===========================
FDH: Output 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 :math:`X = [x_1, x_2, \cdots, x_n]` and output variables :math:`Y = [y_1, y_2, \cdots, y_n]`,
we measure the output oriented efficiency for observation :math:`i` by solving the following MILP problem: 

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

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

Example: Output oriented FDH `[.ipynb] <https://colab.research.google.com/github/ds2010/pyStoNED/blob/master/notebooks/FDH_oo.ipynb>`_
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    .. code:: python
    
        # import packages
        from pystoned import FDH
        from pystoned import dataset as dataset
        from pystoned.constant import ORIENT_OO, 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_OO, yref=None, xref=None)
        model.optimize(OPT_LOCAL)
    
        # display the technical efficiency
        model.display_theta()
    
        # display the intensity variables
        model.display_lamda()