Fuzzy decision making for cross domain sentiment analysis and MABAC approach with complex T-spherical fuzzy operators

Abstract

Cross-domain sentiment analysis is an advanced approach in natural language processing that focuses on transferring sentiment knowledge from one domain to another, where labeled data may be scarce or vague. To overcome the impact of redundant and ambiguous information about the sentiment expressions, we explore a potent approach of a fuzzy framework with robust algorithms of a decision analysis system. This article initiates an advanced decision-making approach of the multi-attributive border approximation area comparison (MABAC) method to aggregate human judgments or opinions precisely and accurately. To achieve the goals of the presentation, we expose the notion of complex t-spherical fuzzy set (CT-SFS), which is used to manage uncertainty and vagueness during the aggregation process. Some feasible operations of Sugeno–Weber t-norm and t-conorm are formulated under the system of complex t-spherical fuzzy information. A family of Sugeno–Weber weighted average and weighted geometric operators is also developed based on a complex t-spherical fuzzy framework. To showcase the validation and strength of the discussed mathematical models, an intelligent decision algorithm of the MABAC method is established to resolve multi-criteria decision-making (MCDM) problems under complex t-spherical fuzzy environments. An experimental case study is discussed to evaluate different sentiment recognition methods under conflicting criteria and aggregation operators. Additionally, a comparative analysis is conducted to highlight the superiority and effectiveness of the diagnosed mathematical terminologies.

Introduction

Sentiment analysis is the analytical examination of people’s opinions, attitudes, and assessments of people, things, events, topics, and products, as well as their characteristics^1,2. This kind of analysis is theoretically difficult, yet it is highly beneficial in many real-world applications. People can openly express their thoughts in a variety of fields since social media has expanded in both its types and applications (e.g., blogs, forum debates, reviews, and social networks)^3–5. However, as the training data for such an endeavor must be annotated for a large number of domains, capturing all of these perspectives comes at a high computational expense. The exploitation of a wealth of information that is shared across domains is prevented by this challenge. In fact, a major problem in cross-domain sentiment analysis is the absence of annotated data, which is essential for precise sentiment categorization. Furthermore, customer reviews, for instance, may cover a variety of services or goods for which different terminology is used, making classification even more difficult. Consequently, in order to address this problem, research efforts have recently turned to creating cross-domain methods.

Cross-domain sentiment analysis is an advanced area of sentiment classification that focuses on identifying emotions or opinions in text data across different domains also discussed in Fig. 1. Unlike traditional sentiment analysis, which is often trained and tested on data from the same context, cross-domain sentiment analysis aims to transfer knowledge from a labeled source domain to an unlabeled or sparsely labeled target domain. Due to variations in vocabulary, expression, and context across domains, this task introduces challenges such as domain shift, data sparsity, and semantic inconsistency. To overcome these, methods like domain adaptation, transfer learning, and adversarial training are used to bridge the gap between domains and enable effective sentiment prediction.

Fig. 1.

Visualization of sentiment recognition methods and their interrelations.

The importance of cross-domain sentiment analysis lies in its ability to build robust, scalable models that generalize well to new, unseen domains with little to no labeled data. This capability significantly reduces the cost and time associated with manual labeling, making it highly practical for real-world applications in e-commerce, healthcare, education, and social media monitoring. It empowers businesses and organizations to understand customer sentiment in diverse sectors, adapt to changing market trends, and make informed decisions. Moreover, it plays a critical role in building adaptive AI systems that can respond intelligently to domain-specific language patterns without retraining from scratch. Stepwise sentiment analysis method is also mentioned in Fig. 2.

Fig. 2.

Stepwise sentiment analysis methods illustrating the sequential process of data preprocessing, feature extraction, and decision interpretation for intelligent opinion analysis.

Cross-domain sentiment analysis using cross-domain text data and decision-making approaches is intricately linked, as both aim to handle uncertainty, variability, and information gaps across different contexts. In cross-domain sentiment analysis, models must interpret sentiment expressions from one domain and transfer that understanding to a new, unfamiliar domain—despite linguistic differences and contextual mismatches. This process often involves evaluating multiple features, such as syntactic patterns, semantic similarities, or contextual cues, which inherently align with the principles of decision-making. The MCDM techniques, fuzzy logic, and probabilistic reasoning can be used to weigh and balance these features, helping the model to make informed predictions in a new domain. These approaches enable more robust sentiment classification even when labeled data is limited or the target domain is dynamic and complex.

Advanced decision-making is the techniques and mechanisms that are more complex than direct, intuitive decision-making. The practices are united in standardized procedures and data-based and systematic analysis of assessments in order to enhance the quality of judgments. The multi attribute group decision making MAGDM is a way of decision making where individuals will rank and even evaluate alternatives based on many aspects. The MAGDM is applied to a situation where a number of stakeholders have to participate in a collaborative or team decision-making process. The MAGDM allows improving the quality of decision making by considering diverse criteria, allowing a more comprehensive analysis of alternatives, improving decision group performance, and doing all of this by integrating and uniting multiple perspectives. All regions are full of uncertainties, imprecision, and ambiguous information. The concept of fuzzy sets (FS) evolved by Zadeh⁶ was aimed at regulating human judgment under conditions of the presentation of unclear and ambiguous information. To be more precise, the theory of intuitionistic fuzzy sets (IFS) was developed by Atanassov⁷, where the non-membership value (NMV) and membership value (MV) are restricted to [0, 1]. However, IFS has its weaknesses, especially when the overall figures exceed this bandwidth. To resolve this, Yager⁸ proposed the pythagorean fuzzy set (PyFS), having its MV and NMV sum of squares taking values within the range [0, 1]. This growth provided a system of dealing with uncertainty that was flexible. Later, Yager⁹ generalized the usefulness of the model to cover the q-rung orthopair fuzzy set (q-ROFS). Atanassov¹⁰ discussed different properties and fundamental operations under the system of an intuitionistic fuzzy model.

Later on, Cuong¹¹ proposed the concept of picture fuzzy sets (PFS) that incorporates MV and NMV with some other dimensions, including abstinence value (AV) and refusal value (RV). These advancements enabled the more advanced modeling of uncertain data. Mahmood et al.¹² also developed broader fuzzy frameworks of spherical and t-spherical fuzzy environments. Later on, many research scholars have applied themselves to resolve different complicated real-life applications with numerical examples.

The subject of what would happen if we changed the FS’s range into a unit disc in a complicated plane was also brought up by other researchers. The theory of complex FS (CFS), investigated by Ramot et al.¹³, provides the grade of complex-valued supports pertaining to the unit disc in a complex plane in the form of polar coordinates and is useful for dealing with this type of problem. Furthermore, Alkouri and Salleh¹⁴ enhanced CFS to investigate complex intuitionistic FS (CIFS), which includes the supporting grade and the supporting against grade as complex numbers in a unit disc, provided that the real part (also for the imaginary part) of the supporting grade and supporting against grade is not greater than a unit interval. Ullah et al.¹⁵ additionally adjusted CIFS to investigate the complex PyFS (CPyFS) under the stipulation that the total of the squares of the real (and imaginary) components of the supporting and opposing grades cannot be greater than one from a unit interval. The complicated q-ROFS (Cq-ROFS)¹⁶ was shown to be an effective tool for defining the susceptibility of MADM issues following the introduction of CPyFS. The Cq-ROFS is further characterized by the participation degree and the non-membership degree, whose sum of the q-powers of the real part (and likewise of the imaginary part) is not exactly equal to 1. As a result, Cq-ROFS is more comprehensive than CIFS and CPyFS. Therefore, Cq-ROFS is more comprehensive than CIFS and CPyFS. The Cq-ROFSs are very powerful and can represent a number of situations with uncertainty; however, they still can’t solve many difficulties because these sets only take on the grades for membership and non-membership. Thus, it makes it possible to include complex abstinence grades as a generalization of complex SFSs (CSFSs) and complex PFSs (CPFSs). Table 1 and Fig. 3 demonstrate a comprehensive overview about the fuzzy frameworks and their extensions.

Table 1.

Overview of different fuzzy frameworks with their mathematical expressions.

Framework	Membership term	Non- membership term	Abstinence value	Phase value of Membership term	Phase value of non-membership term	Phase abstinence value	Mathematical shape	Main limitation
Fuzzy set FS6	✓	✗	✗	✗	✗	✗		Cannot express hesitation or complex relations
Intuitionistic Fuzzy set (IFS)	✓	✓	✗	✗	✗	✗		Lacks higher-dimensional representation
Picture fuzzy set (PFS)11	✓	✓	✓	✗	✗	✗		Inability to handle complex or periodic data
t-spherical fuzzy set (T-SFS)12	✓	✓	✓	✗	✗	✗		High computational complexity and depends upon parameter sensitivity
Complex fuzzy set (CFS)13	✓	✗	✗	✓	✗	✗	and	Real-domain interpretation difficulty and limited handling of hesitation
Complex Intuitionistic Fuzzy set (CIFS)14	✓	✓	✗	✓	✓	✗	and	Limited interpretability of complex-valued membership
Complex Picture fuzzy set (CPFS)39	✓	✓	✓	✓	✓	✓	and	Modified approach with amplitude and phase terms
Complex t-spherical fuzzy set (CT-SFS)41	✓	✓	✓	✓	✓	✓	and	A broader and efficient mathematical framework is used to handle uncertain opinions more precisely

Fig. 3.

illustrates the evolution from classical fuzzy sets to advanced hybrid and complex fuzzy models for enhanced uncertainty handling.

Aggregation operators are fundamental tools in the decision-making and data integration process, serving to combine multiple inputs, such as expert judgments or attribute values, into a single representative output that supports meaningful analysis and comparison. Their primary function is to simplify complex, multi-dimensional information by summarizing it while preserving key relationships and priorities among the inputs. Numerous research scholars and mathematicians developed a variety of aggregation operators and mathematical models. For instance, Ali and Yang¹⁷ designed mathematical approaches to assess different sources of renewable energy enterprises. Ali and Pamucar¹⁸ determined a flexible wireless communication device for a smart grid using the MABAC method with Sugeno–Weber aggregation operators. Ali et al.¹⁹ developed Dombi aggregation models to evaluate artificial intelligence tools by incorporating an intuitionistic fuzzy linguistic decision-making approach. Ali and Hila²⁰ introduced some reliable aggregation operators to manage uncertain and vague type information of human opinion. Dhumras et al.²¹ resolved uncertainty in the field of pattern recognition using similarity measures and a complex picture fuzzy framework. Bansal et al.²² developed hybrid mathematical approaches taking into account the t-spherical fuzzy framework. Singh et al.²³ proposed a robust decision-making approach to evaluate different supplier selection using an appropriate decision analysis system. Aggarwal et al.²⁴ modified properties of similarity measures to derive new aggregation operators and a decision support system. Dhumras and Bajaj²⁵ discussed an application related to medical diagnosis using similarity measures to fix uncertainty in human opinions. Dhumras et al.²⁶ combined two different techniques of clustering analysis and pattern recognition, considering a picture fuzzy framework. Irvanizam et al.²⁷ modified an innovative decision analysis model to investigate suitable banking based on financial offering aids. Ashraf and Chohan²⁸ derived hybrid aggregation operators to integrate risk analysis in industrial circumstances. Irvanizam and Zahara²⁹ discovered new mathematical approaches of the Harmonic and arithmetic means for resolving numerical examples related to the healthcare system. Jameel et al.³⁰ discussed properties of interactive algebraic operations to determine ranking among low-carbon techniques. Farid et al.³¹ proposed a dynamic decision analysis approach under t-spherical fuzzy scenario and aggregation operators. Irvanizam et al.³² put forward the novel approach of the best–worst decision analysis model using properties of the harmonic mean. Ali et al.³³ deduced fuzzy-based Frank aggregation operators to fix uncertainty in human judgments. A family of Copula extended aggregation models and optimization technique of the MEREC method was developed by Ali³⁴. Alia et al.³⁵ combined two different theories of Choquet integral and GRA method to investigate a reliable ranking of preferences. Another decision analysis model of the COPRAS method was proposed by Khan and Ali³⁶. Mustafa et al.³⁷ expanded the theory of complex linear Diophantine fuzzy situation to derive Dombi prioritized mathematical models. Ali et al.³⁸ modified properties of Schweizer-Sklar t-norm and t-conorm to determine water purification under different key criteria.

To manage vagueness and uncertainty during the integration process of an advanced agriculture system, Ma et al.⁴² designed a list of Schweizer-Sklar aggregation operators under a q-rung orthopair fuzzy scenario. Senapati et al.⁴³ discussed prioritization among transportation sharing techniques with power aggregation mathematical models. Senapati et al.⁴⁴ suggested a reliable decision support system to manage uncertainty and vagueness in expert’s judgments during the aggregation process. Akram and Ahmad⁴⁵ discussed a complicated real-life application related to water supply with a decision support system and aggregation operators. Akram and Bibi⁴⁶ explored a novel theory of the PROMETHEE method under consideration of 2-tuple linguistic Fermatean fuzzy frameworks. Hussain et al.⁴⁷ derived a family of robust mathematical approaches of Aczel Alsina t-norm and t-conorms with a decision support system. Akram et al.⁴⁸ applied the theoretical concepts of the spherical fuzzy rough framework to enhance the applicability of aggregation process and decision-making models. Akram et al.⁴⁹ utilized a feasible decision support system of the CODAS method for handling a group of experts opinions with a numerical example. Farid et al.⁵⁰ investigated a suitable smart waste management source based on prioritization among different available alternatives under different conflicting criteria and decision-making models. Hussain et al.⁵¹ developed Muirhead mean models motivated by the significance of Frank t-norm and t-conorm. Riaz and Farid⁵² modified the linear Diophantine fuzzy framework to investigate the ranking of alternatives and decision-making models. Hussain et al.⁵³ fixed uncertainty in the aggregation process to evaluate a suitable solar panel by applying Aczel Alsina and Heronian mean operators with a decision-making model. Riaz and Farid⁵⁴ utilized properties of soft max aggregation operators with linear Diophantine fuzzy environments to assess feasible green supply chain enterprises. Hussain et al.⁵⁵ derived intuitionistic fuzzy Sugeno–Weber mathematical models to investigate feasible digital security techniques. Wang et al.⁵⁶ employed the theoretical concepts of q-rung orthopair fuzzy frameworks with Sugeno–Weber operators with advanced decision-making optimization techniques.

Research gap

The research gap behind developing complex T-spherical fuzzy Sugeno–Weber aggregation operators and the MABAC method of decision analysis lies in the limited exploration of advanced fuzzy environments where multi-dimensional uncertainty, hesitation, and periodic information coexist. Existing research has mostly focused on simple extensions of fuzzy environments, leaving a significant gap in developing models that can simultaneously address computational efficiency, parameter sensitivity, and interpretability when solving multi-criteria decision-making problems under highly uncertain and vague conditions.

On the other hand, in the domain of cross-domain sentiment analysis, most existing models are still constrained by domain-dependency, lack of generalization, and insufficient handling of uncertain linguistic expressions. Sentiment knowledge learned in one domain often fails to transfer effectively to another due to contextual shifts and feature mismatches. While fuzzy-based decision models, such as those incorporating MABAC with advanced aggregation operators, offer the potential to handle vagueness and inconsistency in sentiment data, there has been limited research on integrating such frameworks into cross-domain sentiment analysis.

Motivation and novelty of proposed methodologies

The motivation behind the MABAC method stems from the need for a stable, consistent, and mathematically simple approach to MCDM problems. Traditional MCDM methods often face challenges such as rank reversal, sensitivity to normalization, and limited interpretability when handling complex or uncertain decision environments. Pamučar and Ćirović⁵⁷ initiated a novel theory of the MABAC method to investigate the ranking of preferences under different conflicting criteria. The MABAC method addresses these limitations by introducing the concept of a border approximation area, which serves as a reference point to measure the deviation of each alternative’s performance. This geometric and additive structure allows for intuitive and direct comparisons across multiple criteria, enhancing both the reliability and transparency of the decision-making process. Its simplicity and versatility make it adaptable for integration with fuzzy environments and other aggregation techniques, thereby increasing its applicability in real-world problems.

On the other hand, the use of CT-SFS is motivated by the increasing need to represent and process highly uncertain and imprecise information in modern decision-making scenarios. The CT-SFS is a broader framework of IFSs and T-SFSs incorporate truth, falsity, indeterminacy, and refusal value, with additional complex number components that capture dynamic and oscillating behavior. This extended structure provides a richer mathematical framework to represent human judgment, especially in situations involving incomplete information. Besides the concepts of CT-SFSs, the decision models of the MABAC method offer a more comprehensive and realistic representation of uncertainty in complex domains. The CT-SFSs cover broader and efficient human opinions due to their structure and features. The CT-SFS is an extended fuzzy framework of IFSs and T-SFSs used to fix uncertainty and imprecision in complicated real-life applications. However, the main contributions and primary features of the manuscript are expressed as follows:

1. To expose a novel approach of a complex t-spherical fuzzy framework to fix uncertainty and vagueness in expert’s judgments

2. Formulation of feasible operations of Sugeno–Weber t-norm and t-conorm with viable comparison rules

3. To develop a family of new mathematical approaches of Sugeno–Weber weighted average and Sugeno–Weber weighted geometric operators with appropriate properties

4. To apply the theory of the MABAC method to determine a flexible ranking of preferences under different conflicting criteria, and derive mathematical approaches of Sugeno–Weber t-norms

5. To show the feasibility and effectiveness of the proposed mathematical aggregation operators, we discuss a numerical example to investigate a reliable sentiment analysis method under different conflicting criteria and proposed mathematical methodologies.

6. To exhibit the strength and superiority of the pioneered aggregation operators, compared the obtained results with previous aggregation operators in the comparison section. Furthermore, geometrical interpretations also explore the results of score functions. In Fig. 4, we also mentioned the primary outlines of the manuscript.

Fig. 4.

Visual representation of the key contributions of the manuscript, methodological innovations, and integrated framework developed for enhanced decision-making analysis.

Layout of the manuscript

The remaining sections of the manuscript are explored as follows: Section "Preliminaries" presents the basic overview of Sugeno–Weber t-norm and t-conorm, CT-SFSs with flexible operations and comparison rules. Section "Basic operations of Sugeno–Weber operators" formulates operational laws of Sugeno–Weber t-norm and t-conorm, taking into account CT-SF information. In Section "Sugeno–Weber weighted average aggregation operators", we derive a family of mathematical aggregation operators of weighted average operators using Sugeno–Weber operations. Moreover, a list of geometric operators is modified with operational laws of Sugeno–Weber operations in Section "Sugeno–Weber geometric aggregation operators". In section "Overview of the MABAC method under complex T-spherical fuzzy information", we initiate a novel theory of the MABAC method for resolving complicated real-life challenges under multiple conflicting criteria. Furthermore, a decision algorithm for the MCDM problem is also elaborated to investigate a suitable ranking of preferences. To deliberate on the rationality and efficiency of proposed decision-making models and mathematical approaches, we discuss a numerical example to assess prioritization among teaching materials under different key criteria. To show the strength and applicability of the proposed mathematical models, we compared the obtained results with the results of existing aggregation operators in Section “Result discussion”. Finally, a summary of the manuscript is mentioned in section “Conclusion”.

Preliminaries

This section briefly expresses a fundamental overview of CT-SFS with its fundamental rules. The basic theory of the Sugeno–Weber t-norm and t-conorm is also discussed here.

Definition 1

(Sarkar et al.⁵⁸) The notion of Sugeno–Weber t-norm and t-norm is defined as follows:

and

Definition 2

(Mahmood et al.¹²) Let be a non-empty set and a T-SFS is characterized as follow:

As and indicate membership value (MV), abstinence value (AV), and non-membership value (NMV), respectively, and a T-SFS is expressed with the following mathematical condition:

The refusal index of a T-SFS is given by . A T-spherical fuzzy value (T-SFV) is denoted by .

Definition 3

(Ali et al.⁴¹) A CT-SFS is expressed as follows:

Note that all MV, AV and NMV  =  and partitioned into two components, such as amplitude and phase terms. The mathematical condition of CT-SFS is given by:

and

Furthermore, the refusal value of in a CT-SFS is denoted by . Additionally, a complex t-spherical fuzzy value (CT-SFV) is represented by .

Definition 4

(Ali et al.⁴¹) For any CT-SFV , we can express the score function and accuracy function as follows:

1

and

2

Basic operations of Sugeno–Weber operators

In this section, we deduced the operational laws of Sugeno–Weber triangular norms in the light of complex t-spherical fuzzy theory.

Definition 5

Let a class of CT-SFVs

and with and . Then:

1. 

2. 

3. 

4. 

Sugeno–Weber weighted average aggregation operators

In this section, the authors deliberated a list of new mathematical approaches to express the prioritization among different arguments under the system of t-spherical fuzzy context.

Definition 6

Consider a series of CT-SFVs with degree of weights of such that and . The CT-SFSWWA operator is articulated as follows:

3

Theorem 1

Let a collection of CT-SFVs . Then, the integrated value by the CT-SFSWWA operator is again a CT-SFV, and we have:

4

Proof

See the proof of the theorem 1 in Appendix A.

Theorem 2

Let a class of CT-SFVs , implies that . Then we have:

Proof

We have shifted proof of the theorem 2 in appendix B.

Theorem 3

Let two families of CT-SFVs and . If , such that and we have:

Proof

The proof is straightforward.

Theorem 4

Let a class of CT-SFVs . If and . Then we have:

Sugeno–Weber geometric aggregation operators

Definition 8

Consider a series of CT-SFVs with degree of weights of such that and . The CT-SFSWWG operator is articulated as follows:

5

Theorem 9

Let a collection of CT-SFVs . Then, the integrated value by the CT-SFSWWG operator is again a CT-SFV, and we have:

6

Theorem 10

Let a class of CT-SFVs , implies that . Then we have:

Theorem 11

Let two families of CT-SFVs and . If , such that and we have:

Theorem 12

Let a class of CT-SFVs . If and . Then we have:

Overview of the MABAC method under complex T-spherical fuzzy information

The main impact of this part is to assess the MABAC model while taking into account the suggested operators. Pamucar and Cirovic⁵⁷ initially introduced the MABAC technique model in 2015, along with its applications, including^59–61. The distance metrics are necessary for the study of MABAC under different conflicting criteria. By comparing the distance measures between any two values, we seek to determine the aggregated matrix, from which we can determine the ideal value in the dataset. This subsection presents a brief overview of the MCDM problem by incorporating CT-SFVs and different conflicting criteria. For this purpose, let and are two sets that denote the set of finite alternatives and attributes. Furthermore, the weight of the criteria such that and . The stepwise process of the MCDM problem is mentioned as follows. The primary steps of the MABAC method are given below.

(a) Decision algorithm of the MABAC method

Step 1: Arrange experts opinions in the form of CT-SFVs and arrange them in a decision matrix as follows:

7

Step 2: The Normalization process is needed if more than one type of attribute information such as beneficial and non-beneficial attribute information. The following expression is used for the normalization of the standard decision-matrix as follows:

8

Step 3: Obtained weighted decision matrix using the following expression.

9

Step 4: Aggregate experts information using derived mathematical approaches of CT-SFSWWA and CT-SFSWWG operators as follows:

10

and

11

Step 5: Here, we compute the distance measure using information from the above two matrices, such as:

12

Notice that the distance measure⁶² is obtained using the following expression.

13

Step 6: As a result, we look into the assessment values based on the following formula:

14

Step 7: Lastly, we use the ranking values based on their appraisal values to determine which of the selections is the best choice. In order to increase the value of the developed theory, we lastly demonstrate the stability of the suggested theory by analyzing different optimal options. Figure 5 demonstrates a stepwise decision algorithm of the MABAC method.

Fig. 5.

Stepwise decision algorithm illustrating the systematic implementation of the MABAC method for addressing complex MADM problems, ensuring accurate ranking and efficient decision evaluation.

(b) Decision algorithm of the MADM problem

Step 1: We aim to organize expert’s opinions in the form of CT-SFVs, and the decision matrix is given by:

15

Step 2: This step is necessary if more than one type of attribute, such as beneficial and non-beneficial information.

Step 3: Employed mathematical methodologies of CT-SFSWWA and CT-SFSWWG operators.

Step 4: Demonstrate score or accuracy functions associated with each alternative.

Step 5: Rank alternatives based on computed score or accuracy functions.

Numerical example

Sentiment analysis methods range from traditional machine learning to cutting-edge deep learning and hybrid models, each offering unique advantages for interpreting emotional tone in text data. This real-world application demonstrated the power of advanced sentiment analysis methods to enhance decision-making, improve customer experience, and drive business strategies through intelligent text understanding. Sentiment analysis methods are essential because they unlock the hidden emotions, opinions, and attitudes embedded in text, transforming raw data into meaningful insights. These methods help identify trends, detect customer satisfaction or dissatisfaction, and guide strategic decisions by enabling data-driven emotional intelligence. To achieve the main goals of this experimental case study, we have considered different sentiment analysis methods, which are discussed as follows:

Lexicon-Based Methods

Machine Learning Methods

Deep Learning Methods

Transfer Learning with Pre-trained Models

Domain Adaptation Techniques

Hybrid Approaches

Additionally, Table 2 also explores key features of the different sentiment analysis methods.

Table 2.

Comparison of different sentiment analysis methods.

Method	Cross-domain support	Enhancement for cross-domain
Lexicon-Based	Low	Use domain-adapted lexicons
Hybrid Methods	High (with flexibility)	Combine rules, lexicons, and ML with adaptation modules
Deep Learning	Medium–High	Pre-training + Fine-tuning
Transfer Learning (BERT)	High	Domain-adaptive fine-tuning, adversarial training
Traditional ML	Medium	Feature transformation, instance reweighting
Domain Adaptation	High	TCA, CORAL, DANN, shared representation

The above-discussed sources of learning materials are evaluated under the following key features:

Versatility across industries

Cross-domain sentiment analysis allows models to be applied across various sectors, from healthcare feedback to product reviews, without retraining from scratch, saving time and resources.

Reduced dependence on labeled data

By transferring knowledge from a well-labeled source domain to a low-resource target domain, these methods significantly reduce the need for expensive and time-consuming manual annotations.

Enhanced model intelligence

Cross-domain techniques improve the generalization ability of sentiment models, enabling them to understand context shifts, slang, and emotional cues across different types of text data.

Broader sentiment coverage

They capture a wider variety of expressions, tone, and domain-specific sentiments, making the analysis more comprehensive and reflective of real-world linguistic diversity.

Enhance customer experience

Detect customer satisfaction or dissatisfaction to improve services, personalize responses, and resolve issues proactively.

Understand public opinion

Analyze large volumes of text data (reviews and feedback) to determine public sentiment toward products, services, or events.

The investigation and ranking among different discussed sentiment analysis methods are under consideration of two different decision analysis processes, such as the MABAC method and the MCDM problem.

Evaluation process with the decision algorithm of the MABAC method

In this subsection, we apply the decision algorithm of the MABAC method to integrate flexible ranking of alternatives under different conflicting criteria. The assessment of alternatives based on experts judgments in the form of CT-SFVs. The detailed investigation process of alternatives is under consideration as follows.

Step 1: Table 3 presents expert opinions about different methods discussed in the case study under the system of CT-SF information. In this table, each alternative is under consideration of six different criteria and key features.

Table 3.

Expert opinion about advanced techniques.

Step 2: The decision-maker organizes the same type of criteria and attributes information. So, we ignore the normalization process in this experimental case study and proceed with the given standard decision matrix.

Step 3: Obtained a weighted decision matrix using same weight for all criterion and scalar multiplicative expression of Sugeno–Weber operation in Definiton 5. The Table 4 depicts the results of the weighted decision matrix.

Table 4.

weighted decision matrix.

Step 4: By considering the values of the weighted decision matrix of Table 4, we aggregate expert opinion using derived approaches of CT-SFSWWA and CT-SFSWWG operators, as described in Table 5. Here, decision-maker integrates values of different criteria using weights associated to each criterion.

Table 5.

Integrated decision-matrix obtained from the proposed mathematical approaches of CT-SFSWWA and CT-SFSWWG operators.

	CT-SFSWWA
	0.2274	0.2126	0.9328	0.9298	0.9254	0.2274
	0.2835	0.2277	0.9373	0.9279	0.9255	0.2835
	0.2538	0.2229	0.9309	0.9272	0.9327	0.2538
	0.2995	0.2288	0.9305	0.9278	0.9309	0.2995
	0.2258	0.2509	0.9349	0.9301	0.9313	0.2258
	0.2227	0.2108	0.9278	0.9295	0.9344	0.2227
	CT-SFSWWG
	0.0117	0.0162	0.0157	0.0113	0.0153	0.0117
	0.0341	0.0177	0.0137	0.0172	0.0265	0.0341
	0.0162	0.0177	0.0068	0.0132	0.0136	0.0162
	0.0157	0.0137	0.0068	0.0131	0.0161	0.0157
	0.0113	0.0172	0.0132	0.0131	0.0128	0.0113
	0.0153	0.0265	0.0136	0.0161	0.0128	0.0153

Step 5: Integrate the results of distance measures using the following formula.

We find out distance measures by combining the above two decision matrices, and Table 6 illustrates their results.

Table 6.

Results of distance measures.

	CT-SFSWWA					CT-SFSWWG

Step 6: In order to investigate a suitable optimal option, we computed appraisal values that are described in Table 7. Finally, the ranking of alternatives and are obtained from both mathematical aggregation operators of CT-SFSWWA and CT-SFSWWG operators, respectively.

Table 7.

shows appraisal values.

	0.0342	0.0550
	0.0408	0.0695
	0.0325	0.0532
	0.0339	0.0495
	0.0335	0.0527
	0.0395	0.0677

In order to understand the graphical behavior of the computed results of alternatives, we explore appraisal values in a graphical representation of Fig. 6. From this figure, we can easily observe that the colored bar has the highest appraisal value computed by the derived approaches of CT-SFSWWA and CT-SFSWWG operators.

Fig. 6.

Graphical illustration of the appraisal values for each alternative, highlighting the comparative performance and ranking outcomes in the decision model.

Evaluation with the decision algorithm of the MCDM problem

To highlight the validation and compatibility of derived mathematical models of Sugeno–Weber aggregation operators, we also applied a stepwise decision of the MCDM problem to investigate feasible optimal options under different conflicting criteria.

Step 1: Consider the expert’s judgments listed in Table 3.

Step 2: The normalization process is meaningless because all criteria are of the same type.

Step 3: Aggregate the expert’s opinion using weights of critiera corresponding to each attribute and the proposed mathematical models of CT-SFSWWA and CT-SFSWWG operators, as described in Table 8.

Table 8.

Aggregated results by the proposed operators.

	CT-SFSWWA
	0.4112	0.3848	0.4715	0.4239	0.3315	0.4552
	0.5103	0.4118	0.5297	0.3887	0.3354	0.4117
	0.4581	0.4032	0.4426	0.3755	0.4702	0.3797
	0.5381	0.4136	0.4365	0.3865	0.4420	0.3997
	0.4083	0.4529	0.5004	0.4305	0.4494	0.4449
	0.4028	0.3816	0.3882	0.4199	0.4939	0.4611
	CT-SFSWWG
	0.4059	0.3785	0.4715	0.4239	0.3384	0.4700
	0.4892	0.4075	0.5297	0.3887	0.3363	0.4157
	0.4534	0.3892	0.4426	0.3755	0.4724	0.3842
	0.5054	0.4091	0.4365	0.3865	0.4534	0.4010
	0.4059	0.4490	0.5004	0.4305	0.4538	0.4528
	0.3979	0.3755	0.3882	0.4199	0.5103	0.4714

Step 4: Determined the score functions of each alternative by employing the expression of Definition 4, and the computed results are shown in Table 9.

Table 9.

Results of score functions associated with alternatives.

Step 5: To compute the best optimal option, rank the alternatives, arranging the obtained score functions in descending order. However, the ranking of alternatives is obtained from both the mathematical aggregation operators of CT-SFSWWA and CT-SFSWWG operators. Figure 7 shows the graphical behavior of the computed score function by the MCDM problem and derived aggregation operators.

Fig. 7.

shows the ranking of preferences based on computed score functions. Each column of a different color indicates an alternative with its score values, and the highest score function is the best one.

Sensitivity analysis

This subsection shows the efficiency and validity of proposed decision-making models by setting different parametric variables in step 3 of the MCDM problem. By changing the value of Sugeno–Weber t-norm and t-conorm in the derived mathematical approaches of the CT-SFSWWA and CT-SFSWWG operators. To achieve the goal of this study, we applied different parametric variables of Sugeno–Weber t-norm and t-conorm from to . After analyzing the results of score functions associated with alternatives, we have acquired a ranking of preferences from both developed mathematical approaches of CT-SFSWWA and CT-SFSWWG operators, and the computed results are shown in Table 10.

Table 10.

Determine the ranking of alternatives at different parametric variables of Sugeno–Weber operators.

Parametric variable	Ordering of alternatives	Parametric variable	Ordering of alternatives

From Table 10, the decision-maker can analyze the consistency in the ranking of preferences. This stability in the ranking of preferences shows the authenticity and reliability of the proposed mathematical models.

Statistical test

A statistical test called the Spearman Correlation Coefficient (SCC) approach is used to demonstrate the effectiveness and strength of the alternatives’ ranking that has been examined using the suggested mathematical techniques and decision-making model. The SCC method’s theoretical foundation was created by Ali Abd Al-Hameed⁶³. The following is an illustration of the SCC method’s mathematical form:

Keep in mind that is the number of preference choices in a ranking, and is the distance between two variables. The method’s results must fall between 1 and -1. The ranking of alternatives is totally correlated with the compared values if the results are 1, and we will argue that there is a negative correlation between the ranking of alternatives and the results if they are − 1. If the , then the alternative ranking is not ideal. We used the SCC approach on the data in Table 11 to determine the efficacy and efficiency of the alternative ranking.

Table 11.

Results of preferences obtained by the MADM problem and proposed operators.

Rank Set A	Rank Set B
5	5	0	0
4	4	0	0
2	2	0	0
6	6	0	0
3	3	0	0
1	1	0	0

When we use the SCC method on the information in Table 11, we obtain:

We calculated from the SCC method’s statistical test. This demonstrates the validity of the alternative ranking examined using the suggested mathematical approaches.

Result discussion

The proposed decision algorithms of the MABAC method and MADM problem integrated expert opinions in the form of CT-SF information, which provide a comprehensive insight into the ranking and selection of alternatives under different conflicting criteria. By employing CT-SF data, the method efficiently captures both the magnitude and phase aspects of uncertainty, allowing for more precise evaluation of decision criteria. The MABAC approach calculates the border approximation area for each alternative, reflecting its closeness to the ideal and anti-ideal solutions. The computed score functions and distance measures derived from CT-SF information enable flexible comparison among alternatives while maintaining the complex interrelations between criteria. As a result, the ranking obtained demonstrates improved accuracy, robustness, and interpretability compared to traditional fuzzy frameworks. After integration of human opinion with the MABAC method, the decision maker examined is a more efficient method for analyzing human sentiments. The MABAC method has great capabilities to investigate the ranking of preferences without any effect of biased weights of criteria by the decision-maker. Based on the reliability of the proposed approaches, we can say that the decision-makers may identify the most suitable alternative under uncertain and interdependent environments, confirming the effectiveness and reliability of the proposed CT-SF–based MABAC decision-making framework.

Comparative study

This section has great importance in decision-making problems, in comparison with prevailing or derived mathematical methodologies used to reveal the strength of developed aggregation operators. For this purpose, we used some previous aggregation models based on CT-SFSs and a decision support system. To show the validation and efficiency of the proposed operators, the comparison method is a very effective and dominant technique. To initiate a strong comparative analysis, we employed the following mathematical methodologies developed by different mathematicians and research scholars. Ali and Naeem⁶⁴ proposed a list of Aczel Alsina weighted average and weighted geometric operators under the discussed fuzzy framework. Besides the theoretical concepts of CT-SFSs, Frank prioritized aggregation operators were developed by Ullah et al.⁶⁵. Khan et al.⁶² modified power aggregation operators using algebraic operations and decision-making models. Liu et al.⁶⁶ suggested Aczel Alsina Heronian mean operators to investigate the unknown degree of weights. Ali et al.⁴⁰ put forward the theory of the TOPSIS method to integrate human judgments under different key criteria and a complex t-spherical fuzzy situation. The obtained ranking of preferences by applying existing mathematical models to the considered expert’s opinion is listed in Table 12.

Table 12.

Ranking of preferences obtained by existing aggregation operators.

Operators with authors	Ordering of alternatives
CT-SFSWWA-MABAC method
CT-SFSWWG-MABAC method
CT-SFSWWA
CT-SFSWWG
Ali and Naeem64
Ali and Naeem64
Ullah et al.65
Ullah et al.65
Khan et al.62
Khan et al.62
Ali et al.40
Liu et al.66	Limited structure
Mahmood and Rehman67	Limited structure
Ahmed et al.68	Limited structure

From Table 12, we can observe a few fuzzy frameworks unable to manage expert’s opinions due to limited structure and key features of mathematical terminologies. However, the integration of the MABAC method with Sugeno–Weber aggregation operators under the TCT-SFS represents a significant advancement over existing decision-making methodologies. Traditional models often struggle to manage the high degrees of uncertainty, indeterminacy, and conflicting expert opinions prevalent in real-world problems. The CT-SFSs offer a richer and more flexible mathematical structure by incorporating multiple degrees of membership, non-membership, hesitancy, and complex-valued information. The Sugeno–Weber aggregation operators further enhance this by capturing interactive relationships among criteria and allowing for adjustable sensitivity through a parameter variable. The combined approach not only provides superior discrimination power among alternatives but also demonstrates robustness in handling multi-dimensional uncertainty, making it more effective and practical than conventional MCDM techniques.

Advantages of proposed methodologies.

1. The CT-SFS has extensive information about human judgments, because this fuzzy framework is a broader and efficient mathematical model with four components of membership, abstinence, non-membership, and refusal values.

2. The complex t-spherical fuzzy-based Sugeno–Weber aggregation operators have great capabilities to acquire authentic and smooth aggregated results from the integration of expert’s judgments with the help of numerical examples.

3. The MABAC method offers several advantages that make it an attractive tool in the MADM problem. It provides a simple yet powerful ranking mechanism by evaluating the relative closeness of alternatives to the border approximation area, ensuring consistency and transparency in the decision process.

4. The MABAC method demonstrates clear superiority over many existing decision analysis models due to its balance of computational simplicity and strong discriminatory power in ranking alternatives. Its structure ensures both stability and transparency in decision outcomes, while effectively handling both qualitative and quantitative criteria without significant information loss. Moreover, the method is highly adaptable and integrates smoothly with advanced fuzzy frameworks, enabling decision-makers to better capture uncertainty and hesitancy in real-world problems.

Conclusion

In conclusion, the comprehensive evaluation of various sentiment analysis methods reveals that no single approach universally excels across all contexts, but each holds distinct strengths depending on the data domain and complexity of sentiment expressions. Traditional machine learning models offer simplicity and speed for structured datasets, while deep learning techniques, particularly transformer-based architectures like BERT, demonstrate superior accuracy and contextual understanding in diverse and unstructured environments. Hybrid and ensemble methods further enhance performance by combining interpretability with adaptability. This study highlights the critical role of selecting an appropriate method based on data characteristics and domain variability. Additionally, we modified a comprehensive decision-making structure for computing feasible rankings of alternatives under different conflicting criteria and the MCDM problem. The MABAC method plays a significant role in resolving MCDM problems by providing a systematic and robust framework for evaluating and ranking alternatives based on multiple conflicting criteria. We also constructed a family of Sugeno–Weber aggregation operators under consideration of complex t-spherical fuzzy models with appropriate properties such as idempotency, monotonicity, and boundedness. Moreover, we illustrated an application related to different sentiment analysis methods to highlight the validation and compatibility of the MABAC method under a multi-criteria decision-making model and fuzzy-based mathematical approaches. An experimental case study is discussed to integrate human-based numerical data and also investigate reliable sentiment analysis method based on different conflicting criteria and key features. Additionally, a comparative method is conducted to reveal the superiority and strength of the proposed mathematical models.

Limitations and drawbacks

Although Sugeno–Weber aggregation operators combined with complex T-spherical fuzzy sets and the MABAC method provide a flexible and effective mechanism to capture multi-dimensional uncertainty in decision analysis, several drawbacks and limitations remain. The high computational complexity of handling four-dimensional membership information, coupled with the parameter sensitivity of the Sugeno–Weber operators, can lead to instability and inconsistencies in the final outcomes. Moreover, the MABAC method, while robust in ranking alternatives, may suffer from loss of interpretability when applied to highly complex fuzzy environments, especially when decision-makers face difficulties in parameter selection and preference calibration.

Future directions

In the future, we can expand the developed decision-making model into more sophisticated decision analysis approaches like the EDAS⁶⁹ method, MARCOS⁷⁰ method, TOPSIS method⁷¹, and VIKOR⁷² method. The discussed decision-making models could enhance the adaptability and intelligence of the MCDM problem. Some complex real-life challenges may also be resolved by employing derived mathematical models and the decision-making approach of the MABAC method. The proposed methodologies may also apply to assess the performance of advanced technologies in different fields of life, such as artificial intelligence-based neural networks⁷³, renewable energy sources⁷⁴, and logistics and transportation systems⁷⁵.

Abbreviations

MABAC

Multi-attributive border approximation area comparison

MCDM

Multi-criteria decision-making

IFS

Intuitionistic fuzzy sets

MV

Membership value

q-ROFS

Q-rung orthopair fuzzy set

SFS

Spherical fuzzy set

CFS

Complex fuzzy set

CPyFS

Complex pythagorean fuzzy set

COPRAS

Complex proportional assessment

EDAS

Evaluation based on distance from average solution

MARCOS

Measurement of alternatives and ranking according to the compromise solution

CT-SFS

Complex t-spherical fuzzy set

FS

Fuzzy sets

NMV

Non-membership value

PyFS

Pythagorean fuzzy set

PFS

Picture fuzzy set

T-SFS

T-spherical fuzzy set

CIFS

Complex intuitionistic fuzzy set

CT-SFS

Complex t-spherical fuzzy set

CODAS

Combinative distance-based assessment

VIKOR

VIekriterijumsko KOmpromisno Rangiranje

CSFS

Complex spherical fuzzy set

List of symbols

Sugeno–Weber t-norm

Membership value

Non-membership value

Refusal value

Amplitude term of membership value

Phase term of membership value

Phase term of abstinence value

Element of spherical fuzzy set

Weight vector

Attribute

Accuracy function

Parametric variable

Sugeno–Weber t-conorm

Abstinence value

Non-empty set

Amplitude term of abstinence value

Amplitude term of non-membership value

Phase term of non-membership value

T-spherical fuzzy value

Scalar multiple

Alternatives

Score function

Decision-matrix

Appendix A (Proof of Theorem 1)

Since a class of CT-SFVs and the above expression can be verified by using the induction technique for . So, we have:

Suppose that the above expression is true for .

Next, we can prove for +1.

Appendix B (Proof of Theorem 2)

Since a class of identical CT-SFVs implies that . Then we have:

Author contributions

Yina Qi is fully contributed to this article.

Funding

No external funding has been received for this submission.

Data availability

All data generated or analyzed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.