# The Theoretical Relationship between the CCR Model and the Two-Stage DEA Model with an Application in the Efficiency Analysis of the Financial Industry

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Construction

#### 3.1. Model Explanation

#### 3.1.1. Notation Explanation

- n: the number of evaluated decision-making units (DMUs).
- m: the number of inputs of DMUs.
- s: the number of mediators of DMUs.
- t: the number of outputs of DMUs.
- z
_{hk}: h^{th}output of the k^{th}DMU. - y
_{rk}: r^{th}mediator of the k^{th}DMU. - x
_{ik}: i^{th}input of the k^{th}DMU. - u
_{rk}: the weight for r^{th}mediator of the k^{th}DMU in stage one. - ν
_{ik}: the weight for i^{th}input of the k^{th}DMU in stage one. - u*
_{rk}: the weight for r^{th}mediator of the k^{th}DMU in stage one when V’ replaces V. - u’
_{rk}: the weight for r^{th}mediator of the k^{th}DMU in stage two. - w
_{hk}: the weight for h^{th}output of the k^{th}DMU in stage two. - w*
_{hk}: the weight for h^{th}output of the k^{th}DMU in stage two when U replaces U’. - w’
_{hk}: the weight for h^{th}output of the k^{th}DMU in the overall stage. - ν’
_{ik}: the weight for i^{th}input of the k^{th}DMU in the overall stage. - ε: a arbitrary small positive real number.
- U = (u
_{1k}, u_{2k}, …, u_{sk}), the set of weights for mediator of the k^{th}DMU in stage one. - U’ = (u’
_{1k}, u’_{2k}, …, u’_{sk}), the set of weights for mediator of the k^{th}DMU in stage two. - U* = (u*
_{1k}, u*_{2k}, …, u*_{sk}), the set of weights for the mediator of the k^{th}DMU in stage one when V’ replaces V. - W = (w
_{1k}, w_{2k}, …, w_{tk}), the set of weights for output of the k^{th}DMU in stage two. - W’ = (w’
_{1k}, w’_{2k}, …, w’_{tk}), the set of weights for output of the k^{th}DMU in overall stage. - W* = (w*
_{1k}, w*_{2k}, …, w*_{tk}), the set of weights for output of the k^{th}DMU in stage two when U replaces U’. - V = (ν
_{1k}, ν_{2k}, …, ν_{mk}), the set of weights for input of the k^{th}DMU in stage one. - V’ = (ν’
_{1k}, ν’_{2k}, …, ν’_{mk}), the set of weights for the input of the k^{th}DMU in the overall stage. - θ
_{k}_{(1)}: the relative efficiency score of the k^{th}DMU in stage one, k = 1, 2,…, n. - θ
_{k}_{(2)}: the relative efficiency score of the k^{th}DMU in stage two, k = 1, 2,…, n. - θ
_{k}_{(3)}: the relative efficiency score of the k^{th}DMU in overall stage, k = 1, 2,…, n. - θ*
_{k}_{(1)}: the relative efficiency score of the k^{th}DMU in stage one when V’ replaces V; k = 1, 2,…, n. - θ*
_{k}_{(2)}: the relative efficiency score of the k^{th}DMU in stage two when U replaces U’; k = 1, 2,…, n.

#### 3.1.2. Mathematical Model

^{th}DMU in stage one can be computed as follows:

^{th}DMU in stage two can be computed as follows:

^{th}DMU in the overall stage can be computed as follows:

#### 3.2. Proposed Theoretical Relationship

**Theorem**

**1.**

_{k(3)}< 1), then it is possible that it is efficient in stage one and/or stage two, which means that θ

_{k(1)}= 1 and/or θ

_{k(2)}= 1.

**Proof.**

_{k}

_{(3)}< 1, then, from Equation (3), we get the following:

_{k}

_{(2)}and θ

_{k}

_{(3)}have a certain relationship since they share a common output factor (z

_{hk}). However, since the set of weight W is different from W’, it is also possible that θ

_{k}

_{(2)}= $\sum _{h=1}^{t}{w}_{hk}{z}_{hk}$ = 1. □

**Theorem**

**2.**

_{k(2)}will not be greater than θ

_{k(2)}.

**Proof.**

_{k}

_{(1)}= 1, it means that $\sum _{r=1}^{s}{u}_{rk}{y}_{rk}$ = 1. From Equation (2), we get the following:

_{k}

_{(2)}is obtained by replacing the set of weights U’ with the set of weights U, then we get the following:

_{k}

_{(2)}while U is not in this case. Therefore, from Equations (5) and (6), it is obvious that θ*

_{k}

_{(2)}≤ θ

_{k}

_{(2)}.

_{k}

_{(1)}< 1, it means that $\sum _{r=1}^{s}{u}_{rk}{y}_{rk}$ < 1. When the new efficiency score θ*

_{k}

_{(2)}is obtained by replacing the set of weights U’ with the set of weights U, then we get the following:

_{k}

_{(2)}≤ θ

_{k}

_{(2)}.

**Theorem**

**3.**

_{k(2)}will not be less than θ

_{k(2)}.

**Proof.**

_{k}

_{(2)}can be calculated as follows:

- (a)
- If θ
_{k}_{(1)}= 1, it means that $\sum _{r=1}^{s}{u}_{rk}{y}_{rk}$ = 1 and $\sum _{i=1}^{m}{v}_{ik}{x}_{ik}$ = 1, then:$$\frac{{\displaystyle \sum _{h=1}^{t}{w}_{hk}{z}_{hk}}}{{\displaystyle \sum _{r=1}^{s}{u}_{rk}^{\prime}{y}_{rk}}}=\frac{{\displaystyle \sum _{h=1}^{t}{w}_{hk}{z}_{hk}}}{{\displaystyle \sum _{r=1}^{s}{u}_{rk}{y}_{rk}}},\mathrm{since}{\displaystyle \sum _{r=1}^{s}{u}_{rk}{y}_{rk}}={\displaystyle \sum _{r=1}^{s}{u}_{rk}^{\prime}{y}_{rk}}=1$$It can be concluded that θ_{k}_{(2)}= θ*_{k}_{(2)} - (b)
- If θ
_{k}_{(1)}< 1, it means that $\sum _{r=1}^{s}{u}_{rk}{y}_{rk}$ < 1 and $\sum _{i=1}^{m}{v}_{ik}{x}_{ik}$ = 1, then:$$\frac{{\displaystyle \sum _{h=1}^{t}{w}_{hk}{z}_{hk}}}{{\displaystyle \sum _{r=1}^{s}{u}_{rk}^{\prime}{y}_{rk}}}<\frac{{\displaystyle \sum _{h=1}^{t}{w}_{hk}{z}_{hk}}}{{\displaystyle \sum _{r=1}^{s}{u}_{rk}{y}_{rk}}}\mathrm{since}{\displaystyle \sum _{r=1}^{s}{u}_{rk}{y}_{rk}}{\displaystyle \sum _{r=1}^{s}{u}_{rk}^{\prime}{y}_{rk}}=1$$It can be concluded that θ_{k}_{(2)}< θ*_{k}_{(2)}

**Theorem**

**4.**

_{k(1)}will not be greater than θ

_{k(1)}.

**Proof.**

_{k}

_{(1)}is obtained by replacing the set of weights V with the set of weights V’, then we get the following:

_{k}

_{(1)}, while V’ is not in this case. Therefore, from Equations (9) and (10), it is obvious that θ*

_{k}

_{(1)}≤ θ

_{k}

_{(1)}. □

## 4. Real Case Application

_{1}) and commissions and acquisition expenses (I

_{2}); mediator factors: direct written premiums (M

_{1}) and reinsurance premiums received (M

_{2}); and output factors: net underwriting income (O

_{1}) and investment income (O

_{2}).

#### 4.1. Relationship between the Efficiency Scores

_{6}, DMU

_{9}, DMU

_{15}, and DMU

_{19}; and five DMUs that are efficient in profitability (Stage 2), they are: DMU

_{3}, DMU

_{5}, DMU

_{12}, DMU

_{17}, and DMU

_{20}. It is obvious that none of the DMUs in this example are efficient in both stages: marketability and profitability. Moreover, the results also show that some of these DMUs are not efficient in the overall stage using the original DEA models, such as DMU

_{3}, DMU

_{6}, DMU

_{9}, and DMU

_{17}. These results are solid evidence that reinforces the reliability of Theorem one. From the results, it seems that if a DMU has an efficiency score of 1 in the one-stage DEA model (overall stage), then it could not have an efficiency score of 1 in both stages of the two-stage DEA model, as we can see in Table 2.

_{2}has an efficiency score of 1 using the one-stage DEA and different efficiency score in stage one (score of 0.998) and stage two (score of 0.635). While the difference between the marketability efficiency and the one-stage efficiency is small, there is a significant difference between the efficiency score of profitability and the one-stage efficiency score. Based on this argument, it can be stated that the analysis using the one-stage DEA model is insufficient for reflecting the management of the mediating factors (direct written premiums and reinsurance premiums received) of DMU

_{2}.

_{5}and DMU

_{15}have efficiency scores of 1 using the one-stage DEA model, and it can only be concluded that these two DMUs were performing well compared to the others in the field. In terms of business, this implies that they are better at making profits than the others. Nevertheless, once the two-stage DEA model is adopted for the analysis, the results show something different: while DMU

_{5}is efficient with respect to profitability, DMU

_{15}is efficient with respect to the marketability aspect.

_{19}has an efficiency score of one in the one-stage DEA model, but when the two-stage model is used, it provides a score of 0.455 for profitability, and it is significantly lower than their marketability efficiency, which also has a score of one. This suggests that this DMU suffered big problems with respect to making profits, and the manager of this company should implement solutions to increase productivity using mediating factors (direct written premiums and reinsurance premiums received).

#### 4.2. How the Weights Affect Efficiency Scores

_{k}

_{(2)}will not be greater than θ

_{k}

_{(2)}, which means that: θ*

_{k}

_{(2)}≤ θ

_{k}

_{(2)}. The original scores (refer to Table 2) and new efficiency scores of stage two that correspond to Theorem 2 are summarized in Table 6.

_{k}

_{(2)}will always greater than or equal to θ

_{k}

_{(2)}, which means that: θ

_{k}

_{(2)}≤ θ*

_{k}

_{(2)}. The original scores (refer to Table 2) and new efficiency scores of stage two corresponding to Theorem 3 are summarized in Table 7.

_{k}

_{(2)}are calculated after replacing U’ with U, in Theorem 2, only the new efficiency score θ*

_{k}

_{(2)}is calculated in Theorem 3. With this difference, the new efficiency scores are totally reversed from being less than the original ones in Theorem 2 to be greater than those in Theorem 3. The difference between the original and the new values according to the heorem supports Theorem 3 in that only the DMUs (DMU

_{6}, DMU

_{9}, DMU

_{15}, and DMU

_{19}) that are efficient in stage one retain the same scores after U replaces U’.

_{k}

_{(1)}will not greater than θ

_{k}

_{(1)}. The original scores (refer to Table 2) and new efficiency scores of stage one corresponding to Theorem 4 are summarized in Table 8.

#### 4.3. Application to the Bank

^{3}), fixed assets (FAs)(¥10

^{8}), and expenses (EXs)(¥10

^{8}) are inputs; credit (CR)(¥10

^{8}) and interbank loans (ILs)(¥10

^{8}) are the mediator in the first stage. Outputs loan (Lo)(¥10

^{8}) and profit (PR)(¥10

^{8}) are outputs in the second stage.

## 5. Conclusions and Suggestions

#### 5.1. Conclusions

#### 5.2. Suggestions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Charnes, A.; Cooper, W.W.; Rhodes, E. Measuring the efficiency of decision-making units. Eur. J. Oper. Res.
**1978**, 2, 429–444. [Google Scholar] [CrossRef] - Ruggiero, J.; Vitaliano, D.F. Assessing the efficiency of public schools using data envelopment analysis and frontier regression. Contemp. Econ. Policy
**1999**, 17, 321–331. [Google Scholar] [CrossRef] - Wong, W.P.; Wong, K.Y. Supply chain performance measurement system using DEA modeling. Ind. Manag. Data Syst.
**2007**, 107, 361–381. [Google Scholar] [CrossRef] - Thanassoulis, E. Data envelopment analysis and its use in banking. Interfaces
**1999**, 29, 1–13. [Google Scholar] [CrossRef] - Mariappan, P.; Sreeaarthi, G. Application of data envelopment analysis (DEA) to study the performance efficiency of the scheduled commercial banks of India. Int. J. Mark. Technol.
**2013**, 3, 52–70. [Google Scholar] - Hwang, S.N.; Kao, T.L. Measuring Managerial Efficiency in Non-Life Insurance Companies: An Application of Two-Stage Data Envelopment Analysis. Int. J. Manag.
**2006**, 23, 699–720. [Google Scholar] - Seiford, L.M.; Zhu, J. Profitability and marketability of the top 55 U.S. commercial banks. Manag. Sci.
**1999**, 45, 1270–1288. [Google Scholar] [CrossRef] [Green Version] - Gregoriou, G.N.; Lusk, E.J.; Halperin, M. A Two-Staged Benchmarked Decision Support System Using DEA Profiles of Efficiency. Infor: Inf. Syst. Oper. Res.
**2008**, 46, 177–187. [Google Scholar] [CrossRef] - Deyneli, F. Analysis of relationship between efficiency of justice services and salaries of judges with two-stage DEA method. Eur. J. Law Econ.
**2012**, 34, 477–493. [Google Scholar] [CrossRef] - An, Q.; Chen, H.; Wu, J.; Liang, L. Measuring slacks-based efficiency for commercial banks in China by using a two-stage DEA model with undesirable output. Ann. Oper. Res.
**2015**. [Google Scholar] [CrossRef] - Bian, Y.; Liang, N.; Xu, H. Efficiency evaluation of Chinese regional industrial systems with undesirable factors using a two-stage slacks-based measure approach. J. Clean. Prod.
**2015**, 87, 348–356. [Google Scholar] [CrossRef] - Fatimah, S.; Mahmudah, U. Two-stage data envelopment analysis (DEA) for measuring the efficiency of elementary schools in Indonesia. Int. J. Environ. Sci. Educ.
**2017**, 12, 1971–1987. [Google Scholar] - Raheli, H.; Rezaei, R.M.; Jadidi, M.R.; Mobtaker, H.G. A two-stage DEA model to evaluate sustainability and energy efficiency of tomato production. Inf. Process. Agric.
**2017**, 4, 342–350. [Google Scholar] [CrossRef] - Liu, W.; Zhou, Z.; Ma, C.; Liu, D.; Shen, W. Two-stage DEA models with undesirable input-intermediate-outputs. Omega
**2015**, 56, 74–87. [Google Scholar] [CrossRef] - Chen, Y.; Li, Y.; Liang, L.; Salo, A.; Wu, H. Frontier projection and efficiency decomposition in two-stage processes with slacks-based measures. Eur. J. Oper. Res.
**2016**, 250, 543–554. [Google Scholar] [CrossRef] - Lim, S.; Zhu, J. A note on two-stage network DEA model: Frontier projection and duality. Eur. J. Oper. Res.
**2016**, 248, 342–346. [Google Scholar] [CrossRef] - Lim, S.; Zhu, J. Primal-dual correspondence and frontier projections in two-stage network DEA models. Omega
**2019**, 83, 236–248. [Google Scholar] [CrossRef] - Wu, J.; Yin, P.; Sun, J.; Chu, J.; Liang, L. Evaluating the environmental efficiency of a two-stage system with undesired outputs by a DEA approach: An interest preference perspective. Eur. J. Oper. Res.
**2016**, 254, 1047–1062. [Google Scholar] [CrossRef] - Chen, K.; Zhu, J. Second order cone programming approach to two-stage network data envelopment analysis. Eur. J. Oper. Res.
**2017**, 262, 231–238. [Google Scholar] [CrossRef] - Li, H.; Chen, C.; Cook, W.D.; Zhang, J.; Zhu, J. Two-stage network DEA: Who is the leader? Omega
**2018**, 74, 15–19. [Google Scholar] [CrossRef] - Zhang, L.; Chen, Y. Equivalent solutions to additive two-stage network data envelopment analysis. Eur. J. Oper. Res.
**2018**, 264, 1189–1191. [Google Scholar] [CrossRef] - Guo, C.; Roohollah, A.S.; Ali, A.F.; Zhu, J. Decomposition weights and overall efficiency in two-stage additive network DEA. Eur. J. Oper. Res.
**2017**, 257, 896–906. [Google Scholar] [CrossRef] - Davoodi, A.; Rezai, H.Z. Common set of weights in data envelopment analysis: A linear programming problem. Cent. Eur. J. Oper. Res.
**2012**, 20, 355–365. [Google Scholar] [CrossRef] - Andersen, P.; Petersen, N.C. A Procedure for Ranking Efficiency Units in Data Envelopment Analysis. Manag. Sci.
**1993**, 39, 1179–1297. [Google Scholar] [CrossRef] - Sexton, T.R.; Silkman, R.H.; Hogan, A.J. Data Envelopment Analysis: Critique and Extensions. In Measuring Efficiency: An Assessment of Data Envelopment Analysis; Silkman, R.H., Ed.; Jossey-Bass: San Francisco, CA, USA, 1986. [Google Scholar] [CrossRef]
- Aigner, D.; Lovell, C.A.K.; Schmidt, P. Formulation and estimation of stochastic frontier production function models. J. Econom.
**1977**, 6, 21–37. [Google Scholar] [CrossRef] - Zhu, J. Multi-factor performance measure model with an application to Fortune 500 companies. Eur. J. Oper. Res.
**2000**, 123, 105–124. [Google Scholar] [CrossRef] - Chen, T. Measuring operation, market and financial efficiency in the management of Taiwan’s banks. Serv. Mark. Q.
**2002**, 24, 15–27. [Google Scholar] [CrossRef] - Sexton, T.R.; Lewis, H.F. Two-stage DEA: An application to major league baseball. J. Product. Anal.
**2003**, 19, 227–249. [Google Scholar] [CrossRef] - Fare, R.; Grosskopf, S. Network DEA. Socio-Econ. Plan. Sci.
**2000**, 34, 35–49. [Google Scholar] [CrossRef] - Fare, R.; Grosskopf, S.; Whittaker, G. Network DEA. In Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis; Zhu, J., Cook, W.D., Eds.; Springer: Boston, MA, USA, 2007. [Google Scholar] [CrossRef]
- Kao, C. Network data envelopment analysis: A review. Eur. J. Oper. Res.
**2014**, 239, 1–16. [Google Scholar] [CrossRef] - Li, H.; Xiong, J.; Xie, J.; Zhou, Z.; Zhang, J. A Unified Approach to Efficiency Decomposition for a Two-Stage Network DEA Model with Application of Performance Evaluation in Banks and Sustainable Product Design. Sustainability
**2019**, 11, 4401. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**A two-stage DEA model. (θ

_{(1)}: the relative efficiency of a DMU in stage one. θ

_{(2)}: the elative efficiency of a DMU in stage two. θ

_{(3)}: the relative efficiency of a DMU in the overall stage.)

No. | DMU | Inputs | Mediator | Outputs | |||
---|---|---|---|---|---|---|---|

I_{1} | I_{2} | M_{1} | M_{2} | O_{1} | O_{2} | ||

1 | Taiwan-Fire | 1,178,744 | 673,512 | 7,451,757 | 856,735 | 984,143 | 681,687 |

2 | Chung-Kuo | 1,381,822 | 1,352,755 | 10,020,274 | 1,812,894 | 1,228,502 | 834,754 |

3 | Tai-Ping | 1,177,494 | 592,790 | 4,776,548 | 560,244 | 293,613 | 658,428 |

4 | China-Mriners | 601,320 | 594,259 | 3,147,851 | 371,863 | 248,709 | 177,331 |

5 | Fubon | 6,699,063 | 3,531,614 | 37,392,862 | 1,753,794 | 7,851,229 | 3,925,272 |

6 | Zurich | 2,627,707 | 668,363 | 9,747,908 | 952,326 | 1,713,598 | 415,058 |

7 | Taian | 1,942,833 | 1,443,100 | 10,685,457 | 643,412 | 2,239,593 | 439,039 |

8 | Ming-Tai | 3,789,001 | 1,873,530 | 17,267,266 | 1,134,600 | 3,899,530 | 622,868 |

9 | Central | 1,567,746 | 950,432 | 11,473,162 | 546,337 | 1,043,778 | 264,098 |

10 | First | 1,303,249 | 1,298,470 | 8,210,389 | 504,528 | 1,697,941 | 554,806 |

11 | Kuo-Hua | 1,962,448 | 672,414 | 7,222,378 | 643,178 | 1,486,014 | 18,259 |

12 | Union | 2,592,790 | 650,952 | 9,434,406 | 118,489 | 1,574,191 | 909,295 |

13 | Shing-Kong | 2,609,941 | 1,368,802 | 13,921,467 | 811,343 | 3,609,236 | 223,047 |

14 | South-China | 1,396,002 | 988,888 | 7,396,396 | 465,509 | 1,401,200 | 332,283 |

15 | Cathay-Century | 2,184,944 | 651,063 | 10,422,297 | 749,893 | 3,355,197 | 555,482 |

16 | Allianz-President | 1,211,716 | 415,071 | 5,606,013 | 402,881 | 854,054 | 197,947 |

17 | Newa | 1,453,797 | 1,085,019 | 7,695,461 | 342,489 | 3,144,484 | 371,984 |

18 | AIU | 757,515 | 547,997 | 3,631,484 | 995,620 | 692,371 | 163,927 |

19 | North-America | 159,422 | 182,338 | 1,141,950 | 483,291 | 519,121 | 46,857 |

20 | Federal | 145,442 | 53,518 | 316,829 | 131,920 | 355,624 | 26,537 |

No. | Stage 1 | Stage 2 | Overall Stage |
---|---|---|---|

1 | 0.993 | 0.738 | 0.984 |

2 | 0.998 | 0.635 | 1 |

3 | 0.690 | 1 | 0.988 |

4 | 0.718 | 0.451 | 0.488 |

5 | 0.838 | 1 | 1 |

6 | 1 | 0.466 | 0.602 |

7 | 0.752 | 0.577 | 0.538 |

8 | 0.726 | 0.563 | 0.495 |

9 | 1 | 0.294 | 0.371 |

10 | 0.862 | 0.721 | 0.811 |

11 | 0.741 | 0.416 | 0.333 |

12 | 0.905 | 1 | 1 |

13 | 0.811 | 0.596 | 0.530 |

14 | 0.725 | 0.561 | 0.518 |

15 | 1 | 0.796 | 1 |

16 | 0.907 | 0.432 | 0.499 |

17 | 0.723 | 1 | 0.838 |

18 | 0.794 | 0.389 | 0.468 |

19 | 1 | 0.455 | 1 |

20 | 0.934 | 1 | 1 |

No. | Inputs | Mediators | ||
---|---|---|---|---|

v_{1} | v_{2} | u_{1} | u_{2} | |

1 | 3.21 × 10^{−7} | 9.23 × 10^{−7} | 1.11 × 10^{−7} | 1.91 × 10^{−7} |

2 | 7.24 × 10^{−7} | 1×10^{−11} | 9.86 × 10^{−8} | 5.70 × 10^{−9} |

3 | 3.47 × 10^{−7} | 9.97 × 10^{−7} | 1.20 × 10^{−7} | 2.07 × 10^{−7} |

4 | 1.66 × 10^{−6} | 1 × 10^{−11} | 2.27 × 10^{−7} | 1.31 × 10^{−8} |

5 | 5.17 × 10^{−8} | 1.85 × 10^{−7} | 2.24 × 10^{−8} | 1 × 10^{−11} |

6 | 1 × 10^{−11} | 1.50 ×10^{−6} | 6.80 × 10^{−8} | 3.54 × 10^{−7} |

7 | 5.15 × 10^{−7} | 1 × 10^{−11} | 7.01 × 10^{−8} | 4.06 × 10^{−9} |

8 | 1.09 × 10^{−7} | 3.13 × 10^{−7} | 3.78 × 10^{−8} | 6.49 × 10^{−8} |

9 | 6.38 × 10^{−7} | 1 × 10^{−11} | 8.72 × 10^{−8} | 1 × 10^{−11} |

10 | 7.67 × 10^{−7} | 1 × 10^{−11} | 1.05 × 10^{−7} | 6.05 × 10^{−9} |

11 | 2.57 × 10^{−7} | 7.38 × 10^{−7} | 8.89 × 10^{−8} | 1.53 × 10^{−7} |

12 | 1 × 10^{−11} | 1.54 × 10^{−6} | 9.60 × 10^{−8} | 1 × 10^{−11} |

13 | 1.53 × 10^{−7} | 4.39 × 10^{−7} | 5.29 × 10^{−8} | 9.10 × 10^{−8} |

14 | 7.16 × 10^{−7} | 1 × 10^{−11} | 9.76 × 10^{−8} | 5.65 × 10^{−9} |

15 | 2.21 × 10^{−7} | 7.93 × 10^{−7} | 9.59 × 10^{−8} | 1 × 10^{−11} |

16 | 4.16 × 10^{−7} | 1.19 × 10^{−6} | 1.44 × 10^{−7} | 2.48 × 10^{−7} |

17 | 6.88 × 10^{−7} | 1 × 10^{−11} | 9.40 × 10^{−8} | 1 × 10^{−11} |

18 | 4.29 × 10^{−7} | 1.23 × 10^{−6} | 1.49 × 10^{−7} | 2.55 × 10^{−7} |

19 | 6.27 × 10^{−6} | 1 × 10^{−11} | 8.52 × 10^{−7} | 4.94 × 10^{−8} |

20 | 1 × 10^{−11} | 1.87 × 10^{−5} | 7.70 × 10^{−7} | 5.23 × 10^{−6} |

No. | Mediators | Outputs | ||
---|---|---|---|---|

u′_{1} | u′_{2} | w_{1} | w_{2} | |

1 | 1.10 × 10^{−7} | 2.10 × 10^{−7} | 1.06 × 10^{−7} | 9.30 × 10^{−7} |

2 | 9.98 × 10^{−8} | 1 × 10^{−11} | 3.61 × 10^{−8} | 7.08 × 10^{−7} |

3 | 2.09 × 10^{−7} | 1 × 10^{−11} | 7.57 × 10^{−8} | 1.49 × 10^{−6} |

4 | 2.59 × 10^{−7} | 4.94 × 10^{−7} | 2.51 × 10^{−7} | 2.19 × 10^{−6} |

5 | 2.46 × 10^{−8} | 4.67 × 10^{−8} | 2.37 × 10^{−8} | 2.07 × 10^{−7} |

6 | 7.03 × 10^{−8} | 3.31 × 10^{−7} | 1.47 × 10^{−7} | 5.18 × 10^{−7} |

7 | 7.29 × 10^{−8} | 3.43 × 10^{−7} | 1.52 × 10^{−7} | 5.38 × 10^{−7} |

8 | 4.42 × 10^{−8} | 2.08 × 10^{−7} | 9.23 × 10^{−8} | 3.26 × 10^{−7} |

9 | 7.12 × 10^{−8} | 3.35 × 10^{−7} | 1.49 × 10^{−7} | 5.25 × 10^{−7} |

10 | 9.45 × 10^{−8} | 4.45 × 10^{−7} | 1.97 × 10^{−7} | 6.97 × 10^{−7} |

11 | 9.06 × 10^{−8} | 5.38 × 10^{−7} | 2.80 × 10^{−7} | 1 × 10^{−11} |

12 | 4.56 × 10^{−8} | 4.81 × 10^{−6} | 6.35 × 10^{−7} | 1 × 10^{−11} |

13 | 5.34 × 10^{−8} | 3.17 × 10^{−7} | 1.65 × 10^{−7} | 1 × 10^{−11} |

14 | 1.04 × 10^{−7} | 4.91 × 10^{−7} | 2.18 × 10^{−7} | 7.69 × 10^{−7} |

15 | 7.17 × 10^{−8} | 3.38 × 10^{−7} | 1.50 × 10^{−7} | 5.28 × 10^{−7} |

16 | 1.33 × 10^{−7} | 6.28 × 10^{−7} | 2.78 ×10^{−7} | 9.83 × 10^{−7} |

17 | 1.03 × 10^{−7} | 6.10 × 10^{−7} | 3.18 × 10^{−7} | 1 × 10^{−11} |

18 | 2.75 × 10^{−7} | 1 × 10^{−11} | 9.96 × 10^{−8} | 1.95 × 10^{−6} |

19 | 8.76 × 10^{−7} | 1 × 10^{−11} | 3.17 × 10^{−7} | 6.21 × 10^{−6} |

20 | 3.16 × 10^{−6} | 1 × 10^{−11} | 2.63 × 10^{−6} | 1 × 10^{−11} |

No. | Inputs | Outputs | ||
---|---|---|---|---|

v′_{1} | v′_{2} | w_{1} | w_{2} | |

1 | 8.15 × 10^{−8} | 5.80 × 10^{−8} | 1 × 10^{−11} | 1.44 × 10^{−6} |

2 | 7.24 × 10^{−7} | 1 × 10^{−11} | 7.02 × 10^{−8} | 1.09 × 10^{−6} |

3 | 2.05 × 10^{−7} | 1.28 × 10^{−6} | 1 × 10^{−11} | 1.50 × 10^{−6} |

4 | 1.66 × 10^{−6} | 1 × 10^{−11} | 1 × 10^{−11} | 2.75 × 10^{−6} |

5 | 2.92 × 10^{−8} | 2.28 × 10^{−7} | 3.14 × 10^{−8} | 1.92 × 10^{−7} |

6 | 1 × 10^{−11} | 1.50 × 10^{−6} | 1.58 × 10^{−7} | 7.97 × 10^{−7} |

7 | 3.52 × 10^{−7} | 2.19 × 10^{−7} | 1.38 × 10^{−7} | 5.22 × 10^{−7} |

8 | 2.02 × 10^{−7} | 1.26 × 10^{−7} | 7.91 × 10^{−8} | 2.99 × 10^{−7} |

9 | 4.63 × 10^{−7} | 2.88 × 10^{−7} | 1.82 × 10^{−7} | 6.87 × 10^{−7} |

10 | 7.67 × 10^{−7} | 1 × 10^{−11} | 1.43 × 10^{−7} | 1.02 × 10^{−6} |

11 | 1 × 10^{−11} | 1.49 × 10^{−6} | 2.24 × 10^{−7} | 1 × 10^{−11} |

12 | 1 × 10^{−11} | 1.54 × 10^{−6} | 1.63 × 10^{−7} | 8.18 × 10^{−7} |

13 | 3.03 × 10^{−7} | 1.54 × 10^{−7} | 1.47 × 10^{−7} | 1 × 10^{−11} |

14 | 4.97 × 10^{−7} | 3.10 × 10^{−7} | 1.95 × 10^{−7} | 7.37 × 10^{−7} |

15 | 1 × 10^{−11} | 1.54 × 10^{−6} | 1.76 × 10^{−7} | 7.36 × 10^{−7} |

16 | 2.25 × 10^{−7} | 1.75 × 10^{−6} | 2.42 × 10^{−7} | 1.48 × 10^{−6} |

17 | 4.70 × 10^{−7} | 2.93 × 10^{−7} | 1.84 × 10^{−7} | 6.96 × 10^{−7} |

18 | 9.10 × 10^{−7} | 5.67 × 10^{−7} | 3.57 × 10^{−7} | 1.35 × 10^{−6} |

19 | 6.27 × 10^{−6} | 1 × 10^{−11} | 1.93 × 10^{−6} | 1 × 10^{−11} |

20 | 5.79 × 10^{−6} | 2.94 × 10^{−6} | 2.81 × 10^{−6} | 1 × 10^{−11} |

No. | Original Efficiency Scores θ_{k}_{(2)} | New Efficiency Scores θ*_{k}_{(2)} |
---|---|---|

1 | 0.738 | 0.728 |

2 | 0.635 | 0.633 |

3 | 1 | 0.689 |

4 | 0.451 | 0.309 |

5 | 1 | 0.687 |

6 | 0.466 | 0.444 |

7 | 0.577 | 0.279 |

8 | 0.563 | 0.329 |

9 | 0.294 | 0.198 |

10 | 0.721 | 0.483 |

11 | 0.416 | 0.202 |

12 | 1 | 0.674 |

13 | 0.596 | 0.292 |

14 | 0.561 | 0.283 |

15 | 0.796 | 0.655 |

16 | 0.432 | 0.349 |

17 | 1 | 0.355 |

18 | 0.389 | 0.296 |

19 | 0.455 | 0.454 |

20 | 1 | 0.873 |

No. | Original Efficiency Scores θ_{k}_{(2)} | New Efficiency Scores θ*_{k}_{(2)} | Difference |
---|---|---|---|

1 | 0.738 | 0.744 | 0.006 |

2 | 0.635 | 0.636 | 0.001 |

3 | 1 | 1.449 | 0.449 |

4 | 0.451 | 0.627 | 0.177 |

5 | 1 | 1.194 | 0.194 |

6 | 0.466 | 0.466 | 0 |

7 | 0.577 | 0.767 | 0.190 |

8 | 0.563 | 0.776 | 0.213 |

9 | 0.294 | 0.294 | 0 |

10 | 0.721 | 0.837 | 0.116 |

11 | 0.416 | 0.562 | 0.146 |

12 | 1 | 1.105 | 0.105 |

13 | 0.596 | 0.735 | 0.139 |

14 | 0.561 | 0.774 | 0.213 |

15 | 0.796 | 0.796 | 0 |

16 | 0.432 | 0.476 | 0.044 |

17 | 1 | 1.383 | 0.383 |

18 | 0.389 | 0.490 | 0.101 |

19 | 0.455 | 0.455 | 0 |

20 | 1 | 1.071 | 0.071 |

No. | Original Efficiency Scores θ_{k}_{(1)} | New Efficiency Scores θ*_{k}_{(1)} |
---|---|---|

1 | 0.993 | 0.873 |

2 | 0.998 | 0.982 |

3 | 0.690 | 0.655 |

4 | 0.718 | 0.709 |

5 | 0.838 | 0.761 |

6 | 1 | 1 |

7 | 0.752 | 0.714 |

8 | 0.726 | 0.665 |

9 | 1 | 1 |

10 | 0.862 | 0.859 |

11 | 0.741 | 0.716 |

12 | 0.905 | 0.858 |

13 | 0.811 | 0.759 |

14 | 0.725 | 0.7 |

15 | 1 | 1 |

16 | 0.907 | 0.877 |

17 | 0.723 | 0.681 |

18 | 0.794 | 0.723 |

19 | 1 | 1 |

20 | 0.934 | 0.41 |

No. | DMU | Inputs | Mediator | Outputs | ||||
---|---|---|---|---|---|---|---|---|

I_{1}(EMs) | I_{2}(FAs) | I_{3}(EXs) | M_{1}(CR) | M_{2}(ILs) | O_{1}(Lo) | O_{2}(PR) | ||

1 | Maanshan | 0.478 | 0.526 | 0.385 | 49.917 | 5.461 | 34.990 | 0.843 |

2 | Anqing | 1.236 | 0.713 | 0.555 | 37.495 | 4.083 | 20.601 | 0.486 |

3 | Huangshan | 0.446 | 0.443 | 0.342 | 20.985 | 0.690 | 8.633 | 0.129 |

4 | Fuyang | 1.248 | 0.638 | 0.457 | 45.051 | 1.724 | 9.235 | 0.302 |

5 | Suzhou | 0.705 | 0.575 | 0.404 | 38.163 | 2.249 | 12.017 | 0.314 |

6 | Chuzhou | 0.645 | 0.432 | 0.401 | 30.168 | 2.335 | 13.813 | 0.377 |

7 | Luan | 0.724 | 0.510 | 0.371 | 26.539 | 1.342 | 5.096 | 0.145 |

8 | Chizhou | 0.336 | 0.322 | 0.233 | 16.124 | 0.489 | 5.980 | 0.093 |

9 | Chaozhou | 0.668 | 0.423 | 0.347 | 22.185 | 1.177 | 9.235 | 0.200 |

10 | Bozhou | 0.342 | 0.256 | 0.159 | 13.436 | 0.406 | 2.533 | 0.006 |

No. | Stage 1 | Stage 2 | Overall Stage |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 0.554 | 0.786 | 0.434 |

3 | 0.499 | 1 | 0.293 |

4 | 0.759 | 0.970 | 0.301 |

5 | 0.729 | 0.833 | 0.355 |

6 | 0.736 | 1 | 0.545 |

7 | 0.552 | 0.632 | 0.179 |

8 | 0.533 | 1 | 0.282 |

9 | 0.553 | 1 | 0.328 |

10 | 0.650 | 0.498 | 0.175 |

No. | Original Efficiency Scores θ_{k}_{(2)} | New Efficiency Scores θ*_{k}_{(2)} | |
---|---|---|---|

Recomputed W* | Not Recomputed Weights | ||

1 | 1 | 1 | 1 |

2 | 0.786 | 0.434 | 1.419 |

3 | 1 | 0.293 | 2.003 |

4 | 0.970 | 0.301 | 1.277 |

5 | 0.833 | 0.355 | 1.143 |

6 | 1 | 0.545 | 1.359 |

7 | 0.632 | 0.179 | 1.145 |

8 | 1 | 0.282 | 1.878 |

9 | 1 | 0.328 | 1.809 |

10 | 0.498 | 0.175 | 0.766 |

No. | Original Efficiency Scores θ_{k}_{(1)} | New Efficiency Scores θ*_{k}_{(1)} |
---|---|---|

1 | 1 | 1 |

2 | 0.554 | 0.554 |

3 | 0.499 | 0.499 |

4 | 0.759 | 0.759 |

5 | 0.729 | 0.729 |

6 | 0.736 | 0.736 |

7 | 0.552 | 0.552 |

8 | 0.533 | 0.533 |

9 | 0.553 | 0.553 |

10 | 0.650 | 0.650 |

No. | Stage 1 | Stage 2 | Overall Stage |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 0.554 | 0.786 | 0.434 |

3 | 1 | 1 | 0.587 |

4 | 0.759 | 0.970 | 0.301 |

5 | 0.729 | 0.833 | 0.355 |

6 | 0.736 | 1 | 0.545 |

7 | 0.552 | 0.632 | 0.179 |

8 | 0.533 | 1 | 0.282 |

9 | 0.553 | 1 | 0.328 |

10 | 0.650 | 0.498 | 0.175 |

© 2020 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**

Tsai, M.-C.; Cheng, C.-H.; Nguyen, V.T.; Tsai, M.-I.
The Theoretical Relationship between the CCR Model and the Two-Stage DEA Model with an Application in the Efficiency Analysis of the Financial Industry. *Symmetry* **2020**, *12*, 712.
https://doi.org/10.3390/sym12050712

**AMA Style**

Tsai M-C, Cheng C-H, Nguyen VT, Tsai M-I.
The Theoretical Relationship between the CCR Model and the Two-Stage DEA Model with an Application in the Efficiency Analysis of the Financial Industry. *Symmetry*. 2020; 12(5):712.
https://doi.org/10.3390/sym12050712

**Chicago/Turabian Style**

Tsai, Ming-Chi, Ching-Hsue Cheng, Van Trung Nguyen, and Meei-Ing Tsai.
2020. "The Theoretical Relationship between the CCR Model and the Two-Stage DEA Model with an Application in the Efficiency Analysis of the Financial Industry" *Symmetry* 12, no. 5: 712.
https://doi.org/10.3390/sym12050712