RANKING EXTREME AND NON-EXTREME EFFICIENT DECISION MAKING UNITS IN DATA ENVELOPMENT ANALYSIS

Abstract: In evaluating decision making units (DMU) by using Data Envelopment Analysis (DEA) technique, it happens that more than one unit got efficiency score one. In such a case there should be some criterion for ranking these DMUs. Up to now, all of DEA model could rank only extreme efficient units. In this paper the authors proposed a method for ranking extreme and non extreme efficient units.


INTRODUCTION
Data Envelopment Analysis (DEA) is a non parametric method for evaluating decision making units (DMU).It has been proposed by Charnes et al [8], in assessment of an educational center in USA and extended by Banker et al [5].They used mathematical programming to reach this aim.
One of the main objectives of DEA is to measure the efficiency of Decision Making Units (DMUs) such as schools, public agencies, banks and etc.One of the ways for determining efficiency score of DMUs is to apply the Charnes et.al [8] model (CCR model) that deals with a ratio of multiple outputs and multiple inputs.One of the most interesting research subjects in DEA is to discriminate efficient units.In section 2 we summarized some ranking methods.
The objective of this work is to propose a method for ranking all efficient units, either extreme efficient or non extreme efficient DMUs.In section three the method is illustrated and in section four an example is used to explain our purpose.Our proposed method is based on using a ranking method; for example AP model which is proposed by Anderson and Peterson [4] or other ranking models; and finding strong efficient hyperplane which define production possibility set (PPS).
In evaluating the relative efficiency of each Decision making units (DMUs) by DEA models, we obtain efficiency scores between zero and one.In this way, usually more than one unit may be efficient in DEA models and their efficiency scores are 1.Also, pay attention that the number of efficient units in the Variable return to scale (VRS) models is not less than the Constant return to scale (CRS) models.Therefore, the researchers proposed some methods to discriminant these efficient units.This concept has named Ranking efficient units in DEA.There are lots of ranking methods and each of them has special quality and property to rank efficient units.
Charnes et all [6], counted the number of times that an efficient DMU play the role of benchmark unit for others, and used this norm to rank these units.Since, finding the reference set of a DMU is not easy, their model is not a suitable method.Charnes et all [7], proposed another method to find the benchmark DMUs.They changed the rate of outputs and evaluate the change of efficiency score.However, they didn't distinguish how they can do it.
Sexton et all [17], suggested the cross efficiency method.In this model they used the weights which they had obtained them by solving each of n-linear problems.They evaluate the efficiency of each DMU, n times and store these data in a matrix.Each row of this matrix contains the cross efficiency score of DMUs.They computed the average of these rows and store the result as a ranking measure.It seams this is an acceptable method, but it may has some difficulties.The biggest problem appears when the DEA models have alternative solutions.
Finally, it should be noted that there are some techniques and strategies in DEA which they affect on ranking.For example, Thompson et al [18], used the assurance regions.In their technique, the number of efficient DMUs may be decrease.But it isn't a suitable method because finding suitable weights isn't easy.Adler et al [2], proposed another method to difference between DMUs.In their model they decrease the number of inputs and outputs by component analysis.Therefore, the number of efficient DMUs is decreased.But in general this model couldn't use for a complete ranking.
Anderson and Petersen (AP model) [4], have ranked extreme efficient units by omitting them from Possibility Production Set (PPS), and then Mehrabian et al (MAJ) [15], have modified the AP model.In some circumstances, the mentioned models may be infeasible and specially the AP model may be unstable because of extreme sensitivity to small variations in data, where some DMUs have relatively small values for some of the inputs.Saati et al [16], have modified MAJ model and solved its infeasibility and Jahanshahloo et al [13], have changed the type of data normalization in order to receive a much better result.In order to remove the difficulties from AP and MAJ models, some mathematicians have used specific norms.For instance, Jahanshahloo et al [10], have practiced norm for ranking efficient units.Amirteimoori et al [3], have experienced norm to find the gap between evaluated efficient units and the new PPS.Gradient line and ellipsoid norms have been used by Jahanshahloo et al [14], in order to rank efficient units.Tone [19] and [20], has used SBM model in this way.To review ranking methods see also Adler et al [1].
This paper is organized as follows: section 1 is introduction.Record of ranking will coming in section.We presented our method in section 3 and illustrated a numerical example in section 4. Finally, section 5 is conclusion.

DEA BACKGROUND
In this section we are going to summarize some DEA and ranking models.In this way consider DMUs n, with m inputs and s outputs.The input and output vectors By using the variable return to scale, convexity and possibility postulates, the non-empty production possibility set (PPS) is defined as follows: By the above definition the BCC model proposed by Banker et al [7] is as follows: and its dual is: Clearly, the evaluated p DMU is efficient if and only if 1 = * θ and all slack variables in every optimal solution be zero in problem (1).Equivalently p DMU is efficient if and only if there exists an optimal solution for problem (2) is said to be strong efficient hyperplane when there exists an optimal solution ) , , ( . Anderson and Peterson [4] by omission of efficient units have been ranked these DMUs.Their proposed model is as follows: Clearly, the optimal objective value of AP model is grater than 1 for extreme efficient units and equal 1 for non extreme efficient units.Therefore, AP model does not have any suggestion for ranking non extreme efficient units, and it is the problem of all DEA modes.

PROPOSED METHOD FOR RANKING ALL EFFICIENT UNITS
According to the last section, ranking efficient units has been developed because of existence numbers of efficient units.One of the most important point in ranking models is that there are no DEA models for ranking non extreme efficient DMUs.Therefore, there are no difference between these units.For example by using super efficiency models such as AP and MAJ models, efficiency score 1 is obtained again for these units.In this article we are going to construct a method for ranking all efficient units either extreme or non extreme efficient units by using DEA methods.Before explaining this method it is important to say that Jahanshahloo et.al [12] used mont carlo method for ranking all efficient units.But it is not a DEA method.
This new method is based on AP model and omission of efficient units from production possibility set (Using AP model is arbitrary and one may use other ranking models, especially when AP model occurs infeasibility).Also, strong efficient hyperplane which are binding on extreme efficient units play an important role in this method.To gain this aim the following computational steps should be done: • Use AP model for evaluating all units and suppose it is feasible for all DMUs.Three sets of units may be considered: • Find all strong efficient hyperplane which defines PPS and are binding on members of SE.Then construct a subset of SE which exactly m+s strong efficient hyperplane are binding on them and name the new set by SE .Rearrange members of SE by their descending ranks.Suppose SE be in following format: } ,..., { = , where DMU has the best rank in this set.May be there is no such DMU which m+s strong efficient hyperplane are binding on it.In this case order members of SE by their ranks and continue the process.
At the end of this step note that for finding strong efficient hyperplane the method which is proposed by Huang and Rousseau [9], may be used and it is as follows: • Consider .After above partitioning go to 4.
• Omit • Rank all efficient units by using super efficiency scores of extreme efficient units obtained by step 1 and super efficiency scores of non extreme efficient obtained by the last step.
By using the above procedure all efficient DMUs can be ranked.

EXAMPLE
In this section we constructed 3 examples to use our proposed method for ranking all efficient units.In the first example AP model is used for our proposal an i has one input and one output.1 L norm method is used in second example because of infeasibility of AP model.The third example consists of 2 inputs and 2outputs.

Example1:Ap Model Is Feasible
Data are summarized in table 1 and are shown in figure 1: The results of ranking by AP model are shown in table 2: . Then the set of extreme efficient units which exactly 2 strong efficient hyperplane are binding on them is: , where the members of SE are ordered by their rank.After that it is evident from figure 1 where the members of ' Follow to the last table non extreme efficient unit B has better rank than extreme efficient unit F.

Example2: Ap Model Is Infeasible
Consider the following data set.It is evident that by using AP model that AP model is ifeasible for DMU H. Therefore, we use 1 L norm [10], and it is as ollows:

θ
. By these new scores the rank of extreme efficient units are as follows: Clearly, DMU17 is a non-extreme efficient unite and it is binding on hyperplane no. 3, 4, 5 and 6.Note that there is no extreme efficient unite which it is binding on exactly s m + efficient hyperplane.Therefore, we do the proposed model for DMU3, because it has the best rank in this example.By doing this the new score of DMU17 is: 1.012000 .

CONCLUSION
In this paper we proposed a super efficiency method for ranking extreme and non extreme efficient units.AP model and the proposed model by Huang and Rousseau are used for ranking all efficient units.It is very important to note that in this method discriminant of efficient units is considered.For example unit B is a non extreme efficient units which its super efficiency score obtained by AP model is again 1. Follow to figure 1, unit B is better than extreme efficient units F and H; and it is not considered in AP model.But discriminant of unit B with units F and G is considered in our proposed model.Finally, note that this method may be threatened when non extreme efficient unit is very similar to extreme efficient units.For feature views the authors can remove this weakness and develop it for other ranking models.

θ
is the optimal objective value obtained by AP model in evaluating j DMU .By above definition SE is the set of extreme efficient units, E′ is the set of non extreme efficient units and finally, F is the set of inefficient DMUs.Suppose that ∅

1 i
DMU .Define a subset of E' which its members are binding on strong efficient hyperplane of 1 i DMU and omit it from E' value.The table of data and result of ranking is as follows: Continue these process for other members of SE and store all super efficiency scores.

Table 2 :
The results of ranking

Table 3 :
Results of ranking obtained by proposed model

Table 5 :
The result of ranking

Example3: Multiple Inputs And Outputs
This example consists of 2 inputs and 2 outputs.Table of data and the result of ranking are shown in the following table:

Table 7 :
Coefficients of defining hyperplanes