# A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion
## 1 Introduction
The ensemble DEA model requires an exhaustive search procedure using a binary vector \(\boldsymbol{z}\) whose components indicate whether an input or output is considered in performing DEA analysis. The dimension of this binary vector is \((m+s)\). Figure 1 illustrates the \(\boldsymbol{z}\) vector and exhaustive search tree for two-input-and-one-output datasets. The exhaustive tree is pruned (dotted edges) for models that have either no inputs or no outputs. DEA analysis is then conducted on the remaining models, and the efficiency results of each model for each DMU are averaged and used as ensemble DEA scores. To illustrate the ensemble DEA approach on a two-input-and-one-output dataset, a CCR DEA analysis using partial Cobb-Douglas production function data on US economic growth between 1899 and 1910 [19] is conducted. Table 1 illustrates the results of our DEA analysis and resulting ensemble scores. The two inputs were labor in person-hours worked per year and the amount of capital invested. The output was the total annual production. The results of the analysis show that the traditional DEA with \(\boldsymbol{z}=[111]\) does not provide unique rankings (for the years 1901 and 1902 receive the same efficiency score), but the ensemble DEA model provides unique DMU rankings. [PERSON]'s [PERSON] (1988) study provides a theoretical basis for the reliability of ensemble DEA scores. The maximum entropy (ME) principle has been applied to DEA DMU ranking distribution (Kal  
The ensemble entropy is higher than the benchmark. The highest possible entropy value or upper bound (UB) for a model is given by the following expression:  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]4  
The \(\mathrm{ME}^{\mathrm{UB}}\) for the data in Table 1 is 2.485, and the ensemble entropy is very close to the maximum value. It is important to note that obtaining the maximum value is not always desirable, but it provides a theoretical benchmark estimate for a completely unbiased normalized DMU score distribution. To compute ensemble efficiency scores, an \(n\times m\) matrix \(E\) of DEA efficiency scores is necessary. The rows of such a matrix are the numbers of DMUs, and the columns are the numbers of models given by the numbers of eligible models considered in computing ensemble efficiency scores. This number of eligible models will have an upper bound given by \(N\), computed using Equation (1b). The elements of this matrix will be efficiency scores for each DMU computed for a given model identified by column number. Figure 2 illustrates a five-DMU-and-five-model matrix. The ensemble efficiency score (\(\theta_{i}^{E}\)) for each DMU is computed using the following formula:  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]5  
A few observations can be made about any row \(i\in\{1,\ldots,n\}\) of the ensemble efficiency score matrix. First, all the elements of a given row are an independent computation of efficiency scores by the same DMU under a different model number with its unique set of input(s) and output(s). Second, in all the elements of a given row, the DMU is maximizing its efficiency given its model constraints. Thus, each row represents independent evaluations by a DMU under the maximum decisional efficiency (MDE) principle [24]. The MDE principle was introduced by [PERSON] [25] to develop a function to the aggregate the performance of multiple decision-makers. The underlying assumption of the MDE principle is that all decision-makers seek to maximize their decisional efficiencies. [PERSON] [26] later used the MDE approach to rank DMUs and showed that DMUs deemed efficient under MDE are also efficient when ranked using the DEA. For a linear aggregator function, such as the one used in Equation (2j), [PERSON] [26] illustrated that the decisional efficiencies \(\theta\) can be described by the following probability density function (pdf):  
\[\sum\limits_{i=1}^{m}v_{i}x_{io}=1 \tag{2b}\]6  
The pdf in (2k) is monotone, increasing on its interval with a mode at \(\theta=1\) (see Figure 5 for illustration).