Fuzzy TOPSIS: Robust Decision-Making in Uncertainty

• Fuzzy TOPSIS is a robust multi-criteria decision-making method that uses triangular fuzzy numbers to capture imprecise, subjective evaluations.
• It systematically normalizes, weights, and aggregates fuzzy assessments to compute closeness to ideal solutions for ranking alternatives.
• The method is widely applied in personnel selection, supplier evaluation, risk analysis, and more, demonstrating high accuracy and resilience.

Fuzzy TOPSIS Method

The Fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method is a robust multi-criteria decision-making (MCDM) framework designed to address imprecise, ambiguous, and subjective information commonly arising in real-world evaluation and ranking scenarios. By integrating the classical TOPSIS paradigm with fuzzy set theory—principally through triangular fuzzy numbers (TFNs)—Fuzzy TOPSIS systematically transforms linguistic, uncertain, and interval-valued input into a quantitative, interpretable ranking that preserves the indeterminate nature of expert judgments and automatic assessments. This methodology has been extensively utilized in diverse domains such as personnel selection, supplier evaluation, risk analysis, group decision making, and human resource management, and is the basis of several modern MCDM extensions including neutrosophic, intuitionistic, and Z-number-based approaches.

1. Fundamental Principles and Mathematical Foundations

At its core, Fuzzy TOPSIS generalizes the crisp TOPSIS method by representing both evaluation scores and criterion weights as triangular fuzzy numbers (TFNs) or, in some variants, as higher-order fuzzy, interval-valued, or neutrosophic numbers. A TFN $\tilde{a} = (l, m, u)$ models imprecision by bounding an assessment between pessimistic ($l$), most likely ($m$), and optimistic ($u$) estimates, with the associated membership function

$$\mu_{\tilde a}(x)= \begin{cases} 0, & x < l, \ \frac{x-l}{m-l}, & l\le x\le m, \ \frac{u-x}{u-m}, & m\le x\le u, \ 0, & x>u. \end{cases}$$

This enables explicit handling of vagueness: for example, linguistic assessments (“poor,” “fair,” “excellent”) are mapped to TFNs, capturing both the degree and range of possible values. Weights $\tilde{w}_j = (l^w_j, m^w_j, u^w_j)$ are similarly fuzzified, typically via expert pairwise comparisons or the Analytic Hierarchy Process (AHP), with subsequent fuzzification.

Benefit and cost criteria are normalized using componentwise functions, and aggregation is performed by componentwise multiplication: $$\tilde{v}_{ij} = \tilde{w}_j \otimes \tilde{r}_{ij} = (l^w_j\,l_{ij},\, m^w_j\,m_{ij},\, u^w_j\,u_{ij}).$$ Closeness to the fuzzy positive ideal solution (FPIS) and negative ideal solution (FNIS) is measured using a symmetric root-mean-square Euclidean metric: $$d_F(\tilde a, \tilde b) = \sqrt{\tfrac13\bigl[(l_a-l_b)^2 + (m_a-m_b)^2 + (u_a-u_b)^2\bigr]}.$$ The closeness coefficient $CC_i = S_i^-/(S_i^+ + S_i^-)$, where $S_i^+$ and $S_i^-$ are aggregated distances to the FPIS and FNIS, is used for final ranking. This structure ensures rankings are robust to the inherent ambiguity of linguistic and subjective input, as demonstrated in personnel selection, risk evaluation, and group decision making (Sodhi et al., 2012, Hoque et al., 30 Jan 2026).

2. Standard Fuzzy TOPSIS Workflow

The canonical Fuzzy TOPSIS procedure, as formalized in "When LLM meets Fuzzy-TOPSIS for Personnel Selection through Automated Profile Analysis" (Hoque et al., 30 Jan 2026), consists of the following sequence:

1. Modeling Inputs as TFNs: Map each alternative–criterion evaluation and criterion weight to TFNs using established linguistic-to-fuzzy mappings or by aggregating expert interval data.
2. Normalization: For benefit-type criteria,

$$\tilde r_{ij} = \left( \frac{l_{ij}}{\sqrt{\sum_{i=1}^n u_{ij}^2}},\, \frac{m_{ij}}{\sqrt{\sum_{i=1}^n m_{ij}^2}},\, \frac{u_{ij}}{\sqrt{\sum_{i=1}^n l_{ij}^2}} \right).$$

Similar, domain-appropriate normalization applies for cost-type criteria.

1. Weighted Fuzzy Decision Matrix: Perform componentwise multiplication of normalized TFNs by the corresponding criterion TFN weights.
2. Identification of FPIS and FNIS:

$$\tilde v_j^+ = \bigl(\max_i\,l_{ij}'',\,\max_i\,m_{ij}'',\,\max_i\,u_{ij}''\bigr), \quad \tilde v_j^- = \bigl(\min_i\,l_{ij}'',\,\min_i\,m_{ij}'',\,\min_i\,u_{ij}''\bigr).$$

1. Distance Computation: Calculate $S_i^+$ and $S_i^-$ as the root sum-of-squares of fuzzy Euclidean distances to FPIS and FNIS.
2. Closeness Coefficient and Ranking: Compute $CC_i$ and rank alternatives in descending order.

Each mathematical step propagates the uncertainty encoded in input TFNs, yielding rankings that are robust against ambiguity, as shown empirically in both synthetic and real-world data ('Uyun et al., 2013, Sadeghi et al., 2021, Gunn et al., 2020).

3. Extensions and Advanced Variants

Numerous generalizations of Fuzzy TOPSIS extend the TFN paradigm to cover additional forms of imprecision, higher-level consensus models, and novel aggregation or similarity measures.

• Interval-Valued and IAA-Fuzzy TOPSIS: Constructs piecewise-constant fuzzy numbers from sets of expert interval judgments using the Interval Agreement Approach (IAA), and employs specialized similarity metrics (ensemble of Jaccard and attribute-comparison) for closeness evaluation instead of strictly geometric distance (Gunn et al., 2020).
• Neutrosophic and SVNS TOPSIS: Extends TOPSIS to single-valued neutrosophic sets, modeling truth, indeterminacy, and falsity independently; aggregation uses SVNWA operators and closeness is measured via a (T, I, F)-triplet Euclidean metric (Şahin et al., 2014, Jiang et al., 2018).
• Intuitionistic and IVIFS TOPSIS: Employs intuitionistic fuzzy values or interval-valued intuitionistic fuzzy sets (IVIFS), with closeness determined by inclusion-based distances and group-weight optimization (e.g., expert and criterion weights determined via quadratic programming) (Hu et al., 2023, Wu et al., 2022).
• Fuzzy Exergy–Entropy TOPSIS: Replaces geometric FPIS/FNIS metrics with exergy (combining consensus and quality of ratings) and entropy (dispersion), ranking alternatives by exergy score (Verma et al., 2015).
• Rough/Fuzzy Covering–Based TOPSIS: Integrates fuzzy covering-based rough sets to determine criterion weights objectively via neighborhood operators and rough approximations, with distances computed via fuzzy-cardinality-weighted Euclidean aggregation (Qi et al., 2022).

These extensions preserve the essential steps of normalization, weighting, and ideal-solution-based ranking, but adapt the underlying data structures and similarity metrics to more accurately model different types of uncertainty or heterogeneity in group decision-making.

4. Integration with LLMs and NLP

The LLM-TOPSIS framework exemplifies the integration of transformer-based NLP models with Fuzzy TOPSIS. In the personnel selection system described in (Hoque et al., 30 Jan 2026), fine-tuned DistilRoBERTa classifiers evaluate LinkedIn profiles on target criteria (e.g., Experience, Skills, Education, About), outputting categorical labels mapped to fuzzy intervals, which are then converted to TFNs via deterministic linguistic–TFN mappings (e.g., “Low” $\rightarrow$ $(0.1, 0.3, 0.5)$). This fusion allows fully automated, scalable candidate ranking that exhibits up to 91% attribute-level accuracy and near-perfect agreement with human-expert rankings (MAP = 0.99, MRR = 0.99, NDCG = 0.926, RMSE = 0.0439, MAE = 0.0368, ranking vector cosine similarity = 0.983). By leveraging the interpretability and fuzziness inherent in TOPSIS, the system addresses key HR challenges in bias reduction, explainability, and resilience to ambiguous data.

5. Representative Applications

Fuzzy TOPSIS and its variants have been employed in a wide array of multi-criteria evaluation scenarios, including:

• Personnel and Scholarship Selection: Ranking candidates by aggregating crisp and linguistic inputs on multiple benefit and cost attributes, as in academic/non-academic scholarship allocation ('Uyun et al., 2013), LLM-personnel selection (Hoque et al., 30 Jan 2026).
• Supplier and Vendor Evaluation: Handling both numerical and linguistic criteria, supporting group decision making with conflicting or uncertain assessments (Jiang et al., 2018, Şahin et al., 2014).
• Risk Analysis and Project Management: Prioritizing risk factors in high-tech projects or complex engineering systems under group uncertainty (Sadeghi et al., 2021).
• Medical and Treatment Decision: IVIFS/intuitionistic/neutrosophic enhancements support selection where evidence is interval-valued or inherently indeterminate (Hu et al., 2023).
• Biosynthetic Material Selection: Application of fuzzy covering-based rough set TOPSIS for nanomaterial choice, leveraging objectively calculated weights (Qi et al., 2022).

In all cases, the method’s explicit propagation of ambiguity yields rankings that reflect both the distribution and the quality of uncertainty in the input data.

6. Performance Metrics and Comparative Studies

Empirical validation of Fuzzy TOPSIS methods typically employs ranking similarity measures (Mean Average Precision, Mean Reciprocal Rank, NDCG, cosine similarity), error metrics (RMSE, MAE), and task-specific accuracy (attribute-level classification agreement). Notably, integration with transformer-based NLP in automated HR frameworks demonstrates human-expert alignment exceeding 91% accuracy, MAP/MRR of 0.99, and high Spearman rank correlations with alternative MCDM methodologies (Hoque et al., 30 Jan 2026, Gunn et al., 2020, Qi et al., 2022). Comparative investigations demonstrate that advanced fuzzy/rough/neutrosophic extensions offer higher robustness to interval uncertainty, indeterminacy, and consensus quality compared to both classical TOPSIS and simpler fuzzy implementations.

7. Limitations and Ongoing Research

The effectiveness of Fuzzy TOPSIS relies on the appropriateness of the fuzzy number mappings, the semantic expressivity of the membership functions, and the normalization/scaling selected for each domain. Potential limitations include increased computational complexity (in the case of higher-order fuzzy or neutrosophic extensions), sensitivity to normalization or weighting functions, and subjective specification of linguistic–fuzzy mappings. Current research directions include dataset expansion, enhancement of model interpretability, practical deployment in operational settings, and formal treatment of aggregation, similarity, and consensus metrics for heterogeneous groups (Hoque et al., 30 Jan 2026). Variable performance across mid-ranked alternatives in some benchmarking studies also suggests value in further integrating explainable AI and optimization-driven attribute weighting (Gunn et al., 2020, Hu et al., 2023).