TOPSIS: Multi-Criteria Decision-Making

• TOPSIS is a multi-criteria decision-making method that evaluates alternatives based on their geometric closeness to ideal and nadir points.
• The methodology involves constructing a decision matrix, applying normalization and weighting, and calculating Euclidean distances to derive a closeness coefficient.
• Widely applied in engineering, service quality, and network optimization, TOPSIS balances computational simplicity with adaptability, despite sensitivity to normalization and weight choices.

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a foundational multi-criteria decision-making (MCDM) method based on the principle that the most desirable alternative should have the shortest geometric distance from the ideal solution and the greatest distance from the negative-ideal solution. Developed for quantitative ranking tasks, TOPSIS facilitates decision-making in environments characterized by multiple, potentially conflicting criteria. The method has wide applicability, including industrial optimization, service quality evaluation, intelligent system configuration, and multi-objective optimization across diverse engineering domains.

1. Mathematical Principles and Computational Workflow

TOPSIS is structured as a deterministic, distance-based approach with the following canonical steps:

1. Construction of the Decision Matrix: For $m$ alternatives (e.g., products, strategies) and $n$ criteria (features or objective metrics), the matrix $X = [x_{ij}]$ encapsulates all performance data.
2. Normalization: To address disparate units and ranges among criteria, vector normalization is applied:

$$r_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$$

1. Weighting: Each criterion is assigned a non-negative weight $w_j$ such that $\sum_j w_j = 1$. The normalized matrix is scaled as $v_{ij} = w_j \cdot r_{ij}$.
2. Ideal and Negative-Ideal Solutions: For each criterion $j$, define
   • $v_j^+ = \max_i v_{ij}$ (or $\min_i v_{ij}$ for cost or minimization criteria) as the positive ideal,
   • $v_j^- = \min_i v_{ij}$ (or $\max_i v_{ij}$ for cost) as the negative ideal.
3. Distance Calculation: The Euclidean distance for each alternative $i$ to the ideal and negative-ideal points is computed:

$$D_i^+ = \sqrt{ \sum_{j=1}^n (v_{ij} - v_j^+)^2 }, \quad D_i^- = \sqrt{ \sum_{j=1}^n (v_{ij} - v_j^-)^2 }$$

1. Closeness Coefficient and Ranking: The closeness coefficient is defined as

$C_i = \frac{D_i^-}{D_i^+ + D_i^-}$

Ranking is effected by sorting $C_i$ in descending order; a value closer to 1 indicates greater proximity to the ideal.

2. Recent Algorithmic Innovations and Theoretical Extensions

Multiple studies have sought to extend the classical TOPSIS construct to enhance robustness, adaptability to uncertainty, or group decision-making capabilities.

• Neutrosophic and Fuzzy Generalizations: Group decision-making under vagueness may encode responses as single valued neutrosophic numbers, $(a,b,c)$ denoting membership, indeterminacy, and non-membership degrees. Aggregation, weighting, and distance computation then proceed in a multi-dimensional space, and the final closeness coefficient incorporates Euclidean distances across all components (Şahin et al., 2014).
• Cloud Models and Interval Data: To incorporate epistemic uncertainty, each criterion–alternative evaluation may be cast as a cloud model parameterized by expectation, entropy, and hyper-entropy, allowing both performance and uncertainty to be considered. The aggregation of interval assessments is followed by a bilevel optimization to determine optimal criteria weights. Custom distance metrics compare clouds, and the ranking uses a generalized closeness coefficient (Khorshidi et al., 2020).
• Thermodynamic Analogies: Some research connects TOPSIS to thermodynamical indicators—energy (quantity), exergy (quantity × quality), and entropy (dispersion of expert ratings)—highlighting that classical TOPSIS essentially ranks by an energy-like metric. Incorporating a "quality" factor, as in exergy, provides a more reliable assessment of each alternative's value by penalizing inconsistent ratings (Verma et al., 2015).
• Mean–Standard Deviation Handling in Algorithm Rankings: The A-TOPSIS extension applies the TOPSIS procedure separately to both the means and standard deviations of algorithmic results, allowing the aggregation of these two dimensions via assigned weights prior to final ranking and thus capturing the dual desiderata of high expected performance and low variability (Pacheco et al., 2016).
• Robustification against Degeneracies: To avoid computational failures arising from invariant criteria or universally poor alternatives, variants such as CoCoFISo adopt the vector normalization of TOPSIS, ensuring existence and consistency of the normalized decision matrix even with degenerate input data. Aggregation steps are modified to remain well-defined in such cases (Rasoanaivo et al., 22 Apr 2024).

3. Applications Across Domains

TOPSIS has been employed across a vast landscape of practical MCDM scenarios:

• Intelligent Network and Handover Decision Schemes: In heterogeneous wireless networks, distributed TOPSIS computation swiftly selects the optimal visited network for vertical handover by evaluating alternatives on criteria such as bandwidth, delay, jitter, and cost. TOPSIS enables the offloading of computation to prospective networks, reducing processing delay and supporting seamless connectivity (Savitha et al., 2011).
• Service Quality Assessment: Integrated frameworks (e.g., TOPSIS-VIKOR-AISM) leverage the distance-based evaluation of TOPSIS alongside compromise programming for airline service quality ranking. By combining Euclidean distance measures, compromise solutions, and hierarchical interpretive modeling, the framework supports robust and multi-dimensional evaluation, tiered ranking, and stability analysis (Xie et al., 2022).
• Optimization of Engineering Systems: In process engineering, TOPSIS is often used as an MCDM post-optimizer in NSGA-II-driven multi-objective workflows, as seen in the selection of operating parameters for steam methane reforming reactors or membrane reactors in ethylene production. The immediate result is the selection of Pareto-optimal solutions that are quantitatively closest to all desired objectives, as determined by the calculated closeness coefficient (Nabavi et al., 10 Jul 2025, Nabavi et al., 15 Jul 2024).
• Dynamic Decision-Support in Intelligent Systems: In real-time performance status sensing for mobile devices, TOPSIS combined with entropy-based weight calculation and principal component analysis allows for dynamic device ranking according to real-time criteria such as resource utilization, latency, and throughput, with implications for adaptive power management and user experience optimization (Wang et al., 5 Sep 2024).
• Layout and Resource Optimization: In multi-objective engineering design such as animal shelter layout optimization, the integration of TOPSIS with evolutionary algorithms and graph-theoretic accessibility scoring allows simultaneous optimization over accessibility, capacity, and welfare-linked spatial features (Jalayer et al., 23 May 2024).

4. Strengths, Limitations, and Comparative Methodological Analysis

TOPSIS offers several advantages as an MCDM tool:

• Computational Simplicity: The algorithm is algebraic, deterministic, and computationally efficient, making it suitable for repeated or embedded decision processes.
• Intuitive Rationale: The geometric interpretation (distance from ideal/nadir) aligns closely with practitioner understanding.
• Scalability and Integration: TOPSIS normalizes for units, is robust to the number of alternatives and criteria, and integrates naturally with methods for weighting (entropy, ANP, FAHP) and extensions for group decision-making or uncertainty quantification (Lahby et al., 2012, Lahby et al., 2012, Khorshidi et al., 2020).

However, several limitations are noted:

• Sensitivity to Normalization and Scale: Classical TOPSIS may be sensitive to normalization procedures or weights, with modifications such as that in CoCoFISo addressing specific normalization pathologies (Rasoanaivo et al., 22 Apr 2024).
• Potential for Rank Reversal: In some cases, the removal of low-ranked alternatives may disrupt rank order ("ranking abnormality"), motivating hybridizations with methods such as ANP (Lahby et al., 2012).
• No Direct Aggregation of Extreme Poor Performance: The original method can fail when an alternative is universally poor or if a criterion is perfectly invariant; robust extensions remediate this (Rasoanaivo et al., 22 Apr 2024).
• Linearity Assumptions: The Euclidean distance framework is inherently linear in the criteria, which may not capture more complex nonlinear preference structures.

Comparisons to other MCDM techniques show that while methods such as Simple Additive Weighting (SAW) or PROBID (Preference Ranking on the Basis of Ideal-Average Distance) offer alternatives based on weighted sums or ideal-average distances, TOPSIS's dual focus on both best and worst benchmarks provides a more nuanced “closeness” metric. Nonetheless, these methods generally yield highly correlated rankings when objective normalization and weighting are consistently applied (Nabavi et al., 15 Jul 2024, Nabavi et al., 10 Jul 2025).

5. Integration with Hybrid and Multi-Stage Frameworks

Recent trends emphasize embedding TOPSIS within hybrid frameworks to enhance decision robustness or optimize complex systems:

• Hybridization with Evolutionary Algorithms: TOPSIS can act as a scalarizer within evolutionary or metaheuristic optimization (e.g., genetic algorithms, differential evolution), systematically transforming multi-objective spaces into ranked, single-objective evaluations at each iteration (Krohling et al., 2022, Jalayer et al., 23 May 2024).
• Confirmatory Forecasting in Financial Applications: In stock trading, a hybrid system may first rank opportunities via TOPSIS using technical indicators and then confirm the selection via secondary time series forecasting (EMD + ELM), increasing the reliability of the trading decision (Ebermam et al., 2022).
• Combination with Structural and Interpretive Models: In complex service sector assessment, combining TOPSIS with VIKOR (compromise programming) and AISM (adversarial interpretive structural modeling) facilitates both robust ranking and hierarchical interpretation of alternatives and indicator relationships (Xie et al., 2022).

6. Impact, Prospective Directions, and Generalization

TOPSIS continues to be a preferred MCDM method in operational, engineering, and service evaluation domains due to its interpretability, methodical extensibility, and robust integration prospects with both optimization and AI frameworks. With the expansion into uncertainty-handling, group decision-making, and real-time adaptive applications—such as smart monitoring, resource allocation, and intelligent network management—TOPSIS remains a central construct.

Key areas for further research and application development include:

• Algorithmic adaptation for real-time and edge computing environments, where efficient, parallelizable distance computation is required.
• Generalization for high-dimensional or tensor-valued criteria, particularly in settings involving multidimensional or time-dependent decision inputs.
• Formal frameworks for uncertainty, ambiguity, and aggregation in group decisions, building upon neutrosophic, fuzzy, and cloud-based mathematics.
• Transdisciplinary use in explainable AI by relating distance-based rankings to interpretable performance and sensitivity analyses.

Through ongoing methodological refinement and application, TOPSIS offers a mature yet flexible foundation for high-dimensional, multi-criteria decision analysis in both established and emergent technical fields.