# A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion

## Abstract

**:**

## 1. Introduction

## 2. DEA Preliminaries, Ensemble DEA Model, Entropy Criterion for DEA Model Comparisons and Statistical Distribution of Ensemble Scores

_{j}(j = 1, …, n), the space for the input and output vectors (x

_{j},y

_{j}) ϵ ${\mathrm{\mathbb{R}}}_{+}^{m+s}$. 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:

_{i}and u

_{r}are the weights associated with the ith input and jth output, respectively. The constant ε > 0 is infinitesimally non-Archimedean. The model (2a)–(2d) is often called the primary CCR model [1], and its dual is written as follows:

**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

**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

**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. Pendharkar’s [14] study provides a theoretical basis for the reliability of ensemble DEA scores.

**z**as follows:

^{111}= 2.4768, ME

^{101}= 2.4775 and ME

^{011}= 2.4757. The model with labor as an input and production as an output (

**z**= [101]) has the highest entropy and has the least bias, with a maximum difference between DMU efficiencies for closely ranked DMUs for the years 1901 and 1902. The ensemble entropy is 2.4769, and since it is an average of all

**z**-vector combinations, the comparison benchmark for ensemble entropy is the model with

**z**= [111]. 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:

^{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.

_{α}= α (e

^{α}− 1)

^{−1}. Since each element in a given row of the ensemble efficiency score matrix is an independent evaluation by a decision-maker (i.e., a DMU in an ensemble model) trying to maximize its decisional efficiency ${\theta}_{ij}^{*}$ for j = {1, …, m}, the probability density function for each row (DMU) can be written as:

_{1}= α

_{2}= … = α

_{n}. Under the restrictive assumption that α

_{1}= α

_{2}= … = α

_{n}, the ensemble efficiency scores are guaranteed to asymptotically converge to a normal distribution by the central limit theorem. In practice, however, the ensemble efficiency scores are not entirely random or perfectly identically distributed (due to the slight likely variation of Equation (2l)’s α

_{i}parameters for each row), and each ensemble model does introduce a degree of mild randomization. For mild differences in the row pdf parameters α

_{i}

_{,}where α

_{1}≈ α

_{2}≈ … ≈ α

_{n}, the ensemble efficiency scores are likely to be normally distributed. A reader may note that under ideal conditions, where α

_{1}= α

_{2}= … = α

_{n}and individual DMU scores follow Equation (2l)’s distribution, the entropy of the ensemble scores will be highest and close to the highest upper bound mentioned in Equation (2i) because the distribution in Equation (2i) has a mode of 1 (see Figure 5). Thus, it may be argued that the likelihood of normality of the ensemble scores increases when the entropy of the ensemble scores is closer to its upper bound given by Equation (2i). It is important to note that an entropy equal to the exact value of the upper bound given by Equation (2i) is undesirable because at that value, the distribution is a uniform distribution where all the DMUs are fully efficient for all the models. The entropy of the pdf in Equation (2k) is maximized on the interval [0, 1] when the mean of the distribution is greater than 0.5 [27]. Additionally, another important aspect of the distribution of the ensemble efficiency scores is that both the rows and columns of ensemble efficiency scores (Figure 2) play a role in the pdf of the ensemble efficiency scores because the rows represent sampling from the MDE distributions and the columns represent sampling from the distribution of the sums of independent variables. Larger sample sizes increase the statistical reliability and robustness of the results.

## 3. Comparing Variable Selection Models and Ensemble Model Using Gas Company Data and Entropy Criterion

_{1}), staff (x

_{2}) and cost (x

_{3}). The outputs are customers (y

_{1}), the length of the gas network (y

_{2}), the volume delivered (y

_{3}) and gas sales (y

_{4}). Table 2 lists these data. Table 3 lists the efficiency scores of the ensemble DEA with the CCR and BCC models and models used by Toloo and Babaee [16]. Using formula (1b), a total of 105 unique DEA models were used to compute the DEA ensemble efficiency score.

^{UB}from Equation (2i) is 2.639. Comparing the Ensemble CCR with the Non-Radial and Radial CCR models shows that the Ensemble CCR model has a higher entropy. Only the VRS Ensemble BCC model has a higher entropy than the Ensemble CCR model. The standard deviations of the Ensemble BCC model are mostly lower than the CCR model’s as well. More importantly, the Ensemble CCR model generates unique rankings for the DMUs, whereas the Non-Radial and Radial models generate a tie for three DMUs. The Ensemble BCC model also generates unique rankings, but the differences occur at the third decimal place. The Ensemble BCC efficiency scores for DMU 10, 12 and 13 were 0.960, 0.959 and 0.962, respectively.

_{1}≈ α

_{2}≈ … ≈ α

_{n}was satisfied for the theoretical normal distribution of the ensemble efficiency scores. For these parameters to be similar, the expectation is that a similar number of fully efficient DMUs should exist across all models. Clearly, some DMUs are never fully efficient under any of 105 models and the assumption of identical distributions is violated. While the assumption is violated, Figure 4 illustrates that some DMUs, e.g., 1, 8, 10, 12 and 13, have a somewhat similar number of fully efficient DMUs to 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).

## 4. Summary, Conclusions and Directions for Future Work

## Funding

## Conflicts of Interest

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**Figure 5.**The maximum decisional efficiency (MDE) probability density function (pdf) for α = 5 and α = 10, respectively.

Year | Production | Labor | Capital | DEA Model Efficiencies | Ensemble Score | ||
---|---|---|---|---|---|---|---|

z = [111] | z = [101] | z = [011] | |||||

1899 | 100 | 100 | 100 | 0.681 | 0.681 | 0.665 | 0.676 |

1900 | 101 | 105 | 107 | 0.722 | 0.722 | 0.678 | 0.707 |

1901 | 112 | 110 | 114 | 0.693 | 0.693 | 0.689 | 0.692 |

1902 | 122 | 117 | 122 | 0.693 | 0.681 | 0.693 | 0.689 |

1903 | 124 | 122 | 131 | 0.720 | 0.720 | 0.714 | 0.718 |

1904 | 122 | 121 | 138 | 0.770 | 0.770 | 0.758 | 0.766 |

1905 | 143 | 125 | 149 | 0.793 | 0.710 | 0.793 | 0.765 |

1906 | 152 | 134 | 163 | 0.809 | 0.730 | 0.809 | 0.783 |

1907 | 151 | 140 | 176 | 0.836 | 0.794 | 0.836 | 0.822 |

1908 | 126 | 123 | 185 | 1.000 | 1.000 | 1.000 | 1.000 |

1909 | 155 | 143 | 198 | 0.921 | 0.870 | 0.921 | 0.904 |

1910 | 159 | 147 | 208 | 0.941 | 0.891 | 0.941 | 0.924 |

DMU | x_{1} | x_{2} | x_{3} | y_{1} | y_{2} | y_{3} | y_{4} |
---|---|---|---|---|---|---|---|

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 |

DMU | Ensemble CCR | Ensemble BCC | Non-Radial ^{&} | Radial ^{&} |
---|---|---|---|---|

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 |

^{&}Results taken from Toloo and Babaee’s [16] study.

Kolmogorov–Smirnov | Shapiro–Wilk | |||||
---|---|---|---|---|---|---|

Statistic | df | Sig. | Statistic | df | Sig. | |

Ensemble BCC | 0.196 | 14 | 0.149 | 0.876 | 14 | 0.051 |

Ensemble CCR | 0.182 | 14 | 0.200 | 0.944 | 14 | 0.477 |

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Pendharkar, P.C.
A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion. *Algorithms* **2020**, *13*, 232.
https://doi.org/10.3390/a13090232

**AMA Style**

Pendharkar PC.
A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion. *Algorithms*. 2020; 13(9):232.
https://doi.org/10.3390/a13090232

**Chicago/Turabian Style**

Pendharkar, Parag C.
2020. "A Comparison of Ensemble and Dimensionality Reduction DEA Models Based on Entropy Criterion" *Algorithms* 13, no. 9: 232.
https://doi.org/10.3390/a13090232