DEA: Distance function

Chambers et al. (1996) introduced the directional distance function (DDF) into efficiency measurement, and the inefficient DMUs can be projected to the frontier using direction \(g = (−g_x , g_y) \neq 0_{m+s}\), where \(g_x \in R^m\) and \(g_y \in R^s\).

The VRA and CRS models are presented as follows

  1. CRS

\begin{align*} \underset{\mathbf{\theta},\mathbf{\lambda }}max \quad \theta \\ \mbox{s.t.} \quad X\lambda \le x_o - \theta g_x \\ Y\lambda \ge y_o + \theta g_y\\ \lambda \ge 0 \end{align*}

  1. VRS

\begin{align*} \underset{\mathbf{\theta},\mathbf{\lambda }}max \quad \theta \\ \mbox{s.t.} \quad X\lambda \le x_o - \theta g_x \\ Y\lambda \ge y_o + \theta g_y\\ \sum_{j=1}^{n}\lambda_j = 1 \\ \lambda \ge 0 \end{align*}

Example: DEA with DDF [.ipynb]

# import packages
from pystoned import DEA
from pystoned import dataset as dataset
from pystoned.constant import RTS_VRS, OPT_LOCAL

# import the data provided with Tim Coelli’s Frontier 4.1
data = dataset.load_Tim_Coelli_frontier()

# define and solve the DEA DDF model
model = DEA.DDF(y=data.y, x=data.x, b=None, gy=[1], gx=[0.0, 0.0], gb=None, rts=RTS_VRS, yref=None, xref=None, bref=None)
model.optimize(OPT_LOCAL)

# display the technical efficiency
model.display_theta()

# display the intensity variables
model.display_lamda()