Data Envelopment Analysis (DEA) is a mathematical programming approach to measure the relative efficiency of peer decision making units (DMUs) which use multiple inputs to produce multiple outputs. One of the drawbacks of traditional DEA models is the neglect of internal structures of the DMUs. Network DEA models are able to overcome the shortcoming of the traditional DEA models. In network DEA a DMU is made up of some divisions linked together by intermediate products. An intermediate product has the dual role of output from one division and input to another one. Improving the efficiency of one process may reduce the efficiency of another process. To address the conflict caused by the dual role of intermediate measures, this paper presents a new approach which categorizes the intermediate measures into either input or output type endogenously, while keeping the continuity of link flows between divisions. This categorization allows us to measure the inefficiencies associated with intermediate measures and account their indirect effects on the objective function. In this paper we propose a new Slacksbased measure which includes any nonzero slacks identified by the model and inherits the properties of monotonicity in slacks and units invariance from the conventional SBM approach.
Data Envelopment Analysis (DEA), developed by Charnes et al. [
The simplest structure of network systems is a twostage system composed of two processes connected in series. Besides inputs and outputs, there are a set of intermediate measures that link these two stages together. The intermediate measures play the role of outputs from the first stage and inputs to the second stage at the same time. Several models have been proposed to measure the efficiency of this type of system (see the review of Cook et al. [
Many researchers propose solutions to address the potential conflict caused by the dual role of intermediate measures. There are four types of papers that use various approaches for measuring efficiency of DMUs with twostage processes.
In the first type, two separate DEA runs are applied to the stages to measure the relative efficiency of each stage separately. [
Another type of researches is called “Efficiency Decomposition Methodology,” as in Kao and Hwang [
The third type of modeling called “Game theoretic approaches” originated from the work of Liang et al. [
In the case that there are additional independent inputs to second stage and the second stage has its own inputs not linked with the first stage, “Network DEA” approach is introduced to the literature of DEA. Färe and Grosskopf [
Despotis et al. [
Tone and Tsutsui [
Tone and Tsutsui [
Lozano [
F.h. F. Liu and Y.c. Liu [
As we discussed above the dual role of intermediate products is an issue that needs to be addressed in network DEA. In this paper we propose two new network DEA models in the slacksbased measure (SBM) framework, called Model (I) and Model (II), in which the intermediate products are categorized into either input or output type. The proposed models compute the input excesses and output shortfalls associated with intermediate measures and keep the continuity of link flows between divisions. Model (II) is able to take into account the inefficiency associated with the link variables.
The rest of this paper is structured as follows; Section
In this section the network SBM approaches of Tone and Tsutsui [
In this approach the divisional efficiency is evaluated individually. The weighted average of each division gives the overall efficiency of a DMU. In this case, for evaluating the efficiency of
Suppose that there are a set of
Tone and Tsutsui [
It should be noted that the above model assumes the variable returnstoscale (VRS) for production and by removing the last constraint
Regarding linking constraints, they proposed two possible cases called “
In the next section we propose our new network models.
As we discussed in previous section the linking constraints proposed by Tone and Tsutsui [
Incorporation of the slacks of intermediate measures in objective function allows us to incorporate the inefficiency associated with intermediate measures in efficiency measurement directly.
We present Model (I) as follows:
Note that if
In other words the proposed model classifies the intermediate measures into input or output type. The proposed model also identifies nonzero slacks and uncovers the sources of inefficiency associated with intermediate measures. Since the optimal values of intermediate measures can be equal, above, or below the observed value the proposed model corresponds to the free link case.
Set of constraints (
The outputoriented efficiency of DMU
And the outputoriented divisional efficiency for
Similarly the inputoriented efficiency of
And the inputoriented divisional efficiency for
The projected DMU in Model (I) is overall efficient.
We prove the theorem in the nonoriented case.
Let
And let (
If only one of
Although the slacks of intermediate products in Model (I) are not included in the objective function, their indirect effect on the objective function incorporates inefficiency corresponding to intermediate measures in efficiency measurement. In order to include the inefficiency associated with intermediate measure in the objective function directly, we propose Model (II) that minimizes the objective function (
Neglecting the constraints (
The slack based measure (
To measure the nonoriented divisional efficiency score applying the direct effect of intermediate slacks on efficiency score we use the following formula:
To evaluate the inputoriented efficiency score of DM
The efficiency score in the outputoriented case for DMU_{p} can be evaluated from following model.
The projected DMU in Model (II) is overall efficient.
We prove the theorem in the nonoriented case.
Let
And let
Hence we have the overall efficiency as follows:
It should be noted that, in the case
In this section to illustrate our proposed models, we will use a numerical example and compare the results of our proposed models with some existing approaches in SBM framework. Table
Exhibits data for inputs, outputs, and links of the ten DMUs in their numerical example; data for inputs, outputs, and links of the ten DMUs presented by Tone and Tsutsui [
DMU  Generation process 
Transmission process 
Distribution process 
links  

Input 1 
Input 2 
Output 2 
Input 3 
Output 3 
Link 12 
Link 23 

DMU1  0.838  0.277  0.879  0.962  0.337  0.894  0.362 
DMU2  1.233  0.132  0.538  0.443  0.180  0.678  0.188 
DMU3  0.321  0.045  0.911  0.482  0.198  0.836  0.207 
DMU4  1.483  0.111  0.570  0.467  0.491  0.869  0.516 
DMU5  1.592  0.208  1.086  1.073  0.372  0.693  0.407 
DMU6  0.790  0.139  0.722  0.545  0.253  0.966  0.269 
DMU7  0.451  0.075  0.509  0.366  0.241  0.647  0.257 
DMU8  0.408  0.074  0.619  0.229  0.097  0.756  0.103 
DMU9  1.864  0.061  1.023  0.691  0.380  1.191  0.402 
DMU10  1.222  0.149  0.769  0.337  0.178  0.792  0.187 
Consider the dataset provided by Tone and Tsutsui [
Network structure of vertically integrated electric power companies.
In this section first, we solved the black box model using Inputs 1, 2, and 3 and Outputs 2 and 3 where links were neglected. The column “black box” in Table
SBM scores for black box and proposed models.
DMU  Black Box  Model (I)  Model (II)  

Overall efficiency score  Overall efficiency score 






 
DMU1  1.00  0.385  0.383  0.383  0.389  0.441  0.383  0.659  0.389 
DMU2  0.54  0.433  0.260  0.341  0.652  0.433  0.26  0.341  0.652 
DMU3  1.00  0.968  1.00  1.00  0.919  0.968  1  1  0.919 
DMU4  1.00  0.719  0.297  1.00  1.00  0.719  0.297  1  1 
DMU5  1.00  0.456  0.263  1  0.377  0.456  0.263  1  0.377 
DMU6  0.681  0.484  0.406  0.420  0.593  0.608  0.406  0.643  0.792 
DMU7  1.00  0.778  0.712  0.740  0.863  0.778  0.712  0.74  0.863 
DMU8  1.00  0.969  0.922  1.00  1.00  0.969  0.922  1  1 
DMU9  1.00  0.832  1.00  1.00  0.581  0.832  1  1  0.581 
DMU10  1.00  0.506  0.271  0.338  0.825  0.506  0.271  0.338  0.825 


Average  0.9221  0.602  0.524  0.688  0.637  0.620  0.524  0.738  0.657 
Next, we solved the two proposed models explained in Sections
Throughout this section, we used the inputoriented SBM (slacksbased measure) under the variable returnstoscale (VRS) assumption for efficiency evaluation.
Figure
Comparisons of scores between black box and proposed models.
In this section we compare our proposed model with separation model.
In order to take into account the inefficiency associated with link flows, there is another approach to evaluate divisional efficiency individually called
Table
SBM scores for separation model.
DMU  Separation model  

Overall score 


 
DMU1  0.659  0.633  0.662  0.684 
DMU2  0.657  0.26  0.763  1.00 
DMU3  0.984  1.00  1.00  0.959 
DMU4  0.719  0.297  1.00  1.00 
DMU5  0.547  0.202  1.00  0.665 
DMU6  0.844  1.00  0.635  0.792 
DMU7  0.855  0.712  1.00  0.926 
DMU8  0.893  0.787  0.890  1.00 
DMU9  0.915  1  1  0.786 
DMU10  0.640  0.263  0.672  1 
Model (II) and separation model both take into the account the inefficiency associated with link flows. In proposed models the continuity of link values between divisions is assured whereas in separation model it is not. Figure
Comparisons of scores between proposed models and separation approach.
In this section we aim to compare the scores given by SlacksBased Network DEA Models (proposed by Tone and Tsutsui [
Slacksbased network DEA.
DMU  Free link case  Fixed link case  








 
DMU1  0.385  0.383  0.383  0.389  0.477  0.633  0.339  0.393 
DMU2  0.433  0.260  0.341  0.652  0.740  0. 349  1.000  1.000 
DMU3  0.968  1.000  1.000  0.919  0.968  1.000  1.000  0.919 
DMU4  0.719  0.297  1.000  1.000  0.719  0.297  1.000  1.000 
DMU5  0.456  0.263  1.000  0.377  0.456  0.263  1.000  0.377 
DMU6  0.484  0.406  0.420  0.593  0.719  1.000  0.403  0.596 
DMU7  0.778  0.712  0.740  0.863  0.947  1.000  1.000  0.868 
DMU8  0.969  0.922  1.000  1.000  0.969  0.922  1.000  1.000 
DMU9  0.832  1.000  1.000  0.581  0.832  1.000  1.000  0.581 
DMU10  0.506  0.271  0.338  0.825  0.590  0.287  0.376  1.000 


Avarage  0.653  0.551  0.722  0.720  0.742  0.675  0.812  0.773 
Based on the results shown in Table
The advantage of applying Model (I) instead of free link case is that we can find out the intermediate products are being viewed as inputs or outputs in the system.
The linking constraints in fixed link case are tighter than free link case and the proposed models; hence the overall scores of the fixed link case exceed or are equal to those of the free case and Model (I) for every DMU. Figure
Comparisons of scores between proposed network models and NSBM models proposed by Tone and Tsutsui.
Model (II) takes into account the inefficiency associated with the link variables, whereas the NSBM does not.
Comparing the SBM scores obtained by Model (I) and Model (II) shows that incorporation of intermediate product slacks in efficiency measurement may increase or decrease the divisional or overall efficiency (see Figure
As it can be seen from Table
Table
Exhibition of the role of intermediate measures in proposed models.
DMU  Model (I)  Model (II)  






DMU1  Input to 
Input to 
Input to 
Output from 
DMU2  Output from 
Output from 
Output from 
Output from 
DMU3  Input to 
Input to 
Input to 
Output from 
DMU4  Input to 
Input to 
Input to 
Input to 
DMU5  Input to 
Input to 
Input to 
Output from 
DMU6  Input to 
Input to 
Input to 
Input to 
DMU7  Output from 
Input to 
Output from 
Output from 
DMU8  Input to 
Input to 
Input to 
Output from 
DMU9  Input to 
Input to 
Output from 
Output from 
DMU10  Output from 
Output from 
Output from 
Output from 
Optimum slack variables in Model (I).








 

DMU1 
0.517  0.171  0.588  0.000  0.000  0.000  0.058  0.000  0.007 
DMU2 
0.912  0.087  0.154  0.373  0.016  0.158  0.000  0.019  0.000 
DMU3 
0.00  0.000  0.039  0.000  0.000  0.000  0.000  0.000  0.000 
DMU4 
1.043  0.000  0.00  0.000  0.000  0.000  0.000  0.000  0.000 
DMU5 
1.173  0.000  0.669  0.000  0.015  0.000  0.000  0.000  0.002 
DMU6 
0.469  0.081  0.222  0.112  0.000  0.000  0.130  0.000  0.003 
DMU7 
0.130  0.019  0.050  0.341  0.000  0.189  0.000  0.000  0.000 
DMU8 
0.032  0.000  0.00  0.000  0.000  0.000  0.000  0.000  0.000 
DMU9 
0.00  0.000  0.290  0.000  0.002  0.000  0.000  0.000  0.000 
DMU10 
0.891  0.099  0.059  0.088  0.000  0.029  0.000  0.001  0.000 
Optimum slack variables in Model (II).








 

DMU1 
0.517  0.171  0.588  0.000  0.000  0.000  0.058  0.000  0.000 
DMU2 
0.912  0.087  0.154  0.373  0.016  0.158  0.000  0.019  0.000 
DMU3 
0.000  0.000  0.039  0.000  0.000  0.000  0.000  0.000  0.000 
DMU4 
1.043  0.000  0.000  0.000  0.000  0.000  0.130  0.000  0.000 
DMU5 
1.173  0.000  0.669  0.000  0.015  0.000  0.000  0.000  0.000 
DMU6 
0.469  0.081  0.222  0.112  0.000  0.000  0.000  0.000  0.002 
DMU7 
0.130  0.019  0.050  0.341  0.000  0.189  0.000  0.000  0.000 
DMU8 
0.032  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000 
DMU9 
0.000  0.000  0.290  0.000  0.002  0.000  0.000  0.000  0.000 
DMU10 
0.891  0.099  0.059  0.088  0.000  0.029  0.000  0.001  0.000 
In this paper in order to address the conflict caused by the dual role of intermediate measures we proposed two alternative slacksbased measure models, called Model (I) and Model (II). To resolve this conflict the intermediate measures are categorized into input or output type endogenously. These categorizations allow models to identify nonzero slacks and uncover the sources of inefficiency associated with intermediate measures. In Model (I) the excesses or shortfalls corresponding to intermediate measures are contributed to the optimum objective indirectly. In order to incorporate the direct effect of inefficiency associated with intermediate measures, we proposed Model (II) in which the average reduction or expansion rate of intermediate products has been taken into the account in the objective function. Keeping continuity of link flows between divisions and incorporation of link flows in efficiency measurements at the same time is the clear advantage of our proposed model over other approaches.
To verify our proposed models we provided a numerical example and we compared the results with black box model, separation model, and Slacksbased network DEA (free link and fixed link case). In comparing the scores obtained by proposed network models and black box model, no significant correlation between the efficiency scores was found and the trends of network models were in sharp contrast to that of black box model. It was not unexpected because the internal linking activities in black box approach were neglected.
Overall and divisional efficiency scores obtained by Model (I) in the numerical example are equal to those of free link case. It is quite natural because in both models the continuity of link values between divisions is assured and the target values of the intermediate products are free to be above or below their observed values. A clear advantage of using Model (I) is revealing the role of intermediate product in the system. Since the linking constraints of the fixed link case are tighter than that of Model (I), the scores of the fixed link case tend to be higher than those of Model (I) for every DMU.
The scores of separate model follow a roughly similar trend to those of the proposed network models. However there are some differences between network models and separate model that must be caused by different assumption on the linking activities. The proposed network models keep the continuity of link flows among divisions whereas the separate model does not.
Separate model and Model (II) both take into account the inefficiency associated with the link variables.
Comparing the results of Model (I) and Model (II), we can see how the inclusion of intermediate product slacks in the objective function may change the categorization of the intermediate products and exert influence over the efficiency of each division.
For further research we can suggest the following issues.
The proposed approach can be easily extended to the dynamic network models.
LP formulation of Model (II) could be analyzed and interpreted. The proposed models can be extended to the situation in which some input/output data are fuzzy numbers. Another possible line of research is to include undesirable (or bad) outputs.
See Tables
In Model (I) the objective function is clearly nonlinear. The objective function can be easily transformed to linear form by CharnesCooper transformations.
By introducing a positive scalar variable
Then the model becomes as follows:
The authors declare that they have no conflicts of interest.