DEA-SBM - Slack-Based Measure Data Envelopment Analysis

1. 1. Partition strictly positive observations into DEA inputs X and outputs Y; validate the requested SBM orientation.

   $$X\in {\mathbb{R}}_{>0}^{n\times p},Y\in {\mathbb{R}}_{>0}^{n\times q}$$

2. 2. Solve the selected CRS SBM model per DMU: Charnes-Cooper linearized non-oriented SBM, input-oriented SBM, or output-oriented SBM.

   $${\rho }^{\ast }=\min \frac{1-\frac{1}{p}{\sum }_i{s}_i^{-}/x_{io}}{1+\frac{1}{q}{\sum }_r{s}_r^{+}/y_{ro}};{\rho }_I^{\ast }=\min (1-\frac{1}{p}\sum _i{s}_i^{-}/x_{io});{\rho }_O^{\ast }=1/\max (1+\frac{1}{q}\sum _r{s}_r^{+}/y_{ro})$$

3. 3. Identify SBM-efficient frontier DMUs using the solved orientation-specific efficiency scores.

   $${\mathcal{F}}_{SBM}=\{o:{\rho }_o^{\ast }=1\}$$

4. 4. Rank DMUs by SBM efficiency descending with deterministic ID tie-breaking.

   $$\mathrm{rank}\downarrow {\rho }_o^{\ast }$$