# Generalized Distance Function Models

Chavas and Cox (1999) introduced a generalized distance function efficiency measure that reescales both inputs and outputs toward the frontier technology.

\begin{aligned} & \underset{\delta ,\mathbf{\lambda }}{\mathop{\min }}\,\quad \quad \quad \;\ \delta \\ & \text{subject}\ \text{to} \\ & \quad \quad \quad \quad \quad \ X\mathbf{\lambda } \le \delta^{1 - \alpha} {{\mathbf{x}}_{o}} \\ & \quad \quad \quad \quad \quad \;Y\mathbf{\lambda }\ \ge {{\mathbf{y}}_{o}} / \delta^{\alpha} \\ & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}. \end{aligned}

The measurement of technical efficiency assuming variable returns to scale, VRS, adds the following condition:

$$$\sum\nolimits_{j=1}^{n}\lambda_j=1$$$

In this example we compute the generalized distance function DEA model under variable returns to scale using $0.5$ for the value of $\alpha$:

using DataEnvelopmentAnalysis

X = [5 3; 2 4; 4 2; 4 8; 7 9];

Y = [7 4; 10 8; 8 10; 5 4; 3 6];

deagdf(X, Y, alpha = 0.5, rts = :VRS, slack = false)
Generalized DF DEA Model
DMUs = 5; Inputs = 2; Outputs = 2
alpha = 0.5; Returns to Scale = VRS
─────────────
efficiency
─────────────
1     0.68185
2     1.0
3     1.0
4     0.25
5     0.36
─────────────

### deagdf Function Documentation

DataEnvelopmentAnalysis.deagdfFunction
deagdf(X, Y, alpha)

Compute generalized distance function data envelopment analysis model for inputs X, outputs Y, and alpha.

Optional Arguments

• alpha=0.5: alpha value.
• rts=:CRS: chooses constant returns to scale. For variable returns to scale choose :VRS.
• slack=true: compute input and output slacks.
• 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.