```@meta
CurrentModule = DataEnvelopmentAnalysis
```

# Directional Distance Function Models

*Chambers, Chung and Fare (1996)* introduced a measure of efficiency that projects observation $\left( {{\mathbf{x}_o,\mathbf{y}_{o}}} \right)$
in a pre-assigned  direction  $\mathbf{g}= {\left({-{\mathbf{g_{x}^-},\mathbf{g^{+}_y}}} \right)\neq\mathbf{0}_{m+s}}$, $\mathbf{g^{-}_{x}}\mathbb{\in R}^m$ and  $\mathbf{g^{+}_{y}}\mathbb{\in R}^s$, in a proportion $\beta$. The associated linear program is:

```math
\begin{aligned}
 & \underset{\beta ,\mathbf{\lambda }}{\mathop{\max }}\,\quad \quad \quad \quad \beta  \\
 & \text{subject}\ \text{to} \\
 & \quad \quad \quad \quad \quad \ X\lambda\le {{\mathbf{x}}_{o}} -\beta{{\mathbf{g^-_x}}} \\
 & \quad \quad \quad \quad \quad \  Y\mathbf{\lambda }\ge\ {{\mathbf{y}}_{o}}+\beta {{\mathbf{g^+_y}}}  \\
 & \quad \quad \quad \quad \quad \ \mathbf{\lambda }\ge \mathbf{0}.\\  & \quad 
\end{aligned}
```

The measurement of technical efficiency assuming variable returns to scale, **VRS**, adds the following condition:
```math
\sum\nolimits_{j=1}^{n}\lambda_j=1
```

In this example we compute the directional distance function DEA model under constant returns to scale using ones as directions for both inputs and outputs:
```@example ddf
using DataEnvelopmentAnalysis

X = [5 13; 16 12; 16 26; 17 15; 18 14; 23 6; 25 10; 27 22; 37 14; 42 25; 5 17];

Y = [12; 14; 25; 26; 8; 9; 27; 30; 31; 26; 12];

deaddf(X, Y, Gx = :Ones, Gy = :Ones)
```

To compute the variable returns to scale model, we simply set the `rts` parameter to `:VRS`:
```@example ddf
deaddfvrs = deaddf(X, Y, Gx = :Ones, Gy = :Ones, rts = :VRS)
```

Estimated efficiency scores are returned with the `efficiency` function:
```@example ddf
efficiency(deaddfvrs)
```

The optimal peers, ``λ``, are returned with the `peers` function and are returned as a `DEAPeers` object:
```@example ddf
peers(deaddfvrs)
```

## Directional Distance Function Model in Multiplier Form

The dual to the directional distance function DEA model in envelopment form presented above is the multiplier form.

This example computes the directional distance function models DEA model in multiplier form under variable returns to scale:
```@example ddf
ddfonesm = deaddfm(X, Y, Gx = :Ones, Gy = :Ones, rts = :VRS)
```

Input and output virtual multipliers (shadow prices) are returned with the `multipliers` function:
```@example ddf
multipliers(ddfonesm, :X)
```

```@example ddf
multipliers(ddfonesm, :Y)
```

The value measuring the returns to scale is returned with the `rts` function:
```@example ddf
rts(ddfonesm)
```

### deaddf Function Documentation

```@docs
deaddf
deaddfm
```