presupposes that interactions occur exclusively between the dyads of DMs. This simplification may not fully capture the complexities of higher-order interactions, where three or more agents simultaneously influence one another.

Building on the identified limitations, our future research agenda is two-fold. The primary objective is to develop computationally tractable approximate algorithms for large-scale group quantum interference, thereby enhancing the model’s applicability to larger systems while mitigating the associated computational burden. Furthermore, to ascertain the model’s real-world viability, its robustness and scalability will be rigorously tested by applying it to diverse and critical domains, including healthcare resource allocation and renewable energy planning.