VIKOR: Multi-Criteria Compromise Decision Method

• VIKOR is a multi-criteria decision-making approach that seeks a compromise solution by balancing group utility and individual regret.
• It utilizes a normalization process comparing best and worst reference values to ensure consistent and fair evaluation across conflicting criteria.
• The method integrates modular strategies and advanced normalization techniques, such as logarithmic normalization, to enhance decision robustness in complex environments.

VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is a reference-type multi-criteria decision-making (MCDM) method that seeks a compromise solution when ranking alternatives with conflicting criteria. Originating from the work of Opricovic (1998), VIKOR is distinguished by its emphasis on balancing overall group benefit and individual regret, making it particularly appropriate in situations where a solution with universally optimal performance is infeasible. VIKOR’s procedure operates over an alternatives-criteria matrix (ACM), systematically comparing each alternative to positive and negative reference solutions to form a robust, rank-ordered compromise (Wang et al., 22 Aug 2025).

1. Principles and Methodological Foundations

VIKOR is built on the premise that decision problems often involve multiple, conflicting criteria for which no alternative simultaneously excels. The method introduces two key performance measures:

• Group Utility (S): Aggregated normalized deviation from the ideal performance across all criteria, representing overall benefit.
• Individual Regret (R): Maximum normalized deviation from the ideal across all criteria, capturing the worst-case shortfall.

Formally, for a set of alternatives $A_1, ..., A_m$ and criteria $C_1, ..., C_n$ with weights $w_j$, the steps are (Wang et al., 22 Aug 2025):

1. Reference values:
   • For each criterion $j$:
     • Maximization: $f_j^+ = \max_i \{f_{ij}\}$; $f_j^- = \min_i \{f_{ij}\}$
     • Minimization: $f_j^+ = \min_i \{f_{ij}\}$; $f_j^- = \max_i \{f_{ij}\}$
2. Normalization of deviations:

$F_{ij} = \frac{f_j^+ - f_{ij}}{f_j^+ - f_j^-}$

1. Aggregation:
   • Group utility:

   $S_i = \sum_{j=1}^{n} w_j F_{ij}$

• Individual regret:

  $R_i = \max_{j} \{ w_j F_{ij} \}$

1. Compromise index:

$$Q_i = v\,\frac{S_i-S^+}{S^- - S^+} + (1-v)\,\frac{R_i-R^+}{R^- - R^+}$$

Where: - $S^+ = \min_i S_i$ - $S^- = \max_i S_i$ - $R^+ = \min_i R_i$ - $R^- = \max_i R_i$ - $v \in [0,1]$ (often $v=0.5$ for balanced compromise)

An alternative with the smallest $Q_i$ is deemed the best compromise option.

2. Reference Solutions and Normalization

VIKOR ranks alternatives by comparing normalized performance to the best and worst (reference) solutions derived from the ACM. The normalization ensures that alternative performances are transformed onto the same scale, accounting for both extremes in the observed data (Wang et al., 22 Aug 2025). This process accommodates both benefit and cost criteria, providing a unified basis for aggregation and comparison.

In contrast to other methods (e.g., TOPSIS, which uses Euclidean distances to both positive and negative ideals), VIKOR’s approach is single-directional but retains the influence of both best and worst values via the normalization denominator.

3. Compromise Ranking: Group Utility and Individual Regret

The dual emphasis of VIKOR lies in its balance between group utility ($S_i$) and individual regret ($R_i$). By combining these into the $Q_i$ compromise index, with the user-selected $v$ parameter, the method allows risk preference tuning (Wang et al., 13 Jul 2024, Levin, 2012). A lower $v$ increases the weight on the worst-case criterion (regret), which is suitable in risk-averse decision environments. Conversely, a higher $v$ prioritizes aggregated performance.

The concept of compromise is operationalized through acceptable advantage and stability assessments. If the top-ranked alternative’s $Q_i$ is not markedly superior, VIKOR may recommend a set of alternatives as plausible compromises (Wang et al., 22 Aug 2025).

4. Modular and Composite Strategy Integration

VIKOR’s modularity enables its use as a component within composite ranking frameworks (Levin, 2012). In such architectures (e.g., Hierarchical Morphological Multicriteria Design, HMMD), VIKOR can be inserted at specific stages—typically the final ranking phase (U stage) or aggregated with other candidate methods (Stage X).

A composite DSS (Decision Support System) can support flexible deployment of VIKOR, offering:

• Interactive module selection for end-users
• Automatic algorithm choice predicated on problem features requiring compromise
• Aggregation of ranking outputs from multiple techniques (including VIKOR) via multiset interval estimates, optimizing robustness and adaptability

Interval multiset estimates, with components $(\eta_1, \eta_2, ..., \eta_l)$ summing votes or satisfaction levels over ordinal scales, are used for quality evaluation of each module and the overall composite strategy:

$$e^I = e^1 \biguplus e^2 \biguplus ... \biguplus e^n$$

Composite strategy synthesis maximizes both total quality and module compatibility.

5. Computational Variants and Enhancements

Normalization is fundamental in VIKOR. Recent research demonstrates the impact of Logarithmic Normalization (LN) compared to traditional linear or vector methods (Zolfani et al., 2020). LN yields normalized values summing to unity:

$$f_{ij} = \frac{\ln(x_{ij})}{\sum_{i=1}^{n} \ln(x_{ij})}$$

LN improves discrimination among near-equal alternatives and enhances rank stability. In sensitivity analyses, LN-based VIKOR produced higher Spearman’s correlation coefficients (SCC) and showed resistance to rank reversal under dynamic changes to the decision matrix.

Hybrid frameworks have been proposed integrating VIKOR with other methods, including TOPSIS and Adversarial Interpretive Structural Modeling (AISM) (Xie et al., 2022). Such integrations exploit strengths of each approach; VIKOR’s compromise ranking ($Q_i$) feeds into interpretable hierarchical relationships (via AISM), while TOPSIS contributes additional distance-based metrics.

6. Comparative Perspective and Limitations

VIKOR is one of several reference-type MCDM methods. Table 7.6 in (Wang et al., 22 Aug 2025) lists characteristic reference treatments among nine approaches. VIKOR stands out for:

• Explicit normalization to both best and worst values
• Twin aggregation measures ($S_i$ and $R_i$)
• Controlled risk preference via parameter $v$

Advantages include computational simplicity, flexibility in risk tuning, and balanced assessment of both overall and criterion-specific performance. However, VIKOR can be susceptible to rank reversal when alternatives are added or removed due to its dependence on extreme values for normalization (Zolfani et al., 2020, Wang et al., 22 Aug 2025). The subjective choice of $v$ also affects results; small changes may shift rankings, especially in closely-matched alternatives.

Compared to distance-based approaches (TOPSIS, GRA, CODAS) and hypervolume-based methods (HVAS) (Deveci et al., 2022), VIKOR offers a clear compromise perspective but shares some limitations in reference dependency and sensitivity to ranking perturbations.

7. Practical Applications and Case Studies

VIKOR has been employed in numerous decision-making contexts, including:

• Resilient supplier selection, energy planning, and product/system optimization (Jiang et al., 2018, Bhatia et al., 2023, Wang et al., 13 Jul 2024)
• Industrial pattern recognition, e.g., identifying optimal melting profiles in induction furnaces with quantifiable improvements in cost and environmental impact (Howard et al., 9 Jan 2024)
• Big data privacy research impact quantification, where VIKOR highlights discrepancies between research output and infrastructure investment (Rebello et al., 2018)
• Evaluation of airline service quality using integrated TOPSIS-VIKOR-AISM frameworks for robust, interpretable ranking (Xie et al., 2022)

VIKOR’s robustness varies with normalization method, weight assignment, and the presence of conflicting objectives. In modular DSS environments, strategy synthesis using VIKOR alongside other ranking techniques enhances decision adaptability and robustness under uncertainty (Levin, 2012).

8. Conclusion

VIKOR is a well-established compromise solution MCDM method, characterized by its dual attention to group utility and individual regret and its flexible incorporation into composite decision frameworks. Its hybridization with normalization advances (such as LN) and synergy with other MCDM approaches make it suitable for a broad array of ranking and sorting tasks with heterogeneous, conflicting criteria. While susceptible to rank reversal and subjective tuning of compromise, VIKOR remains a core approach in reference-type decision analytics, enabling systematic examination of alternatives with both holistic and risk-sensitive perspectives.