MABAC: Multi-Attributive Border Approximation

• MABAC is a multi-criteria decision-making method that constructs a geometric mean-based border using weighted normalized scores to rank alternatives.
• It adapts to uncertainty by incorporating rough numbers and IT2 fuzzy extensions, thereby enhancing evaluation robustness in imprecise environments.
• The method’s practical applications span medical tourism, staff selection, and chemical engineering, showcasing its versatility in diverse decision-making scenarios.

The Multi-attributive Border Approximation Area Comparison (MABAC) method is a reference-type multi-criteria decision-making (MCDM) approach that ranks alternatives by comparing their aggregate performances with a specifically constructed border approximation area, typically derived as the geometric mean of weighted normalized criterion scores. The method is frequently applied in contexts characterized by uncertainty or imprecise evaluation, such as expert-based assessments and decision environments where linguistic or interval-valued data prevail. Owing to its structured, interpretable algorithm and robustness to certain types of perturbations in decision data, MABAC has seen adoption and further extension for various application domains, including medical tourism, candidate selection, and chemical engineering optimization.

1. Foundational Principles and Theoretical Basis

MABAC centers on the construction of a "border approximation area" that serves as a reference solution for all alternatives under each criterion. Unlike methods such as TOPSIS—which rely on positive (ideal) and negative (anti-ideal) solutions—or EDAS, which uses the average solution, MABAC’s reference point is computed as the geometric mean of each criterion’s weighted normalized scores across all alternatives. This geometric mean acts as a “midpoint” or border, allowing an explicit delineation between alternatives performing above or below the established baseline.

Each alternative’s performance is evaluated by aggregating its signed distances from the border under each criterion. A positive difference signals membership in the upper approximation area (above the benchmark), while a negative difference denotes the lower approximation area (below the benchmark). The sum of these distances across criteria yields the final performance score for each alternative, which determines the ranking order (Wang et al., 22 Aug 2025).

2. Canonical Algorithm and Mathematical Framework

The standard MABAC procedure consists of systematic steps:

1. Normalization: For each alternative $i$ and criterion $j$, raw scores $f_{ij}$ are normalized to $F_{ij} \in [0,1]$ using Max–Min scaling. For benefit (maximization) criteria, $$F_{ij} = \dfrac{f_{ij} - \min_k f_{kj}}{\max_k f_{kj} - \min_k f_{kj}}$$; for cost (minimization) criteria, $$F_{ij} = \dfrac{\max_k f_{kj} - f_{ij}}{\max_k f_{kj} - \min_k f_{kj}}$$.
2. Weighted Normalized Matrix: To every normalized value, add 1 (to avoid zero entries) and multiply by its criterion weight $w_j$: $V_{ij} = (1 + F_{ij}) \cdot w_j$.
3. Border Approximation Area Calculation: For each criterion, compute the geometric mean across alternatives: $b_j = \left(\prod_{i=1}^m V_{ij}\right)^{1/m}$, for $m$ alternatives.
4. Performance Score: For each alternative $i$, calculate the cumulative signed deviation from the border: $P_i = \sum_{j=1}^n (V_{ij} - b_j)$.

The highest $P_i$ corresponds to the most preferred alternative. This calculation approach ensures computational simplicity and interpretability while using the geometric mean border as the performance baseline (Wang et al., 22 Aug 2025).

3. Treatment of Uncertainty: Rough Numbers and Fuzzy Extensions

MABAC is adaptable to decision environments characterized by imprecision in the evaluation process. Two main uncertainty modeling strategies have been demonstrated:

• Rough Number-Based MABAC: Decision makers may provide assessments as intervals ("rough numbers"), $\left[x_{ij}^L, x_{ij}^U\right]$, to capture lower and upper judgments for each alternative-criterion pair. All arithmetic operations (normalization, weighting, geometric mean calculation, and score aggregation) are performed on interval-valued data, relying on rough set theory for interpretation. This method was successfully demonstrated in selecting medical tourism destinations, with both criteria weights (obtained via rough AHP) and performance data expressed as rough numbers. The resulting framework enables aggregation of interval-valued and subjective data without recourse to membership functions as required by fuzzy set theory (Roy et al., 2016).
• IT2 Trapezoidal Fuzzy Number-Based MABAC: Uncertainty and vagueness can also be handled by using interval type-2 trapezoidal fuzzy numbers (IT2TrFNs) for both criteria weights and performance evaluations. Decision makers’ linguistic evaluations are mapped to corresponding IT2TrFNs, and all computational steps (including weighted normalization and geometric mean computation) are implemented with appropriate IT2TrFN arithmetic. The geometric Bonferroni mean operator is used for aggregation. This approach further enhances the model’s ability to represent both fuzziness and the footprint of uncertainty, as evidenced in candidate selection problems (Roy et al., 2016).

These extensions preserve the interpretability of the border approximation concept while increasing the method’s applicability in subjective, data-sparse, or inherently ambiguous assessment contexts.

4. Application Scenarios and Illustrative Cases

MABAC, including its rough and fuzzy extensions, has been deployed in a variety of real-world settings:

• Medical Tourism Site Selection: In the assessment of nine Indian cities as potential medical tourism destinations, a hybrid rough AHP–MABAC framework was employed. Expert linguistic judgments were converted to rough intervals, and the cities were ranked by proximity (via Euclidean distance) to the rough border approximation area. The process enabled the diagnosis of specific criteria contributing to each city’s performance gap, supporting actionable improvement recommendations (Roy et al., 2016).
• Staff Selection: An extended MABAC using IT2 trapezoidal fuzzy numbers was used to select a system analysis engineer out of several candidates based on multiple soft-skill attributes. Decision makers’ linguistic ratings were systematically converted to IT2TrFNs, with intermediate performance diagnostics provided for each criterion. The method was benchmarked against alternatives (TOPSIS, methods by Gong et al.), showing consistent outcomes with simpler computations (Roy et al., 2016).
• Chemical Engineering Optimization: MABAC has been analyzed for its robustness in ranking Pareto-optimal solutions in process optimization, including applications such as co-gasification, olefin production, and material selection. Its resistance to unit-conversion (linear transformation of objectives), objective reformulation, and alternative removal—compared to other MCDM methods—was found to be significant when using appropriate weighting schemes (Nabavi et al., 18 Mar 2024).

These cases underscore the method’s versatility and transparent diagnostic capability, particularly in contexts where explicit visualization of the margin by which each alternative exceeds or falls short of a calibrated baseline is critical.

5. Sensitivity, Robustness, and Comparative Performance

The stability and reliability of MABAC rankings were extensively evaluated under systematic modifications of the decision matrix:

• Linearity and Robustness: MABAC typically exhibits minor rank changes in response to linear transformations of objectives (e.g., change of measurement units) and removal of alternatives (e.g., sampling or incomplete data). The average absolute fractional deviation (AAFD) for top alternatives remains low, and Spearman’s $\rho$ between original and perturbed rankings approaches unity under such changes (Nabavi et al., 18 Mar 2024).
• Objective Reformulation: MABAC is more sensitive to reciprocal transformations (e.g., switching from maximization to minimization using $y = 1/(x + A)$). Although this sensitivity is a universal effect across reference-type methods, MABAC was shown to be comparatively less sensitive, particularly with entropy or CRITIC weighting.
• Comparison with Other Methods: MABAC’s ranking response under perturbations closely aligns with reference-type methods like CODAS and GRA when using robust weighting. Its use of the geometric mean border often imparts greater robustness to outliers compared to methods that rely exclusively on extreme values (e.g., TOPSIS).

This suggests MABAC is suitable for applications where ranking stability is paramount and minor modifications of the evaluation matrix are expected, although the possibility of rank reversal persists if substantially different or additional alternatives are introduced (Wang et al., 22 Aug 2025).

6. Advantages, Limitations, and Methodological Context

Attribute	MABAC	Commentary
Reference solution	Geometric mean border	Intermediate, less extreme
Computational steps	Simple, arithmetic	No distance computation required
Interpretability of criteria margin	High; direct above/below border	Visualization of performance gap
Sensitivity to outliers	Moderate	Less than min-max-based methods
Susceptibility to rank reversal	Present	As with most reference methods

Significant advantages:

• Streamlined calculation without the need for Euclidean distance computation or handling of positive/negative ideal solutions (Wang et al., 22 Aug 2025).
• Diagnostic insight into which criteria drive final rankings via the explicit evaluation of distances from the border area.
• High adaptability to settings with interval or fuzzy-valued data, supporting nuanced modeling of decision environments marked by subjectivity or limited data (Roy et al., 2016, Roy et al., 2016).

Principal limitations:

• Abstractness of the border approximation area, which may lack the intuitive appeal of explicit best/worst-case referents for practitioners.
• Geometric mean’s continued—though mitigated—sensitivity to outlier performance values.
• Rank reversal risk, notably if the set of alternatives changes, as the geometric mean border is recalculated and may shift the relative distances (Wang et al., 22 Aug 2025).

7. Extensions, Implications, and Future Directions

MABAC’s development has been paralleled by efforts to integrate advanced weighting schemes (e.g., entropy, CRITIC, AHP, DEMATEL) and uncertainty modeling tools (rough numbers, IT2TrFNs), expanding its scope to more complex and high-stakes settings. A plausible implication is that future enhancements may involve hybridization with other MCDM schemes or further aggregation strategies for both reference area and alternative scoring.

The method’s computational transparency, combined with proven robustness to standard matrix modifications, positions it advantageously for engineering design, resource allocation, and expert-driven decision scenarios where diagnostic clarity regarding the performance of alternatives is critical. Nevertheless, the persistent possibility of rank reversal mandates controlled application, possibly in conjunction with supplementary MCDM methods to validate decision robustness across reference perspectives (Nabavi et al., 18 Mar 2024, Wang et al., 22 Aug 2025).