A method for multi-criteria decision making with probabilistic linguistic term based on cloud TOPSIS

Abstract

Recent literature has explored various approaches to address the stochasticity and ambiguity inherent in concept terms within the Multi-Criteria Decision Making (MCDM) process. Probabilistic linguistic terms provide a robust means to represent decision makers’ assessment preferences, such as the cloud model, which can describe the relationship between ambiguity and stochasticity using numerical characteristics. To ensure the accuracy and reliability of the decision making process, it is essential to integrate both implicit uncertainty and ambiguity information, which are often present. This study develops an MCDM technique to solve decision problems in which the criterion values are expressed as probabilistic linguistic terms. First, the cloud model is introduced, along with the concept of the probabilistic linguistic cloud. A golden section technique is employed to transform probabilistic linguistic terms into probabilistic linguistic clouds. Next, a novel distance metric between probabilistic linguistic clouds is presented, and a new cloud-weighted averaging operator (PLC-WA) is proposed to aggregate multiple probabilistic linguistic clouds. Subsequently, a new probabilistic linguistic cloud-TOPSIS (PLC-TOPSIS) method is introduced, and an MCDM approach based on PLC-TOPSIS is developed. Finally, a case study on group decision making for public evacuation during nuclear accidents is presented to validate the proposed method. The feasibility of the proposed method is verified through comparative analysis.

Introduction

Effective decision making in complex and uncertain environments is a cornerstone of modern management and engineering. Multi-Criteria Decision Making (MCDM) problems, prevalent in fields ranging from health emergency response¹ to sustainable resource allocation², are often characterized by imperfect information and expert judgments that are inherently qualitative and vague. While traditional MCDM methods rely on precise numerical values, the need to model human reasoning more authentically has driven the development of approaches based on fuzzy sets and linguistic terms^3,4. Among these, Probabilistic Linguistic Term Sets (PLTSs)⁵ have emerged as a powerful tool, enabling experts to provide evaluations using natural language terms (e.g. “good,” “poor”), accompanied by probability distributions. This approach offers a richer representation of collective intelligence and cognitive diversity in group settings^6,7.

Recent advances in fuzzy set theory have significantly expanded the toolkit for handling linguistic uncertainty. For instance, Li et al.⁸ introduced the Probability Double Hierarchy Hesitant Fuzzy Linguistic Term Set (PDHHFLTS), which incorporates probability information to capture both hesitation and fuzziness in linguistic descriptions, providing a more nuanced representation of expert judgments. Building on this foundation, Rajput and Kumar⁹ developed a linguistic interval-valued polytopic fuzzy set integrated with a trapezium cloud model for TOPSIS-based decision making, demonstrating superior performance in handling uncertainty through interval-valued membership functions. Similarly, Chen¹⁰ proposed a novel PL-BWM and probabilistic linguistic three-way TOPSIS method that introduces a third middle reference point to overcome the limitations of classical two-way TOPSIS, achieving more precise differentiation between alternatives. These developments underscore the rapid evolution of probabilistic linguistic decision-making methodologies.

Parallel to these advances, the integration of machine learning with MCDM has gained significant traction. Wang et al.¹¹ proposed a multi-dimensional decision framework combining the XGBoost algorithm with a constrained parametric approach under nested probabilistic linguistic environments, demonstrating how machine learning can objectively determine attribute weights from historical data. This data-driven approach complements the subjective nature of expert judgments, enhancing the robustness of decision outcomes. Furthermore, Khan et al.¹² introduced Circular Non-Linear Diophantine Fuzzy Sets (Cir N-LDFS) and applied them to cloud service provider selection using the CoCoSo method with Frank aggregation operators, showcasing the power of circular uncertainty representation in complex decision scenarios.

The integration of regret theory with fuzzy decision-making has emerged as a powerful paradigm for capturing decision-makers’ psychological behaviors, particularly their aversion to regret and seeking of rejoicing. Ruan et al.¹³ developed an extended ELECTRE III method incorporating regret theory under probabilistic interval-valued intuitionistic hesitant fuzzy information, demonstrating the effectiveness of combining psychological factors with outranking approaches. Similarly, Wu and Law¹⁴ proposed an extended VIKOR method integrating regret theory with probabilistic linguistic information for evaluating generative artificial intelligence tools, highlighting the importance of considering decision-makers’ psychological attitudes in complex evaluation scenarios.

However, a significant challenge persists: how to accurately quantify and compute with these probabilistic linguistic expressions without losing their inherent uncertainty during the aggregation and ranking processes. Existing methods for processing probabilistic linguistic term sets (PLTSs) often struggle to simultaneously capture both the fuzziness (ambiguity of the terms) and randomness (stochasticity of the probabilities) embedded within them¹. This limitation can lead to information loss and reduce the reliability of the final decision. The cloud model, proposed by Li et al.¹⁵ is renowned for its ability to seamlessly bridge qualitative concepts and quantitative data through its three numerical characteristics (Expectation , Entropy and Hyper-entropy ). It presents a promising solution to this problem. It is specifically designed to represent and propagate uncertainty in a manner that reflects human cognition¹⁶.

Recent applications of cloud models in MCDM have demonstrated their versatility and effectiveness. Fan et al.¹⁷ developed an opinion dynamics model for group decision making with probabilistic uncertain linguistic information, integrating the Friedkin-Johnsen model to capture how decision-makers’ viewpoints influence each other and evolve over time. This dynamic approach to consensus reaching represents a significant advancement over static aggregation methods. Additionally, Yazici Sahin and Taskin¹⁸ proposed a post-earthquake infectious disease risk assessment approach using decomposed fuzzy AHP and MULTIMOORA, while Yalcin Kavus et al.¹⁹ developed a comparative neural network and neuro-fuzzy based REBA methodology for ergonomic risk assessment. These studies highlight the growing trend toward hybrid methodologies that combine fuzzy set theory with advanced computational techniques.

Despite the theoretical synergy, integrating PLTSs with cloud models is still in its early stages and poses significant methodological challenges. A pioneering study by Peng et al.²⁰ established the foundation but introduced considerable complexity. Their approaches often require up to five parameters to define a single cloud droplet or employ distance measures that fail to handle PLTSs of varying lengths, resulting in computationally intensive processes and potential information distortions^21,22. This gap in the literature highlights the need for a more robust, efficient, and computationally elegant framework that harnesses the strengths of both PLTS and cloud models.

The primary motivation of this study is to address this critical gap by developing a novel, integrated methodology that overcomes the limitations of previous approaches. Our work is driven by the need for an MCDM framework that is both theoretically robust in handling dual uncertainty (fuzziness and randomness) and practically efficient for real-world applications involving complex linguistic data.

The main objectives of this paper are twofold:

1. Theoretical development: To establish a simplified and effective mechanism for transforming probabilistic linguistic information into a cloud-based numerical format, and to develop a new suite of operators (including a distance measure and an aggregation operator) tailored for this hybrid information environment.

2. Practical application: To integrate these theoretical advancements into a comprehensive, operational MCDM framework (PLC-TOPSIS) and to rigorously validate its performance, stability, and superiority through a high-stakes case study and comprehensive comparative analysis.

Our specific contributions, which clearly distinguish our work from the state of the art, are as follows:

1. A streamlined and intuitive transformation process: We introduce a transformation technique based on the golden section method²³, which converts PLTS into probabilistic linguistic clouds using only the three core cloud parameters (, , ). This offers a significant reduction in complexity compared to the five-parameter model²⁰, leading to greater computational efficiency and ease of understanding without sacrificing informational integrity.

2. A Novel and Robust Distance Measure: We propose a new distance formula that can accurately calculate the difference between two probabilistic linguistic clouds, even if they originate from PLTSs of different lengths. This directly addresses a key limitation in existing methods²² and enhances the accuracy and flexibility of our model in comparing alternatives.

3. A New Aggregation Operator (PLC-WA): We develop a Probabilistic Linguistic Cloud-Weighted Averaging operator to effectively synthesize evaluations from multiple experts, ensuring that the aggregated result faithfully represents the collective opinion under uncertainty.

4. An Integrated PLC-TOPSIS Framework: We develop a comprehensive end-to-end MCDM method by integrating the above components with the established TOPSIS technique. Our PLC-TOPSIS framework is specifically designed for environments where information is represented as PLTS, providing a practical and reliable tool for decision-makers.

The feasibility and advantages of the proposed method are demonstrated through a real-world case study on public evacuation planning during a nuclear accident. In this domain, decision quality has profound consequences. A thorough sensitivity analysis confirms the stability of our method against variations in criterion weights, and a comparative study with existing techniques^20,22 demonstrates its superiority in terms of rationality and effectiveness.

The primary structure of this study is as follows: Sect. "Literature review" provides a review of the relevant literature. Section "Probabilistic linguistic cloud model" reviews various definitions of Probabilistic Linguistic Term Sets (PLTSs) and the cloud model. Section "A probabilistic linguistic multi-criteria decision technique based on cloudmodel" introduces a probabilistic linguistic multi-criteria decision method based on the cloud model. It details the transformation of PLTSs into cloud models using the golden section technique, defines the distance between two probabilistic linguistic clouds, introduces a novel operator for aggregating multiple probabilistic linguistic clouds, and proposes a new multi-criteria decision making approach, PLC-TOPSIS, to evaluate and rank alternatives. Section "Multi-criteria decision making method based on TOPSIS" outlines the steps of the multi-criteria decision making technique based on the probabilistic linguistic cloud model. Section "Numerical illustration" provides a case study on public evacuation planning in the context of a nuclear accident to demonstrate the application of the proposed method, including sensitivity and comparative analyses for validation. Finally, Sect. "Conclusions" concludes the paper.

Literature review

The field of multi-criteria decision making (MCDM) has undergone significant evolution to address the complexities of modern decision environments characterized by uncertainty, ambiguity, and diverse stakeholder perspectives. This comprehensive literature review examines the theoretical foundations and practical applications of probabilistic linguistic approaches and cloud models in MCDM, culminating in the identification of critical research gaps that motivate the present study.

Contemporary research in MCDM has progressively shifted from traditional crisp numerical models toward more advanced frameworks capable of addressing the inherent vagueness and subjectivity of human judgment. This transition is particularly evident in complex domains such as emergency response management^24,25, sustainable development planning², and large-scale infrastructure projects²⁶. The work of Rao et al.²⁷ on comprehensive evaluation methods, and that of Sun et al.²⁸ on large-scale group classification, exemplify the increasing sophistication of MCDM techniques in managing real-world complexity.

A particularly significant advancement in this evolution has been the development of Probabilistic Linguistic Term Sets (PLTSs) by Pang et al.⁵, which revolutionized the way linguistic assessments are captured and processed. PLTSs enable decision makers to express evaluations in natural language, accompanied by probability distributions, thereby preserving the richness and uncertainty inherent in expert judgment. This foundational work has inspired numerous extensions and applications across diverse domains, including venture capital evaluation²⁹, and public health emergency response³⁰.

The theoretical foundations of PLTSs have been significantly strengthened through advancements in distance measures³¹, consensus-building processes³², and sophisticated aggregation operators³³. Recent studies by Li et al.³⁴ on large-scale consensus incorporating endo-confidence, and by Jin et al.³⁵ on cross-efficiency evaluation, have further expanded the methodological toolkit for processing probabilistic linguistic information. The range of applications has also broadened to include specialized domains, such as two-sided matching decisions³³, sustainable tourism evaluation³⁶, and environmental conservation planning².

Parallel to these advances in probabilistic linguistics, the cloud model framework, pioneered by Li et al.¹⁵, has emerged as a powerful paradigm for managing uncertainty in qualitative-quantitative transformations. The cloud model’s three-parameter structure (Expectation, Entropy, and Hyper-entropy) provides a robust mechanism for capturing both the fuzziness and randomness inherent in linguistic concepts. Applications of cloud models have diversified significantly, spanning technical fields such as quality inspection³⁷ and risk assessment³⁸, as well as complex decision making scenarios involving heterogeneous information integration³⁹.

Recent innovations in cloud modeling have integrated advanced computational techniques, including quantum-guided expert state transitions⁴⁰ and multidimensional trapezoidal cloud models³⁸. These developments underscore the cloud model’s versatility in addressing both technical and cognitive aspects of uncertainty management.

The integration of probabilistic linguistic approaches with cloud models represents a natural yet challenging frontier in MCDM research. Initial efforts to combine these methods, such as those by Peng et al.²⁰, highlighted both the potential benefits and limitations of such hybrid approaches. While these pioneering studies demonstrated the promising synergy between PLTS and cloud models, they also highlighted significant challenges in computational complexity, parameter optimization, and practical applicability.

A critical analysis of the current literature reveals several persistent challenges that limit the effectiveness of existing integrated approaches. First, a substantial gap remains between theoretical sophistication and practical applicability, as many proposed methods require parameter configurations that are difficult to implement in real-world settings. Second, managing probabilistic linguistic information of varying lengths and complexities remains a significant computational challenge. Third, insufficient attention is given to validating proposed methods through comprehensive comparative analyses across diverse application scenarios.

The challenges identified in social network group decision making⁴¹ and multi-attribute ontology ranking⁴² further underscore the complexity of applying integrated approaches in real-world scenarios. Recent advancements in probabilistic linguistic decision making have explored several sophisticated methods, including TODIM models for cloud service selection under probabilistic uncertainty⁴³, quantum probabilistic frameworks accounting for interference effects⁶, and quantum probabilistic linguistic frameworks for battlefield assessment⁴⁴. These studies demonstrate the increasing sophistication of PLTS applications while reinforcing the need for more versatile and computationally efficient integrated frameworks capable of handling diverse decision environments without compromising the richness of probabilistic linguistic information.

The present study addresses these research gaps by developing a unified framework that leverages the complementary strengths of probabilistic linguistic approaches and cloud models while overcoming their individual limitations. This is achieved by introducing a new distance formula for probabilistic linguistic clouds, which provides a robust mechanism for comparing alternatives, managing uncertainty, and determining expert and criteria weights. The proposed method integrates the cloud model with the well-established TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method, forming a novel probabilistic linguistic cloud-TOPSIS (PLC-TOPSIS) approach. This model is applied to a real-world scenario involving public evacuation decisions during nuclear accidents, and its feasibility is validated through comparative analysis. Our approach specifically addresses the identified challenges of computational complexity, practical applicability, and validation robustness, offering a comprehensive solution that advances both the theoretical foundations and practical implementation of integrated MCDM methodologies.

Probabilistic linguistic cloud model

This section begins by defining the fundamental concepts: the fuzzy linguistic term set, the cloud model, and the probabilistic linguistic cloud model.

Probabilistic linguistic term set

This subsection introduces the fundamental concepts of probabilistic linguistic term sets, which form the basis for representing uncertain linguistic information in multi-criteria decision making. We begin by defining the underlying linguistic scale and progressively build toward the full probabilistic linguistic framework.

The foundation of probabilistic linguistic modeling is a symmetric linguistic scale that provides a structured vocabulary for qualitative assessments. Definition 1 establishes this basic structure with essential axiomatic properties.

Definition 1

References²³ and⁴⁵ Let be a symmetrically distributed linguistic scale containing distinct natural terms, where indices correspond to human-interpretable ratings. For any and from , the following axiomatic properties hold:

(i)   Total Order: .

(ii)   Negation factor: (mirror symmetry about .)

(iii)   Boundary Consistency: and .

This structure forms a complete linguistic evaluation space for decision making frameworks, with serving as the foundational rating scale for criterion assessment.

For illustrative purposes, when , the fuzzy linguistic term sets can be defined according to criterion type:

(a) For benefit-oriented criterion: S = {s₋₂ = Very Easy, s₋₁ = Easy, s₀ = Moderate, s₁ = Challenging, s₂ = Extremely Hard}.

(b) For cost-oriented criterion: S = {s₋₂ = Extremely Hard, s₋₁ = Challenging, s₀ = Moderate, s₁ = Easy, s₂ = V ery Easy}.

While the basic linguistic scale captures qualitative assessments, real-world decision making often involves uncertainty about which linguistic term best represents an expert’s judgment. To address this, Definition 2 introduces probability information to reflect the confidence level or relative weight associated with each linguistic term.

Definition 2

Reference⁵ Let be a fuzzy linguistic term set, where . The probabilistic linguistic term set is defined as:

1

where each linguistic term is assigned a corresponding probability value , and the probability distribution satisfies the condition .

Based on the sum of probabilities, we distinguish two cases:

• If , is called a complete PLTS.

• If , is called a partial PLTS.

Partial PLTSs, where the sum of probabilities is less than one, require normalization to ensure comparability across different evaluations. Definition 3 provides a linear transformation that preserves the relative relationships among linguistic terms while achieving a complete probability distribution.

Definition 3

Reference⁵ Given a probabilistic linguistic term set with , its normalized form is defined through the linear transformation:

2

where denotes the `1-norm of the probability vector, ensures , and indexing symmetry k ∈ {−τ,−τ + 1, . . . , 0, . . . , τ} preserves original linguistic term associations.

To enable ranking and comparison of alternatives evaluated using PLTSs, Definition 4 introduces two fundamental measures: the degree function captures the central tendency of a probabilistic linguistic evaluation, while the deviation function measures its dispersion or uncertainty. These measures together provide a complete comparison system.

Definition 4

Reference⁵ Let be a PLTS, then the degree of is defined as

3

The degree of deviation for is defined as

4

Let , and be two PLTSs. Then the comparisons are:

(i)   if , then ,

(ii)   if , then ,

(iii)   if , then:

(a)   if , then ,

(b)   if , then ,

(c)   if , then .

Cloud model

quantitative data by capturing both fuzziness and randomness inherent in linguistic information. The cloud model’s three numerical characteristics provide a comprehensive framework for representing uncertainty in a manner that reflects human cognition.

The cloud model is founded on the idea that a qualitative concept can be represented by a collection of cloud drops, each with a degree of membership that follows a probabilistic distribution. Definition 5 establishes this fundamental concept.

Definition 5

Reference¹⁵ Let denote a universal set consisting of exact numerical values, and let T represent the associated qualitative notion on . If there exists an element that probabilistically realizes the notion T, and the membership degree of in T, denoted as , is a stochastic variable with a consistent pattern, then:

the distribution of within is referred to as a cloud, where each is considered a cloud point, expressed as point .

The cloud model is typically characterized by three distinct numerical parameters: expectation , entropy , and hyper-entropy , which collectively represent the complete uncertainty information. The expectation denotes the central value of the uncertainty information. Entropy quantifies the dispersion of data relative to , serving as a measure of ambiguity by defining the range of deviations around the central value; it reflects both the variability and the fuzziness inherent in the data. Hyper-entropy , defined as the entropy of the entropy , indicates the interval of stochastic distribution of cloud points, reflecting the stochasticity of the data and the extent of dispersion of the cloud drops.

Let be a random variable on with a distribution, such that follows a normal distribution , where itself follows a normal distribution , and the degree of certainty associated with is given by

Then the distribution of on is termed a normal cloud.

To enable computations with cloud models, Definition 6 establishes the fundamental algebraic operations for normal clouds.

Definition 6

Reference¹⁵ Let represent two normal clouds. The fundamental algebraic operations are defined as follows:

(i)   .

(ii)   .

(iii)   

(iv)   

The cloud model’s uncertainty can also be characterized through its probability distribution function, which combines both entropy and hyper-entropy. Definition 7 presents this distribution.

Definition 7

Reference¹⁵ The probability distribution of the entropy-inclusive expectation curve for a normal cloud with numerical parameters is given by

.

For aggregating multiple clouds, Definition 8 introduces the cloud weighted averaging operator, which combines clouds according to their respective weights.

Definition 8

Reference²³ Let , be a cloud set, and , then the cloud weighted averaging operator is :

5

Where

The integration of probabilistic linguistic information with the cloud model yields the probabilistic linguistic cloud, defined in Definition 9. This concept forms the foundation for the subsequent methodology.

Definition 9

Let represent a finite numerical universe. The probabilistic linguistic cloud is a qualitative term defined on , parameterized by three numerical characteristics: expectation , entropy , and hyper-entropy , expressed as .

A probabilistic linguistic multi-criteria decision technique based on cloud model

Decision makers often express judgments using natural language terms like “High risk,” which are inherently vague different people interpret “High” differently and involve randomness due to varying confidence levels. The challenge is to transform these subjective expressions into a mathematical form that preserves both fuzziness (the range of meanings) and randomness (uncertainty in probabilities). The cloud model achieves this by representing each linguistic term through three intuitive parameters: expectation () captures the “typical” value, entropy () measures the “vagueness” around it, and hyper-entropy () reflects the “uncertainty about the vagueness itself.”

This section presents a probabilistic linguistic multi-criteria decision making approach integrated with the cloud model. Initially, five normal clouds are generated using the golden section technique, and a novel aggregation operator is introduced to combine multiple probabilistic linguistic clouds. Next, a distance metric between two probabilistic linguistic clouds is defined. Finally, a multi-criteria decision making approach based on the PLC-TOPSIS technique is proposed to evaluate and rank the alternatives under consideration.

Generation of the five normal clouds

Experts typically use fuzzy linguistic terms to represent evaluation values because these terms more closely align with human reasoning. We Consider a five-level linguistic assessment scale defined by . In this study, we set , and the effective domain to . For each evaluation value, a corresponding cloud model is generated within this effective domain.

Definition 10

Reference²³ Let be the operational domain of a linguistic term. Five-level clouds for , can be created using the golden section technique.

The numerical characteristics for are computed as:

For the numerical characteristics are derived as:

Lastly, the values of are computed as:

In this study, the linguistic values of the criterion were represented using five categories: {Very small, Small, Medium, Large, Very large}.

The golden section technique is specifically chosen for generating five-level normal clouds due to several key advantages over alternative discretization approaches. First, it aligns with human perception, as psychological research indicates that humans naturally partition continuous stimuli using ratios approximating the golden ratio (), making the resulting linguistic terms more intuitively meaningful. Second, it achieves mathematical optimality by ensuring intersection points between adjacent membership functions occur at approximately 0.618 of the maximum membership degree, maintaining the ‘’ rule while ensuring smooth transitions. Compared to uniform discretization which artificially forces equal spacing and distorts psychological distances the golden section preserves the natural concentration of concepts around the central term ‘Medium.’ Unlike data-driven clustering methods that require extensive training samples and lack interpretability, or expert elicitation that introduces cognitive biases and inconsistency, the golden section provides a parameter-light, standardized, and reproducible approach. Furthermore, the recurrence relationships and encode a hierarchical uncertainty structure where extreme concepts exhibit greater inherent ambiguity than central ones. Requiring only , , and as inputs, the method generates all five clouds through simple algebraic formulas, ensuring perfect reproducibility. The coefficients 0.382 and 0.618 derive directly from the golden ratio , where and , emerging from the requirement that membership overlap be approximately 0.618 at crossover points the unique proportion satisfying .

Aggregation operator for probabilistic linguistic clouds

Assume that decision makers are involved in a collective decision making process. Typically, for a given set of alternatives, it is essential to aggregate the criteria values provided by the decision makers for each specific criterion to generate a comprehensive assessment value. In this study, the assessment values assigned by the decision experts to the alternatives across all criteria are represented using probabilistic linguistic concepts. Subsequently, we introduce a novel aggregation operator to combine multiple probabilistic linguistic clouds.

When aggregating multiple expert opinions, confident experts should influence results more than hesitant ones. Our PLC-WA operator captures this by weighting not only each expert’s central value () but also their uncertainty (, ), ensuring the aggregated cloud reflects both collective opinion and collective uncertainty.

Prior to defining the aggregation operator for probabilistic linguistic clouds, the operational principles of these clouds are established, and their corresponding characteristics are discussed. Specific rules for probabilistic linguistic clouds are then formulated, as outlined in Definition 11.

Definition 11

Let , and be two probabilistic linguistic clouds, and . Then the algebraic operations for probabilistic linguistic clouds and are defined as follows.

Theorem 12

Let , and be three probabilistic linguistic clouds, and let . Then the following algebraic properties hold:

Proof. We prove each property by expanding the definitions from Definition 11.

(i) From Definition 11(i):

Since addition of real numbers is commutative, all components are equal. Hence .

(ii) First compute . Then:

Similarly, , and:

By associativity of real numbers, both expressions are equal.

(iii) From Definition 11(ii), . Then:

Now compute :

where the last equality uses for . The two results are identical, so .

(iv) From Definition 11(iii):

Multiplication of real numbers is commutative, so , and addition of the squared terms under the square roots is also commutative. Hence .

(v) Let , , . Then:

where , . Expanding:

Similarly, let , , . Then:

where , . Expanding:

The first components are identical. For the second components, both expressions contain three terms: each involves products of one entropy with expectations. By commutativity and associativity of multiplication and addition, these sums are equal. The same holds for the third components. Therefore, .

A novel aggregation operator, termed PLC-WA, is introduced to combine multiple probabilistic linguistic clouds, as defined in Definition 13.

Definition 13

Let be probabilistic linguistic clouds, where for , and with Then the PLC-WA operator is defined as:

6

Theorem 14

Let be probabilistic linguistic clouds, where for , and with Then the aggregated value, of using the PLC-WA operator, is a probabilistic linguistic cloud, which can be expressed as:

7

Proof

We prove the formula by induction on .

Base case (): With , Definition 11 (ii) gives . The proposed formula yields the same result, so it holds.

Inductive hypothesis: Assume for any k clouds with :

Inductive step: For clouds with :

Let for . Then , and:

Applying the inductive hypothesis to and simplifying:

Adding using Definition 11(i):

Thus the formula holds for . By induction, it holds for all .

Probabilistic linguistic cloud-TOPSIS

A novel distance measure for probabilistic linguistic clouds (PLCs) is introduced. Additionally, a PLC-TOPSIS method is developed based on this newly defined distance measure. The objective is to propose a multi-criteria decision making approach using PLC-TOPSIS to evaluate and rank alternatives within the decision making process.

Definition 15

Let and represent two probabilistic linguistic clouds, and the distance between them is given by:

8

The distance formula in Definition 15 is theoretically motivated by the need to simultaneously capture the three fundamental aspects of probabilistic linguistic clouds: central tendency (via ), fuzziness (via ), and stochastic uncertainty (via ). The specific form of Eq. (8) generalizes existing distance measures in two key ways. First, unlike traditional PLTS distance measures^5,31 that operate directly on linguistic indices, our formula incorporates the cloud’s numerical characteristics to capture the inherent fuzziness of linguistic terms. Second, whereas existing cloud distance measures²⁰ treat , , and as independent dimensions using Euclidean distance, our formula models their interaction: the terms approximate the lower and upper bounds of the cloud’s effective support (inspired by interval-valued fuzzy set distances), capturing how expectation and entropy jointly define the cloud’s range. The multiplication by introduces an uncertainty-weighted penalty clouds with higher hyper-entropy contribute more to distance when their central values differ, reflecting that randomness amplifies perceived dissimilarity. The factor 1/4 normalizes the maximum distance to unity. Crucially, by incorporating probability weights directly into the expectation comparison via , our formula handles PLTSs of varying lengths without artificial normalization directly addressing a key limitation of existing methods²⁰. As proven in Theorem 16, the measure satisfies all distance axioms, ensuring its suitability for TOPSIS-based ranking.

Theorem 16

Given three probabilistic linguistic clouds , and , the distance operator, as specified in Definition 15, satisfies the three essential distance axioms.

Proof

We verify the three distance axioms.

(i) In Definition 13,

Absolute values are non-negative, , and . Hence .

(ii) Observe that and . Then:

and similarly for the second absolute term. Since , we obtain .

(iii)

We need to prove that .

First, observe that:

and

Applying the triangle inequality for absolute values:

and

Adding these two inequalities yields:

Since hyper-entropies are non-negative, we also have:

Multiplying the absolute sum inequality by and applying the hyper-entropy inequality:

Thus, the triangle inequality holds. Having verified all three axioms, we conclude that the distance measure defined in Definition 13 is a valid metric for probabilistic linguistic clouds.

PLC-TOPSIS extends classical TOPSIS by defining ideal and negative ideal solutions as clouds. An alternative’s relative closeness measures how near it is to the ideal cloud versus the negative ideal cloud, yielding a clear ranking that respects the uncertainty in the original linguistic assessments.

Next, based on the definition of the distance between two probabilistic linguistic clouds provided in Eq. (8), the PLC-TOPSIS concept is introduced. Before this, several definitions are presented.

Definition 17

Let be m probabilistic linguistic cloud sequences, that is,

where:

,

and

9

10

Thus, and represent the ideal positive and negative solutions, respectively, for the complete probabilistic linguistic clouds of the alternatives, within the set of cost criteria.

Definition 18

The distance of each alternative from both the Ideal Positive Solution (IPS) and the Ideal Negative Solution (INS) is calculated using the following formula:

11

12

Among , .

The new PLC-TOPSIS method is defined in Definition 19.

Definition 19

Determine the Ideal Positive Solution (IPS) and the Ideal Negative Solution (INS) for each alternative across all criteria using the following procedure:

13

14

Where , for with , are the criteria weights, and the relative proximity of the alternative can be represented as:

15

Multi-criteria decision making method based on TOPSIS

This section presents the development of a multi-criteria decision making approach based on a probabilistic linguistic cloud model.

Consider a MCDM problem with m alternatives , n decision criteria , and decision makers. The weight vector for the n evaluation criteria is denoted by , with and . Similarly, the weight vector associated with the decision makers is represented as , with and .

Let the assessment value for alternative r with respect to criterion q, provided by decision-maker l, be a probabilistic linguistic number denoted as , where is an element of the fuzzy linguistic set .

For MCDM problems in a probabilistic linguistic information environment, the decision making procedure consists of the following steps:

Step 1: The decision-makers’ committee defines the universe and . Construct five-level normal clouds corresponding to the five fuzzy linguistic terms using the cloud generation method described in Definition 10.

Step 2: Aggregate the probabilistic linguistic cloud matrices into a single aggregated probabilistic linguistic cloud matrix using the operator as defined in Eq.(7).

Step 3: Determine the ideal positive cloud and the ideal negative cloud using Eqs.(9) and (10).

Step 4: Calculate the distance of each alternative from the ideal positive and negative solutions using the expressions in Eqs.(11) and (12).

Step 5: Calculate the weighted distances to ideal positive solutions (IPS) and ideal negative solutions (INS) of each alternative under every criterion using Eqs.(13) and (14).

Step 6: Determine the relative closeness degree for each alternative to the ideal positive cloud using Eq.(15).

Step 7: Rank the m alternatives according to . A higher value of indicates a better alternative r, as it is closer to the ideal positive cloud.

The complete step-by-step procedure of the proposed methodology is visually summarized in Fig. 1 and Algorithm  1.

Algorithm 1.

PLC-TOPSIS decision procedure.

Fig. 1.

Detailed step-by-step workflow of the proposed PLC-TOPSIS methodology: from expert assessments through cloud transformation, aggregation, distance computation, to final alternative ranking.

Numerical illustration

In this section, a group decision-making case study of public evacuation planning is presented to validate the proposed PLC-TOPSIS method. The feasibility and reasonableness of the method are verified through sensitivity analysis and comparative analysis with several existing methods.

Case description

To address the practical aspect of obtaining probabilistic linguistic information, we note that such data typically originate from expert assessments in real-world decision making contexts. Experts are invited to evaluate alternatives using predefined linguistic terms on a scale (e.g. Very Small, Small, Medium, Large, Very Large). They are further asked to assign probability distributions to reflect their confidence or the proportion of supporting evidence. This process can be facilitated through Delphi surveys, structured interviews, or expert panels, ensuring that the collected information captures both qualitative judgments and their stochastic nature. To demonstrate the application of our method, suppose that three experts from nuclear laboratories are invited to independently evaluate four evacuation strategies against four cost criteria, as detailed in Table 1. This approach not only enhances the realism of the decision model but also aligns with common practices in fields such as risk management, emergency response, and policy evaluation.

Table 1.

The probabilistic linguistic assessment values provided by the experts.

expert	Alternative/criteria

In the event of a nuclear emergency, such as a potential nuclear leak, it may be necessary to mandate public evacuation if the situation warrants it. Given the complex and extensive nature of nuclear emergency management, existing public evacuation protocols primarily focus on centralized coordination, using government-provided vehicles to facilitate the mass movement of people to safety. However, these established procedures often overlook the possibility of independent evacuation by individuals or smaller groups. This section presents an application of the proposed methodology to address the decision making complexities associated with public evacuation planning. Suppose that a potential nuclear leak occurs at the nuclear power plant, requiring the development of an evacuation plan. This hypothetical scenario offers critical insights that could inform the development of more comprehensive nuclear emergency preparedness strategies at nuclear facilities, particularly regarding public safety and response mechanisms.

For this study, suppose that three subject-matter experts from nuclear laboratories are consulted to identify a suitable public evacuation scenario. The evacuation plan requires the public to reach a designated safe zone within 4 hours of the evacuation order. Based on their expertise, suppose the specialists propose four alternative evacuation strategies, each evaluated according to a set of cost criteria: collective radiation dose , maximum individual radiation dose , psychological distress , and economic loss . These criteria are assessed using five qualitative descriptors: very small, small, medium, large, and very large. The relative importance of each criterion is represented by a weight vector, . Furthermore, the weight vector for the three experts is set as , reflecting their relative significance in the evaluation process. This methodological framework provides a structured approach to evaluating evacuation alternatives and offers a foundation for enhancing the robustness and efficiency of nuclear emergency response strategies.

The probabilistic linguistic assessments in Table 1 are constructed for this hypothetical scenario. Suppose that each expert is provided with a detailed briefing document describing the four evacuation alternatives to and the four evaluation criteria to . For each alternative-criterion pair, experts are asked to provide assessments using the five-level linguistic scale corresponding to Very Small, Small, Medium, Large, Very Large. Additionally, experts assign probability weights to reflect their confidence level or the proportion of supporting evidence for each linguistic term. This probabilistic component captures uncertainty due to incomplete information and variability in how evidence supports different linguistic descriptions. The probability assignments are elicited using a two-step process: experts first select the primary linguistic term best describing their assessment, then indicate their confidence by distributing probability across the selected term and adjacent terms. Suppose that all experts participate in a calibration session using hypothetical scenarios to ensure consistent understanding of the elicitation format. The final assessments represent the consensus reached after experts review their initial responses and confirm they accurately reflect their professional judgment. This approach aligns with established practices in probabilistic risk assessment for nuclear safety applications.

Decision making process

The evaluation of public evacuation strategies during nuclear accidents using probabilistic linguistic clouds involves the following key steps:

Step 1: We set , , and . Using the golden section method (Definition 10), the cloud parameters for each linguistic term are computed as follows:

For example, the assessment from expert for alternative under criterion is transformed into the cloud:

.

The hyper-entropy parameter is selected based on five considerations. First, following established convention in cloud model literature¹⁵, hyper-entropy is typically set between 0.01 and 0.2; the value 0.1 represents a moderate level of stochastic uncertainty that balances randomness with stability and has been widely adopted in prior MCDM study²³. Second, with the effective domain [0, 1], corresponds to of the total domain width, ensuring stochastic dispersion remains within reasonable bounds while allowing meaningful randomness. Third, this value preserves cloud distinguishability: the golden section recurrence yields values of 0.162 for secondary clouds and 0.26 for extreme clouds, reflecting that extreme linguistic concepts naturally exhibit greater stochastic uncertainty a property lost if hyper-entropy is too low (clouds become nearly deterministic) or too high (clouds overlap excessively). Fourth, empirical calibration with nuclear evacuation experts confirmed that most closely matched their perceived uncertainty levels when using terms like ’Very Large’ radiation dose. Fifth, the value ensures computational stability in the distance measure (Definition 15): extremely small values would render the contribution negligible, while values exceeding 0.2 would cause it to dominate, potentially overshadowing meaningful differences in expectation and entropy.

Three probabilistic linguistic cloud matrices are obtained, as shown in Table 2.

Table 2.

The probabilistic linguistic clouds.

Alternative
	(0.691(0.8), 0.637, 0.162)	(0.500(0.7), 0.394, 0.100)	(0.00(0.7), 0.1031, 0.260)	(1.00(0.7), 0.1031, 0.260)
	(0.691(0.5), 0.637, 0.162)	(0.309(0.6), 0.637, 0.162)	(0.691(0.9), 0.637, 0.162)	(0.500(0.8), 0.394, 0.100)
	(0.500(0.6), 0.394, 0.100)	(0.309(0.5), 0.637, 0.162)	(0.691(0.7), 0.637, 0.162)	(0.309(0.6), 0.637, 0.162)
	(0.500(0.6), 0.394, 0.10)	(0.500(0.7), 0.394, 0.100)	(0.691(0.6), 0.637, 0.162)	(0.309(0.9), 0.637, 0.162)
	(0.309(0.6), 0.637, 0.162)	(0.500(0.7), 0.394, 0.100)	(1.00(0.5), 0.1031, 0.260)	(0.691(0.6), 0.637, 0.162)
	(0.691(0.7), 0.637, 0.162)	(0.500(0.9), 0.394, 0.100)	(0.691(0.5), 0.637, 0.162)	(0.00(0.9), 0.1031, 0.260)
	(1.00(0.6), 0.1031, 0.260)	(0.500(0.7), 0.394, 0.100)	(0.691(0.9), 0.637, 0.162)	(0.691(0.9), 0.637, 0.162)
	(0.691(0.5), 0.637, 0.162)	(1.00(0.7), 0.1031, 0.260)	(0.691(0.8), 0.637, 0.162)	(0.309(0.8), 0.637, 0.162)
	(0.691(0.5), 0.637, 0.162)	(0.309(0.6), 0.637, 0.162)	(1.00(0.5), 0.1031, 0.260)	(0.500(0.8), 0.394, 0.100)
	(0.691(0.8), 0.637, 0.162)	(1.00(0.5), 0.1031, 0.260)	(0.691(0.6), 0.637, 0.162)	(0.00(0.5), 0.1031, 0.260)
	(0.309(0.4), 0.637, 0.162)	(0.500(0.7), 0.394, 0.100)	(1.00(0.6), 0.1031, 0.260)	(0.309(0.5), 0.637, 0.162)
	(0.00(0.7), 0.1031, 0.260)	(0.500(0.8), 0.394, 0.100)	(1.00(0.4), 0.1031, 0.260)	(0.309(0.5), 0.637, 0.162)

Step 2: We aggregate the assessments of the three experts for each alternative-criterion pair using the Probabilistic Linguistic Cloud Weighted Averaging () operator, as defined in Eq. (7). The weights assigned to the three experts are 0.4, 0.3,  and 0.3 respectively.

For example, for alternative under criterion :

Thus, the aggregated cloud for is (0.380(0.65), 0.637, 0.162).

The aggregated probabilistic linguistic cloud matrix shown in Table 3.

Table 3.

Aggregated probabilistic linguistic cloud matrix.

Alternative
	(0.380(0.65), 0.637, 0.162)	(0.301(0.67), 0.480, 0.122)	(0.30(0.58), 0.1031, 0.260)	(0.524(0.70), 0.415, 0.195)
	(0.449(0.65), 0.637, 0.162)	(0.359(0.66), 0.461, 0.184)	(0.477(0.69), 0.637, 0.162)	(0.160(0.74), 0.262, 0.211)
	(0.337(0.54), 0.433, 0.179)	(0.272(0.62), 0.505, 0.128)	(0.560(0.73), 0.536, 0.197)	(0.307(0.66), 0.637, 0.162)
	(0.224(0.60), 0.433, 0.179)	(0.470(0.73), 0.334, 0.165)	(0.452(0.60), 0.536, 0.197)	(0.232(0.75), 0.637, 0.162)

Step 3: The ideal positive cloud and the ideal negative cloud using Eqs.(9) and (10) are:

Step 4: Calculate the distance of each alternative from the ideal positive and negative solutions using the formulas in Eqs. (11) and (12).

For example, we compute the distance between the aggregated cloud and the positive ideal cloud :

This distance value of 0.0330 represents the separation between the aggregated assessment and the ideal solution for criterion and alternative .

The resulting individual distances to the ideal positive and negative solutions are presented in Tables 4 and 5.

Table 4.

Solving individual distance to the ideal positive solution.

Alternative
	0.0330	0.0430	0.0387	0.0441
	0.0128	0.0194	0.0348	0.0118
	0.0696	0.0430	0.0000	0.0492
	0.0134	0.0374	0.0334	0.0607

Table 5.

Solving individual distance to the ideal negative solution.

Alternative
	0.0215	0.0228	0.0724	0.0398
	0.0236	0.0126	0.0780	0.0570
	0.1172	0.0278	0.0989	0.0338
	0.0460	0.0480	0.1127	0.0128

Step 5: The weight vector for these four assessment criteria is specified as , and the ideal positive solutions(IPS) and the ideal negative solutions (INS) of each alternative under every criterion using Eqs.(13) and (14) are estimated:

Step 6: Estimate the relative closeness degree for each alternative to the ideal positive cloud using Eq. (15), we obtain:

.

Step 7: In this case, the rank of the alternatives in descending order is .

Sensitivity analysis

A reliable MCDM method must ensure that the final rankings remain invariant and are not influenced by changes in the criteria weights²⁷. Therefore, a sensitivity analysis of the criteria weights is conducted to assess their impact on the ranking outcomes and to validate the stability of the proposed MCDM approach (Sect. "Multi-criteria decision making method based on TOPSIS") under parametric variability.

In the sensitivity analysis, optimization models are constructed for each criterion weight to calculate its lower and upper bounds, ensuring that the ranking outcomes of potential public evacuation plans in response to a possible nuclear leak at the nuclear power plant remain consistent. This process determines the variation interval for each criterion weight, along with the corresponding range length. The weight sensitivity determination method is based on the principle that a wider range of weight variations corresponds to reduced sensitivity in the ranking outcomes. Optimization models are formulated independently for each case.

The functional representations for the nearness degree in relation to the criteria weights are derived using the steps outlined in Sect. "Multi-criteria decision making method based on TOPSIS".

The closeness degrees corresponding to the four alternatives are obtained from the ranking results in step 6.

16

consider the variation range of the weight associated with criteria q as , which preserves the ranking order in Eq. (16). Therefore, the following optimization problems must be satisfied:

17

and.

18

Lingo ver. 21.0 is employed to solve Eqs. (17) and (18), resulting in the following change intervals for the four criteria weights: .

Let the length of the change interval be defined as . Therefore, the following values are obtained:

.

Based on this outcomes, for the rankings in Eq. (16) to remain unchanged, the lengths are required to satisfy the following condition.

.

According to this ranking, the criterion (collective radiation dose) shows the largest range, , indicating that it is the least sensitive to changes in weights and has minimal impact on the overall ranking. Conversely, the criterion (psychological distress) has the smallest range, , indicating that it is the most sensitive to weight changes and has the greatest influence on the ranking outcome. Moreover, the smallest range among the four criteria is , which is approximately 0.3. This indicates that minor variations in the weights of most criteria (less than 0.4) do not significantly affect the ranking. This suggests that the decision making method is robust. The final rankings remain largely stable despite small changes in criteria weights, demonstrating the high stability and reliability of the proposed approach.

To assess how hyper-entropy affects the results, we varied across [0.05, 0.2] while keeping all other parameters constant. The analysis reveals three key insights. First, the top two alternatives ( and ) remain stable across the entire range, confirming that the method’s primary recommendations are robust to hyper-entropy variations. Second, a threshold effect occurs at , where the relative order of and swaps. This occurs because higher hyper-entropy amplifies distance contributions for clouds with larger values (extreme terms), benefiting alternatives that rely less on extreme assessments. Third, for , the complete ranking remains fully stable. Our chosen value falls comfortably within this robust region, confirming its appropriateness.

Comparative analysis

In this subsection, the feasibility and validity of the proposed PLC-TOPSIS method are rigorously evaluated through a comprehensive comparative analysis with four established methods from the existing literature.

The comparison encompasses two categories of methods: (i) cloud-based approach that share our fundamental framework of integrating probabilistic linguistic terms with cloud models, and (ii) advanced PLTS-based methods that represent the state of the art in probabilistic linguistic decision making without cloud aggregation.

Comparison with cloud-based approach

To validate the effectiveness of our proposed methodology, we compare it with existing cloud-based MCDM approach.

We selected Peng et al.²⁰ as it represents the most direct cloud-based predecessor to our work, utilizing a five-parameter cloud representation and the Heronian mean (PLICWHM) aggregation operator. Comparing against this method allows us to demonstrate the advantages of our simplified three-parameter approach. Applying their method to our nuclear evacuation case study yields the ranking as shown in Table 6. This result differs significantly from our proposed ranking .

Table 6.

Comparison results of different MCDM methods.

Method	Type	Ranking of alternatives
Peng et al.20	Cloud-based
Chen et al.22	PLTS-based (PROMETHEE II)
Zhang et al.36	PLTS-based (TODIM)
Sahin et al.18	Subjective (pairwise comparison)
Kavus et al.19	Subjective (direct rating)
Proposed PLC-TOPSIS	Cloud-based

The discrepancy arises from three specific mechanisms in Peng et al.’s approach. First, their five-parameter representation treats entropy and hyper-entropy as intervals rather than precise values. For alternative , which contains extreme assessments (e.g. from Expert 1 under ), this interval expansion artificially amplifies uncertainty, potentially inflating its perceived performance. Second, their Heronian mean aggregation emphasizes interactions between criteria through parameters p, q, which can disproportionately favor alternatives with balanced performance a property that may benefit differently than our neutral weighted averaging. Third, their method involves generating actual cloud drops (N drops) for comparison, introducing stochastic variability into the ranking process, whereas our algebraic approach ensures deterministic results.

Comparison with advanced PLTS-based approaches

To provide a more comprehensive validation, we compare our method with two advanced PLTS-based approaches that utilize fundamentally different methodological frameworks.

We selected Chen et al.²² because it operates directly on PLTS without cloud transformation, using PROMETHEE II an outranking approach rather than a distance based technique. The exact convergence between our results and this fundamentally different method would provides strong external validation for our ranking outcomes. Notably, applying their method yields the ranking , as shown in Table 6, which is identical to the ranking produced by our proposed PLC-TOPSIS method. This exact convergence between two fundamentally different methodologies provides compelling evidence for the validity, robustness, and reliability of our ranking results.

We selected Zhang et al.³⁶ because it incorporates decision-makers’ psychological factors through prospect theory and uses Wasserstein distance within a TODIM framework, representing yet another distinct decision making paradigm. Comparing against it tests whether our distance based approach produces results consistent with psychologically grounded methods. Application of their method yields the ranking , as presented in Table 6. While this result fully agrees with our method regarding the best alternative () and the second-best alternative (), minor differences appear in the lower ranks. These discrepancies can be attributed to the TODIM method’s consideration of psychological factors such as loss aversion. Nevertheless, the consensus on the optimal alternatives reinforces the practical effectiveness of our method.

Comparison with existing subjective methods

To further validate our PLC-TOPSIS framework, we compare it with two subjective approaches: Analytic Hierarchy Process (AHP) and expert-based risk assessment, applied to the same nuclear evacuation case study. The complete comparison results are presented in Table 6.

Using pairwise comparisons from the same three experts on Saaty’s 1-9 scale, AHP produced the ranking , differing from our PLC-TOPSIS ranking (). While both methods identify as optimal, AHP ranks second whereas our method places it last. This discrepancy arises because AHP compresses probabilistic linguistic information into crisp judgments, facing challenges with “the precision and consistency of the assigned expert judgments” as noted by Yazici Sahin and Taskin¹⁸ in their work on decomposed fuzzy AHP, which demonstrates how fuzzy extensions can address such limitations.

Experts rated alternatives on 1-10 scales, yielding , as shown in Table 6. This partial agreement (same top two) contrasts with our full ranking. As Kavus et al.¹⁹ observe in their comparative study of neural network and neuro-fuzzy methods, subjective assessment “heavily relies on the subjective judgments of the assessor, leading to inconsistencies in results, and lacks sensitivity in detecting small changes in risk factors.” Their research further demonstrates that integrating fuzzy logic with traditional assessment methods yields “more accurate” results and provides “greater flexibility in defining which member belongs to which risk level cluster”.

Our PLC-TOPSIS addresses these limitations by preserving probability distributions, incorporating confidence weights, and capturing both fuzziness and randomness through its three-parameter cloud representation.

Discussion and insights

The comparative analysis reveals several key insights:

(i) Methodological validation: The exact correspondence between our proposed method and the PROMETHEE II approach by Chen et al. provides strong external validation for our results. This convergence is especially significant given the fundamentally different mathematical foundations of the two methods, indicating that our cloud-based approach effectively captures the essential information contained in probabilistic linguistic evaluations.

(ii) Addressing limitations of existing cloud methods: The discrepancies observed in the cloud-based methods of Peng et al. and Gong et al. highlight specific limitations that our method successfully addresses: (a) simplifying the cloud transformation process (3 parameters versus 5 parameters), and (b) developing a distance measure capable of handling probabilistic linguistic clouds of varying lengths.

(iii) Practical effectiveness: The agreement on the top alternatives using the psychologically grounded TODIM method demonstrates that our approach yields practically meaningful results consistent with other advanced decision making paradigms.

(iv) Computational efficiency: Our method achieves robust results with significantly reduced computational complexity compared to other approaches, particularly by avoiding the complex calculations required by the Heronian mean operator, dual expectation models, and Wasserstein distance measures.

In conclusion, this comprehensive comparative analysis demonstrates that, although cloud-based methods may yield varying results due to their specific implementation details, our proposed PLC-TOPSIS method produces rankings consistent with those obtained by other advanced PLTS processing techniques. This consistency, coupled with the computational advantages of our method, strongly supports its validity and practical utility for solving complex MCDM problems under probabilistic linguistic uncertainty.

Limitations and future research directions

Despite the advantages of the proposed PLC-TOPSIS framework, several limitations should be acknowledged. First, the method involves subjective parameter selection for domain boundaries and hyper-entropy , which, while justified and tested through sensitivity analysis, still requires domain expertise. Second, the golden section assumption underlying linguistic term transformation, though psychologically grounded, remains a cognitive approximation that may not hold uniformly across all decision contexts. Third, the PLC-WA operator assumes independence among expert assessments, potentially overlooking correlations arising from shared information or group dynamics. Fourth, the distance measure, while mathematically rigorous, introduces complexity that may challenge practitioner understanding. Fifth, the static nature of the framework limits applicability in dynamic environments requiring real-time updates. Finally, empirical validation remains limited to a single case study with three experts, four alternatives, and four criteria; broader testing across diverse domains is needed.

Future research directions are numerous and promising, including: (i) integration with linguistic hypersoft sets for multi-argument approximation of complex criteria interactions; (ii) development of OWA hypersoft set-based aggregation operators to capture optimistic/pessimistic attitudes through ordered weighted averaging; (iii) extension to dynamic decision environments via time-dependent PLTS that update rankings as new information emerges; (iv) automated parameter optimization using machine learning to reduce subjectivity in cloud parameter selection; (v) incorporation of correlated expert assessments through Choquet integrals and non-linear aggregation; (vi) adaptation for large-scale group decision making with consensus-reaching mechanisms and expert clustering; (vii) development of interactive visualization tools to enhance practitioner interpretability; and (viii) cross-domain validation across healthcare, supply chain management, environmental policy, and other fields to test generalizability.

Conclusions

This study successfully develops a novel Probabilistic Linguistic Cloud-TOPSIS (PLC-TOPSIS) framework that effectively bridges the gap between the expressive richness of Probabilistic Linguistic Term Sets (PLTSs) and the computational robustness of cloud models. Our research directly addresses critical limitations identified in the existing literature by introducing a streamlined transformation mechanism based on the golden section technique, which efficiently converts PLTSs into probabilistic linguistic clouds using only three intuitive parameters (). This approach significantly reduces the complexity of the previous five-parameter models while maintaining mathematical rigor. Furthermore, the development of a novel distance measure capable of handling probabilistic linguistic clouds of varying lengths resolves a fundamental constraint that previously restricted the applicability of integrated approaches.

The practical viability and robustness of our proposed framework have been rigorously validated through comprehensive testing, including a high-stakes nuclear emergency evacuation case study that demonstrates its effectiveness in real-world decision making scenarios. Extensive sensitivity analysis confirms the method’s stability against parameter variations, with the largest weight variation interval () indicating minimal sensitivity to input changes. Notably, the expanded comparative analysis, which includes four state-of-the-art methods, provides compelling evidence of our approach’s validity and reliability. The exact convergence between our results and those obtained using the PROMETHEE II method by Chen et al.²², despite fundamentally different mathematical foundations, offers particularly strong validation of our methodological soundness.

However, several limitations should be acknowledged. The method involves subjective parameter selection for domain boundaries and hyper-entropy, relies on the golden section assumption about linguistic perception, and assumes independence among expert assessments. The distance measure, while rigorous, introduces complexity that may challenge practitioner understanding, and the static framework limits applicability in dynamic environments. Furthermore, empirical validation remains limited to a single case study, and broader testing across diverse domains is needed.

Beyond its theoretical contributions, the PLC-TOPSIS framework offers significant practical advantages for decision-makers operating in complex, uncertain environments across diverse domains, including emergency management, healthcare, environmental planning, and supply chain management. Its ability to simultaneously address both fuzziness and randomness in linguistic assessments, coupled with greater computational efficiency compared to existing methods, makes it especially valuable for real-time decision support applications where accuracy and speed are critical.

Future research directions are numerous and promising, including: (i) integration with linguistic hypersoft sets for multi-argument approximation of complex criteria interactions; (ii) development of OWA hypersoft set-based aggregation operators to capture optimistic/pessimistic attitudes through ordered weighted averaging; (iii) extension to dynamic decision environments via time-dependent PLTS that update rankings as new information emerges; (iv) automated parameter optimization using machine learning to reduce subjectivity in cloud parameter selection; (v) incorporation of correlated expert assessments through Choquet integrals and non-linear aggregation; (vi) adaptation for large-scale group decision making with consensus-reaching mechanisms and expert clustering; (vii) development of interactive visualization tools to enhance practitioner interpretability; and (viii) cross-domain validation across healthcare, supply chain management, environmental policy, and other fields to test generalizability. Addressing these limitations will further strengthen the framework’s theoretical foundations and practical utility, advancing both the science and practice of multi-criteria decision-making under uncertainty.

Author contributions

Abdulrahman Almandeel: Methodology, Investigation, Software, Resources, Writing original draft. Congjun Rao: Methodology, Conceptualization, Supervision, Formal analysis. Xiaolong Zhang: Methodology, Visualization, Validation, Writing - review & editing. Hui Qi: Methodology, Validation, Investigation.

Funding

This work was supported by the Planning Fund Project of the Ministry of Education’s Humanities and Social Sciences Research (25YJAZH138), and the Fujian Provincial Natural Science Foundation of China (Nos. 2024J01903, 2025J01393).

Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Declarations

Competing interests

The authors declare no competing interests.

Informed consent

The hypothetical scenario presented in Sect. "Numerical illustration" uses simulated data for methodological demonstration purposes only. No real experts were consulted in this study, and no personal or sensitive information was collected or disclosed. All assessments and evaluations are simulated. As this study involves no human participants, real-world data collection, or personal information processing, informed consent was not applicable.