A multi-criteria approach to ESG-based portfolio optimization incorporating historical performance, forward-looking insights, and credibilistic CVaR: a case study on the DJIA

Abstract

This paper presents a new framework for portfolio management that incorporates sustainability considerations in the form of environmental, social, and governance risks (ESG) alongside the impact of historical and projected company performance, as well as a fuzzy environment to account for expected return uncertainty. To account for market and investor uncertainties, we integrate the conditional value-at-risk (CVaR) risk measure with credibility theory. Unlike traditional portfolio optimization methods, which rely heavily on probabilistic assumptions and may fail to capture real-world uncertainty, our model uses fuzzy logic principles to more realistically represent uncertainty when historical data is incomplete or expert opinions are inaccurate. To assess historical and projected company performance, we use data extracted from quarterly company reports and employ advanced text analytics tools such as FinBERT sentiment analysis and the NotebookLM platform. These tools enable us to extract subtle insights and sentiment trends that are critical for predicting future performance. Empirical validation of the proposed framework is conducted using historical stock return data from the DJIA. A diversified portfolio of assets is selected and the optimal stock allocation is obtained under the proposed credibilistic CVaR (CCVaR) validation approach. The results show that the portfolios optimized under the CCVaR framework offer superior adverse risk control and greater resilience to market volatility compared to traditional approaches. At the end of the paper, the proposed model is compared with the traditional equal weight strategy and the results are presented. This study provides valuable practical insights for risk-averse investors and portfolio managers seeking stronger and more stable investment strategies in uncertain financial environments.

Introduction

Investing in the stock market is essential for building long-term wealth and securing financial stability. Unlike traditional savings accounts, stocks offer the potential for higher returns that can outpace inflation over time^1–4. The stock market offers opportunities for portfolio diversification, risk hedging, and benefiting from company growth. However, the risks of equity investing are significant, and managing them effectively is crucial. Forming a diversified portfolio is one way to reduce investment risk⁵. Diversifying a stock portfolio reduces non-systematic risk by spreading investments across sectors, regions, and asset types. This strategy minimizes volatility and potential losses from individual investments. While systematic risk remains unavoidable, diversification helps stabilize returns and strengthens the portfolio against company-specific events^6–9.

Traditional portfolio optimization models often rely on precise and deterministic inputs to assess risk and return, a methodology that assumes a clear understanding of market conditions¹⁰. However, stock market data inherently contains a high degree of uncertainty due to factors such as market volatility, incomplete information, investor sentiment, and unpredictable external events. To address this challenge, fuzzy set theory¹¹ has emerged as a powerful tool, offering a flexible framework for modeling imprecise, vague, or ambiguous information Fuzzy set theory offers a flexible way to handle uncertainty in expected returns by capturing imprecise and ambiguous information that classical probability cannot. By representing returns as fuzzy numbers, it reflects both expected values and their uncertainty, aligning more closely with investor perceptions. This approach also incorporates subjective and linguistic information, making it especially valuable when data are scarce or markets are volatile, ultimately improving the robustness and interpretability of financial decision-making^12,13.

In addition, another thing that can increase investor confidence and reduce investment risk is examining the company’s financial and operational performance and its future forecast (hereinafter abbreviated as HFP) and also considering sustainability risks (ESG). By considering these factors along with risk and return, a more reliable investment approach can be provided. In this study, we are looking at how these four factors can be expressed together and by considering the constraints of real-world conditions such as cardinality constraints, etc. in the form of a mathematical model. Second, by implementing this proposed model on real-world data, we can measure its performance and see if this model works properly or not?

In this study, we use the CVaR risk measure within the framework of credibility theory to calculate risk. CVaR is a risk measure in portfolio optimization that goes beyond VaR by capturing the average losses beyond a given threshold. By minimizing CVaR, investors can reduce exposure to extreme downside risks, making it a more robust tool for managing portfolio risk¹⁴. CCVaR offers significant improvements over traditional CVaR in the area of portfolio optimization, especially when dealing with imprecise, fuzzy, or uncertain financial data¹⁵. While CVaR is fundamentally based on probability theory and relies on the distribution of historical data to assess risk, CCVaR leverages credibility theory, which is better suited to modeling financial uncertainties that cannot be adequately measured by conventional probabilistic methods¹⁶. One of the key advantages of CCVaR is its ability to handle situations where data is scarce, unreliable, or subject to severe market volatility. In traditional CVaR-based models, overreliance on historical data can lead to inaccurate risk estimates, especially during financial crises or periods of rapid market change. In contrast, CCVaR offers more flexibility by incorporating fuzzy set theory, allowing it to more effectively account for uncertain and ambiguous data¹⁷.

Two metrics that can shape portfolio profitability while controlling risk and mitigating future risks are the incorporation of ESG considerations and HFP performance metric of companies. Incorporating ESG factors as sustainability metrics in portfolio optimization is increasingly recognized as a critical factor in creating resilient, high-performance investment strategies¹⁸. ESG considerations not only reflect a company’s commitment to sustainability and ethical practices, but also provide important indicators of long-term financial health and risk management^19,20. The financial and operational performance of companies serves as a key criterion in evaluating a company’s overall status and can play a critical role in investment decision-making. In many previous studies, company performance has been incorporated by using fundamental financial indicators, such as the P/E ratio, ROA, and similar metrics^21,22. These performance measures are often applied either as filters during the pre-selection stage of asset screening or as objective functions within multi-objective portfolio optimization models, helping to balance financial returns with other investment goals^10,23,24.

Many investment models have been proposed, but very few have considered these four factors together. This study is an improvement on the authors’ previous work¹⁵. Overall, in this paper, we present a comprehensive model for quarterly investment that simultaneously optimizes four key metrics: risk, return, sustainability considerations (via ESG metrics), and the company’s HFP. To assess performance and prospects, we extract information from companies’ quarterly financial reports that provide important insights into past performance and future strategies. For this purpose, we use advanced artificial intelligence (AI) models, in particular FinBERT²⁵, a sentiment analysis model trained on financial text data. FinBERT allows us to systematically assess the sentiment and tone of financial disclosures. In addition, we use the NotebookLM²⁶ tool to extract detailed insights into companies’ historical performance and future prospects, enriching the dataset used in the model. And the Analytical Hierarchy process (AHP)²⁷ method is used to transform the output results of the FinBERT model into a score. Finally, the proposed model is applied to real-world data from DJIA-listed companies. The results of this implementation are presented and discussed, highlighting the real-world usability of the model and its potential value for investors.

The structure of the rest of the paper is as follows: Section “Literature review and research gaps” offers an overview of the relevant literature and previous studies conducted in the field. Section “Model formulation” presents a detailed explanation of the modeling framework and the proposed approach. Section “Data and Experiments” focuses on the implementation of the proposed model using actual-world data and is divided into two main parts: the description of the required data and the presentation of the results. Section “Discussion” offers a discussion of the findings, providing deeper insights into the performance and implications of the proposed model. Finally, Section “Conclusion” summarizes the key conclusions and findings of the study and offers recommendations for future research directions.

Literature review and research gaps

This section is divided into two main parts. The first part provides a comprehensive review of the literature on the subject, reviewing existing research and highlighting key findings, methodologies and theoretical frameworks that have shaped understanding of the topic. The review covers a variety of studies, from seminal works to more recent contributions, to provide a comprehensive view of the current state of knowledge in this area. The second part of this section focuses on identifying research gaps. By highlighting these gaps, we aim to demonstrate the importance of the current study and its potential contribution to advancing the field. Both parts will be discussed in detail below, providing a thorough analysis of the body of work available and the need for further research.

Literature review

In this article, we employ the CVaR risk measure within the framework of credibility theory under a fuzzy environment for risk modeling. Building upon this foundation, we incorporate a set of additional assumptions to further enhance the robustness and realism of the proposed model. CVaR is a widely used risk measure that captures the expected losses occurring beyond a certain confidence level, offering a more comprehensive assessment of tail risk compared to traditional VaR^14,28,29. Unlike VaR, which only provides a threshold value, CVaR accounts for the severity of losses in extreme scenarios, making it particularly valuable for financial decision-making. In portfolio optimization, CVaR is often employed as an objective function or constraint to ensure that portfolios are robust against rare but severe market downturns. By minimizing CVaR, investors can design portfolios that not only aim for attractive returns but also maintain greater stability under adverse market conditions, thus balancing the trade-off between risk and reward more effectively^15,30,31.

In this article, we use the CVaR measure in the context of credibility theory and in a fuzzy environment. Credibility theory, first introduced by Liu¹⁶, offers a robust and flexible mathematical framework for analyzing and modeling phenomena characterized by ambiguity and uncertainty. Later, Liu et al.³² herself presented the CVaR model in the form of credibility theory in a study. Ghanbari et al.¹⁷ propose an innovative method for optimizing cryptocurrency portfolios within the framework of CVaR, effectively addressing the extreme of unpredictability and extreme volatility characteristic of digital asset markets. Credibility theory has been used in various studies, including:^15,33–35. One of the criteria that we have used in the proposed model is the sustainability criterion, which we will briefly review in the following.

Sustainability has also been widely recognized as a key criterion in investment decisions, with several studies addressing its impact. Notable among these are: Utz et al.³⁶ proposed a framework for inverse optimization within the traditional Markowitz portfolio model, expanding it by introducing a third criterion centered on social responsibility (SR). In this approach, social responsibility factors are incorporated into the asset allocation process after the initial screening phase. Xidonas and Essner²⁰ introduced a multi-objective min–max portfolio optimization model aimed at simultaneously maximizing risk-adjusted performance across three ESG investment objectives. Grewal et al.³⁷ examined the relationship between shareholder proposals and their effects on ESG performance and market valuation, comparing firms with high versus low ESG performance.

Chen et al.¹⁸ developed a three-stage framework that integrates ESG performance into portfolio optimization. Unlike approaches that rely on aggregated ESG rating scores, they employed a DEA model with quadratic and cubic terms to more accurately capture the multidimensional nature and interactions among environmental, social, and governance factors. Varmaz et al.³⁸ proposed a portfolio optimization approach based on a multi-indicator model that integrates ESG factors. The growing investment in ESG-related assets has attracted significant attention from both academic researchers and investment funds. Other studies that have addressed ESG considerations in portfolio discussions include the following^39–42. Next, we will look at studies that consider the performance of companies under various headings.

In previous studies, company performance has been incorporated into portfolio optimization in two main ways: (1) during the pre-selection stage for portfolio formation, and (2) as an objective function within multi-objective optimization models. For instance: Larni-Fouik et al.²³ employed a range of significant volatility risk measures to evaluate the impact of risk on portfolio optimization. They incorporated regret as a key risk metric to guide decision-making and help avoid actions that could lead to unfavorable investment outcomes. Additionally, they utilized fundamental indicators to screen and select stocks for inclusion in the investment portfolio. Azmi and Tamiz⁴³ proposed a multi-objective optimization model in which fundamental financial criteria were incorporated as objective functions. The model was solved using the ideal programming method, aiming to achieve a balanced solution that simultaneously satisfies multiple investment goals. Since in this article we use the sentiment analysis method of financial texts to extract company performance from quarterly reports and predict their future performance, it is necessary to review the literature in this field. Sentiment analysis (SA) is a branch of NLP that focuses on detecting and extracting subjective information such as emotions, opinions, and attitudes from written text⁴⁴.

Over the years, SA techniques have evolved significantly and can now be broadly categorized into two main groups: lexicon-based approaches and machine learning-based approaches⁴⁵. Lexicon-based methods rely on predefined dictionaries of words associated with sentiment scores to infer the overall sentiment of a text. These approaches are often straightforward to implement and interpretable, but they may struggle with context, sarcasm, and domain-specific language. In contrast, machine learning-based techniques leverage statistical models trained on labeled datasets to automatically detect patterns and nuances in sentiment⁴⁶.

Many studies have been conducted in this field in recent years including: Shobayo et al.⁴⁷ carried out a comprehensive study evaluating the forecasting performance of FinBERT, GPT-4, and logistic regression (LR) models, focusing specifically on the NGX All-Share Index dataset. Shen and Zhang⁴⁸ explored the application of large language models (LLMs) and FinBERT for financial sentiment analysis (FSA), evaluating their effectiveness on news articles, financial indicators, and corporate disclosures. Their research highlighted the benefits of fast-evolving model technologies, particularly through zero-shot and few-shot learning methods, in improving sentiment classification accuracy. Jun Gu and et al.⁴⁹ developed a novel FinBERT-LSTM model that integrates deep learning techniques with historical stock prices and market data derived from financial, business, and technical news articles to forecast stock prices. Their findings showed that incorporating weighted news categorization into the model significantly enhanced prediction accuracy.

Colasanto et al.⁵⁰ integrated sentiment scores referred to as poles into the Black-Litterman framework to improve stock price forecasting. These sentiment measures, reflecting both positive and negative events impacting individual stocks, were calculated using statements extracted from articles published in the Financial Times, a leading international financial publication. Kim et al.⁴⁵ Analyzed the effect of news sentiment on changes in stock prices by collecting articles from the New York Times website. They compared the accuracy of stock price forecasts with and without the incorporation of sentiment analysis. Employing the FinBERT model, which is tailored for financial language analysis they extracted sentiment information to assess its effect on predictive performance. The study focused on determining whether sentiment analysis could enhance the precision of stock price predictions. Leow et al.⁵¹ developed two models, SAW and SMPT, that incorporate sentiment analysis from Twitter to reflect real-time market trends. The sentiment data is processed using the BERT model, known for its ability to grasp the subtleties of social media content. These models use sentiment indicators to optimize portfolio strategies and adjust investment approaches dynamically. Table 1 shows a summary of the literature review conducted.

Table 1.

Summary of the review of existing research.

No	Article	Model type		Constraints	ESG	SA	Data type	References
		Portfolio	AI Methods
1	Salahi et al. (2013)	CVaR	–	Others	–	–	Robust	52
2	Sharma et al. (2016)	Omega-CVaR	–	Others	–	–	Probabilistic	53
3	Ban et al. (2018)	MV, CVaR	KFCV	Others	–	–	–	54
4	Liu et al. (2018)	Mean-CVaR	–	–	–	–	Fuzzy	55
5	Fang et al. (2019)	MV	–	BC	–	–	Deterministic	56
6	Vercher and Bermúdez (2016)	MASD	–	BC, QC	–	–	Fuzzy	33
7	Ghanbari et al. (2024)	CVaR	–	BC, CC, QC	–	–	Fuzzy	17
8	MirAboalhassani et al. (2022)	CVaR	–	BC, LC, QC, CC	–	–	Fuzzy	34
9	Zhang et al. (2010)	MV	–	BC	–	–	Fuzzy	35
10	Wu et al. (2024)	–	BERT	–	–	–	–	57
11	Fatouros et al. (2023)	–	ChatGPT	–	–	–	–	58
12	Lara-Moreno et al. (2024)	MV	–	BC	✓	–	Deterministic	59
13	Leow et al. (2021)	MV	VADER-FinBERT	BC	–	✓	Deterministic	51
14	Larni-Fooeik et al. (2024)	SAD	–	BC, CC, QC	–	–	Stochastic	23
15	Colasanto et al. (2022)	BL	FinBERT	BC	–	✓	Deterministic	50
16	Ślusarczyk and Ślepaczuk (2025)	MV	–	BC	–	–	Deterministic	60
Our study		CCVaR	FinBERT	CC, QC, BC	✓	✓	Fuzzy

CC, cardinality constraints, QC, quantitative constraints, BC, budget constraint, LC, liquidity constraint, BL, black litterman, MV, mean–variance, CVaR, conditional value at risk, KFCV, K-fold cross-validation, SAD, semi absolute deviation, MASD, mean absolute semi deviation, VADER, valence aware dictionary and sEntiment reasoner.

Research gaps

According to the studies conducted in the previous section, it is clear that despite the significant amount of research in the field of portfolio optimization and investment strategies, there are still opportunities to help investors by examining this topic from different perspectives. Building on previous research and extending it, this study introduces a robust portfolio optimization model formulated in the fuzzy logic framework. The integration of fuzzy logic provides a more flexible and adaptable approach to risk assessment and portfolio management, and addresses the uncertainties and imprecision inherent in financial data. Furthermore, with the rapid development of artificial intelligence tools and their models in recent years, these technologies have proven to be valuable assets in financial decision-making. This study uses artificial intelligence tools in a novel way to improve the portfolio optimization process, enabling more accurate and data-driven decision-making. By combining AI-based sentiment analysis and performance prediction, we can further refine the optimization process and provide a more comprehensive approach to portfolio management. Among the implications of the current study are the following:

• Presenting a comprehensive investment model considering four criteria: risk, return, sustainability considerations, and performance criteria in a fuzzy environment, taking into account real-world constraints such as cardinality and box constraints.

• Calculating investment risk using the CVaR measure and combining it with credibility theory.

• Financial and operational performance of companies, as well as predicting future performance from companies’ quarterly reports using the NotebookLM AI tool and the FinBERT sentiment analysis model.

• Converting qualitative data from financial texts into quantitative data using the AHP method and the Finbert model.

• Implementing the proposed model using real data from companies in the DJIA and comparing its results with the same weight allocation strategy.

Overall, the model presented in this study is unique compared to studies conducted in this field and approaches the topic from a new and innovative perspective. While previous research has made significant contributions to portfolio optimization and investment strategies, our model stands out by integrating fuzzy logic with advanced AI tools such as sentiment analysis and performance prediction, as well as sustainability considerations. This combination provides a more comprehensive and adaptive approach to risk assessment, enabling investors to make more informed decisions in uncertain market conditions. By approaching the topic from this new angle, the model offers a new solution that increases the robustness and flexibility of portfolio management and paves the way for more effective financial decision-making.

Model formulation

This section outlines the mathematical structure of the proposed portfolio optimization model, which integrates four principal objectives: (1) minimizing risk through CVaR under the framework of credibility theory, (2) maximizing the portfolio’s expected return, (3) incorporating environmental, social, and governance (ESG) sustainability metrics, and (4) maximizing overall company performance. The utility function is optimized under several practical constraints, including a total budget constraint, cardinality limits (which restrict the number of assets selected), minimum and maximum investment thresholds for each asset, and a no-short-selling condition. Given the inherent unpredictability of financial markets, the expected return is modeled as an uncertain quantity. To capture this uncertainty effectively, the model employs fuzzy set theory an approach that enables a nuanced and adaptive treatment of imprecise financial data. This results in a more resilient and realistic model that better reflects the complexity of real-world investment environments. The complete formulation of the utility function is presented as follows:

1

In Eq. (1), represents the criteria of efficiency, performance, risk, and sustainability, respectively, and denotes the weight associated with each objective, reflecting the investor’s individual priorities and strategic focus. To maintain a balanced trade-off among the objectives, the weights are constrained such that their sum equals one . Regarding the choice of , we have considered five different investment profiles for the sample and solved the model and we obtain these weights using pairwise comparisons and the AHP method, which is discussed in detail in Section “Data and Experiments”. In these profiles, each one assigns a greater weight to a criterion determined based on the investor’s opinion, and one profile assigns the same weight to all criteria.

The upcoming sections begin by outlining the key modeling prerequisites, setting the foundation for the subsequent formulation. This is followed by an in-depth analysis of the four fundamental components that constitute the utility function, alongside a discussion of the model’s constraints. Each component is systematically examined to highlight its significance and role within the overall framework. Through this detailed evaluation, the individual and combined influence of these elements on the optimization process will be made clear. The section concludes with the presentation of the final version of the proposed model, offering a coherent and transparent structure for the optimization approach.

Preliminaries

Given that uncertainty in expected returns is addressed through fuzzy set theory, the next section outlines the fundamental definitions and conceptual underpinnings of this approach. This is followed by a detailed explanation of how the CVaR model is adapted within the framework of credibility theory to handle risk under uncertainty.

Fuzzy set theory

Traditional set theory assesses elements using a strict Dichotomous logic, where each item either fully belongs to a set or does not belong at all. This rigid framework assumes sharp boundaries between categories, leaving no room for ambiguity or partial inclusion. However, this model proves inadequate for representing many real-world concepts that are inherently vague or gradational such as “tall,” “warm,” or “similar” which cannot be precisely defined using binary classifications. To address this limitation, fuzzy set theory offers a more adaptable mathematical structure. Instead of fixed membership, it uses a membership function that assigns each element a value between 0 and 1, indicating the degree to which it belongs to a set. A value of 1 denotes full membership, 0 denotes non-membership, and any value in between reflects partial inclusion. This allows for more accurate modeling of complex, imprecise, or subjective phenomena.

Definition 1

(Fuzzy set) Let X represent a universal set, with x denoting a generic element membership to this universe. A fuzzy set A within X is characterized as a set of ordered pairs, mathematically defined in the following manner:

2

Here, denotes the membership function of the fuzzy set A, assigning each element x a membership grade ranging between 0 and 1.

A widely used representation of fuzzy numbers is the trapezoidal form, which is particularly effective for modeling uncertainty and imprecision in various quantitative analyses. Trapezoidal fuzzy numbers are favored in practical applications due to their structural simplicity and their capacity to capture ambiguous or incomplete information. In this study, following the guidance of financial experts, trapezoidal fuzzy numbers are employed to represent uncertain variables. This representation offers enhanced flexibility in modeling a range of potential values and is formally defined as follows:

Definition 2

(Trapezoidal fuzzy numbers) A trapezoidal fuzzy number, symbolized by , can be identified by four parameters , where and are actual numbers satisfying . The membership function for is defined as follows:

3

The trapezoidal fuzzy number has a membership function shaped like a trapezoid. It reaches full membership (value of 1) between b and c, and decreases linearly to zero as it approaches the outer limits a and d. This shape represents maximum certainty in the middle and gradually decreasing confidence towards the edges. Figure 1 shows this membership function graphically.

Fig. 1.

The membership function of trapezoidal fuzzy number.

Credibility theory

Credibility theory was first put forward by Liu¹⁶, and later extended and improved in his later works⁶¹. The theory provides a robust and flexible mathematical framework for analyzing and modeling situations involving ambiguity and uncertainty. Over time, the theory has become a cornerstone of fuzzy mathematics and has played a particular role in the development of fuzzy reliability programming. This methodology enables decision makers to assess their confidence in achieving specific goals or constraints even in the presence of uncertain or ambiguous information. Credibility theory provides a systematic and consistent framework for handling various types of fuzzy data, including the common triangular and trapezoidal fuzzy numbers. These fuzzy numbers are often used to represent imprecise or unpredictable information in the process of making decisions. Through the integration of such fuzzy numbers, the theory provides an accurate and practical representation of uncertainty, a feature that we regularly encounter in real-world challenges in fields such as engineering, economics, and operations research. According to⁶², the computation of the credibility measure is carried out in the following manner:

4

In Eq. (4), denotes the possibility measure of the event , reflecting how feasible or achievable the event is within a fuzzy context. Conversely, refers to the necessity measure of the same event, expressing the level of certainty or the inevitability of its occurrence under the specified conditions.

• Possibility: This metric assesses the highest level of membership in the fuzzy set A, indicating the utmost likelihood or plausibility with which the event may occur. It is formally defined as:

  5
• Necessity: This metric quantifies the assurance or unavoidability of the event by evaluating the complement of the highest membership degree within the complement set . Its mathematical formulation is given as:

  6

It is important to highlight that, since and Necessity {} = , the credibility measure can alternatively be computed using the following formulation:

7

Formulation 7 illustrates that the credibility criterion unifies the concepts of possibility and necessity for the event , resulting in a thorough evaluation of its probability within a fuzzy logic framework. This integration allows the credibility measure to present a more balanced and insightful understanding of both the feasibility and certainty of the event when faced with uncertain or ambiguous conditions. When examining a particular fuzzy event described by , with representing a real number, the credibility measure is formulated as:

8

Equation (8) serves as a tool to assess the credibility of a fuzzy variable will have a value that is either less than or equal to a certain real number . This measure combines the maximum degree of membership within the interval and the complementary degree of membership in the interval . By integrating these components, it offers a balanced and complete view of the event’s likelihood, effectively capturing both its possibility and necessity within the fuzzy logic context. The expected value of a fuzzy variable is a fundamental concept in credibility theory, reflecting the “average” or central tendency of while incorporating its inherent uncertainty. It can be determined with the following formula:

9

Formulation 9 integrates the credibility measure over the entire set of real numbers, yielding a thorough computation of the expected value of by ensuring an equal distribution of contributions from both optimistic and pessimistic scenarios. The first integral assesses the credibility that is greater than or equal to , while the second considers the credibility that is less than or equal to . This dual approach ensures a symmetric and accurate representation of ’s expected value, capturing its fuzzy nature across all potential outcomes. The following analysis focuses on applying the validity criterion to specific types of fuzzy variables, with an emphasis on trapezoidal fuzzy numbers. These fuzzy numbers are commonly used because of their simplicity and clarity in representing uncertainty. By applying the validity framework to trapezoidal fuzzy numbers, a systematic assessment of the credibility and their expected values becomes possible, providing a useful approach for decision making in fuzzy situations.

Definition 3

(Trapezoidal fuzzy variable) A fuzzy variable can be precisely described by a quadruple (a, b, c, d) of real numbers satisfying a ≤ b ≤ c ≤ d. Applying the general formula for the credibilistic expected value (see Eq. 8), the credibilistic expected value of a trapezoidal fuzzy variable is given by:

10

Subsequently, the credibility measure for a trapezoidal fuzzy number is defined in the following manner:

11

Similarly, the credibility measure for a trapezoidal fuzzy number can be determined using the following equation:

12

Figure 2 presents a visual representation of these credibility measures, illustrating the probabilities linked to various events for a trapezoidal fuzzy variable. In the remainder of this paper, Eqs. (11) and (12) serve as prerequisites for deriving the CVaR risk measure within the credibility theory framework. Specifically, these equations are used to link the statistical properties of the underlying loss distribution with the credit-adjusted estimator, thereby allowing the expression of CVaR in a form that incorporates both risk sensitivity and credit weighting.

Fig. 2.

Credibility evaluation of trapezoidal fuzzy variables.

Expected return

Given that the proposed method operates within a fuzzy environment, the expected return of an investment portfolio is computed by aggregating the average expected returns of individual assets, evaluated through the lens of fuzzy set theory. This analytical framework is especially effective as it integrates the uncertainty inherent in financial markets into the estimation process. By accounting for data imprecision and vagueness, fuzzy theory enhances the robustness and credibility of the forecasted outcomes. Consequently, the model offers a more detailed and realistic evaluation of the portfolio’s potential performance. The mathematical expression for the objective function representing the portfolio’s expected return is formulated as follows:

13

In Eq. (13), n represents the total number of assets, while x_i denotes the share of the portfolio assigned to the ith asset. The parameters , , , and correspond to the fuzzy estimates of the expected return for the ith company, derived from expert evaluations.

Credibilistic-CVaR

CVaR is a risk measure that quantifies the potential loss in the tail of a distribution beyond a specified VaR threshold. In contrast to VaR, which only considers the worst loss at a given confidence level, CVaR takes into account the magnitude of losses that exceed this threshold, giving a more detailed understanding of risk exposure. This makes CVaR particularly valuable in risk management, as it helps financial institutions, investment managers, and regulators assess the extent of potential extreme losses, which are often the most significant in terms of impact. By integrating CVaR into risk analysis, firms can better understand the potential for catastrophic financial outcomes, allowing for more effective strategies to mitigate such risks. As markets become increasingly volatile, CVaR has gained importance in stress testing, portfolio optimization, and regulatory compliance. For a random variable , the CVaR at a confidence level β is mathematically defined as:

14

In Eq. (14), the value corresponds to ,, which is if and 0 if . Thus, η stands for the VaR threshold, demonstrates the confidence level, refers to the loss function, and is the possibility distribution function of . This method employs linear programming methods to effectively compute CVaR, enabling it especially valuable for financial optimization applications. The following section explores the integration of the CVaR model within the framework of credibility theory.

By applying credibility theory to CVaR calculation, it becomes possible to for a more flexible and advanced treatment of uncertainty. Particularly effective in scenarios involving fuzzy or vague data, credibility theory provides a unique alternative to traditional probabilistic methods. For a fuzzy variable and a confidence level β ∈ (0,1], the VaR based on credibility theory is characterized as:

15

This concept establishes the upper bound of x such that the credibility measure remains less than or equal to β, serving as the fuzzy equivalent of the classical VaR metric. Likewise, a substitute representation for VaR within the credibility theory framework can be formulated as:

16

where denotes the aggregated credibility distribution function. CVaR within the context of credibility theory, represented as , is derived through the integration of the credibility-based VaR function over the defined confidence interval:

17

Given the above, the CVaR for a trapezoidal fuzzy variable distinguished by the specifications and a confidence level β  (0,1] may be computed using as outlined below expression:

18

Performance

To assess the performance objective function, this study utilizes data obtained from the quarterly financial reports of the selected companies. These documents serve as reliable and detailed sources of information, offering view of each company’s fiscal health, operational productivity, and strategic vision during the past fiscal quarter. Beyond historical performance, the reports frequently contain forward-looking statements, aiding in the projection of future trends. By leveraging this robust dataset, the model grounds its performance evaluation in verifiable financial data. The methodology for calculating the performance score is thoroughly outlined in Section “Data and experiments”, and the mathematical representation of the objective function is presented below.

19

Here, the variable denotes the computed score for the ith company, reflecting a combination of its historical financial performance and projected operational and financial outlook. This parameter is extracted from companies’ quarterly reports with the help of the NotebookLM tool, the FinBERT model, and the AHP method, which is explained in detail in Section “Performance data”.

Sustainability

Integrating ESG factors into portfolio optimization enhances the sustainability and long-term viability of investment strategies. ESG criteria enable investors to assess not only the fiscal results of assets but also their broader influence on communities and ecological systems. By including ESG metrics, portfolio managers can identify risks and opportunities that traditional financial models may overlook, such as exposure to climate change, labor practices, or corporate governance issues. This approach helps reduce downside risk, improve risk-adjusted returns, and align investments with responsible and ethical standards. Moreover, ESG-based portfolios tend to attract long-term investors and institutional capital, fostering market stability. As regulatory pressure and stakeholder awareness grow, incorporating ESG considerations into the optimization process is becoming essential for building resilient, future-ready portfolios. This objective function is expressed mathematically as follows:

20

In Eq (20), the variable denotes the environmental risk associated with the ith asset, while and represent the social and governance risk components, respectively. Together, these three variables encapsulate the essential dimensions of ESG (Environmental, Social, and Governance) risk. Their inclusion is crucial for assessing the sustainability profile and ethical implications of investment choices, providing a more comprehensive risk assessment framework beyond traditional financial metrics.

Practical constraints

Optimizing investment portfolios in real-world conditions requires considering a wide range of practical constraints to guarantee that the model faithfully represents actual investment situations. These limitations may include budget constraints, minimum and maximum allocation constraints (floor and ceiling constraints), cardinality limitations on asset quantity, and short-selling prohibitions. Such Key drivers impact the formation of investment choices decisions and plans of action. Their integration increases the realism and applicability of the investment the process of choosing portfolio assets and ensuring alignment the optimization model with practical investment requirements. As a result, the framework is more robust and reflects real-world complexities, leading to more practical, reliable, and implementable investment portfolio solutions. This alignment ultimately increases the relevance of the model as a decision-making tool in real-world financial environments. We will examine these limitations further.

• Budget constraints: The budget constraint is a fundamental condition in portfolio optimization that ensures the total investment across all selected assets does not exceed the available capital. Mathematically, it is typically formulated as:

  21

  This constraint guarantees that the investor fully allocates available resources without overspending or underinvesting.

• Cardinality constraint: The cardinality limitation in portfolio optimization restricts the total count of assets included in the final portfolio, allowing for better control over diversification and transaction costs. Mathematically, it is typically formulated as:

  22

  This constraint turns the optimization into a mixed-integer programming (MIP) problem, making it more complex but also more realistic for practical portfolio management.

• Box limitations: The box limitations in portfolio optimization specify the minimum and maximum proportion of the total budget that can be invested in each asset. These constraints help manage diversification, concentration risk, and compliance with investment policies or regulatory limits. Mathematically, it is typically formulated as:

  23

  where and are the floor and ceiling of investment in ith asset, respectively. The floor ensures that if an asset is selected, a minimum meaningful investment is made (e.g., to avoid excessive fragmentation of capital). The ceiling prevents overexposure to a single asset, promoting diversification and limiting risk.
• Short-selling constraints: The short-selling constraints in portfolio optimization prohibits allocating negative weights to assets, meaning investors cannot sell assets they do not own in anticipation of buying them back at a lower price. This constraint is commonly applied in practice due to regulatory restrictions, risk aversion, or investment policy limitations.

  24

  This constraint ensures that all portfolio weights are non-negative, reflecting long-only investment strategies. It simplifies portfolio management, reduces exposure to potential unlimited losses from short positions, and aligns with the policies of many institutional investors.

Final model

This section introduces the final version of the proposed model, developed to optimize portfolio selection within a trapezoidal fuzzy variable framework. The model integrates sentiment analysis derived from companies’ quarterly reports to incorporate qualitative insights into the investment decision process. Furthermore, it includes ESG considerations as a sustainability factor, aligning the model with realistic trading environments. Risk is assessed using the CCVaR, a credibility theory-based metric that offers a robust means of evaluating portfolio risk. By merging both quantitative data and qualitative factors, the model provides a more holistic approach to portfolio optimization. It addresses the uncertainty and ambiguity present in financial markets by utilizing fuzzy variables, which offer a more accurate representation Based on empirical data as opposed to classical probability-based approaches. Through the use of trapezoidal fuzzy numbers, the model delivers a consistent and practical framework for evaluating risk and making informed investment choices. In the context of the CVaR risk metric, the value of β plays a crucial role in determining the threshold for the worst-case scenario. When β is (0,0.5], it emphasizes more extreme tail risks, leading to a more conservative risk assessment. Conversely, for β values (0.5,1), the focus shifts towards less severe outcomes, reducing the sensitivity to tail risks and reflecting a more optimistic outlook. The model is presented for both β range case. As such, the model serves as a reliable tool for investors focused on minimizing downside risk. Key constraints incorporated into the model include budget limits, cardinality restrictions, upper and lower bounds on asset allocation, and short-selling limitations. These constraints enhance the model’s realism and ensure its practical relevance to real-world portfolio management. The final mathematical formulation of the proposed approach is presented as follows:

Case 1: When

For the first case, when the β value is less or equal than 0.5, by putting together Eqs. (1) to (24), the final model is as follows:

25

Subject to:

26

27

28

29

30

Equation (25) represents the investor utility-based objective function of the proposed model, which aims to maximize overall utility by addressing four critical components: maximizing expected returns, increasing the performance score of firms, minimizing unsystematic risk through CCVaR, and reducing sustainability-related risk. Equations (26)–(30) determine the constraints of the model: Eq. (26) specifies the budget constraint and ensures that the total investment capital is fully allocated among the selected assets. Equation (27) imposes the cardinality constraint and limits the total count of assets in the portfolio to K, which helps manage transaction costs and simplifies portfolio monitoring. Equation (28) introduces allocation boundaries and sets lower and upper thresholds for asset weights. The binary variable z_i indicates whether a particular asset is included in the portfolio or not. Equations (29) and (30) emphasize that z_i must be a binary variable (0 or 1), and x_i, which represents the allocation to asset i, must be nonnegative, thus ruling out the possibility of short selling.

Case 2: When

In the second case, when the β value is greater than 0.5, by putting Eqs. (1) to (24) together, the final model is exactly the same as the case model, and only the objective function of the problem changes. According to Eq. (18), which shows the CCVaR risk measure for two different cases, the objective function is as follows, and the model constraints do not change compared to the first case:

31

Data and experiments

To apply and evaluate the portfolio optimization model presented in this study, we have chosen the constituent companies of the DJIA as the basis for our analysis. The selection of companies in the DJIA as the case study is justified by both methodological and practical considerations. The DJIA, as one of the most established and widely recognized indices worldwide, consists of 30 large and financially robust corporations that represent a broad range of industries and serve as a benchmark for U.S. and global market performance. These firms are characterized by high market capitalization, strong liquidity, and the availability of consistent historical data, which ensures the reliability and replicability of empirical analysis. Their multinational presence and influence across economic cycles provide broader relevance, as the results extend beyond domestic markets and hold significance for international investors and policymakers. Moreover, the long-standing stability, industry diversity, and benchmark role of DJIA constituents make them an appropriate and meaningful sample for testing predictive models and portfolio optimization strategies, thereby reinforcing both the academic rigor and the practical applicability of the study.

Compared to other indices, the DJIA has the advantage of historical depth, being one of the oldest stock market benchmarks with over a century of data. It consists of highly reputable and financially strong firms, which enhances the reliability of empirical studies. Unlike sector-specific indices, its diversified composition provides a broader perspective of economic performance. Moreover, its global recognition and use as a reference point by investors, policymakers, and researchers make it more impactful than many alternative indices.

The companies that make up the DJIA index include UnitedHealth Group (UNH), Goldman Sachs Group (GS), Microsoft Corporation (MSFT), Home Depot (HD), Caterpillar (CAT), Amgen Corporation (AMGN), McDonald’s Corporation (MCD), Salesforce Corporation (CRM), Visa Corporation (V), American Express Corporation (AXP), Travelers Corporation (TRV), Apple Corporation (AAPL), International Business Machines Corporation (IBM), JPMorgan Chase & Co. (JPM), Honeywell International Corporation (HON), and Amazon.com Corporation (Amazon.com). (AMZN), Procter & Gamble Company (PG), Johnson & Johnson (JNJ), Boeing Company (BA), Chevron Company (CVX), 3M Company (MMM), Merck & Co. (MRK), The Walt Disney Company (DIS), Nike Company (NKE), Walmart Company (WMT), The Coca-Cola Company (KO), Cisco Systems Company (CSCO), Dow Company (DOW), Verizon Communications Company (VZ), and Intel Company (INTC).

In analyzing the stock price data set, several statistical measures were calculated to understand the price distribution and its volatility. The mean price provides an average value that indicates the central tendency of the stock over the observed period. The standard deviation shows the extent to which the price fluctuates around the mean and provides insights into the stock’s volatility. In addition, the maximum and minimum prices highlight the highest and lowest points reached, respectively, which helps identify the range of price movements in the data set. Table 2 shows these statistical characteristics for the selected companies in this study.

Table 2.

Descriptive statistics of the chosen assets for quarterly timeframe.

No	Ticker	Mean	SD	Min	Max	No	Ticker	Mean	SD	Min	Max
1	UNH	480.08	33.18	399.57	528.77	16	AMZN	134.70	28.51	86.08	180.97
2	GS	325.09	41.07	272.23	404.58	17	PG	140.67	9.84	115.98	155.74
3	MSFT	301.36	56.19	220.86	421.35	18	JNJ	155.29	7.11	144.66	168.23
4	HD	302.70	32.83	266.69	359.10	19	BA	194.81	34.64	129.79	251.76
5	CAT	226.97	54.19	165.12	356.63	20	CVX	139.43	19.32	94.72	161.08
6	AMGN	234.58	29.23	184.99	286.35	21	MMM	95.03	19.73	69.68	130.52
7	MCD	252.39	22.35	220.11	288.35	22	MRK	95.16	17.11	70.58	127.22
8	CRM	212.22	50.10	139.51	300.50	23	DIS	110.34	29.07	78.35	172.28
9	V	222.81	25.20	180.29	276.22	24	NKE	111.57	20.35	83.91	149.87
10	AXP	162.77	24.58	134.08	224.37	25	WMT	48.44	5.23	40.26	59.33
11	TRV	169.63	21.55	145.85	224.19	26	KO	56.04	3.51	49.15	59.20
12	AAPL	160.64	19.26	128.13	190.16	27	CSCO	47.26	4.91	37.34	56.17
13	IBM	129.62	21.05	107.25	183.65	28	DOW	48.49	4.28	38.66	54.89
14	JPM	137.63	25.98	99.38	193.30	29	VZ	36.97	5.11	28.00	44.02
15	HON	188.10	15.21	162.53	203.74	30	INTC	38.36	8.83	24.47	51.70

As shown in Table 2, UNH recorded the highest average share price, indicating its relatively strong market capitalization compared to the other stocks analyzed. In terms of price volatility, MSFT showed the highest standard deviation, indicating significant fluctuations in its share price over the observed period. On the other hand, the lowest average price was observed for VZ, while KO showed the lowest standard deviation, indicating a more stable price trend. When examining the overall price range, which represents the difference between the highest and lowest prices recorded, MSFT again stood out with the widest price range, while VZ had the narrowest price range, indicating relatively limited price movement over the period. It is worth noting that this data is on a quarterly basis from 2021 to the end of the first quarter of 2024.

Data

In this study, we adopt a comprehensive framework that simultaneously addresses four key dimensions in investment analysis: risk, return, sustainability, and overall performance. To effectively assess these aspects, we use three distinct and complementary types of data. First, historical financial data are used to estimate traditional metrics such as expected returns and associated risks. These data allow us to model past market behavior and employ fuzzy logic techniques to reflect the uncertainty and volatility inherent in financial markets. Second, we use textual data from company’s quarterly reports to extract qualitative insights about company performance. Sentiment analysis is applied to this unstructured data to interpret public and investor perceptions, which play an important role in influencing market movements. Third, we use ESG data to assess the sustainability aspect of each portfolio component and align investment strategies with long-term ethical and environmental goals. By integrating these three data sources quantitative historical trends, sentiment-based qualitative indicators, and ESG metrics our approach creates a comprehensive model that bridges traditional financial analysis with modern data science methods. This hybrid strategy not only improves the reliability and depth of analysis, but also strengthens decision-making in portfolio optimization and risk management. It enables a richer, more multifaceted understanding of market dynamics and provides a stronger foundation for investors seeking to balance profitability with responsible investing.

In this study, we have chosen the companies that make up the DJIA as a case study. Selecting the DJIA as a case study is well justified given the distinctive features of its constituent companies. These firms represent some of the world’s largest and most valuable corporations, characterized by high market capitalization and strong liquidity, which ensures reliable market signals. The index also encompasses a diverse range of industries including technology, finance, healthcare, energy, and consumer goods providing a broad perspective on economic dynamics. With a strong international presence, these companies both influence and reflect global economic trends. Their long-standing histories and relative stability offer rich datasets for empirical research, while the DJIA itself serves as a widely recognized benchmark for assessing overall market and economic conditions.

Historical data

Historical expected returns represent the average returns that investors have earned on an asset or portfolio over a specified period in the past. These returns are often used as a measure of future performance, although they do not guarantee future results. By analyzing historical data, investors can gain insight into the potential risks and rewards associated with various investments. However, when relying on historical returns to guide investment decisions, it is important to consider changing economic conditions, market fluctuations, and unforeseen events. Since the model presented in the fuzzy environment is trapezoidal, four numbers are required for each symbol to describe it, and it should be noted that to calculate the amount of trapezoidal fuzzy numbers, we used the opinions of 5 financial experts from the Faculty of Industrial Engineering, Iran University of Science and Technology, and the calculation was based on the results of their opinions. This data is shown in Table 3.

Table 3.

Fuzzy trapezoidal dataset representing asset returns.

No	Ticker	a	b	c	d	No	Ticker	a	b	c	d
1	UNH	− 0.09	− 0.03	0.0762	0.131	16	AMZN	− 0.26	− 0.12	0.14	0.258
2	GS	− 0.18	− 0.08	0.144	0.269	17	PG	− 0.14	− 0.06	0.13	0.247
3	MSFT	− 0.11	− 0.03	0.1725	0.3	18	JNJ	− 0.1	− 0.04	0.07	0.132
4	HD	− 0.21	− 0.09	0.1039	0.188	19	BA	− 0.25	− 0.12	0.33	0.641
5	CAT	− 0.19	− 0.05	0.2475	0.411	20	CVX	− 0.16	− 0.05	0.2	0.341
6	AMGN	− 0.11	− 0.03	0.1282	0.21	21	MMM	− 0.19	− 0.11	0.12	0.269
7	MCD	− 0.13	− 0.05	0.0959	0.168	22	MRK	− 0.08	− 0.01	0.19	0.32
8	CRM	− 0.18	− 0.08	0.1954	0.37	23	DIS	− 0.28	− 0.15	0.16	0.34
9	V	− 0.1	− 0.04	0.1069	0.187	24	NKE	− 0.18	− 0.11	0.2	0.433
10	AXP	− 0.2	− 0.08	0.1653	0.29	25	WMT	− 0.18	− 0.07	0.09	0.146
11	TRV	− 0.11	− 0.03	0.1465	0.242	26	KO	− 0.13	− 0.05	0.1	0.171
12	AAPL	− 0.13	− 0.05	0.1498	0.272	27	CSCO	− 0.16	− 0.08	0.11	0.21
13	IBM	− 0.15	− 0.05	0.138	0.224	28	DOW	− 0.19	− 0.08	0.14	0.256
14	JPM	− 0.21	− 0.09	0.1748	0.311	29	VZ	− 0.26	− 0.13	0.13	0.259
15	HON	− 0.12	− 0.06	0.118	0.233	30	INTC	− 0.3	− 0.15	0.17	0.335

ESG data

In the past few years, the use of ESG data in the context of portfolio optimization has gained substantial popularity among investors increasingly seek strategies that align with ethical values and long-term sustainability. ESG indicators provide important insights into how companies manage their environmental responsibilities, social impact, and corporate governance structures, all of which can impact financial performance over time. For example, companies with strong environmental practices may be better positioned to comply with climate regulations, while companies with sound corporate governance are often less exposed to scandals or operational inefficiencies. By incorporating ESG data into portfolio selection and optimization models, investors can screen for companies that demonstrate both financial stability and responsible behavior. This approach not only facilitates the shift toward a more sustainable economy, but also reduces non-financial risks that can affect returns. Furthermore, empirical studies have shown that investment portfolios constructed with ESG considerations can achieve competitive or even superior performance compared to traditional investment portfolios, especially in times of market stress. Thus, ESG integration improves the decision-making process by balancing profit objectives with social and environmental responsibility, leading to more flexible and future-proof investment strategies. These data are presented for selected companies in Table 4, and this data was collected from the YahooFinance website. It should be noted that the ESG data was normalized using the equation. Be noted that we have used this normalization method to scale ESG data with three other data categories, namely risk, reward, and HFP.

Table 4.

Sustainability data.

No	Ticker	ER	SR	GR	ESG	ESGn	No	Ticker	ER	SR	GR	ESG	ESGn
1	UNH	0.1	11.9	4.6	17	0.25	16	AMZN	8.3	11.1	6.8	26.1	0.55
2	GS	0.8	13.5	11	25	0.79	17	PG	9.7	10.5	5.6	25.8	0.53
3	MSFT	3.6	9	4.8	17	0.30	18	JNJ	0.9	13	5.9	19.9	0.30
4	HD	3.4	6.5	2.7	13	0.00	19	BA	7.9	22.2	5.9	36	0.95
5	CAT	10	11.8	6.4	28	0.99	20	CVX	19.9	9.6	7.8	37.3	1.00
6	AMGN	1.8	14.6	6.1	23	0.62	21	MMM	1.7	5.5	19	26.2	0.55
7	MCD	9.9	13	5.6	29	1.00	22	MRK	2.9	11.6	5.7	20.2	0.31
8	CRM	3.7	10.3	4.1	18	0.35	23	DIS	0.1	10	5.8	15.9	0.13
9	V	1.7	5.5	19	26	0.86	24	NKE	3	10	5.4	18.4	0.23
10	AXP	0.1	9.7	8.4	18	0.36	25	WMT	7.2	11.7	6.4	25.3	0.51
11	TRV	0.9	10.6	8.3	20	0.45	26	KO	9.2	10.3	4.3	23.9	0.46
12	AAPL	2	8.4	8.4	19	0.39	27	CSCO	0.4	7.6	5.1	13.1	0.02
13	IBM	1.6	6.7	5	13	0.04	28	DOW	12.2	3.4	4.3	20	0.30
14	JPM	2.4	14	10.8	27	0.92	29	VZ	4.5	10.1	5.1	19.7	0.29
15	HON	13	9.1	6.3	28	0.97	30	INTC	7.7	6.9	4.6	19.2	0.27

ER, environment risk, SR, social risk, GR, governance risk, ESGn, ESG normalized.

Performance data

The company’s quarterly financial and operational reports, here for the first quarter of 2024, were used as the primary source of textual data. These documents were initially processed through the AI-based tool NotebookLM, which was deployed to generate concise summaries that reflected both the financial results and the forecasted outlook for the second quarter. Sentiment analysis was then performed on the resulting summaries to quantify the tone and outlook of each company. For this purpose, the FinBERT model, an adaptation of the BERT architecture tailored to the financial context, was implemented. FinBERT is tuned using financial-specific data, enabling it to interpret the unique nuances and contextual dependencies of financial language. This customization increases the model’s accuracy in analyzing texts such as earnings announcements, financial news, and market commentary. Through this process, sentiment scores were created that provide valuable input for downstream tasks such as investment strategy development, risk management, and predictive analytics and it should be noted that we have used NotebookLM version 1.9 and the ProsusAI/finbert¹ model version Finbert to calculate the performance score of companies. This process is graphically illustrated in Fig. 3.

Fig. 3.

Graphical display of performance score extraction steps.

The value of S_i is computed using the AHP in conjunction with outputs from the FinBERT sentiment analysis model. AHP is a structured, multi criteria decision-making method that facilitates the ranking and evaluation of alternatives by incorporating diverse criteria. Originally developed by Saaty²⁷, AHP is especially useful in scenarios involving complex trade-offs among interrelated factors. In this study, AHP is applied to assign relative importance (weights) to three sentiment categories (positive, neutral, negative) based on expert judgment. These weights are then combined with FinBERT’s sentiment scores to compute S_i, the sentiment-based score for the ith asset. Expert defined preference ratios are as follows: positive to negative = 8, positive to neutral = 4, and neutral to negative = 4. From these comparisons, the resulting weight vector for the sentiment categories is (0.707, 0.223, 0.07) for positive, neutral, and negative, respectively. The Consistency Ratio (CR), a key indicator in AHP that evaluates the logical consistency of pairwise comparisons, is calculated at 5.6%. Since this value is When the value falls beneath the 10% threshold, the comparisons are regarded as consistent and the calculated weights dependable. Letting SA_i denote the sentiment vector for the ith asset and Q the weight vector, the sentiment score S_i is calculated using the element-wise product: S_i = SA_i Q. An illustrative example follows:

Finally, the performance scores of the selected companies are shown in Table 5.

Table 5.

HFP data.

No	Symbol	Score ()	No	Symbol	Score ()
1	UNH	0.662669	16	AMZN	0.663918
2	GS	0.269123	17	PG	0.671865
3	MSFT	0.666463	18	JNJ	0.256387
4	HD	0.223990	19	BA	0.223990
5	CAT	0.234407	20	CVX	0.238735
6	AMGN	0.263906	21	MMM	0.499202
7	MCD	0.631965	22	MRK	0.632305
8	CRM	0.592853	23	DIS	0.231052
9	V	0.653879	24	NKE	0.326711
10	AXP	0.681337	25	WMT	0.621835
11	TRV	0.189312	26	KO	0.233000
12	AAPL	0.225186	27	CSCO	0.231606
13	IBM	0.661309	28	DOW	0.270740
14	JPM	0.297343	29	VZ	0.256462
15	HON	0.677192	30	INTC	0.472885

Results and experiments

This section consists of two main parts for both cases: the first is computational results and the second is validation results, each of which is discussed in detail below.

Experimental results

The proposed portfolio optimization model is evaluated under multiple weighting scenarios, each designed to reflect distinct strategic preferences based on the weights obtained from the AHP method. The results for different profile are presented in Table 6. This multi-scenario approach facilitates a comparative analysis of how changing the emphasis on returns, risk, ESG considerations, and past and outlook performance affect the optimal portfolio structure. To maintain consistency and allow for meaningful comparisons between scenarios, the model is solved using a uniform set of parameters: Lower bound:, Upper bound: , Number of assets allocated: K = 3,5,7. The results obtained from solving the model for different scenarios are presented below, and the optimal asset allocation is illustrated in great detail. These results provide valuable insights into how different prioritizations affect portfolio composition and highlight the trade-offs involved in balancing multiple investment objectives.

Table 6.

Criteria weights.

No	Profile	W	CR (%)
1	Profile1	(0.25,0.25,0.25,0.25)	0
2	Profile2	(0.57,0.17,0.12,0.14)	2.2
3	Profile3	(0.19,0.57,0.12,0.12)	2.2
4	Profile5	(0.2,0.12,0.57,0.12)	2.2
5	Profile5	(0.2,0.14,0.1,0.56)	4.5

Based on the above data, we have solved the model for both cases discussed in this article. It should be noted that for the proposed model, we have used the GAMS software and the CPLEX solver to solve MIP problems.

Case 1: When

If we consider the beta value to be 0.05, the results will be as shown in Tables 7, 8 and 9. Table 7 shows the outcomes of solving the suggested model in various situations. The findings highlight the selection of three specific companies, designated here as UNH, MSFT, and IBM, and the strategic allocation of budgets to them. This budget allocation reflects the optimization process of the model, which aims to maximize performance or efficiency under the constraints defined in each scenario.

Table 7.

Results from solving the model under different scenarios with K = 3. (Case 1).

K	W				UF	UF with equal weight
k = 3	(0.25,0.25,0.25,0.25)	=0.1	=0.5	=0.4	0.13	0.1219
	(0.57,0.17,0.12,0.14)	=0.1	=0.5	=0.4	0.124	0.121298
	(0.19,0.57,0.12,0.12)	=0.1	=0.4	=0.5	0.364	0.357161
	(0.2,0.12,0.57,0.12)	=0.1	=0.4	=0.5	0.057	0.053418
	(0.2,0.14,0.1,0.56)	=0.4	=0.5	=0.1	0.046	0.036047

Table 8.

Results from solving the model under different scenarios with K = 5. (Case 1).

K	W						UF	UF with equal weight
k = 5	(0.25,0.25,0.25,0.25)	=0.1	=0.1	=0.1	=0.5	=0.2	0.126	0.107906
	(0.57,0.17,0.12,0.14)	=0.1	=0.1	=0.1	=0.5	=0.2	0.116	0.107222
	(0.19,0.57,0.12,0.12)	=0.1	=0.2	=0.1	=0.5	=0.1	0.361	0.3546
	(0.2,0.12,0.57,0.12)	=0.1	=0.2	=0.1	=0.2	=0.5	0.049	0.040602
	(0.2,0.14,0.1,0.56)	=0.1	=0.2	=0.5	=0.1	=0.1	0.031	0.001092

Table 9.

Results from solving the model under different scenarios with K = 7. (Case 1).

K	W								UF	UF with equal weight
k = 7	(0.25,0.25,0.25,0.25)	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	0.11	0.087252
	(0.57,0.17,0.12,0.14)	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	0.105	0.088139
	(0.19,0.57,0.12,0.12)	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	0.351	0.337668
	(0.2,0.12,0.57,0.12)	=0.1	=0.1	=0.1	=0.1	=0.1	=0.1	=0.4	0.037	0.025222
	(0.2,0.14,0.1,0.56)	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	=0.1	0.01	− 0.01711

When k = 5, it is natural that the utility obtained is lower, as shown in Table 8. In this case, the results obtained are lower than when k is equal to 3, but it reflects the trade-off made by diversifying the risk of a single company⁶³. In this case, the budget has been allocated to the UNH, MSFT, AXP, IBM, MRK, HD, CRM, DIS and CSCO symbols in difference Scenarios.

Table 9 shows the results of solving the model when k is equal to 7. In this case, the budget is allocated to the UNH, MSFT, CRM, AXP, IBM, MRK, CSCO, NKE, PG, DIS and HD symbols in difference scenarios.

As demonstrated in Tables 7 through 9, the utility values derived from the model reach their highest levels when the performance criterion is given the greatest weight among all considered factors. For the different values of K analyzed, the maximum utility values obtained are 0.364, 0.361, and 0.351, respectively. These findings suggest that prioritizing performance can lead to more favorable investment outcomes, particularly under the conditions modeled. This highlights the model’s sensitivity to the weighting scheme applied to the criteria, and reinforces the importance of performance as a critical driver of portfolio optimization. Moreover, one of the key strengths of the proposed model lies in its adaptability to various investor profiles. By adjusting the weights of the criteria such as return, risk, sustainability, and performance the model can accommodate different risk appetites and strategic preferences. This allows for the design of tailored investment strategies that align with individual investor goals, whether they prioritize returns, seek to minimize risk, or place value on sustainability and operational performance. Figure 4 illustrates the best-performing portfolios under varying values of K, providing a visual representation of how portfolio composition changes with different configurations of the model. Notably, the model does not operate under the traditional binary focus of risk and return. Instead, it adopts a more holistic approach by integrating two additional criteria, which significantly influence the final outcomes. For instance, although companies such as INTC and BA exhibit high expected returns, they are consistently excluded from the optimal portfolios. This exclusion can be attributed to their subpar evaluations in sustainability and performance dimensions, which diminish their overall desirability in a multi-criteria context. Such results validate the comprehensive nature of the model and emphasize its capacity to produce balanced and ethically conscious investment decisions, rather than solely profit-driven ones.

Fig. 4.

Optimal portfolios under different scenarios (case 1).

Case 2: When

When the beta parameter is set to 0.95, the corresponding results are reported in Tables 10, 11 and 12. Specifically, Table 10 shows the model solution in different scenarios. The analysis identifies three selected companies from the selected set of companies MRK, AXP, MSFT, JPM, NKE, CSCO, CRM, BA, and IBM, along with how the investment resources are distributed among them. This distribution reflects the optimization logic of the proposed framework, which seeks to increase performance or efficiency while satisfying the constraints imposed in each case.

Table 10.

Results from solving the model under different scenarios with K = 3. (Case 2).

K	W				UF	UF with equal weight
k = 3	(0.25,0.25,0.25,0.25)	=0.4	=0.5	=0.1	0.203	0.165
	(0.57,0.17,0.12,0.14)	=0.1	=0.4	=0.5	0.159	0.142
	(0.19,0.57,0.12,0.12)	=0.4	=0.1	=0.5	0.399	0.381
	(0.2,0.12,0.57,0.12)	=0.1	=0.5	=0.4	0.282	0.154
	(0.2,0.14,0.1,0.56)	=0.4	=0.5	=0.1	0.072	0.055

Table 11.

Results from solving the model under different scenarios with K = 5. (Case 2).

K	W						UF	UF with equal weight
k = 5	(0.25,0.25,0.25,0.25)	=0.2	=0.1	=0.1	=0.5	=0.1	0.199	0.160
	(0.57,0.17,0.12,0.14)	=0.1	=0.1	=0.1	=0.2	=0.5	0.154	0.135
	(0.19,0.57,0.12,0.12)	=0.1	=0.2	=0.1	=0.5	=0.1	0.395	0.379
	(0.2,0.12,0.57,0.12)	=0.1	=0.2	=0.5	=0.2	=0.2	0.275	0.199
	(0.2,0.14,0.1,0.56)	=0.1	=0.2	=0.5	=0.1	=0.1	0.059	0.024

Table 12.

Results from solving the model under different scenarios with K = 7. (Case 2).

K	W								UF	UF with equal weight
k = 7	(0.25,0.25,0.25,0.25)	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	0.187	0.158
	(0.57,0.17,0.12,0.14)	=0.1	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	0.145	0.124
	(0.19,0.57,0.12,0.12)	=0.1	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	0.386	0.370
	(0.2,0.12,0.57,0.12)	=0.1	=0.1	=0.1	=0.4	=0.1	=0.1	=0.1	0.260	0.197
	(0.2,0.14,0.1,0.56)	=0.1	=0.1	=0.1	=0.4	=0.4	=0.1	=0.1	0.039	0.006

As observed in Table 11, when k = 5, the utility achieved is naturally lower compared to when k = 3. This reduction in utility is expected, but it reflects the trade-off made by diversifying the risk of a single company. In this scenario, the budget is distributed across the companies UNH, MSFT, HD, V, CRM, HD, IBM, MRK, NKE, and CSCO, depending on the specific conditions of each scenario.

Table 12 presents the outcomes of solving the model when k = 7. In this scenario, the budget is distributed among the companies UNH, MSFT, CRM, AXP, IBM, BA, MRK, CSCO, NKE, PG, DIS, and HD, across different scenarios.

As illustrated in Tables 10, 11 and 12, the utility values for the investor remain identical to the initial case, with the only difference being the adjusted weights and types of certain companies. The maximum utility values obtained for the analyzed values of K are 0.399, 0.395, and 0.386, respectively. These results suggest that placing greater emphasis on performance can lead to better investment outcomes, particularly under the given model conditions. Furthermore, the second case emphasizes the model’s sensitivity to the weighting approach used for the criteria and reinforces the importance of performance as a key factor in portfolio optimization.

Figure 5 shows the best portfolios based on the utility function. The difference compared to the first case is in the number of weights assigned to the companies. In the second case, companies IBM and AXP have received the best weights, which according to the data are in a good position in terms of various criteria.

Fig. 5.

Optimal portfolios under different scenarios (case 2).

Next, in Fig. 6, the results of the sensitivity analysis performed on the parameter for both cases are presented. This analysis examines how changes in the parameter affect the utility function and provides a comparative view of both cases. The figure highlights the differences and trends in the utility function that emerge when adjusting the parameter.

Fig. 6.

Sensitivity analysis on parameter.

As previously mentioned in the context of CVaR and more specifically in this CCVaR study, the parameter represents the confidence level or threshold at which risk is assessed. This parameter defines the quantile of the loss distribution beyond which risk is considered for analysis. Essentially, this parameter determines the degree of certainty required to estimate potential losses. As increases, the risk threshold becomes more precise and focuses on a more extreme tail of the loss distribution. As shown in Fig. 7, the utility function also increases with increasing for both cases. Another point that can be seen from this figure is that with increasing K the utility function takes a smaller value, which is also reasonable considering the risk distribution.

Fig. 7.

Comparison of the proposed model with the equal weight investment strategy (case 1).

Validation

In this section, we compare the results obtained from both cases with a baseline scenario in which equal weight is assigned to each company. This comparison allows us to evaluate the impact of different weighting strategies on the overall outcomes. By analyzing the performance under these varied conditions, we can better understand the effectiveness of prioritizing certain companies or factors. The results of this comparison are presented in Figs. 7 and 8, where the utility values, allocation of budgets, and other relevant metrics are visually displayed for a clearer understanding of the differences between the approaches.

Fig. 8.

Comparison of the proposed model with the equal weight investment strategy (case 2).

And for the case 2, the comparisons are shown in Fig. 8.

As demonstrated in Figs. 7 and 8, the results clearly show that our proposed model consistently outperforms the equal weight strategy across various investment profiles. This superior performance is observed in both case and , indicating that our model offers a more effective and efficient approach to portfolio management compared to the equal weight strategy, regardless of the specific investment profile being analyzed. The figures highlight the advantages of our model, suggesting its potential for better risk-adjusted returns and more optimized asset allocation strategies tailored to different investor needs and preferences.

Discussion

This section is organized into three main parts. First, a comprehensive summary of the study is provided, highlighting its core findings along with the managerial and theoretical implications derived from the results. This part emphasizes how the outcomes contribute to both academic literature and practical decision-making in financial markets. Second, a critical discussion of the state-of-the-art aspects of the proposed model is presented, illustrating its contributions relative to existing approaches, as well as the novel insights it offers to the field of stock price forecasting. Finally, the third part outlines the study’s limitations, acknowledging the constraints that may affect the generalizability and robustness of the findings, and providing a basis for identifying potential directions for future research.

Summary and managerial and theoretical insights

This research presents a robust portfolio model that integrates ESG factors, historical and forward-looking performance metrics, and CCVaR to enhance traditional portfolio management approaches. Leveraging tools such as the FinBERT model and the LM Notebook, the model extracts firm performance insights and predicts future sentiment trends. Empirical evaluation using real-world data specifically, the DJIA constituent companies demonstrates the value of incorporating ESG and performance indicators alongside conventional financial metrics. The results indicate that portfolios constructed with ESG and performance considerations not only achieve improved risk-return optimization, but also align with ethical and environmental priorities, thereby enabling more informed and responsible investment decisions. Compared to traditional models that exclude these elements, the proposed framework yields more stable, resilient, and efficient portfolios. Moreover, integrating fuzzy logic within the CCVaR framework enhances the model’s ability to investigates the inherent unpredictability and imprecision in financial markets. By representing uncertain asset returns with trapezoidal fuzzy numbers, the model captures both ambiguity and subjectivity in financial data, resulting in more accurate risk assessment and robust portfolio decisions. This dual-focus approach balancing quantitative performance goals with qualitative ethical standards offers a compelling and effective strategy for investors pursuing both financial competitiveness and sustainable investing.

The practical implications of this research are particularly relevant for both individual and institutional investors aiming to optimize their portfolios by incorporating financial performance alongside sustainability criteria. By explicitly integrating ESG indicators, performance metrics, and fuzzy-based risk measures into the portfolio optimization process, the proposed model offers a more holistic evaluation of investment opportunities. This enables the construction of portfolios that are not only financially robust but also aligned with long-term environmental and ethical objectives. For investment professionals, the use of CCVaR introduces a more realistic and flexible risk management framework that better captures the uncertainty inherent in asset returns. The integration of performance evaluation within the model further enhances the decision-making process by guiding more informed adjustments in asset allocation. Moreover, asset and fund managers gain strategic insights into effectively balancing financial returns with ethical values, which is increasingly important in today’s investment landscape. In practical terms, this approach allows firms to design portfolios that appeal to a growing segment of investors who prioritize sustainable and responsible investing.

From a theoretical standpoint, this study advances the literature on sustainable finance and portfolio optimization by leveraging the FinBERT model alongside the NotebookLM AI tool. By integrating fuzzy set theory with credibility theory, the research introduces a novel framework for managing portfolios under uncertainty. The incorporation of ESG criteria and performance indicators into the optimization process marks a critical evolution in financial decision-making, recognizing that investment outcomes must be evaluated in tandem with environmental and social dimensions. Additionally, the replacement of traditional risk metrics like CVaR with CCVaR introduces a more refined and adaptive method for assessing risk in volatile and ambiguous market environments. This conceptual contribution not only enriches the theoretical understanding of risk and sustainability in finance but also lays a foundation for future studies. It opens avenues to further investigate the interplay between fuzzy logic, credibility theory, AI-driven analysis, and ESG integration, ultimately supporting the development of more resilient and ethically grounded portfolio models.

Despite its valuable contributions to sustainable finance, the proposed portfolio optimization model presents several limitations that warrant consideration. First, the model’s dependence on fuzzy set theory and credibility theory necessitates expert judgment to define fuzzy numbers for expected returns. This reliance introduces a degree of subjectivity, potentially affecting the accuracy of the risk assessment. Although fuzzy logic enhances the model’s flexibility in capturing uncertainty, the reliability of outcomes is highly contingent on the precision and credibility of expert input, which may also hinder the model’s scalability due to the resource-intensive nature of expert elicitation. Second, while the model effectively integrates ESG factors, performance metrics, and risk measures, it assumes that both ESG scores and transaction costs remain constant over time. In practice, ESG performance is dynamic, influenced by evolving regulatory frameworks, market forces, and company-specific developments. This assumption may reduce the model’s responsiveness in fast-changing market environments where timely portfolio rebalancing is essential. Lastly, the model’s computational complexity poses practical challenges, particularly when applied to large-scale portfolios that involve numerous assets and evaluation criteria. The integration of diverse metrics such as ESG indicators, sentiment-driven performance data, and fuzzy risk measures increases the computational demand, potentially limiting the model’s suitability for real-time decision-making. Future research should focus on enhancing the model’s scalability and responsiveness, possibly through algorithmic optimizations or machine learning techniques that can automate and streamline parameter estimation. Moreover, validating the model across different asset classes and market conditions would further improve its robustness and practical relevance. These refinements will help ensure that the model remains a viable and effective tool for sustainable investment in real-world financial environments.

State-of-the-Art

In this study, we aim to develop a new framework by integrating four investment-impacting criteria in the form of a utility function and employing artificial intelligence tools and models. Our model considers risk, return, sustainability, and performance as investment-impacting criteria. The proposed model for calculating investment risk also considers the value-at-risk measure in the form of credibility theory. This model includes textual and numerical price data. For the numerical component, price data is considered using expert opinion in the form of trapezoidal fuzzy numbers, while for the textual component, we collected quarterly reports of DJIA companies from Yahoo Finance. Using the NotebookLM tool, we extracted forward-looking statements about the companies past performance and next-quarter outlook. To convert these textual outlooks into quantitative criteria, we used the FinBERT model to perform sentiment analysis. Since FinBERT produces outputs in the form of vectors (positive, neutral, negative), we used the AHP method to translate these vectors into a single numerical score, known as the “outlook score,” as shown in Fig. 3. Therefore, our approach represents an attempt to provide an advanced model. Unlike most previous studies that typically relied on sentiment extracted from social media platforms such as Twitter, websites, or online forums and then used those results as auxiliary predictors for price prediction, our approach leverages the official quarterly reports of companies and provides a distinct and innovative perspective alongside the new utility function. Examples of previous research in this area include:

Colasanto et al.⁵⁰ incorporated sentiment measures particularly polarity scores into stock price forecasting by treating them as an additional view within the Black-Litterman framework. These scores were derived from textual analysis of Financial Times articles and linked to a range of positive and negative events influencing specific stocks. The distinguishing feature of the present research lies in the use of the FinBERT model, which not only provides predictive features but also extracts forward-looking insights from corporate disclosures. Rather than employing sentiment solely as a direct forecasting input, our approach leverages FinBERT to evaluate firms’ future outlooks, thereby adding a qualitative and interpretive dimension to investment decision-making. Leow et al.⁵¹ proposed two innovative models, SAW and SMPT, that incorporate real-time market sentiment sourced from Twitter data and analyzed using BERT. These models dynamically adjust portfolio allocation weights according to sentiment signals, with optimization guided by genetic algorithms designed to maximize cumulative returns and reduce volatility. By contrast, the current study emphasizes sentiment derived from companies’ quarterly reports to evaluate forward-looking outlooks, offering a more fundamental and document-driven basis for sentiment analysis. Day and Li⁶⁴ investigated the role of diverse financial information sources in shaping investment decisions and assessed the value of deep learning techniques in improving financial news classification accuracy. Their results revealed that the nature of the information source significantly influences investor behavior and outcomes, while classification performance is enhanced by deep learning approaches. Unlike the present work, however, their study did not address portfolio optimization but instead concentrated exclusively on the relationship between information quality, classification, and decision-making.

Taheripour et al.¹⁵ presented a model in a study in which the CCVaR measure was used for model risk, but they did not consider the sustainability factor in their study, which we considered in our proposed model. Fatouros et al.⁵⁸ conducted one of the earliest explorations into the use of large language models specifically ChatGPT-3.5 for financial sentiment analysis within the FOREX market. Employing a goal-free prompting approach, they tested multiple ChatGPT prompts on a large dataset of FOREX-related news headlines. Model evaluation was performed using conventional metrics such as precision, recall, F1-score, and MAE. Importantly, this study did not propose a structured investment model but rather assessed the feasibility of LLMs for sentiment detection. Seshakagari et al.⁶⁵ evaluated three state-of-the-art models FinBERT, GPT-4, and T5 on the task of financial sentiment classification. Their comparative analysis, based on precision, recall, and F1-score, provides valuable guidance for selecting the most appropriate model for different applications in financial sentiment research. However, similar to many other contributions in this field, their work remained focused on classification tasks and did not extend to portfolio optimization or investment strategy design.

Nakagawa et al.⁶⁶ proposed an RM-CVaR model combining multiple CVaR levels with L1 regularization, which improves portfolio optimization and creates more stable portfolios. Experiments show that this model outperforms traditional methods in risk-adjusted returns and controlling for capital loss. Our model differs from this study in that it considers expected returns in a fuzzy manner, uses artificial intelligence tools, and also considers a sustainability criterion. Liang et al.⁶⁷ introduced a news-based approach to online portfolio selection in non-stationary markets by integrating sentiment analysis with CVaR-sensitive adjustments. Their model, built on three modules trend, scale, and filter adapts portfolio changes to market fluctuations while avoiding noisy signals. The difference from our proposed model is that our model is defined in a fuzzy environment and considers a sustainability criterion. In a study, Ghanbari et al.^17,68 presented a CCVaR model for crypto investment portfolio optimization in which AI tools are not included in the sustainability criteria.

Limitations and future research

A key limitation of the current study is the static treatment of ESG and sentiment factors, which may restrict its applicability in settings requiring more frequent updates. Future research could address this by utilizing time-series ESG datasets that capture intra-year variations and by incorporating real-time textual data, such as financial news and social media, through advanced sentiment analysis models. Moreover, extending the framework into a multi-period dynamic optimization would allow ESG and sentiment measures to evolve in parallel with financial returns. Such enhancements would enrich the model’s predictive power and practical relevance.

Another potential limitation of this study is the reliance on AI-based models and tools such as FinBERT and NotebookLM, which may be subject to the risk of AI error⁶⁹. While these models can effectively extract and process large volumes of financial text, they may sometimes produce outputs that are not fully consistent with the underlying data or financial context, thereby introducing noise or bias into the results. Although we employed a well-established model trained https://github.com/ProsusAI/finBERT, we acknowledge that complete elimination of such risks is not guaranteed.

A further limitation concerns the lack of explainability and interpretability inherent in such AI tools. The internal decision-making processes of these models remain largely opaque, which makes it difficult to trace how specific outputs are generated and may reduce confidence in the derived inputs used in our framework. Future research could address these challenges by integrating explainable AI techniques, additional human expert validation, or hybrid modeling approaches to strengthen both the reliability and transparency of AI-driven financial analysis.

Conclusion

In this study, we developed a comprehensive and innovative investment strategy model tailored to uncertain (fuzzy) environments that incorporates four key evaluation criteria: risk, return, historical performance, future outlook, and ESG factors. The proposed framework models expected returns using trapezoidal fuzzy numbers, which lets for a more realistic representation of the unpredictability inherent in financial markets. To quantify portfolio risk, we used the CVaR measure within the credibility theory framework, which provides a robust approach to risk management under fuzzy conditions. Portfolio sustainability was assessed using ESG risk scores, which indicate companies’ alignment with long-term sustainability and resilience principles. To assess historical performance and future outlook, we used data extracted from companies’ quarterly financial reports. The data was analyzed using advanced AI tools, including Google’s FinBERT language model and NotebookLM, allowing us to extract sentiment and insights from unstructured financial texts. The effectiveness of the proposed model was demonstrated through its application to real-world data, specifically focusing on DJIA-listed companies. The results confirmed the feasibility and practical efficiency of the model, indicating its potential as a decision support tool for investors operating in complex and uncertain market environments.

As demonstrated in Tables 7, 8, 9, 10, 11 and 12, the highest portfolio utility was consistently achieved when greater weight was assigned to the performance criterion. Furthermore, the variation of optimal portfolios under different cardinality levels (K = 3, 5, 7) was quantitatively analyzed and incorporated into the findings. The exclusion of high-return but low-score stocks (e.g., INTC and BA) also emerged as a natural outcome of the multi-criteria optimization, underscoring the comprehensiveness of the proposed approach. Finally, the conclusions highlight both the practical implications (e.g., decision-support for risk-averse investors aligning ethical screening with risk/return optimization) and the limitations of the model (such as static inputs and computational burden).

While the proposed fuzzy-based investment strategy model has shown promising results, there are several opportunities for improvement and development that could significantly improve its robustness, consistency, and practical relevance. One potential avenue is to explore alternative representations of fuzzy numbers. Although trapezoidal fuzzy numbers were used in this study due to their computational simplicity and interpretability, future work could explore the use of more complex forms such as Gaussian fuzzy numbers, which may provide smoother and more realistic modeling of uncertain returns and risk factors. These numbers can better capture the imprecise nature of financial forecasts and investor expectations. Another valuable improvement involves the inclusion of additional decision criteria. While the current model considers risk, return, ESG factors, historical performance, and outlook, future models could include fundamental financial indicators such as liquidity ratios, profitability measures, leverage ratios, and valuation multiples. These financial ratios provide measurable measures of a company’s operational health and may enhance the investment selection process. Additionally, investor sentiment and behavioral data derived from news articles, social media platforms, or analyst reports can provide qualitative insights that complement traditional financial indicators. Sentiment analysis tools and NLP models can be integrated to systematically measure market psychology and incorporate it into investment portfolio decisions. Advances in artificial intelligence and machine learning (ML) also offer significant opportunities. Future models can adopt supervised or unsupervised learning methods such as RF, SVM, gradient boosting, or deep neural networks to increase forecast accuracy. In addition, reinforcement learning can be used to adaptively adjust the investment portfolio over time based on market feedback and evolving data streams. From a modeling perspective, extending the current single-period optimization framework to a multi-period environment significantly increases its practical applicability. A dynamic, time-series-based model allows investors to plan and balance their investment portfolio over multiple time horizons, taking into account time dependencies and evolving constraints. In addition, practical investment constraints such as transaction costs, liquidity constraints, turnover constraint, tax implications, and regulatory constraints should be incorporated to align the model with real-world implementation scenarios. Considering such limitations increases the feasibility and attractiveness of the model for institutional investors. Finally, examining alternative and hybrid risk measures could provide more comprehensive insight into portfolio risk. Beyond CVaR, future work could consider measures such as EVaR, TVaR, omega ratio, and downside deviation, each of which offers distinct advantages in assessing risk in uncertain or asymmetric return distributions. Collectively, these future directions could help build smarter, more flexible, and more practical investment decision-making systems that are capable of performing effectively in today’s complex and data-rich financial environments. Researchers can also develop the proposed model with more data and in the form of a multi-period model for a larger number of companies, such as the Nasdaq index.

Author contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Esmaeil Taheripour, Seyed Jafar Sadjadi and Babak Amiri. The manuscript was written by Esmaeil Taheripour, Seyed Jafar Sadjadi and Babak Amiri. All authors read and approved the final manuscript.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data availability

The data that support the findings of this study are available in the Researchgate at: [https://www.researchgate.net/publication/396269888_Dow_Jones_index_Company].

Declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical approval

Not applicable to this research.

Consent to participate

Not applicable to this research.

Consent to publish

This work has not been published before.