Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets

____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Neutrosophic Sets and Systems, Vol. 94, 2026 University of New Mexico Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Mohanned H. Alharbi Finance and Business Sector, Institute of Public Administration, P.O.Box 205, Riyadh 11141, Saudi Arabia Email: [email protected] Abstract This study proposes a new framework to predict corporate financial distress in emerging markets, with a focus on Saudi Arabia. Traditional models such as the Altman Z-score and Ohlson O-score assume that financial data are complete, precise, and reliable. However, in emerging markets, information is often missing, noisy, or conflicting, and many important factors are qualitative. To address this problem, the paper develops a Neutrosophic α-Discounted Cognitive Mapping (Nα-FDM) model that can represent both risk and uncertainty. Financial indicators are expressed as neutrosophic triples that measure the degrees of distress, indeterminacy, and financial health. Expert judgments about the relative importance of key criteria, such as liquidity, solvency, and profitability, are combined using the α-discounting method to obtain consistent weights even when initial preferences are inconsistent. These elements are integrated into a neutrosophic cognitive map, which produces a Neutrosophic Financial Distress Index (NFDI) and a scalar score for each firm. A case study on four Saudi non-financial firms, using the current ratio, debt-to-equity ratio, and return on assets, shows that the model can distinguish clearly distressed, clearly healthy, and grey-zone firms characterized by high uncertainty. The results suggest that the Nα-FDM framework is a useful decision-support tool for investors, managers, and regulators in uncertainty-rich environments. Keywords : Neutrosophic sets; α-discounting; cognitive maps; financial distress prediction; emerging markets; Saudi Arabia . 1. Introduction 1.1 Background and Motivation Predicting corporate financial distress is a critical aspect of risk management, investment decisions, and regulatory oversight, especially in volatile economic landscapes [1, 2]. Since the pioneering discriminant analysis for bankruptcy prediction [1], numerous models have been developed to forecast firm failure using financial ratios, market indicators, and macroeconomic factors [3–5]. These models help stakeholders identify early warning signals of insolvency, allowing for loss mitigation and efficient resource Neutrosophic Sets and Systems, Vol. 94, 2026 448 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets allocation [6]. However, in emerging markets characterized by data scarcity, regulatory inconsistencies, and external shocks, traditional models often underperform due to their dependence on precise and complete datasets [7–9]. Saudi Arabia exemplifies these challenges as a rapidly evolving emerging market. Driven by Vision 2030 reforms, diversification away from oil dependency, and growing foreign investment, the Saudi economy has seen increased corporate activity in non-financial sectors like petrochemicals, logistics, and utilities [10–12]. Yet, firms in these sectors face unique distress risks from global commodity price fluctuations, governance opacity, and sectoral concentration, particularly in regions such as the Eastern Province [13, 14]. Studies on Saudi firms indicate that conventional predictors like liquidity and leverage ratios are influenced by local elements, including Sharia-compliant financing and government subsidies, introducing ambiguities not adequately addressed by crisp statistical methods [15–17]. 1.2 Limitations of Classical and Advanced Distress Prediction Models Classical models, such as the Altman Z-score [1] and Ohlson O-score [2], rely on deterministic inputs and linear relationships, often leading to overconfident predictions in noisy data environments [18, 19]. Machine learning extensions, including support vector machines and neural networks, enhance accuracy but frequently lack interpretability and fail to model indeterminacy explicitly [3, 20–22]. In emerging markets, these shortcomings are exacerbated: panel data analyses reveal that missing values, reporting delays, and conflicting signals from qualitative factors (e.g., governance quality) result in high misclassification rates [23–25]. For example, research on Middle Eastern markets demonstrates that models calibrated on developed economies perform poorly in high macroeconomic volatility contexts [26, 27]. Moreover, interactions among distress drivers, such as feedback loops between profitability and solvency, are rarely modeled explicitly, treating indicators as independent [28, 29]. This limitation is particularly acute in Saudi emerging markets, where structural reforms and energy transitions create interdependent risks [30, 31]. Borderline cases, where firms show mixed signals, pose additional challenges, as classical scores do not distinguish between true ambiguity and simple intermediacy [32, 33]. 1.3 Neutrosophic Sets for Handling Indeterminacy To address these gaps, this study employs neutrosophic sets, which extend fuzzy and intuitionistic fuzzy sets by incorporating three independent components: truth (supporting evidence), indeterminacy (ambiguity or conflict), and falsity (contradicting evidence) [4, 34–36]. Neutrosophic theory allows memberships that do not sum to one, Neutrosophic Sets and Systems, Vol. 94, 2026 449 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets enabling representation of conflicting data without forced resolution [37, 38]. Applications in risk assessment and multi-criteria decision-making (MCDM) highlight its effectiveness in uncertain settings, such as supplier selection and project evaluation [39– 41]. In financial domains, neutrosophic approaches have been applied to credit scoring and investment analysis, quantifying vagueness in qualitative assessments [42, 43]. However, their integration into corporate distress prediction, particularly in emerging markets, is limited [44, 45]. This paper extends neutrosophic representations to encode financial indicators as triples, capturing not only distress signals but also inherent uncertainties common in Saudi data [46]. 1.4 Cognitive Maps and Causal Structures Financial distress arises from networked interactions among factors like liquidity, profitability, and governance [47]. Cognitive maps, especially fuzzy cognitive maps (FCMs), model these causal relationships as directed graphs with weighted edges [6]. Neutrosophic cognitive maps (NCMs) enhance FCMs by assigning neutrosophic triples to edges and nodes, accommodating uncertain causal influences. Previous applications of NCMs include healthcare and energy planning, but rarely financial distress [47]. In this framework, nodes represent distress drivers, and edges propagate neutrosophic states, enabling dynamic simulation of shock effects [47]. This is especially pertinent for Saudi firms, where macroeconomic stress (e.g., oil price volatility) interacts with firmlevel metrics. 1.5 α-Discounting in Multi-Criteria Decision Making Assigning weights to criteria often involves inconsistent expert judgments, particularly in diverse markets [7]. The α-discounting method resolves this by perturbing linear preference relations to achieve consistency while minimizing deviation from original inputs [7]. When integrated with neutrosophic MCDM, it yields robust weights and an inconsistency measure. Applications in engineering and environmental decisions underscore its utility [7], yet its pairing with NCMs for financial prediction is novel. 1.6 Research Gaps and Contributions Despite progress, gaps remain: (i) classical models overlook explicit indeterminacy in emerging markets ; (ii) few studies combine neutrosophic representations with cognitive maps for distress dynamics; (iii) α-discounting is underutilized in financial MCDM; and (iv) Saudi-specific applications are scarce, emphasizing aggregate rather than firm-level nuances. Neutrosophic Sets and Systems, Vol. 94, 2026 450 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets This paper bridges these gaps by proposing the Nα-FDM model, contributing: a neutrosophic financial distress index separating truth, indeterminacy, and falsity; an integrated approach merging neutrosophic encoding, α-discounted weights, and NCM propagation; and a Saudi case study on non-financial firms, differentiating clear and ambiguous cases. 2. Theoretical Background and Notation This section introduces the main mathematical tools used in the paper: neutrosophic sets and numbers, their use in representing financial distress information, cognitive maps (fuzzy and neutrosophic), and the α-discounting method for deriving criteria weights from expert preferences [3,7]. 2.1 Neutrosophic sets and single-valued neutrosophic numbers Let 𝑈be a non-empty universe of discourse. A neutrosophic set 𝐴in 𝑈is characterized by three functions, 𝑇𝐴,𝐼𝐴,𝐹𝐴:𝑈 →[0,1], where, for each element 𝑥 ∈𝑈, a. 𝑇𝐴(𝑥)is the degree of truth that “𝑥 belongs to 𝐴”, b. 𝐼𝐴(𝑥)is the degree of indeterminacy, c. 𝐹𝐴(𝑥)is the degree of falsity [34,35]. Classical and fuzzy sets are recovered as special cases when indeterminacy is forced to zero, and the sum 𝑇𝐴(𝑥)+𝐹𝐴(𝑥)is constrained. In the neutrosophic setting, the three components 𝑇𝐴(𝑥), 𝐼𝐴(𝑥), and 𝐹𝐴(𝑥)are allowed to vary independently in [0,1], and they are not required to sum to one. This additional flexibility makes it possible to express: 1. high truth and high falsity simultaneously (conflicting evidence), 2. high indeterminacy with low truth and low falsity (lack of information), 3. low indeterminacy with dominant truth or falsity (clear signal). In this paper, we work with single-valued neutrosophic numbers (SVNNs). A singlevalued neutrosophic number is a triple, 𝑥 =(𝑇𝑥,𝐼𝑥,𝐹𝑥) with 𝑇𝑥∈[0,1],𝐼𝑥∈[0,1],𝐹𝑥∈[0,1]. We denote the set of all SVNNs by 𝒩 ={(𝑇,𝐼,𝐹) ∣𝑇,𝐼,𝐹 ∈ [0,1]}. For later use, we define a few basic operations on SVNNs. Let 𝑎 =(𝑇𝑎,𝐼𝑎,𝐹𝑎)and 𝑏 = (𝑇𝑏,𝐼𝑏,𝐹𝑏)be elements of 𝒩, and let 𝜆 ≥0be a scalar [35]. 1. Scalar multiplication (bounded): 𝜆⊗𝑎:=(min {1,𝜆𝑇𝑎}, min {1,𝜆𝐼𝑎}, min {1,𝜆𝐹𝑎}). (1) 2. Weighted sum aggregation. Let 𝑎𝑘=(𝑇𝑘,𝐼𝑘,𝐹𝑘) ∈𝒩for 𝑘 =1,…,𝑚, and let 𝜔1,…,𝜔𝑚≥0be weights satisfying Neutrosophic Sets and Systems, Vol. 94, 2026 451 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets ∑ 𝑚 𝑘=1 𝜔𝑘=1. (2) The weighted sum (or convex aggregation) of the 𝑎𝑘is defined as ∑ 𝑚 𝑘=1 𝜔𝑘⊗𝑎𝑘:=(∑ 𝑚 𝑘=1 𝜔𝑘𝑇𝑘, ∑ 𝑚 𝑘=1 𝜔𝑘𝐼𝑘, ∑ 𝑚 𝑘=1 𝜔𝑘𝐹𝑘). (3) Because each 𝑇𝑘,𝐼𝑘,𝐹𝑘 lies in [0,1]and the weights sum to one, each component of (3) also lies in [0,1]. 3. Complement. The neutrosophic complement of 𝑎is defined by swapping truth and falsity while keeping indeterminacy: 𝑎𝑐≔(𝐹𝑎,𝐼𝑎,𝑇𝑎). (4) These operations are sufficient for the constructions used in this paper. More elaborate operations (such as neutrosophic union and intersection) are not required for the proposed model and are therefore omitted here. 2.2 Neutrosophic representation of financial distress information We now connect neutrosophic numbers to the financial distress context. Suppose there are 𝑁firms, indexed by 𝑗 =1,…,𝑁. For each firm 𝑗, consider the proposition: 𝑃𝑗: “Firm 𝑗 is financially distressed.” Our goal is to represent the available information regarding the truth of 𝑃𝑗 in neutrosophic form. Assume that we observe 𝑚indicators for each firm, such as financial ratios or qualitative scores. Let 𝑋𝑗𝑘 ∈ℝ,𝑗 =1,…,𝑁, 𝑘 =1,…,𝑚, Denote the value of the 𝑘-th indicator for firm 𝑗. Examples of indicators include: a. liquidity ratios (e.g., current ratio), b. solvency ratios (e.g., debt-to-equity), c. profitability ratios (e.g., return on assets), d. governance indices, e. market-based measures (e.g., volatility, credit spreads). For each indicator 𝑘, we define three membership functions: 𝜇𝑘 𝑇:ℝ→[0,1],𝜇𝑘 𝐼:ℝ→[0,1],𝜇𝑘 𝐹:ℝ→[0,1]. These functions are designed to encode, for each possible indicator value 𝑥, a. 𝜇𝑘 𝑇(𝑥): the degree to which 𝑥supports the truth of 𝑃𝑗(i.e., supports distress), b. 𝜇𝑘 𝐼(𝑥): the degree of indeterminacy associated with 𝑥, c. 𝜇𝑘 𝐹(𝑥): the degree to which 𝑥supports the falsity of 𝑃𝑗(i.e., supports financial health). Given an observed value 𝑋𝑗𝑘, we obtain the indicator-level neutrosophic number for firm 𝑗and indicator 𝑘: 𝑥𝑗𝑘 =(𝑇𝑗𝑘,𝐼𝑗𝑘,𝐹𝑗𝑘)=(𝜇𝑘 𝑇(𝑋𝑗𝑘), 𝜇𝑘 𝐼(𝑋𝑗𝑘), 𝜇𝑘 𝐹(𝑋𝑗𝑘))∈𝒩. (5) Neutrosophic Sets and Systems, Vol. 94, 2026 452 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets The choice of membership functions 𝜇𝑘 𝑇, 𝜇𝑘 𝐼, and 𝜇𝑘 𝐹 is part of the model design and depends on the economic interpretation of each indicator. For example, for a liquidity ratio where low values are associated with distress: a. Small values of the ratio should yield high 𝜇𝑘 𝑇and low 𝜇𝑘 𝐹, b. Large values should yield low 𝜇𝑘 𝑇and high 𝜇𝑘 𝐹, c. Intermediate values should be associated with a higher 𝜇𝑘 𝐼, reflecting ambiguity. The explicit shapes of these functions will be defined later in the methodology and case study sections. Ultimately, the neutrosophic financial distress index for firm 𝑗will be constructed by aggregating the indicator-level neutrosophic numbers {𝑥𝑗𝑘}𝑘=1 𝑚 across indicators and across criteria, using weights derived from expert preferences via the αdiscounting method. 2.3 Cognitive maps: fuzzy and neutrosophic The previous subsection focused on single indicators. Financial distress, however, is the outcome of a system of interacting factors. To represent this interaction structure, we use cognitive maps. 2.3.1 Fuzzy cognitive maps (FCMs) A cognitive map is a directed graph whose nodes represent concepts and whose edges represent causal influences between concepts. Let 𝒞 ={𝐶1,𝐶2,…,𝐶𝑛} Denote the set of concepts. In the context of financial distress, important concepts may include: 1. 𝐶1: short-term liquidity risk, 2. 𝐶2: long-term solvency pressure, 3. 𝐶3: profitability weakness, 4. 𝐶4: cash-flow volatility, 5. 𝐶5: governance quality, 6. 𝐶6: macroeconomic stress, 7. 𝐶7: market-based distress signals, 8. 𝐶8: overall financial distress status. In a fuzzy cognitive map (FCM), each directed edge from a concept 𝐶𝑖to concept 𝐶𝑗carries a real-valued weight 𝑤𝑖𝑗 ∈[−1,1], which represents the sign and strength of the causal influence: a. 𝑤𝑖𝑗 >0: an increase in 𝐶𝑖tends to increase 𝐶𝑗, b. 𝑤𝑖𝑗 <0: an increase in 𝐶𝑖tends to decrease 𝐶𝑗, c. 𝑤𝑖𝑗 = 0: no direct influence. The state of the system at discrete time 𝑡is represented by the vector 𝑆(𝑡) =(𝑠1 (𝑡),𝑠2 (𝑡),…,𝑠𝑛 (𝑡))⊤,(6) Neutrosophic Sets and Systems, Vol. 94, 2026 453 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets where 𝑠𝑖(𝑡) ∈[0,1]denotes the activation level of the concept 𝐶𝑖at time 𝑡. The typical FCM update rule is 𝑠𝑗(𝑡+1) =𝑓(∑ 𝑛 𝑖=1 𝑠𝑖(𝑡)𝑤𝑖𝑗), (7) Where 𝑓:ℝ→[0,1]is an activation function (for example, a sigmoid or a threshold function). Under suitable conditions, iterating (7) can lead to a fixed point or a limit cycle that represents the system’s long-run behaviour. 2.3.2 Neutrosophic cognitive maps (NCMs) In a neutrosophic cognitive map (NCM), uncertainty and indeterminacy in causal relationships are modelled explicitly. Instead of a single real weight 𝑤𝑖𝑗, each edge from 𝐶𝑖to 𝐶𝑗is assigned a neutrosophic weight 𝑤𝑖𝑗 =(𝑇𝑖𝑗,𝐼𝑖𝑗,𝐹𝑖𝑗)∈𝒩, (8) Where: a. 𝑇𝑖𝑗is the degree of truth of the statement “𝐶𝑖 has a positive causal influence on 𝐶𝑗”, b. 𝐼𝑖𝑗is the degree of indeterminacy about this influence, c. 𝐹𝑖𝑗is the degree of falsity (evidence against the influence). Similarly, the state of each node 𝐶𝑖 at time 𝑡is represented by a neutrosophic number 𝑥𝑖(𝑡) =(𝑇𝑖(𝑡),𝐼𝑖(𝑡),𝐹𝑖(𝑡))∈ 𝒩. (9) To update the system over time, we need a rule that aggregates incoming neutrosophic influences. There are many possible choices. In this paper, we adopt a linear aggregation followed by component-wise bounding, which is simple, transparent, and consistent with the operations in Section 2.1. For a given time 𝑡and node 𝐶𝑗, define the intermediate (unbounded) values 𝑇 𝑗(𝑡+1) =∑ 𝑛 𝑖=1 (𝑇𝑖(𝑡)𝑇𝑖𝑗 −𝐹𝑖(𝑡)𝐹𝑖𝑗), 𝐼󰆻𝑗(𝑡+1) =∑ 𝑛 𝑖=1 (𝐼𝑖(𝑡)𝑇𝑖𝑗 +𝑇𝑖(𝑡)𝐼𝑖𝑗), 𝐹 𝑗(𝑡+1) =∑ 𝑛 𝑖=1 (𝐹𝑖(𝑡)𝑇𝑖𝑗 −𝑇𝑖(𝑡)𝐹𝑖𝑗). (10) The intuition is as follows: 1. Truth at node 𝐶𝑗 increases when “true” antecedents transmit true positive influence (𝑇𝑖(𝑡)𝑇𝑖𝑗), and decreases when “false” antecedents support the nonoccurrence of the consequent (𝐹𝑖(𝑡)𝐹𝑖𝑗); 2. indeterminacy at 𝐶𝑗 accumulates when indeterminate or ambiguous nodes influence 𝐶𝑗(𝐼𝑖(𝑡)𝑇𝑖𝑗) or when true antecedents have uncertain influence (𝑇𝑖(𝑡)𝐼𝑖𝑗); Neutrosophic Sets and Systems, Vol. 94, 2026 454 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets 3. falsity at 𝐶𝑗 increases when false antecedents transmit influence and decreases when true antecedents support the absence of the effect. We then apply a clipped linear activation to ensure that the updated components remain in [0,1]: 𝑇𝑗(𝑡+1) =𝑓𝑇(𝑇 𝑗(𝑡+1)),𝐼𝑗(𝑡+1) =𝑓𝐼(𝐼󰆻𝑗(𝑡+1)),𝐹𝑗(𝑡+1) =𝑓𝐹(𝐹 𝑗(𝑡+1)), (11) where, for each component 𝑋 ∈{𝑇,𝐼,𝐹}, 𝑓𝑋(𝑧) =min {1, max {0, 𝑧}}. (12) Equations (10)–(12) define a discrete-time dynamical system on 𝒩𝑛. A state vector (𝑥1 (𝑡),…,𝑥𝑛 (𝑡))is said to reach a fixed point if there exists 𝑡0such that 𝑥𝑖(𝑡+1) =𝑥𝑖(𝑡)for all 𝑖 and for all 𝑡 ≥𝑡0. (13) In the context of financial distress, one of the nodes, say 𝐶8, will represent overall distress. The neutrosophic state of this node at a fixed point will be used as a dynamic version of the firmlevel distress index. 2.4 The α-discounting method for multi-criteria decision making The neutrosophic framework and cognitive maps describe how to represent and propagate information. We still need a principled way to assign weights to the high-level criteria (such as liquidity, solvency, profitability, and so on). Expert judgement is an important source of such information, but expert statements are often inconsistent. The α-discounting method for multi-criteria decision making is a procedure for deriving a consistent weight vector from a set of possibly inconsistent linear relations between weights. 2.4.1 General formulation Consider 𝑚criteria 𝐾1,𝐾2,…,𝐾𝑚, with unknown weights 𝑤 =(𝑤1,𝑤2,𝑤3,…,𝑤𝑚)⊤, (14) satisfying 𝑤𝑘≥ 0for all 𝑘,∑ 𝑚 𝑘=1 𝑤𝑘=1. (15) Experts express preferences in the form of linear relations, such as. “𝐾1 is twice as important as 𝐾2” or “𝐾2 and 𝐾3have the same importance”. Each such statement can be written as a linear equation in the weights. Suppose that, in total, we obtain 𝐿equations: ℓℓ(𝑤1,…,𝑤𝑚)=0,ℓ =1,…,𝐿. (16) These can be written compactly as 𝐴𝑤 =0, (17) Neutrosophic Sets and Systems, Vol. 94, 2026 455 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Where 𝐴is an 𝐿×𝑚matrix whose coefficients are determined by the expert statements. If the system (17) admits a non-trivial solution 𝑤 ≠0with non-negative components, we can normalise it to satisfy (15), and we are done. In practice, however, the system (17) may be inconsistent, so that the only solution is 𝑤 =0. The α-discounting method addresses this by introducing discounting parameters that adjust the equations just enough to regain consistency. 2.4.2 α-discounting and the fairness principle For each equation in (16), consider a representation of the form (left-hand side) =(right-hand side). We introduce a positive scalar 𝛼ℓ> 0and multiply one side (typically the right-hand side) by 𝛼ℓ. The α-discounted system then has the form 𝐴(𝛼)𝑤 =0, (18) Where 𝐴(𝛼)is an 𝐿×𝑚matrix whose entries depend linearly on the discounting parameters 𝛼 =(𝛼1,…,𝛼𝐿)⊤. A pair (𝛼⋆,𝑤⋆)is called a consistent α-discounted solution if: 1. 𝐴(𝛼⋆)𝑤⋆=0, 2. 𝑤𝑘 ⋆≥0for all 𝑘, 3. ∑𝑚 𝑘=1 𝑤𝑘 ⋆=1, 4. 𝛼ℓ ⋆>0for all ℓ. There may be many such pairs. To select a particular solution, a natural rule is the fairness principle, which sets all discounting parameters equal: 𝛼1=𝛼2=⋯=𝛼𝐿=𝛼. (19) Under this principle, the original system is adjusted uniformly, and the single scalar 𝛼measures the overall inconsistency of the expert preference system: the further 𝛼is from 1, the stronger the inconsistency that had to be corrected. 2.4.3 Worked example with three criteria We now illustrate the α-discounting method step by step in a simple example with three criteria: 𝐾1, 𝐾2, 𝐾3 and weights, 𝑤1, 𝑤2, 𝑤3. Suppose experts state: 1. “𝐾1 is twice as important as 𝐾2”: 𝑤1=2𝑤2.(20) 2. “𝐾2 is three times as important as 𝐾3”: 𝑤2=3𝑤3.(21) 3. “𝐾3 is as important as 𝐾1”: 𝑤3=𝑤1. (22) Equations (20)–(22) are inconsistent. To see this, substitute 𝑤3= 𝑤1from (22) into (21): Neutrosophic Sets and Systems, Vol. 94, 2026 462 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Because each 𝛾ℎ𝑘 ≥0and sums to 1 over ℐℎ, and each 𝑇𝑗𝑘,𝐼𝑗𝑘,𝐹𝑗𝑘 ∈[0,1], the components 𝑇𝑗ℎ (𝑐),𝐼𝑗ℎ (𝑐),𝐹𝑗ℎ (𝑐) also lie in [0,1]. 3.5 Criteria weights from α-discounting Section 2.4 introduced the α-discounting method and provided a worked example with three criteria. In the context of Nα-FDM, we apply that procedure to the criteria 𝐾1,…,𝐾𝐻. Let 𝑤 =(𝑤1,…,𝑤𝐻), With, 𝑤ℎ≥0for all ℎ,∑𝐻 ℎ=1 𝑤ℎ=1. Experts express qualitative preferences about the relative importance of the criteria. These preferences are encoded as linear equations in the weights. If the resulting system is inconsistent, α-discounting is used to obtain a consistent solution 𝑤∗and a scalar 𝛼∗That measures inconsistency. In the three-criterion example used later in this section (liquidity, solvency, profitability), we will work with the specific α-discounted weight vector: 𝑤1 ∗≈0.4067,𝑤2 ∗≈0.3695,𝑤3 ∗≈0.2238. The method for obtaining these numbers has already been detailed in Section 2.4; here we treat them as given and use them to build the financial distress index. 3.6 Firm-level Neutrosophic Financial Distress Index The Neutrosophic Financial Distress Index (NFDI) for firm 𝑗is defined by aggregating the criterion-level neutrosophic evaluations 𝑐𝑗ℎusing the α-discounted weights 𝑤ℎ ∗.For each firm 𝑗, define: 𝑇𝑗= ∑ 𝐻 ℎ=1 𝑤ℎ ∗ 𝑇𝑗ℎ (𝑐), 𝐼𝑗= ∑ 𝐻 ℎ=1 𝑤ℎ ∗ 𝐼𝑗ℎ (𝑐), 𝐹𝑗= ∑ 𝐻 ℎ=1 𝑤ℎ ∗ 𝐹𝑗ℎ (𝑐). We then set NFDI𝑗=(𝑇𝑗,𝐼𝑗,𝐹𝑗). Because each 𝑇𝑗ℎ (𝑐),𝐼𝑗ℎ (𝑐),𝐹𝑗ℎ (𝑐)lies in [0,1]and the 𝑤ℎ ∗sum to 1, all components 𝑇𝑗,𝐼𝑗,𝐹𝑗 lie in [0,1]. The sum 𝑇𝑗+𝐼𝑗+𝐹𝑗need not equal 1 and may be greater than 1, which is acceptable in neutrosophic theory. To facilitate classification, we often use a normalised version of the NFDI when the sum 𝑇𝑗+𝐼𝑗+𝐹𝑗is positive: Neutrosophic Sets and Systems, Vol. 94, 2026 463 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets 𝑇𝑗∗=𝑇𝑗 𝑇𝑗+𝐼𝑗+𝐹𝑗,𝐼𝑗∗=𝐼𝑗 𝑇𝑗+𝐼𝑗+𝐹𝑗,𝐹𝑗∗=𝐹𝑗 𝑇𝑗+𝐼𝑗+𝐹𝑗. These normalized components satisfy: 𝑇𝑗∗,𝐼𝑗∗,𝐹𝑗∗∈[0,1],𝑇𝑗∗+𝐼𝑗∗+𝐹𝑗∗=1. We then define a scalar distress score: 𝑆𝑗=𝑇𝑗∗−𝐹𝑗∗. The value of 𝑆𝑗 shows the overall direction of the evidence about a firm’s financial condition. When 𝑆𝑗 is large and positive; the evidence strongly points toward financial distress. When 𝑆𝑗 is large and negative, the evidence strongly supports that the firm is financially healthy. When 𝑆𝑗 is close to zero, the signals of distress and health are roughly balanced. In this case, the level of uncertainty can be examined through 𝐼𝑗∗. In the case study, specific thresholds on 𝑆𝑗 are used to classify firms as distressed, healthy, or belonging to a grey zone where the situation is ambiguous. 3.7 Integration into a neutrosophic cognitive map The NFDI defined above is static: it aggregates information observed at a point in time. Financial distress, however, is dynamic and depends on how shocks propagate among criteria such as liquidity, solvency, profitability, and governance. To capture this dimension, we embed the criteria into a neutrosophic cognitive map as follows: a. Each criterion 𝐾ℎis associated with a concept 𝐶ℎ. b. A final concept 𝐶𝐷represents overall financial distress. c. Directed edges between concepts are assigned neutrosophic weights 𝑤𝑖𝑗 = (𝑇𝑖𝑗,𝐼𝑖𝑗,𝐹𝑖𝑗)that encode the strength and uncertainty of causal links. For firm 𝑗, we initialise the concepts at time 𝑡 =0as: a. 𝑥ℎ (0)(𝑗)=𝑐𝑗ℎfor each criterion node 𝐶ℎ, b. 𝑥𝐷 (0)(𝑗)=(0,1,0)for the distress node, representing a neutral prior (fully indeterminate) before propagation. We then iterate the neutrosophic cognitive map update rule (as defined in Section 2.3) until the state of the distress node 𝐶𝐷converges or stabilises. Denote the limiting state for firm 𝑗by 𝑥𝐷 (∞)(𝑗)=(𝑇 𝑗,𝐼󰆻𝑗,𝐹 𝑗). We call NFDI𝑗 dyn =(𝑇 𝑗,𝐼󰆻𝑗,𝐹 𝑗) Neutrosophic Sets and Systems, Vol. 94, 2026 464 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets In this paper, the numerical examples will focus on the static NFDI to keep the arithmetic fully transparent. The dynamic NFDI is conceptually straightforward once the map and its weights are specified and can be used in further empirical work for stress testing. 3.8 NFDI for a single firm We now give a complete numerical example for a single hypothetical firm. This example illustrates all the components of the Nα-FDM in the simplest non-trivial setting. 3.8.1 Setup We consider: Three criteria: 1. 𝐾1: liquidity, 2. 𝐾2: solvency, 3. 𝐾3: profitability. One indicator per criterion: a. 𝑋𝑗1 =CR𝑗(current ratio), b. 𝑋𝑗2 =DE𝑗(debt-to-equity ratio), c. 𝑋𝑗3 =ROA𝑗(return on assets). α-discounted weights: 𝑤1 ∗≈0.4067,𝑤2 ∗≈0.3695,𝑤3 ∗≈0.2238. Local indicator weights: 𝛾11 =𝛾22 =𝛾33 =1, since each criterion has only one indicator. Assume the firm has the following observed values: a. CR𝑗=1.5, b. DE𝑗= 2.5, c. ROA𝑗=0.03(3%). We use the calibration specified in Section 3.3.3: a. 𝑎CR =1.0,𝑏CR =2.0,𝑚CR =1.5, b. 𝑐DE =1.0,𝑑DE =3.0,𝑚DE =2.0, c. 𝑝ROA =0.00,𝑞ROA =0.08,𝑚ROA =0.04. 3.8.2 Neutrosophic encoding of indicators (a) Liquidity (CR = 1.5) Since 1.0< 1.5<2.0, we are in the transition zone for CR: Truth: 𝜇CR 𝑇(1.5)= 𝑏CR−1.5 𝑏CR−𝑎CR =2.0−1.5 2.0−1.0 =0.5 1.0 =0.5. Falsity: 𝜇CR 𝐹(1.5)= 1.5−𝑎CR 𝑏CR−𝑎CR =1.5−1.0 2.0−1.0 =0.5 1.0 =0.5. Indeterminacy: 𝜇CR 𝐼(1.5)=2 ∣1.5−𝑚CR∣ 𝑏CR−𝑎CR =2 ∣1.5−1.5∣ 2.0−1.0 =0 1.0 =0.0. So, 𝑥𝑗1 =(𝑇𝑗1,𝐼𝑗1,𝐹𝑗1)=(0.5,0.0,0.5). Neutrosophic Sets and Systems, Vol. 94, 2026 465 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets (b) Solvency (DE = 2.5) Since 1.0< 2.5<3.0, we are in the transition zone for DE: Truth: 𝜇DE 𝑇(2.5)= 2.5−𝑐DE 𝑑DE−𝑐DE =2.5−1.0 3.0−1.0 =1.5 2.0 =0.75. Falsity: 𝜇DE 𝐹(2.5)= 𝑑DE−2.5 𝑑DE−𝑐DE =3.0−2.5 3.0−1.0 =0.5 2.0 =0.25. Indeterminacy: 𝜇DE 𝐼(2.5)=2 ∣2.5−𝑚DE∣ 𝑑DE−𝑐DE =2 ∣2.5−2.0∣ 3.0−1.0 =2×0.5 2.0 =1.0 2.0 =0.5. So, 𝑥𝑗2 =(𝑇𝑗2,𝐼𝑗2,𝐹𝑗2)=(0.75,0.5,0.25). (c) Profitability (ROA = 0.03) Since 0.00<0.03<0.08, we are in the transition zone for ROA: Truth: 𝜇ROA 𝑇(0.03)= 𝑞ROA−0.03 𝑞ROA−𝑝ROA =0.08−0.03 0.08−0.00 =0.05 0.08 =0.625. Falsity: 𝜇ROA 𝐹(0.03)= 0.03−𝑝ROA 𝑞ROA−𝑝ROA =0.03−0.00 0.08−0.00 =0.03 0.08 =0.375. Indeterminacy: 𝜇ROA 𝐼(0.03)=2 ∣0.03−𝑚ROA∣ 𝑞ROA−𝑝ROA =2 ∣0.03−0.04∣ 0.08−0.00 =2×0.01 0.08 =0.02 0.08 =0.25. So, 𝑥𝑗3 =(𝑇𝑗3,𝐼𝑗3,𝐹𝑗3)=(0.625,0.25,0.375). All components of 𝑥𝑗1,𝑥𝑗2,𝑥𝑗3lie in [0,1], as required. 3.8.3 Criterion-level values Each criterion has a single indicator, so the criterion-level neutrosophic triples are identical to the indicator-level triples: 𝑐𝑗1 =𝑥𝑗1 =(0.5,0.0,0.5), 𝑐𝑗2 =𝑥𝑗2 =(0.75,0.5,0.25), 𝑐𝑗3 =𝑥𝑗3 =(0.625,0.25,0.375). 3.8.4 NFDI aggregation using α-discounted weights Using the weights 𝑤1 ∗,𝑤2 ∗,𝑤3 ∗we compute: Truth component 𝑇𝑗 𝑇𝑗=𝑤1 ∗𝑇𝑗1 (𝑐) +𝑤2 ∗𝑇𝑗2 (𝑐) +𝑤3 ∗𝑇𝑗3 (𝑐) =0.4067×0.5+0.3695×0.75+0.2238×0.625. We compute each product: a. 0.4067×0.5 =0.20335, b. 0.3695×0.75=0.3695×3 4=0.277125, c. 0.2238×0.625=0.2238×5 8=0.139875. Summing: 𝑇𝑗=0.20335+0.277125+0.139875=0.62035. Indeterminacy component 𝐼𝑗 𝐼𝑗= 𝑤1 ∗𝐼𝑗1 (𝑐) +𝑤2 ∗𝐼𝑗2 (𝑐) +𝑤3 ∗𝐼𝑗3 (𝑐) =0.4067×0.0+0.3695×0.5+0.2238×0.25. Neutrosophic Sets and Systems, Vol. 94, 2026 466 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Products: a. 0.4067×0.0 =0.00000, b. 0.3695×0.5=0.18475, c. 0.2238×0.25=0.05595. Summing: 𝐼𝑗=0.00000+0.18475+0.05595=0.24070. Falsity component 𝐹𝑗: 𝐹𝑗=𝑤1 ∗𝐹𝑗1 (𝑐) +𝑤2 ∗𝐹𝑗2 (𝑐) +𝑤3 ∗𝐹𝑗3 (𝑐) =0.4067×0.5+0.3695×0.25+0.2238×0.375. Products: a. 0.4067×0.5 =0.20335, b. 0.3695×0.25=0.092375, c. 0.2238×0.375=0.083925. Summing: 𝐹𝑗=0.20335+0.092375+0.083925=0.37965. The sum of components is 𝑇𝑗+𝐼𝑗+𝐹𝑗=0.62035+0.24070+0.37965=1.24070. The unnormalized NFDI is therefore, NFDI𝑗=(𝑇𝑗,𝐼𝑗,𝐹𝑗)= (0.62035,0.24070,0.37965). 3.8.5 Normalization and scalar score We normalize the components by dividing by the sum 𝑇𝑗+𝐼𝑗+𝐹𝑗= 1.24070: Normalized truth 𝑇𝑗∗=0.62035 1.24070. Since 2×0.62035=1.24070, this gives: 𝑇𝑗∗=0.5. Normalized indeterminacy 𝐼𝑗∗=0.24070 1.24070≈0.1940. Normalized falsity 𝐹𝑗∗=0.37965 1.24070. Using the fact that the three normalized components must sum to 1: 𝐹𝑗∗=1−𝑇𝑗∗−𝐼𝑗∗≈ 1−0.5−0.1940= 0.3060. Thus, the normalized NFDI is approximately: NFDI𝑗 ∗=(𝑇𝑗∗,𝐼𝑗∗,𝐹𝑗∗)≈(0.50,0.19,0.31). The scalar distress score is, 𝑆𝑗=𝑇𝑗∗−𝐹𝑗∗≈0.50−0.31=0.19. This firm, therefore, has more evidence supporting distress than supporting health (𝑆𝑗> 0), but falsity remains significant, and indeterminacy is non-negligible. In other words, it is a borderline firm with an explicit quantification of ambiguity. Neutrosophic Sets and Systems, Vol. 94, 2026 467 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets 4. Case Study: Evidence from Saudi Emerging Markets This section applies the Nα-FDM framework to a concrete, fully worked case study. All data here are stylized but realistic and are meant to represent non-financial firms operating in Saudi emerging markets, with a focus on firms located in the Eastern Province. We use the same three core criteria and indicators introduced in Section 3: a. 𝐾1: Liquidity, measured by the current ratio (CR). b. 𝐾2: Solvency, measured by the debt-to-equity ratio (DE). c. 𝐾3: Profitability, measured by return on assets (ROA). The α-discounted weights are those already derived: 𝑤1 ∗≈0.4067,𝑤2 ∗≈0.3695,𝑤3 ∗≈0.2238, with 𝑤1 ∗+𝑤2 ∗+𝑤3 ∗≈1.0000. 4.1 Study context and sample We consider a small, illustrative cross-section of 𝑁 =4 non-financial firms operating in Saudi emerging markets. To keep the focus on the method and avoid disclosure issues, we label the firms generically: a. Firm A b. Firm B c. Firm C d. Firm D All firms are assumed to be located in the Eastern Province and to belong to sectors typical of the region (for example, petrochemical manufacturing, industrial services, logistics, and utilities). The actual sector names are not needed for the numerical calculations; what matters is the financial ratios. 4.2 Indicators and raw financial data For each firm 𝑗, we observe: 1. Current ratio (CR) — short-term liquidity: CR𝑗=current assets of firm 𝑗 current liabilities of firm 𝑗. 2. Debt-to-equity ratio (DE) — long-term solvency: DE𝑗=total debt of firm 𝑗 total equity of firm 𝑗. 3. Return on assets (ROA) — profitability: ROA𝑗=net income of firm 𝑗 total assets of firm 𝑗. Neutrosophic Sets and Systems, Vol. 94, 2026 468 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Table 1 reports current ratio (CR), debt-to-equity ratio (DE), and return on assets (ROA) for four non-financial firms operating in Saudi emerging markets (Eastern Province). Table 1. Raw financial indicators for four Saudi firms (stylized data) Firm Sector (generic description) CR𝑗 DE𝑗 ROA𝑗 A Capital-intensive petro-industrial 2.50 1.00 0.09 (9%) B Mixed industrial services 1.50 2.50 0.03 (3%) C Distressed manufacturing 0.80 3.50 −0.02 (−2%) D Moderately leveraged logistics 1.10 1.80 0.06 (6%) These values will now be transformed into neutrosophic numbers using the membership functions specified in Section 3.3. For completeness, we restate the parameter values in the next subsection. 4.3 Membership function parameters We treat each ratio as follows: 1. CR: lower values indicate worse (more distressed) liquidity. 2. DE: higher values indicate worse (more distressed) solvency. 3. ROA: lower values indicate worse profitability. 4.3.1 Current ratio (CR): “lower is worse.” Parameters: 𝑎CR =1.0,𝑏CR =2.0,𝑚CR =1.0+2.0 2=1.5. The current ratio (CR) is used to interpret a firm’s liquidity position. When 𝐶𝑅 ≤ 1.0, the firm’s liquidity is clearly distressed, meaning it may struggle to meet its short-term obligations. When 𝐶𝑅 ≥2.0, the liquidity position is clearly safe, indicating a comfortable cushion of current assets over current liabilities. For values between 1.0 and 2.0, the firm is in a transition zone, where liquidity is neither clearly safe nor clearly distressed. In this range, some degree of indeterminacy or uncertainty about the true liquidity condition may be present. The membership functions are: Truth (distress-supporting): 𝜇CR 𝑇(𝑥) ={ 1, 𝑥 ≤1.0, 2.0−𝑥 2.0−1.0, 1.0< 𝑥 <2.0, 0, 𝑥 ≥2.0; Falsity (health-supporting): 𝜇CR 𝐹(𝑥) ={ 0, 𝑥 ≤1.0, 𝑥−1.0 2.0−1.0, 1.0< 𝑥 <2.0, 1, 𝑥 ≥2.0; Neutrosophic Sets and Systems, Vol. 94, 2026 469 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Indeterminacy: 𝜇CR 𝐼(𝑥) ={ 0, 𝑥 ≤1.0, 2 ∣𝑥−1.5∣ 2.0−1.0 , 1.0<𝑥 < 2.0, 0, 𝑥 ≥2.0. 4.3.2 Debt-to-equity (DE): “higher is worse.” Parameters: 𝑐DE =1.0,𝑑DE =3.0,𝑚DE =1.0+3.0 2=2.0. The DE is used to describe a firm’s solvency position. When 𝐷𝐸 ≤1.0, solvency is clearly safe, meaning the firm is not heavily dependent on debt and its capital structure is relatively conservative. When 𝐷𝐸 ≥3.0, solvency is clearly distressed, indicating that the firm relies strongly on debt financing and may face higher financial risk. For values between 1.0 and 3.0, the firm is in a transition zone. In this range, solvency is neither clearly safe nor clearly distressed, and there may be some indeterminacy or uncertainty about the true level of financial risk. Membership functions: Truth (distress-supporting): 𝜇DE 𝑇(𝑥) ={ 0, 𝑥 ≤1.0, 𝑥−1.0 3.0−1.0, 1.0< 𝑥 <3.0, 1, 𝑥 ≥3.0; Falsity (health-supporting): 𝜇DE 𝐹(𝑥) ={ 1, 𝑥 ≤1.0, 3.0−𝑥 3.0−1.0, 1.0< 𝑥 <3.0, 0, 𝑥 ≥3.0; Indeterminacy: 𝜇DE 𝐼(𝑥) ={ 0, 𝑥 ≤1.0, 2 ∣𝑥−2.0∣ 3.0−1.0 , 1.0<𝑥 < 3.0, 0, 𝑥 ≥3.0. 4.3.3 Return on assets (ROA): “lower is worse.” Parameters: 𝑝ROA =0.00,𝑞ROA =0.08,𝑚ROA =0.00+0.08 2=0.04. ROA is used to evaluate a firm’s profitability. When ROA≤ 0.00, profitability is clearly distressed, meaning the firm is generating zero or negative profit from its assets. This signals weak performance and potential financial problems. Neutrosophic Sets and Systems, Vol. 94, 2026 470 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets When ROA≥0.08, profitability is clearly healthy, showing that the firm is earning a strong return on its assets. For values between 0.00 and 0.08, the firm is in a transition zone. In this range, profitability is neither clearly distressed nor clearly healthy, and there may be some uncertainty about the firm’s true financial performance. Membership functions: Truth (distress-supporting): 𝜇ROA 𝑇(𝑥)={ 1, 𝑥 ≤0.00, 0.08−𝑥 0.08−0.00, 0.00<𝑥 <0.08, 0, 𝑥 ≥0.08; Falsity (health-supporting): 𝜇ROA 𝐹(𝑥)={ 0, 𝑥 ≤0.00, 𝑥−0.00 0.08−0.00, 0.00<𝑥 <0.08, 1, 𝑥 ≥0.08; Indeterminacy: 𝜇ROA 𝐼(𝑥)={ 0, 𝑥 ≤0.00, 2 ∣𝑥−0.04∣ 0.08−0.00 , 0.00<𝑥 <0.08, 0, 𝑥 ≥0.08. All three ratios now have fully specified membership functions with no missing parameters. 4.4 Neutrosophic encoding of the four firms For each firm 𝑗and each indicator, we compute the neutrosophic triple 𝑥𝑗𝑘 =(𝑇𝑗𝑘,𝐼𝑗𝑘,𝐹𝑗𝑘)=(𝜇𝑘 𝑇(𝑋𝑗𝑘),𝜇𝑘 𝐼(𝑋𝑗𝑘),𝜇𝑘 𝐹(𝑋𝑗𝑘)). We show the computations explicitly. 4.4.1 Liquidity (CR) Using the CR membership functions: Firm A: CR𝐴=2.50≥2.0 𝜇CR 𝑇(2.5)=0,𝜇CR 𝐼(2.5)=0,𝜇CR 𝐹(2.5)=1. So 𝑥𝐴,1 =(0.0,0.0,1.0). Firm B: CR𝐵=1.50, with 1.0<1.5<2.0 𝜇CR 𝑇(1.5)=2.0−1.5 2.0−1.0=0.5 1.0=0.5, 𝜇CR 𝐹(1.5)=1.5−1.0 2.0−1.0=0.5 1.0=0.5, 𝜇CR 𝐼(1.5)=2 ∣1.5−1.5∣ 2.0−1.0 =0. So 𝑥𝐵,1 =(0.5,0.0,0.5). Neutrosophic Sets and Systems, Vol. 94, 2026 471 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets Firm C: CR𝐶=0.80≤1.0 𝜇CR 𝑇(0.8)=1,𝜇CR 𝐼(0.8)=0,𝜇CR 𝐹(0.8)=0. So 𝑥𝐶,1 =(1.0,0.0,0.0). Firm D: CR𝐷=1.10, with 1.0<1.1<2.0 𝜇CR 𝑇(1.1)=2.0−1.1 2.0−1.0=0.9, 𝜇CR 𝐹(1.1)=1.1−1.0 2.0−1.0=0.1, 𝜇CR 𝐼(1.1)=2 ∣1.1−1.5∣ 2.0−1.0 =2×0.4=0.8. So 𝑥𝐷,1 =(0.9,0.8,0.1). 4.4.2 Solvency (DE) Using the DE membership functions: Firm A: DE𝐴= 1.00≤1.0 𝜇DE 𝑇(1.0)=0,𝜇DE 𝐼(1.0)=0,𝜇DE 𝐹(1.0)=1. So 𝑥𝐴,2 =(0.0,0.0,1.0). Firm B: DE𝐵=2.50, with 1.0<2.5< 3.0 𝜇DE 𝑇(2.5)=2.5−1.0 3.0−1.0=1.5 2.0=0.75, 𝜇DE 𝐹(2.5)=3.0−2.5 3.0−1.0=0.5 2.0=0.25, 𝜇DE 𝐼(2.5)=2 ∣2.5−2.0∣ 3.0−1.0 =2×0.5 2.0 =0.5. So 𝑥𝐵,2 =(0.75,0.5,0.25). Firm C: DE𝐶=3.50≥3.0 𝜇DE 𝑇(3.5)=1,𝜇DE 𝐼(3.5)=0,𝜇DE 𝐹(3.5)=0. So 𝑥𝐶,2 =(1.0,0.0,0.0). Firm D: DE𝐷=1.80, with 1.0< 1.8<3.0 𝜇DE 𝑇(1.8)=1.8−1.0 3.0−1.0=0.8 2.0=0.4, 𝜇DE 𝐹(1.8)=3.0−1.8 3.0−1.0=1.2 2.0=0.6, 𝜇DE 𝐼(1.8)=2 ∣1.8−2.0∣ 3.0−1.0 =2×0.2 2.0 =0.2. So 𝑥𝐷,2 =(0.4,0.2,0.6). 4.4.3 Profitability (ROA) Using the ROA membership functions: Firm A: ROA𝐴=0.09≥0.08 Neutrosophic Sets and Systems, Vol. 94, 2026 478 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets 2. Firms B and D resemble companies that would fall in a classical grey band, but the neutrosophic model further clarifies how they are ambiguous: B has balanced truth and falsity with moderate 𝐼𝐵 ∗, while D exhibits substantial indeterminacy 𝐼𝐷 ∗alongside a smaller truth–falsity imbalance. Thus, Nα-FDM can be viewed as a generalization of classical ratio-based models, where a scalar distress measure is retained through 𝑆𝑗, but is systematically accompanied by an explicit decomposition into truth, falsity, and indeterminacy components. 5.2 Reading the neutrosophic outputs: four firm profiles The four firms in the Saudi case study illustrate four qualitatively distinct configurations: 1. Firm A — robust health NFDI𝐴 ∗=(0,0,1),𝑆𝐴=−1.0. Liquidity, solvency, and profitability all lie in clearly safe regions. The model attributes full weight to the falsity of distress and zero to both truth and indeterminacy. This corresponds to a firm that is far into the “safe” region of any classical distress scale and for which the available information is internally coherent. 2. Firm C — clear distress NFDI𝐶 ∗=(1,0,0),𝑆𝐶=1.0. All three ratios strongly support distress: liquidity is very weak, leverage is high, and profitability is negative. The model reflects this unanimity by assigning full mass to truth and none to falsity or indeterminacy. There is no ambiguity: both the level and direction of the signal are clear. 3. Firm B — symmetric but leaning towards distress NFDI𝐵 ∗≈(0.50,0.19,0.31),𝑆𝐵≈0.19. This firm has moderately weak liquidity, high leverage, and low but positive profitability. The triple shows that evidence in favour of distress is stronger than evidence in favour of health, yet indeterminacy is not negligible. In practical terms, this is a firm that should be treated as elevated risk but not as unequivocally distressed. The model makes this explicit instead of forcing a sharp binary classification. 4. Firm D — high indeterminacy and mild distress tilt NFDI𝐷 ∗≈(0.38,0.34,0.28),𝑆𝐷≈0.09. Here, all three components are substantial. Liquidity is weak, but solvency and profitability are more acceptable. The model signals a slight leaning towards distress, but, more importantly, a large indeterminate component. From a decision-making perspective, D is not merely average; it is structurally ambiguous. This is an important distinction: two firms with similar 𝑆𝑗can have very different 𝐼𝑗∗, leading to different policy responses. Neutrosophic Sets and Systems, Vol. 94, 2026 479 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets These four patterns illustrate the key strength of the neutrosophic representation: the ability to distinguish “clearly safe”, “clearly distressed”, “borderline but relatively well understood” (B), and “borderline and fundamentally uncertain” (D). 5.3 Role of α-discounting and expert inconsistency The α-discounting method connects expert knowledge to the quantitative model in a way that is both systematic and transparent. In the three-criterion example (liquidity, solvency, profitability), experts provide a set of relations among the weights, such as “liquidity is more important than solvency” and “solvency is more important than profitability”. These relations, taken together, can be mutually incompatible. Rather than discarding statements or solving an approximate least-squares system without interpretation, the α-discounting method: a. introduces a single scalar 𝛼(under the fairness principle), b. adjusts each relation multiplicatively by 𝛼, c. finds a weight vector 𝑤∗That satisfies the adjusted relations exactly. The outcome is twofold: 1. A weight vector 𝑤∗That respects the overall pattern of preferences and is guaranteed to be non-negative and normalised. 2. A consistency indicator 𝛼∗Whose deviation from 1 reflects how strongly the original system had to be altered. In the present setting, the resulting weights :𝑤1 ∗≈0.4067,𝑤2 ∗≈0.3695,𝑤3 ∗≈0.2238 assign slightly more importance to liquidity than to solvency, and clearly more to both than to profitability. This aligns with the intuition that in Saudi emerging markets, shortterm liquidity and leverage constraints can trigger distress quickly, while profitability is important but somewhat less immediate as a default driver. Importantly, if future expert assessments or regulatory priorities change (for example, giving more weight to profitability in highly competitive sectors), the same α-discounting machinery can be applied to generate a new weight vector, without altering the neutrosophic encoding of indicators or the structure of the cognitive map. 5.4 Managerial and regulatory implications for Saudi emerging markets In the context of Saudi emerging markets, and particularly in regions like the Eastern Province where firms are exposed to sectoral concentration and macro-financial shocks, the Nα-FDM outputs suggest several practical uses: 1. Screening and prioritization of supervisory effort Neutrosophic Sets and Systems, Vol. 94, 2026 480 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets a. Firms like C, with 𝑆𝐶close to +1 and 𝐼𝐶 ∗near zero, are clear candidates for immediate supervisory attention, restructuring plans, or stricter credit conditions. b. Firms like A, with 𝑆𝐴close to −1 and zero indeterminacy, require only routine monitoring. c. Firms like B and D fall in the grey zone; here, the distinction between moderate and high indeterminacy is crucial. Firm D, with larger 𝐼𝐷 ∗, would merit closer qualitative analysis, additional disclosures, or scenario-based stress tests, even though its scalar score 𝑆𝐷is lower than 𝑆𝐵. 2. The neutrosophic framework is designed to handle indicators that may be qualitative or subjectively scored (such as governance quality or exposure to specific regional risks). In the Saudi setting, this is very relevant: data on governance practices, ownership structures, or project-specific guarantees can be incorporated as additional criteria without breaking the structure of the model. Their uncertain nature is naturally captured through non-zero indeterminacy components. 3. Regulators and lenders frequently need to justify decisions to stakeholders. Expressing firm-level risk as a triple (𝑇𝑗∗,𝐼𝑗∗,𝐹𝑗∗)rather than only as a single probability allows for a more nuanced narrative: a. “The firm appears risky, but information is incomplete” (high 𝑇𝑗∗and high 𝐼𝑗∗), versus b. “The firm appears risky, and the evidence is consistent” (high 𝑇𝑗∗, low 𝐼𝑗∗). In emerging markets, where data gaps are common, this explicit separation can improve transparency and trust. 4. By extending the static NFDI to a dynamic version via the neutrosophic cognitive map, one can simulate how shocks (for example, a drop in oil prices or a tightening of credit conditions) propagate through liquidity, solvency, and profitability to affect overall distress. This opens the door to stress-testing frameworks tailored to Saudi market conditions, while preserving the explicit representation of indeterminacy. 5.5 Limitations and methodological reflections Although the model is internally consistent and the case study is numerically complete, several limitations must be acknowledged: 1. The case study uses stylised data for four firms and three criteria. A full empirical implementation would require a larger sample, more criteria (e.g., governance, cashflow volatility, market-based indicators), and actual historical distress events. 2. The membership function parameters are chosen to be plausible and economically sensible, but they have not been statistically calibrated. Calibration using historical Saudi firm data would refine the boundaries between “distressed”, “ambiguous”, and “safe” regions in indicator space. Neutrosophic Sets and Systems, Vol. 94, 2026 481 ____________________________________________________________________________________ Mohanned H. Alharbi, Neutrosophic α-Discounted Cognitive Mapping for Financial Distress Prediction: Evidence from Saudi Emerging Markets 3. The neutrosophic cognitive map has been described conceptually. The static NFDI already provides useful insight, but a dynamic implementation—where edge weights are estimated from data or expert elicitation and the map is iterated to convergence— would increase realism at the cost of additional modelling effort. 4. Finally, like any model, Nα-FDM does not remove uncertainty; instead, it makes uncertainty visible. Its value lies in exposing where the information set is strong, where it is weak, and where experts disagree. Despite these limitations, the case study demonstrates that the Nα-FDM approach is capable of capturing both the level of distress risk and the structure of ambiguity surrounding it. For emerging markets such as Saudi Arabia, where incomplete and conflicting information is routine rather than exceptional, this explicit handling of indeterminacy is a key advantage. 6. Conclusion and Future Research This paper proposed a Nα-FDM framework for financial distress prediction in emerging markets, with an application to Saudi non-financial firms. The model combines: 1. neutrosophic encoding of indicators into triples (𝑇,𝐼,𝐹), 2. α-discounted multi-criteria weighting of high-level factors, 3. construction of a firm-level Neutrosophic Financial Distress Index NFDI𝑗and scalar score 𝑆𝑗. Compared to classical ratio-based models, Nα-FDM does not compress all uncertainty into a single scalar. Instead, it separates supporting evidence, contradicting evidence, and indeterminacy, and directly incorporates expert inconsistency through α-discounting. The Saudi case study shows that the model behaves intuitively in extreme cases and provides meaningful nuance for borderline firms by highlighting where ambiguity dominates. Future research directions include: I. empirical calibration on larger Saudi datasets and comparison with machinelearning models [3], II. extension of the criteria set to include governance, market-based signals, and macroeconomic stress, III. Implementation of a dynamic neutrosophic cognitive map for stress testing, IV. hybrid models where neutrosophic outputs serve as inputs to advanced classifiers. The framework developed here is mathematically rigorous, conceptually transparent, and tailored to the reality of emerging markets where indeterminacy is not an exception, but a central feature of the information environment. Neutrosophic Sets and Systems, Vol. 94, 2026 482 ____________________________________________________________________________________ Mohanned H. 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