# A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion
## 1 Introduction
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DMU** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & \(\mathbf{y_{1}}\) & \(\mathbf{y_{2}}\) & \(\mathbf{y_{3}}\) & \(\mathbf{y_{4}}\) \\ \hline
1 & 177,430 & 401 & 528,325 & 801 & 41,675 & 77,564 & 201,529 \\
2 & 221,338 & 1094 & 1,186,905 & 803 & 34,960 & 44,136 & 840,446 \\
3 & 267,806 & 1079 & 1,323,325 & 251 & 24,461 & 27,690 & 832,616 \\
4 & 160,912 & 444 & 648,685 & 816 & 23,744 & 45,882 & 251,770 \\
5 & 177,214 & 801 & 909,539 & 654 & 36,409 & 72,676 & 443,507 \\
6 & 146,325 & 686 & 545,115 & 177 & 18,000 & 19,839 & 341,585 \\
7 & 195,138 & 687 & 790,348 & 695 & 31,221 & 40,154 & 233,822 \\
8 & 108,146 & 152 & 236,722 & 606 & 23,889 & 37,770 & 118,943 \\
9 & 165,663 & 494 & 523,899 & 652 & 25,163 & 28,402 & 179,315 \\
10 & 195,728 & 503 & 428,566 & 959 & 43,440 & 63,701 & 195,303 \\
11 & 87,050 & 343 & 298,696 & 221 & 9689 & 17,334 & 106,037 \\
12 & 124,313 & 129 & 198,598 & 565 & 21,032 & 30,242 & 61,836 \\
13 & 67,545 & 117 & 131,649 & 152 & 10,398 & 14,139 & 46,233 \\
14 & 47,208 & 165 & 228,730 & 211 & 9391 & 13,505 & 42,094 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Iranian gas company dataset. \begin{table} \begin{tabular}{c c c c c} \hline \hline
**DMU** & **Ensemble CCR** & **Ensemble BCC** & **Non-Radial \& Radial \&** \\ \hline
1 & 0.87 (0.15) & 0.95 (0.11) & 0.98 & 0.75 \\
2 & 0.75 (0.30) & 0.77 (0.28) & 1 & 1 \\
3 & 0.61 (0.36) & 0.62 (0.36) & 0.9 & 0.82 \\
4 & 0.71 (0.19) & 0.8 (0.19) & 0.79 & 0.63 \\
5 & 0.77 (0.22) & 0.82 (0.21) & 0.95 & 0.83 \\
6 & 0.58 (0.27) & 0.64 (0.27) & 0.76 & 0.64 \\
7 & 0.54 (0.16) & 0.57 (0.14) & 0.57 & 0.47 \\
8 & 0.98 (0.08) & 0.99 (0.04) & 1 & 1 \\
9 & 0.57 (0.14) & 0.6 (0.14) & 0.61 & 0.46 \\
10 & 0.86 (0.18) & 0.96 (0.11) & 0.85 & 0.77 \\
11 & 0.47 (0.12) & 0.63 (0.14) & 0.55 & 0.46 \\
12 & 0.93 (0.15) & 0.96 (0.11) & 1 & 1 \\
13 & 0.63 (0.13) & 0.96 (0.09) & 0.68 & 0.51 \\
14 & 0.6 (0.24) & 0.86 (0.17) & 0.56 & 0.51 \\ \hline \hline \end{tabular}  \({}^{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\texttexttext{\text{\texttexttexttexttexttexttext{\text{\text{\texttext{\texttexttexttexttexttexttext{                     \text{\texttext{\texttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttexttext      \textto others. These ensemble scores of these DMUs may be considered as normalized random sums generated from identical distributions (such as Distribution 1). All of these DMUs have ensemble efficiency scores greater than 0.95. Similarly, DMUs 5, 6 and 11, in Figure 4, have no fully efficient scores, and these may also be considered as random normalized sums generated from identically distributed pdfs (such as Distribution 2). The ensemble scores for this dataset appear to be random normalized sums from two or more pdfs of the forms given in Equation (2k). Given that these are independent random normalized sums, it can be easily shown that the product of two or more independent MDE pdfs is also an MDE pdf. Figure 5 illustrates two sample MDE pdfs for two different values of alpha. The entropy of an MDE pdf is maximized when the mean of a distribution is greater than 0.5 [27]. For the ensemble BCC model, from Table 3, this criterion is satisfied. The lowest value of the ensemble BCC score is 0.57, which is greater than the mean of 0.5 required to maximize entropy and higher than the lowest values for the efficiency scores for the radial, non-radial and ensemble CCR models. As a result, the ensemble BCC model appears to maximize its entropy slightly better than the ensemble CCR model. Figure 4: Number of times a DMU is fully efficient in Ensemble BCC models. Figure 3: Number of times a DMU is fully efficient in Ensemble CCR models. While ensemble scores have a minor violation of an identical distribution for some DMUs, a formal test of the normality of the distribution of the ensemble efficiency scores was conducted. Table 4 illustrates the results of these tests.