Evaluation and benchmarking of bank branches on two-level inputs and outputs: An application of MOLP and DEA

Efficiency evaluation of homogenous decision making units (DMUs), is one of the primary objectives of data envelopment analysis (DEA), in which the large number of inputs or outputs compared with the number of DMUs leads to decreased evaluation precision of DEA models. Overcoming this problem is the main issue in two-level DEA. In the present study, we use the multi objective linear programming (MOLP) method proposed by Sumpsi et al., which is based on constructing the pay-off matrix and using goal programming (GP), to investigate this problem. Furthermore, we discuss the enhanced Russell's model for evaluating and benchmarking DMUs with two-level inputs and outputs. The proposed method is applied to 24 branches of an Iranian commercial bank.


Introduction
Data envelopment analysis (DEA) is a scientific method for the performance analysis of different organizations in private and state-run sectors.Considering this, the organizations under evaluation, such as bank branches, schools, hospitals, and universities, are called decision making units (DMUs).What all these units have in common is their homogeneity.In other words, all units use similar inputs to produce similar outputs.The efficiency of each DMU is determined by a production process which uses multiple inputs to produce multiple outputs.A unique characteristic of DEA is that it compares the performance of a certain DMU against that of others in order to determine its efficiency level.However, if the number of inputs and outputs is large compared to the number of DMUs, the analytical power of the DEA models is reduced.
The simplest method to deal with this shortcoming is to eliminate some indices (inputs or outputs) and then perform the evaluation process with a smaller number of indices.However, since DMUs are willing to perform a comprehensive evaluation of the DMUs under their supervision, taking into account all indices, this is not feasible from the managerial point of view.Therefore, another approach for managers is to reduce the number of indices by combining some of them into a specific index.In fact, in two-level DEA method, the indices are categorized into tow levels and the indices assigned to the second level are considered as subindices to those in level one.It should be noted that these subindices should be representative of different characteristics of a particular index, in a manner that if all such subindices are combined, they characterize that particular index.Thus, we can deal with the problem by two approaches: firstly, calculating subindex weights and combining them using the sum of weighted subindices (Shang et al. (1995), Tseng et al. (2009), Korhonen et al. (2001), Yang et al. (2003) ) , and secondly, combining subindices with variable weights, and then evaluating each DMU and calculating the weights in a single model (Meng et al. (2008), Kao et al. (2008) ).
In this paper, the first approach is selected and one of the multi objective linear programming (MOLP) methods, proposed by Sumpsi et al. ( Sumpsi et al. (1997), Andre et al. (2010) ) which is based on constructing the pay-off matrix and goal programming, is employed.Moreover, by introducing the secondary goal function, we address the issue of presence of multiple optimal solutions in single-objective linear programming models used for constructing the pay-off matrix (Liang et al. (2008) ).Considering the type of objective function and the particular feasible region in these models, only one model is required for calculating the weights of subindices.
Another issue to be addressed in such two-level input and output structures is benchmarking, which is beyond a simple comparison, since it includes knowledge of a better condition and approval of changes.In fact, benchmarking, in essence, is an attempt for improvement, and is not necessarily used for solving existing problems; rather, it can be employed for accepting better performances before introducing novel processes.
The ultimate goal of evaluating DMUs is to determine how changing inputs and outputs can improve performance.Thus, with such a two-level structure of indices, one should select a model which can provide an appropriate interpretation for determining the changes that a subindex undergoes in a combined index.To this end, the enhanced Russell Model (Pastor et al. (1999) ) is used in this paper.It should be noted that DEA models for evaluating DMUs with a large number of inputs and outputs are large-scale models, and an alternative approach for solving such models is to break them down into problems of smaller dimensions.For more information on this issue, see (Korhonen et al. (2009) ) .This paper is organized as follows.Section 2 introduces the two-level structure of DMUs , inputs and outputs, and the calculation of the corresponding weights of subindices by MOLP.Benchmarking DMUs with combined data is discussed in Section 3. Section 4 includes a case study application of the proposed method to 24 branches of an Iranian commercial bank.The conclusions are provided in Section 5.

Calculation of subindex weights using goal programming
Consider n DMU with two-level inputs and outputs.To this end, we assume m inputs and s outputs in level one, such that some of the indices (inputs or outputs) are representative of some other indices which are classified as subindices in level two.From the MOLP point of view, these inputs and outputs are the criteria against which the DMUs are assessed.In fact, inputs and outputs are the goals whose minimization and maximization, respectively, is the aim of the DM.Moreover, since the DM deal with technical constraints, maximization and minimization of these objectives should be carried out under such constraints or, in other words, in the DM ' s desired feasible region.
Here, the following production possibility set (PPS) is considered as the feasible region, and is denoted by

 
Constructing the pay-off matrix is the first step to determine the weights of inputs and outputs subindices using the MOLP proposed by Sumpsi et al. (1997) Here, the rank of the square pay-off matrix is equal to the total number of subindices and is represented by nsub, which is calculated as follows: If maximization is desired, as is the case with output subindices, the following singleobjective linear programming problem should be solved corresponding to each of them.
If minimization is desired, as is the case with input subindices, the following singleobjective linear programming problem should be solved corresponding to each of them.
Thus, Model (1) should be solved  O r r s , times, i.e., equal to the number of output subindices, and Model (2) should be solved  I i i m times, i.e., equal to the number of input subindices, to construct the pay-off matrix.A part of the optimal solution of each model, i.e., the corresponding value of the subindices in level two of the ordered vector ) , ( * *

Y X
, is an n sub -component ordered vector.These vectors will constitute n sub -columns of the pay-off matrix.
To show these columns, consider the ordered set I O  .In this case, for any desired subindex l , the following relation holds: Now, the values obtained from the optimal solution of Model (1) or Model (2) corresponding to a second-level subindex, say the lth subindex, constitute the lth column of the pay-off matrix, as denoted below: )).
In the above vector, the superscript " * " denotes the optimal solution and subscript l represents the corresponding solution vector of the lth subindex.It should be noted that the presence of multiple optimal solutions in Models ( 1) and ( 2) is possible, which will lead to the multiplicity of the pay-off matrix.To resolve this issue, a secondary goal is introduced as follows: The DM considers an ideal for inputs and outputs, which is demonstrated by the Vector  .
,..., , , ,..., Recall that for each , of the corresponding subindices values considered by the DM.The secondary goal function is defined such that the sum of the absolute values of the difference between the inputs and outputs corresponding to the subindices, obtained in the optimal solutions of Models ( 1) or ( 2), and the respective goals desired by the DM are minimized.In other words, the following function is to be minimized.

 
Moreover, the feasible region is that of Model (1) or Model (2), i.e., TS v , plus an additional constraint which guarantees the optimal objective value corresponding to the subindex for which Model (1) or Model (2) was solved to remain unchanged.In fact, in the case of multiple optimal solutions for Models (1) and (2), Model (3) provided below selects one of them that is closer to the desired goals of the DM regarding inputs and outputs.Hence, Model (3) is solved by solving Models (1) and ( 2) corresponding to each subindex and by assuming the objective optimal value to be constant.For instance, suppose that Model (1) is solved for the q th output subindex of the p th output index, ). , ( , If the objective optimal value of Model ( 1) is * q p y , the following model is proposed to select the optimal solution with the least difference from the DM's ideals (this similarly holds for input subindices,as well).

 
To linearize the above model, the following alterations of variables are employed.3) is transformed to a linear form as follows: The constraints of Model (3), ).
Therefore, the above model can be employed as the phase-II problem, after each instance of solving Models (1) and ( 2) for each subindex, such that the columns of the constructed pay-off matrix have the closest values to the DM's desired objectives.The next step is to calculate the weights corresponding to subindices.To obtain positive weights, goal programming is used as follows.
are the corresponding values of input and output subindices which are determined by the DM and is the corresponding optimal solution vector of the lth subindex.If l is an output subindex, it is obtained by solving Model (1), and if it is an input subindex, it will be obtained by solving Model (2) Repeating the above procedure for each I i  and O r  , i.e., the combination of all subindices, all inputs and outputs will be of the same level and each DMU will have m inputs and s outputs in level one.
The following points are noteworthy in the above-mentioned method: Note 1: The input and output values of all DMUs are normalized first so that the DM will not encounter the combination of data of different dimensions.This is achieved as follows: Note 2: In the evaluation of DMUs, where the indices (objective functions) are the inputs and outputs, it is not required to solve Models (1) and ( 2) to construct the pay-off matrix.Because, by the property of TS v and considering the type of objective function, the vector (X*, Y*) for instance, obtained from the optimal solution of the maximization problem corresponding to the output subindex ) , ( contains the inputs and outputs corresponding to the DMU with the highest output value k r y among the DMUs.That is, the * j  corresponding to this DMU in the optimal solution is equal to one.
Similarly, the vector (X*, Y*) obtained from the optimal solution of the minimization problem corresponding to the input subindex ) , ( contains the inputs and outputs corresponding to the DMU with the lowest input value k i x among the DMUs.If some DMUs have the largest value of an output subindex or the smallest input subindex in common (in the presence of alternative optimal solutions in Models ( 1) or ( 2)), then the inputs and outputs of the DMU for which the sum of the absolute values of the difference between the input and output values and those desired by the DM are minimal, are considered as the optimal solution.Therefore, by the property of TS v , it is not necessary to solve Model (3) and its optimal solution will be obtained via a simple computational process.
Thus, the pay-off matrix is constructed only by comparing the inputs and outputs of the DMUs, and it is required to solve only Model (5) for determining the weights by this method.

Benchmarking by enhanced Russell Model after combining data
Using the weights obtained from goal programming, the weighted sum of each group of the subindices is considered as a level-one index.Therefore, by the notation given in Section 2, in this step we have n DMUs, each having m inputs and s outputs.
In this section, the calculation of efficiency and the issue of benchmarking are described.Since some data used in this step are obtained from the combination of several data item, we use a DEA model in whose optimal solution there are no non-zero slack.So, it is suggested that the enhanced Russell Model ( is the optimal solution vector for Model (6) is the efficient benchmark for DMU o .Now, to explain how these factors affect the weighted elements in an item of combined data, consider k such that O k  (this can be similarly stated for inputs).Thus, the kth output index of all DMUs is obtained from the weighted sum of Sk output subindices.If In this connection, we consider some presuppositions.
1. DM can change every subindex so that the amount of its change is proportionate to its corresponding weight.
2. Because of determining efficient benchmark on combined data, it may be occured that in the particular DMU, we encounter the decrease of output subindex with low weight and the increase of output subindex with high weight.Similarly, this situation can occur for some input subindices.

Application
In this section, the proposed method is applied to 24 branches of an Iranian commercial bank.Each branch has three inputs: paid profit, personnel, and arrears; and eight outputs: long-term savings, current savings, interestfree savings, short-term savings, other resources, loans, received profit and interest received.The first five outputs are subindices to the main index "savings".Fig. 3 displays the two level structure of the outputs.
The normalized data corresponding to the inputs and outputs of each branch are provided in Table 1 In order to determine the weights of the five output subindices in the second level, the pay-off matrix is first constructed as described in Section 2. The construction of the first column of this matrix, for instance, is explained here.Consider the first output subindex, i.e., long-term savings.Branch 5, among the 24 branches, has the largest amount of long-term savings.So, the first column contains the values corresponding to the five subindices in Branch 5; that is, columns 5-9 of the fifth row of Table 1 constitute the first column of the pay-off matrix.The other columns are constructed in the same manner with regard to the other subindices.
Now, the following model is solved for calculating the weights corresponding to these subindices.
with variable returns to scale assumption of technology that it has been generated by indices and subindices.
The corresponding weights of subindices are determined by) the optimal solution of Model(5).For instance, assume that ) ,..., ( corresponding weights of the subindices of(5) the ith index.Then, the weighted sum of these subindices provides the ith combined index for each DMU; i.e., the value of the ith input for DMU j is: weights of the subindices of the rth output index, then the rth output for DMU j is calculated as follows:


By applying the simple variable change, the above model is transformed into the linear enhanced Russell's Model.In evaluating DMU o by the enhanced Russell Model (Model the efficient benchmark is introduced by contraction of inputs and expansion of outputs.

.Definition 3 . 1 . 1 .
weight vector of these subindices obtained from the goal programming discussed in Section 2, we will have DMU o and the efficient benchmark introduced for it.It is obvious that the kth output of DMU o a way that the weighted sum of the altered components is equal to There are various methods to achieve this aim.For instance, one way is to multiply all the elements by * k  .In this study, a geometric concept is used to explain how subindices are changed.The perpendicular projection of vector A onto vector B is a vector Moreover, as was stated earlier, the following relation should hold. of the subindices of DMU o in the bench-marking process (the superscript "*"denotes the subindex values in the benchmark).introducing efficient benchmarks for the combined-data DMU under evaluation, the altered values of the original data (subindices) are calculated by the weight vector employed for the combination of subindices.
Pastor et al. (1999)) be used under variable returns to scale, since this model has zero slacks in the optimal solution.
in evaluating DMU o .

Table 1 .
. The values desired by the DM for each input and output are given in the last row of the table.The normalized data corresponding to the inputs and outputs The second column of Table2demonstrates the data obtained by combining the five output