/** # Data envelopment analysis From: Section 1.6.4 of Sierksma and Zwols, *Linear and Integer Optimization: Theory and Practice*. Let the decision making units (DMUs) be labeled $k = 1,\ldots,N$. For each $k=1,\ldots, N$, the *relative efficiency* $RE(k)$ of DMU $k$ is defined as: \[ RE(k) = \frac{\mbox{weighted sum of output values of DMU $k$}}{\mbox{weighted sum of input values of DMU $k$}}, \] where $0 \leq RE(k) \leq 1$. Let $m \geq 1$ be the number of inputs, and $n \geq 1$ the number of outputs. For each $i = 1,\ldots,m$ and $j = 1,\ldots,n$, define: * $x_i$ = the weight of input $i$; * $y_j$ = the weight of output $j$; * $u_{ik}$ = the (positive) amount of input $i$ to DMU $k$; * $v_{jk}$ = the (positive) amount of output $j$ to DMU $k$. The relative efficiency of DMU $k$ can then be formulated as follows:
$RE^*(k) = $ | $\max$ | $v_{1k}y_1 + \ldots + v_{nk}y_n$ |
$\mbox{subject to}$ | $u_{1k}x_1 + \ldots + u_{mk}x_m = 1$ | |
$v_{1r}y_1 + \ldots + v_{nr}y_n - u_{1r}x_1 - \ldots - u_{mr}x_m \leq 0$ for $r = 1,\ldots,N$ | ||
$x_1,\ldots, x_m, y_1, \ldots, y_n \geq \varepsilon$ |