Modeling bank performance: A novel fuzzy two-stage DEA approach

Evaluating the banks' performance has always been of interest due to their crucial role in the economic development of each country. Data envelopment analysis (DEA) has been widely used for measuring the performance of bank branches. In the conventional DEA approach, decision making units (DMUs) are regarded as black boxes that transform sets of inputs into sets of outputs without considering the internal interactions taking place within each DMU. Two-stage DEA models are designed to overcome this shortfall. Thus, this paper presented a new two-stage DEA model based on a modification on Enhanced Russell Model. On the other hand, in many situations, such as in a manufacturing system, a production process or a service system, inputs, intermediates and outputs can be given as a fuzzy variable. The main aim of this paper is to build and present a new fuzzy two-stage DEA model for measuring the efficiency of 15 branches of Melli bank in Hamedan province.


Introduction
Banks and financial and credit institutions play a very important role in the economic development of each country. Currently, due to the significant growing the number of banks and financial and credit institutions in Iran and due to the privatization process of state banks and the conversion of credit cooperatives and credit institutions to the bank, their performance evaluation has become very important. One of the well-known methods in assessing the performance of firms is data envelopment analysis that was developed by Charnes et al. [1] as CCR model, see ( [2][3][4][5][6]; [7]) for more details. CCR model is a radial model. Radial models have some disadvantages like failure to recognize weak efficient DMUs, see [8,9]. Another kind of DEA models are non-radial DEA models [10]. One of the important non-radial DEA model is Enhanced Russell Model (ERM model) that proposed by [11]. This model has some useful properties. One of them is ability to recognize the weak efficient DMUs. On the other hand this model has some disadvantages like failure to rank efficient DMUs. Izadikhah et al. [12] proposed a modified version of ERM model that enables ERM model to rank efficient DMUs. Our proposed DEA methodology is an extension of their model that considers internal structures of DMUs.
DEA has been widely used for assessing the performance of banks. Chortareas et al. [13] presented a good survey in this topic. Also, the readers are referred to [14] for a complete review for applications of DEA in industries.
Typically, a single stage production process is assumed to transform inputs to final outputs and is treated as a black box. In contrast to the black-box approach, real-world production systems often have network structure [15]. There is an increasing literature body that is devoted to efficiency assessment in multistage production processes. Castelli et al. [16] provided a comprehensive review of models and methods developed for different multi-stage production structures. In recent years, many researchers studied various DEA models for evaluating efficiencies of two-stage systems especially for evaluating efficiencies of banking systems. Firstly, Seiford and Zhu [17] presented the first two-stage DEA model to evaluate the marketability and profitability of the U.S. commercial banks. For an in-depth review in multi stage DEA model, see ([9,18-21]). In this study, we propose a new two-stage DEA model based on the modified ERM model. However, in many situations, such as in a manufacturing system, a production process or a service system, inputs and outputs are volatile and complex so that it is difficult to measure them in an accurate way. Instead, the data can be given as a fuzzy variable. The concept of fuzzy theory was initialized in Zadeh [22]. After that many fuzzy approaches have been introduced in the DEA literature. Sengupta [23] applied principle of fuzzy set theory to introduce fuzziness in the objective function and the right-hand side vector of the conventional DEA model and developed the tolerance approach that was one of the first fuzzy DEA models. Many authors have combined DEA and Fuzzy modeling to study the efficiency of banking systems, see [24] and for a review on fuzzy DEA modelling see ( [25][26][27][28][29][30][31]; [32]; [33,34]). Conventional DEA needs accurate measurement of inputs and outputs. However, the values of the input and output data in banking systems are sometimes imprecise or uncertain and since the quantity of some of our data in this paper was not known exactly the data was stated as fuzzy data. Thus, we extend our proposed two-stage DEA model in fuzzy environment. Then we present a method for solving the This paper unfolds as follows: Section 2 briefly reviews the possibility approach. Section 3 proposes our new DEA methodology. In Section 4, a case study is presented and final conclusion is appeared in Section 5.

Preliminaries
Zadeh [22] presented the concept of possibility approach in terms of fuzzy set theory. Now, we review the definition of possibility space, ( [22,35] Considering the fuzzy theory, there is a lemma that can be very useful to interpret the possibility function. Now, this lemma is represented.
The above lemma is very useful to defuzzification of the fuzzy DEA model's constrains.

Proposed Methodology
In this section, we first present our proposed two-stage DEA model, and then we bring our new fuzzy two-stage DEA model alongside the procedure for solving its program.

Proposed two-stage DEA model
Let's consider DMUs have an internal production structure. So, in this section we extend black-box production structure and performance measures to a two-stage production process. Here, assume that there are n DMUs (j=1,…,n) consisting of two divisions and

A new Fuzzy two-stage modified ERM model
The classic DEA models can only be used for cases where the data are precisely measured while in real-world situations, the observed values of the input and output data are sometimes inexact, incomplete, vague or ambiguous. These kinds of uncertainty data can be represented as linguistic variables characterized by fuzzy numbers for reflecting a kind of general sense or experience of experts. The concept of fuzzy set theory was first developed by [22] to deal with the issue of uncertainty in systems modeling. Fuzzy DEA is a powerful tool for evaluating the performance of DMUs in uncertainty environments.
In this section, we propose a new fuzzy DEA model for evaluating a set of DMUs with fuzzy inputs, intermediates and outputs. Hence, we extend model (1) to a fuzzy model. two-stage DEA model for calculating the overall efficiency of is as follows:

Justification of the fuzzy model
This model is a fuzzy version of model (1) that the fuzzy numbers are incorporated into the model (1). This fuzzy integrated DEA model cannot be solved like a crisp model. It is needed to design a procedure to solve that model.

Solving Procedure for the proposed fuzzy model
As it is mentioned before the proposed fuzzy DEA model cannot be solved like a crisp model. So, in order to solve it one can apply a possibility approach formulated in terms of fuzzy set theory proposed by [22]. This procedure converts the fuzzy integrated DEA model to the standard linear programming (LP) by α-cut technique. In this case, each fuzzy coefficient can be viewed as a fuzzy variable and each constraint can be considered as a fuzzy event, see [35].
Therefore the proposed model converted as follows: In model (3)   let us assume that 1 = 2 = 3 = 4 = 5 = . By these transformations our model for evaluating DMUp and measuring its overall efficiency becomes as follows: In addition, w1 and w2 are weights assigned to the efficiency scores in the first and second stages, respectively. Several approaches such as point allocation, paired comparisons, trade-off analysis, and regression estimates can be used to specify the weights ( [37,38]). Alternatively, pairwise comparisons and eigenvalue theory proposed by [39] can be used to determine suitable weights for efficiency scores of the two stages, [9].
For each value of Clearly, for each p we have 01 p   . And thus we can rank DMUs according to decreasing order of the stochastic closeness coefficient of overall efficiency. By a same manner we can obtain the stochastic closeness coefficient of stage efficiencies.  Fig. 3, the two-stage performance measurement system of branches of Melli bank is comprised of two stages. Stage 1 represents the profitability and Stage 2 represents marketability banking.

Data
The inputs to the first stage are:    x : Staff 2 2 x : Facilities

Results and Analysis
The results of solving model (5)   From the Table 4-6 we can see there are some DMUs which are efficient in the first stage but they are inefficient in the second stage. One reason for this issue is that since these DMUs are efficient in the first stage, they produce a large amount of output and these outputs are inputs for the second stage. Thus, they consume a large amount of inputs for the second stage to produce outputs in the second stage. This point leads to decrease in their efficiency score in the second stage. Similar reason holds for inefficient DMUs in the first stage, [9].   To better comparison, these results are provided in Fig. 2. This figure shows the efficiency scores of bank branches at the different levels of in an integrated form. According to these tables and figure, it seems that, the performance of DMUs in the first stage is relatively better than second stage. From Fig. 2 it is clear that DMU #8, i.e. "Takhti" has been recognized for its best performance in the first stage in the different levels of (It has maximum efficiency in different values of ). Also, we can see DMU #12, i.e.
"Pasdaran" has been recognized for its best performance in the second stage in the different levels of and because of big difference between its efficiency and others' efficiencies, this DMU has been recognized as the best overall performance, (Fig. 2a).   Table 7. From Table 7 we can see the DMU #12 (Pasdaran) and DMU #10 (BabaTaher) have the best and the worst overall performances among all DMUs, respectively. A slight note to the last row of Table 7 shows that in average the performances of DMUs in the first stage are better than their performances in the second stages. This result had been stated from Fig. 2, too. branch, which is "Pasdaran" branch, that worked efficiently in both stage one and two in all values of , and therefore, is recognized as the only overall efficient branch among these 15 branches.

Fig. 3: Comparison among all stochastic closeness coefficients
Refer to Fig. 3 for an illustrative comparison among the results. Fig. 3 illustrates the graphical representations of the stochastic closeness coefficient results using fuzzy inputintermediate-output data; and it is clear that the first stage efficiency score of DMU "Takhti" is the highest value among all efficiency scores and the next place is for DMU "Pasdaran" for its second stage efficiency score. Finally, the poor performance of DMU "BabaTaher" in all situations is quite clear.

Comparison with other methods
By examining the literature, it can be seen that there are a number of articles on the evaluation of decision-making units with fuzzy two-stage DEA models. Table 8 provides a comparison among the proposed model and some of the existing fuzzy two-stage DEA models. For this comparison, some important criteria such as novelty in two-stage model, non-parametric form, ability to efficiency decomposition, providing an integrated index and application in banking industry are considered.  According to Table 8, the method proposed in this paper has several advantages over the existing models. On the other hand, the method proposed in this study is more general compared with the existing models. The non-parametric nature of the proposed model is a very important feature that makes the evaluation and ranking not dependent on values. Additionally, the proposed model provides an integrated index that helps decision maker to make robust decisions.

Sensitivity analysis on stages' weights
This study is done by considering the equal values for weights of stages. In this section, we check the importance of weight change and its effect on the final ranking of DMUs. For this purpose, we consider four extra cases for weights as Table 9. These four cases have been selected due to consider the large difference as well as the small difference between the weights. The proposed Model is run for all cases and the obtained rankings are illustrated in Table 10. The case of equal weights is called the main case in these tables. In order to check the relation among the results we employ the Spearman's correlation coefficient.  [24]; [37,49]). Also, there are several studies that show the deficiency of traditional DEA models (e.g., [50], [51], [52], [53,54]).
Unlike the traditional DEA models, the two-stage DEA model can examine the structure and processes within DMUs. This helps managers to identify the inefficiency sources within DMUs ( [55]; [56]; [57]; [58] The result of stochastic closeness coefficient (6) are shown in Table 11. Based on Table 11, the average of obtained scores by model (7) is more than the fuzzy two-stage DEA model. This fact indicates that there are some inefficiencies that the single-stage model cannot recognize them. The results of Table 11 show that the overall performance of DMUs, in their single stage form is very far from the results obtained by the proposed two-stage DEA model.

Managerial Insights
Not only performance measurement in bank branches is essential but it also plays a critical role as an element of productive banking industry operations on strategic and operational levels. Mathematical models and analytic approaches are powerful tools in the performance measurement of bank branches and they enable managers to obtain helpful information to inform strategic and operational decisions. One of the popular and rigorous approaches used by managers to evaluate the efficiency in banking industry is DEA model. The findings of this study can also increase operations managers' confidence in the right decision-making for performance measurement of bank branches.
Classical DEA models consider each DMU as a black box and do not care about internal structure of DMUs. Also, primary DEA models assume that data are known exactly.
However, in real world, there might be stochastic data. Our proposed two-stage DEA model can evaluate the bank branches in presence of fuzzy data. Moreover, the feature of using two-stage structure can help managers to identify any inefficient resources in each stage of a banking operation and address these by making the right decisions. Another issue is that since the initial investment in banking industries under uncertain conditions can be costly, time-consuming and risky, powerful performance measurement techniques, including the fuzzy two-stage DEA model presented in this study, can serve as appropriate decision support system tools.
In the case study section, the proposed model has been applied to evaluate the efficiency of 15 bank branches in Hamedan. According to the derived results, only one bank branch was recognized as efficient during the examined period 2014-2015 at all significant levels. This fact provides managers with information about which branches need to be actively developed so as to trigger innovation and growth. It also allows managers to identify productive investment and appropriate management activities. In the first stage, the sub-process of profitability measurement, eight branches were identified as efficient (at all significant levels). In the second stage, the sub-process of marketability measurement, three branches were recognized to perform efficiently (at all significant levels). From a statistical viewpoint, the efficiency of the first stage must be higher than that of the second stage. This indicates that the low efficiency scores obtained for the twostage processes were mainly due to the low efficiency scores of the corresponding second stages, that is, the marketability efficiency values.

Conclusion
The assessment of the banks' performance has always been of interest due to their crucial role in most economic activities and the maintenance of the health of monetary markets and economic conditions. Data envelopment analysis has been found to be a well-known methodology for measuring performance of bank branches. However, conventional DEA model makes difficult, if not impossible, to understand what are the sub-processes and the interactions causing the inefficiency of a DMU. In addition, in the banking environment, there is always a need for tools that allow one to uncover the inefficiencies that can affect the different components of an operation. Moreover, many banking operations take the form of two-stage processes. In this paper, we considered a novel two-stage DEA model based on the modified version of ERM method. The aim of this study was evaluation the 15 branches of Melli bank in Hamedan province. After determining the input, intermediate and output variables, it is realized that generally they weren't precisely known and as a result they couldn't be considered in exact form. For this reason, the data was stated as fuzzy data. We used triangular fuzzy data to state the complexity of data. Therefore, we further extended our proposed two-stage DEA model to deal with fuzzy data. After that we presented a method for solving the proposed fuzzy DEA model based on the concept of alpha cut and possibility approach.
In the case study section, the proposed model was applied to evaluate the We conclude with a few possible research directions towards which to extend the results of this study. Our approach could be extended to consider other kinds of data such as dual-role data, stochastic data and so on. In this paper, we applied our proposed model for evaluating the performance of bank branches. It seems that our proposed model can be used in other problems such as evaluating the sustainability of suppliers, regional R&D processing, evaluating non-life insurance companies, efficiency evaluation of production lines, efficiency evaluation of hospitals which have many wards interacting with each other and have network structure, and so on. Developing a fuzzy dynamic two-stage DEA model will be another interesting research topic. The proposed model may be extended to cases where the intermediate products could be lost or added from external sources.