A NOVEL RANKING APPROACH WITH COMMON WEIGHTS: AN IMPLEMENTATION IN THE PRESENCE OF INTERVAL DATA AND FLEXIBLE MEASURES

. In this paper a ranking method using common weights methodology is presented. The goal of the method is enhancing the decision maker (DM)’s influence in the ranking procedure. Although DM’s preference information is an important element in our method, the approach can also be modified to be used in the absence of it. Since we aim to implement the approach on an empirical instance, the model is modified to deal with the properties of the sample, so it is developed in the presence of the interval data and flexible measures. Finally, the results are discussed


Introduction
Data envelopment analysis (DEA) uses mathematical programming based techniques to evaluate the performance of a set of homogeneous decision making units (DMUs).The first DEA model is known as CCR that was developed by Charnes et al. [7].Classical DEA models calculate the maximum relative efficiency per DMU, and accordingly DMUs are classified in two groups of efficient and inefficient ones.So far, lots of ranking methods have been presented by researches that each can be applied regarding its conditions and the insight behind it.We mention some of them briefly as follows: Blas et al. [15] suggested a ranking method that used measures of dominance derived from social network analysis in combination with DEA.Oukil [38] suggested a new perspective for ranking DMUs under a DEA peer-evaluation framework.He exploited the property of multiple weighting schemes generated over the cross evaluation process in developing a methodology that yields not only robust ranking patterns but also more realistic sets of weights for the DMUs.Also see Shahbazifar et al. [45], Ghadami et al. [23], Hosseinzadeh Lotfi et al. [25], and etc.Now, we focus on some methodologies more relevant to our study, in the followings.

Weight restrictions
Extra information or special conditions may impose some weight restrictions to DEA models.Assurance region (AR) and cone ratio are two well-known approaches in this field.AR approach was firstly introduced by Thompson et al. [50].In the approach, the weights of some indicators are relatively restricted.Cone ratio approach is a more generalized version introduced by Charnes et al. [8] in which the input and output weights are restricted to belong to the related polyhedral convex cones each of which spanned by a finite number of admissible non-negative direction vectors.For further information see Ebrahimi and Khalili [19], Ennen and Batool [21], Podinovski and Bouzdine-Chameeva [39], Chen et al. [9], and Luptáčik and Nežinský [35].

Common weights methodology
In classical DEA models, the best relative efficiency of each DMU is obtained based on individual set of weights.Common weights methodology, introduced by Cook et al. [13] and developed by Roll et al. [41], can provide the opportunity to assess performance of DMUs fairly under common base.In this study, we propose a linear programming to generate a common set of weights.Before describing our method, we review some methods in the context.The idea behind the models generating a CSW can be different.A considerable number of the models, directly or indirectly, search a CSW with the purpose of maximizing the efficiencies (of all DMUs or a subset of them).See the following works in this regard: Chiang and Tzeng [10] presented a multi objective model to find a CSW such that the efficiencies of all DMUs become maximum.In order to solve the multi objective problem, they used max-min approach and presented a non-linear programming to choose a set of weights which its minimum efficiency is maximum in the feasible region.Kao and Hung [29] used compromise solution approach to determine a CSW.They solved the standard DEA models for DMUs and considered them as the ideal solution to achieve.Then, they searched a CSW that its vector of efficiency scores would be the closest to the ideal solution.Liu and Peng [34] searched one common set of weights to maximize group efficiencies.They proposed a linear programming to generate a CSW by using CWA-methodology.Some modifications on their work are presented by Ramezani et al. [40].Chiang et al. [11] used a linear programming to generate a CSW with the aim of maximizing efficiencies.They applied their method to rank some countries based on the gained medals in 2008 Beijing Olympic Games.For further works, see Jahanshahloo et al. [28], Wang et al. [62], Tavana and Santos-Arteaga [49], Khalili-Damghani and Fadaei [31], Shirdel and Ramezani-Tarkhorani [46], Yekta et al. [64], Hatami Marbini et al. [24] and etc.However, maximizing the efficiencies is not always the case to produce a common set of weights.Besides, it may be along with some difficulties, as follows: -Inappropriate treatment.
Generating a CSW with the aim of increasing the efficiency scores of the DMUs may cause the unrealistic imagination of individual performance of the DMUs.So, it may lead to wrong judgments and policies to be made by DM. -Diminution of distinction.
In addition to the importance of the purpose behind each ranking method, its power to distinct the performance of DMUs is important, too.The more the distinctive power of a ranking method, the easier DM can judge the performance of DMUs.In many of the models used to generate a CSW, the benchmark level for efficiency scores is 1, and generating a CSW with the aim of maximizing the efficiencies may increase the probability of having some DMUs with the same efficiency scores of 1.Hence, using more ranking criteria for more distinction may be required.(for DMUs with the same efficiency scores less than 1, it usually can be removed by increasing the decimal precision) Common weights methodology is also utilized in concept of determining most efficient DMUs.Toloo et al. [59] presented an integrated model for determining most BCC-efficient DMU by solving one linear programming.Toloo [54] presented an epsilon-free basic integrated LP model to identify the most efficient candidate unit(s).
In some cases the model can even find the most efficient DMU.Toloo [55] developed a supplier selection approach based on DEA for the case of suppliers with imprecise data.They presented an integrated mixed integer programming-data envelopment analysis (MIP-DEA) model for finding the most efficient suppliers in that condition.Toloo and Salahi [58] suggested a model that lets the efficiency score of only a single unit be strictly greater than one.Toloo and Mirbolouki [57] used common weights methodology in project selection problem.They developed a DEA approach with the aim of finding a composite project with the highest average CCR-efficiency score of indivisdual involved proposals.Selection-based problems are one of the most interesting applications of the common weights methodology in DEA.For example see, Toloo [53], Ertay [22], Toloo [56] and Kresta [32].

The motivation and the purpose of the new method
In our approach a common set of weights is generated with a distinct perspective rather than the mentioned methods.In fact, the approach is to be utilized in the situation in which, for some environmental conditions or DM's objectives, some of the indicators may become significantly important for DM so that he/she prefers to influence their affect in evaluation of DMUs as much as possible.In other words, the adopted criterion to generate a common set of weights is based on a special category of DM's preference information.Thus, it makes us consider another category and another way to deal with DM's preferences.

Overview of our approach
We present a linear programming in which the objective function concentrates on the mentioned DM's preference information.Actually, the DM's preference information is divided into two distinct groups to be used in the body of the model.The CSW is obtained based on considering both of the information.Although attending to DM's preferences is a main element in this ranking approach, but the method can be also developed when no prior information is available.That is, this method can be used in either the presence or absence of DM's preference information.In the first case, DM's information is an essential element; in the second, the CSW is obtained indirectly for maximizing the efficiencies of DMUs.

An implementation of the proposed approach
At first, we briefly mention two necessary concepts in this regard: -Imprecise data envelopment analysis (IDEA) extends DEA models to deal with imprecise data, that is when some input or output data are not exact.For more information, we refer to Cooper et al. [14], Salahi et al.
We aim to implement our approach on a data set regarding to a number of bank branches.The case contains some properties: Firstly, some input and output data are not exact and lie in some bounded intervals.Secondly, there are some flexible measures in the evaluation.Hence, we developed our approach to deal with interval data and flexible measures.We also implement another method with different point of view, and finally the results are discussed.
The rest of this paper is as follows.
In Section 2, some preliminary concepts are stated.In Section 3, the suggested approach is presented.In Section 4, the method is considered in the presence of interval data and flexible measures, and the related modifications are conducted on the model.In Section 5, for comparison purpose, a model to generate a CSW in interval DEA is modified in presence of flexible measures.In Section 6, finally, we are ready to implement our method on the considered case, and the results are investigated.Section 7 presents the conclusion.

Preliminaries
Here, a brief mention of the elementary notions is presented.

CCR model
Consider a set of  homogeneous DMUs that consume  inputs and produce  outputs.Let   ( = 1, . . ., ) and   ( = 1, . . ., ) denote, respectively, the amounts of the th input consumed and the th output produced by DMU  ( = 1, . . ., ).Model (2.2) is run separately for each DMU, and so the extracted set of optimal weights are individual per DMU.The optimal value of 1 for Model (2.2) means that DMU  has the opportunity to gain the best relative efficiency among the group of DMUs.Moreover, if there is also an optimal solution with totally positive weights, then DMU  is called a CCR-efficient DMU.

Multi-objective programming problem
A multi-objective programming problem (MOP) can be written as follows: where   (),  = 1, . . ., , are real valued functions on , and  ⊆ R  .Usually, there is no optimal solution to Model (2.3), and so a Pareto-optimal solution to Model (2.3), that is defined as follows, is searched instead.

The approach
In this section, at first, we explain how to consider DM's preference information, and then the proposed method to generate a CSW is presented.

Categorization of DM's preference information
The DM's preferences is an important element in the proposed method.The initial assumption of this method is that DM assumes more importance for some indicators, so that he/she tends these indicators to have relatively more impact on the assessment of DMUs.We call such indicators worthy indicators.The set of indices of worthy input indicators is denoted by Φ  and that of worthy output indicators by Φ  .The assumption of this method is Φ  ∪ Φ  ̸ = ∅ (so, may Φ  = ∅ or Φ  = ∅).We call the information about the worthy indicators as preference information of type I.In other hand, DM may have more information about the relative importance of some indicators, too.We call such information, preference information of type II (it means the conventional weight restrictions which is not a new notion).In our method, there is no necessarily for DM to state the type II exactly, and it is sufficient to state only the range of their expected values.Now, we introduce the other notations used in this regard in our approach.let  = {1, . . ., } and  = {1, . . ., } where  and  are the number of input and output indicators, respectively.Γ  ⊆  ×  denotes the set of ordered pairs of input indicators which there is some information about their relative importance.Then, Γ  is assumed to satisfy the following conditions: -If (, ) ∈ Γ  then at least one of the maximum or minimum of expected value for relative importance of the th input indicator to the th indicator is given by DM (it's evident that if DM has determined an exact value, the minimum and maximum are the same).
Also, Γ  ⊆  ×  is defined in a similar way.Γ  ⊆  ×  is a set of all (, ) for which there is some information, about the relative importance of the th input indicator to the th output indicator, as mentioned above.Γ  ⊆  ×  is defined in a similar way.In our method, it's assumed that if (, ) ⊆ Γ  then (, ) / ∈ Γ  .It is noticeable that all of Γ  , Γ  , Γ  and Γ  may be equal with ∅.

The suggested model
Here, we present our proposed model to generate a CSW.In the model, it is intended to generate a CSW satisfying the relative importance of the indicators in which the weights of the worthy indicators are maximum.Hence, DM's preference information of type I is used in the objective function, and the DM's preference information of type II is used as the weight restrictions, if there are any.
Suppose that Φ  , Φ  , Γ  , Γ  , Γ  and Γ  are defined as it was discussed in the previous section.Then, the initial model to generate a CSW, in general form, is considered as follows: is a non-Archimedean infinitesimal constant which prevents the weights to become zero.In fact, as it is seen later, the absolute efficiencies calculated based upon the generated CSW have a pivotal affection on the ranking scores, and so we would like to involve all of the input and output indicators in the assessment.In other words, if some of the weights in the generated CSW is equal to zero, the role of the corresponding indicator would be ignored in the in the continuation of the ranking process.Also, since the last batch constraints guarantee that the input and output weights are positive, the model can be equivalently rewritten as the MOLP (3.2), because of the non-zero denominators: According to Theorem 2.3, an optimal solution to the linear programming problem (3.3) is a Pareto-optimal solution to Model (3.2).
max ∑︁ where the parameters    > 0 ( ∈ Φ  ) and    > 0 ( ∈ Φ  ) specify the priority among the indicators of Φ  ∪ Φ  .If DM assumes no priority among the indicators, then all of these parameters are set to 1.The constraint (i) makes the absolute efficiencies of the DMUs to be less than or equal to 1.The constraint (ii) is a normalizing constraint which, as we show later, also guarantees that Model (3.3) has a finite optimal value.If The optimal solution to Model (3.3) is accepted as the common set of weights.It is considerable to state a general issue encountering ranking methods using common weights: Since the ranking methods are usually dependent on the generated CSW, having alternative optimal solutions may lead to different ranking results.Thus, an extra model can be solved to ensure that only one of the optimal solutions is considered as the CSW.For example, see Liu and Peng [34] and Sun et al. [48].To this aim, we use the criterion presented by Sun et al. [48] to choose one of the optimal solutions of Model (3.3) as the CSW.
As it is discussed in the following, in the absence of DM's preference information of type II, Model (3.3) is feasible for a small enough .It is clear that the feasibility of Model (3.3) depends on the weight restrictions, too.
, where all of the components of 1  ∈ R  and 1  ∈ R  are equal to 1.
Then ( ,  , ) is a feasible solution to (3.3) in the absence of DM's preference information of type II. Proof.
It's easily seen that ( ,  , ) satisfies the rest of the constraints to Model (3.3) in the absence of DM's preference information of type II, and we are done.
3) in the absence of DM's preference information of type II.
Proof.Let S be the feasible region of Model (3.3) and ( ,  ) ∈  be an arbitrary feasible solution to it, where   ∈ R  and   ∈ R  .As a consequence of the constraint (ii) and the constraints (iv) and (v), and the assumption that at least one input indicator and one output indicator exist in evaluation process:

The developed model in the absence of DM's preference information
As it is mentioned earlier, Model (3.3) is based upon a pivot assumption of the presence of DM's preference information of type I.However, in the absence of such information, the model can be slightly modified to be used indirectly with respect to the conventional objective of generating a CSW maximizing the efficiencies of DMUs.To this aim we set Φ  = {1, . . ., } and Φ  = ∅.That is Φ = Φ  .Model (3.5) is the modified model to deal with the circumstance lacking of any DM's preference information.
With regard to Theorem 2.3, Model (3.5) can be rewritten as follows: The explanations about the constraints is similar to those of Model (3.3) so it is ignored.It is evident that if there are any weight restrictions, they can be added to the set of the constants to Model (3.4).

The ranking approach
In our method, DM's preference information is also regarded in the ranking procedure.
For ranking the DMUs, firstly, the absolute efficiency of each DMU is calculated based upon the generated CSW.Let ( * ,  * ) be the generated CSW.The absolute efficiency of DMU  ( = 1, . . ., ), based upon the CSW is: The greater value of  *  , the greater ranking score of DMU  .It is notable that in the case of existing two DMUs with the same efficiency of less than 1, it is usually removed through increasing decimal accuracy.However, we present the following general approach.
Considering Φ ̸ = ∅, if there are two DMUs with the same efficiencies, we utilize a second ranking criterion to help to prioritize of them.To this aim, we concentrate on the state of the units in terms of the worthy indicators for DM.Since the DM's preference information of type I may contain only some input indicators, only some output indicators, or both of them, the definition of the second criterion depends on the indicators existing in this type of information, as follows: (I) If Φ  = ∅ then Φ = Φ  ⊂ {1, . . ., }.In this case, if it is required, the second criterion for assessment of DMU  ( = 1, . . ., ) is defined as: (II) If Φ  = ∅, Φ = Φ  ⊆ {1, . . ., }.In this case, if it is required, the second criterion for assessment of DMU  ( = 1, . . ., ) is defined as: (III) If Φ  ̸ = ∅,Φ  ̸ = ∅, Φ contains some input and some output indicator indices.In this case, if it is required, the criterion for assessment of DMU  ( = 1, . . ., ) is defined as: In the last case,   is a ratio that can be interpreted as the absolute efficiency of DMU  based on the worthy indicators.It is significant that, unlike  *  , 1 is not an upper bound for   .Besides, it is noticeable that ∑︀ ∈Φ   *    may be equal to zero.In this case, actually, DMU  consumes none of the worthy inputs, that it is great.In our method, if this statement occurs for only one of two units under comparison, the unit is preferred to the other.If the situation occurs for both of them, they are compared in terms of the valuable output indicators, and the second criterion is defined as Eventually, considering the above discussion, our ranking rules can be stated as follows: Let ,  ∈ {1, . . ., } and   and   denotes the ranking scores of DMU  and DMU  , respectively. Then:

The development in encountering with the interval data and flexible measures
We aim to implement our approach on a sample presented in Section 6.To this aim, it is required that the approach is developed in encountering with interval data and flexible measures.

Dealing with interval data
Let {DMU 1 , . . ., DMU  } be  homogeneous DMUs each of which associated with  inputs and  outputs.The amounts of the th input and the th output for DMU  ( = 1, . . ., ) are generally denoted by   and   , respectively.Besides, it is assumed that their value lie in the intervals of [︀    , where the parameters    > 0 ( ∈ Φ  ) and    > 0 ( ∈ Φ  ) are as already explained.Since, for any 0 < (,  ), we have: The constraint ∑︀  =1      − ∑︀  =1      ≤ 0 ( = 1, . . ., ) allows the most extensive search for the weights placed on the objective function compared to other activity levels.It is also concluded that the absolute efficiencies of each DMU, considering all of its activity levels, based upon each feasible solution do not exceed one.Matrix  is defined similar to what was explained in Model (3.3).For determining one CSW the criterion used in Sun et al. [48] is utilized on optimal solutions to Model (4.1).
In this case, the ranking criterion is the absolute efficiency of DMUs based upon the average data of them and extracted CSW.Hence we set: and calculate where ( * ,  * ) is the generated CSW.In the case of Φ ̸ = ∅, we can consider the same discussions as we had in Section 3.4.Here,   is calculated similar to   , but based on the average data.Hence, the developed ranking method in presence of interval data can be presented as the following.
Let ,  ∈ {1, . . ., } and   and   denote the ranking scores of DMU  and DMU  , respectively.Then: In the absence of DM's preference information, there is a similar discussion as presented in Section 3.3, so we ignore the details.

Dealing with flexible measures
In the sample, which shall be discussed in Section 6, we face with a bank assessment case.The deposit is considered in the assessment, but, in general, there is two points of view to involve it.First, since the deposit should be invested within special periods, it can be regarded as an input for a bank branch.In the other hand, considering the fact that deposit is gained through personnel and advertisement, it can also be considered as an output.In classic models, it is assumed that the nature of the indicators, including inputs and output ones are already clear; however, regarding empirical issues, in some circumstance, there are indicators in the assessment upon the nature of which there is disagreement.Therefore, it is important to consider "flexible measures" as well.Now, assume that there are  determined input indicators and  determined output indicators.Also, assume that  shows the number of the indicators which are flexible and each can be used as an input or output indicator in the evaluation process.In Model (3.3), the role assignment of the flexible indicators is done in such a way that the best result is obtained for the target weights in the objective function.These indicators can appear in the preference information, so the new symbols are defined as follows.
Assume that  = {1, . . .,  }, where  ∈ N, is the set of flexible indicators.We define: Φ  : The set of valuable indicators which are important for the DM in assessment but their natures are not determined.
We call them valuable flexible indicators.
where the parameters    > 0 ( ∈ Φ  ),    > 0 ( ∈ Φ  ) and    > 0 ( = 1, . . .,  ) are positive, and there is a similar explanation as before.In the first constraint, the value 0 for the binary variable   ( = 1, . . .,  ) causes the th flexible indicator to be considered as an input indicator, and the value 1 for this variable causes that the indicator is considered as an output indicator.In the normalizing constraint After obtaining a CSW and the corresponding  * , the state of each flexible measure is determined as an input or output indicator.Afterwards, accordingly, the ranking approach presented in Section 3 can be used.Also, in this case, in the absence of DM's preference information, we can use Model (4.5)After obtaining a CSW and the corresponding  * , the flexible measures are classified as input or output indicators, accordingly.Afterwards, the ranking approach presented in Section 4.1 can be used in this regard.

A model for comparison
Shirdel et al. [47] proposed a method using common weights methodology in interval DEA.They presented a linear programming whose aim was generally to maximize possible efficiencies of DMUs.We aim to utilize the model in Section 6, for comparison purpose, so it is developed to deal with flexible measures, too: where    and    ( = 1, . . ., ,  = 1, . . ., ) are respectively the lower and the upper bounds for possible values of the th flexible indicator for DMU  .In general, Shirdel et al.'s model [47] searches a set of weights for maximizing the minimum possible efficiencies of DMUs.The utilized criterion to the aim, is reducing the sum of the differences between the virtual input and the virtual output of each DMU.Considering the binary variables, Model (5.1) results in the role of the flexible indicators such that the best value of the objective function is obtained in that regard.

An implementation
In this section, we implement our method on a data set regarding with 20 bank branches.In this assessment, there are 5 indicators such that one of which, the number of employees, is considered as the input indicator and two of which, overdue claims and facilities are considered as output indicators.However, the nature of the indicators called long term deposit and short term deposit are not certainly determined in the evaluation and each can be assumed as either input or output indicator.As Tables 1 and 2 show, all data, except the number of employees, lie on intervals.In this assessment, the relative importance of the indicators are as follows.The relative importance of the overdue claims rather than the bank facilities is considered between 1 and 1.5, that of the short term deposit rather than the bank facilities is between 2 and 3, and that of the bank facilities rather than the long term deposit is between 0.5 and 1.
The overdue claims indicator has an important role in bank performance assessment because a bank branch can improve its resources and capital by collecting overdue claims, and so it would be able to give better and more facilities and services to its customers.Regarding the given information in this instance, we use Model (4.6).The weight restrictions to be applied in the model are considered as follows: One can refer to Tables 1 and 2 for getting the information about the lower and upper bounds of the input and the output data per DMU.
Beside applying our approach, we also intend to survey another situation when there is no DM's preference information available.To this aim, our approach in the absence of DM's preference information is used, too.Plus, for comparison intention in the situation, we use the 3rd method.Actually, both of the methods can be considered in the category of the methods that generate a CSW with the aim of maximizing efficiencies, but with two different points of view in this regard.There are two flexible measures in this sample, both of which are of deposit type.Hence, we aim to assign them the same roles, both as input indicators or as output indicators.So, regarding with this sample, only one binary variable is required in the Models (4.6)-(5.1).
For convenience, we call the utilized methods as follows: The 1st method: our approach in the presence of DM's preference information.The 2nd method: our approach in the absence of DM's preference information.The 3rd method: the developed approach based on what is presented at Shirdel et al. [47].
At first, we implement the methods on the set of normalized data, and then they are implemented on the original data set.

Implementing the methods on the set of normalized data
Since the indicators has significantly noted in our method, in order to have a better sense of the impact of them, we also consider the set of normalized data.Besides, for the purpose of comparison, we consider both the original data and the normalized data to be used in the implementation of our method, separately.Here, for normalizing the vectors, for example the first output's, we act as the following: where,   1 ∈ R  and   1 ∈ R  are, respectively, the vector of the lower bounds and the vector of the upper bounds for the first output indicator's values, and ‖.‖ ∞ is the infinity norm.We set: Thus, after the normalization, ]︀ is considered as the interval of the possible normalized values of the first output of DMU  .See Tables 3 and 4.
For each method, the generated CSW and the optimal value of the binary variable, indicating the state of the flexible measures in the optimality, are shown in Table 5.As Table 5 shows, in our approach, the results are obtained while the deposits are considered as input of the bank branches.The CSWs extracted from the 1st and 3rd methods are obtained under the conditions when the flexible measures are assumed as input ones whereas the mentioned measures have a different role, as output, in extracting the CSW of the 2nd method.In the 1st method, satisfying the weight restrictions, the weight of the first output indicator, as the only worthy indicator in this instance, is 0.263286, that is greater compared to when the mentioned indicators function as output indicators in evaluation process, that is 0.214243.Although,  * 1 is greater in the generated CSW from the 2nd method, one should envisage that this weight has been obtained under more restrictions in 1st method rather than the 2nd method.In other words, the CSW extracted through the 2nd method is not even a feasible solution to the Model (4.6).
By fixing the role of the flexible measures according to the results form Table 5, the minimum, the maximum and the average of the possible absolute efficiencies based upon the generated CSW are shown in Table 6.
Finally, Table 7 shows the ranking scores of all DMUs obtained from the considered approaches.As Table 6 shows, the amount of the average of the lower and the upper efficiencies per branch is quite distinct, so there is no need to use extra ranking criterion (The efficiencies have been calculated with six decimal places).
The ranking results have been compared visually in Figure 1; both the ranks of different branches in one method and the rank of one branch in different methods have been illustrated.

Implementing the methods on the original data set
Here, the 1st, 2nd, and 3rd methods were implemented on the original data set.Tables 8, 9, and 10 have similar description as those for Tables 5, 6, and 7, respectively.For brevity, the repetitive explanations are ignored.In the absence of DM's preference information, the 2nd and 3rd methods almost have similar purposes of generating a CSW for maximizing efficiencies.We calculate the average for the lower, upper, and average efficiencies in both the methods; the ratio of the obtained averages in the 3rd method compared to the 2nd method is 0.974169, 1.701211, and 1.364175, respectively.

Some further discussion on the results
In this sample, although the first output indicator, overdue claims, gets involved in the weight restrictions, it's desired to influence its affect in the performance assessment as much as possible.As it is seen in Table 5,  * 1 in the 1st method is significantly greater than the 3rd method, but  * 1 has gained the largest value in the 2nd method.The results were obtained when the long term deposit and the short term deposit are considered as inputs in the 1st and 3rd methods and as outputs in the 2nd method.It is considerable that, although, here, the purpose of the 1st method is to produce a CSW maximizing  1 , the weight restrictions of the model is more 0.019971 0.015886 0.001717 0.009441 0.005579 0.020621 0.032275 0.026448 Branch6 0.029491 0.034736 0.032114 0.026635 0.03161 0.029122 0.035194 0.047552 0.041373 Branch7 0.074257 0.105096 0.089677 0.005472 0.006991 0.006231 0.179533 0.37888 0.279206 Branch8 0.06118 0.067967 0.064573 0.060052 0.065146 0.062599 0.046957 0.062663 0.05481 Branch9 0.09748 0.122616 0.110048 0.027232 0.029817 0.028524 0.372545 0.550435 0.46149 Branch10 0.01495 0.022967 0.018958 0.000514 0.001183 0.000848 0.048096 0.090375 0.069235 Branch11 0.013072 0.014953 0.014013 0.004024 0.005791 0.004908 0.021093 0.025288 0.023191 Branch12 0.013572 0.028587 0.02108 0.002273 0.023199 0.012736 0.035498 0.061384 0.048441 Branch13 0.012481 0.01469 0.013586 0.001562 0.003734 0.002648 0.027675 0.035649 0.031662 Branch14 0.010827 0.017181 0.014004 0.002229 0.003799 0.003014 0.023142 0.042042 0.032592 Branch15 0.011659 0.014963 0.013311 0.001735 0.002653 0.002194 0.022236 0.033238 0.027737 Branch16 0.00538 0.009878 0.007629 0.000165 0.006539 0.003352 0.011164 0.01527 0.013217 Branch17 0.006913 0.009767 0.00834 0.000887 0.002631 0.001759 0.014078 0.023744 0.018911 Branch18 0.013615 0.016279 0.014947 0.001559 0.002367 0.001963 0.044855 0.080198 0.062526 Branch19 0.005663 0.007511 0.006587 0.000437 0.001579 0.001008 0.015922 0.021716 0.018819 Branch20 0.023844 0.029308 0.026576 0.002084 0.005821 0.003952 0.025887 0.03607 0.030978 than those of the 2nd method.In Table 8, there is a similar discussion, but the long term deposit and the short term deposit are considered as inputs in all of the methods.As seen, the values of  * 1 obtained by using the normalized data are generally greater than those obtained by using the original data.Now, we are interested in surveying the state of the branches in terms of the overdue claims indicator and their ranking scores.To this aim, we use a roughly criterion as follows: At first, the average of the lower and upper bounds of overdue claims is calculated per branch.The quantity for DMU  is indicated by AVEO1  ,  = 1, . . ., 20.Then, the average of AVEO1  ,  = 1, . . ., 20, is calculated.It is indicated by AVEO1.We set   = AVEO1 AVEO1 ,  = 1, . . ., 20. Figure 2 illustrates the trends of changes in the ranking scores (obtained by using the normalized data) by increasing   .The improvement in the ranking scores by increasing   is more perceptible in the 2nd method rather that of the others.The values of   ,  = 1, . . ., 20, are arranged in ascending order.
As Figure 2 illustrates, the Branch19 has adopted the worst ranking score in all of the methods, and  19 is also minimum.Besides, the Branch4, has gained the greatest ranking score in all of the methods, and  20 is maximum, too.However, it is not a general rule.Now, we concentrate on the original data, and compare the obtained efficiencies from the methods, specially the 2nd and the 3rd method.To this aim, we consider the columns of Table 9.Given a method, the average of the obtained lower efficiencies and the average of the obtained upper efficiencies for the branches are indicated, respectively, by AVELE and AVEUE.As Figure 2 illustrates, in the absence of the DM's preference information of type II, the 3rd method has a better state than the 2nd method in terms of maximizing the efficiencies.According to Table 8, for both methods, the efficiencies have been obtained under the situation that the flexible indicators are considered as inputs.The output weights from the 2nd method are greater than or equal to the corresponding ones in the 3rd method.However, it is not a contradiction and confirms that the magnitude of the obtained efficiencies is not only dependent on the magnitude of the output weights.In fact, as expected,   the summation of the input weights (that includes 3 indicators in the optimality) in the 3rd method is greater than that of the 2nd method.But, in the 3rd method,  * 1 is significantly larger than the other weights whereas the values of the first input indicator, number of employees, are considerably less than the others'.However, in the 2nd method,  * 1 (that in the optimality acts as the weight of the second input indicator) is significantly larger than the other weights, and the values of the corresponding indicator is also considerable rather than the others'.Thus, the large value of  * 1 in the 2nd method has been generally more effective in the obtained amounts of the efficiencies rather than the value of  * 1 in the 2nd method.As Figure 3 illustrates, the 1st method has a relatively good state in terms of maximizing the efficiencies.Of course, this is not the concern of the 1st method but shows that concentrating on the weights may also lead to relatively acceptable results in terms of maximizing the efficiencies.Ultimately, the ranking scores obtained from the 1st method (as the main approach) in two cases of considering the normalized data and the original data are compared visually in Figure 4.

Conclusion
In this paper, we propose a new ranking method using common weights with a different point of view as it is conventional.DM's preference information has a substantial role in this method.The information is categorized into two groups to be used in the model structure.Albeit, the model can be also developed to be used in the absence of the preference information through some modifications.In fact, we study another perspective to generate a CSW.The goal of our approach is to emphasize on the influence of some valuable indicators (in DM's viewpoint) as much as possible in the evaluation process.The approach is implemented on a data set regarding with a number of bank branches.The instance includes interval data and some flexible measures.Hence, the proposed approach is developed to encounter with the issues.Considering two cases of the original data and the normalized data, the results are discussed.It is noticeable that the structure of the feasible region of the proposed model is not a directly consequence of the considered criterion to select a CSW.Hence, the model can not be interpreted as one that makes the feasible region originally regarding its objective.
) is a matrix corresponding with the weight restrictions, where0 <  ≤ 2 (|Γ  | + |Γ  | + |Γ  | + |Γ  |).Ifwe have the weight restriction      −   ≤ 0, (, ) ∈ Γ  , in the model, there is corresponding column     + −  + , and if we have the weight restriction   −      ≤ 0, (, ) ∈ Γ  , in the model, then there is corresponding column  + −     + in Matrix .Other columns of Matrix  are also based upon the weight restrictions related to Γ  , Γ  and Γ  similar to what was stated about Γ  .It is trivial that in the absence of the weight restrictions, the constraints (iii) is omitted.

Figure 1 .
Figure 1.The Comparison of the obtained ranking scores from all methods by using the normalized.

Figure 2 .
Figure 2. The trends of changes in the ranking scores (using the normalized data) by increasing   .

Figure 3 .
Figure 3.Comparison of the averages of the efficiencies (by using the original data).

Figure 4 .
Figure 4.The ranking scores obtained from the 1st method using the normalized and the original.
According to(3.4)and the positivity of    ( ∈ Φ  ) and    ( ∈ Φ  ):  is an upper bound for the set of objective values of Model(3.3).Therefore, in feasibility, Model (3.3) has a finite optimal objective value.

Table 1 .
The lower bounds of the input and the output data for 20 bank branches.

Table 2 .
The upper bounds of the input and the output data for 20 bank branches.

Table 3 .
The lower bounds of the normalized input and output data for 20 bank branches.

Table 4 .
The upper bounds of the normalized input and output data for 20 bank branches.

Table 5 .
The produced CSWs by using the normalized data.

Table 7 .
The obtained ranking scores by using the normalized data.

Table 8 .
The produced CSWs by using the original data set.

Table 10 .
The obtained ranking scores by using the original data.