# Enhanced Russell Graph Slack Based Measure

Based on the data matrix $(X,Y)$, we calculate the Enhanced Russell Graph Measure, ERG, (Pastor et al., 1999) – also known as the Slack Based Measure, SBM, Tone (2001) – of each observation o by solving $n$ times the following linear programming problem:

\begin{aligned} & \underset{\beta, t_i^-, t_r^+ ,\mu_j }{\mathop{\min }}\,\quad \quad \quad \;\ \beta - \frac{1}{m} \sum_{i=1}^{m}{\frac{t_i^-}{x_{io}}} \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ \beta + \frac{1}{s} \sum_{r=1}^{s}{\frac{t_r^+}{y_{ro}}} = 1 \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\mu_j x_{ij} }\ = \beta {x}_{io} - t_i^- \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ \sum_{j=1}^{n}{\mu_j y_{rj} }\ = \beta {y}_{ro} + t_r^+ \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \beta \ge 0 \\ & \quad \quad \quad \quad \quad \ t_i^- \ge 0 \qquad i = 1,...,m \\ & \quad \quad \quad \quad \quad \ t_r^+ \ge 0 \qquad r = 1,...,s \\ & \quad \quad \quad \quad \quad \ \mu_j \ge 0 \qquad j = 1,...,n. \end{aligned}

with the following condition when assuming variable returns to scale:

$$$\sum\nolimits_{j=1}^{n}\mu_j=\beta$$$

After solving the model, input and output slacks are recovered through the following expressions:

\begin{aligned} & s_i^- = \frac{t_i^-}{\beta} \qquad i = 1,...,m \\ & s_r^+ = \frac{t_r^+}{\beta} \qquad r = 1,...,s. \end{aligned}

In this example we compute the Enhanced Russell Graph DEA model under variable returns to scale:

using DataEnvelopmentAnalysis

X = [2; 4; 8; 12; 6; 14; 14; 9.412];

Y = [1; 5; 8; 9; 3; 7; 9; 2.353];

deaergvrs = deaerg(X, Y, rts = :VRS)
Enhanced Russell Graph Slack Based Measure DEA Model
DMUs = 8; Inputs = 1; Outputs = 1
Orientation = Graph; Returns to Scale = VRS
───────────────────────────────────────
efficiency    beta  slackX1  slackY1
───────────────────────────────────────
1    1.0       1.0     0.0        0.0
2    1.0       1.0     0.0        0.0
3    1.0       1.0     0.0        0.0
4    1.0       1.0     0.0        0.0
5    0.4       0.6     2.0        2.0
6    0.47619   1.0     7.33333    0.0
7    0.857143  1.0     2.0        0.0
8    0.2       0.4706  5.412      2.647
───────────────────────────────────────

Estimated efficiency scores are returned with the efficiency function:

efficiency(deaergvrs)
8-element Vector{Float64}:
1.0
1.0
1.0
1.0
0.39999999999999997
0.47619047619047616
0.8571428571428574
0.2

Estimated $\beta$'s are returned with the efficiency function using :beta as the second argument:

efficiency(deaergvrs, :beta)
8-element Vector{Float64}:
1.0
1.0
1.0
1.0
0.6
0.9999999999999998
0.9999999999999998
0.4706000000000001

### deaerg Function Documentation

DataEnvelopmentAnalysis.deaergFunction
deaerg(X, Y)

Compute data envelopment analysis Enhanced Russell Graph Slack Based Measure for inputs X and outputs Y.

Optional Arguments

• rts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.
• Xref=X: Identifies the reference set of inputs against which the units are evaluated.
• Yref=Y: Identifies the reference set of outputs against which the units are evaluated.
• names: a vector of strings with the names of the decision making units.
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