Interval-valued intuitionistic fuzzy three-way conflict analysis based on cumulative prospect theory

Abstract

This paper makes a significant contribution to the field of conflict analysis by introducing a novel Interval-Valued Intuitionistic Fuzzy Three-Way Conflict Analysis (IVIFTWCA) method, which is anchored in cumulative prospect theory. The method’s key innovation lies in its use of interval-valued intuitionistic fuzzy numbers to represent an agent’s stance, addressing the psychological dimensions and risk tendencies of decision-makers that have been largely overlooked in previous studies. The IVIFTWCA method categorizes conflict situations into affirmative, impartial, and adverse coalitions, leveraging the evaluation of the closeness function and predefined thresholds. It incorporates a reference point, value functions and cumulative weight functions to assess risk preferences, leading to the formulation of precise decision rules and thresholds. The method’s efficacy and applicability are demonstrated through detailed examples and comparative analysis, and its exceptional performance is confirmed through a series of experiments, offering a robust framework for real-world decision-making in conflict situations.

Keywords

Three-way decision conflict analysis interval-valued intuitionistic fuzzy sets cumulative prospect theory

1 Introduction

Amidst the intricate landscape of complex scenarios, uncertainty arises from its complicated nature, diverse origins, and the impact of subjective human factors. This poses considerable challenges to conventional artificial intelligence fields and technologies. Since its inception by Professor Yao Yiyu in 2010 [33], Three-Way Decision (TWD) has emerged as a pivotal research methodology and tool in uncertainty reasoning and granular computing. It has provided a practical and innovative approach to address intricate and uncertain problems [20 , 39]. The foundational notion of TWD entails partitioning the realm of discourse into three distinct domains. Each domain corresponds to one of three decision courses of action: acceptance, rejection, and postponement. In scenarios where information is insufficient for an immediate decision, TWD introduces a postponement option in addition to the conventional binary decision-making process. This mirrors the standard human decision-making process and has attracted considerable interest from researchers in the domain of decision theory [5 , 41]:

In the realm of theoretical approaches, TWD places its primary emphasis on the refinement of models and concepts, along with the advancement of sequential TWD models and methodologies. Wang et al. [21] put forth a TWD approach incorporating a risk strategy founded on uncertain fuzzy decision information structures. Yang et al. [33] delved into a composite data fusion framework grounded in a sequential TWD, considering subjective and objective dynamics. Zhang et al. [42] introduced an optimal granularity selection technique for sequential TWD, considering cost sensitivity.

Regarding practical applications, TWD has found extensive use in recommendation systems, image processing, medical decision-making, and various other domains. Zhang et al. [40] tailored a three-way naive Bayes collaborative filtering recommendation model specifically for smart cities. Chen et al. [3], approaching the problem from a TWD standpoint, integrated shadow sets into image classification, resulting in a two-stage approach. Yu and Yang [35] examined the challenges encountered in intelligent decision-making within the realm of industrial big data and provided illustrative examples of TWD applications in industrial big data contexts. Yue et al. [36] seamlessly merged evidence theory with deep neural networks, culminating in developing a medical image TWD method centered around an evidence-based deep neural network well-suited for medical image classification tasks.

Conflict, an inherent aspect of human society and daily life [45], is a state of opposition that requires systematic analysis, known as Conflict Analysis (CA) [16]. In practical decision-making scenarios, decision-makers must consider the dynamics of cooperation and conflict within a given context. For instance, consider the leader of an audit team: when team members maintain positive and collaborative relationships, task execution efficiency is markedly enhanced. Conversely, when conflicts persist among team members, the efficacy of work and collaboration is significantly compromised. Due to the unity of the three distinct attitudes in CA–agreement, opposition, and neutrality–with the trisecting concept of TWD theory, numerous scholars have integrated TWD theory and methodologies into CA research, conducting multidimensional studies and yielding noteworthy results [13 , 32]. Li et al. [12] delved into research on Three-Way CA (TWCA) based on a triangular fuzzy Information System (IS). Lang et al. [10] embarked on a study regarding three-way group CA rooted in Pythagorean fuzzy set theory. Shi et al. [17] introduced a CA method based on an object-oriented three-way concept lattice. Gong and Wang [7] introduced the concept of the importance weight vector of conflict attributes in the CA IS, establishing a weighted continuous CA model.

In the initial stages of TWCA theories and methodologies, there was limited consideration for decision emotional aspects and hazard inclinations. Nonetheless, in practical decision-making contexts, emotional facets and risk inclinations play a pivotal role in assessing the severity of conflicts within a team. Psychological studies and experimental economics research have demonstrated that decision-makers are not solely driven by rationality in practical decision-making processes. They are also influenced by the intricate psychological mechanisms that govern human behavior [14]. To refine the assumption of decision-makers being perfectly rational, as posited in traditional decision theory, some researchers have incorporated insights from cognitive psychology into the study of individual behavior under conditions of uncertainty, leading to the emergence of behavioral decision theory. Among these, the Cumulative Prospect Theory (CPT) [18] adeptly captures decision-makers’ reference points and their varying risk preferences regarding gains and losses and delineates decision-makers’ estimations of conditional probabilities through cumulative decision weights [24 , 28]. For instance, Chai et al. [2] have developed an innovative fuzzy multi-criteria decision-making approach that combines intuitionistic fuzzy sets, interval-valued fuzzy sets, and CPT to select the most sustainable supplier. Wang et al. [27] conducted an analysis of multi-criteria fuzzy portfolio selection based on TWDs and CPT. Hence, to provide a more inclusive representation of decision analysts’ individual risk stances and inclinations in the CA process, this study integrates the CPT into the structure of CA. This integration allows for the advantage of a more nuanced and comprehensive representation of individual decision-making behaviors, providing valuable insights into the complex interplay between risk and decision outcomes.

Moreover, as emphasized in reference [12], it’s been recognized that representing an agent’s stance towards an event solely with a real number is often impractical in CA research. In real-world decision-making contexts, presenting this perspective in the form of intervals may better align with the complexities of general CA situations. The concept of an Interval-Valued Intuitionistic Fuzzy Set (IVIFS), introduced by Atanassov and Gargov [1], stands as an extension of the conventional intuitionistic fuzzy set [30]. By defining membership and non-membership degrees as interval values, it exhibits a robust capacity to address ambiguity and uncertainty within the decision-making process [8, 19]. Dong and Wan [6] have pioneered a novel approach, the IVIF best-worst method, which maintains additive consistency. Chen and Huang [4] have proposed a multiattribute decision-making technique based on nonlinear programming methodology, the score function of IVIF values, and the dispersion degree of score values. Yao and Guo [34] explored multiattribute team decision-making involving an innovative scoring mechanism that considers both the angle and fuzzy details in an environment characterized by IVIFS. Li et al. [11] have devised an innovative method for product ranking by extracting insights from online reviews, employing an IVIF technique for order preference based on similarity to an ideal solution. In light of these advancements and to refine the characterization of CA ISs in uncertain environments, this paper introduces the theory of IVIFSs into the study of three-way CA.

Drawing from the preceding analysis, this paper’s principal contributions and primary merits can be briefly outlined as follows:

At the inception of our study, we introduced the theoretical framework of the IVIFS as a novel approach to delineate the IS of CA. The IVIFS is a mathematical tool that allows for the representation of uncertainty and incomplete information, which are common characteristics in conflict situations [1, 8]. By employing IVIFSs, we aimed to capture the nuances and complexities of conflict scenarios, providing a more robust foundation for analysis and decision-making.

Within the broader context of CA, we focused on the delineation of three distinct regions that are critical for understanding and managing conflicts effectively. These regions correspond to different levels of conflict intensity or potential outcomes, and they help to classify conflicts in a way that can guide appropriate interventions. We discussed the importance of identifying these regions and how they can inform strategies for conflict prevention, mitigation, and resolution.

Moving forward, we derived decision rules specific to TWCA by integrating insights from both the CPT and the loss function. CPT is a behavioral model that accounts for how people make decisions under uncertainty, considering factors such as risk aversion and the framing of outcomes [24 , 28]. By combining CPT with loss functions, which quantifies the potential negative consequences of different decisions, we were able to establish a set of decision rules that are both theoretically grounded and practical for CA.

To validate and demonstrate the practical utility of our IVIFTWCA method, we provided a series of illustrative examples. These examples showcased the method’s ability to handle real-world conflict scenarios with the necessary complexity and uncertainty. Additionally, we conducted a comparative analysis to highlight the advantages of our approach over traditional conflict analysis methods. This analysis revealed that our method offered improved sensitivity to the dynamic nature of conflicts and provided decision-makers with more nuanced insights into the potential outcomes of their choices. In conclusion, our findings underscored the efficacy of the IVIFS-based TWCA method rooted in CPT as a promising tool for conflict management and decision support in various domains.

These contributions significantly augment comprehension and application of CA within uncertain environments. They furnish decision-makers grappling with intricate and conflicting scenarios with a valuable framework.

The article is organized as follows. Section 2 provides a concise introduction to the foundational principles of TWCA, IVIFSs, and CPT. Section 3 focuses on CA within the IVIF ISs (IVIFISs). Moving to Section 4, we delve into the decision rules and threshold calculations for the IVIFTWCA method, drawing from the principles of CPT. Section 5 employs CA among team members during the audit process as a contextual backdrop, conducting a comprehensive examination of the proposed TWCA method. Section 6 conducts several experiments to validate the performance. Finally, Section 7 presents the concluding remarks and briefly summarizes the key findings.

2 Preliminary

Having outlined the scope and objectives of our study in the introduction, we now turn to the preliminary section to establish the essential background and notation that will underpin our subsequent analysis and development of the IVIFTWCA method. The following concisely overview three pivotal concepts: TWCA, IVIFSs, and CPT.

2.1 TWCA

Definition 1. [16] An IS can be outlined as a quadruple denoted by $S = (O, A, V, F)$ . Here, $O = {o_{1}, o_{2}, \dots, o_{mitsc}}$ represents a finite collection of objects or agents, $A = {a_{1}, a_{2}, \dots, a_{nitsc}}$ signifies a finite set of issues or attributes, $A = {A_{aitsc} | a \in A}$ encompasses the range of potential values for these attributes. $F$ is a function that maps from $O \times A$ to $V$ . In cases where an object o_i holds one of three different attitudes (approval, neutrality, and opposition) towards an attribute a_j, the values of $F (o_{iitsc}, a_{jitsc})$ are 1, 0, and -1, respectively.

Example 1. Table 1 shows the IS for the Middle East conflict [16]. $O = {o_{1}, o_{2}, \dots, o_{6}}$ denote six countries, including Israel, Egypt, Palestine, Jordan, Syria and Saudi Arabia respectively. $A = {a_{1}, a_{2}, \dots, a_{5}}$ denote five specific issues. In this IS, $F (o_{1}, a_{1}) = - 1$ represents that the country o₁ holds the negative opinion on the issue a₁. On the contrary, $F (o_{1}, a_{2}) = + 1$ represents the country o₁ holds the positive opinion on the issue a₂. Moreover, $F (x_{2}, c_{2}) = 0$ illustrates that the country o₂ is neutral about the issue a₂.

Table 1
IS for the Middle East conflict

$O$ a ₁ a ₂ a ₃ a ₄ a ₅

o ₁ -1 +1 +1 +1 +1

o ₂ +1 0 -1 -1 -1

o ₃ +1 -1 -1 -1 0

o ₄ 0 -1 -1 0 -1

o ₅ +1 -1 -1 -1 -1

o ₆ 0 +1 -1 0 +1

$O$	a ₁	a ₂	a ₃	a ₄	a ₅
o ₁	-1	+1	+1	+1	+1
o ₂	+1	0	-1	-1	-1
o ₃	+1	-1	-1	-1	0
o ₄	0	-1	-1	0	-1
o ₅	+1	-1	-1	-1	-1
o ₆	0	+1	-1	0	+1

Definition 2. [9] Consider an IS represented by $S = (O, A, V, F)$ . Let α and β denote two thresholds, ensuring 0 ≤ α, β ≤ 1. $ρ_{A} (o_{1}, o_{2})$ denotes a distance function on $A$ . For $o_{1} \in O$ , we define its conflict set as ${CO}_{β}^{α} (o_{1})$ , neutral set as ${NE}_{β}^{α} (o_{1})$ , and alliance set as ${AL}_{β}^{α} (o_{1})$ as follows: $\begin{matrix} {CO}_{β}^{α} (o_{1}) = {o_{2} | o_{2} \in O, ρ_{A} (o_{1}, o_{2}) \geq α} \\ {NE}_{β}^{α} (o_{1}) = {o_{2} | o_{2} \in O, β < ρ_{A} (o_{1}, o_{2}) < α} \\ {AL}_{β}^{α} (o_{1}) = {o_{2} | o_{2} \in O, ρ_{A} (o_{1}, o_{2}) \leq β} \end{matrix}$ (1)

Definition 3. [9] A set of loss functions is characterized by a triplet $λ = {Ω, A, R}$ , where $Ω = {{CO}_{β}^{α} (o_{1}), {AL}_{β}^{α} (o_{1})}$ represents two states. $A = {a_{C}, a_{N}, a_{A}}$ denotes three distinct actions, corresponding to placing o₂ in the conflict set ${CO}_{β}^{α} (o_{1})$ , neutral set ${NE}_{β}^{α} (o_{1})$ , and alliance set ${AL}_{β}^{α} (o_{1})$ , respectively. $R = {λ_{CC}, λ_{NC}, λ_{AC}, λ_{CA}, λ_{NA}, λ_{AA}}$ encompasses six loss functions. Among these, λ_CC, λ_NC, λ_AC represent the loss functions when taking actions a_C, a_N, a_A, respectively, where $o_{2} \in {CO}_{β}^{α} (o_{1})$ . Additionally, λ_CA, λ_NA, λ_AA signify the loss functions when taking actions a_C, a_N, a_A, respectively, where $o_{2} \in {AL}_{β}^{α} (o_{1})$ . Following this, the expected losses for each of these three activities can be calculated as follows: $\begin{matrix} R^{o_{1}} (a_{C} | o_{2}) = λ_{CC} ρ_{A} (o_{1}, o_{2}) + λ_{CA} (1 - ρ_{A} (o_{1}, o_{2})) \\ R^{o_{1}} (a_{N} | o_{2}) = λ_{NC} ρ_{A} (o_{1}, o_{2}) + λ_{NA} (1 - ρ_{A} (o_{1}, o_{2})) \\ R^{o_{1}} (a_{A} | o_{2}) = λ_{AC} ρ_{A} (o_{1}, o_{2}) + λ_{AA} (1 - ρ_{A} (o_{1}, o_{2})) \end{matrix}$ (2)

Utilizing the Bayesian minimum risk principle, the thresholds α and β can be determined as follows: α = (λ_CA - λ_NA)/(λ_CA - λ_NA + λ_NC - λ_CC), β = (λ_NA - λ_AA)/(λ_NA - λ_AA + λ_AC - λ_CC).

2.2 IVIFSs

Definition 4. [1] X is a non-empty set. We denote B ={ 〈x, M_B (x) , N_B (x) 〉|x ∈ X } as an IVIFS on X. Here, $M_{B} (x) = [r_{B}^{L} (x), r_{B}^{U} (x)]$ and $N_{B} (x) = [s_{B}^{L} (x), s_{B}^{U} (x)]$ represent the intervals for membership and non-membership values of x in B, respectively. These intervals adhere to the conditions: M_B (x) , N_B (x) ⊂ [0, 1], and $r_{B}^{U} (x) + s_{B}^{U} (x) \leq 1$ , ∀x ∈ X. For simplicity, an Interval-Valued Intuitionistic Fuzzy Number (IVIFN) is denoted as $\tilde{b} = ([r^{L}, r^{U}], [s^{L}, s^{U}])$ .

Definition 5. [31] Consider $\tilde{b} = ([r^{L}, r^{U}], [s^{L}, s^{U}])$ as an IVIFN. The function $S (\tilde{b}) = (r^{L} + r^{U} - s^{L} - s^{U}) / 2$ is termed the score function of $\tilde{b}$ , while $H (\tilde{b}) = (r^{L} + r^{U} + s^{L} + s^{U}) / 2$ is known as the accuracy function of $\tilde{b}$ .

Definition 6. [31] Let ${\tilde{b}}_{1} = ([r_{1}^{L}, r_{1}^{U}], [s_{1}^{L}, s_{1}^{U}])$ and ${\tilde{b}}_{2} = ([r_{2}^{L}, r_{2}^{U}], [s_{2}^{L}, s_{2}^{U}])$ be two IVIFNs:

(1) If $S ({\tilde{b}}_{1}) < S ({\tilde{b}}_{2})$ , then ${\tilde{b}}_{1} < {\tilde{b}}_{2}$ .

(2) If $S ({\tilde{b}}_{1}) = S ({\tilde{b}}_{2})$ , then: if $H ({\tilde{b}}_{1}) < H ({\tilde{b}}_{2})$ , then ${\tilde{b}}_{1} < {\tilde{b}}_{2}$ .

Example 2. For two IVIFNs ${\tilde{b}}_{1} = ([0.6, 0.7], [0.1, 0.2])$ and ${\tilde{b}}_{2} = ([0.4, 0.5], [0.0, 0.1])$ , we try to utilize Definitions 5 and 6 to rank them. By calculating score functions of two IVIFNs, we obtain: $S ({\tilde{b}}_{1}) = 0.5 > S ({\tilde{b}}_{2}) = 0.4$ such that we acquire ${\tilde{b}}_{1} > {\tilde{b}}_{2}$ in this example.

Definition 7. [43] Consider ${\tilde{b}}_{1} = ([r_{1}^{L}, r_{1}^{U}], [s_{1}^{L}, s_{1}^{U}])$ and ${\tilde{b}}_{2} = ([r_{2}^{L}, r_{2}^{U}], [s_{2}^{L}, s_{2}^{U}])$ as two IVIFNs. The distance between them is calculated as: $d ({\tilde{b}}_{1}, {\tilde{b}}_{2}) = (| r_{1}^{L} - r_{2}^{L} | + | r_{1}^{U} - r_{2}^{U} | + | s_{1}^{L} - s_{2}^{L} | + | s_{1}^{U} - s_{2}^{U} |) / 4$ .

2.3 CPT

The CPT is chiefly characterized by three key components [18]. First and foremost, it hinges on the decision-maker’s framing of gains and losses about a designated reference point. Secondly, the decision-maker typically exhibits a penchant for risk aversion when confronted with gains. Conversely, when dealing with losses, there is a tendency towards risk-seeking behavior, and notably, the sensitivity to losses outweighs gains. Lastly, the theory employs cumulative weight functions to adjust conditional probabilities nonlinearly.

Suppose z_t denotes the t-th outcome and z^r represents the chosen reference point by the decision-maker. According to the CPT, when z_t ≥ z^r, the decision-maker regards the outcome as a gain. Conversely, when z_t < z^r, he/she interprets it as a loss. The corresponding value function is expressed as follows:

$v (z_{t}) = {\begin{matrix} {(z_{t} - z^{r})}^{δ}, & z_{t} \geq z^{r} \\ - ϑ {(z^{r} - z_{t})}^{τ}, & z_{t} < z^{r} \end{matrix}$ (3) where: δ = τ = 0.88, and ϑ = 2.25.

Within the framework of the CPT, where p_t signifies the probability of z_t, the computation of the cumulative weighting function necessitates considering the aggregate probabilities instead of individual probability values. Let there be a total of l + q + 1 outcome values, arranged in ascending order: z_-l < ⋯ < z₀ < ⋯ < z_q. The corresponding probability combination is denoted as p = (p_-l, ⋯ , p_q). For the sorted z_t, the expression for its cumulative decision weighting function π_t is given as follows: $π_{t} = {\begin{matrix} w^{+} (p_{t} + \dots + p_{q}) \\ - w^{+} (p_{t + 1} + \dots + p_{q}), & t \geq 0 \\ w^{-} (p_{- l} + \dots + p_{t}) \\ - w^{-} (p_{- l} + \dots + p_{t - 1}), & t < 0 \end{matrix}$ (4) where: w⁺ (p) = p^σ/(p^σ + (1 - p) ^σ) ^1/σ, w^- (p) = p^ɛ/(p^ɛ + (1 - p) ^ɛ) ^1/ɛ, σ = 0.61, and ɛ = 0.69.

When t = - l and t = q, the cumulative decision weights are computed as follows: $π_{t} = {\begin{matrix} w^{+} (p_{n}), & t = q \\ w^{-} (p_{- m}), & t = - l \end{matrix}$ (5)

Utilizing the value function and cumulative weighting function as a foundation, we articulate the corresponding prospect value function as: $V = \sum_{t = - l}^{q} v (z_{t}) π_{t}$ (6)

3 CA in IVIFISs

With the foundational concepts and notations established in the preliminary section, we are now equipped to explore the application of CA within the context of IVIFISs, which will be the focus of the following chapter.

Definition 8. An IVIFIS can be defined by a quadruple $S = (O, A, V, F)$ . $O = {o_{1}, o_{2}, \dots, o_{m}}$ signifies a finite set of objects or agents. $A = {a_{1}, a_{2}, \dots, a_{nitsc}}$ denotes a finite set of issues or attributes. $V = {V_{aitsc} | a \in V}$ represents the set of value ranges for attributes, all of which are IVIFNs. $F$ is a function that maps $O \times A$ to $V$ , and is represented as:

$F (o_{i}, a_{j}) = ([r_{ij}^{L}, r_{ij}^{U}], [s_{ij}^{L}, s_{ij}^{U}])$ (7) where:

(1) When $r_{ij}^{L} + r_{ij}^{U} > s_{ij}^{L} + s_{ij}^{U}$ , the sum of the lower and upper bounds of the positive ratings ( $r_{ij}^{L}$ , $r_{ij}^{U}$ ) is greater than the sum of the lower and upper bounds of the negative ratings ( $s_{ij}^{L}$ , $s_{ij}^{U}$ ). This indicates that, on balance, the positive ratings outweigh the negative ones, leading to an overall positive viewpoint of of o_i on a_j.

(2) When $r_{ij}^{L} + r_{ij}^{U} = s_{ij}^{L} + s_{ij}^{U}$ , the sums of the lower and upper bounds for both positive and negative ratings are equal. This suggests that the positive and negative aspects are balanced, resulting in a neutral viewpoint of o_i on a_j.

(3) When $r_{ij}^{L} + r_{ij}^{U} < s_{ij}^{L} + s_{ij}^{U}$ , the sum of the lower and upper bounds of the negative ratings ( $s_{ij}^{L}$ , $s_{ij}^{U}$ ) is greater than the sum of the lower and upper bounds of the positive ratings ( $r_{ij}^{L}$ , $r_{ij}^{U}$ ). This implies that the negative ratings dominate, leading to an overall negative viewpoint of o_i on a_j.

The IVIFIS can be viewed as an enhancement of the general intuitionistic fuzzy IS. In contrast to the general system, it provides a more effective means of delineating the perspectives of diverse objects on distinct issues in CA, utilizing IVIFNs. For ease of representation, the matrix form of an IVIFIS is as follows:

$M (S) = [\begin{matrix} \begin{matrix} {\tilde{b}}_{11} & {\tilde{b}}_{12} \\ {\tilde{b}}_{21} & {\tilde{b}}_{22} \end{matrix} & \begin{matrix} \dots & {\tilde{b}}_{1 n} \\ \dots & {\tilde{b}}_{2 n} \end{matrix} \\ \begin{matrix} \dots & \dots \\ {\tilde{b}}_{m 1} & {\tilde{b}}_{m 2} \end{matrix} & \begin{matrix} \dots & \dots \\ \dots & {\tilde{b}}_{mn} \end{matrix} \end{matrix}]$ (8)

Example 3. Table 2 showcases an IVIFIS expressed in the form of IVIFNs. Here,

O = {o_{1}, o_{2}, \dots, o_{6}}

represents six distinct member agents, and

A = {a_{1}, a_{2}}

denotes two distinct issues or attributes. Analyzing Table 2 yields the following insights:

Table 2

The IVIFIS in Example 3

U	a ₁	a ₂
o ₁	([1, 1] , [0, 0])	([0.7, 0.8] , [0.1, 0.2])
o ₂	([0.7, 0.8] , [0.1, 0.2])	([0.4, 0.5] , [0.4, 0.5])
o ₃	([0.0, 0.1] , [0.8, 0.9])	([0.0, 0.1] , [0.7, 0.8])
o ₄	([0.4, 0.5] , [0.4, 0.5])	([0.1, 0.2] , [0.7, 0.8])
o ₅	([0.7, 0.9] , [0, 0.1])	([0.3, 0.4] , [0.4, 0.6])
o ₆	([0, 0] , [1, 1])	([0.8, 0.9] , [0, 0.1])

From $F (o_{1}, a_{1}) = ([1, 1], [0, 0])$ , it’s evident that, as 1 + 1 >0 + 0, member o₁ holds a supportive stance on issue a₁. Conversely, from $F (o_{3}, a_{1}) = ([0.0, 0.1], [0.8, 0.9])$ , it’s clear that, as 0.0 + 0.1 < 0.8 + 0.9, agent o₃ adopts an opposing stance on issue a₁. Furthermore, based on $F (o_{4}, a_{1}) = ([0.4, 0.5], [0.4, 0.5])$ , given that 0.4 + 0.5 = 0.4 + 0.5, o₄ maintains a neutral stance on issue a₁.

The information from the IVIFIS in Table 2 can be represented as an IVIF matrix:

\begin{matrix} M (S) = \\ [\begin{matrix} \begin{matrix} ([1, 1], [0, 0]) \\ ([0.7, 0.8], [0.1, 0.2]) \\ ([0.0, 0.1], [0.8, 0.9]) \end{matrix} & \begin{matrix} ([0.7, 0.8], [0.1, 0.2]) \\ ([0.4, 0.5], [0.4, 0.5]) \\ ([0.0, 0.1], [0.7, 0.8]) \end{matrix} \\ ([0.4, 0.5], [0.4, 0.5]) & ([0.1, 0.2], [0.7, 0.8]) \\ \begin{matrix} ([0.7, 0.9], [0, 0.1]) \\ ([0, 0], [1, 1]) \end{matrix} & \begin{matrix} ([0.3, 0.4], [0.4, 0.6]) \\ ([0.8, 0.9], [0, 0.1]) \end{matrix} \end{matrix}] \end{matrix}

Definition 9. Consider $S = (O, A, V, F)$ as an IVIFIS. $W = {ω_{1}, ω_{2}, \dots, ω_{n}}$ denotes the weights assigned to attributes in the IVIFIS, subject to the condition: $\sum_{j = 1}^{n} ω_{j} = 1$ . Consequently, the comprehensive stance of agent o_i towards all attributes can be articulated as follows: $\begin{matrix} C (o_{i}) = & ([\sum_{j = 1}^{n} ω_{j} * r_{ij}^{L}, \sum_{j = 1}^{n} ω_{j} * r_{ij}^{U}]) \\ + [\sum_{j = 1}^{n} ω_{j} * s_{ij}^{L}, \sum_{j = 1}^{n} ω_{j} * s_{ij}^{U}] \end{matrix}$ (9)

As per Definition 9, it is feasible to compute the comprehensive stance of any object towards all attributes. Nonetheless, since the resultant value remains an IVIFN, employing it directly for CA is challenging. This paper draws inspiration from the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method [44] and defines a closeness function tailored to IVIFNs.

Definition 10. Consider $\tilde{b} = ([r^{L}, r^{U}], [s^{L}, s^{U}])$ as an IVIFN. Let ${\tilde{b}}^{+} = ([1, 1], [0, 0])$ denote the positive ideal IVIFN and ${\tilde{b}}^{-} = ([0, 0], [1, 1])$ represent the negative ideal IVIFN. With this, the closeness function for IVIFNs is defined as follows:

$ϵ (\tilde{b}) = d (\tilde{b}, {\tilde{b}}^{-}) / (d (\tilde{b}, {\tilde{b}}^{-}) + d (\tilde{b}, {\tilde{b}}^{+}))$ (10)

Proposition 1. For $\tilde{b} = ([r^{L}, r^{U}], [s^{L}, s^{U}])$ , the closeness function $ϵ (\tilde{b})$ takes values within the interval [0, 1].

Proof. As per Definition 7, it is evident that $d (\tilde{b}, {\tilde{b}}^{-}) \geq 0$ and $d (\tilde{b}, {\tilde{b}}^{+}) \geq 0$ . When $d (\tilde{b}, {\tilde{b}}^{+}) = 0$ , $ϵ (\tilde{b})$ reaches its maximum value of 1, representing its highest point. Conversely, when $d (\tilde{b}, {\tilde{b}}^{-}) = 0$ , $ϵ (\tilde{b})$ achieves its minimum value of 0, signifying its lowest point. Thus, $ϵ (\tilde{b})$ always falls within the range of [0, 1]. □

In accordance with Definition 10, this paper can further provide the expression for the closeness function $ϵ (\tilde{b})$ for $o \in O$ as follows: $ϵ (o) = \frac{d (C (o), {\tilde{b}}^{-})}{d (C (o), {\tilde{b}}^{-}) + d (C (o), {\tilde{b}}^{+})}$ (11)

Furthermore, by leveraging ϵ (o) and two thresholds α and β, it becomes possible to effectively categorize affirmative coalition, impartial coalition, and adverse coalition on $O$ in CA scenarios.

Definition 11. Consider $S = (O, A, V, F)$ representing an IVIFIS. Let α and β denote two thresholds, with values satisfying 0 ≤ α, β ≤ 1. Subsequently, the affirmative coalition $P_{(α, β)}$ , impartial coalition $T_{(α, β)}$ , and adverse coalition $N_{(α, β)}$ in CA scenarios are defined as follows: $\begin{matrix} P_{(α, β)} = {o \in O | ϵ (o) \geq α} \\ T_{(α, β)} = {o \in O | β < ϵ (o) < α} \\ N_{(α, β)} = {o \in O | ϵ (o) \leq β} \end{matrix}$ (12)

In accordance with Definition 11, this paper partitions the universe $O$ into the affirmative coalition $P_{(α, β)}$ , impartial coalition $T_{(α, β)}$ , and adverse coalition $N_{(α, β)}$ based on two thresholds. This partitioning adheres to the condition $P_{(α, β)} \cup T_{(α, β)} \cup N_{(α, β)} = O$ .

Example 4. Assuming that the weights of attributes a₁ and a₂ in Table 2 are equal, i.e., ω₁ = ω₂ = 0.5, and let the values of the two thresholds be α = 0.6 and β = 0.4. According to Definition 9, we can compute the comprehensive stances of the six members in Table 2 towards all attributes as follows: $\begin{matrix} C (o_{1}) = ([0.85, 0.90], [0.05, 0.10]) \\ C (o_{2}) = ([0.55, 0.65], [0.25, 0.35]) \\ C (o_{3}) = ([0.00, 0.10], [0.75, 0.85]) \\ C (o_{4}) = ([0.25, 0.35], [0.55, 0.65]) \\ C (o_{5}) = ([0.50, 0.65], [0.20, 0.35]) \\ C (o_{6}) = ([0.40, 0.45], [0.50, 0.55]) \end{matrix}$

By Definition 10, we can compute the closeness function values for the comprehensive stances of all objects: ϵ (o₁) = 0.9, ϵ (o₂) = 0.65, ϵ (o₃) = 0.125, ϵ (o₄) = 0.35, ϵ (o₅) = 0.65, ϵ (o₆) = 0.45. Furthermore, by employing the thresholds α and β, and as per Definition 11, we can ascertain the categorization into affirmative coalition $P_{(α, β)}$ , impartial coalition $T_{(α, β)}$ , and adverse coalition $N_{(α, β)}$ in the CA scenario: $\begin{matrix} P_{(α, β)} = {o_{1}, o_{2}, o_{5}} \\ T_{(α, β)} = {o_{6}} \\ N_{(α, β)} = {o_{3}, o_{4}} \end{matrix}$

The CA results for Example 4 indicate that members o₁, o₂, and o₅ maintain a supportive stance. Agent o₆ remains neutral, while agents o₃ and o₄ hold opposing stances. This outcome leads to dividing the overall domain into three regions based on the attitudes held by different objects.

4 TWCA based on CPT

As per Definition 3, it is established that the thresholds α and β in TWCA can be determined utilizing loss functions and Bayesian risk theory. However, in practical decision-making scenarios, the selection of thresholds must also account for the decision-maker’s risk attitude and preferences. In this section, we will investigate the decision rules and threshold calculations for the IVIFTWCA method, drawing upon the principles of CPT.

Definition 12. The IVIF loss functions are defined by the triplet $\tilde{λ} = {\tilde{Ω}, \tilde{A}, \tilde{R}}$ . $\tilde{Ω} = {P_{(α, β)}, N_{(α, β)}}$ represents the set of states and $\tilde{A} = {a_{P}, a_{T}, a_{N}}$ represents the set of actions. These actions correspond to placing o into the affirmative coalition $P_{(α, β)}$ , the impartial coalition $T_{(α, β)}$ , and the adverse coalition $N_{(α, β)}$ , respectively. The set $\tilde{R} = {{\tilde{λ}}_{PP}, {\tilde{λ}}_{TP}, {\tilde{λ}}_{NP}, {\tilde{λ}}_{PN}, {\tilde{λ}}_{TN}, {\tilde{λ}}_{NN}}$ encompasses six IVIF loss functions. Specifically, ${\tilde{λ}}_{PP}, {\tilde{λ}}_{TP}, {\tilde{λ}}_{NP}$ represent the loss functions when o belongs to $P_{(α, β)}$ and actions a_P, a_T, a_N are taken respectively. Similarly, ${\tilde{λ}}_{PN}, {\tilde{λ}}_{TN}, {\tilde{λ}}_{NN}$ represent the loss functions when o belongs to $N_{(α, β)}$ and actions a_P, a_T, a_N are taken respectively. These are outlined in Table 3.

Table 3
IVIF loss functions

$P_{(α, β)}$ $N_{(α, β)}$

a _P ${\tilde{λ}}_{PP}$ ${\tilde{λ}}_{PN}$

a _T ${\tilde{λ}}_{TP}$ ${\tilde{λ}}_{TN}$

a _N ${\tilde{λ}}_{NP}$ ${\tilde{λ}}_{NN}$

	$P_{(α, β)}$	$N_{(α, β)}$
a _P	${\tilde{λ}}_{PP}$	${\tilde{λ}}_{PN}$
a _T	${\tilde{λ}}_{TP}$	${\tilde{λ}}_{TN}$
a _N	${\tilde{λ}}_{NP}$	${\tilde{λ}}_{NN}$

Suppose there are a total of g decision-makers denoted as E ={ e₁, e₂, …, e_g }. According to the CPT, a decision-maker’s interpretation of gains and losses hinges on their chosen reference points. Let the loss function reference point chosen by decision-maker e_k be ${\tilde{λ}}_{k}^{r} = ([r_{k}^{L}, r_{k}^{U}], [s_{k}^{L}, s_{k}^{U}])$ . Given the nature of the loss function, it is evident that when ${\tilde{λ}}_{• \circ} \leq {\tilde{λ}}_{k}^{r}$ (where • = P, T, N and ∘ = P, N), ${\tilde{λ}}_{• \circ}$ is considered as a gain. Conversely, when ${\tilde{λ}}_{• \circ} > {\tilde{λ}}_{k}^{r}$ , ${\tilde{λ}}_{• \circ}$ is interpreted as a loss. Given that this manuscript represents loss functions using IVIFNs, it is suggested to utilize the distance function of IVIFNs to quantify gains and losses post comparison with the reference point.

Definition 13. A collection of gain-loss functions is denoted by $z = {\tilde{Ω}, \tilde{A}, Z}$ . In this context, $\tilde{Ω}$ and $\tilde{A}$ respectively denote the sets of states and actions. $Z = {z_{PP}^{k}, z_{TP}^{k}, z_{NP}^{k}, z_{PN}^{k}, z_{TN}^{k}, z_{NN}^{k}}$ represents a set of six gain-loss functions. Specifically, $z_{PP}^{k}, z_{TP}^{k}, z_{NP}^{k}$ pertain to the gain-loss functions for decision-maker e_k when $o \in P_{(α, β)}$ and actions a_P, a_T, a_N are taken, respectively. Similarly, $z_{PN}^{k}, z_{TN}^{k}, z_{NN}^{k}$ indicate the gain-loss functions for decision-maker e_k when $o \in N_{(α, β)}$ and actions a_P, a_T, a_N are taken, respectively. The gain-loss function $z_{• \circ}^{k}$ is defined as:

$z_{• \circ}^{k} = {\begin{matrix} d ({\tilde{λ}}_{• \circ}, {\tilde{λ}}_{k}^{r}), {\tilde{λ}}_{• \circ} \leq {\tilde{λ}}_{k}^{r} \\ - d ({\tilde{λ}}_{• \circ}, {\tilde{λ}}_{k}^{r}), {\tilde{λ}}_{• \circ} > {\tilde{λ}}_{k}^{r} \end{matrix}$ (13) where $d (z_{• \circ}, {\tilde{λ}}_{k}^{r}) = (| r_{• \circ}^{L} - r_{k}^{L} | + | r_{• \circ}^{U} - r_{k}^{U} | + | s_{• \circ}^{L} - s_{k}^{L} | + | s_{• \circ}^{U} - s_{k}^{U} |) / 4$ . The meaning expressed by this formula is that when the objective loss function is less than the reference point of the loss function, the value can be considered as a gain; conversely, if it is greater, it is considered as a loss. The calculation of loss and gain is generated by the distance function of the IVIFNs.

Let us assume a collection of value functions is denoted by $v = {\tilde{Ω}, \tilde{A}, V}$ . In this context, $\tilde{Ω}$ and $\tilde{A}$ respectively represent the sets of states and actions. V ={ v_PP, v_TP, v_NP, v_PN, v_TN, v_NN } encompasses six value functions. Specifically, v_PP, v_TP, v_NP pertain to the value functions when $o \in P_{(α, β)}$ and actions a_P, a_T, a_N are taken, respectively. Similarly, v_PN, v_TN, v_NN indicate the value functions when $o \in N_{(α, β)}$ and actions a_P, a_T, a_N are taken, respectively, as depicted in Table 4. The value functions can be subsequently calculated in accordance with formula (3):

$v_{• \circ}^{k} = {\begin{matrix} {(z_{• \circ})}^{δ}, z_{• \circ} \geq 0 \\ - ϑ {(- z_{• \circ})}^{τ}, z_{• \circ} < 0 \end{matrix}$ (14)

Table 4

Value functions of TWCA

	$P_{(α, β)}$	$N_{(α, β)}$
a _P	$v_{PP}^{k}$	$v_{PN}^{k}$
a _T	$v_{TP}^{k}$	$v_{TN}^{k}$
a _N	$v_{NP}^{k}$	$v_{NN}^{k}$

Drawing on the insights from reference [9], the closeness function ϵ (o) utilized in CA comprehensively reflects the attitude of object o, accounting for all attributes. Building upon the TWD model rooted in the CPT as outlined in reference [24], this paper proceeds to furnish the cumulative weighting functions integral to the TWCA method: $π_{•}^{k} (ϵ (o)) = {\begin{matrix} w^{+} (ϵ (o)), & 0 \leq v_{• N}^{k} \leq v_{• P}^{k} \\ 1 - w^{+} (1 - ϵ (o)), & 0 \leq v_{• P}^{k} \leq v_{• N}^{k} \\ 1 - w^{-} (1 - ϵ (o)), & v_{• N}^{k} \leq v_{• P}^{k} < 0 \\ w^{-} (ϵ (o)), & v_{• P}^{k} \leq v_{• N}^{k} < 0 \\ w^{+} (ϵ (o)), & v_{• N}^{k} < 0 \leq v_{• P}^{k} \\ w^{-} (ϵ (o)), & v_{• P}^{k} < 0 \leq v_{• N}^{k} \end{matrix}$ (15) $π_{•}^{k} (1 - ϵ (o)) = {\begin{matrix} 1 - w^{+} (ϵ (o)), & 0 \leq v_{• N}^{k} \leq v_{• P}^{k} \\ w^{+} (1 - ϵ (o)), & 0 \leq v_{• P}^{k} \leq v_{• N}^{k} \\ w^{-} (1 - ϵ (o)), & v_{• N}^{k} \leq v_{• P}^{k} < 0 \\ 1 - w^{-} (ϵ (o)), & v_{• P}^{k} \leq v_{• N}^{k} < 0 \\ w^{-} (1 - ϵ (o)), & v_{• N}^{k} < 0 \leq v_{• P}^{k} \\ w^{+} (1 - ϵ (o)), & v_{• P}^{k} < 0 \leq v_{• N}^{k} \end{matrix}$ (16)

Proposition 2. When ϵ (o) = 0, $π_{•}^{k} (ϵ (o)) = 0$ and $π_{•}^{k} (1 - ϵ (o)) = 1$ . When ϵ (o) = 1, $π_{•}^{k} (ϵ (o)) = 1$ and $π_{•}^{k} (1 - ϵ (o)) = 0$ .

Proof. When ϵ (o) = 0, w⁺ (ϵ (o)) = w^- (ϵ (o)) = 0 and w⁺ (1 - ϵ (o)) = w^- (1 - ϵ (o)) = 1. When ϵ (o) = 1, w⁺ (ϵ (o)) = w^- (ϵ (o)) = 1 and w⁺ (1 - ϵ (o)) = w^- (1 - ϵ (o)) = 0. Hence, the proposition is proven. □

Using the previously computed value functions and cumulative weighting functions, we can now proceed to calculate the value functions corresponding to three distinct actions: $\begin{matrix} V^{k} (a_{P} | o) = v_{PP}^{k} * π_{P}^{k} (ϵ (o)) + v_{PN}^{k} * π_{P}^{k} (1 - ϵ (o)) \\ V^{k} (a_{T} | o) = v_{TP}^{k} * π_{T}^{k} (ϵ (o)) + v_{TN}^{k} * π_{T}^{k} (1 - ϵ (o)) \\ V^{k} (a_{N} | o) = v_{NP}^{k} * π_{N}^{k} (ϵ (o)) + v_{NN}^{k} * π_{N}^{k} (1 - ϵ (o)) \end{matrix}$ (17)

Adhering to the principle of maximizing prospect value, we outline the decision rules for the TWCA method grounded in CPT as follows: $\begin{matrix} (P) If & V^{k} (a_{P} | o) \geq V^{k} (a_{T} | o) and \\ V^{k} (a_{P} | o) \geq V^{k} (a_{N} | o), then o \in P_{(α, β)} \\ (T) If & V^{k} (a_{T} | o) \geq V^{k} (a_{P} | o) and \\ V^{k} (a_{T} | o) \geq V^{k} (a_{N} | o), then o \in T_{(α, β)} \\ (N) If & V^{k} (a_{N} | o) \geq V^{k} (a_{P} | o) and \\ V^{k} (a_{N} | o) \geq V^{k} (a_{T} | o), and o \in N_{(α, β)} \end{matrix}$

Denote α_k as the intersection of $V^{k} (a_{P} | o)$ and $V^{k} (a_{T} | o)$ ; β_k as the intersection of $V^{k} (a_{T} | o)$ and $V^{k} (a_{N} | o)$ ; and γ_k as the intersection of $V^{k} (a_{P} | o)$ and $V^{k} (a_{N} | o)$ . As stipulated in reference [24], if the value functions adhere to the following hierarchy: $v_{NP}^{k} < v_{TP}^{k} < v_{PP}^{k}$ , $v_{PN}^{k} < v_{TN}^{k} < v_{NN}^{k}$ , $v_{PN}^{k} < v_{PP}^{k}$ , and $v_{NP}^{k} < v_{NN}^{k}$ , then the values of thresholds α_k, β_k, and γ_k are uniquely determined. Consequently, the decision rules (P) - (N) can be further simplified and deduced.

Proposition 3. When ϵ (o) = 0, $V^{k} (a_{P} | o) < V^{k} (a_{T} | o) < V^{k} (a_{N} | o)$ . When ϵ (o) = 1, $V^{k} (a_{P} | o) > V^{k} (a_{T} | o) > V^{k} (a_{N} | o)$ .

Proof. If ϵ (o) = 0, then $V^{k} (a_{P} | o) = v_{PN}^{k}$ , $V^{k} (a_{T} | o) = v_{TN}^{k}$ , $V^{k} (a_{N} | o) = v_{NN}^{k}$ . In this case, $V^{k} (a_{P} | o) < V^{k} (a_{T} | o) < V^{k} (a_{N} | o)$ holds true. Similarly, it can be proven that when ϵ (o) = 1, $V^{k} (a_{P} | o) > V^{k} (a_{T} | o) > V^{k} (a_{N} | o)$ . □

Proposition 4. If $v_{NP}^{k} < v_{TP}^{k} < v_{PP}^{k}$ , $v_{PN}^{k} < v_{TN}^{k} < v_{NN}^{k}$ , $v_{PN}^{k} < v_{PP}^{k}$ , and $v_{NP}^{k} < v_{NN}^{k}$ , then the CA decision rule can be further simplified to: $\begin{matrix} (P 1) If ϵ (o) \geq α_{k} and ϵ (o) \geq γ_{k}, then o \in P_{(α, β)} \\ (T 1) If ϵ (o) \leq α_{k} and ϵ (o) \geq β_{k}, then o \in T_{(α, β)} \\ (N 1) If ϵ (o) \leq γ_{k} and ϵ (o) \leq β_{k}, then o \in N_{(α, β)} \end{matrix}$

Proof. In accordance with Proposition 3, if ϵ (o) ≥ α_k, then $V^{k} (a_{P} | o) \geq V^{k} (a_{T} | o)$ ; likewise, if ϵ (o) ≥ γ_k, then $V^{k} (a_{P} | o) \geq V^{k} (a_{N} | o)$ . Under these conditions, in accordance with decision rules (P) - (N), it can be concluded that $o \in P_{(α, β)}$ . Similarly, it can be demonstrated that decision rules (T1) and (N1) are valid. □

Proposition 5. When α_k > β_k, α_k > γ_k > β_k; when α_k ≤ β_k, α_k ≤ γ_k ≤ β_k.

Proof. Proposition 5 can be demonstrated through a proof by contradiction. Let us assume γ_k > α_k > β_k. In this scenario, the decision rules (P1) - (N1) are as follows: if ϵ (o) ≥ γ_k, then $o \in P_{(α, β)}$ ; if β_k ≤ ϵ (o) ≤ α_k, then $o \in T_{(α, β)}$ ; if ϵ (o) ≤ β_k, then $o \in N_{(α, β)}$ . However, under these conditions, it becomes evident that $P_{(α, β)} \cup T_{(α, β)} \cup N_{(α, β)} \neq O$ , which renders the initial assumption false. By a similar rationale, it can be established that none of the other conditions, except Proposition 5, are met. □

Based on Proposition 5, we can streamline the decision rules for the TWCA based on CPT. When α_k > β_k, the value of γ_k is smaller than α_k such that ϵ (o) ≥ α_k can always derive ϵ (o) ≥ γ_k in decision rule P1. Similarly, ϵ (o) ≤ β_k can always derive ϵ (o) ≤ γ_k in decision rule N1. Therefore, the CA domain for decision-maker e_k can be categorized as follows:

$\begin{matrix} P_{(α, β)} = {o \in O | ϵ (o) \geq α_{k}} \\ T_{(α, β)} = {o \in O | β_{k} < ϵ (o) < α_{k}} \\ N_{(α, β)} = {o \in O | ϵ (o) \leq β_{k}} \end{matrix}$ (18)

If α_k ≤ β_k, the value of γ_k is bigger than α_k according to Proposition 5 such that ϵ (o) ≥ γ_k can always derive ϵ (o) ≥ α_k in decision rule P1. Similarly, ϵ (o) ≤ γ_k can always derive ϵ (o) ≤ β_k in decision rule N1. Therefore, the CA domain for decision-maker e_k can be segmented as follows: $\begin{matrix} P_{(α, β)} = {o \in O | ϵ (o) \geq γ_{k}} \\ T_{(α, β)} = \emptyset \\ N_{(α, β)} = {o \in O | ϵ (o) < γ_{k}} \end{matrix}$ (19)

Algorithm 1 An algorithm for deriving IVIFTWCA results based on CPT

Input: The IVIFIS in Table 2 and the IVIF loss functions in Table 3.

Output: The TWCA decisions for different decision-makers

1: for i ∈ [1, m]

2: According to formula (9), calculate the comprehensive attitude $C (o_{i})$ of any object o_i across all attributes;

3: Calculate the closeness function ϵ (o_i) of the comprehensive attitude $C (o_{i})$ for any object o_i according to formula (10);

4: end for

5: for k ∈ [1, g]

6: For any decision-maker e_k, select the corresponding loss function reference point ${\tilde{λ}}_{k}^{r} = ([r_{k}^{L}, r_{k}^{U}], [s_{k}^{L}, s_{k}^{U}])$ ;

7: Based on formula (13), calculate the gain-loss function $z_{• \circ}^{k}$ using the IVIF loss function and the reference point chosen by the decision-maker;

8: Calculate the value function $v_{• \circ}^{k}$ according to equation (14), and further compute the numerical solutions for the thresholds α_k, β_k, and γ_k;

9: for i ∈ [1, m]

10: if α_k > β_k

11: if ϵ (o_i) ≥ α_k

12: Decision: $o_{i} \in P_{(α, β)}$ ;

13: else if ϵ (o_i) ≤ β_k

14: Decision: $o_{i} \in N_{(α, β)}$ ;

15: else

16: Decision: $o_{i} \in T_{(α, β)}$ ;

17: end if

18: else

19: if ϵ (o_i) ≥ γ_k

20: Decision: $o_{i} \in P_{(α, β)}$ ;

21: else

22: Decision: $o_{i} \in T_{(α, β)}$ ;

23: end if

24: end if

25: end for

26: end for

Subsequent to the aforementioned analysis, the precise algorithmic implementation of the TWCA approach put forth in this document is delineated below. As shown in Algorithm 1, it is evident that the algorithm complexity for the IVIFTWCA method based on the CPT is O (m²) if m ≈ g. Otherwise, if m is much larger than g, the time complexity of the overall algorithm will be O (m). The specific decision steps are outlined as follows:

Step 1: Begin with an IVIFIS and an associated IVIF loss function matrix.

Step 2: Proceed to compute the comprehensive attitudes and corresponding closeness values for every object within the IS.

Step 3: For each decision-maker, based on their chosen reference point, evaluate the gain and loss functions alongside the value functions.

Step 4: Calculate the numerical solutions for the thresholds. Use these values to determine the ultimate decision outcomes for the TWCA through comparison.

5 Illustrative example and comparative analysis

To demonstrate the efficacy of the proposed method, this study applies the TWCA technique within the context of conflicts arising among team members during the execution of an audit project. Conducting an audit project entails extensive collaboration among the project team members. Differing viewpoints can lead to conflicts, potentially causing delays or disruptions in the project’s progress. Table 5 showcases an IVIFIS tailored for CA in audit execution. All elements in the table are expressed using IVIFNs. In this table, $O = {o_{1}, o_{2}, \dots, o_{6}}$ represents the six members of the audit team, while $A = {a_{1}, a_{2}, \dots, a_{5}}$ denotes five specific issues, each assigned equal attribute weights.

Table 5
The IVIFIS of audit execution CA

a ₁ a ₂ a ₃ a ₄ a ₅

o ₁ ([0.5, 0.6] , [0.3, 0.4]) ([0.6, 0.7] , [0.1, 0.3]) ([0.6, 0.7] , [0.2, 0.3]) ([0.7, 0.8] , [0.1, 0.2]) ([1, 1] , [0, 0])

o ₂ ([0.0, 0.2] , [0.6, 0.8]) ([0.1, 0.2] , [0.7, 0.8]) ([0.1, 0.2] , [0.7, 0.8]) ([0.4, 0.5] , [0.4, 0.5]) ([0.1, 0.2] , [0.7, 0.8])

o ₃ ([0.1, 0.2] , [0.6, 0.7]) ([0.1, 0.2] , [0.6, 0.7]) ([0.0, 0.1] , [0.8, 0.9]) ([0.0, 0.1] , [0.7, 0.8]) ([0.0, 0.1] , [0.8, 0.9])

o ₄ ([0.4, 0.6] , [0.3, 0.4]) ([1, 1] , [0, 0]) ([0.6, 0.7] , [0.2, 0.3]) ([0.8, 0.9] , [0, 0.1]) ([1, 1] , [0, 0])

o ₅ ([0.2, 0.3] , [0.6, 0.7]) ([0.1, 0.2] , [0.5, 0.6]) ([0.2, 0.3] , [0.3, 0.5]) ([0.3, 0.4] , [0.4, 0.6]) ([0.7, 0.9] , [0, 0.1])

o ₆ ([0.2, 0.3] , [0.6, 0.7]) ([0.1, 0.2] , [0.7, 0.8]) ([0.3, 0.5] , [0.4, 0.5]) ([0, 0.1] , [0.8, 0.9]) ([0, 0] , [1, 1])

	a ₁	a ₂	a ₃	a ₄	a ₅
o ₁	([0.5, 0.6] , [0.3, 0.4])	([0.6, 0.7] , [0.1, 0.3])	([0.6, 0.7] , [0.2, 0.3])	([0.7, 0.8] , [0.1, 0.2])	([1, 1] , [0, 0])
o ₂	([0.0, 0.2] , [0.6, 0.8])	([0.1, 0.2] , [0.7, 0.8])	([0.1, 0.2] , [0.7, 0.8])	([0.4, 0.5] , [0.4, 0.5])	([0.1, 0.2] , [0.7, 0.8])
o ₃	([0.1, 0.2] , [0.6, 0.7])	([0.1, 0.2] , [0.6, 0.7])	([0.0, 0.1] , [0.8, 0.9])	([0.0, 0.1] , [0.7, 0.8])	([0.0, 0.1] , [0.8, 0.9])
o ₄	([0.4, 0.6] , [0.3, 0.4])	([1, 1] , [0, 0])	([0.6, 0.7] , [0.2, 0.3])	([0.8, 0.9] , [0, 0.1])	([1, 1] , [0, 0])
o ₅	([0.2, 0.3] , [0.6, 0.7])	([0.1, 0.2] , [0.5, 0.6])	([0.2, 0.3] , [0.3, 0.5])	([0.3, 0.4] , [0.4, 0.6])	([0.7, 0.9] , [0, 0.1])
o ₆	([0.2, 0.3] , [0.6, 0.7])	([0.1, 0.2] , [0.7, 0.8])	([0.3, 0.5] , [0.4, 0.5])	([0, 0.1] , [0.8, 0.9])	([0, 0] , [1, 1])

Since the weights of the five attributes are equal, we have ω₁ = ω₂ = ω₃ = ω₄ = ω₅ = 0.2. Using Definition 9, we can calculate the comprehensive attitudes of the six members of the audit execution team towards the five specific issues outlined in Table 5: $\begin{matrix} C (o_{1}) = ([0.68, 0.76], [0.14, 0.24]) \\ C (o_{2}) = ([0.14, 0.26], [0.62, 0.74]) \\ C (o_{3}) = ([0.04, 0.14], [0.70, 0.80]) \\ C (o_{4}) = ([0.76, 0.84], [0.10, 0.16]) \\ C (o_{5}) = ([0.30, 0.42], [0.36, 0.50]) \\ C (o_{6}) = ([0.12, 0.22], [0.70, 0.78]) \end{matrix}$ According to Definition 10, we can proceed to compute the closeness functions for the comprehensive attitudes of the six members: ϵ (o₁) = 0.765, ϵ (o₂) = 0.26, ϵ (o₃) = 0.17, ϵ (o₄) = 0.835, ϵ (o₅) = 0.465, and ϵ (o₆) = 0.215.

In order to perform CA within the audit team, we utilize the loss function matrix provided in Table 6, which is described using IVIFNs. We proceed to calculate the values of $S ({\tilde{λ}}_{• \circ})$ as follows: $S ({\tilde{λ}}_{PP}) = - 1$ , $S ({\tilde{λ}}_{TP}) = - 0.15$ , $S ({\tilde{λ}}_{NP}) = 1$ , $S ({\tilde{λ}}_{PN}) = 0.8$ , $S ({\tilde{λ}}_{TN}) = 0$ , $S ({\tilde{λ}}_{NN}) = - 1$ . Let’s consider six decision-makers denoted as E ={ e₁, e₂, …, e₆ }. The loss function reference points ${\tilde{λ}}_{k}^{r}$ for these decision-makers are provided in Table 7, along with the corresponding $S ({\tilde{λ}}_{k}^{r})$ . Using formula (13), we calculate the gain-loss function $z_{• \circ}^{k}$ based on the values of $S ({\tilde{λ}}_{• \circ})$ and $S ({\tilde{λ}}_{k}^{r})$ . Further, employing formula (14), we determine the value function $v_{• \circ}^{k}$ , which is summarized in Table 8. It is observed from Table 8 that the value function $v_{• \circ}^{k}$ adheres to the condition: $v_{NP}^{k} < v_{TP}^{k} < v_{PP}^{k}$ , $v_{PN}^{k} < v_{TN}^{k} < v_{NN}^{k}$ , $v_{PN}^{k} < v_{PP}^{k}$ , $v_{NP}^{k} < v_{NN}^{k}$ . Therefore, the thresholds α_k, β_k, and γ_k have unique values, and the decision rules (P) - (N) can be further simplified.

Table 6

IVIF loss functions of audit team CA

	$P_{(α, β)}$	$N_{(α, β)}$
a _P	([0, 0] , [1, 1])	([0.8, 0.9] , [0, 0.1])
a _T	([0.3, 0.4] , [0.4, 0.6])	([0.4, 0.5] , [0.4, 0.5])
a _N	([1, 1] , [0, 0])	([0, 0] , [1, 1])

Table 7

Reference points of six decision-makers

	${\tilde{λ}}_{k}^{r}$	$S ({\tilde{λ}}_{k}^{r})$
e ₁	([0, 0] , [1, 1])	-1
e ₂	([0.1, 0.2] , [0.7, 0.8])	-0.6
e ₃	([0.3, 0.4] , [0.5, 0.6])	-0.2
e ₄	([0.5, 0.6] , [0.3, 0.4])	0.2
e ₅	([0.7, 0.8] , [0.5, 0.6])	0.6
e ₆	([1, 1] , [0, 0])	1

Table 8

Value functions of audit team CA

	$v_{PP}^{k}$	$v_{TP}^{k}$	$v_{NP}^{k}$	$v_{PN}^{k}$	$v_{TN}^{k}$	$v_{NN}^{k}$
e ₁	0	-1.06	-2.25	-2.051	-1.223	0
e ₂	0.243	-0.606	-1.849	-1.644	-0.78	0.243
e ₃	0.447	-0.088	-1.435	-1.223	-0.3	0.447
e ₄	0.638	0.216	-1.005	-0.78	0.132	0.638
e ₅	0.822	0.422	-0.546	-0.297	0.347	0.822
e ₆	1	0.615	0	0.132	0.543	1

As per formula (17), we have calculated the numerical solutions for the thresholds α_k, β_k, and γ_k in the CPT-based IVIFTWCA method. The results are presented in Table 9. Utilizing the numerical solutions from Table 9 in conjunction with the closeness function ϵ (o) for the comprehensive attitudes of the six object members, we derive the final decision results for this audit team CA problem, as outlined in Table 10. The results in Tables 8 and 9 distinctly illustrate that the selection of reference points by decision-makers significantly impacts the values of the thresholds in the TWCA, ultimately determining the conclusive decision result of the CA problem. This underscores the effectiveness of the IVIFTWCA method based on the CPT. It adeptly addresses uncertainty in intricate environments and accurately encapsulates the risk attitudes and preferences of decision-makers during CA.

Table 9

Thresholds of TWCA for six decision-makers

	α_k	β_k	γ_k
e ₁	0.3366	0.5732	0.4665
e ₂	0.4615	0.4621	0.4501
e ₃	0.7063	0.2848	0.4546
e ₄	0.7960	0.1824	0.4426
e ₅	0.7624	0.1937	0.4179
e ₆	0.6725	0.2555	0.4210

Table 10

TWCA decision results for six decision-makers

	$P_{(α, β)}$	$T_{(α, β)}$	$N_{(α, β)}$
e ₁	{o₁, o₄}	∅	{o₂, o₃, o₅, o₆}
e ₂	{o₁, o₄, o₅}	∅	{o₂, o₃, o₆}
e ₃	{o₁, o₄}	{o₅}	{o₂, o₃, o₆}
e ₄	{o₄}	{o₁, o₂, o₅, o₆}	{o₃}
e ₅	{o₁, o₄}	{o₂, o₅, o₆}	{o₃}
e ₆	{o₁, o₄}	{o₂, o₅}	{o₃, o₆}

In conclusion, this study conducts a comparative analysis of the proposed CPT-based IVIFTWCA with methods outlined in references [10] and [9]. The results of this comparative analysis are summarized in Table 11, where √ signifies “yes” and × denotes “no”. The findings presented in Table 11 unmistakably demonstrate that, in comparison to the TWCA methods outlined in references [10] and [9], the approach presented in this paper, based on CPT and IVIF logic, adeptly captures decision-makers’ risk attitudes and preferences in the context of CA problems. Furthermore, by leveraging IVIFSs, this approach adeptly addresses uncertainty and fuzziness inherent in decision-making contexts. This endows the CPT-based IVIFTWCA method proposed in this paper with distinct advantages and enhanced efficacy.

Table 11

Comparative analysis of three CA methods

	[10]	[9]	Proposed method
Loss function	√	√	√
Fuzzy environment	×	√	√
Interval-valued fuzzy	×	×	√
Reference point	×	×	√
Risk attitude	×	×	√

6 Experiments

In this section, our primary focus is on conducting experiments to validate and thoroughly analyze the effectiveness of the proposed model and methods. All these experiments were conducted on a sophisticated desktop computer running Windows 11. The hardware configuration includes a 13th generation Intel(R) Core(TM) i5-13400F processor with a clock frequency of 2.50 GHz and 16.0 GB RAM, operating under a 64-bit operating system. The experimental software environment is Matlab R2019b, ensuring a robust method evaluation and comparison platform.

Our experiments delve into the comprehensive analysis and validation of the IVIFIS conversion process embedded within the TWCA framework. The critical validation task involves meticulously assessing the efficiency of transforming intuitive fuzzy information into proximity functions. We strategically manipulated the number of agents and issues in our experimental design to identify variations in the conversion process duration and completion status. The range of agent quantity (m) started at 100, rising by 100, up to 1000, while the attribute quantity (n) ranged from 100, increasing to 500 in increments of 100. For each combination of m and n values, the experiment was run 100 times, and the average run time was recorded for further analysis. The running time measurements were recorded in seconds, and the graphical representation of the experimental results is shown in the figure below.

Fig. 1

Running time of IVIFIS transformation with m from 100 to 1000 and n from 100 to 500.

From these experimental results, several key conclusions were drawn. Even with a gradual increase in the number of agents (m) and issues (n), the time requirements for the entire IVIFIS conversion process significantly decreased. This observation validates the practicality and effectiveness of the proposed method. Furthermore, as the number of agents and attributes increased, the overall conversion process exhibited consistent linear growth in run time. This linear correlation indicates that the time required for the proximity function acquisition process scales proportionally with increased agents and attributes.

These insights confirm the robustness of our proposed method and emphasize its practicality and adaptability under different conditions. This provides valuable guidance for the real-world application of our process, laying a foundation for its practical use in diverse scenarios.

7 Conclusion

1) Introduction:

The paper presents a novel approach to TWCA using IVIFSs grounded in CPT.

2) Methodology Overview:

The methodology involves classifying CA scenarios into affirmative, impartial, and adverse coalitions through a closeness function and predefined thresholds. CPT is integrated to effectively capture decision-makers’ risk attitudes and preferences in CA situations.

3) Decision Rules and Thresholds:

Decision reference points from CPT are integrated into the approach to establish CA decision rules and thresholds aligned with decision-makers’ risk preferences.

4) Validation Through Case Studies and Experiments:

Illustrative case studies, comparative analyses and experiments are conducted to demonstrate the superior efficacy of the proposed IVIFTWCA method, emphasizing its ability to capture decision-makers’ risk attitudes and preferences.

5) Significance and Future Research:

The methodology presented holds both theoretical significance and practical relevance in the field of TWCA. Future research efforts will be directed toward investigating the dynamic evolution and granularity adjustments of IVIFISs in the context of TWCA scenarios.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundations of China (Nos. 62206129, 62276136, 72274098, 72371132, and 62176116) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Nos. 22KJB520019 and 20KJA520006).

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Interval-valued intuitionistic fuzzy three-way conflict analysis based on cumulative prospect theory

Abstract

Keywords

1 Introduction

2 Preliminary

2.1 TWCA

Table 1 IS for the Middle East conflict O a 1 a 2 a 3 a 4 a 5 o 1 -1 +1 +1 +1 +1 o 2 +1 0 -1 -1 -1 o 3 +1 -1 -1 -1 0 o 4 0 -1 -1 0 -1 o 5 +1 -1 -1 -1 -1 o 6 0 +1 -1 0 +1

2.3 CPT

Table 3 IVIF loss functions P ( α , β ) N ( α , β ) a P λ ˜ PP λ ˜ PN a T λ ˜ TP λ ˜ TN a N λ ˜ NP λ ˜ NN

Footnotes

Acknowledgments

References

Table 1
IS for the Middle East conflict

$O$ a ₁ a ₂ a ₃ a ₄ a ₅

o ₁ -1 +1 +1 +1 +1

o ₂ +1 0 -1 -1 -1

o ₃ +1 -1 -1 -1 0

o ₄ 0 -1 -1 0 -1

o ₅ +1 -1 -1 -1 -1

o ₆ 0 +1 -1 0 +1

Table 3
IVIF loss functions

$P_{(α, β)}$ $N_{(α, β)}$

a _P ${\tilde{λ}}_{PP}$ ${\tilde{λ}}_{PN}$

a _T ${\tilde{λ}}_{TP}$ ${\tilde{λ}}_{TN}$

a _N ${\tilde{λ}}_{NP}$ ${\tilde{λ}}_{NN}$