FDH: Input orientation

The FDH estimator was first proposed by Deprins et al. (1984), and the mixed integer linear program (MILP) formulation of FDH was introduced by Tulkens (1993).

Given the input variables \(X = [x_1, x_2, \cdots, x_n]\) and output variables \(Y = [y_1, y_2, \cdots, y_n]\), we measure the input oriented efficiency for observation \(i\) by solving the following MILP problem:

\begin{align*} \underset{\mathbf{\phi},\mathbf{\lambda}}min \quad \phi_i \\ \mbox{s.t.} \quad X\lambda \le \phi_i x_i \\ Y\lambda \ge y_i \\ \sum \lambda = 1 \\ \lambda_j \in \{0, 1\}, \forall j \end{align*}

where \(\lambda = [\lambda_1, \lambda_2, \cdots, \lambda_n]\) is the vector of intensity weights. The efficiency of observation \(i\) is \(\phi^*_i\). The corresponding calculation processes are as follow:

Example: Intput oriented FDH [.ipynb]

# import packages
from pystoned import FDH
from pystoned import dataset as dataset
from pystoned.constant import 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 FDH model
model = FDH.FDH(data.y, data.x, 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()