Radial model: Input orientation

A set of \(j= 1,2,\cdots,n\) observed DMUs transform a vector of \(i = 1, 2,\cdots,m\) inputs \(x \in R^m_{++}\) into a vector of \(i = 1, 2, \cdots, s\) outputs \(y \in R^s_{++}\) using the technology represented by the following CRS production possibility set: \(P_{crs} = \{(x, y) |x \ge X\lambda, y \le Y\lambda, \lambda \ge 0\}\), where \(X = (x)_j \in R^{s \times n}\), \(Y =(y)_j \in R^{m \times n}\) and \(\lambda = (\lambda_1, . . . , \lambda_n)^T\) is a intensity vector.

Based on the data matrix \((X, Y)\), we measure the input oriented efficiency of each observation o by solving n times the following linear programming problems:

\begin{align*} \underset{\mathbf{\phi},\mathbf{\lambda }}min \quad \phi \\ \mbox{s.t.} \quad \phi x_o \ge X\lambda \\ Y\lambda \ge y_o \\ \lambda \ge 0 \end{align*}

The measurement of technical efficiency assuming VRS considers the following production possibility set \(P_{vrs} = \{ (x, y) |x \ge X\lambda, y \le Y\lambda, e\lambda = 1, \lambda \ge 0. \}\). Thus, the only difference with the CRS model is the adjunction of the condition \(\sum_{j=1}^{n}\lambda_j = 1\).

\begin{align*} \underset{\mathbf{\phi},\mathbf{\lambda }}min \quad \phi \\ \mbox{s.t.} \quad \phi x_o \ge X\lambda \\ Y\lambda \ge y_o \\ \sum_{j=1}^{n}\lambda_j = 1 \\ \lambda \ge 0 \end{align*}

Example: Intput oriented DEA [.ipynb]

In the following code, we calculate the VRS radial model with pyStoNED.

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

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

# define and solve the DEA radial model
model = DEA.DEA(data.y, data.x, rts=RTS_VRS, orient=ORIENT_IO, yref=None, xref=None)
model.optimize(OPT_LOCAL)

# display the technical efficiency
model.display_theta()

# display the intensity variables
model.display_lamda()