Super Efficient DEA Model

• The paper introduces super-efficient DEA models that refine rankings by resolving ties among CCR or SBM efficient DMUs with a min-second-best approach.
• It details mathematical formulations, dual equivalences, and numerical illustrations that enhance the discriminatory power of standard DEA techniques.
• The study emphasizes practical insights including algorithm implementation, handling outliers, and addressing infeasibility issues for robust efficiency analyses.

Super Efficient Data Envelopment Analysis (DEA) models provide refined, discriminatory rankings among decision-making units (DMUs) that are efficient under standard DEA, resolving ties and addressing interpretational ambiguities found in classical CCR, BCC, and original SBM frameworks. The following sections detail mathematical formulations, model equivalences, theoretical properties, numerical illustrations, implementation guidance, and recent advances in continuous and directional super-efficiency approaches.

1. Mathematical Formulations of Super-Efficiency Models

Super-efficient DEA models derive from the need to differentiate among CCR- or SBM-efficient DMUs, whose classical DEA scores are uniformly 1. The output-oriented “minimize second-best” model (Kitahara et al., 2024) posits:

$$\begin{aligned} \min_{u,v,t}\quad & t\ \text{s.t.}\quad & x_{o}^{T}\,u \;=\; t,\ & y_{o}^{T}\,v \;=\; 1,\ & Y_{-o}\,v \;\le\; X_{-o}\,u,\ & u\;\ge0,\ v\;\ge0. \end{aligned} \tag{P}$$

This program selects input and output weights $(u,v)$ that retain ${\rm DMU}_o$ efficient (score = 1) while forcing the largest score for all others to be minimized. The optimal $t^*$ equals the highest efficiency (ratio $y^T v / x^T u$) among all peer DMUs, and its reciprocal $\eta^* = 1/t^*$ is the Andersen–Petersen super-efficiency score.

The dual (Kitahara et al., 2024):

$$\begin{aligned} \max_{\lambda,\eta}\quad & \eta\ \text{s.t.}\quad & X_{-o}\,\lambda \;\le\; x_{o},\ & Y_{-o}\,\lambda \;\ge\; \eta\,y_{o},\ & \lambda\;\ge0, \end{aligned} \tag{S}$$

where DMU o is omitted from the reference set, aligns directly with the classical Andersen–Petersen super-efficiency model.

Slacks-based super-efficiency (SBM) models generalize this to non-radial settings (Bolos et al., 2024, Liao et al., 2023):

$$\delta^*(x,y) = \min_{\lambda\ge0,\;t^-\ge0,\;t^+\ge0} \frac{1+\tfrac1m\sum_{i=1}^m \frac{t^-_i}{x_i}} {1-\tfrac1s\sum_{r=1}^s \frac{t^+_r}{y_r}} \quad \text{s.t.} \begin{cases} x + t^- \ge X\lambda, \ 0 < y - t^+ \le Y\lambda. \end{cases}$$

Directional super-efficiency models, such as Generalized Directional Super-Efficiency (GDSE) (Mehdiloozad et al., 2014), resolve infeasibility and offer complete ranking via:

$$\begin{aligned} \min_{\tau,\{t_i\},\{u_r\},\{\lambda_j\}} & 1+\frac{1}{m}\sum_{i=1}^m t_i + \frac{1}{s}\sum_{r=1}^s u_r \ \text{s.t.} & \sum_{j \in J} \lambda_j x_{ij} \leq x_{i0} + \tau g^-_i - t_i g^-_i, \ & \sum_{j \in J} \lambda_j y_{rj} \geq y_{r0} - \tau g^+_r + u_r g^+_r, \ & L \leq \sum_{j \in J} \lambda_j \leq U, \ \lambda_j \geq 0,\ t_i \geq 0,\ u_r \geq 0,\ \tau \in \mathbb{R}. \end{aligned}$$

2. Equivalence, Uniqueness, and Discriminatory Power

The “minimize second-best” model's primal (P) and its dual (S) are exactly equivalent by LP duality (Kitahara et al., 2024). At optimality, $t^* = 1/\eta^*$, with $(u^*,v^*)$ and $(\lambda^*,\eta^*)$ providing the same discriminative ranking.

Super-efficiency breaks the score tie present in CCR/BCC models by “pushing down” the nearest competitor and enabling strict ranking across all efficient DMUs. The super-efficiency score quantifies the highest permissible ratio that still makes the DMU dominate peers, providing actionable separation for strategic management.

Uniqueness of weights is generally not guaranteed; however, minimizing the second-best tends to yield more focused, sometimes unique, weight solutions, critical for managerial interpretation.

3. Key Properties and Theoretical Insights

Feasibility: Both output-oriented super-efficiency programs are feasible iff the target DMU is CCR-efficient. For CCR-inefficient units, enforcing efficiency dominance induces infeasibility ($t^* > 1$) (Kitahara et al., 2024).

Discrimination: While CCR and SBM assign efficiency score $=1$, super-efficiency scores $\eta > 1$ strictly rank efficient DMUs. The “second-best” score quantifies the largest efficiency score achievable by any competitor under the selected weights, thus signaling the true “gap” between top performer and runner-up (Kitahara et al., 2024).

Invariance and Monotonicity: Directional and SBM-based super-efficiency models demonstrate monotonicity, unit, and translation invariance under proper choice of direction vector and weighting (Mehdiloozad et al., 2014, Bolos et al., 2024).

Pitfalls: Super-efficiency is sensitive to outliers, may suffer infeasibility under VRS, and can overestimate efficiency when projections are weakly efficient (SBM); composite scores have been proposed to regain continuity and correct ranking (Bolos et al., 2024).

4. Application: Banking Case Studies and Domain Extensions

Multiple studies exemplify super-efficient DEA in financial institutions (Kitahara et al., 2024, Liao et al., 2023):

Study	Context	Model	Key Outputs
Kitahara & Tsuchiya	Japanese banks, 2016	Min-second-best (super-eff)	$\eta^*=1.389$ for Bank of Yokohama
Wang et al	Chinese listed banks	Super-SBM-UND-VRS	State-owned banks most super-efficient

In the Japanese bank dataset, applying the “min-second-best” approach led to unique weight allocation and discriminated the top DMU (Yokohama) with $\eta^*>1$ while all other banks scored below $0.720$, clearly displaying a performance gap (Kitahara et al., 2024).

Other sectors (healthcare, education, supply chain) commonly leverage super-efficiency for peer ranking, stress testing (“stretch factor” $\eta$ analysis), and sensitivity studies on input/output adjustments (Kitahara et al., 2024).

5. Implementation: Algorithms, Software, and Model Variants

Practitioners implement super-efficient DEA models using standard LP solvers:

• Generic LP: Gurobi, CPLEX, XPRESS, GLPK, COIN-OR (CBC, CLP)
• DEA-specific: R (“Benchmarking”, “deaR” (Bolos et al., 4 Jun 2025)), Python (“pyDEA”)
• Algorithmic Steps:

1. Formulate either the primal (min-second-best) or dual (Andersen–Petersen) super-efficiency program. 2. Use fractional-to-linear transformation (Charnes–Cooper) when required (SBM, directional, super-SBM-UND-VRS). 3. Solve per DMU, exclude the candidate from the peer set. 4. For SBM/UND: handle undesirable outputs via dedicated constraints penalizing excess bad output (Liao et al., 2023).

Two-stage DEA routines—compute the radial super-efficiency score, then maximize slacks (if needed) for target determination (Bolos et al., 4 Jun 2025). For SBM or direction-based super-efficiency, solve a single LP (using Charnes–Cooper if needed).

For infeasible cases (extreme DMUs, VRS boundary), two-stage relaxation (Fang et al.'s remedy (Liao et al., 2023)) or model composite scores (Bolos et al., 2024) are recommended.

6. Advances: Directional and Continuous Super-Efficiency

Directional distance function-based models (GDSE) (Mehdiloozad et al., 2014) guarantee feasibility, monotonicity, unit and translation invariance, extending super-efficiency to all DMUs and supporting preference weighting. Linear and fractional GDSE formulations subsume classical super-efficiency approaches as special cases.

Continuous SBM-based models (Bolos et al., 2024) address the discontinuity and overestimation observed with classical super-SBM at weakly efficient projections. The composite SBM score,

$$\gamma(x,y) = \delta^*(x,y) \times \max_{(\bar{x},\bar{y}) \in \bar{P}} \rho^*(\bar{x},\bar{y}),$$

ensures continuous transition between SBM efficiency and super-efficiency regimes, correctly ranking DMUs whether on the strong or weakly efficient frontier.

7. Extensions, Pitfalls, and Future Research

• Orientation and Returns to Scale: Both input/output and non-oriented super-efficiency models may be constructed, with CR, VR, and generalized returns to scale; each affects feasibility and discriminatory power (Liao et al., 2023, Bolos et al., 4 Jun 2025, Mehdiloozad et al., 2014).
• Handling Undesirable Outputs: Explicit constraints penalize DMUs reliant on undesirable outputs (e.g., NPL in banking) (Liao et al., 2023).
• Scaling and Outliers: Data scaling and diagnostic checks (influence diagnostics) are essential, as super-efficiency is sensitive to extremes and may lose robustness in low-variance datasets (Bolos et al., 4 Jun 2025).
• Computation and Continuity: Composite super-efficiency models (SBM) introduce nonlinear minimax programs; improving tractability and exploring monotonic composite indices remain open research directions (Bolos et al., 2024).
• Preference Weighting, Negative Data: Directional approaches permit arbitrary preference settings and negative data, provided well-definedness checks are applied (Mehdiloozad et al., 2014).
• Applications: Super-efficient DEA supports panel regressions, Malmquist index catch-up/frontier-shift analysis, and multi-stage benchmarking for strategic decision processes (Liao et al., 2023).

Super-efficient DEA models thus represent a mathematically rigorous, managerially interpretable, fully discriminatory ranking solution for advanced efficiency analysis, integrating contemporary advances from slacks-based, fractional/directional, and composite continuous scoring frameworks.