Developing Common Set of Weights with Considering Nondiscretionary Inputs and Using Ideal Point Method

Data envelopment analysis (DEA) is used to evaluate the performance of decision making units (DMUs) with multiple inputs and outputs in a homogeneous group. In this way, the acquired relative efficiency score for each decision making unit lies between zero and one where a number of them may have an equal efficiency score of one. DEA successfully divides them into two categories of efficient DMUs and inefficient DMUs. A ranking for inefficient DMUs is given but DEA does not provide further information about the efficient DMUs. One of the popular methods for evaluating and ranking DMUs is the common set of weights (CSW) method.We generate a CSWmodel with considering nondiscretionary inputs that are beyond the control of DMUs and using ideal point method. The main idea of this approach is to minimize the distance between the evaluated decision making unit and the ideal decision making unit (ideal point). Using an empirical example we put our proposed model to test by applying it to the data of some 20 bank branches and rank their efficient units.


Introduction
Data envelopment analysis (DEA) which was first proposed by Charnes et al. [1] and developed by Banker et al. [2] is a nonparametric technique for measuring the efficiency of a homogeneous group of decision making units (DMUs) on the basis of multiple inputs and outputs based on observed data [3][4][5][6][7].DEA provides weights that are DMU-specific and permits individual circumstances of operation of the DMUs and for each DMU, it provides efficiency scores in the form of a ratio of a weighted sum of the outputs to a weighted sum of the inputs [8].
This method was applied to evaluate productivity and performance of airports, efficiency of air force maintenance units, hospitals, university departments, schools, industries, banks, products and services, strategic decision making, and technologies [9].
On the basis of various assumptions, a number of different models have been developed.The DEA models may be generally classified into radial and nonradial models.The radial models include the CCR and the BCC models, and the nonradial models include the additive model, the multiplication model, the range-adjusted measure (RAM), and the slack-based measure (SBM) [5,10,11].
Fundamental assumptions of the original DEA models [12] are that inputs and outputs are measured by exact values or are factual and definite factors [4] and assume that the assessed units (DMUs) are homogeneous.In other words, they perform the same tasks with similar objectives, consume similar inputs and similar outputs, and operate in similar operational environments and generally called discretionary factors [6].However, in the real world situations and in many applications of the efficiency evaluation of the units, the assumption of homogeneous environments may be violated and the factors that describe the differences in the environments may need to be included in the analysis.These factors as well as other factors that are beyond the control of the DMU's management, frequently called "exogenously fixed" or nondiscretionary, also need to be considered.Some examples of nondiscretionary factors in the DEA literature are the

CCR Model
Using the traditional denotations in DEA, we assume that there are a set of  DMUs, and each DMU  ( = 1, . . ., ) produces  different outputs using  different inputs which are denoted by   ( = 1, . . ., ) and   ( = 1, . . ., ), respectively.Here   and   are all positive [1].For any evaluated DMU  , the efficiency score  can be calculated by the following CCR model according to following hypotheses:  is the number of decision making units (DMUs) being compared in the DEA analysis, DMU  the th decision making unit,  the efficiency rating of the decision making unit being evaluated by DEA,   the amount of output  used by decision making unit ,   the amount of input  used by decision making unit ,  the number of inputs used by the DMUs,  the number of outputs generated by the DMUs,   the coefficient or weight assigned to output  by DEA, and V  the coefficient or weight assigned to input  by DEA.Also, And the dual of model ( 1) is ≥   ,  = 1, . . ., ; ≥ 0,  = 1, 2, . . ., . ( A characteristic of the above DEA model can be used to evaluate the relative efficiency of its favorable weights in order to calculate its maximum efficiency score for each decision making unit.We note that these efficiency scores usually lie in (0, 1].The DEA successfully divides them into two categories: efficient DMUs and inefficient DMUs.A ranking for inefficient DMUs is given; however, DEA does not provide sufficient information about the efficient DMUs.It is noteworthy that one of the popular methods for evaluating and ranking DMUs is the common set of weights (CSW) method [7,43].

Common Set of Weights
As the mathematical models in DEA are run separately for each DMU, the set of weights will be different for the various DMUs, and in some cases it is unacceptable that the same factor is accorded widely differing weights.This flexibility in selecting the weights deters the comparison among DMUs on a common base.A possible answer to this difficulty lies in the specification of a common set of weights, which was first introduced by Cook et al. [44] and Roll et al. [21] in the context of applying DEA to evaluate highway maintenance units.
In other words, the major purpose for generating a common set of weights is to provide a common base for ranking the DMUs [18,34].
It is argued by Kao and Hung [31] that using different sets of weights to classify the DMUs as efficient or inefficient is acceptable to the practitioners; however, if different sets of weights are used for ranking, most practitioners may not agree.To reduce the flexibility in selecting input and output weights, common weights have been suggested instead of variable weights for assessing the performances of DMUs.The use of common weights makes it possible to compare and rank the performances of the DMUs on the same basis [39].
Table 1 gives a brief summary of some relevant research on DEA to find CSW.

Common Set of Weights by Comparing with Ideal DMU (Ideal Point)
. DEA was initially developed as a methodology for assessing the comparative efficiencies of organized units.In conventional DEA models each DMU in turn maximizes the efficiency score, under the constraint that none of the DMUs efficiency scores is allowed to exceed 1.0.Decision maker always intuitively takes the maximal efficiency score 1.0 as the common benchmark level for DMUs.Liu and Hsuan Peng [18] have taken advantage of this benchmark level to concretely describe the concept of the generation of common weights.By the definition of the efficiency score, the common benchmark level is one straight line with slope 1.0 that passes through the origin.Some might argue that it cares too much about distance of each DMU's input and output itself and wonder why not consider the distance between the evaluated decision making unit and the ideal decision making unit.
Here we attempt to rank efficiency of DMUs with common weights by comparison of ideal point (ideal DMU: DMU).Definition 1. Assume that there are a set of  DMUs.Each DMU  ( = 1, . . ., ) has  different inputs and  different outputs, which are denoted by   ( = 1, 2, . . ., ) and   ( = 1, 2, . . ., ), respectively.All the data are required to be positive just like the traditional DEA model [1].The input data of all DMUs form an  by  matrix and output data form an  by  matrix.The smallest data of each row of input matrix is selected to be the input of the virtual ideal DMU, and the biggest data of each row of output matrix is selected to be the output of the virtual ideal DMU.
In Figure 1, the vertical and horizontal axes are set to be the weighted sum of  outputs and the weighted sum of  inputs, respectively.Line "ox" is an ideal line representing that all the points on the line must satisfy the constraint that the weighted sum of  outputs equals the weighted sum of  inputs and so DMU = (∑  =1   V  , ∑  =1   ú  ) is an ideal DMU.Given one set of weights ú  ( = 1, . . ., ) and V  ( = 1, . . ., ), the virtual gaps, between points  and DMU on the horizontal axis and vertical axis, are denoted by =1     , respectively.Similarly, for points  and , the gaps will be calculated (Δ   , Δ   and Δ   , Δ   ).Observing that there exists a total virtual gap to the ideal point, we aim to determine an optimal set of weights  *  ( = 1, . . ., ) and V *  ( = 1, . . ., ) such that both points  * and  * below the ideal line could be as possibly close to their ideal point (DMU) on the ideal line.In other words, by adopting the optimal weights, the total virtual gaps to the ideal point are the shortest to DMU.As for the constraint, the numerator is the weighted sum of outputs plus the vertical gap ∑  =1 Δ   and the denominator is the weighted sum of inputs minus the horizontal virtual gap ∑  =1 Δ   .The equations in constraints are equal to 1.0, meaning that the projection point (ideal DMU) is reached.Therefore we have the following model: V  ,   ≥  ≥ 0,  = 1, . . .., ,  = 1, . . .., .
(3) Used a general unbounded DEA model to obtain different sets of weights and then taking their average or weighted average with DEA efficiencies as the weights, maximizing the average efficiency of DMUs, maximizing the number of DEA efficient units, and ranking various factors by some order of importance and then assigning low weights to less important factors and maximal feasible weights to important ones Roll et al. [21] Roll [22] 1991 1993 3 Considered the common weights for all the units, by maximizing the sum of efficiency ratios of all the units, in order to rank each unit as well as suggesting a potential use of the common weights for ranking DMUs.
Ganley and Cubbin [ Introduced a minimum weight restriction and as a side effect, common weights are also achieved.Imposed weight restrictions to incorporate value judgment are widely researched within DEA but as these methods originally do not necessarily and purposefully provide a full ranking, they are not explicitly discussed here.Used an approach to minimize the deviations of the CSW from the DEA profiles of weights without zeros of the efficient DMUs.This minimization reduces, in particular, the differences between the DEA profiles of weights that are chosen, so the CSW proposed is a representative summary of such DEA weights profiles.Several norms to the measurement of such differences are used.

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Proposed two models considering ideal and anti-ideal DMU to generate common weights from the view of multiple criteria decision analysis (MADA), for performance evaluation and ranking.
(4) Definition 3. The performance of DMU  is better than that of From the model (4), it is found that the distance between DMU and DMU  is defined as Note that the purpose of the model ( 4) is to obtain an optimal solution (V *  ,  *  ) to make the total distances between all DMUs and DMU as short as possible [6,42].
Next we find efficiency of each DMU with optimal weights.If a DMU  is on ideal point then we use definition of the CSW efficiency score of DMU  that was defined by the following equation (e.g., see [18,45]):

. , 𝑠).
There are some models that incorporate nondiscretionary inputs into DEA models.Banker and Morey [45] provided the first model by modifying the constraints on the fixed factors within the DEA model.This model differs from the original CCR DEA model by breaking the link between nondiscretionary inputs and efficiency: We note that there is a great similarity between fixed factor constraints and constraints on the nondiscretionary inputs while both constraints are modified.This modification is used to take the fixed factors of production under control in order to break the link between the efficiency and the fixed factors [7,46].Although this model allows each DMU to measure the efficiency with its favorable weights, to calculate its efficiency it may not be compared and ranked on the same basis.Moreover, some of the efficient DMUs may have their efficiency scores equal to one because of the flexibility in the selection of weights.

Proposed Model
To address the problems mentioned above, using ideal point (ideal DMU) method, we propose a new model to extend the existing nondiscretionary DEA model for generating common weights.This model allows us to obtain and compare the efficiency scores from multiple different angles.
Then, according to models (4) and ( 6) we can construct the following model The dual of model ( 7

Numerical Examples
In this section we provide an empirical study of bank performance evaluation in order to demonstrate the robustness of our method as well as having a better understanding of the performance of our proposed model.

Empirical Example.
To illustrate the proposed model consider 20 bank branches in Iran with 2 discretionary inputs and 1 nondiscretionary input and 2 outputs.In Tables 2, 3, 4, and 5, we apply models ( 6) and ( 7) to evaluate efficient DMUs.The second, third, seventh, and eighth columns of Table 4 report the model ( 6) efficiency scores with nondiscretionary inputs and its rankings, respectively.This model allows DMUs to measure their efficiencies with various weights.Thus, the efficiencies of 20 DMUs obtained by 20 sets of weights may not be possible to be compared and ranked on the same basis, and so a common set of weights method in model ( 7) is utilized.The efficiencies of the 20 bank branches of model (7) with optimal weights and using (5) are shown in the fourth, fifth, ninth, and tenth columns of Table 4, respectively.Table 5 shows the weight results of the proposed model.It is evident that the new models can all be used for generating common weights.We emphasize that they all offer more reasonable results than the conventional DEA models.Table 4 shows the efficiencies of all bank branches from two models.The column of model (6) gives the CCR efficiency scores with nondiscretionary inputs.Observe that there are 5 efficient DMUs with different selection weights.It is not possible to give them a full ranking.
In order to solve this problem, we propose a common set of weights model considering nondiscretionary inputs to calculate a set of optimal weights (see Table 5) for all DMUs.Using these and ( 5), all efficiency scores for all DMUs are calculated and ranked.The results are shown in the fourth, fifth, ninth, and tenth columns of Table 4.
Using models ( 7) and ( 5), clearly we can find an optimal set of weights for evaluating each DMU and calculate efficiency scores to rank all the bank branches completely which is preferable to that of using model (6).
The above empirical example shows that the new proposed DEA model can successfully acquire a full ranking for the DMUs.This method may be a good way for full ranking DMUs with various data since they are accustomed to a good unit of comparison.

Conclusions and Future Research
General DEA models are used to evaluate the relative efficiency with its favorable weights in order to calculate the efficiency score of each decision making unit.These obtained scores are between zero and one, with a possibility of some having an equal efficiency score of one (efficient DMUs) which is due to the flexibility in the selection of weights.DEA successfully divides DMUs into two categories: efficient DMUs and inefficient DMUs.Ranking of DMUs in DEA is an important phase for efficiency evaluation of DMUs.In DEA techniques, a ranking for inefficient DMUs is given.However, generally, DEA does not provide adequate information about the efficient DMUs and does not rank them.
One of the popular methods for evaluating and ranking efficiency and inefficiency DMUs is common set of weights (CSW) method, that is, the most favorable in determining the absolute efficiency for all of DMUs.
The conventional DEA methodology requires the inputs and the outputs of the DMUs to be discretionary.Nevertheless, in reality, many observations are nondiscretionary in nature.We generated a nondiscretionary version of a CSW model, that is, beyond the control of DMUs, and for this purpose we used ideal point method.The idea of this approach is to minimize the distance between the evaluated decision making unit and the ideal decision making unit (ideal point).Ranking DMUs determines the input and output weights by minimizing the distance of all DMUs and the point (ideal DMU) to get the best efficiency score.The optimal solution of this model was considered as a set of weights for all DMUs.Then DMUs were ranked according to (5).To validate our model, we used an empirical example in ranking DMUs using our proposed model.
We hope that this paper will inspire future researchers to explore relevant ideas in developing CSW method to consider various data such as nondiscretionary inputs and stochastic data.

Figure 1 :
Figure 1: An illustration of the model for gap analysis, showing DMUs below the virtual ideal DMU.

Table 2 :
Labels of inputs and outputs.

Table 3 :
The data of 20 bank branches.