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:
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 \(\sum_{j=1}^{n}\lambda_j = 1\).
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()