Data envelopment analysis (DEA) is a nonparametric method for evaluating the relative efficiency of a set of decision-making units (DMUs) with multiple inputs and outputs. As an extension of the DEA, a multiplicative two-stage DEA model has been widely used to measure the efficiencies of two-stage systems, where the first stage uses inputs to produce the outputs, and the second stage then uses the first-stage outputs as inputs to generate its own outputs. The main deficiency of the multiplicative two-stage DEA model is that the decomposition of the overall efficiency may not be unique because of the presence of alternate optima. To remove the problem of the flexible decomposition, in this paper, we maximize the sum of the two-stage efficiencies and simultaneously maximize the two-stage efficiencies as secondary goals in the multiplicative two-stage DEA model to select the decomposition of the overall efficiency from the flexible decompositions, respectively. The proposed models are applied to evaluate the performance of 10 branches of China Construction Bank, and the results are compared with the results of the existing models.

Data envelopment analysis (DEA) that was introduced by Charnes et al. is a nonparametric method for evaluating the relative efficiency of a set of decision-making units (DMUs) with multiple inputs and outputs [

Halkos et al. classified the existing two-stage DEA models into four categories: independent two-stage DEA, connected two-stage DEA, relational two-stage DEA, and game theoretic two-stage DEA models [

In the additive model, Chen et al. and Guo et al. modeled the overall efficiency as the weighted sum of the two individual efficiencies [

In the multiplicative model, Kao and Hwang defined the overall efficiency of all of the DMUs as the product of the two individual efficiencies [

To obtain the decomposition of the overall efficiency, Kao et al. and Liang et al. both suggested obtaining an optimal solution, which produces the largest individual efficiency score in first- or second-stage while maintaining the overall efficiency score that is unchanged [

The rest of the paper is organized as follows: Section

There are two types of relational two-stage DEA models, including the multiplicative two-stage DEA model proposed by Kao and Hwang and the additive two-stage DEA model proposed by Chiou et al. [

Consider a two-stage process with _{k} (_{k} in the first and second stages can be measured independently using equations (

In the multiplicative two-stage DEA model, Kao and Hwang define the overall efficiency of the DMU_{k} as the product of the two individual efficiencies, and their model for measuring the overall efficiency of the DMU_{k} is given as follows [_{k}. By applying Charnes and Cooper’s transformation, model (

After the optimal solutions _{k} is computed subsequently as

However, the optimal solutions of model (

Once the individual maximal achievable efficiency score

The main problem of the DEA model is the nonuniqueness of the optimal solutions; therefore, it is recommended that secondary goals be introduced in the DEA model to be used for choosing weights from the optimal solutions of the model. In the one-stage DEA model, various forms of secondary goals are proposed to deal with the nonuniqueness issue, for example, selecting symmetric weights as a secondary goal, optimizing the rank position of the DMU as secondary goal [

As mentioned before, in the multiplicative two-stage DEA model (

The existing secondary goals in DEA models.

References | Secondary goal |
---|---|

Dimitrov and Sutton [ | Promoting symmetric weight selection |

Contreras [ | Optimizing the rank position of the DMU |

Liang et al. [ | Minimizing the total deviation from the ideal point, minimizing the maximum d-efficiency score, minimizing the mean absolute deviation |

Wu et al. [ | Selecting weights based on both desirable and undesirable cross-efficiency targets |

Kao and Hwang [ | Maximizing one stage’s efficiency |

Zhou et al. [ | Minimizing the individual efficiency of the other party |

An et al. [ | Making the ratio of decomposed divisional efficiencies close to the reference point |

Fang [ | Assessing the stage efficiencies by considering the different and DMU-specific degree of priority given to the stages. |

The choice of secondary goals depends on two aspects: the relationship between two stages and decision maker’s preference in the particular application. If one stage is the leader and the other one is a follower, then Kao and Hwang’s decomposition method should be used; if we view the two stages as two parties, then the Nash bargaining game model proposed by Zhou et al. should be used [

We first maximize the sum of two individual efficiencies as a secondary goal in the multiplicative two-stage DEA model. Given the optimal multiplicative efficiency

Let

Let

The maximal and minimal achievable efficiency scores

It is obvious that the objective function in model (

Step 1: find the maximal achievable efficiency score

Solve model (

Step 2: find the minimal achievable efficiency score

Solve model (

Step 3: compare the objective function values to obtain the optimal solution.

To obtain the optimal solution of model (

If

If the relative importance or the contribution of the performances is considered, the secondary goal in model (

We then can determine the optimal solutions

Similar to the analysis in Section

Model (

We use the fuzzy programming method to solve model (

In model (

The expressions

The procedure for finding the fuzzy efficient solution of model (

Step 1: construct the membership functions.

In model (

Step 2: build an auxiliary crisp model.

Introducing an auxiliary variable

The optimal solution to model (

Step 3: solve model (

Model (

The above max–min’s approach for transforming model (

Model (

In this section, a numerical example is used to illustrate the proposed method. In order to compare the results determined by the proposed model with the results of the existing model, a numerical example used by Yang et al. [

Dataset of 10 bank branches (reproduced from Zhou et al., 2013).

DMU | Inputs ( | Intermediate ( | Outputs ( | |||||
---|---|---|---|---|---|---|---|---|

No. | Branch | EM (10^{3}) | FA (￥10^{8}) | EX (￥10^{8}) | CR (￥10^{8}) | IL (￥10^{8}) | LO (￥10^{8}) | PR (￥10^{8}) |

1 | Maanshan | 0.526 | 0.4775 | 0.3848 | 49.9174 | 5.4613 | 34.9897 | 0.8430 |

2 | Anqing | 0.713 | 1.2363 | 0.5547 | 37.4954 | 4.0825 | 20.6013 | 0.4864 |

3 | Huangshan | 0.443 | 0.4460 | 0.3419 | 20.9846 | 0.6897 | 8.6332 | 0.1288 |

4 | Fuyang | 0.638 | 1.2481 | 0.4574 | 45.0508 | 1.7237 | 9.2354 | 0.3019 |

5 | Suzhou | 0.575 | 0.7050 | 0.4036 | 38.1625 | 2.2492 | 12.0171 | 0.3138 |

6 | Chuzhou | 0.432 | 0.6446 | 0.4012 | 30.1676 | 2.3354 | 13.813 | 0.3772 |

7 | Luan | 0.510 | 0.7239 | 0.3709 | 26.5391 | 1.3416 | 5.0961 | 0.1453 |

8 | Chizhou | 0.322 | 0.3363 | 0.2334 | 16.1235 | 0.4889 | 5.9803 | 0.0928 |

9 | Chaohu | 0.423 | 0.6678 | 0.3471 | 22.1848 | 1.1767 | 9.2348 | 0.2002 |

10 | Bozhou | 0.256 | 0.3418 | 0.1594 | 13.4364 | 0.4064 | 2.5326 | 0.0057 |

In this example, Zhou et al. have calculated the overall efficiency scores and the maximal and minimal achievable efficiency scores of the two stages for each DMU using models (_{1} performed efficiently in both stages and in the whole system. For the remaining nine DMUs, because the maximal and minimal achievable efficiency scores for each stage are not the same, some flexibility exists in the efficiency decomposition between the two stages for those DMUs. Therefore, it is necessary to use a secondary goal to select the reasonable decomposition of the overall efficiency from the flexible decompositions.

Efficiency scores and ranks of 10 bank branches determined by Kao and Hwang’s models.

DMU | Model ( | Model ( | Model ( | ||
---|---|---|---|---|---|

1 | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) |

2 | 0.4344 (3) | 0.5541 (6) | 0.7838 (2) | 0.5526 (2) | 0.7861 (8) |

3 | 0.2930 (7) | 0.4991 (10) | 0.5869 (5) | 0.2930 (9) | 1.0000 (1) |

4 | 0.3013 (6) | 0.7593 (2) | 0.3968 (8) | 0.3209 (7) | 0.9389 (4) |

5 | 0.3549 (4) | 0.7289 (4) | 0.4869 (7) | 0.4304 (4) | 0.8246 (7) |

6 | 0.5448 (2) | 0.7359 (3) | 0.7404 (3) | 0.5448 (3) | 1.0000 (1) |

7 | 0.1788 (9) | 0.5516 (8) | 0.3242 (9) | 0.2881 (10) | 0.6206 (9) |

8 | 0.2818 (8) | 0.5325 (9) | 0.5291 (6) | 0.3052 (8) | 0.9232 (5) |

9 | 0.3282 (5) | 0.5526 (7) | 0.5939 (4) | 0.3845 (5) | 0.8535 (6) |

10 | 0.1747 (10) | 0.6498 (5) | 0.2689 (10) | 0.3722 (6) | 0.4695 (10) |

Mean | 0.3892 | 0.6564 | 0.5711 | 0.4492 | 0.8416 |

SD | 0.2411 | 0.1537 | 0.2236 | 0.2163 | 0.1771 |

Min | 0.1747 | 0.4991 | 0.2689 | 0.2881 | 0.4695 |

Max | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

We first apply model (

Efficiency scores and ranks of 10 bank branches by the proposed models (

DMU | Proposed model ( | Proposed model ( | ||||||
---|---|---|---|---|---|---|---|---|

1 | 1.0000 (1) | 1.0000 (1) | 2.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 2.0000 (1) | 1.0000 (1) |

2 | 0.5526 (3) | 0.7861 (8) | 1.3387 (3) | 0.4344 (3) | 0.5534 (4) | 0.7850 (3) | 1.3384 (3) | 0.4344 (3) |

3 | 0.2930 (9) | 1.0000 (1) | 1.2930 (4) | 0.2930 (7) | 0.3961 (10) | 0.7398 (4) | 1.1359 (6) | 0.2930 (7) |

4 | 0.3209 (7) | 0.9389 (4) | 1.2598 (5) | 0.3013 (6) | 0.5401 (5) | 0.5579 (8) | 1.0980 (7) | 0.3013 (6) |

5 | 0.4304 (5) | 0.8246 (7) | 1.2550 (6) | 0.3549 (4) | 0.5797 (3) | 0.6123 (7) | 1.1919 (4) | 0.3549 (4) |

6 | 0.5448 (4) | 1.0000 (1) | 1.5448 (2) | 0.5448 (2) | 0.6404 (2) | 0.8508 (2) | 1.4911 (2) | 0.5448 (2) |

7 | 0.2881 (10) | 0.6206 (9) | 0.9087 (10) | 0.1788 (9) | 0.4199 (8) | 0.4259 (9) | 0.8457 (10) | 0.1788 (9) |

8 | 0.3052 (8) | 0.9232 (5) | 1.2284 (8) | 0.2818 (8) | 0.4189 (9) | 0.6728 (6) | 1.0916 (8) | 0.2818 (8) |

9 | 0.3845 (6) | 0.8535 (6) | 1.2380 (7) | 0.3282 (5) | 0.4686 (7) | 0.7005 (5) | 1.1690 (5) | 0.3282 (5) |

10 | 0.6498 (2) | 0.2689 (10) | 0.9187 (9) | 0.1747 (10) | 0.5110 (6) | 0.3419 (10) | 0.8529 (9) | 0.1747 (10) |

Mean | 0.4769 | 0.8216 | 1.2985 | 0.3892 | 0.5528 | 0.6687 | 1.2215 | 0.3892 |

SD | 0.2231 | 0.2280 | 0.3096 | 0.2411 | 0.1756 | 0.1954 | 0.3354 | 0.2411 |

Min | 0.2881 | 0.2689 | 0.9087 | 0.1747 | 0.3961 | 0.3419 | 0.8457 | 0.1747 |

Max | 1.0000 | 1.0000 | 2.0000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 1.0000 |

Comparing the proposed model (_{10} and the results from Kao and Hwang’s model (

Comparing the proposed model (_{1}, DMU_{5}, DMU_{7}, and DMU_{10}, and the ranks of the second-stage efficiency scores determined by the proposed model (_{5}, DMU_{6}, the ranks of the first-stage efficiency score determined by the proposed model (

In the following, to obtain some interesting and valuable findings, we further compare the results determined by the proposed models with the results of the existing models.

The common feature of the proposed model (

Efficiency scores and ranks of 10 bank branches determined by the existing models.

DMU | Chiou et al.’s model | Zhou et al.’s model | ||||||
---|---|---|---|---|---|---|---|---|

1 | 1.0000 (1) | 1.0000 (1) | 2.0000 (1) | 1.0000 (1) | 1.0000 (1) | 1.0000 (1) | 2.0000 (1) | 1.0000 (1) |

2 | 0.5526 (3) | 0.7861 (8) | 1.3387 (3) | 0.4344 (3) | 0.5534 (4) | 0.7850 (3) | 1.3384 (3) | 0.4344 (3) |

3 | 0.2930 (9) | 1.0000 (1) | 1.2930 (5) | 0.2930 (7) | 0.3824 (10) | 0.7661 (4) | 1.1485 (6) | 0.2930 (7) |

4 | 0.3070 (6) | 0.9699 (5) | 1.2769 (6) | 0.2977 (5) | 0.4936 (5) | 0.6104 (8) | 1.1040 (7) | 0.3013 (6) |

5 | 0.4268 (5) | 0.8293 (7) | 1.2561 (7) | 0.3539 (4) | 0.5601 (3) | 0.6337 (7) | 1.1938 (4) | 0.3549 (4) |

6 | 0.5448 (4) | 1.0000 (1) | 1.5448 (2) | 0.5448 (2) | 0.6332 (2) | 0.8605 (2) | 1.4937 (2) | 0.5448 (2) |

7 | 0.2801 (10) | 0.6316 (9) | 0.9117 (10) | 0.1769 (9) | 0.3987 (9) | 0.4485 (9) | 0.8472 (9) | 0.1788 (9) |

8 | 0.3052 (7) | 0.9232 (6) | 1.2284 (8) | 0.2818 (8) | 0.4032 (8) | 0.6989 (6) | 1.1021 (8) | 0.2818 (8) |

9 | 0.2968 (8) | 0.9988 (4) | 1.2957 (4) | 0.2965 (6) | 0.4610 (7) | 0.7119 (5) | 1.1729 (5) | 0.3282 (5) |

10 | 0.6498 (2) | 0.2689 (10) | 0.9187 (9) | 0.1747 (10) | 0.4918 (6) | 0.3553 (10) | 0.8471 (10) | 0.1747 (10) |

Mean | 0.4656 | 0.8408 | 1.3064 | 0.3854 | 0.5377 | 0.6870 | 1.2247 | 0.3892 |

SD | 0.2306 | 0.2353 | 0.3082 | 0.2425 | 0.1811 | 0.1886 | 0.3351 | 0.2411 |

Min | 0.2801 | 0.2689 | 0.9117 | 0.1747 | 0.3824 | 0.3553 | 0.8471 | 0.1747 |

Max | 1.0000 | 1.0000 | 2.0000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 1.0000 |

Comparing the proposed model (_{1}, DMU_{2}, DMU_{3}, DMU_{6}, DMU_{8}, and DMU_{10}, the results of the optimal efficiency decomposition as determined by the proposed model (_{4} is 7/4 in the proposed model (_{4} is 5/6 in the proposed model (

To further compare the results determined by the proposed model (

Spearman’s rank correlation coefficients tests further show that the multiplicative and additive models are two different models but that they are equally valid ways of aggregating the components of a two-stage process, as Chen et al. explained [

The comparisons above offer some valuable insights for the choice of the models: (i) If the two-stage DEA model is used to determine the efficiencies scores of all of the DMUs, then the choice of the models depends on the decision maker’s preference in the particular application. (ii) If the two-stage DEA model is used to determine the efficiencies ranks of all of the DMUs, then the multiplicative model should be selected. The rank that is determined by the multiplicative model and additive model has the same direction, and the multiplicative model is easy to solve to optimality.

The main purpose of the proposed model (

Comparing the proposed model (_{1} is relatively efficient and the remaining nine DMUs are relatively inefficient in both stages and in the whole system. Particularly, the efficiency scores for the second stage of DMU_{3} and DMU_{6} are less than 1, which is different from Chiou et al.’s model and the proposed model (_{7} is eight in the proposed model (_{10} is nine in the proposed model (

Spearman’s rank correlation coefficient test is also used to assess the rank correlations of the efficiency scores for two models, and the result shows that the rank that is determined by the proposed model (

Kao and Hwang proposed a multiplicative two-stage DEA model by taking into account the series relationship of the two subprocesses within the whole production process, and the overall efficiency is the product of the efficiencies of the two subprocesses. However, as Kao and Hwang mentioned, the optimal weights that are solved from their model may not be unique. Consequently, some flexibility exists in the efficiency decomposition between the two stages [

To solve this problem, we maximize the sum of the two individual efficiencies and simultaneously maximize the two individual efficiencies, respectively, as a secondary goal in the multiplicative two-stage DEA model to select the decomposition of the overall efficiency from the flexible decompositions. The proposed models with their different secondary goals are applied to a numerical example for the efficiency decomposition of 10 bank branches, and the results are compared with the results of the existing models.

The contributions of the paper are summarized as follows: (i) the proposed methods provide more alternative ways for the decomposition of the overall efficiency in the multiplicative two-stage DEA model. (ii) The proposed models are easy to solve to optimality and can be easily extended to the case for which the weight of the two stages is considered in the secondary goal without increasing the difficulty for solving it. (iii) Some interesting and valuable findings for the choice of the models are obtained by the comparisons of the results.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this article.