# Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures

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## Abstract

**:**

## 1. Introduction

## 2. Solution Procedure

#### 2.1. Cross-Efficiency Intervals

_{ij}and Y

_{rj}denote the i-th input, i = 1, …, m, and r-th output, r = 1, …, s, respectively, of the j-th DMU, j = 1, …, n. The DEA model proposed by Charnes et al. [1] for calculating the efficiency of DMU d under the assumption of constant returns-to-scale, referred to as the CCR model, is:

#### 2.2. Entropy of Cross-Efficiency Intervals

## 3. Examples

#### 3.1. Academic Departments in a University

#### 3.2. Chinese City

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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DMU | 1 | 2 | … | n |
---|---|---|---|---|

1 | [${E}_{11}^{L}$, ${E}_{11}^{U}$] | [${E}_{12}^{L}$, ${E}_{12}^{U}$] | … | [${E}_{1n}^{L}$, ${E}_{1n}^{U}$] |

2 | [${E}_{21}^{L}$, ${E}_{21}^{U}$] | [${E}_{22}^{L}$, ${E}_{22}^{U}$] | … | [${E}_{2n}^{L}$, ${E}_{2n}^{U}$] |

… | … | … | … | … |

n | [${E}_{n1}^{L}$, ${E}_{n1}^{U}$] | [${E}_{n2}^{L}$, ${E}_{n2}^{U}$] | … | [${E}_{nn}^{L}$, ${E}_{nn}^{U}$] |

Average | [${\overline{E}}_{1}^{L}$, ${\overline{E}}_{1}^{U}$] | [${\overline{E}}_{2}^{L}$, ${\overline{E}}_{2}^{U}$] | … | [${\overline{E}}_{n}^{L}$, ${\overline{E}}_{n}^{U}$] |

Department | Inputs | Outputs | CCR Efficiency | ||||
---|---|---|---|---|---|---|---|

${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | ||

1 | 12 | 400 | 20 | 60 | 35 | 17 | 1 |

2 | 19 | 750 | 70 | 139 | 41 | 40 | 1 |

3 | 42 | 1500 | 70 | 225 | 68 | 75 | 1 |

4 | 15 | 600 | 100 | 90 | 12 | 17 | 0.8197 |

5 | 45 | 2000 | 250 | 253 | 145 | 130 | 1 |

6 | 19 | 730 | 50 | 132 | 45 | 45 | 1 |

7 | 41 | 2350 | 600 | 305 | 159 | 97 | 1 |

Dep. | Dep. 1 | Dep. 2 | Dep. 3 | Dep. 4 | Dep. 5 | Dep. 6 | Dep. 7 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

L | U | L | U | L | U | L | U | L | U | L | U | L | U | |

1 | 1.0000 | 1.0000 | 0.3347 | 0.9850 | 0.5181 | 1.000 | 0.0686 | 0.6836 | 0.3314 | 1.0000 | 0.5143 | 1.0000 | 0.1514 | 1.0000 |

2 | 0.6856 | 0.9361 | 1.0000 | 1.0000 | 0.7346 | 0.8481 | 0.6868 | 0.8197 | 0.6620 | 0.9208 | 0.9506 | 1.0000 | 0.6044 | 1.0000 |

3 | 0.7933 | 1.0000 | 0.5533 | 0.8584 | 1.0000 | 1.0000 | 0.1515 | 0.4695 | 0.3148 | 0.7081 | 0.8213 | 1.0000 | 0.1509 | 0.2941 |

4 | 0.6874 | 0.6874 | 1.0000 | 1.0000 | 0.7349 | 0.7349 | 0.8197 | 0.8197 | 0.7649 | 0.7649 | 0.9506 | 0.9506 | 1.0000 | 1.0000 |

5 | 0.4904 | 1.0000 | 0.6990 | 0.9703 | 0.5505 | 0.8285 | 0.2417 | 0.6721 | 1.0000 | 1.0000 | 0.7799 | 1.0000 | 0.5252 | 1.0000 |

6 | 0.6449 | 1.0000 | 0.6954 | 1.0000 | 0.7488 | 1.0000 | 0.2136 | 0.7718 | 0.4778 | 1.0000 | 1.0000 | 1.0000 | 0.2460 | 1.0000 |

7 | 0.6336 | 1.0000 | 0.5564 | 1.0000 | 0.4175 | 0.7719 | 0.2063 | 0.8197 | 0.7558 | 1.0000 | 0.6107 | 1.0000 | 1.0000 | 1.0000 |

Ave. | 0.7050 | 0.9462 | 0.6884 | 0.9734 | 0.6720 | 0.8834 | 0.3412 | 0.7223 | 0.6153 | 0.9134 | 0.8039 | 0.9929 | 0.5254 | 0.8992 |

Department | Aggressive | Benevolent | HAI | Entropy |
---|---|---|---|---|

1 | 0.7050 (2) | 0.9462 (3) | 35.06 (3) | 1.5805 (2) |

2 | 0.6884 (3) | 0.9734 (2) | 60.95 (2) | 1.5632 (3) |

3 | 0.6720 (4) | 0.8834 (6) | 25.33 (6) | 1.4771 (4) |

4 | 0.3412 (7) | 0.7223 (7) | 6.21 (7) | 0.8701 (7) |

5 | 0.6153 (5) | 0.9134 (4) | 27.01 (5) | 1.4171 (5) |

6 | 0.8039 (1) | 0.9929 (1) | 86.06 (1) | 1.7232 (1) |

7 | 0.5254 (6) | 0.8992 (5) | 29.36 (4) | 1.2197 (6) |

DMU | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ |
---|---|---|---|---|---|

1 | 2874.8 | 16,738 | 160.89 | 80,800 | 5092 |

2 | 946.3 | 691 | 21.14 | 18,172 | 6563 |

3 | 6854.0 | 43,024 | 375.25 | 144,530 | 2437 |

4 | 2305.1 | 10,815 | 176.68 | 70,318 | 3145 |

5 | 1010.3 | 2099 | 102.12 | 55,419 | 1225 |

6 | 282.3 | 757 | 59.17 | 27,422 | 246 |

7 | 17,478.3 | 116,900 | 1029.09 | 351,390 | 14,604 |

8 | 661.8 | 2024 | 30.07 | 23,550 | 1126 |

9 | 1544.2 | 3218 | 160.58 | 59,406 | 2230 |

10 | 428.4 | 574 | 53.69 | 47,504 | 430 |

11 | 6228.1 | 29,842 | 258.09 | 151,356 | 4649 |

12 | 697.7 | 3394 | 38.02 | 45,336 | 1555 |

13 | 106.4 | 367 | 7.07 | 8236 | 121 |

14 | 4539.3 | 45,809 | 116.46 | 56,135 | 956 |

15 | 957.8 | 16,947 | 29.20 | 17,554 | 231 |

16 | 1209.2 | 15,741 | 65.36 | 62,341 | 618 |

17 | 972.4 | 23,822 | 54.52 | 25,203 | 513 |

18 | 2192.0 | 10,943 | 25.24 | 40,627 | 895 |

DMU | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |||||||||

L | U | L | U | L | U | L | U | L | U | L | U | L | U | L | U | L | U | |

1 | 0.469 | 0.469 | 1.000 | 1.000 | 0.249 | 0.251 | 0.461 | 0.463 | 0.629 | 0.631 | 1.000 | 1.000 | 0.306 | 0.309 | 0.491 | 0.496 | 0.556 | 0.561 |

2 | 0.032 | 0.469 | 1.000 | 1.000 | 0.006 | 0.278 | 0.031 | 0.502 | 0.061 | 0.631 | 0.034 | 1.000 | 0.013 | 0.358 | 0.059 | 0.496 | 0.073 | 0.658 |

3 | 0.468 | 0.468 | 0.999 | 1.000 | 0.278 | 0.278 | 0.502 | 0.502 | 0.586 | 0.586 | 1.000 | 1.000 | 0.358 | 0.358 | 0.414 | 0.414 | 0.628 | 0.628 |

4 | 0.468 | 0.468 | 1.000 | 1.000 | 0.278 | 0.278 | 0.502 | 0.502 | 0.586 | 0.586 | 1.000 | 1.000 | 0.358 | 0.358 | 0.414 | 0.414 | 0.628 | 0.628 |

5 | 0.446 | 0.469 | 0.999 | 1.000 | 0.239 | 0.249 | 0.447 | 0.461 | 0.631 | 0.631 | 1.000 | 1.000 | 0.292 | 0.306 | 0.484 | 0.496 | 0.556 | 0.560 |

6 | 0.138 | 0.469 | 0.107 | 1.000 | 0.126 | 0.278 | 0.228 | 0.502 | 0.482 | 0.631 | 1.000 | 1.000 | 0.128 | 0.358 | 0.195 | 0.496 | 0.422 | 0.658 |

7 | 0.468 | 0.468 | 1.000 | 1.000 | 0.278 | 0.278 | 0.502 | 0.502 | 0.586 | 0.586 | 1.000 | 1.000 | 0.358 | 0.358 | 0.414 | 0.414 | 0.628 | 0.628 |

8 | 0.469 | 0.469 | 1.000 | 1.000 | 0.249 | 0.249 | 0.460 | 0.461 | 0.631 | 0.631 | 0.999 | 1.000 | 0.306 | 0.306 | 0.496 | 0.496 | 0.556 | 0.557 |

9 | 0.183 | 0.183 | 0.999 | 1.000 | 0.133 | 0.133 | 0.267 | 0.267 | 0.629 | 0.629 | 1.000 | 1.000 | 0.146 | 0.146 | 0.265 | 0.265 | 0.658 | 0.658 |

10 | 0.058 | 0.469 | 0.173 | 1.000 | 0.041 | 0.249 | 0.079 | 0.461 | 0.319 | 0.631 | 0.437 | 1.000 | 0.036 | 0.306 | 0.141 | 0.496 | 0.223 | 0.658 |

11 | 0.469 | 0.469 | 0.999 | 1.000 | 0.249 | 0.249 | 0.461 | 0.461 | 0.631 | 0.631 | 0.999 | 1.000 | 0.306 | 0.307 | 0.496 | 0.496 | 0.556 | 0.557 |

12 | 0.439 | 0.439 | 1.000 | 1.000 | 0.210 | 0.210 | 0.408 | 0.408 | 0.582 | 0.582 | 0.875 | 0.875 | 0.261 | 0.261 | 0.490 | 0.490 | 0.481 | 0.481 |

13 | 0.439 | 0.439 | 0.999 | 1.000 | 0.210 | 0.210 | 0.408 | 0.408 | 0.582 | 0.582 | 0.875 | 0.875 | 0.261 | 0.261 | 0.489 | 0.490 | 0.481 | 0.481 |

14 | 0.469 | 0.469 | 1.000 | 1.000 | 0.249 | 0.249 | 0.461 | 0.461 | 0.631 | 0.631 | 1.000 | 1.000 | 0.306 | 0.307 | 0.496 | 0.496 | 0.556 | 0.557 |

15 | 0.469 | 0.469 | 0.999 | 1.000 | 0.249 | 0.249 | 0.461 | 0.461 | 0.631 | 0.631 | 1.000 | 1.000 | 0.306 | 0.306 | 0.496 | 0.496 | 0.556 | 0.557 |

16 | 0.437 | 0.439 | 0.990 | 1.000 | 0.210 | 0.211 | 0.407 | 0.409 | 0.581 | 0.583 | 0.875 | 0.877 | 0.260 | 0.262 | 0.487 | 0.490 | 0.479 | 0.482 |

17 | 0.468 | 0.468 | 0.999 | 1.000 | 0.277 | 0.278 | 0.502 | 0.502 | 0.586 | 0.586 | 1.000 | 1.000 | 0.357 | 0.358 | 0.414 | 0.416 | 0.627 | 0.628 |

18 | 0.430 | 0.442 | 0.962 | 1.000 | 0.208 | 0.215 | 0.402 | 0.414 | 0.578 | 0.588 | 0.868 | 0.889 | 0.258 | 0.266 | 0.482 | 0.490 | 0.475 | 0.490 |

Ave. | 0.379 | 0.446 | 0.901 | 1.000 | 0.208 | 0.244 | 0.388 | 0.453 | 0.552 | 0.610 | 0.887 | 0.973 | 0.257 | 0.305 | 0.401 | 0.464 | 0.508 | 0.579 |

DMU | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | |||||||||

L | U | L | U | L | U | L | U | L | U | L | U | L | U | L | U | L | U | |

1 | 0.981 | 1.000 | 0.299 | 0.301 | 0.749 | 0.763 | 0.708 | 0.725 | 0.662 | 0.666 | 0.670 | 0.674 | 0.678 | 0.682 | 0.686 | 0.690 | 0.693 | 0.697 |

2 | 0.079 | 1.000 | 0.016 | 0.301 | 0.048 | 0.787 | 0.035 | 0.751 | 0.354 | 0.353 | 0.351 | 0.350 | 0.348 | 0.347 | 0.345 | 0.344 | 0.343 | 0.341 |

3 | 0.661 | 0.661 | 0.273 | 0.273 | 0.521 | 0.522 | 0.429 | 0.430 | 0.449 | 0.442 | 0.434 | 0.427 | 0.420 | 0.413 | 0.405 | 0.398 | 0.391 | 0.383 |

4 | 0.661 | 0.661 | 0.273 | 0.273 | 0.521 | 0.522 | 0.429 | 0.430 | 0.449 | 0.442 | 0.435 | 0.427 | 0.420 | 0.413 | 0.405 | 0.398 | 0.391 | 0.383 |

5 | 0.999 | 1.000 | 0.290 | 0.301 | 0.724 | 0.763 | 0.702 | 0.725 | 0.661 | 0.665 | 0.669 | 0.673 | 0.677 | 0.681 | 0.685 | 0.689 | 0.693 | 0.697 |

6 | 0.598 | 1.000 | 0.121 | 0.301 | 0.157 | 0.763 | 0.258 | 0.725 | 0.535 | 0.540 | 0.545 | 0.550 | 0.555 | 0.560 | 0.565 | 0.570 | 0.575 | 0.580 |

7 | 0.661 | 0.661 | 0.273 | 0.273 | 0.521 | 0.522 | 0.429 | 0.430 | 0.449 | 0.442 | 0.435 | 0.427 | 0.420 | 0.413 | 0.405 | 0.398 | 0.391 | 0.384 |

8 | 1.000 | 1.000 | 0.301 | 0.301 | 0.763 | 0.763 | 0.725 | 0.725 | 0.668 | 0.672 | 0.677 | 0.681 | 0.685 | 0.689 | 0.693 | 0.697 | 0.702 | 0.706 |

9 | 0.999 | 1.000 | 0.142 | 0.142 | 0.223 | 0.224 | 0.296 | 0.296 | 0.396 | 0.392 | 0.387 | 0.383 | 0.378 | 0.374 | 0.369 | 0.365 | 0.360 | 0.356 |

10 | 1.000 | 1.000 | 0.061 | 0.301 | 0.120 | 0.787 | 0.206 | 0.751 | 0.015 | 0.138 | 0.013 | 0.187 | 0.044 | 0.470 | 0.013 | 0.303 | 0.025 | 0.197 |

11 | 0.999 | 1.000 | 0.301 | 0.301 | 0.762 | 0.763 | 0.724 | 0.725 | 0.138 | 0.138 | 0.186 | 0.187 | 0.465 | 0.465 | 0.303 | 0.303 | 0.185 | 0.185 |

12 | 1.000 | 1.000 | 0.284 | 0.284 | 0.787 | 0.787 | 0.751 | 0.751 | 0.124 | 0.124 | 0.174 | 0.174 | 0.470 | 0.470 | 0.270 | 0.270 | 0.197 | 0.197 |

13 | 1.000 | 1.000 | 0.283 | 0.284 | 0.786 | 0.787 | 0.751 | 0.751 | 0.124 | 0.124 | 0.174 | 0.175 | 0.470 | 0.470 | 0.269 | 0.270 | 0.197 | 0.197 |

14 | 0.999 | 1.000 | 0.301 | 0.301 | 0.762 | 0.763 | 0.724 | 0.725 | 0.138 | 0.138 | 0.187 | 0.187 | 0.465 | 0.465 | 0.303 | 0.303 | 0.184 | 0.185 |

15 | 1.000 | 1.000 | 0.301 | 0.301 | 0.762 | 0.763 | 0.725 | 0.725 | 0.138 | 0.138 | 0.187 | 0.187 | 0.465 | 0.465 | 0.303 | 0.303 | 0.185 | 0.185 |

16 | 1.000 | 1.000 | 0.283 | 0.284 | 0.784 | 0.787 | 0.751 | 0.751 | 0.123 | 0.124 | 0.174 | 0.175 | 0.470 | 0.470 | 0.269 | 0.270 | 0.196 | 0.197 |

17 | 0.661 | 0.666 | 0.273 | 0.273 | 0.521 | 0.525 | 0.429 | 0.434 | 0.136 | 0.136 | 0.160 | 0.161 | 0.295 | 0.298 | 0.306 | 0.306 | 0.102 | 0.103 |

18 | 0.992 | 1.000 | 0.281 | 0.286 | 0.777 | 0.784 | 0.746 | 0.749 | 0.122 | 0.125 | 0.169 | 0.176 | 0.460 | 0.470 | 0.258 | 0.273 | 0.195 | 0.195 |

Ave. | 0.849 | 0.925 | 0.242 | 0.282 | 0.572 | 0.687 | 0.546 | 0.645 | 0.316 | 0.322 | 0.335 | 0.344 | 0.455 | 0.479 | 0.381 | 0.397 | 0.333 | 0.343 |

**Table 7.**CCR-efficiency, cross-efficiencies, entropies, and ranks (in parentheses) of the 18 Chinese cities.

DMU | CCR | Aggressive | Benevolent | Entropy |
---|---|---|---|---|

1 | 0.469 (11) | 0.379 (12) | 0.446 (11) | 1.147 (11) |

2 | 1.000 (1) | 0.901 (1) | 1.000 (1) | 2.684 (1) |

3 | 0.278 (15) | 0.208 (18) | 0.244 (18) | 0.630 (18) |

4 | 0.502 (8) | 0.388 (10) | 0.453 (10) | 1.178 (10) |

5 | 0.631 (7) | 0.552 (5) | 0.610 (6) | 1.653 (6) |

6 | 1.000 (1) | 0.887 (2) | 0.973 (2) | 2.632 (2) |

7 | 0.358 (12) | 0.257 (16) | 0.305 (16) | 0.780 (16) |

8 | 0.496 (9) | 0.401 (9) | 0.464 (9) | 1.218 (9) |

9 | 0.658 (6) | 0.508 (7) | 0.579 (7) | 1.542 (7) |

10 | 1.000 (1) | 0.849 (3) | 0.925 (3) | 2.514 (3) |

11 | 0.301 (14) | 0.242 (17) | 0.282 (17) | 0.733 (17) |

12 | 0.787 (4) | 0.572 (4) | 0.687 (4) | 1.738 (4) |

13 | 0.751 (5) | 0.546 (6) | 0.645 (5) | 1.655 (5) |

14 | 0.138 (18) | 0.316 (15) | 0.322 (15) | 0.880 (15) |

15 | 0.187 (17) | 0.335 (13) | 0.344 (13) | 0.924 (13) |

16 | 0.470 (10) | 0.455 (8) | 0.479 (8) | 1.266 (8) |

17 | 0.306 (13) | 0.381 (11) | 0.397 (12) | 1.053 (12) |

18 | 0.195 (16) | 0.333 (14) | 0.343 (14) | 0.923 (14) |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lu, T.; Liu, S.-T.
Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures. *Entropy* **2016**, *18*, 452.
https://doi.org/10.3390/e18120452

**AMA Style**

Lu T, Liu S-T.
Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures. *Entropy*. 2016; 18(12):452.
https://doi.org/10.3390/e18120452

**Chicago/Turabian Style**

Lu, Tim, and Shiang-Tai Liu.
2016. "Ranking DMUs by Comparing DEA Cross-Efficiency Intervals Using Entropy Measures" *Entropy* 18, no. 12: 452.
https://doi.org/10.3390/e18120452