Convex hull models

Convex hull as an optimization problem

In the previous posts, the construction of a convex hull played a significant role: it was an easy way to reduce the size of the data sets, often by a large amount. Somehow, it escaped me that we can try to formulate this as what turns out to be a conceptually rather simple optimization problem. This has probably little or no practical value. But it remains an interesting exercise.

Solving DEA Models with GAMS

Data Envelopment Analysis (DEA) models are somewhat special. They typically consist of small LPs, of which a whole bunch have to be solved. The CCR formulation (after [1]), for the \(i\)-th DMU (Decision Making Unit), can be stated as [2]:

CCR LP Model

\[\begin{align} \max \>& \color{darkred}{\mathit{efficiency}}_i=\sum_{\mathit{outp}} \color{darkred}u_{{\mathit{outp}}} \cdot \color{darkblue}y_{i,{\mathit{outp}}} \\ & \sum_{\mathit{inp}} \color{darkred}v_{{\mathit{inp}}} \cdot \color{darkblue}x_{i,{\mathit{inp}}} = 1 \\ & \sum_{\mathit{outp}} \color{darkred}u_{{\mathit{outp}}} \cdot \color{darkblue}y_{j,{\mathit{outp}}} \le \color{darkred}v_{{\mathit{inp}}} \cdot \color{darkblue}x_{j,{\mathit{inp}}} && \forall j \\ & \color{darkred}u_{{\mathit{outp}}} \ge 0, \color{darkred}v_{{\mathit{inp}}} \ge 0 \end{align}\]