# A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion
## 1 Introduction
In particular, the ensemble efficiency scores maximize entropy, meaning that the DMU ranking distribution generated by the ensemble efficiency scores has a lower bias when compared to some competing radial and non-radial variable selection models recently reported in the literature, and second, the ensemble efficiency scores may be normally distributed under certain restrictive assumptions. The normal distribution of the efficiency score feature is particularly attractive because returns-to-scale hypothesis testing may be conducted by using traditional difference-in-means parametric statistical procedures. Both of these features are tested using data and models reported in a published study [16]. The rest of the paper is organized as follows: In Section 2, the basic DEA radial and non-radial models, ensemble DEA model and Entropy criterion for comparing different DEA models are described. In Section 3, using Iranian gas company data, the results of ensemble DEA models are compared with the results of variable selection models used in [PERSON] and [PERSON]'s [16] study. Additionally, in Section 3, the properties of the ensemble DEA scores are investigated in terms of the entropy criterion and their statistical distributions. In Section 4, the paper concludes with a summary and directions for future research. DEA Preliminaries, Ensemble DEA Model, Entropy Criterion for DEA Model Comparisons and Statistical Distribution of Ensemble Scores  
The basic DEA model assumes \(n\) DMUs, with each DMU consisting of \(m\) different inputs that produce \(s\) different outputs. The input and output vectors are semi-positive, and for DMU\({}_{j}\) (\(j=1,\ldots,\)\(n\)), the space for the input and output vectors \((x_{j}\),\(y_{j})\in\mathbb{R}_{+}^{m+s}\). For a DMU\({}_{o}\), its relative efficiency may be computed by using the linear programming model under the constant returns-to-scale assumption. This efficiency is computed by solving the following model:  
\[\max\sum\
olimits_{r=1}^{s}u_{r}y_{ro},\] (2a) subject to: \[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\] \[\sum\limits_{r=1}^{s}u_{r}y_{rj}-\sum\limits_{i=1}^{m}v_{i}x_{ij}\leq 0 \quad\text{for all }j=1,\ldots,n \tag{2c}\]\[v_{i},u_{r}\geq\varepsilon\;\;\text{for all}\;i=1,\ldots,m\;\text{and}\;r=1,\ldots,s \tag{2d}\]  
where \(v_{i}\) and \(u_{r}\) are the weights associated with the \(i\)th input and \(j\)th output, respectively. The constant \(\varepsilon\)\(>0\) is infinitesimally non-Archimedean. The model (2a)-(2d) is often called the primary CCR model [1], and its dual is written as follows:  
\[\text{minimize}\;\theta-\varepsilon\!\left(\sum\
olimits_{i=1}^{m}s_{i}^{-}+ \sum\
olimits_{r=1}^{s}s_{r}^{+}\right)\;\ldots\ldots. \tag{2e}\]  
subject to:  
\[\sum\limits_{j=1}^{n}\lambda_{j}x_{ij}+s_{i}^{-}=\theta x_{i0},\;\;\;i=1, \ldots,m \tag{2f}\]  
\[\sum\
olimits_{j=1}^{n}\lambda_{j}y_{rj}-s_{r}^{+}=y_{r0},\;\;\;r=1,\ldots,s, \;\text{and} \tag{2g}\]  
\[\lambda_{j},s_{i}^{-},s_{r}^{+}\geq 0\;\text{for all}\;i=1,\ldots,m;j=1, \ldots,n;r=1,\ldots,s \tag{2h}\]  
The VRS BCC model augments the system (2e)-(2h) by adding the following constraint:  
\[\sum\limits_{j=1}^{n}\lambda_{j}=1\]  
The aforementioned models are radial DEA models that are criticized for not providing input or output projections (for inefficient DMUs) that satisfy Pareto optimality conditions [17]. [PERSON] and [PERSON] [18] independently proposed radial DEA models that allow for input or output reductions at variable rates. The radial version of the CCR model is mathematically represented in the following dual form:  
\[\text{minimize}\;\frac{1}{m}\sum\limits_{i=1}^{m}\theta_{i}\]  
subject to:  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]0  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]1  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]2  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]3  
[PERSON] [14] proposed an ensemble DEA model based on the popularity of ensemble models in machine learning literature.