Smart decision framework for financial planning and investment optimization

Abstract

In today’s volatile financial landscape, investment planning has become increasingly complex due to inflationary pressures, market instability, and geopolitical uncertainties. Traditional decision-making models, which rely solely on static optimization and quantitative data, often fail to incorporate investors’ subjective judgments, leading to biased evaluations and less adaptive outcomes. To overcome these limitations, this study proposes an intelligent decision-making framework for financial planning and investment optimization that integrates both objective and subjective perspectives. The logarithmic percentage change-driven objective weighting (LOPCOW) method is employed to derive objective weights based on data variability and informational entropy, ensuring transparency and data-driven rigor. Simultaneously, the ranking comparison (RANCOM) technique captures subjective weights from expert evaluations, reflecting strategic insights, behavioral preferences, and market experience. These complementary approaches are combined to create a balanced and holistic weighting structure. The evaluation process is further enhanced through an enhanced preference ranking organization method for enrichment of evaluations (EPROMETHEE-II) that integrates nonlinear distance and similarity measures, facilitating superior modeling of uncertainty, enhanced discrimination among proximate alternatives, and more reliable and precise ranking outcomes compared to the conventional PROMETHEE-II methodology. A real-world case study on financial planning and investment optimization validates the framework, demonstrating its capability to support balanced, expert-informed, and data-driven portfolio decisions, with the ESG-Focused Portfolio () identified as the top-performing alternative due to its adaptive learning and strong ESG integration. The framework provides superior adaptability, robustness, and interpretability, aiding investors and policymakers in making resilient, informed, and context-aware financial decisions.

Introduction

The increasing global trend toward a more digitalized economy has dramatically increased the complexity of the financial markets through their connection to each other, data, and investors. There have never been more opportunities for investors, nor as many challenges as we see today because of the unprecedented rate of inflation, the ever-increasing levels of volatility, the rapidly changing nature of geopolitics, and the rapid technological advancements that continue to emerge. These factors have further complicated the process of financial planning and developing portfolios; thus, sophisticated decision-making frameworks are required to handle the ambiguity of information from multiple sources with different conclusions². The current way of doing investment research that is mainly based on static optimization and quantitative financial ratios is inadequate for describing complex investor behavior including investor sentiment, sustainability restrictions, and risk behavior tolerance³ and therefore often too rigid and short-term as criminal environments vary. Today’s financial situation has moved away from the traditional accounting and budgetary systems to the more advanced, flexible, and science-based systems. Artificial intelligence (AI), machine learning (ML) and predictive analytics techniques are becoming increasingly popular in the context of contemporary financial planning to aid in decision-making. The use of AI, ML and Predictive Analysis has also improved the future predictions and performance measurements allowing for a better way to handle changing global conditions. Analogous to developments and aid from governments which has precipitated this and furthered the growth of intelligent financial systems, together with research that has thereby aided economic growth⁴. The development of AI and big data aspects of financial theory has brought about the possibility for financial institutions to model complex nonlinear relationships between variables affecting investment results and business growth.

Even while innovations in digital technology are developed and implemented into daily decision making, traditional decision making techniques are not sufficient to adapt to the changes faced in integrated and volatile financial systems, requiring that not only should behavioral adjustments be modeled but sustainability issues and market assessments be made also, that rather many consumer are faced with rigid mental patterns and fixed optimization techniques⁵. The problems occur in part as a result of conflicting factors, which must be taken into consideration at one and the same time, being the need for assessing investment return, the risk of the investor, the level of market euphoria, and the social responsibility’s relation to the market perception by way of example⁶. In addition, it appears not only that uncertainty exists as to the data and expert opinion but it seems that this leads to both erratic and biased assessments being made, leading to the lack of credit we see in the conventional financial phenomena⁷. A recent investigation, however, suggests that predictive analysis, complex algorithms, and expert opinion can be brought together in a synthesis to assist in better financial decision-making when the future is uncertain. By means of these data-based techniques, improved financial assessments can be produced and, it seems are capable of being made, and if a rapid updating of such financial assessments is to occur, when the variable indicators of the market alter⁸. AI enhanced predictive models have shown themselves to be significant means for making improved forecasting assessments, and also for improving the operational processes in a company and also increasing the sustainability of the corporate issues pertaining to operational processes^9,10. The validation of the use of data systems, especially in its relation to the emergence of big data technologies, has produced significant models of being an entrepreneur, particularly suitable as applied to sustainability in the contexts of investment and energy finance. These also indicate a need for intelligent systems for decision-making in the various dimensions of society¹¹. Furthermore, studies on strategic financial management indicate a requirement for far better adaptation of the corporate performance to acceptable loyal analytics and which indicates at the same time continuity and robustness in the reaction to volatility in the markets^5,6. Evidence at the firm level indicates that between the assessment systems and policy performance, there exists a close dependence on the size of the firm, indicating very readily a need for flexible and adaptable decision-making in the situation of the different decision-making⁷.

Despite the substantial improvements in financial analysis caused by the growth in both analytical technology and analytic approaches, there are still some important gaps. Current frameworks rarely achieve a thorough merging of subjective expert opinion and objective data examination. They often ignore such things as uncertainty, nonlinearity, and correlation of the various elements in the criteria, which can have a gross effect on the real investment results as well. Many of the traditional models make use either of heuristics or statistical weighting techniques, which do not take into account the synergetic effect of the combination of human expertise and data analysis. It has been noted that the results of the financial decision process are made unreliable by uncertainty arising from both limited knowledge, biased experts and market volatility. This is an indication of the severity of this problem and the need for hybrid intelligent decision support systems that may be used to integrate cognitive reasoning, objective analysis and uncertainty modeling into one transparent decision making process. A hybrid decision support system is proposed here as a means to address some of the limitations associated with financial planning and investment optimization. Combining LOPCOW and the RANCOM method , the proposed method balances both analytical accuracy and managerial skill through a synergistic approach of combining two methods that are otherwise complementary. EPROMETHEE-II is utilized during the enhanced evaluation phase to improve uncertainty management and enhance the accuracy of results from each preference model using non-linear similarity and distance measures.

Objectives of the study

Ultimately, the goal of this research is to build an intelligible and transparent decision support system to enhance the productivity, clarity and flexibility of both financial market planning and investment optimization. Specifically, the objective of this research is to:

• Build a complete, flexible decision-making framework for both financial planning and investment evaluation combining quantitative metrics of performance with qualitative insight into management.

• To provide a complete, flexible framework capable of illustrating uncertainty, dependency, and complexity when making financial decisions; ultimately, providing investment results that are based on data and more dependable.

• Encourage openness, fairness, and flexibility in the evaluation of some financial metrics, such as profitability, liquidity, risk, and sustainability, to help organizations make better decisions.

• Support financial managers and policymakers in figuring out how to invest wisely so that short-term performance and long-term wealth creation are balanced, even when the market is changing.

• Improve strategic decision-making by combining data-driven reasoning with managerial judgment to move forward with financial analytics and decision-support research.

Literature review

In 1965, Zadeh introduced fuzzy sets to address the ambiguity frequently encountered in daily life¹². Fuzzy set theory provides a robust framework for defining situations characterised by ambiguous or unclear data. Fuzzy sets assign a degree of membership to an element within a set to address such circumstances. An individual may perceive that an object x is only partially included in a set A, however they may lack total certainty. The extent of x’s membership in A is ambiguous. Fuzzy sets have been essential in addressing numerous challenges in applied mathematics, information science, and decision-making. Atanassov’s pioneering study developed the concept of “intuitionistic fuzzy sets (IFSs)” as an evolution of conventional fuzzy sets. IFS introduces an additional parameter termed “non-membership” (N-MS) to traditional fuzzy sets, which solely consider membership degrees. This novel concept enables the representation of ambiguity and uncertainty in sets more sophisticatedly. Since its conception, IFSs have been widely employed in various domains necessitating advanced approaches to manage uncertainty, such as pattern recognition, decision support systems, medical diagnostics, and control systems. Atanassov’s research laid the groundwork for the following investigations into IFS, promoting deeper exploration of uncertainty modeling and the development of novel operations defined on IFSs¹⁴.

Ye et al.¹⁵ introduced a novel similarity metric for IFCSs and applied it in MCDM to evaluate the performance of industrial robots. The authors have shown the applicability of credibility theory to create more expressive and accurate If models. For example, Qiyas et al.¹⁶ developed an intuitionistic fuzzy (IF) credibility Dombi aggregation operator that they used to select train locomotives in Pakistan through a credibility based methodology. The results clearly demonstrated how using credibility enhanced operators would reduce decision maker’s doubt and trust issues to create a higher level of reliability and consistency in transportation planning. Du et al.¹⁷ expanded on this idea to create the first trapezoidal fuzzy neutrosophic credibility interval aggregation operators for decision making in landslip risk assessment and management. They showed that credibility theory can be applied in conjunction with high order fuzzy and neutrosophic models to evaluate very uncertain engineering alternatives. Zhang et al.¹⁸ established a credibility based evaluation framework for complex simulation systems by combining perceptual computing with Type-2 fuzzy sets. This is a new application of integrating uncertainty modeling with human cognition to analyze large scale simulation environments in which both trust and ambiguity are only partially present. Yahya et al.¹⁹ also developed multiple criteria decision support models that used fuzzy credibility ratings as input to identify environmentally friendly supplier candidates. Their research highlighted the need to combine fuzzy credibility with rough set theory to provide a viable solution to addressing ambiguous evaluations and differing levels of expertise in environmental sustainability supply chains.

Banerjee et al.²⁰ provided a detailed overview of the first decade of Z-number research, including theoretical foundations, techniques for calculation, and applications. Z-numbers define uncertain information by expressing both value and trustworthiness, thus giving an excellent model of human thought. Sari and Kahraman²¹ introduced the concept of intuitionistic fuzzy Z-numbers (IFZNs), which extend classical Z-numbers by introducing intuitionistic fuzzy sets into both components of the Z-number pair. This allows the simultaneous representation of membership, non-membership, and hesitation, and improves the modeling of complex uncertainty and human indecision. Jaini²² applied the intuitionistic fuzzy Z-number model to the problem of supplier selection and found that IFZNs outperform classical fuzzy and intuitionistic fuzzy models by better capturing the uncertainty and evaluative trust in expert judgment, thus leading to better quality and more stable decisions, more suitably interpreted. Recently, Sergi and Sarı²³ extended this technique using interval valued intuitionistic fuzzy Z-numbers (IVIFZNs) to evaluate learning management systems, combining interval-valued intuitionistic reasoning with Z-number logic to address linguistic evaluations more suitably when meeting the aspects of imprecision and uncertainty. The net result of these developments is the evolution of Z-numbers from a classical point to an advanced intuitionistic fuzzy Z-number model or interval-valued intuitionistic fuzzy Z-number model, greatly enhancing the modeling of uncertainty, hesitation, and confidence in multi-criteria decision-making systems. In recent years, the primary focus of studies on MCDM methods has been on improving objectivity in the criterion evaluation and dealing with the uncertainty present in decision-making by means of fuzzy logic and hybrid methods. For example, Kumar and Gandotra²⁴ used a Pythagorean fuzzy entropy measure and the TOPSIS method to get a measure of preferences on advertising agencies, thereby reducing ambiguity in the decision-making data. Wang et al.²⁵ developed an objective weighing measure based on the concept of entropy, aimed at measuring the sustainable performance of third-party logistics, which emphasised the need for objective reliability. Setiawansyah²⁶ and Wang et al.²⁷ introduced the measure of entropy into the methods of EDAS and MABAC to obtain added reliability in decision-making in customer service assessment and selection of e-commerce platforms.

Recent studies have looked into the amalgamation of several different objective weighting systems. Van Dua et al.²⁸ showed the unification of a variety of objective weighting systems under a material selection exercise to improve consistency and reliability. Hashemkhani Zolfani²⁹ did comparisons between three different methods in order to show the advantages and disadvantages of each method when looking at various criteria weights involved in different fields. Shunmugapriya et al.³⁰ applied the MEREC method in a fuzzy environment in order to improve the effectiveness of the decision process when faced with ambiguity. Dündar³¹ used LOPCOW and CRADIS to assess the success of Turkish university projects and show the versatility of the method. Van Dua³² devised hybrid methods that applied PSI to SAW and MARCOS to enhance objectivity and ranking accuracy. Fuzzy MCDM models have been extensively analysed to tackle uncertainty. Derse³³ employed a fuzzy DEMATEL–FUCOM–SWARA methodology to rank obstacles in green reverse logistics. Conversely, Nezhad et al.³⁴ proposed a fuzzy SWARA-based readiness model for the deployment of IoT in the banking industry. Arslan and Cebi³⁵ improved the WASPAS approach by integrating decomposed fuzzy sets to tackle ambiguity in decision parameters more effectively.

Recent advancements in fuzzy operators and decision-making frameworks have attracted considerable attention. Ying³⁶ and Xue³⁷ utilised sophisticated fuzzy operators, namely circular q-rung orthopair and picture fuzzy Muirhead mean, alongside the MOORA approach to evaluate mental health and educational quality in higher education, respectively. Biswas et al.³⁸ introduced the ERUNS technique, which combines relative utility and nonlinear standardisation for assessing business performance in the energy sector. The IDOCRIW and CRADIS techniques have been employed in performance evaluation contexts. Nestić et al.³⁹ determined weights for commercial investment projects using IDOCRIW, whereas Arman et al.⁴⁰ assessed digital innovation performance in European nations with a hybrid IDOCRIW-CRADIS technique. The studies collectively demonstrate the growing emphasis on objectivity, robustness, and flexibility in MCDM applications across many sectors and decision-making contexts. Furthermore, Farid et al. introduced the circular intuitionistic fuzzy framework for assessing smart robotics technologies⁴¹ and for prioritizing sustainable methods in the smart waste management of automotive fuel cells in road freight trucks within extended fuzzy contexts. The Table 1 represented the limitations of the traditional method, and it was covered by the proposed framework.

Table 1.

Comparative evaluation of conventional decision-making techniques and the proposed framework.

Method	Underlying principle	Identified shortcomings	Advancements in the proposed framework
AHP42	Hierarchical pairwise comparisons to establish relative importance among criteria based on expert judgment.	Highly subjective; consistency issues; lacks adaptability with complex datasets.	RANCOM sharpens the participating expert interpretation by means of ranked logic contrived in the development, while LOPCOW imbues objective verification to dampen bias and irregularity.
Entropy27	Determines weights from the level of information dispersion across criteria.	Purely data-dependent; overlooks human perception and contextual relevance.	The entropy of LOPCOW is extended by means of logarithmic percentage change, which increases the sensitivity of data fluctuation, while it is entirely complementary to the experts’ views.
CRITIC29	Assigns objective weights through contrast intensity and correlation analysis.	Excludes qualitative opinions; fails to capture subjective or behavioral factors.	The integrated form of LOPCOW and RANCOM balances the dispersion of statistics with the experts’ logic, and well warrants the comprehensive procedure of development, therefore of weights.
SWARA33	Sequential expert-driven weighting based on comparative importance.	Heavily relies on expert consistency; absence of an objective verification mechanism.	The expert-derived ranking from RANCOM enters into the formulation of LOPCOW so that the results of the rankings are re-aligned objectively, thus perfecting the coherence and reliability of the weights.
TOPSIS50	Evaluates the proximity of each alternative to ideal and anti-ideal solutions.	Sensitive to normalization; less reliable under uncertain or fuzzy environments.	The EPROMETHEE-II includes in its development non-linear measures of similarity and of uncertainty measures which will establish steadiness of the decision with different measures of scale.
VIKOR43	Provides a compromise solution minimizing group dissatisfaction and individual regret.	Parameter-dependent; unstable under complex or high-dimensional uncertainty.	The computation of the net flow based on similarity measures in EPROMETHEE-II removes the objection of dependence on arbitrary compromise coefficients.
CoCoSo44	Fuses additive and multiplicative aggregation principles to form a compromise score.	Weighting sensitivity and parameter tuning can distort rankings; limited interpretability.	The integrated method of weighting using LOPCOW and RANCOM brings the results of aggregation into a steady state, and also the same is capable of clear interpretation under uncertain data environments.
PROMETHEE-II45	Outranking model relying on preference functions to compute positive and negative flows.	Linear preference assumptions cannot effectively represent fuzzy or hesitant judgments.	Non-linear distance-similarity functions are introduced in EPROMETHEE-II as being able to advance the dimensions of robustness and to treat uncertainties.

Identification of research gaps

To position the suggested framework within the existing literature, several pertinent areas of research were concerned with exploring the ongoing gaps in the literature that deal with financial planning and investment decision-making. These research areas can be seen in Table 2, showing the critical weaknesses identified in the previous literature and the solutions proposed by the framework to overcome them. Table 2 lists the main drawbacks of current financial decision-making models and the proposed framework’s procedures. To address these issues, the algorithm operates sequentially and modularly. The IFZN environment models uncertainty in expert judgments and dynamic market information, allowing imprecise and confusing assessments to be mathematically recorded and processed reliably. The process of strengthening input data before weighting or ranking is next used in the development of the framework hybrid weights for criteria importance. LOPCOW objectively calculates weights using logarithmic percentage changes, dispersion, and informational entropy, eliminating human bias and giving a greater emphasis to criteria that have better discriminative power, to use in objective comparisons. The RANCOM module also uses expert-driven ranking comparisons to measure investor sentiment, strategic foresight, and risk preferences. These objective and subjective weights create a balanced and consistent priority structure that balances quantitative evidence and domain expertise.

Table 2.

Key research gaps and corresponding methodological contributions.

Focus area	Research gap	Proposed solution
Integration of subjective and behavioral dimensions	Most existing decision-making frameworks primarily emphasize numerical indicators, while human-related aspects such as expert judgment, behavioral tendencies, and strategic intuition are often insufficiently addressed.	The proposed RANCOM module explicitly incorporates expert evaluations, allowing behavioral preferences and professional insights to be systematically integrated alongside quantitative data.
Weighting of financial and strategic criteria	Many prior studies depend either on purely subjective weighting schemes or fully data-driven techniques, which may result in biased or unstable weight distributions.	The LOPCOW method is employed to derive objective weights by analyzing data variability and informational contribution, thereby improving consistency while still supporting informed decision-making.
Preference modeling and alternative ranking	Conventional EPROMETHEE-II approaches often show limited responsiveness to nonlinear relationships and minor preference differences among competing alternatives.	An enhanced EPROMETHEE-II mechanism is adopted to better capture preference intensity, leading to more reliable, transparent, and meaningful ranking results.
Portfolio optimization and risk-responsive management	Traditional portfolio selection models frequently overlook ESG considerations, adaptive investment behavior, and the growing complexity of modern financial assets.	The proposed framework evaluates investment alternatives in a comprehensive manner, enabling risk-aware and ESG-integrated portfolio optimization under dynamic market conditions.
Modeling and managing uncertainty	Uncertainty arising from expert assessments and fluctuating market environments is not adequately represented in many existing decision models.	Intuitionistic fuzzy Z-number (IFZN) modeling is utilized to better reflect uncertainty and reliability in expert opinions, thereby strengthening the robustness of the evaluation process.
Decision support in evolving financial contexts	Several available decision-support tools lack flexibility and fail to adapt to rapidly changing financial and economic environments.	By combining objective weighting, expert judgment, and uncertainty handling, the proposed decision-support framework offers an adaptive and context-aware solution for complex financial decision-making.

Motivation

Modern financial markets are highly dynamic and unpredictable due to rapid technological advancements, globalization, and frequent economic disruptions. In such situations, investment planning requires decision-making systems that are able to cope with numerous conflicting factors, that combine various information sources, and are capable of adapting to the modern era. Methods of the traditional nature, which are based mainly on static optimization and quantitative models, often do not consider the complex nature of the relationships between risk, return, and investor behaviour, and do not allow for the consideration of qualitative issues such as investor sentiment and overall strategic foresight and behavioural risk tolerance, leading to biased, inflexible, and poor portfolio alternatives. Recent research has demonstrated that traditional multi-criteria decision-making approaches such as EPROMETHEE-II, TOPSIS⁵⁰, and VIKOR⁴³ are unable to accurately model the nonlinear relationships between alternatives under consideration, and the subtle differences in performance scores will continue to exist for all alternatives considered so making some alternatives not so beneficial in terms of risk and return. Further, these traditional approaches cannot cope with the uncertainty of expert judgments concerning the criteria weightings or the evaluation of the alternatives themselves. Today’s financial decision-making also increasingly seeks to include sustainable investment approaches, not just financial rejuvenation, but also ethical and environmental factors via ESG integration. In general, the current multi-criteria decision-making approaches fail as they all fail to incorporate fully adequately or successfully financial performance with ethical and environmental concerns. AI methods can lead to portfolio optimization that incorporates an ability to adapt, but frequently are not very good at dealing with the methods of structured expert judgment required in such more sophisticated multi-criteria decision-making approaches, making them complex and difficult to understand and utilize in practice. The paper presents a new decision-making approach, which, as far as is known, is the first to unify the objective weightings obtained from LOPCOW, the subjective evaluations of alternatives using RANCOM, and finally deals with managing the uncertainty of expert judgment via IFZNs. The current EPROMETHEE-II model is improved through the use of new distance and similarity metrics, which, if employed correctly, should lead to improvement in preference modeling sensitivity and robustness of rankings. The system can therefore be termed unique as it unifies issues of objective data, structured expert evaluation, and uncertainty modeling, as well as including ESG issues and the capability of AI adaptation. The outcome will offer an extremely powerful, clear, and simple decision support system for investment planning as it incorporates both quantitative and qualitative factors. It will offer investors or decision makers a system through which better-informed and flexible decisions may take place, reaching portfolio alternatives more in line with the current business climate. The research has major implications for the development and extension of both MCDM theory as well as extending the application to better financial decision-making systems.

Contributions of the study

This study makes several significant contributions to the field of financial planning and investment decision-making:

1. The proposed method of analysis takes advantage of both the LOPCOW method for objectivity and the RANCOM method for subjectivity. Therefore, the LOPCOW and RANCOM methods provide an equal balance of both quantitative and qualitative aspects when comparing investment alternatives.

2. The original EPROMETHEE-II method has been improved upon through new metrics of similarity and distance. These metrics enable the identification of subtle differences between alternative investments and improve the handling of non-linear interactions as well.

3. The research utilizes IFZN to express uncertainty within both expert evaluations and financial data, thereby providing a method of increasing the reliability and robustness of the decision making process in volatile markets.

4. The proposed methodology will provide a wide variety of means to compare different investment strategies by incorporating ESG criteria into the decision making process, utilizing Adaptive AI Learning, and Risk Management techniques; therefore, creating a comprehensive tool for selecting portfolios in today’s world.

5. The proposed method will combine all three viewpoints (objective, subjective, and uncertain), and produce a decision support system that provides a clear, understandable, and flexible means for investors and policymakers to develop solutions to their complex financial problems.

6. The proposed method will be demonstrated through a real-world example to illustrate its feasibility and effectiveness in comparison to traditional methodologies, specifically regarding the ability to be adaptable, consistent in ranking, and precise in decision making.

Structure of the paper

This paper is organized as follows: Sect. 2 presents the fundamentals of intuitionistic fuzzy Z-numbers for addressing uncertainty. Section 3 outlines the proposed algorithm, which integrates LOPCOW and RANCOM-EPROMETHEE-II, and Sect. 4 provides a case study. Section 5 presents the results and analysis, encompassing comparative and sensitivity assessments. Section 6 concludes the paper and delineates potential avenues for future research.

Intuitionistic fuzzy Z-numbers

Definition 2.1

Atanassov¹³ an IFS in a set X is defined by , , where and are the membership degree (MD) and non-membership degree (NMD) of the element such that .

The MD and NMD of IFS serve as personal assessments of DMs to evaluate various objects based on one or multiple criteria. To improve decision makers’ preferences in the MCDM process, Ye et al.²¹ introduced the concept of IFZN, incorporating credibility degree (CD) into both MD and NMD of IFS.

Definition 2.2

Sari²¹ let X be a set. Then, an IFZN in X is defined as

where is a pair of the MD and CD of the element , and is a pair of the NMD and CD for , along with the condition and .

For the convenient representation, the component in R is simply denoted as , which is named IFZN.

Definition 2.3

Sari²¹ let

be two IFZNs, and let . Then, the operational relations are defined as follows:

1. Inclusion:

   
2. Equality:

   
3. Union:

   
4. Intersection:

   
5. Complement:

   
6. Algebraic Sum:

   
7. Algebraic Product:

   
8. Scalar Multiplication:

   
9. Power Operation:

   

Definition 2.4

Sari²¹ the score function of an intuitionistic fuzzy Z-number is defined as, the score function can be written as:

or equivalently,

The score value indicates the overall preference strength of the IFZN.

Definition 2.5

Sari²¹ the accuracy function of an intuitionistic fuzzy Z-number quantifies the determinacy or precision level of information and is expressed as: By substituting, it becomes:

A higher H(Z) value indicates a greater degree of accuracy and lower hesitation in the IFZN.

Definition 2.6

Given two IFZNs and , their ranking is determined as follows:

In the event of identical score values, the accuracy function serves as a secondary ranking criterion.

IFZN distance measures

First, we introduce some distance measures between two IFZN and .

Definition 2.7

Let us consider two IFZN and given as

then the distance measures between two IFZN and , in the universe of discourse , are defined as follows.

(i) Hamming distance:

(ii) Euclidean distance:

(iii) Hamming distance including hesitation degree:

(iv) Euclidean distance including hesitation degree:

(v) Hausdorff distance between and :

(vi) Hausdorff distance between and including hesitation degree:

Definition 2.8

For two IFZN and as defined in Definition 2.7, the Minkowski distance of order is defined as

(ii) The weighted Minkowski distance is defined as

where and

IFZN similarity measures

This section presents novel similarity metrics for IFZN to more precisely depict the proximity of uncertain data. These measurements establish a robust basis for the formulation of sophisticated ambiguity-driven decision-making procedures. Let and denote two IFZN as specified in Definition 2.7. The following symbols and relations will be employed to determine the similarity measures based on the reference parameters of IFZN.

where and be the hesitation degree with respect to and . Based on the weighted Minkowski distance between the two IFZN defined in Definition 2.8, and considering the preference rates , we propose the first similarity measure for IFZN which is an extension of⁴⁶ as follows.

The Hamming distance contributes as the secondary similarity metric for IFZN, building upon the research of⁴⁷ as outlined below.

The third similarity measure for IFZN is established by augmenting the similarity measure for IFSs⁴⁸ as detailed below.

Developing a decision support system using IFZNS environment

The proposed decision support framework compares several alternatives that are denoted by against b criteria, which are denoted as . The assessments are conducted by a group of experts in the field f, which are denoted as , where each one is given a weight which reflects both the weight that each one has in his/her field and/or expertise, normalized to guaranty that . ISFZNs are used in order to express the uncertainty and indecision associated with the answers of the experts. This is an adequate means of expressing the decision adequately due to the fact that they express the membership, non-membership, and degrees of uncertainty about the concept that these categories clearly involve. The logic behind the determination of criteria weights arises out of a hybrid procedure. The weights that are objectively derived are obtained by the procedure called LOPCOW, while the subjective ones are derived from the method called RANCOM based on the subjective data that has been developed by the experts.

Algorithm

Step 1: A decision-making scenario comprises a collection of criteria, , and a collection of alternatives, . Each alternative is evaluated against every criterion utilizing IFZNs. The resultant assessments are compiled in an IFZN matrix , where denotes the evaluation of the -th alternative about the -th criterion.

Table 3 delineates definitions for the seven linguistic words of IFZNs employed by decision-makers.

Table 3.

Linguistic variables and their corresponding IFZNs.

Linguistic terms	IFZNs
Exceptional (VH)
Strong (H)
Fairly Strong (MH)
Balanced (M)
Fairly Weak (ML)
Weak (L)
Critical (VL)

Softmax function

Step 2: Each item in the IFZN sets is an evaluation given to a specific criterion in the decision-making framework. Equation 1 shows how to measure each element’s contribution using a scoring function, which is then used to compare the performance of each option, and Equation (2) is used to score value for every fuzzy number supplied by DMs.

1

where,

2

Step 3: Evaluate each DM listed in Table 4 and use Equation (3) to calculate their weights. The tasks and duties that are assigned to each DM during the decision-making process determine the scores.

3

where is a scaling parameter that controls the dispersion and sensitivity of the weight distribution.

Table 4.

Characteristics of DMs in the case study.

DM	Expertise	Evaluation Focus
DM1	Institutional Investor	Prefers stability, consistent returns, and strong liquidity with minimal volatility.
DM2	Financial Analyst	Focuses on quantitative efficiency, maximizing risk-adjusted returns and expected gains.
DM3	ESG Compliance Officer	Prioritizes sustainability, ethical responsibility, and inflation protection.
DM4	AI Portfolio Strategist	Emphasizes predictive accuracy, technological exposure, and dynamic adaptability.

Step 4:The combined IFZNs are the sum of the assessments from each DM. Equation 4 defines the aggregated Intuitionistic Fuzzy Z-number Dombi Weighted Average (IFZNDWA), which is created by combining the separate decision tables into a single group based on the DMs’ judgments.

4

Step 5: The aggregated decision matrix’s score function can be determined with Equation (5).

5

LOPCOW method

Step 6: The normalized matrix was created by altering the score matrix in this manner:

6

where and represent the cost-type and benefit-type criteria, respectively.

Step 7: Calculating the percentage values for each criterion is accomplished through the utilization of Equation 7.

7

8

Step 8: In order to ascertain the percentage values for every criterion, the equation 7 is commonly employed.

9

Step 9: One by one, the criteria’s positions are compared in order to produce the ranking comparison matrix ().

Where,

10

and is the significance function of jth criteria.

Step 10: Use Equation (11) to calculate the summand criteria weights ().

11

Step 11: The subjective weights obtained by using Equation 12.

12

Step 12: The criteria weights from the RANCOM and the objective weighting method LOPCOW are calculated using equation (13). The final criteria weights are then established, as shown in Equation (14).

13

14

Step 13: A variety of techniques are employed to evaluate the distinctions between alternatives. Equations (18)–(21) describe the use of similarity-based measures, whereas Equations (22)–(25) compute distance-based measures.

15

and the hesitation degrees

16

Define the component-wise differences

17

18

19

20

21

(i) Hamming distance:

22

(ii) Euclidean distance:

23

(iii) Hamming distance with degree of hesitation included:

24

(iv) Euclidean distance with degree of hesitation included:

25

Step 14: The preference index, is determined through the application of similarity and distance measures. The similarity measures are determined through the application of Eq. 26 while the distances are determined as in Eq. 27 covering all the various criteria and taking into account the different weights of those criteria. In the interpretation, higher values for similarity indicate stronger preference while lower values show closeness and more preferability of alternatives. The computed through the preference index is used to rank the alternatives so that the preferred alternative is that which has the highest similarity or the lowest distance number.

26

for distances, use this equation.

27

Step 15: In the PROMETHEE-II method a preference matrix is established through which the performances are compared for all alternatives. The outflow gives the way of leaving the flow as shown in Eq. (28) indicates the better this alternative is than all the rest. The inflow as shown in Equation (29) indicates the less importance of an alternative as compared with others. Equation (30) gives the net outflow , which is the total strength of an alternative in that it indicates the degree of the total flow (the sum of both outflow and inflow). Mathematically, these flows are defined as:

28

29

30

The workflow of the proposed PROMETHEE-II-based algorithm is presented in Fig. 1, demonstrating the sequential process of evaluation and ranking of alternatives.

Figure 1.

Framework of the proposed algorithm incorporating the PROMETHEE-II.

Case study

Investment planning and portfolio optimization have become more complex in today’s financial world, which is characterized by volatility, uncertainty, and rapid change in the areas of technology, economics, and geopolitics. Traditional methods for making decisions, in all their varieties, often based on static models, historical data, and only quantitative analysis, are not well-suited for dealing with the complexities of today’s financial markets. These traditional methods are essentially quantitative, focusing on numbers, meaning expected return, risk, and liquidity, but not on the very important qualitative and behavioral factors that are necessary for market-specific decisions: investor sentiment, strategic foresight, and individual risk factors. Consequently, investment decisions based on these methodologies will not only result in mistakes but are likely to be strongly biased and not adequately related to real investment objectives. Also, uncertainty is always a barrier to financial decision-making. The data from the market tends to be a mass of noise, with gaps and quick changes in data, while expert assessments differ greatly because of the factor of bias, scarcity of information, and differences in strategic opinion. Traditional models almost never include successful models to deal with the uncertainties of the situation, and thereby fail in their adaptability, reliability, and forecasting abilities in practical applications. Furthermore, new investment fashions such as ESG-Focused Portfolio, AI-driven financial instruments, and sustainable investment strategies require new models that bring ethical and adaptive learning thinking into decision-making processes. Modern methods generally do not consider contrasting criteria, combine rational and emotional criteria, or deal efficiently with problems of uncertainty. Therefore, their productive application is limited in complex conditions that are demanded in real worlds of finance.

The absence of complete systems designed to cope with the above drawbacks exposes investors and planners to excessive financial risk of finance, instability of portfolios, and ineffective investment of resources. There is a need for a fully developed decision support system that employs quantitative analysis, expert opinion, and uncertainty modeling, and is capable of rapidly adapting itself to the fast-varying systems found in the world of finance. Such systems should provide a complete picture of investment potentialities available, facilitate the making of comparative rankings, and lay down explanatory statements which will assist investors in making intelligent, competent, and situation-conscious decisions regarding their portfolios. This thesis provides an intelligent and flexible decision-making framework in financial planning and investment optimization, intended to cope with the foregoing problems, which can be successfully applied to the complexities of developing investment programs. The system provides for objective evaluation, subjective expert opinion, and consideration of uncertainty factors, and improves upon conventional ranking procedures by the introduction of an improved series of distance measures and similarity measures. This combination makes portfolio decisions stronger, more open, and more situation-conscious, thus eliminating the deficiencies incurred in the application of conventional methods. The need for this type of decision support system will be correspondingly developed in an actual case study of financial planning and investment optimization to prove its practical efficiency. In the model, a number of differential investment alternatives will be evaluated employing an extensive series of financial, economic, and conservation types of criteria which include both a quantization and qualitative aspect of decision-making, as shown in Table 5. The criteria have been arranged through a method of selection practised by the several types of experts to obtain that which would best conform to the numerous desired ends of the investor, namely profit, marketability of shares, stability of portfolios, and survivability, as illustrated in Table 6.

Table 5.

Investment strategies and their responses to financial planning challenges.

Alternative	Investment strategy description	How it covers current challenges
: Cryptocurrency Diversification Fund	A balanced investment in top-performing cryptocurrencies combined with stablecoins to reduce volatility.	Mitigates extreme volatility using a hybrid crypto-stablecoin balance, addressing the digital asset revolution while managing risk exposure.
: Hybrid Pension Optimization Plan	Integrates traditional retirement funds with high-yield short-term digital investments.	Balances long-term stability and short-term liquidity, addressing inflation and pension fund underperformance.
: Real Estate Tokenization Model	Uses blockchain-based tokenization to fractionalize real estate assets, increasing accessibility for small investors.	Overcomes entry barriers in real estate investment, enhances liquidity, and supports transparent asset tracking.
: Impact Investment Fund	Targets projects that generate measurable social and environmental impact alongside financial returns.	Aligns with global SDGs, combats inequality, and provides stable long-term returns under ESG regulatory frameworks.
: Quantitative Hedge Strategy	Applies machine learning-based predictive analytics to hedge against macroeconomic and interest rate uncertainties.	Provides data-driven risk management against inflation, policy shifts, and global market disruptions.
: AI-Driven Robo-Advisory System	An automated AI platform that provides personalized investment advice using real-time market data and behavioral analytics.	Handles information overload, reduces human bias, and enables adaptive portfolio management through continuous learning from market fluctuations.
: Sustainable Infrastructure Bonds	Focuses on investing in green and smart infrastructure bonds backed by government or multilateral organizations.	Provides low-risk stable returns while addressing the infrastructure financing gap and sustainability transition needs.
: ESG-Focused Portfolio	A portfolio that integrates Environmental, Social, and Governance (ESG) metrics into investment decisions.	Addresses sustainability demands, mitigates regulatory risks, and aligns with global trends toward green investments, helping investors achieve both financial and ethical goals.

Table 6.

Evaluation criteria for the proposed smart decision framework in financial planning and investment optimization.

Criterion	Description	Impact on decision	Nature
: Expected return (ER)	Represents the anticipated annual return of an investment portfolio, estimated using historical trends, economic indicators, and forecasting models. It reflects the potential growth of invested capital under prevailing market conditions.	Higher expected returns suggest stronger growth prospects and support the selection of portfolios aimed at long-term wealth accumulation.	Benefit
: Portfolio risk (PR)	Describes the variability of portfolio returns caused by market volatility, asset interdependence, and systemic uncertainty. It indicates the level of exposure to unfavorable market movements.	An increase in portfolio risk implies greater uncertainty and a higher probability of losses, which may reduce portfolio stability and investor confidence.	Cost
: Value-at-risk (VaR)	Estimates the maximum potential loss of a portfolio over a specified time horizon at a given confidence level. It is commonly derived using simulation and stress-testing techniques.	Lower VaR values enhance downside protection and improve the portfolio’s ability to withstand adverse market scenarios.	Cost
: Liquidity ratio (LR)	Indicates the ease with which portfolio assets can be converted into cash without substantial price concessions. It reflects the portfolio’s responsiveness to changing market conditions.	Higher liquidity enables timely rebalancing and reduces exposure to forced losses during periods of market stress.	Benefit
: Inflation resilience (IR)	Measures the capacity of a portfolio to preserve real value during inflationary periods by accounting for inflation-sensitive assets and instruments.	Greater inflation resilience helps protect purchasing power and ensures more stable real returns over time.	Benefit
: ESG score (ESG)	Evaluates the degree to which an investment adheres to environmental, social, and governance principles, including sustainability practices, social responsibility, and corporate governance quality.	A higher ESG score supports sustainable investment objectives, regulatory compliance, and alignment with socially responsible investment strategies.	Benefit
: Diversification index (DI)	Reflects the extent of asset diversification across sectors, asset classes, and geographic regions based on correlation and allocation structure.	Improved diversification reduces concentration risk and contributes to more consistent portfolio performance across market cycles.	Benefit
: Sharpe ratio (SR)	Assesses risk-adjusted performance by comparing excess portfolio returns to overall volatility, providing a measure of return efficiency.	A higher Sharpe ratio indicates more effective risk utilization and supports performance comparison among competing investment strategies.	Benefit
: Transaction cost (TC)	Represents the total costs associated with trading activities, management fees, and operational expenses incurred during portfolio maintenance.	Higher transaction costs diminish net returns and may negatively affect portfolio efficiency, particularly in active trading environments.	Cost
: AI confidence index (AIC)	Captures the reliability of AI-driven financial predictions based on validation performance, robustness, and uncertainty estimation.	Higher confidence levels enhance trust in automated recommendations and support more dependable investment decisions.	Benefit

Results and discussion

Step 1: Table 3 shows the linguistic terms used to evaluate the criteria and alternatives. Table 7 shows the assessments made by different decision makers using IFZNs. This study utilized MATLAB to produce expert-elicited matrices via a regulated and data-driven randomization methodology. Instead of making random synthetic values, the evaluations that are made are based on real-world conditions and expert judgments that are usually seen in the financial market. Historical factors like changes in risk perception, levels of volatility, and sectoral preference distributions help set these limits.

Table 7.

Expert evaluation matrices using linguistic values.

Decision Maker 1
	MH	H	M	H	VH	M	H	VH	H	E
	H	VH	MH	H	VH	MH	E	H	VH	H
	M	MH	M	MH	H	M	MH	H	H	MH
	VH	E	H	VH	E	H	E	VH	VH	E
	L	M	MH	M	MH	M	H	MH	M	H
	MH	H	VH	MH	E	H	VH	E	VH	E
	M	MH	H	MH	H	MH	VH	H	MH	VH
	VH	E	VH	E	E	VH	E	E	VH	E

Decision Maker 2
	H	VH	MH	VH	E	MH	VH	E	VH	H
	MH	H	M	H	VH	MH	E	H	VH	VH
	M	MH	M	MH	H	MH	MH	H	MH	H
	VH	E	VH	E	E	VH	E	VH	E	E
	L	M	MH	M	H	MH	H	MH	M	H
	MH	VH	E	VH	E	VH	E	E	VH	E
	M	MH	H	MH	H	MH	VH	H	MH	VH
	VH	E	VH	E	E	VH	E	E	VH	E

Decision Maker 3
	MH	H	M	MH	VH	MH	H	VH	H	H
	H	VH	MH	H	VH	H	E	VH	VH	E
	M	MH	M	MH	H	M	MH	H	H	MH
	VH	E	VH	E	E	VH	E	VH	VH	E
	L	M	MH	M	H	M	H	MH	M	H
	MH	VH	E	VH	E	VH	E	E	VH	E
	M	MH	H	MH	H	MH	VH	H	MH	VH
	VH	E	VH	E	E	VH	E	E	VH	E

Decision Maker 4
	MH	H	M	H	VH	MH	H	VH	H	H
	H	VH	MH	H	VH	MH	E	VH	VH	E
	M	MH	M	MH	H	M	MH	H	H	MH
	VH	E	VH	E	E	VH	E	VH	VH	E
	L	M	MH	M	H	M	H	MH	M	H
	MH	VH	E	VH	E	VH	E	E	VH	E
	M	MH	H	MH	H	MH	VH	H	MH	VH
	VH	E	VH	E	E	VH	E	E	VH	E

Step 2: Equation (1) is employed to calculate the score values and matrices for each matrix that decision makers provide.

Step 3: Using Equation (2), the softmax function for every matrix supplied by DMs is calculated. Equation (3) is used to normalise the results, taking into account .

Step 4: The expert-derived combined IFZNs Table is developed by combining the data from each table into a single group. This integration is performed using the IFZNDWA operator and equation (4). Table 8 displays the findings.

Table 8.

Aggregated decision matrix.

Step 5: The aggregated score matrix is calculated using Equation (5).

Step 6: With the use of Equation 6, the normalized matrix is provided below.

Step 7: Equation 6 was used to derive the normalized matrix from the score matrix (Table 9).

Table 9.

Standard deviation.

SD
	0.3801	0.3365	0.3534	0.3260	0.3279	0.3464	0.3435	0.3574	0.4049	0.4075
	89.5404	64.6250	92.4394	94.4871	90.8658	66.0543	78.6323	72.2404	75.4447	68.9636

Step 8: Table 10 displays the objective weights of the criteria, which are calculated using Equation 9.

Table 10.

Objective weights of criteria.

Weights
	0.1129	0.0815	0.1165	0.1191	0.1145	0.0833	0.0991	0.0911	0.0951	0.0869

Step 9: Pairwise comparisons of the criteria locations are used to create the ranking comparison matrix, as illustrated below.

31

Step 10: Equation (11) provides the formula used to calculate the summand criteria weights ().

32

Step 11: Equation (12) is used to calculate the subjective weight of the criteria.

33

Step 12: Equation (13) is used to calculate the combined weights from the objective and subjective approaches, with set to 0.5.

34

Step 13: The differences between alternatives are evaluated using a variety of techniques. Equations (18)–(21) describe the use of similarity-based measures, whereas Equations (22)–(25) compute distance-based measures. When taken as a whole, these metrics provide a comprehensive assessment of how close or far apart options are. the results of pairwise hamming distances and preference values from to are shown in Table 11 and the pairwise Hamming distances and preference Values from to are shown in Table 12.

Table 11.

Pairwise hamming distances and preference values from to .

Comparison											Preference
vs	0.900	0.92409	0.94842	0.93475	0.86495	1.00000	0.99223	0.95475	0.93000	0.95475	0.39938
vs	0.77252	0.82334	0.84843	0.87251	0.92387	0.96669	0.93000	0.88475	0.83000	0.90318	0.28741
vs	0.76000	0.68502	0.76093	0.84864	0.90000	0.96487	0.90612	0.82727	0.74250	0.82727	0.36404
vs	0.76560	0.59752	0.67343	0.74540	0.86495	0.87943	0.87644	0.90318	0.74250	0.81475	0.25812
vs	0.90000	0.87252	0.98347	0.93475	0.78954	0.93831	1.00000	0.95475	0.93000	0.96486	0.30795
vs	0.90000	0.94252	0.94842	0.87251	0.78520	0.84989	0.95796	0.95475	1.00000	0.95475	0.30357
vs	0.76512	0.87252	0.84843	0.98749	0.90000	0.98989	0.93000	0.81475	0.93000	0.81475	0.17880
vs	0.90000	0.92409	0.94842	0.93475	0.86495	1.00000	0.99223	0.95475	0.93000	0.95475	0.63555
vs	0.87252	0.89925	0.90000	0.93777	0.94108	0.96669	0.93777	0.93000	0.90000	0.94842	0.37704
vs	0.86000	0.76092	0.81250	0.91389	0.96495	0.96487	0.91389	0.87252	0.81250	0.87252	0.45560
vs	0.86560	0.67342	0.72500	0.81065	1.00000	0.87943	0.88421	0.94842	0.81250	0.86000	0.35254
vs	1.00000	0.94842	0.96496	1.00000	0.92459	0.93831	0.99223	1.00000	1.00000	0.98989	0.49661
vs	1.00000	0.98158	1.00000	0.93777	0.92025	0.84989	0.95020	1.00000	0.93000	1.00000	0.71400
vs	0.86512	0.94842	0.90000	0.92223	0.96495	0.98989	0.93777	0.86000	1.00000	0.86000	0.36202
vs	0.77252	0.82334	0.84843	0.87251	0.92387	0.96669	0.93000	0.88475	0.83000	0.90318	0.58731
vs	0.87252	0.89925	0.90000	0.93777	0.94108	0.96669	0.93777	0.93000	0.90000	0.94842	0.54646
vs	0.98749	0.86168	0.91250	0.97612	0.97613	0.93155	0.97612	0.94251	0.91250	0.92409	0.39276
vs	0.99308	0.77418	0.82500	0.87288	0.94108	0.91274	0.94644	0.98158	0.91250	0.91158	0.39106
vs	0.87252	0.95082	0.86496	0.93777	0.86566	0.97163	0.93000	0.93000	0.90000	0.93832	0.45889
vs	0.87252	0.88083	0.90000	1.00000	0.86132	0.88321	0.88796	0.93000	0.83000	0.94842	0.56040
vs	0.99261	0.95082	1.00000	0.86000	0.97613	0.97679	1.00000	0.93000	0.90000	0.91158	0.59103
vs	0.76000	0.68502	0.76093	0.84864	0.90000	0.96487	0.90612	0.82727	0.74250	0.82727	0.45421
vs	0.86000	0.76092	0.81250	0.91389	0.96495	0.96487	0.91389	0.87252	0.81250	0.87252	0.41529
vs	0.98749	0.86168	0.91250	0.97612	0.97613	0.93155	0.97612	0.94251	0.91250	0.92409	0.54455
vs	0.99441	0.91250	0.91250	0.89676	0.96495	0.84430	0.97032	0.92409	1.00000	0.98749	0.58079
vs	0.86000	0.81250	0.77746	0.91389	0.88954	0.90318	0.90612	0.87252	0.81250	0.86241	0.41790
vs	0.86000	0.74250	0.81250	0.97612	0.88520	0.81476	0.86409	0.87252	0.74250	0.87252	0.40616
vs	0.99488	0.81250	0.91250	0.83612	1.00000	0.95476	0.97612	0.98749	0.81250	0.98749	0.54359

Table 12.

Pairwise Hamming distances and preference values from to .

Comparison											Preference
vs	0.7656	0.5975	0.6734	0.7454	0.8650	0.8794	0.8764	0.9032	0.7425	0.8148	0.5224
vs	0.8656	0.6734	0.7250	0.8107	1.0000	0.8794	0.8842	0.9484	0.8125	0.8600	0.5676
vs	0.9931	0.7742	0.8250	0.8729	0.9411	0.9127	0.9464	0.9816	0.9125	0.9116	0.5109
vs	0.9944	0.9125	0.9125	0.8968	0.9650	0.8443	0.9703	0.9241	1.0000	0.9875	0.4574
vs	0.8656	0.7250	0.6900	0.8107	0.9246	0.9411	0.8764	0.9484	0.8125	0.8499	0.4947
vs	0.8656	0.6550	0.7250	0.8729	0.9203	0.9705	0.8344	0.9484	0.7425	0.8600	0.3952
vs	0.9995	0.7250	0.8250	0.7329	0.9650	0.8895	0.9464	0.9116	0.8125	1.0000	0.5226
vs	0.9000	0.8725	0.9835	0.9348	0.7895	0.9383	1.0000	0.9548	0.9300	0.9649	0.7163
vs	1.0000	0.9484	0.9650	1.0000	0.9246	0.9383	0.9922	1.0000	1.0000	0.9899	0.8789
vs	0.8725	0.9508	0.8650	0.9378	0.8657	0.9716	0.9300	0.9300	0.9000	0.9383	0.4597
vs	0.8600	0.8125	0.7775	0.9139	0.8895	0.9032	0.9061	0.8725	0.8125	0.8624	0.4419
vs	0.8656	0.7250	0.6900	0.8107	0.9246	0.9411	0.8764	0.9484	0.8125	0.8499	0.3446
vs	1.0000	0.9300	0.9650	0.9378	0.9957	0.9116	0.9580	1.0000	0.9300	0.9899	0.6056
vs	0.8651	1.0000	0.8650	0.9222	0.8895	0.9484	0.9300	0.8600	1.0000	0.8499	0.5618
vs	0.9000	0.9425	0.9484	0.8725	0.7852	0.8499	0.9580	0.9548	1.0000	0.9548	0.7155
vs	1.0000	0.9816	1.0000	0.9378	0.9203	0.8499	0.9502	1.0000	0.9300	1.0000	0.6432
vs	0.8725	0.8808	0.9000	1.0000	0.8613	0.8832	0.8880	0.9300	0.8300	0.9484	0.4553
vs	0.8600	0.7425	0.8125	0.9761	0.8852	0.8148	0.8641	0.8725	0.7425	0.8725	0.4374
vs	0.8656	0.6550	0.7250	0.8729	0.9203	0.9705	0.8344	0.9484	0.7425	0.8600	0.4389
vs	1.0000	0.9300	0.9650	0.9378	0.9957	0.9116	0.9580	1.0000	0.9300	0.9899	0.5415
vs	0.8651	0.9300	0.9000	0.8600	0.8852	0.8600	0.8880	0.8600	0.9300	0.8600	0.3320

Step 14: Both distance and similarity metrics are used to evaluate the preference index . While distance measures are obtained from Eq. 27, which aggregates results across all criteria with their associated weights, similarity measurements are computed using Eq. 26. Reduced distance values indicate closer and more desirable options, whilst higher similarity values indicate a stronger preference. The generated values make it easier to rank the options, with the option with the lowest distance score or the highest similarity being the most desired.

Step 15: The PROMETHEE-II method employs a matrix of preferences to evaluate the performances of the alternatives considered in this project. The flow of leaving is given by equation (28), which measures the extent to which an alternative prevails over the others available. The flow of entering is explained in equation (29), which shows the extent to which an alternative is less favored than the other alternatives. From the sum of the flows of leaving and entering is obtained the flow given by equation (30). The total flow is a measure of the power of an alternative. The flows , and for the alternatives considered are given in Table 13. The values determine the ranking; the greater the , the more favourable the alternative. These results naturally give the alternatives a rank with respect to the problem of decisions.

Table 13.

The final ranking of alternatives.

Alternative				Rank
	0.2799	0.6460	− 0.3662	8
	0.4686	0.6047	− 0.1360	7
	0.4833	0.4272	0.0561	5
	0.5089	0.4400	0.0689	4
	0.5445	0.4670	0.0775	3
	0.5825	0.4619	0.1206	2
	0.5213	0.4864	0.0349	6
	0.5253	0.3810	0.1443	1

Sensitivity analysis

also presents an examination of the variations in the input parameters and as the scaling factor is manipulated in different variations. The results indicate that the same ordering of preferences is retained in one form or another under every set of combinations, with always being the most preferred one for all of the variations in the values of the parameters. This is an excellent indication of the stability and efficiency of the proposals, since the orderings of preferences remain the same under many combinations of the values for the parameter . The efficiency and stability of the proposed model are also evident since the systems of preference do not vary very widely for many values of the parameter . It is to be noted that such differences in values do not upset the relative order of preferences. The examination shows that the relationships are preserved if either the parameter or is kept constant, averaging in all possible combinations from the required information. It is thus shown that the order of preference remains the same for some combinations, with always being the most preferred alternative. This is a direct indication of the strength and stability of the proposed model, since the order of preferences is retained for a variation of structural values of the and effectively. These results indicate the advantages of and also the fact that alterations in the values of moderate or large extent in an adjustment or confidence parameter do not alter the ultimate decision. The outcome of the sensitivity examination indicates that the stability of the proposed scheme is one of its recommendations, since the order is not dependent upon parameter characteristics, and enables the working out of values for all of the alternative methods of determining these preferences, which give results sometimes the same, and at all times, a reproducible order of preference under variations of the parameters and These results are all in harmony with the stable behaviour of the decision support model in actual decision-making situations, in which some data will have to be made use of or which the model is prepared to interpret, with the reproducible decision result. The resilience of the model and the stability of the ranking results are highlighted by Fig.  2, which shows how changes in the parameter and the selected distance measures have a substantial impact on performance scores. The combined impact of and on the evaluated options is also examined in Fig.  3, which demonstrates the consistency of the final rankings and validates the decision-making framework’s resilience to parameter changes (Tables 14 and 15).

Figure 2.

Influence of the parameter and distance metrics on the performance scores of the evaluated alternatives.

Figure 3.

Effect of varying and parameters on the overall performance of alternatives.

Table 14.

The impact of the parameterand distances on the decision result.

										Ranking
		− 0.3192	− 0.1021	0.0492	0.0510	0.0923	0.1032	0.0042	0.1252
		− 0.3203	− 0.1030	0.0482	0.0500	0.0915	0.1024	0.0034	0.1244
Euclidean distance based PROMETHEE-II		− 0.3198	− 0.1025	0.0486	0.0504	0.0918	0.1027	0.0037	0.1247
		− 0.3191	− 0.1020	0.0493	0.0511	0.0924	0.1033	0.0043	0.1253
		− 0.3190	− 0.1019	0.0494	0.0512	0.0925	0.1034	0.0044	0.1254
		− 0.1310	− 0.0623	0.0006	0.0144	0.0264	0.0361	0.0330	0.0859
		− 0.1313	− 0.0626	0.0003	0.0141	0.0261	0.0358	0.0327	0.0856
Hamming distance based PROMETHEE-II		− 0.1319	− 0.0632	-0.0003	0.0135	0.0255	0.0352	0.0321	0.0850
		− 0.1316	− 0.0629	0.0000	0.0138	0.0258	0.0355	0.0324	0.0853
		− 0.1308	− 0.0621	0.0008	0.0146	0.0266	0.0363	0.0332	0.0861
		0.0218	0.0131	0.0496	0.0864	0.0964	0.1161	0.0373	0.1359
		0.0210	0.0123	0.0488	0.0856	0.0956	0.1151	0.0332	0.1351
Hausdorff distance based PROMETHEE-II		0.0213	0.0126	0.0491	0.0859	0.0959	0.1156	0.0725	0.1354
		0.0216	0.0129	0.0494	0.0862	0.0962	0.1159	0.0328	0.1357
		0.0220	0.0133	0.0498	0.0866	0.0966	0.1163	0.0532	0.1361
		0.0193	0.0064	0.0720	0.1202	0.1233	0.1495	0.0346	0.1967
		0.0181	0.0152	0.1006	0.1192	0.1223	0.1485	0.0316	0.1957
Similarity measure based PROMETHEE-II		0.0186	0.0057	0.0912	0.1196	0.1227	0.1489	0.0820	0.1961
		0.0494	0.0365	0.0921	0.1203	0.1234	0.1496	0.0527	0.1968
		0.0895	0.0366	0.1022	0.1204	0.1235	0.1497	0.0928	0.1969

Table 15.

The impact of the parameterandon the decision result.

										Ranking
		− 0.0820	− 0.0308	0.1139	0.1254	0.1385	0.1485	− 0.0224	0.1908
		− 0.0816	− 0.0304	0.1143	0.1258	0.1389	0.1489	− 0.0220	0.1912
		− 0.0824	− 0.0312	0.1135	0.1250	0.1381	0.1481	− 0.0228	0.1904
		− 0.0821	− 0.0309	0.1140	0.1255	0.1386	0.1486	− 0.0225	0.1909
		− 0.0819	− 0.0307	0.1138	0.1253	0.1384	0.1484	− 0.0223	0.1907
		− 0.0815	− 0.0303	0.1144	0.1259	0.1390	0.1490	− 0.0219	0.1913
		− 0.0825	− 0.0313	0.1134	0.1249	0.1380	0.1480	− 0.0229	0.1903
		− 0.0822	− 0.0310	0.1141	0.1256	0.1387	0.1487	− 0.0226	0.1910
		− 0.0818	− 0.0306	0.1137	0.1252	0.1383	0.1483	− 0.0222	0.1906
		− 0.0814	− 0.0302	0.1145	0.1260	0.1391	0.1491	− 0.0218	0.1914
		− 0.0820	− 0.0308	0.1139	0.1254	0.1385	0.1485	− 0.0224	0.1908
		− 0.0817	− 0.0305	0.1142	0.1257	0.1388	0.1488	− 0.0221	0.1911
		− 0.0823	− 0.0310	0.1136	0.1251	0.1382	0.1482	− 0.0227	0.1905
		− 0.0819	− 0.0307	0.1140	0.1255	0.1386	0.1486	− 0.0223	0.1909
		− 0.0821	− 0.0309	0.1138	0.1253	0.1384	0.1484	− 0.0225	0.1907
		− 0.0816	− 0.0304	0.1143	0.1258	0.1389	0.1489	− 0.0220	0.1912
		− 0.0824	− 0.0311	0.1135	0.1250	0.1381	0.1481	− 0.0228	0.1904
		− 0.0818	− 0.0306	0.1141	0.1256	0.1387	0.1487	− 0.0222	0.1910
		− 0.0822	− 0.0310	0.1137	0.1252	0.1383	0.1483	− 0.0226	0.1906
		− 0.0815	− 0.0303	0.1144	0.1259	0.1390	0.1490	− 0.0219	0.1913

Comparative analysis

To evaluate the robustness and consistency of the proposed framework, its ranking results were compared with various established MCDM methodologies referenced in the literature (Table 16). The comparative analysis employed methodologies such as DEMATEL-based evaluation⁴⁹, TOPSIS⁵⁰, WASPAS⁵¹, COPRA⁵², ELECTRE⁵³, and the Delphi method⁵⁴. The ESG-focused portfolio () has always been the best choice for all benchmarking methods, showing that ranking results are reliable even when the methods are different. This consistency shows how well the alternative does in terms of sustainability alignment, following the rules, and being financially stable in the long term. The minor differences in the intermediate ranking positions, especially between , , and , are due to changes in the models’ normalisation methods, aggregation operators, and weighting determinations. The DEMATEL-based model⁴⁹ emphasised causal relationships among criteria, while TOPSIS⁵⁰ and WASPAS ⁵¹ focused on distance-based and multiplicative aggregation techniques, respectively. Similarly, COPRAS⁵² evaluated proportional significance, while ELECTRE⁵³ employed concordance and discordance principles to ascertain outranking flows. Despite these methodological inconsistencies, the proposed hybrid EPROMETHEE-II strategy demonstrated enhanced discriminative capability, effectively integrating objective weight assessment (via LOPCOW) and relative preference articulation (through RANCOM). So, it gives a more advanced and adaptable way to evaluate that fits with complex investment environments that are full of uncertainty and inter-criterion interdependence. As a result, the new model is more stable, easier to understand, and more transparent in its decision-making than the benchmark MCDM techniques. This shows that it can be trusted to evaluate sustainable financial investments.

Table 16.

Comparison of the proposed approach with existing methods when .

Authors	Methodology	Ranking of alternatives	Optimal alternative
Govindan et al.49	DEMATEL-based evaluation
Yazdi50	TOPSIS
Stanujkić and Karabašević51	WASPAS
Kumari and Mishra52	COPRAS technique
Rouyendegh53	ELECTRE-based MCDM method
Roy and Garai54	Delphi approach
Proposed	LOPCOW and RANCOM based EPROMETHEE-II

Results and discussion

The proposed structure for making decisions uses the LOPCOW method to find objective weights based on how data changes and how much information it contains, as well as the RANCOM method to combine subjective expert opinions that include strategic insights, behavioural tendencies, and market experience. To make a fair evaluation system, different ways of weighting are combined. The new version of the EPROMETHEE-II algorithm uses nonlinear distance and similarity measures to make better predictions about preferences and rankings. We looked at eight different investment opportunities using ten different criteria, including money, behaviour, and the environment. The results show that the options are very different from each other, which highlights the trade-offs between growth, stability, and ESG compliance. The ESG-focused Portfolio superior ranking is financially validated by its comprehensive ESG integration, which improves stability under high market volatility, reduces regulatory and downside risks, and enhances governance quality, thereby resulting in stronger risk-adjusted performance. In contrast, the AI-driven Portfolio is more sensitive to market fluctuations as a result of its concentrated growth exposure, which results in increased volatility and less consistent returns. Table 17 gives a summary of each alternative’s strengths, minor limitations, and major criteria impacts. This gives investors and portfolio managers clear advice and information.

Table 17.

Ranking of investment alternatives with key observations and ranking justification.

Rank	Alternative	Score	Discussion points	Strengths/Key advantages	Marginal limitations	Notable criteria impact	Recommended investor type	Ranking observation
1	ESG-focused portfolio ()	0.1443	Highest-ranking due to alignment with sustainability trends and regulatory compliance.	Strong ESG integration, risk mitigation, ethical investing	Moderate short-term returns compared to tech-focused options	ESG Score, Expected Return, Regulatory Alignment	Growth-oriented, sustainability-focused investors	Ranks top due to strong sustainability alignment, stable long-term returns, and regulatory advantages.
2	AI-Driven robo-advisory system ()	0.1206	Adaptive AI enables personalized and data-driven investment decisions.	Reduces human bias, dynamic portfolio adjustment, real-time analytics	Complexity of AI algorithms, dependence on data quality	AI Confidence Index, Expected Return, Risk-Adjusted Performance	Tech-savvy, growth-oriented investors	High rank due to innovative AI-driven adaptability and dynamic portfolio management, despite moderate complexity.
3	Quantitative hedge strategy ()	0.0775	Effective risk management against macroeconomic uncertainty and market volatility.	Data-driven predictive analytics, hedging against inflation	Requires advanced knowledge, may incur higher management costs	Expected Return, Portfolio Risk, Inflation Hedge	Risk-conscious, professional investors	Ranks third because of strong macroeconomic risk mitigation, though less innovative than AI-driven strategies.
4	Impact investment fund ()	0.0689	Combines financial returns with measurable social/environmental impact.	Aligns with SDGs, ethical and sustainable returns	Lower liquidity, slower capital growth	ESG Score, Social Impact Metrics, Expected Return	Socially conscious, long-term investors	Moderate rank due to combined social impact and financial return, but slower growth and lower liquidity.
5	Real estate tokenization model ()	0.0561	Democratizes real estate investment and enhances liquidity via blockchain.	Fractional ownership, transparent transactions, accessibility	Regulatory uncertainty, platform dependency	Liquidity Ratio, Asset Diversification, Blockchain Transparency	Moderate-risk, tech-savvy investors	Middle rank due to blockchain-enabled accessibility and liquidity, but constrained by regulatory and platform risks.
6	Sustainable infrastructure bonds ()	0.0349	Low-risk stable returns; supports sustainable infrastructure projects.	Predictable returns, ESG alignment, low volatility	Lower growth potential, interest rate sensitivity	Expected Return, ESG Score, Portfolio Risk	Conservative, sustainability-focused investors	Lower rank due to limited growth potential, despite low risk and ESG benefits.
7	Hybrid pension optimization plan ()	-0.1360	Balances long-term retirement stability with short-term digital investments.	Stability with moderate liquidity, inflation hedge	Moderate returns, limited growth potential	Expected Return, Liquidity Ratio, Inflation Resilience	Conservative to moderate investors	Ranks low because of moderate returns and limited growth potential relative to other alternatives.
8	Cryptocurrency diversification fund ()	-0.3662	High potential returns but significant volatility; suited for risk-tolerant investors.	Exposure to emerging digital assets, diversification across cryptos	High volatility, regulatory and security risks	Expected Return, Portfolio Risk, Transaction Cost	High-risk tolerant, speculative investors	Lowest rank due to extreme volatility and speculative nature, despite high upside potential.

Limitations

Even though the proposed framework for making decisions is strong and well-thought-out, it does have some flaws that should be kept in mind when looking at the results. Even though it can successfully combine objective and subjective points of view, it still has some features that are unique to it.

• The RANCOM technique, even though it includes subjective expert opinions, could still be influenced by the biases, inconsistencies, or lack of knowledge of the people making the decisions.

• The accuracy of objective weights obtained through LOPCOW depends on high-quality, up-to-date financial data. The ranking results may be less reliable if data is missing, incomplete, or distorted.

• The combination of LOPCOW, RANCOM, and an improved version of EPROMETHEE-II that uses nonlinear distance and similarity metrics increases the amount of computing power needed, especially when there are a lot of options or criteria.

• The model works well in normal market conditions, but it may not work well when there are rare events, black swan events, or sudden geopolitical changes that are very different from what has happened in the past.

• The framework includes both objective and subjective inputs. However, changes in what investors want mean that it needs to be re-evaluated, which can be time-consuming and costly.

• Linguistic evaluations show how much risk people are willing to take and what strategies they prefer, but they may not fully capture the complex ways people make decisions. The model can handle uncertainty, but dealing with a lot of uncertain inputs and many decision-makers is hard and may require advanced computational optimisation.

Practical Implications

The proposed way of making decisions can provide large-scale practical advantages to bank employees, investment managers and analysts. As these employees are looking for ways to improve their portfolios performance as they deal with an ever-changing stock market. This method is able to combine two methods of evaluation; objective and subjective (using both data analysis and expert intuition) to create a much larger system of investment strategies that will allow professionals to be responsive to new information in the markets and make decisions that are based on current market conditions and the lesvel of informational uncertainty. RANCOM, which focuses on the behavioral and strategic insights of the decision-maker, the review system should be more contextualized and balanced to allow for more accurate investment choices by portfolio managers. The enhanced EPROMETHEE-II model also includes non-linear distance and similarity metrics to enable reliable rankings in unstable and unpredictable environments. Therefore, there will be a higher degree of trust in the priorities of investments and the allocations of capital. Financial analysts using this approach will have access to adaptable dashboards and smart decision-making tools capable of adjusting the priority of their portfolios based upon new information received through the markets. In addition, the integration of advanced financial technologies (e.g., machine-learning driven forecasting systems; blockchain-based transparency platforms; etc.) may provide for faster and more accurate decision-making. The use of these technologies allows for an automation process that provides decisions that are both understandable and supervised by humans. The proposed model provides a practical, intelligent and scalable means of improving the efficiency, consistency and long-range vision in the development of financial plans and the optimization of investment.

Conclusion

The study presented an extensive LOPCOW and RANCOM-EPROMETHEE-II methodology for evaluating and prioritizing sustainable investment alternatives amid uncertainty. The hybrid methodology skilfully combines data-driven objective weighing (LOPCOW), expert-oriented subjective refining (RANCOM), and a complete preference-based ranking system (EPROMETHEE-II). This methodological synthesis ensures analytical rigour and interpretive clarity, making it suitable for complex financial decision-making contexts that involve multiple quantitative and qualitative factors. The empirical study demonstrated that the ESG-focused portfolio attained the highest ranking due to its superior integration of sustainability, compliance with regulations, and stability of long-term returns. The AI-powered robo-advisory system and quantitative hedging method came next, showing how important smart, data-driven processes are becoming for getting adaptive portfolio performance. The bitcoin diversification fund came in last because it was very volatile and had a lot of exposure to regulatory uncertainty. This shows how risky it is to invest in digital assets. A comparative analysis employing benchmark MCDM techniques, such as DEMATEL, TOPSIS, WASPAS, COPRAS, and ELECTRE, confirmed the consistency and robustness of the proposed strategy, as all methods identified as the optimal alternative. This convergence validates the dependability and uniformity of the integrated framework across various methodological paradigms. The proposed model offers decision-makers, fund managers, and policymakers a comprehensive and comprehensible tool for enhancing sustainable investment portfolios. The framework includes both the ethical and economic sides of investment alternatives, which fits with the global trend towards responsible and data-driven financial planning.

Future research in financial planning and investment optimization may take advantage of the rapid evolution of machine learning, deep learning, and artificial intelligence to further improve decision-making approaches. By integrating predictive analysis, reinforcement learning, and neural network-centered models, it is possible to optimize portfolios in real time, evaluate risk in a manner adaptable to changing market opportunities, and rebalance strategies as a function of market instability. Deep learning techniques, such as recurrent neural networks (RNN) and long short-term memory (LSTM) models, can be used to detect time dependencies in financial time series, increase the accuracy of forecasts, and discover non-linear correlations that are often overlooked by many conventional models. Furthermore, hybrid models that utilize fuzzy logic, Z-number-based uncertainty evaluation, and AI-driven optimization algorithms can better manage the subjective evaluations of expert evaluators, investor sentiments, and market uncertainties. Explainable AI (XAI) modeling techniques will serve to assist the automation of investment decisions, thus providing transparency, clarity, and trust in the automated results. Furthermore, modern ensemble learning and multi-agent reinforcement learning models will continue to improve the multi-criteria decision-making of complex and larger financial systems. Finally, future research may also be concerned with the use of alternative data systems, such as social media sentiments, data analytics, ESG metrics, and derivatives design, to develop greater adaptive, responsive, advanced financial support systems that are capable of effective application in uncertain or very dynamic market environments.

Author contributions

X.C. and A.S. wrote the main manuscript text, and A.S. prepared figures. All authors reviewed the manuscript.

Funding

No funding

Data Availability

The datasets generated and analyzed during the current study, including all MATLAB input matrices, and associated code, are deposited in the Zenodo repository and are publicly available at: https://doi.org/10.5281/zenodo.18815072

Declarations

Institutional Review

Not applicable.

Informed Consent

Not applicable.

Conflicts of Interest

The authors declare no conflicts of interest.

Competing interests

The authors declare no competing interests.