A Cross-Efficiency Based Ranking Method for Finding the Most Efficient DMU

In many applications of DEA, ranking of DMUs and finding the most efficient DMU are desirable, as reported by Toloo (2013). In this paper, after introducing an improvement to the measure of cross-efficiency by Jahanshahloo et al. (2011), we develop a new ranking method under the condition of variable returns to scale (VRS). Numerical example illustrates the effectiveness of the proposed cross-efficiency based ranking method and demonstrates the advantages of our proposal, against the other ranking approaches.


Introduction
Data envelopment analysis (DEA) provides a relative efficiency measure to evaluate decision making units (DMUs) with multiple inputs and multiple outputs.While it is an effective approach in identifying the best practice frontier, its flexibility in selecting the input/output weights and its nature of self-evaluation may result in a relatively high number of efficient DMUs.The lack of discrimination between efficient DMUs has been considered as an important problem in DEA models and subgroups of papers have been developed in this field in which many researchers have sought to improve the differential capabilities of DEA and to fully rank both efficient and inefficient DMUs.
Since these ranking methods have been developed based on some of the aspects of production possibility set (PPS), in certain cases, different calculations are reached in applying the alternative ranking methods.Furthermore, for each method, there are problematic areas, for example, infeasibility and instability of the proposed model.Hence, whilst each ranking technique is useful in a specialist area, no methodology can be prescribed as the complete solution to the question of ranking.
To increase the discrimination power of DEA models and make its weights more realistic, cross-efficiency evaluation has been suggested by Sexton et al. [1] and was later investigated by Doyle and Green [2,3].The basic idea of the crossefficiency evaluation is to evaluate the overall efficiencies of the DMUs through both self-and peer-evaluations and can usually provide a full ranking for the DMUs to be evaluated.Therefore, it has found a significant number of applications in various fields; see Green et al. [4], Sun and Lu [5], Bao et al. [6], Wu et al. [7,8], and Yang et al. [9].
This paper develops a new ranking method under the condition of VRS, which is based on introducing an improvement to Jahanshahloo et al. 's measure of crossefficiency [10].That is, the proposed cross-evaluation method is developed as a BCC extension tool that can be utilized to rank the DMUs using cross-efficiency scores and to identify the most BCC efficient DMU.
The rest of the paper is organized as follows.Section 2 gives a brief introduction to the DEA models.The mathematical foundation of our method to propose a secondary objective function in VRS cross-evaluation is discussed in Section 3. In Section 4, numerical example illustrates the effectiveness of the proposed cross-efficiency based ranking method.Finally, Section 5 is devoted to concluding remarks.

DEA Background
Consider  DMUs that are evaluated in terms of  inputs and  outputs.Let   and   ,  = 1, 2, . . ., ,  = 1, 2, . . ., , be their input and output values for DMU   = 1, 2, . . ., .The CCR efficiency of  DMUs is measured by where  and V are input and output weight vectors and  0 and  0 are the input and output vectors for the DMU under evaluation, respectively.Model (1) can be transformed to the following LP format, called input oriented CCR model: Note that model CCR in oriented output is as follows: Previous models have constant returns to scale (CRS) characteristic.Banker et al. [32] proposed BCC model which has variable returns to scale (VRS).The LP form of this model in oriented output is as follows: The optimal objective value of this model is BCC efficiency score of DMU 0 in oriented output.However, in classical DEA models, no preference information is needed and the weights are allowed total flexibility to obtain maximum efficiency.Thompson et al. [33] were the first who studied the role of weight restrictions in DEA models.Note that the above DEA models have alternative optimal solutions and free selection of weights may lead one DMU to have two types of weights.In one type, all of positive weights are on one group of variables and another with its weights symmetrically allocated to all variables.Dimitrov and Sutton [34] proposed symmetry measure which was the relative weight of each output dimension to all other output dimensions as follows: where   is the difference in symmetry between output dimension  and dimension  for all DMUs under evaluation.They minimized the sum of all the   values (∑ ,   =   , where  = [1, 1, . . ., 1]  is an -dimensional vector) and then effectively rewarded symmetry with asymmetry scaling factor  ≥ 0. By adding this constraint to model (3), we have But model ( 6) is not an LP model.Fortunately, as we are minimizing   , we may change the equality to ≤ as any optimal solution will have the equality constraint satisfied.With this observation, Dimitrov and Sutton [34] rewrite (6) as follows, where its objective function is ∈ [1, ∞) and has the same feasibility region as LP model (3): The resulting objective function value of the LP ( 7) is referred to as the SWAT score with smaller scores being more desirable.Here, we want to extend the above model to BCC model and variable returns to scale environment.It is straightforward that the SWAT model can easily adapt to situation with variable returns to scale in BCC model as follows: where the objective function is ∈ [1, ∞) and the  factor is a nonnegative factor, which determines how much a particular DMU will be penalized for an asymmetric selection of virtual weights.Note that  = 0 is equivalent to the original oriented output BCC model and  → ∞ is equivalent to the situation that all DMUs select equal virtual weights for all outputs.However, ideal  value can be determined by the decision maker (DM).

BCC Cross-Efficiency Evaluation
Jahanshahloo et al. [10], based on model (7), suggested a secondary goal for cross-efficiency evaluation in CRS environment.In this section and based on model ( 8), we propose an improvement to their method to propose a secondary objective function in VRS cross-evaluation.

Cross-Efficiency for CCR Model.
Let  * 0 ( = 1, 2, . . ., ) and V * 0 ( = 1, 2, . . ., ) be the optimal solution to model (7).Then,  * 00 = ∑  =1  * 0  0 is referred to as the CCR efficiency of DMU 0 by self-evaluation.If  * 00 = 1, DMU 0 is CCR efficient and otherwise it is CCR inefficient.Moreover, is referred to as DMU  efficiency peer-evaluated by DMU 0 ( = 1, 2, . . ., ,  ̸ = 0).As a result, each DMU has one CCR efficiency and ( − 1) peer-efficiencies and CCR cross-efficiency score of DMU 0 is calculated as follows: It is noticed that DEA models may have multiple optimal solutions.This nonuniqueness of input/output optimal weights would damage the use of cross-efficiency concept due to the ambiguity in using weights for execution of final results.To resolve this ambiguity, alternative secondary goals in cross-efficiency evaluation have been introduced.

Cross-Efficiency for BCC Model. Let 𝑢 *
0 ,  = 1, 2, . . ., , V * 0 ,  = 1, 2, . . ., , V * 0 , be the optimal solution of model ( 4).Then,  * 00 = ∑  =1 V 0  0 − V * 0 is BCC efficiency The average of all  * 0 is called BCC cross-efficiency score which is calculated as follows: Note that, because of using oriented output of BCC model, the DMU with lower cross-efficiency score has better rank.However, the weights obtained from model ( 4) are usually not unique.As a result, the cross-efficiency is determined depending on the optimal solution arising from the particular LP software in use.As discussed before, Jahanshahloo et al. [10] suggested the use of symmetric weights as a secondary goal in CCR cross-efficiency evaluation.Here, to propose a secondary objective function in VRS cross-evaluation, we propose the following algorithm.
Step 2. To select the best weight from alternative optimal weights of model ( 4), based on the concept of symmetric weights introduced by model ( 8), solve LP model (13) to select suitable weight between alternative solutions.Consider We add constraints (), () for selecting symmetric weights in BCC model by minimizing the sum of difference in symmetry between output dimension  and dimension  for DMU 0 in objective function.Selecting symmetric weights is desirable, because, in application, centralization of weights in one group of variables may not be acceptable.
Step 3. Compute the BCC cross-efficiency score of DMU  based on the formulations ( 11) and (12), where ( * 0 , V * 0 , V  * 0 ) is optimal solution of model (13).Finally, DMU 0 with the lowest cross-efficiency score is the most BCC efficient DMU.
Note that we can use model ( 8) instead of Steps 1 and 2 of the above algorithm.As discussed before,  is nonnegative factor where  = 0 results in the classical BCC model in oriented output and, for  = 1, model ( 8) is equivalent to model (13).
However, the ideal value for parameter  can be determined by DM.In other words, as it is discussed in Dimitrov and Sutton [34], instead of (8) and for greater flexibility, we can use the following model which puts greater burden on the DM to define the appropriate   values: In the next section, we use the proposed BCC crossevaluation approach for ranking of DMUs in a numerical example and finding the most BCC efficient DMU.To demonstrate the advantages of our proposal, against the other approaches, we compare the results to the results of using the approach proposed by Toloo and Nalchigar [35].

Numerical Example
Six nursing homes are evaluated in terms of two inputs and two outputs.The data set is reported in Table 1.Input variables are  1 : staff hours per day including nurses and physicians,  2 : supplier per day, measured in thousands of dollars.
Output variables are  1 : total medicative plus medicated reimbursed patient days,  2 : total privately paid patient days.By using mixed integer linear problem proposed by Toloo and Nalchigar [35], the most BCC efficient DMU is not unique and, based on the results depicted in Table 2, all of BCC efficient DMUs can choose as most BCC efficient DMU.This is due to the fact that the model proposed by Toloo and Nalchigar [35] has alternative optimal solutions.
Results of the proposed model (13) in this paper are presented in Table 3.The second column of this table shows BCC efficiency, and BCC cross-efficiency scores produced by our proposed method are depicted in the third column.Based on our procedure, DMU 2 is the most BCC efficient unit, alone.Moreover, to compare the obtained results with BCC model, in BCC model, five of six units are most efficient and choosing single most efficient unit is impossible.
Moreover, Table 4 shows cross-efficiency score based on model (14) with different amounts for .Rank of DMUs is depicted in parentheses.It represents that DMU 2 is the most efficient DMU with three types of , where, as mentioned before, factor  shows how a particular DMU will be penalized for an asymmetric choice of virtual weights.

Conclusion
In this paper, we proposed BCC cross-efficiency with model based on symmetric weight selection and by using it we discussed a procedure for finding the most BCC efficient DMU.Considerably, advantages of our cross-efficiency model are as

Table 1 :
Data set for six nursing homes.the DMU 0 is called BCC efficient and otherwise it is called BCC inefficient.Moreover, the BCC peer-evaluated efficiency of DMU  ( = 1, 2, . . ., ) by DMU 0 is calculated by

Table 3 :
Efficiency scores based on BCC model and BCC crossefficiency model.
follows: it is a linear program which is always feasible, it gives full ranking for DMUs, and it can find single most efficient unit.