CSV readers mutilating my data

R and CSV files

When I deal with regional codes such as FIPS[1] and HUC[2], CSV file readers often mutilate my regions. Here is an example in R: 

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

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

Multiple Solutions in Minimum Spanning Tree example

In [1], I discussed some LP and MIP formulations for the Minimum Spanning Tree (MST) problem. 

Minimum Spanning Tree visualized through Google Maps

Here, I focus on two formulations: a multicommodity network approach (this can be solved as a large LP) and a MIP formulation based on techniques we know from the Traveling Salesman Problem (TSP). The main issue I want to discuss is the presence of multiple optimal solutions.

N-queens and solution pool

In [1], I described some chess-related problems. Here, I want to reproduce the \(n\)-queens problem. The single solution problem, placing as many queens on the chess board as possible so they don't attack each other, is pretty standard. I want to focus on the more complex question: How many different ways can we place those queens? In other words: what are all the optimal solutions? We can do this by adding a no-good constraint that forbids the previously found solution. However, as this problem has more than a handful of different solutions, I want to use the Cplex solution pool.

Single Solution Model

We define the decision variables as: \[\color{darkred}x_{i,j} = \begin{cases} 1 & \text{if we place a queen on the square $(i,j)$} \\ 0 & \text{otherwise}\end{cases}\] 

Chess Board