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

Authors: Reza Kiani Mavi; Sajad Kazemi; Jay M. Jahangiri
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

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 CSW model 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.


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## Body

## 1. 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 [3–7]. 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 number of competitors in a restaurant chain and snowfall or weather in evaluating the efficiency of maintenance units and so forth [6, 9, 13].The efficiency scores of decision making units, when DEA models are used, are between zero and one inclusively [3, 5]. DEA successfully divides them into two categories: efficient DMUs and inefficient DMUs. A ranking for inefficient DMUs is given; however, efficient DMUs cannot be ranked [7]. In order to differentiate these efficient units, a variety of methods or “ranking efficient units” in the DEA are proposed [3]. For example, Anderson and Petersen [14] and Mehrabian et al. [15] introduced two of the most popular methods, namely, AP and MAJ. Cook et al. [16] divided efficient units with equal scores on the boundary, by imposing the restrictions on the weights in a DEA analysis. Jahanshahloo et al. [17] introduced L1-norm approach that removes some deficiencies arising from AP and MAJ but fails to rank nonextreme DMUs. Liu and Hsuan Peng [18] introduced a common set of weights (CSW) to create the best efficiency score of one group composed of efficient DMUs. They then used this common set of weights to evaluate the absolute efficiency of each efficient DMU in order to rank it [5, 7].Here in this paper, we use a method to rank the efficiency of DMUs and obtain the common set of weights model, that is, extended with nondiscretionary inputs to evaluate the absolute efficient DMUs. Using this common set of weights, the efficiency scores for DMUs are also obtained. In Section2, we propose our model according to the CCR model which was initially proposed by Charnes, Cooper, and Rhodes in 1978, and we briefly review a general CSW method while the CSW with ideal point method is described in Section 3. Nondiscretionary version of the DEA model is extended in Section 4. Section 5 includes the extended proposed mode and in Section 6, we apply the model in empirical example. Finally Section 7 includes the paper’s conclusions and future research ideas.


## 2. CCR Model

Using the traditional denotations in DEA, we assume that there are a set ofn DMUs, and each DMUj(j=1,…,n)produces s different outputs using m different inputs which are denoted by xij(i=1,…,m) and yrj(r=1,…,s), respectively. Here xij and yrj are all positive [1]. For any evaluated DMUj, the efficiency score H can be calculated by the following CCR model according to following hypotheses:j
is the number of decision making units (DMUs) being compared in the DEA analysis,DMUj  the jth decision making unit,θ the efficiency rating of the decision making unit being evaluated by DEA,yrj the amount of output r used by decision making unit j,xij the amount of inputiused by decision making unit j,i the number of inputs used by the DMUs,r the number of outputs generated by the DMUs,ur the coefficient or weight assigned to output r by DEA, andvi the coefficient or weight assigned to input i by DEA.Also,(1)MaxH=∑r=1suryro,subjectto∑r=1suryrj-∑i=1mvixij≤0,j=1,2,…,n,∑i=1mvixio=1,ur≥0,vi≥0,00000000000000r=1,…,s,i=1,…,m.
And the dual of model (1) is
(2)MinD=θosubjectto∑j=1nλjxij≤θoxio,i=1,…,m.∑j=1nλjyrj≥yio,r=1,…,s;hhhhhhhhhhhhhhλj≥0,j=1,2,…,n.
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].


## 3. 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].Table1 gives a brief summary of some relevant research on DEA to find CSW.Table 1
Number
CSW methods developed
Authors
Year
1
Provided a subjective ordinal preference ranking by developing common weights through a series of bounded DEA runs, by closing the gap between the upper and lower limits of the weights.
Cook and Kress [19, 20]
1990, 1991
2
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 [23]
1992
4
Developed a two-stage linear discriminate analysis approach to generate the common weights
Sinuany-Stern, et al. [24]
1994
5
Developed a maxi-min efficiency ratio model which also creates common weights for evaluation
Troutt [25]
1995
6
Used the canonical correlation analysis to provide a single weight vector for inputs and outputs, respectively, common to all DMUs.
Friedman and Sinuany-Stern [26]
1997
7
Presented a nonlinear discriminate analysis to provide the common weights for all DMUs.
Sinuany-Stern and Friedman [27]
1998
8
Presented the multiple objectives max-min model to determine CSW
Chiang and Tzeng [28]
2000
9
Minimizes a convex combination of these deviations measured in terms of a couple of distances in such family
Despotis [29]
2002
10
Proposed a DEA-CP (compromise programming) model which aims at seeking a common set of weights across the DMUs by combining the DEA and the compromise programming.
Hashimoto and Wu [30]
2004
11
Based on multiple objective nonlinear programming and by using compromise solution approach, proposed a method to generate a common set of weights for all DMUs which are able to produce a vector of efficiency scores closest to the efficiency scores calculated from the standard DEA model (ideal solution)
Kao and Hung [31]
2005
12
Based on multiple objective nonlinear programming and maximization of the minimum value of the efficiency scores, proposed a method to generate a common set of weights for all DMUs.
Jahanshahloo et al. [32]
2005
13
Developed a goal-programming model for this setting that seeks to derive such a common-multiplier set. The important feature of this multiplier set is that it minimizes the maximum discrepancy among the within-group scores from their ideal levels. And deal with these distances but relax the objective to groups of DMUs which operate in similar circumstances
Cook and Zhu [33]
2007
14
Used a multiple objective linear programming (MOLP) approach for generating a common set of weights in the DEA framework.
Makui et al. [34]
2008
15
Proposed a common weights analysis (CWA) methodology to search for a common set of weights for DMUs.
Liu and Peng [18]
2008
16
Dealt with deviations regarding the total input virtual and the total output virtual
Franklin Liu and Peng [35]
2009
17
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.
Wang et al. [36]
2009
18
Proposed two approaches to obtain the set of common weights for ranking efficient DMUs by comparing with an ideal line and the special line.
Jahanshahloo et al. [6]
2010
19
Proposed a CSW as the average of the profiles of weights provided by the so-called ‘‘neutral’’ model used in the cross-efficiency evaluation.
Wang and Chin [37]
2010
20
Proposed a common weight MCDA-DEA method with a more discriminating power over the existing ones that enable us to construct CIs using a set of common weights.
Hatefi and Torabi [38]
2010
21
Used methods based on regression analysis to seek a common set of weights that are easy to estimate and can produce a full ranking for DMUs.
Wang et al. [39]
2011
22
A separation method is proposed for locating a set of weights, also known as a common set of weights (CSW), in the data envelopment analysis (DEA).
Chiang et al. [40]
2011
23
Extended a common-weights DEA approach involving a linear programming problem to gauge the efficiency of the DMUs with respect to the multiobjective model.
Davoodi and Rezai. [41]
2012
24
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.
Ramón et al. [8]
2012
25
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.
Sun et al. [42]
2013


### 3.1. 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 ofn DMUs. Each DMUj(j=1,…,n)hasmdifferent inputs and s different outputs, which are denoted by xij(i=1,2,…,m) andyrj(r=1,2,…,s), respectively. All the data are required to be positive just like the traditional DEA model [1]. The input data of all DMUs form an m by n matrix and output data form an s by n 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.
The virtual ideal DMU is a DMU with minimized inputs of all of DMUs as its input and maximized outputs of all of DMUs as its output. Generally, if we show ideal DMU withDMU-=(X-,Y-), then we have x-i=min{xij∣j=1,…,n}(i=1,…,m) and y-r=max{yrj∣j=1,…,n}(r=1,…,s).Definition 2.
An ideal level is one straight line that passes through the origin and ideal DMU with slope 1.0.In Figure1, the vertical and horizontal axes are set to be the weighted sum of s outputs and the weighted sum of m 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 s outputs equals the weighted sum of m inputs and so DMU-=(∑i=1mx-iv´i,∑r=1sy-ru´r)is an ideal DMU. Given one set of weights u´r(r=1,…,s)and v´i(i=1,…,m), the virtual gaps, between points M and DMU- on the horizontal axis and vertical axis, are denoted by ΔMI=∑j=1nvMxMj-∑j=1nvMxmin and ΔMR=∑j=1nuMymax-∑r=1suMyMj, respectively. Similarly, for points N and L, the gaps will be calculated (ΔNI,ΔNRandΔLI,ΔLR). Observing that there exists a total virtual gap to the ideal point, we aim to determine an optimal set of weights ur*(r=1,…,s) and vi*(i=1,…,m) such that both points M* and N* 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 gapsΔMI+ΔMR+ΔNI+ΔNR+ΔLI+ΔLRto the ideal point are the shortest to DMU-.Figure 1
An illustration of the model for gap analysis, showing DMUs below the virtual ideal DMU.As for the constraint, the numerator is the weighted sum of outputs plus the vertical gap∑j=1nΔjI and the denominator is the weighted sum of inputs minus the horizontal virtual gap ∑j=1nΔjR. The equations in constraints are equal to 1.0, meaning that the projection point (ideal DMU) is reached. Therefore we have the following model:
(3)MinΔ*=∑j=1n(ΔjI)+∑j=1m(ΔjR)subjectto∑i=1mvixij-∑r=1suryrj≥0,j=1,2,…,n∑i=1mvixmin=1;∑r=1surymax=1,vi,ur≥ε≥0,i=1,….,m,hhhhhhhhhhhhhhhhhhh.hhhhhr=1,….,s.
Then, if we let ΔjI be ∑i=1mvixij-∑i=1mvixmin and let ΔjR be ∑r=1surymax-∑r=1suryrj, model (3) is then simplified to the following linear programming (4):
(4)MinΔ*=∑j=1n(∑i=1mvixij-∑i=1mvixmin)+∑j=1n(∑r=1surymax-∑r=1suryrj),subjectto∑i=1mvixij-∑r=1suryrj≥0,j=1,2,…,n,∑i=1mvixmin=1;∑r=1surymax=1,hhhhhhhhhhhhhhvi,ur≥ε≥0,i=1,…,m,hhhhhhhhhhhhhhhhhhhhhhhhhhr=1,…,s.Definition 3.
The performance ofDMUj is better than that of DMUj ifΔj<Δi.From the model (4), it is found that the distance between DMU- and DMUj is defined as ΔjIDMU=(∑i=1mvixij-∑i=1mvixmin)+(∑r=1surymax-∑r=1suryrj). Note that the purpose of the model (4) is to obtain an optimal solution(vi*,ur*)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 aDMUj is on ideal point then we use definition of the CSW efficiency score of DMUj that was defined by the following equation (e.g., see [18, 45]):
(5)μj*=∑r=1syrjur*-∑i=1nzijv´i*∑i=1nxijvi*.


## 3.1. 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 ofn DMUs. Each DMUj(j=1,…,n)hasmdifferent inputs and s different outputs, which are denoted by xij(i=1,2,…,m) andyrj(r=1,2,…,s), respectively. All the data are required to be positive just like the traditional DEA model [1]. The input data of all DMUs form an m by n matrix and output data form an s by n 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.
The virtual ideal DMU is a DMU with minimized inputs of all of DMUs as its input and maximized outputs of all of DMUs as its output. Generally, if we show ideal DMU withDMU-=(X-,Y-), then we have x-i=min{xij∣j=1,…,n}(i=1,…,m) and y-r=max{yrj∣j=1,…,n}(r=1,…,s).Definition 2.
An ideal level is one straight line that passes through the origin and ideal DMU with slope 1.0.In Figure1, the vertical and horizontal axes are set to be the weighted sum of s outputs and the weighted sum of m 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 s outputs equals the weighted sum of m inputs and so DMU-=(∑i=1mx-iv´i,∑r=1sy-ru´r)is an ideal DMU. Given one set of weights u´r(r=1,…,s)and v´i(i=1,…,m), the virtual gaps, between points M and DMU- on the horizontal axis and vertical axis, are denoted by ΔMI=∑j=1nvMxMj-∑j=1nvMxmin and ΔMR=∑j=1nuMymax-∑r=1suMyMj, respectively. Similarly, for points N and L, the gaps will be calculated (ΔNI,ΔNRandΔLI,ΔLR). Observing that there exists a total virtual gap to the ideal point, we aim to determine an optimal set of weights ur*(r=1,…,s) and vi*(i=1,…,m) such that both points M* and N* 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 gapsΔMI+ΔMR+ΔNI+ΔNR+ΔLI+ΔLRto the ideal point are the shortest to DMU-.Figure 1
An illustration of the model for gap analysis, showing DMUs below the virtual ideal DMU.As for the constraint, the numerator is the weighted sum of outputs plus the vertical gap∑j=1nΔjI and the denominator is the weighted sum of inputs minus the horizontal virtual gap ∑j=1nΔjR. The equations in constraints are equal to 1.0, meaning that the projection point (ideal DMU) is reached. Therefore we have the following model:
(3)MinΔ*=∑j=1n(ΔjI)+∑j=1m(ΔjR)subjectto∑i=1mvixij-∑r=1suryrj≥0,j=1,2,…,n∑i=1mvixmin=1;∑r=1surymax=1,vi,ur≥ε≥0,i=1,….,m,hhhhhhhhhhhhhhhhhhh.hhhhhr=1,….,s.
Then, if we let ΔjI be ∑i=1mvixij-∑i=1mvixmin and let ΔjR be ∑r=1surymax-∑r=1suryrj, model (3) is then simplified to the following linear programming (4):
(4)MinΔ*=∑j=1n(∑i=1mvixij-∑i=1mvixmin)+∑j=1n(∑r=1surymax-∑r=1suryrj),subjectto∑i=1mvixij-∑r=1suryrj≥0,j=1,2,…,n,∑i=1mvixmin=1;∑r=1surymax=1,hhhhhhhhhhhhhhvi,ur≥ε≥0,i=1,…,m,hhhhhhhhhhhhhhhhhhhhhhhhhhr=1,…,s.Definition 3.
The performance ofDMUj is better than that of DMUj ifΔj<Δi.From the model (4), it is found that the distance between DMU- and DMUj is defined as ΔjIDMU=(∑i=1mvixij-∑i=1mvixmin)+(∑r=1surymax-∑r=1suryrj). Note that the purpose of the model (4) is to obtain an optimal solution(vi*,ur*)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 aDMUj is on ideal point then we use definition of the CSW efficiency score of DMUj that was defined by the following equation (e.g., see [18, 45]):
(5)μj*=∑r=1syrjur*-∑i=1nzijv´i*∑i=1nxijvi*.


## 4. Nondiscretionary Model

Assume that there aren DMUs, where each DMUj(j=1,2,…,n), uses t different discretionary inputs, xij(i=1,…,t),  and k different nondiscretionary inputs, zij(i=1,…,k) for t+k=m, to produce s different outputs yrj(r=1,…,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:
(6)Maxω=∑r=1suryro-∑i=1kv´iziosubjectto∑r=1suryrj-∑i=1kv´izij-∑i=1tvixij≤0,j=1,2,…,n,∑i=1tvixio=1,vi≥ε,i=1,2,…,t,000000..000000000000000ur≥ε,r=1,2,…,s,00000000000000000000000v´i≥0,i=1,2,…,k.
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.


## 5. 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.Assume that there aren DMUs, where each DMUj(j=1,2,…,n) usestdifferent discretionary inputs, xij(i=1,…,t) and k different nondiscretionary inputs, zij(i=1,…,k) for t+k=m, to produce s different outputs yrj(r=1,…,s).Definition 1.
The virtual ideal DMU is a DMU with minimized inputs of all of the DMUs as its input and maximized outputs of all of the DMUs as its output. Generally, if we show ideal DMU withDMU-=(X-,Y-) then we have x-i=min{xij∣j=1,…,n}(i=1,…,m) and y-r=max{yrj∣j=1,…,n}(r=1,…,s). HereX-consists of two parts, comprehensive minimum discretionary and nondiscretionary inputs and one maximum output for ideal DMU.For this reason, we haveDMU-=(X-,Y-)that x-i1=min{xij,∣j=1,…,n}(i=1,…,m), x-i2=min{zij,∣j=1,…,n}(i=1,…,m), and y-r=max{yrj∣j=1,…,n}(r=1,…,s).Then, according to models (4) and (6) we can construct the following model
(7)MinE=∑j=1n[∑i=1mvixij-∑i=1mvix-1min]+∑j=1n[∑i=1mv´izij-∑i=1mv´ix-2min]+∑j=1n[∑r=1sury-max-∑r=1suryrj]subjectto∑i=1mvixij+∑i=1mv´izij-∑r=1suryrj≥0,hhhhhhhhhhhhhhhhhhhhhhhj=1,2,…,n,∑i=1mvix-1min=1,∑r=1sury-max-∑i=1mv´ix-2min=1,hhhhhhhhhhvi,ur,v´i≥ε>0,i=1,…,m,r=1,…,s.The dual of model (7) is
(8)MaxT=α+β+ε∑i=1mγi+ε∑i=1mγ~i+ε∑r=1sδrsubjectto∑j=1nλjxij+αx-1min+γi≤n(∑i=1mxij-x-1min),i=1,2,…,t.∑j=1nλjzij-βx-2min+γ~i≤n(∑i=1mzij-x-2min),i=1,2,…,t.βy-max-∑j=1nλjyrj+δr≤n(y-max-∑r=1syrj),r=1,2,…,s,λj,γi,γ~i,δr≥0,j=1,2,…,n,i=1,2,…,m,r=1,2,…,s,α,β→free.


## 6. 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.


### 6.1.  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 Tables2, 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 2
Labels of inputs and outputs.
Inputs
Description
Outputs
Description
X1
The number of personnel
Y1
The number of insurance policies
X2
The total number of computers
Y2
The received total sum (income)
Z1
Distance of each branch to city centreTable 3
The data of 20 bank branches.
DMUs
Discretionary inputs
Nondiscretionary inputs
Outputs
X1
X2
Z1
Y1
Y2
1
96
86
64
30
145
2
75
88
1.2
0.001
175
3
77
85
0.4
11
113
4
91
93
2.3
10
128
5
89
83
68
9
101
6
102
97
0.8
7
82
7
96
90
6.5
47
154
8
85
92
2.3
11
54
9
106
84
20.1
43
179
10
107
95
1.4
9
117
11
94
78
49
81
37
12
78
89
1.7
11
124
13
102
107
0.7
30
185
14
82
92
1.4
28
51
15
77
92
1.7
6
28
16
89
85
23.5
15
85
17
84
104
5.4
15
109
18
94
91
1.9
13
72
19
97
95
3.5
13
129
20
82
100
1.2
29
150Table 4
The results of 20 bank branches from models (5)–(7).
DMUs
Model (6)
Rank
Models (5) and (7)
Rank
DMUs
Model (6)
Rank
Models (5) and (7)
Rank
1
0.8193072
7
0.814882
5
11
1.0000000
1
0.274558
19
2
1.0000000
1
1.000001
2
12
0.7129804
9
0.703550
7
3
0.6949586
11
0.667146
9
13
0.9295474
6
0.864070
3
4
0.6982259
10
0.677215
8
14
0.7662059
8
0.293495
18
5
0.5892561
13
0.586333
11
15
0.1801541
20
0.157402
20
6
0.4343827
16
0.409140
15
16
0.4927510
15
0.489511
14
7
1.0000000
1
0.845831
4
17
0.5596400
14
0.541312
13
8
0.3169876
18
0.296808
17
18
0.4135209
17
0.388773
16
9
1.0000000
1
1.000017
1
19
0.2405702
19
0.663105
10
10
0.6294556
12
0.585199
12
20
1.0000000
1
0.777212
6Table 5
The weights of model (6).
Weights
Model (7)
Weights
Model (7)
Weights
Model (7)
v
1
*
0.3080772
E
-
02
v
´
1
*
0.1000000
E
-
05
u
1
*
0.7204692
E
-
03
v
2
*
0.9858232
E
-
02
u
2
*
0.6277616
E
-
02Table4 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 Table5) 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.


## 6.1.  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 Tables2, 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 2
Labels of inputs and outputs.
Inputs
Description
Outputs
Description
X1
The number of personnel
Y1
The number of insurance policies
X2
The total number of computers
Y2
The received total sum (income)
Z1
Distance of each branch to city centreTable 3
The data of 20 bank branches.
DMUs
Discretionary inputs
Nondiscretionary inputs
Outputs
X1
X2
Z1
Y1
Y2
1
96
86
64
30
145
2
75
88
1.2
0.001
175
3
77
85
0.4
11
113
4
91
93
2.3
10
128
5
89
83
68
9
101
6
102
97
0.8
7
82
7
96
90
6.5
47
154
8
85
92
2.3
11
54
9
106
84
20.1
43
179
10
107
95
1.4
9
117
11
94
78
49
81
37
12
78
89
1.7
11
124
13
102
107
0.7
30
185
14
82
92
1.4
28
51
15
77
92
1.7
6
28
16
89
85
23.5
15
85
17
84
104
5.4
15
109
18
94
91
1.9
13
72
19
97
95
3.5
13
129
20
82
100
1.2
29
150Table 4
The results of 20 bank branches from models (5)–(7).
DMUs
Model (6)
Rank
Models (5) and (7)
Rank
DMUs
Model (6)
Rank
Models (5) and (7)
Rank
1
0.8193072
7
0.814882
5
11
1.0000000
1
0.274558
19
2
1.0000000
1
1.000001
2
12
0.7129804
9
0.703550
7
3
0.6949586
11
0.667146
9
13
0.9295474
6
0.864070
3
4
0.6982259
10
0.677215
8
14
0.7662059
8
0.293495
18
5
0.5892561
13
0.586333
11
15
0.1801541
20
0.157402
20
6
0.4343827
16
0.409140
15
16
0.4927510
15
0.489511
14
7
1.0000000
1
0.845831
4
17
0.5596400
14
0.541312
13
8
0.3169876
18
0.296808
17
18
0.4135209
17
0.388773
16
9
1.0000000
1
1.000017
1
19
0.2405702
19
0.663105
10
10
0.6294556
12
0.585199
12
20
1.0000000
1
0.777212
6Table 5
The weights of model (6).
Weights
Model (7)
Weights
Model (7)
Weights
Model (7)
v
1
*
0.3080772
E
-
02
v
´
1
*
0.1000000
E
-
05
u
1
*
0.7204692
E
-
03
v
2
*
0.9858232
E
-
02
u
2
*
0.6277616
E
-
02Table4 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 Table5) 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.


## 7. 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.



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*Source: 906743-2013-12-17.xml*