Cross-Efficiency DEA Analysis

• Cross-Efficiency DEA is a robust performance evaluation method that integrates self-appraisal with peer-based efficiency scoring to provide discriminatory and actionable rankings.
• It employs advanced linear programming formulations to mitigate weight multiplicity and identify maximal closest reference sets for realistic target setting.
• Extensions such as fuzzy, interval, and two-stage DEA enhance its applicability by addressing uncertainty and ensuring fair benchmarking under diverse operational conditions.

Cross-Efficiency Data Envelopment Analysis (DEA) is a robust performance evaluation methodology extending classical DEA by incorporating peer-appraisal based efficiency scoring. Unlike standard DEA, which only uses self-selected weights to calculate an individual decision making unit’s (DMU's) efficiency, cross-efficiency DEA systematically applies optimal multipliers derived from every DMU to all DMUs in the sample, providing a comprehensive, discriminative, and practically actionable benchmark structure for relative efficiency, target setting, and ranking.

1. Conceptual Foundation of Cross-Efficiency DEA

Cross-efficiency analysis generalizes the classical DEA framework by supplementing self-appraisal with mutual appraisals. In a classical input-oriented multiplier DEA model, each DMU selects input and output weights to maximize its own efficiency score, which is always 1 for efficient DMUs. This structure inhibits discrimination, since multiple DMUs can receive a maximal score without peer comparison.

Cross-efficiency addresses this limitation by evaluating every DMU using the optimal weights of every other DMU. For a set of n DMUs, the resulting cross-efficiency matrix $E$ has entries $E_{ok}$ evaluating DMU k using weights derived from DMU o. The average cross-efficiency score aggregates the peer evaluations to yield a more discriminative ranking and benchmarking landscape.

Mathematically, for DMU $o$ (refer to (Bolos et al., 4 Jun 2025)), the cross-efficiency score for DMU $k$ is computed as:

$$E_{ok} = \frac{\mathbf{u}_o^*\,\mathbf{y}_k + L\,\xi_{Lo}^* + U\,\xi_{Uo}^*}{\mathbf{v}_o^*\,\mathbf{x}_k}$$

where $\mathbf{u}_o^*, \mathbf{v}_o^*$ are optimal outputs and input multipliers, $\xi_{Lo}^*, \xi_{Uo}^*$ are returns-to-scale constraint multipliers, and $L, U$ specify RTS modeling choices.

The overall cross-efficiency score for DMU $k$ is:

$e_k = \frac{1}{n} \sum_{o=1}^{n} E_{ok}$

The Maverick index $M_k$ quantifies deviation of self-appraisal from peer mean:

$M_k = \frac{E_{kk} - e_k}{e_k}$

This framework enables both peer-based discrimination and internal consistency analysis of DMUs.

2. Linear Programming Models and Methodological Advances

Recent contributions have expanded the theoretical and computational foundations of cross-efficiency DEA. Notably, alternative LP formulations have been proposed to handle multiplicity and aggressivity/benevolence in weight assignment (Bolos et al., 4 Jun 2025, Roshdi et al., 2014). For instance, Method II (aggressive input-oriented) constrains the optimization to minimize peer DMUs' input products:

$$\begin{array}{rl} \text{minimize} & -\mathbf{v}\sum_{k\neq o}\mathbf{x}_k + \mathbf{u}\sum_{k\neq o}\mathbf{y}_k + (n-1)(L \xi_L + U \xi_U) \ \text{subject to} & \mathbf{v}\mathbf{x}_o = 1,\ -\mathbf{v}X + \mathbf{u}Y + (\xi_L + \xi_U)\mathbf{e} \leq \mathbf{0},\ & \mathbf{u}\mathbf{y}_o + L\xi_L + U\xi_U = E_{oo},\ \mathbf{v}, \mathbf{u} \geq 0,\ \xi_L \geq 0,\ \xi_U \leq 0, \end{array}$$

Benevolent formulations reverse the objective’s sign and enforce peer-benefiting multiplier selection.

Alternative robust cross-efficiency frameworks have been proposed to address non-uniqueness in multipliers, including interval-based cross-efficiency (Akbarian, 2014), use of strong complementary slackness conditions in reference set identification (Roshdi et al., 2014), and the virtual gap analysis model (Liu et al., 2023), which redefines benchmarking via systematically determined virtual weights and incorporates scale corrections for incomplete slack aggregation.

3. Maximal Closest Reference Set (MCRS) and its Role in Cross-Efficiency

An important recent development is the linear programming-based identification of maximal closest reference sets (MCRS) (Roshdi et al., 2014). Traditional DEA reference sets often identify "furthest" reference DMUs, leading to impractically large improvement targets. In contrast, MCRS seeks the maximal set of closest reference DMUs requiring minimal input reduction and output augmentation:

• Stage 1: Solve an adapted additive LP to find closest Pareto-efficient projection $P = (x^P, y^P) = (x_0 - S^*_{-}, y_0 + S^*_{+})$ for inefficient DMU.
• Stage 2: Apply a supporting hyperplane LP (Model 4) to select all DMUs lying exactly on the hyperplane through $P$ via positive multipliers $\lambda_j > 0$.

This approach ensures that the cross-efficiency analysis, when performed using MCRS, delivers reference sets more relevant for benchmarking, more sensitive to minor operational changes, and robust against over-ambitious efficiency targets.

4. Extensions: Fuzzy, Interval, and Two-Stage Cross-Efficiency Analysis

Advanced DEA variants extend cross-efficiency analysis to uncertain environments and multi-stage processes.

• Fuzzy cross-efficiency, as implemented in the deaR package (Bolos et al., 4 Jun 2025), handles uncertainty in DMU data by adopting possibilistic and interval-based multipliers, facilitating analysis where inputs/outputs are expert estimates or imprecise.
• Interval cross-efficiency (ICE) and interval AHP (IAHP) integrate uncertainty bounds into the evaluation and ranking matrix, providing robust efficiency intervals for each DMU (Akbarian, 2014).
• In two-stage production systems, a transformation splits each DMU into stage-based sub-DMUs, yielding a $2n \times 2n$ cross-efficiency matrix with block diagonal structure (Wang et al., 13 Sep 2024). Cooperative game theory is then used for stage-wise revenue allocation, leveraging Shapley value, nucleolus, and least core solution concepts. This structure ensures fairness and invariance in allocation across direct and secondary modes.

5. Computational Improvements and Scalability

Efficient large-scale cross-efficiency DEA computation is supported by reference-searching algorithms (Chen, 2017). These methods iteratively search for minimal supporting DMU subsets (references), solving small-scale LPs whose cardinality never exceeds dimension $m + n$. This reduces computational complexity by up to 3–3.8x in large, high-density scenarios. The process uses dual screening conditions to guarantee correctness and uniqueness in multiplier selection, which is particularly advantageous in cross-efficiency contexts requiring multiple peer evaluations.

6. Benchmarking, Ranking, and Practical Implications

Cross-efficiency DEA produces peer-driven, discriminative rankings, mitigating the limitation of maximal self-appraisal scores (ties). Applications span faculty appraisal (Oukil, 2021), mutual fund benchmarking (Chopra, 2020), resource-performance trade-off assessment in natural LLMs (Zhou et al., 2022), and fairness evaluation in healthcare allocation (Kaazempur-Mofrad et al., 18 Sep 2024). Benchmarking via cross-efficiency is sensitive to orientation, returns-to-scale assumptions, and peer sample composition.

The approach allows:

• Identification of mavericks (DMUs with significant self-peer deviation)
• Dynamic peer group benchmarking (adjusting efficiency as the sample changes)
• Realistic target setting for inefficient units, especially by leveraging MCRS
• Robust evaluation under uncertainty (fuzzy/interval/fuzzy DEA)
• Structured allocation and incentive schemes in cooperative, multi-stage organizations

7. Future Directions

Current research is targeting:

• Extension of MCRS and virtual gap analysis to variable returns-to-scale settings
• Integration of cross-efficiency with hybrid metaheuristics and multi-objective evolutionary algorithms (e.g., COA-DEA (Gorjestani et al., 2015))
• Advanced sensitivity, stability, and ranking systems for DEA-based efficiency analysis
• Deeper application in multi-criteria, distributed decision-making frameworks, and further development of benchmarking protocols in large networked organizations

Cross-efficiency DEA remains at the forefront of evaluating and ranking performance across heterogeneous, multi-agent environments where fairness, discrimination, and peer-based benchmarking are operationally critical.