# A Cross-Efficiency Evaluation Method Based on Evaluation Criteria Balanced on Interval Weights

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## Abstract

**:**

## 1. Introduction

## 2. The Efficiency Evaluation

_{j}($j=1,\dots ,n$), whose efficiencies are defined as follows. Consider a DMU, say, DMU

_{k}, $k\in \{1,\dots ,n\}$, whose efficiency relative to the other DMUs can be measured by the following CCR model (Charnes et al. [1]):

_{k}. The Charnes and Cooper transformation can be equivalently transformed into the linear program (LP) below for the solution:

_{k}, which is the best relative efficiency that DMU

_{k}can achieve, and reflects the self-evaluated efficiency of DMU

_{k}. As such, ${\theta}_{jk}={\displaystyle {\sum}_{r=1}^{s}{u}_{rk}^{\ast}{y}_{rj}}/{\displaystyle {\sum}_{i=1}^{m}{v}_{ik}^{\ast}{x}_{ik}}$ is referred to as a cross-efficiency of DMU

_{j}and reflects the peer evaluation of DMU

_{k}to DMU

_{j}($j=1,\dots ,n;j\ne k$).

_{k}is $\frac{1}{n}{\displaystyle \sum _{k=1}^{n}{\theta}_{jk}}$ ($j=1,\dots ,n$), where ${\theta}_{jk}$ ($k=1,\dots ,n;j=k$) are the CCR-efficiencies of the n DMUs, that is, ${\theta}_{jk}={\theta}_{kk}^{\ast},j=k$.

## 3. Evaluation Criteria Balanced on Interval Weights of N DMUs

#### 3.1. The DEA Modes of Interval Weights

_{k}. The maximum attainable value of ${u}_{rk}$ or ${v}_{ik}$ of the DMU

_{k}can be obtained by solving the following LP model:

_{k}, this method leads to an interval weight $\left[{u}_{rk}^{-},{u}_{rk}^{+}\right]$ $\left(or\left[{v}_{ik}^{-},{v}_{ik}^{+}\right]\right)$, and then n DMUs lead to n interval weights $\left[{u}_{r1}^{-},{u}_{r1}^{+}\right],\dots \left[{u}_{rj}^{-},{u}_{rj}^{+}\right],\dots ,\left[{u}_{rn}^{-},{u}_{rn}^{+}\right]$ for ${u}_{r}^{}\left(r=1,2,\dots ,s\right)$, and interval weights of ${v}_{i}^{}\left(i=1,2,\dots ,m\right)$ for the same thing.

#### 3.2. Evaluation Criteria Based on the Eclectic Decision-Making Method

#### 3.3. Evaluation Criterion Based on Weighted Mathematical Expectation

_{j}$,j=1,2,\dots ,n$, then solving model (5) n times would lead to n sets of optimal solutions available for n DMUs, which in turn would form an interval weight matrix (IWM). An IWM is shown as follows:

_{j}, ${\overline{\mu}}_{rj}$ and ${\overline{\eta}}_{ij}$ may not be a real weight, and this represents the compromise decision for the decision-makers, as an objective evaluation criterion. There are n sets of mathematical expectations representing the evaluation criteria for n DMUs. As the importance of each DMU is different in cross-evaluation, let ${p}_{j}$ be the weight of DMU

_{j}, which embodies the position of DMU

_{j}. Therefore, the evaluation criterion based on weighted mathematical expectation (ECWME) is calculated as follows:

## 4. DEA Models for Cross-Efficiency Evaluation Based on Evaluation Criteria

_{k}, then the cross-efficiency of a given DMU

_{j}with the profile of weights provided by DMU

_{k}will be obtained as follows:

_{j}is the average of these cross-efficiencies:

_{k}, and let $({u}_{1t}^{CCR},{u}_{2t}^{CCR},\dots ,{u}_{st}^{CCR},{v}_{1t}^{CCR},{v}_{2t}^{CCR},\dots ,{v}_{mt}^{CCR})$, solved by model (2), be the optimal weights of inefficient DMU

_{t}. Suppose there are $n1$ efficient DMUs and $n2$ inefficient DMUs. The CSW based on evaluation criteria is obtained as follows:

## 5. Numerical Examples

**Example 1.**

- Input 1: Total number of academic staff (x
_{1}) - Input 2: Academic staff salaries in thousands of pounds (x
_{2}) - Input 3: Support staff salaries in thousands of pounds (x
_{3}) - Output 1: Total number of undergraduate students (y
_{1}) - Output 2: Total number of postgraduate students (y
_{2}) - Output 3: Total number of research papers (y
_{3})

_{1}= p

_{2}= p

_{3}= p

_{4}= p

_{5}= p

_{6}= p

_{7}and 3p

_{1}= 2 p

_{2}= 6p

_{3}= 6p

_{4}= 6 p

_{5}= 6 p

_{6}= 6 p

_{7}. The cross-efficiency scores and rankings based on ECWME under the two combinations of p are shown in Table 12.

**Example 2:**

- x
_{1}: R&D (research and development) personnel (ratio of R&D personnel to total personnel); - x
_{2}: Total expenditure on scientific and technological activities in the current year (in increments of $10,000); - x
_{3}: Total expenditure on R&D of enterprises (in increments of $10,000); - x
_{4}: Number of senior technicians and technicians at the end of the year (persons); - y
_{1}: Sales revenue of new products (services or processes) of enterprises in this year (in increments of 10,000 yuan); - y
_{2}: Added value of enterprises in this year (in increments of 10,000 yuan); - y
_{3}: Total profits realized by enterprises in this year (in increments of 10,000 yuan); - y
_{4}: Total labor productivity of enterprises in this year (in increments of 10,000 yuan/person).

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

DMU | v_{1} | v_{2} | v_{3} | v_{4} | u_{1} | u_{2} | u_{3} | u_{4} |
---|---|---|---|---|---|---|---|---|

DMU4 | (0, 0.0824) | (0, 0.0058) | (0.029, 0.082) | (0, 0.55915) | (0, 0.0003) | (0, 0.0032) | (0, 0.00185) | (0, 0.4348) |

DMU5 | (0, 0.4453) | (0, 12.676) | (0, 0.0157) | (0, 0.55335) | (0, 0.0016) | (0, 0.0083) | (0, 0.04655) | (0, 1.35135) |

DMU6 | (0, 0.2795) | (0.013,0.027) | (0, 0.0125) | (0, 0.01635) | (0, 0.0003) | (0, 0.0018) | (0.0048, 0.00665) | (0, 0.3408) |

DMU7 | (1.116, 1.40) | (0, 0.0016) | (0.008, 0.016) | (0, 0.0995) | (0.0005, 0.0008) | (0, 0.00115) | (0, 0.0008) | (0.982, 1.263) |

DMU11 | (0, 0.38425) | (0, 0.0086) | (0, 0.0084) | (0.076, 2.424) | (0, 0.00075) | (0.0005, 0.0012) | (0, 0.0092) | (0, 0.5782) |

DMU13 | (0, 0.5) | (0, 0.04005) | (0, 0.0094) | (0, 0.30675) | (0, 0.0004) | (0, 0.00295) | (0, 0.01035) | (0, 0.5804) |

DMU14 | (0.332, 0.507) | (0, 0.0023) | (0, 0.0025) | (0, 0.00485) | (0, 0.0002) | (0, 0.0004) | (0, 0.00135) | (0.276, 0.404) |

DMU15 | (0, 0.30715) | (0, 0.0053) | (0, 0.0054) | (0, 0.56935) | (0, 0.00035) | (0, 0.0006) | (0, 0.0022) | (0, 0.2597) |

DMU16 | (0, 0.4337) | (0, 0.025) | (0, 0.0248) | (0, 1.04525) | (0, 0.0009) | (0, 0.003) | (0, 0.00775) | (0.072, 0.439) |

DMU19 | (0, 0.27505) | (0, 0.0182) | (0, 0.0182) | (19.93, 20.03) | (0, 0.00175) | (0, 0.0138) | (0, 0.0641) | (0, 1.515) |

DMU22 | (0, 0.9381) | (0, 0.0149) | (0, 0.12195) | (0, 1.235) | (0, 0.0009) | (0, 0.01005) | (0, 0.00505) | (0, 1.465) |

DMU26 | (0.444, 0.687) | (0, 0.003) | (0, 0.003) | (0, 0.1071) | (0, 0.0003) | (0, 0.0009) | (0, 0.002) | (0, 0.418) |

**Table A2.**Eclectic decision-making evaluation criterion (ECED) under different values of parameter α.

ECED | v_{1} | v_{2} | v_{3} | v_{4} | u_{1} | u_{2} | u_{3} | u_{4} |
---|---|---|---|---|---|---|---|---|

A (α = 1) | 0.1648 | 0.0032 | 0.0054 | 0.0097 | 0.0004 | 0.0008 | 0.0016 | 0.5194 |

B (α = 0.9) | 0.2599 | 0.01288 | 0.00779 | 2.01145 | 0.00041 | 0.00077 | 0.00192 | 0.56565 |

C (α = 0.8) | 0.355 | 0.02256 | 0.01018 | 4.0132 | 0.00042 | 0.00074 | 0.00224 | 0.6119 |

D (α = 0.7) | 0.4501 | 0.03224 | 0.01257 | 6.01495 | 0.00043 | 0.00071 | 0.00256 | 0.65815 |

E (α = 0.6) | 0.5452 | 0.04192 | 0.01496 | 8.0167 | 0.00044 | 0.00068 | 0.00288 | 0.7044 |

F (α = 0.5) | 0.6403 | 0.0516 | 0.01735 | 10.01845 | 0.00045 | 0.00065 | 0.0032 | 0.75065 |

G (α = 0.4) | 0.7354 | 0.06128 | 0.01974 | 12.0202 | 0.00046 | 0.00062 | 0.00352 | 0.7969 |

H (α = 0.3) | 0.8305 | 0.07096 | 0.02213 | 14.02195 | 0.00047 | 0.00059 | 0.00384 | 0.84315 |

I (α = 0.2) | 0.9256 | 0.08064 | 0.02452 | 16.0237 | 0.00048 | 0.00056 | 0.00416 | 0.8894 |

J (α = 0.1) | 1.0207 | 0.09032 | 0.02691 | 18.02545 | 0.00049 | 0.00053 | 0.00448 | 0.93565 |

Evaluation Criterion | p_{4} | p_{5} | p_{6} | p_{7} | p_{11} | p_{13} | p_{14} | p_{15} | p_{16} | p_{19} | p_{22} | p_{26} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

DMU-4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-5 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-6 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-7 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-11 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-13 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

DMU-14 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

DMU-15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

DMU-16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

DMU-19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

DMU-22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

DMU-26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

ECWME | v_{1} | v_{2} | v_{3} | v_{4} | u_{1} | u_{2} | u_{3} | u_{4} |
---|---|---|---|---|---|---|---|---|

DMU-4 | 0.0824 | 0.0058 | 0.08205 | 0.55915 | 0.0003 | 0.0032 | 0.00185 | 0.4348 |

DMU-5 | 0.4453 | 12.6763 | 0.0157 | 0.55335 | 0.0016 | 0.0083 | 0.04655 | 1.35135 |

DMU-6 | 0.2795 | 0.0274 | 0.0125 | 0.01635 | 0.0003 | 0.0018 | 0.00665 | 0.3408 |

DMU-7 | 1.39955 | 0.0016 | 0.01605 | 0.0995 | 0.0008 | 0.00115 | 0.0008 | 1.26305 |

DMU-11 | 0.38425 | 0.0086 | 0.0084 | 2.42355 | 0.00075 | 0.0012 | 0.0092 | 0.5782 |

DMU-13 | 0.5 | 0.04005 | 0.0094 | 0.30675 | 0.0004 | 0.00295 | 0.01035 | 0.58035 |

DMU-14 | 0.50725 | 0.0023 | 0.0025 | 0.00485 | 0.0002 | 0.0004 | 0.00135 | 0.4041 |

DMU-15 | 0.30715 | 0.0053 | 0.0054 | 0.56935 | 0.00035 | 0.0006 | 0.0022 | 0.2597 |

DMU-16 | 0.4337 | 0.025 | 0.0248 | 1.04525 | 0.0009 | 0.003 | 0.00775 | 0.4391 |

DMU-19 | 0.27505 | 0.0182 | 0.0182 | 20.0272 | 0.00175 | 0.01375 | 0.0641 | 1.51515 |

DMU-22 | 0.9381 | 0.01485 | 0.12195 | 1.23525 | 0.0009 | 0.01005 | 0.00505 | 1.4647 |

DMU-26 | 0.6865 | 0.003 | 0.00305 | 0.1071 | 0.0003 | 0.0009 | 0.002 | 0.41845 |

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**Figure 1.**Comparison of cross-efficiency scores of 27 machinery manufacturing enterprises based on 10 evaluation criteria based on eclectic decision-making (ECED). DMU, decision-making unit.

**Figure 3.**Comparison of cross-efficiency scores of 27 machinery manufacturing enterprises based on 12 ECWMEs.

x_{1} | x_{2} | x_{3} | y_{1} | y_{2} | y_{3} | CCR-Efficiency | |
---|---|---|---|---|---|---|---|

DMU1 | 12 | 400 | 20 | 60 | 35 | 17 | 1 |

DMU2 | 19 | 750 | 70 | 139 | 41 | 40 | 1 |

DMU3 | 42 | 1500 | 70 | 225 | 68 | 75 | 1 |

DMU4 | 15 | 600 | 100 | 90 | 12 | 17 | 0.819 |

DMU5 | 45 | 2000 | 250 | 253 | 145 | 130 | 1 |

DMU6 | 19 | 730 | 50 | 132 | 45 | 45 | 1 |

DMU7 | 41 | 2350 | 600 | 305 | 159 | 97 | 1 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | (0, 79.59) | (0, 2.50) | (0, 50.00) | (0, 15.58) | (1.84, 28.57) | (0, 39.71) |

DMU2 | (0, 52.33) | (0, 1.33) | (0, 2.52) | (5.24, 7.19) | (0, 6.21) | (0, 6.79) |

DMU3 | (0, 11.60) | (0, 0.37) | (6.27, 14.28) | (0, 4.44) | (0, 3.01) | (0, 13.33) |

DMU5 | (0, 22.22) | (0, 0.50) | (0, 1.69) | (0, 1.63) | (0, 6.89) | (0, 7.69) |

DMU6 | (0, 52.22) | (0, 1.37) | (0, 14.27) | (0, 7.57) | (0, 10.99) | (0, 22.22) |

DMU7 | (7.32, 24.39) | (0, 0.29) | (0, 0.53) | (0, 3.28) | (0, 6.29) | (0, 5.41) |

**Table 3.**Eclectic decision-making evaluation criterion (ECED) of all DMUs. UBIW, upper bound of the interval weights; LBIW, lower bound of the interval weights.

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

Minimum of UBIW | 11.602 | 0.3 | 0.53 | 1.63 | 3.01 | 5.41 |

Maximum of LBIW | 7.32 | 0 | 6.28 | 5.24 | 1.85 | 0 |

Cross-evaluation criterion | 9.46 | 0.15 | 3.40 | 3.43 | 2.43 | 2.71 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 0.94603 | 0.20587 | 0.31493 | 0.34347 | 2.13693 | 0.27055 |

DMU2 | 4.79055 | 0.00002 | 0.12812 | 0.61358 | 0.09490 | 0.27055 |

DMU3 | 0.94603 | 0.00557 | 0.74154 | 0.34347 | 0.03571 | 0.27055 |

DMU4 | 6.4150 | 0.0062 | 0 | 0.910 | 0 | 0 |

DMU5 | 2.06488 | 0.00001 | 0.02828 | 0.15625 | 0.17446 | 0.27055 |

DMU6 | 4.43379 | 0.00002 | 0.31493 | 0.34347 | 0.09490 | 1.11982 |

DMU7 | 2.12799 | 0.00001 | 0.02124 | 0.22803 | 0.09490 | 0.15837 |

**Table 5.**Expectation of each DMU and the evaluation criterion based on weighted mathematical expectation (ECWME) of all DMUs.

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 39.795 | 1.25 | 25 | 7.79 | 15.205 | 19.855 |

DMU2 | 26.165 | 0.665 | 1.26 | 6.215 | 3.105 | 3.395 |

DMU3 | 5.8 | 0.185 | 10.275 | 2.22 | 1.505 | 6.665 |

DMU5 | 11.11 | 0.25 | 0.845 | 0.815 | 3.445 | 3.845 |

DMU6 | 26.11 | 0.685 | 7.135 | 3.785 | 5.495 | 11.11 |

DMU7 | 15.855 | 0.145 | 0.265 | 1.64 | 3.145 | 2.705 |

evaluation criterion | 20.81 | 0.53 | 7.46 | 3.74 | 5.32 | 7.93 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 2.08100 | 0.15027 | 0.74600 | 0.37400 | 1.83083 | 0.79300 |

DMU2 | 4.93131 | 0.00002 | 0.08991 | 0.56205 | 0.53200 | 0.00157 |

DMU3 | 1.13735 | 0.00001 | 0.74600 | 0.36162 | 0.00002 | 0.24848 |

DMU4 | 6.4150 | 0.0062 | 0 | 0.910 | 0 | 0 |

DMU5 | 1.62016 | 0.00001 | 0.10837 | 0 | 0.53200 | 0.17585 |

DMU6 | 2.08100 | 0.03173 | 0.74600 | 0.30587 | 0.53200 | 0.79300 |

DMU7 | 2.08100 | 0.00001 | 0.02445 | 0 | 0.53200 | 0.15889 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 0 | 2.5 | 0 | 0 | 28.5714 | 0 |

DMU2 | 0 | 1.3333 | 0 | 7.1942 | 0 | 0 |

DMU3 | 0 | 0.3738 | 6.2762 | 4.4444 | 0 | 0 |

DMU4 | 64.1504 | 0.0629 | 0 | 9.1082 | 0 | 0 |

DMU5 | 0 | 0.5 | 0 | 0 | 4.3165 | 2.8777 |

DMU6 | 0 | 1.2756 | 1.3759 | 7.5758 | 0 | 0 |

DMU7 | 9.9403 | 0.2521 | 0 | 0 | 6.2893 | 0 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 1.99 | 0.08 | 0 | 0.33 | 0.91 | 0.28 |

DMU2 | 2.87 | 0.07 | 0 | 0.56 | 0.65 | 0 |

DMU3 | 0 | 0.03 | 0.74 | 0 | 0.24 | 1.04 |

DMU4 | 5.38 | 0 | 0 | 0.76 | 0 | 0 |

DMU5 | 2.47 | 0.10 | 0 | 0.41 | 1.12 | 0.35 |

DMU6 | 2.07 | 0.08 | 0 | 0.34 | 0.94 | 0.30 |

DMU7 | 2.53 | 0.10 | 0 | 0.42 | 1.15 | 0.36 |

v_{1} | v_{2} | v_{3} | u_{1} | u_{2} | u_{3} | |
---|---|---|---|---|---|---|

DMU1 | 0 | 0 | 0.88 | 0 | 0.50 | 0 |

DMU2 | 0 | 0.11 | 0.12 | 0.68 | 0 | 0 |

DMU3 | 0 | 0 | 0.92 | 0.29 | 0 | 0 |

DMU4 | 4.98 | 0.01 | 0 | 0.76 | 0 | 0 |

DMU5 | 4.29 | 0 | 0.40 | 0 | 0 | 2.26 |

DMU6 | 0.97 | 0 | 0.75 | 0 | 0 | 1.24 |

DMU7 | 6.58 | 0 | 0 | 0 | 1.70 | 0 |

ECWME | Rank | ECED (α = 0.5) | Rank | CCR | Rank | Benevolent | Rank | Aggressive | Rank | |
---|---|---|---|---|---|---|---|---|---|---|

DMU1 | 0.91 | 3 | 0.85 | 3 | 1 | 1 | 0.97 | 2 | 0.97 | 1 |

DMU2 | 0.94 | 2 | 0.91 | 2 | 1 | 1 | 0.93 | 3 | 0.72 | 4 |

DMU3 | 0.81 | 5 | 0.79 | 5 | 1 | 1 | 0.8 | 6 | 0.77 | 3 |

DMU4 | 0.59 | 7 | 0.57 | 7 | 0.89 | 7 | 0.58 | 7 | 0.39 | 7 |

DMU5 | 0.84 | 4 | 0.82 | 4 | 1 | 1 | 0.91 | 4 | 0.66 | 5 |

DMU6 | 0.99 | 1 | 0.97 | 1 | 1 | 1 | 0.99 | 1 | 0.84 | 2 |

DMU7 | 0.79 | 6 | 0.75 | 6 | 1 | 1 | 0.9 | 5 | 0.52 | 6 |

Score (α = 0.2) | Rank | Score (α = 0.5) | Rank | Score (α = 0.8) | Rank | |
---|---|---|---|---|---|---|

DMU1 | 0.88 | 3 | 0.81 | 4 | 0.77 | 6 |

DMU2 | 0.93 | 2 | 0.92 | 2 | 0.94 | 2 |

DMU3 | 0.80 | 5 | 0.79 | 5 | 0.80 | 5 |

DMU4 | 0.58 | 7 | 0.59 | 7 | 0.62 | 7 |

DMU5 | 0.82 | 4 | 0.82 | 3 | 0.87 | 3 |

DMU6 | 0.99 | 1 | 0.97 | 1 | 0.99 | 1 |

DMU7 | 0.77 | 6 | 0.77 | 6 | 0.83 | 4 |

p_{1} = p_{2} = , …, = p_{7} | p_{1} = 0.2, p_{2} = 0.3, p_{3} = …, = p_{7} = 0.1 | |||
---|---|---|---|---|

Score | Rank | Score | Rank | |

DMU1 | 0.89 | 2 | 0.92 | 3 |

DMU2 | 0.84 | 3 | 0.93 | 2 |

DMU3 | 0.76 | 5 | 0.80 | 5 |

DMU4 | 0.45 | 7 | 0.56 | 7 |

DMU5 | 0.82 | 4 | 0.83 | 4 |

DMU6 | 0.93 | 1 | 0.99 | 1 |

DMU7 | 0.72 | 6 | 0.79 | 6 |

CV Based on ECED | CV Based on ECWME | CCR | ||||
---|---|---|---|---|---|---|

α = 0.8 | α = 0.5 | α = 0.2 | p_{1} = , …, = p_{7} | p_{1} = 0.2, p_{2} = 0.3, p_{3} = …, = p_{7} = 0.1 | ||

DMU1 | 0.86 | 1.11 | 1.70 | 0.76 | 1.74 | 2.26 |

DMU2 | 0.07 | 0.20 | 0.001 | 0.12 | 0.0004 | 0.96 |

DMU3 | 0.43 | 0.28 | 0.34 | 0.33 | 0.31 | 2.14 |

DMU5 | 0.10 | 0.07 | 0.08 | 0.19 | 0.09 | 1 |

DMU6 | 1.43 | 1.58 | 0.76 | 0.57 | 0.63 | 1.87 |

DMU7 | 0.36 | 0.33 | 0.23 | 0.33 | 0.57 | 2.65 |

Mean of CV | 0.54 | 0.60 | 0.52 | 0.38 | 0.56 | 1.81 |

sum of CV | 3.25 | 3.57 | 3.12 | 2.3 | 3.34 | 10.87 |

Score and Rank Under ECWME | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

α = 0.8 | α = 0.5 | α = 0.2 | p_{1} = , …, = p_{7} | p_{1} = 0.2, p_{2} = 0.3, p_{3} = …, = p_{7} = 0.1 | ||||||

Score | Rank | Score | Rank | Score | Rank | Score | Rank | Score | Rank | |

DMU1 | 0.75 | 5 | 0.78 | 5 | 0.78 | 5 | 0.88 | 2 | 0.80 | 5 |

DMU2 | 0.94 | 2 | 0.96 | 2 | 0.96 | 2 | 0.86 | 3 | 0.94 | 2 |

DMU3 | 0.74 | 6 | 0.76 | 6 | 0.76 | 6 | 0.76 | 5 | 0.81 | 4 |

DMU4 | 0.67 | 7 | 0.67 | 7 | 0.67 | 7 | 0.47 | 7 | 0.61 | 7 |

DMU5 | 0.78 | 4 | 0.79 | 3 | 0.79 | 3 | 0.78 | 4 | 0.82 | 3 |

DMU6 | 0.95 | 1 | 0.97 | 1 | 0.97 | 1 | 0.94 | 1 | 0.99 | 1 |

DMU7 | 0.79 | 3 | 0.78 | 4 | 0.78 | 4 | 0.58 | 6 | 0.77 | 6 |

Enterprise Number | Inputs | Outputs | CCR Efficiency | ||||||
---|---|---|---|---|---|---|---|---|---|

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

PT1 (DMU1) | 15 | 1361 | 222 | 27 | 3012 | 926.51 | 89 | 4.2 | 0.44 |

SM4 (DMU2) | 12.2 | 520 | 435 | 205 | 2144 | 1146 | 318 | 5.6 | 0.46 |

QZ15 (DMU3) | 16.54 | 226.31 | 226.31 | 7 | 2799.76 | 1118.97 | 158.63 | 8.4 | 0.67 |

LY1 (DMU4) | 40 | 595 | 74.16 | 7 | 198 | 1554.96 | 220.35 | 11.5 | 1 |

SM1 (DMU5) | 11.18 | 0.396 | 317.3 | 9 | 3100 | 603.6 | 107.37 | 3.7 | 1 |

ND1 (DMU6) | 12.61 | 224.6 | 224.8 | 119 | 3436.8 | 581.6 | 1177 | 4.9 | 1 |

NP1 (DMU7) | 4.91 | 349 | 187 | 6 | 3801 | 1404 | 193 | 4.3 | 1 |

PT6 (DMU8) | 100 | 273 | 285 | 198 | 2533 | 1716 | 167 | 8.7 | 0.69 |

QZ1 (DMU9) | 12.22 | 398.89 | 398.89 | 8 | 5192.85 | 1955.78 | 125.79 | 8.8 | 0.78 |

QZ8 (DMU10) | 11 | 532.86 | 532.86 | 391 | 3472.98 | 3550.89 | 528.22 | 9.1 | 0.90 |

FZ14(DMU11) | 11.87 | 566 | 566 | 2 | 2937 | 5197 | 400 | 0.2 | 1 |

QZ22 (DMU12) | 15.72 | 788.73 | 488.73 | 15 | 13,287.54 | 2517.07 | 1682.58 | 11 | 0.98 |

ND1 (DMU13) | 10 | 100 | 485 | 15 | 12,932 | 1688 | 483 | 5.6 | 1 |

PT7 (DMU14) | 14.66 | 1044.5 | 961.62 | 232 | 11,142 | 4362 | 827.14 | 18.8 | 1 |

FZ7 (DMU15) | 16.28 | 886 | 886 | 8 | 15,344 | 8327 | 1442 | 10.2 | 1 |

LY1 (DMU16) | 11.48 | 173.71 | 173.7 | 4 | 1975 | 1508.96 | 43.8 | 12.4 | 1 |

SM2 (DMU17) | 24.56 | 346.04 | 223.3 | 11 | 2605.94 | 771 | 173.67 | 13.5 | 0.80 |

LY4 (DMU18) | 21.18 | 179.39 | 179.39 | 2 | 1858.3 | 504 | 70 | 5.9 | 0.88 |

LY3 (DMU19) | 18.18 | 275 | 275 | 0 | 2841 | 364 | 78 | 3.3 | 1 |

ZZ4 (DMU20) | 10.32 | 454.71 | 454.71 | 31 | 3300.33 | 1382.91 | 532.36 | 4.9 | 0.60 |

QZ9 (DMU21) | 14.62 | 275.7 | 252.6 | 57 | 4435.23 | 1301 | 111.55 | 7.6 | 0.73 |

ND94 (DMU22) | 5.33 | 328 | 41 | 4 | 5549.02 | 368 | 987 | 2.5 | 1 |

ND14 (DMU23) | 24.64 | 462 | 461 | 18 | 6610 | 1748 | 129 | 6.3 | 0.56 |

QZ35 (DMU24) | 10.78 | 547.99 | 518.68 | 11 | 5837 | 2619.39 | 285.45 | 6.4 | 0.69 |

QZ30 (DMU25) | 15.48 | 613.72 | 573.81 | 13 | 4820.71 | 2621.4 | 404.94 | 10.4 | 0.67 |

QZ36 (DMU26) | 10.76 | 825.62 | 825.62 | 16 | 10502 | 5389.64 | 717.38 | 10.5 | 1 |

QZ12 (DMU27) | 12.55 | 615.88 | 316.79 | 11 | 9018.71 | 384.22 | 64.99 | 1.5 | 0.66 |

© 2019 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**

Shi, H.; Wang, Y.; Zhang, X.
A Cross-Efficiency Evaluation Method Based on Evaluation Criteria Balanced on Interval Weights. *Symmetry* **2019**, *11*, 1503.
https://doi.org/10.3390/sym11121503

**AMA Style**

Shi H, Wang Y, Zhang X.
A Cross-Efficiency Evaluation Method Based on Evaluation Criteria Balanced on Interval Weights. *Symmetry*. 2019; 11(12):1503.
https://doi.org/10.3390/sym11121503

**Chicago/Turabian Style**

Shi, Hailiu, Yingming Wang, and Xiaoming Zhang.
2019. "A Cross-Efficiency Evaluation Method Based on Evaluation Criteria Balanced on Interval Weights" *Symmetry* 11, no. 12: 1503.
https://doi.org/10.3390/sym11121503