An Enhanced Russell Measure of Super-Efficiency for Ranking Efficient Units in Data Envelopment Analysis

Problem statement: Conventional Data Envelopment Analysis (DEA) helps decision makers to discriminate between efficient and ineffi cient Decision Making Units (DMUs). However, DEA does not provide more information about the eff icient DMUs. Super-efficiency DEA model can be used in ranking the performance of efficient DMU s. Because of the possible infeasibility of radial super-efficiency DEA model, the ranking has been li mited to the model under the assumption of Constant Returns to Scale (CRS). Approach: This study proposes a super-efficiency model based on the Enhanced Russell Measure (ERM) of efficiency. T his is a non-radial measure and appropriate for ranking the efficient DMUs when inputs and outputs may change non-proportionally. Results: Theoretical results show that the new super-efficie ncy model is always feasible under the assumption of non-CRS. Also, numerical examples from the liter a u e are provided to test the new super-efficiency approach. Conclusion: This study provides a non-radial measure of superefficiency based on the ERM model to discriminate among the efficient DMUs resulting different efficiency scores greater than one. Unlike the traditional radial super-effic iency models, the proposed method is always feasibl e.


INTRODUCTION
Data Envelopment Analysis (DEA) is a mathematical programming technique that can be used to distinguish between efficient and inefficient Decision Making Units (DMUs). However, the conventional DEA models lack the ability to rank the efficient DMUs. For this purpose, many models (called super-efficiency DEA models) have been proposed to identify the classification of efficient DMUs. The super-efficiency models can be applied in many areas such as industries, financial institutions; education and health care (Nahra et al., 2009). Andersen and Petersen (1993) developed the first radial super-efficiency model (AP model hereafter) for ranking the efficient DMUs by excluding the efficient DMU from the reference set of all the other DMUs in such a way that the efficiency scores for efficient DMUs can be greater than one. Then, many authors proposed various models for ranking the efficient DMUs. For more details see (Zhu, 2001;Tone, 2002;Chen, 2004;Li et al., 2007;Liu and Peng, 2008) among others. In some cases, the radial super-efficiency models can be infeasible. For example, see discussions in (Seiford and Zhu, 1999;Chen, 2005;Li et al., 2007). Due to the infeasibility of the super-efficiency model, ranking has been limited to the radial model under the assumption of Constant Returns to Scale (CRS).
In this study, we propose a super-efficiency model based upon the Enhanced Russell Measure (ERM) model developed by Pastor et al. (1999) for ranking the efficient DMUs. The ERM model is non-radial and deal with inputs/outputs individually, unlike the radial DEA models that the variations of inputs/outputs are proportional. In other words in ERM model the inputs/outputs are allowed to decrease/increase at different rates. It is demonstrated that the proposed super-efficiency model is always feasible under both CRS and Variable Returns to Scale (VRS) assumptions. Then, we apply two numerical examples to test the new super-efficiency model.

MATERIALS AND METHODS
Enhanced Russell measure: Suppose there are n DMUs, where each DMU j (j=1,…,n) consumes m inputs x ij (i=1,…,m) to generate s outputs y rj (r=1,…,s). We assume that all inputs and outputs are positive. Vectors x j = (x 1j , x 2j ,…, x mj ) T and y j = (y 1j , y 2j ,…, y sj ) T represent input and output of DMU j , respectively. We denote the DMU j by (x j , y j ). The production possibility set P C under the CRS assumption is defined as: Assuming CRS, the non-radial Enhanced Russell Measure (ERM) model to measure the relative efficiency of DMU k (k=1,…,n), introduced by Pastor et al. (1999), is given as follows: s, 1,..., r m, 1,..., i n, 1,..., j (1) where, the numerator expresses the average efficiency of the inputs and the denominator expresses the average efficiency of the outputs. Therefore, the objective function can be interpreted as the ratio between the average efficiency of the inputs and the average efficiency of the outputs. From model (1), it holds 0<ρ≤1. Note that the ERM model satisfies properties such as unit invariance and monotone decreasing for any increase in input usage or any decrease in output production. ERM Super-efficiency: Suppose that DMU k =(x k , y k ) is ERM-efficient, i.e. ρ * =1. In an effort to evaluate the ranking of efficient DMUs, first we remove the DMU k from the reference set of model (1). Therefore the production possibility set for the remaining DMUs is as follows: Based on the new production possibility set, model (1) needs to be modified because in order for the DMU k to reach the new frontier, we need to increase its inputs and decrease its outputs. For this purpose, the following model is introduced: Note that in this model, θ i ≥1and 0<φ r ≤1 instead of, 0≤θ≤1and φ r ≥1. From the objective function, this is evident that δ≥1.
It is remarkable that the model (2) is proposed under CRS assumption. The proposed model holds under the assumption of VRS by adding the convexity constraint, namely n j j 1, k The model (2) is a nonlinear programming problem that can be converted into a linear programming problem by using the Cooper et al. (2007) transformation as follows:

RESULTS
We demonstrate the characteristics of our model, theoretically by the following theorems. Theorem 1 indicates that unlike the radial superefficiency DEA models, the ERM super-efficiency model is always feasible under the both CRS and VRS assumptions.
Relationship with the AP model: Andersen and Petersen (1993) developed the first radial superefficiency model for ranking the efficient DMUs by excluding the efficient DMU from the reference set of all the other DMUs to reach an efficiency score greater than or equal to one. The AP model is presented as follows: where, ε>0 is a non-Archimedean infinitesimal constant. For an efficient DMU, θ * is not less than one.  The relationship between the ERM superefficiency model and the AP model is demonstrated by the following theorem.
Theorem 3: The super-efficiency score δ* is not greater than the super-efficiency score θ*.

DISCUSSION
In this section, two examples are provided to test the proposed super-efficiency model. First a numerical example is applied for comparing the new superefficiency model with AP model. Then an application of the ERM super-efficiency model for ranking the efficient DMUs, is shown by using an empirical example A numerical example: Consider six efficient DMUs with four inputs (I 1 , I 2 , I 3 and I 4 ) and two outputs (O 1 and O 2 ) taken from Tofallis (1996). The data set is shown in the left hand side of Table 1. The super-efficiency scores obtained by models (2) and (5) are displayed in columns δ* and θ* of Table 1, respectively. According to Theorem 3, the super-efficiency score obtained by the ERM super-efficiency model becomes lower than that of the AP model. The ranking results of these two models, as reported in the right side of Table 1, are quite similar.

Empirical example:
The proposed model is used to rank the efficient branches of 20 bank branches in Iran provided by Amirteimoori and Kordrostami (2005). The data set is presented in Table 2. Each bank is associated with three inputs: staff (x 1 ), computer terminals (x 2 ) and space (x 3 ) and three outputs: deposits (y 1 ), loans (y 2 ) and charges (y 3 ). Table 2 under the heading ρ* reports the ERM efficiency scores calculated by model (1), where there are seven fully efficient DMUs. Our goal is to rank these seven DMUs. The ERM super-efficiency scores and ranking of these seven efficient DMUs measured by model (2), are displayed in the far right columns of Table 2 under the heading δ*. Thus we have the top-ranked bank branch 15 followed by bank branches 4, 17, 7, 12, 20 and 1, respectively. Therefore, the new super-efficiency model can successfully rank the efficient DMUs.

CONCLUSION
This study proposed a super-efficiency model based on the Enhanced Russell Measure (ERM) model to rank the efficient DMUs in Data Envelopment Analysis (DEA). As an ERM model, the ERM superefficiency model is non-radial and appropriate for ranking the efficient DMUs when inputs and outputs may change non-proportionally. As a result, a complete ranking of efficient DMUs can be obtained by using the ERM super-efficiency model. We demonstrated that unlike the radial super-efficiency models, the proposed super-efficiency model is always feasible under both CRS and VRS assumptions. Further, the relationship between our model and the traditional radial measure of super-efficiency, i.e., the AP model, was described.
Two numerical examples have been examined using the proposed super-efficiency model. It has been shown that the super-efficiency model can successfully differentiate among the efficient DMUs.
Recently, Chen et al. (2010) proposed a new measurement of efficiency based on the Russell measure that determines the best weight for each input (output). Application of Chen et al.'s model for ranking the efficient DMUs would be a future research subject