Ranking DMUs using a novel combination method for integrating the results of relative closeness benevolent and relative closeness aggressive models

Article history: Received: March 21, 2021 Received in revised format: April 25, 2021 Accepted: May 3, 2021 Available online: May 3, 2021 In this paper, a novel combination method is offered to integrate the results of two new relative closeness models, called relative closeness benevolent (RCB) and relative closeness aggressive (RCA) models, for ranking all DMUs. To prove the applicability of the proposed method, it is examined in three numerical examples, performance assessment problem, six nursing homes and fourteen international passenger airlines. Firstly, RCB and RCA models were formulated in order to generate the cross-efficiency intervals matrix (CEIM). After obtaining CEIM, the RC index was utilized to generate a combined cross-efficiency matrix (combined CEM). In combined CEM, target DMUs were viewed as criteria and DMUs were viewed as alternatives. After that, the weights of each criterion were generated using a new weighting method based on standard deviation technique (MSDT). Finally, all DMUs were evaluated and ranked. Comparison with existing cross-efficiency models indicates the more reliable results through the use of the proposed method. © 2021 by the authors; licensee Growing Science, Canada.


Introduction
The data envelopment analysis (DEA) is a linear programming model that was first described by Farrel (1957), but a mathematical model was first introduced in the Charnes et al. paper (Charnes, Cooper, & Rhodes, 1979). This non-parametric model evaluates the relative efficiency of decision making units (DMUs) with multiple inputs and outputs (Zerafat Angiz, Mustafa, & Kamali, 2013). One of the advantages of this method is that the weights for inputs and outputs are not required (Sun, Wu, & Guo, 2013). In the DEA model, the input weights and output weights are obtained by maximizing the ratio of the sum of weighted outputs to the sum of weighted inputs. It is well known that the efficiency score of each DMU cannot be greater than 1, and DMUs can be defined as being efficient DMUs if their efficiency scores are 1; otherwise the DMUs are inefficient (Wang, Chin, & Luo, 2011;Wichapa, Khokhajaikiat, & Chaiphet, 2021). Over the past four decades, DEA has been widely applied in efficiency evaluation and benchmarking in many fields (Amalnick & Saffar, 2017;Durga Prasad, Kambagowni, & Prasad, 2017;Lesik et al., 2020). However, a frequently discussed problem in DEA is that efficient DMUs cannot be discriminated between. Hence, various methods have been carried out to overcome this main drawback and to improve the discrimination power of DEA. These methods can be classified into seven groups as follows (Hosseinzadeh Lotfi et al., 2013). The first group is the cross-efficiency method. The second group is the DEA optimal weights. The third group is the super-efficiency method. The fourth group is the benchmarking idea. The fourth group is the multivariate statistical tools. The sixth group is the multi-criteria decision making (MCDM) method. The final group is various ranking methods. Although there are many methods for ranking DMUs, the cross-efficiency method is one of the most popular ranking methods in DEA. The crossefficiency method, first introduced by Sexton et al. (1986), has long been suggested as a power tool for the ranking of DMUs based on the cross-efficiency concept. Based on this idea, a combination of self-evaluation and peer-evaluation was suggested for overcoming the weakness of DEA's discrimination power. So, the weights of all DMUs can be obtained by averaging the best weights of DMUs. Finally, all DMUs can be ranked by their average cross-efficiency scores (ACE scores) in the crossefficiency matrix. Although the cross-efficiency method has been widely studied and used all over the world, there is still one drawback. The optimal weights are not unique, so this possibly reduces the usefulness of cross-efficiency evaluation to help decision makers improve their performance (Si & Ma, 2019). To overcome the main drawback above, Sexton et al. (1986) first suggested using a secondary-goal model in the cross-efficiency method and later, Doyle and Green (1994) investigated the aggressive and benevolent formulations to deal with the non-uniqueness issue. Even though aggressive and benevolent formulations are most commonly used for ranking all DMUs, a question arises: which one is the best? Usually, the performance evaluation results obtained from aggressive and benevolent formulations may not be the same for similar problems, because each of the formulations has a different viewpoint. Certainly, each of the above viewpoints should not be ignored. Hence, it is wise to try alternative formulations and combine the results of benevolent and aggressive formulations for ranking all DMUs. Wang and Luo (2006) have first proposed the concept of efficiency assessment using virtual DMUs (ideal and anti-ideal DMUs) in the DEA literature. The ideal DMU considers the minimum input with the maximum output, while the anti-ideal DMU is the unit consuming the maximum input to produce the minimum output. Jahanshahloo et al. (2010) proposed a ranking method based on positive ideal DMUs with a common set of weights for the efficient DMUs. Wang et al. (2011) exhibited an effective model of four models based on the ideas of cross-efficiency and virtual DMUs, namely the relative closeness model (RC model), for ranking all DMUs. Sun, Wu and Gou (2013) have presented two models based on the viewpoint of virtual DMUs with common weights for performance ranking of all DMUs. Recently, Naseri and Kiaei (2016) have proposed cross-efficiency evaluation based on ideas of ideal and anti-ideal virtual DMUs' assessment. Hou, Wang, and Zhou (2018) have proposed a new formulation by the use of ideal and anti-ideal virtual DMUs' assessment for ranking all DMUs. Nasseri and Kiaei (2019) proposed the new neutral models based on an ideal DMU in cross-efficiency. Inspired by the above ideas, it is wise to try new formulations and combine the ideas of benevolent, aggressive and RC models for ranking all DMUs, because neither view should be ignored. It is well known that determination of criteria weights is an important issue in the multi-criteria decision-making problem (MCDM problem). The final ranking results are highly dependent on the weights of each criterion. Several weighting methods have been presented to obtain criteria weights. These methods are divided into three categories (Keshavarz-Ghorabaee, Amiri, Zavadskas, Turskis, & Antucheviciene, 2021), including subjective weighting method, objective weighting method and hybrid weighting method. In the subjective method, decision makers must determine the criteria weights. The main drawback of the subjective method is that it is a difficult task for decisionmakers, and the accuracy of their preferences can be decreased by increasing the number of criteria (Alfares & Duffuaa, 2016). In the objective weighting method, the decision-makers have no role in determining criteria weights. In this method, the criteria weights are obtained using a specific computational process based on the initial information or decision matrix (Kao, 2010). There are some popular methods in this category that are often used to determine the criteria weights in the literature, such as the Entropy method (Lu & Liu, 2016), the Standard deviation method (Anitha & Das, 2019), CRITIC (Criteria Importance Through Inter-criteria Correlation) (Wichapa et al., 2021) and SECA (Simultaneous Evaluation of Criteria and Alternatives) (Keshavarz-Ghorabaee, Govindan, Amiri, Zavadskas, & Antucheviciene, 2019). The last category of weighting methods is the hybrid weighting method. This method uses a combination of different views of subjective and objective weighting methods. The various methods in this category have no distinctive characteristics, and they borrow the prominent points of other weighting methods. The hybrid weighting method could give more realistic weights because this method can use the decision-makers' preferences and the information of the decision matrix (Liu, Hu, Zhang, Li, & Liu, 2020). This paper proposes a new objective weighting method based on the standard deviation technique (MSDT) for determining criteria weights. Unlike the other methods mentioned above, MSDT converts the standard deviation of each criterion in a decision matrix for estimating the criteria weights.
To this end, this research provides a novel combination method to aggregate the ideas of benevolent, aggressive and RC models for ranking all DMUs. The proposed method has been adapted from Wang's RC model (Wang et al., 2011) in the following ways: (1) we formulate two new RC models, called RCB and RCA models, based on ideas of benevolent, aggressive and Wang's RC models (Wang et al., 2011). The RCB and RCA models are utilized to generate the RC benevolent crossefficiency matrix (RCB-CEM) and the RC aggressive cross-efficiency matrix (RCA-CEM) respectively, and then a combined CEM is generated using the RC index. (2) In the combined CEM, a target DMU and a DMU in the decision matrix are viewed as criterion and alternative respectively, and the MSDT is used to generate the criteria weights of the decision matrix (combined CEM) for evaluating the final weights of each DMU.
The remainder of this article is organized in the following manner. In the next section some mathematical models used in this article are presented. The proposed method and numerical examples are presented in Section 3 and Section 4 respectively. Finally, the conclusion is given in Section 5.

CCR model
The mathematical model of the CCR model was first proposed by Charnes, Cooper and Rhodes (1979). The classic CCR model is used to evaluate the efficiency score of DMUs with multiple inputs and outputs. Many researchers have carried out the application of the CCR model in various fields (Al-Faraj, Alidi, & Bu-Bshait, 1993;Fancello, Carta, & Serra, 2020;Liang, Yang, Cook, & Zhu, 2006;Weber, 1996;Wei, Chen, Li, & Tsai, 2011), which proves that the CCR model is an effective method for measuring performance of all DMUs. Consider a set of n DMUs that is measured in terms of m inputs to produces s outputs. Let xij (i = 1,. . ., m) and yrj (r = 1,. . ., s) be the values of inputs and outputs of DMUj (j = 1,. . .,n). Let urk and vik be the weights of outputs and weights of inputs respectively. For any evaluated DMUk (1 ≤ k ≤ n), the efficiency score kk can be evaluated by the CCR model as follows: For DMUk (k = 1, 2, 3, ..., n), a set of optimal weights can be obtained by solving the CCR model in Equation (1). In the CCR model, DMUs are self-evaluated and termed efficient if and only if the optimal objective function is equal to 1.

Cross-efficiency method
The main drawback of the CCR model is that efficient DMUs cannot be fully discriminated from each other. Hence, many researchers (Sexton et al., 1986) have proposed various cross-efficiency methods to provide a full ranking for all DMUs. The cross-efficiency formulations are given below.
Then the average cross-efficiency score (ACE) of each DMU is defined as follows.

Benevolent and aggressive models
The main drawback of the cross-efficiency method is that the optimal weights of all DMUs obtained from the CCR model in Equation (1) may be not unique, which clearly cannot provide results to help decision makers improve their performance (Si & Ma, 2019;Wu, Sun, Zha, & Liang, 2011). To overcome this drawback, Doyle and Green (Doyle & Green, 1994) have proposed the well-known aggressive and benevolent models to identify the optimal weights of all DMUs. The benevolent and aggressive formulations are as follows.
subject to: Eq. (4) and Eq. (5) represent the benevolent and aggressive models, which aim to maximize and minimize respectively the cross efficiency of the integrated unit consisting of the other DMUs while maintaining the self-evaluation efficiency of a particular DMU under evaluation. Since both models optimize the weights of inputs and outputs from two different viewpoints, the same ranking orders are not guaranteed. Thus, the idea of generating the alternative models for ranking all DMUs is attractive.

RC model
Based on the ideas of Wang, Chin and Luo (Wang et al., 2011), An IDMU and an ADMU are defined as follows: Definition 1. A virtual DMU can be defined as an IDMU if it consumes the least inputs to generate the most outputs. While a virtual DMU can be defined as ADMU if it consumes the most inputs only to produce the least outputs.
Definition 3. The relative closeness of DMUk (RCk) with respect to IDMU and ADMU is defined as The relative closeness model (RC model) for cross efficiency evaluation is as follows (Wang et al., 2011): subject to:

The proposed method
There are many different DEA cross-efficiency models that have been proposed for ranking DMUs. Unfortunately, the ranking results obtained for each cross-efficiency model may differ for similar problems. Hence, it is wise to try the new effective methods that provide more reliable results in ranking DMUs effectively. In this section, a new hybrid method is offered for ranking DMUs. The framework for our method is shown in Fig. 1.

Generating the cross-efficiency intervals matrix (CEIM) using the RCB and RCA models
According to the ideas of Wang's RC model (Wang et al., 2011) in Eq. (4) and the concept of the well-known aggressive and benevolent models (Doyle & Green, 1994), the relative closeness benevolent (RCB) and relative closeness aggressive (RCA) models can be determined as Eq. (7) and Eq. (8) subject to:  End subject to: the same constraints as in Eq. (7) According to the RCB and RCA models, two evaluation matrices can be obtained as follows: RCB cross-efficiency matrix (RCB-CEM) is

Generating the combined CEM using the relative closeness index
The distance measures for alternative i (DMUi ) and criterion j (target DMUj ) can use the relative closeness formula to generate a combined CEM. There are three calculation steps for generating the combined CEM as follows.
(1) Calculate the distance from positive ideal solution for alternative i and criterion j ( ij d + ) Let positive ideal solution (PIS) and negative ideal solution (NIS) be 1 and 0 respectively. Then the distance from PIS for alternative i and criterion j ( ij d + ) can be defined as follows.
(3) Then a relative closeness index for alternative i and criterion j (rcij) can be determined as follows.

Generating the criteria weights in the combined decision matrix using MSDT
In Table 1, consider a combined CEM (X), X = [rcij]nxn, where rcij is the relative closeness index with respect to alternative i (DMUi) and criterion j (target DMUj) and n is the number of DMUs respectively. The new weighting method is formulated for addressing the ranking problem in this paper. Since a higher standard deviation means that the factor is worse, the standard deviation must be converted to a value with an opposite trend for calculating the final weight in the next step.
After obtaining weighting factors, the higher value of weighting factor is the better criterion.

Calculating the final weights of DMUs and ranking all DMUs
After obtaining j ω , the final weights of each DMU (wi) can be obtained by multiplying the weight of each criterion by the corresponding combined CEM using Eq. (16).
After obtaining wi, all DMUs can be ranked so that a higher value of wi means that the DMU's ranking is higher.

Numerical examples
In this section, the validity of the proposed method is examined with three numerical examples. The first is the performance assessment problem investigated by Andersen and Petersen (1993), the second is six nursing homes investigated by Sexton et al. (1986), and the third is fourteen international passenger airlines by Tofallis (1997a). Details of the calculation steps are shown in Sections 4.1, 4.2 and 4.3 respectively.

Performance assessment problem
Consider a performance assessment problem, investigated by Andersen and Petersen (Andersen & Petersen, 1993), which has five DMUs to be evaluated in the light of two inputs (x1 and x2) and one output (y1). The data set of the performance assessment problem is provided in Table 2, together with the CCR-efficiency scores of the five DMUs. Step1: Generate the cross-efficiency intervals matrix (CEIM) using the RCB and RCA models The efficiencies based on the CCR model must be evaluated first. After that, the RCB model as in Equation (7) and the RCA model as in Equation (8) were coded using LINGO software. Based on Equation (9) and Equation (10), the results of the RCA and RCB models can be obtained as RCB-CEM and RCA-CEM, as listed in Table 3 and Table 4 respectively.  After obtaining the RCB-CEM and RCA-CEM, the cross-efficiency intervals matrix (CEIM) can be generated using Eq. (11) as listed in Table 5. Step2: Generate the combined CEM using the relative closeness index The details of the distance from PIS for alternative i and criterion j are shown in Table 6.  (3) Then the relative closeness index for alternative i and criterion j (rcij) can be obtained using Equation (14). For example, rc11=2/(2.000+0.000) = 1.0000, rc12 = 1.7143/ (1.7143+0.2857) = 0.8571 and rc13 = 1.1982/(1.1982+0.8018) = 0.5991. Finally, the combined CEM will be generated as in Table 8. Step3: Calculate the criteria weights in combined CEM using MSDT In Table 8, j σ must be calculated first. Then the weights of each criterion in the combined CEM were determined using Eq.  Table 8.

Step4: Calculate the final weights of each DMU and rank all DMUs
After obtaining the j ω , the wi is obtained by multiplying the weight value by the corresponding combined CEM using Equa-  Table 9.

Table 9
The rating and ranking of proposed method for performance assessment problem Next a rating and ranking comparison of the proposed methods is shown in Table 10. Finally, spearman's rank correlation was used for testing the correlation of each method (rs). The details of each rs value are shown in Table 11.

Table 10
The rating and ranking comparison of the proposed method for a performance assessment problem As seen in Table 10, the rating and ranking comparison of the proposed method for a performance assessment problem was obtained. The proposed method, benevolent model, aggressive model and Wang's RC model as in Equation (6) (Wang et al., 2011) assess that DMU2> DMU3 > DMU4 > DMU1 > DMU5. As seen in Table 11, the correlation coefficients for the proposed method and all models are the same value with rs = 1.000, 1.000, 1.000 respectively.

Six nursing homes
In Table 12, the six nursing homes, proposed by Sexton et al. (Sexton et al., 1986), have two inputs (x1 and x2) and two outputs (y1 and y2). Let x1 be staff hours per day, including nurses, physicians, etc. Let x2 be the supplies per day, measured in thousands of dollars, y 1 be the total Medicare-plus-Medicaid reimbursed patient days, and y 2 be the total privately paid patient days. Step1: Generate the CEIM using the RCB and RCA models Based on the same calculation steps in Section 4.1, the RCB and RCA models were coded using LINGO software. Based on Eq. (7) to Eq. (10), the results of RCB-CEM and RCA-CEM can be obtained as listed in Table 13 and Table 14 respectively.  After obtaining the RCB-CEM and RCA-CEM, the cross-efficiency intervals matrix (CEIM) of six nursing homes can be generated using Eq. (11) as listed in Table 15. Step 2: Generate the combined CEM using the relative closeness index (1) Based on the same calculation steps as in Section 4.1, ij d + were obtained using Eq.n (12). The details of each ij d + are shown in Table 16. (2) Based on the same calculation steps as in Section 4.1, ij d − were obtained using Eq. (13). The details of each ij d − are shown in Table 17. (3) Then RCij can be obtained using Equation (14). As a result, the combined CEMs were generated as in Table 18. Step3: Generate the criteria weights in combined CEM using MSDT Based on the same calculation steps in Section 4.1, the weights of each criterion in the combined CEM were determined using Eq. (15). Details are shown in Table 18.
Step 4: Calculate the final weights of each DMU and rank all DMUs After obtaining the j ω , the wi is obtained by multiplying the weight value by the corresponding combined CEM using Eq.
(16). As a result, all DMUs are ranked as listed in Table 19.

Table 19
The rating and ranking of the proposed method for six nursing homes The rating and ranking comparison of the proposed method is shown in Table 20. Finally, Spearman's rank correlation was used for testing the correlation of each method (r s ). The details of each r s value are shown in Table 21.  As seen in Table 20, the rating and ranking comparisons of the proposed method for six nursing homes were obtained. The proposed method, aggressive model and Wang's RC model assess that DMU1> DMU4 > DMU2 > DMU5 > DMU3 > DMU6.
As seen in Table 21, the correlation coefficients for the proposed method and the benevolent model, the aggressive model and Wang's RC model have values of rs = 0.986, 1.000, 1.000, 1.000 respectively. This is a guarantee that the proposed method is highly reliable.

Fourteen international passenger airlines
In Table 22, the data set of fourteen international passenger airlines, proposed by Tofallis (Tofallis, 1997b), has three inputs (x1, x2 and x3) and two outputs (y1 and y2). Let x1 be the aircraft capacity in ton kilometers, x2 be operating cost and x3 be nonflight assets such as reservation systems, facilities and current assets. Let y1 be passenger kilometers and y2 be non-passenger revenue. Based on the same calculation steps as in Section 4.1 and Section 4.2, the weights of each criterion in the combined CEM are obtained as in Table 23.  After obtaining the combined CEM, DMUs ratings (wi) were obtained, and all DMUs were ranked as listed in Table 24. Finally, Spearman's rank correlation was used for testing the correlation (rs). The details of each rs value are shown in Table  25.

Table 24
The rating and ranking comparison for fourteen international passenger airlines  As seen in Table 24, the proposed method was used to calculate the rating and ranking of all DMUs. The proposed method and the benevolent model agree that DMU11 is the best DMU, but the aggressive model and Wang's RC model indicate that DMU5 is the best DMU. All of the methods agree that DMU2 is the worst DMU.
As seen in Table 25, the Spearman's rank correlation coefficients for the proposed method and the benevolent efficiency value, aggressive efficiency and RC efficiency values are calculated as rs = 0.965, 0.982 and 0.991 respectively. Based on the calculation results from the three numerical examples above, this is a guarantee that the proposed method is highly reliable.
As seen in Fig. 2, the correlation coefficient of the proposed method and Wang's RC efficiency for each problem were most closely related, while the correlation coefficient of the proposed method and benevolent were least related. Undoubtedly, the proposed method can contribute to achieving a more reliable result than the method which is based on a stand-alone existing model.

Conclusions
This research presents a novel aggregation method offered to combine the ideas of benevolent, aggressive and relative closeness (RC) models for ranking all DMUs. In this paper, the proposed method was tested with three numerical examples. We first formulated the new RC models, namely relative closeness benevolent model (RCB model) and relative closeness aggressive model (RCA model), to evaluate the rating of all DMUs. The results of the RCB and RCA models were utilized to generate a cross-efficiency intervals matrix (CEIM). Secondly, the relative closeness index was used to generate a combined cross-efficiency matrix (combined CEM). In the combined CEM, a set of target DMUs was viewed as criteria and a set of DMUs was viewed as alternatives. Finally, all DMUs were ranked. In a comparative analysis, the proposed method shows potential in ranking DMUs, which differ from other models in the literature. Based on the results of this research, the proposed model can lead to achieving a more reliable result than the other existing methods.
For future research, in order to enhance the validity of the research output further, application of the proposed method should be tested with more cases. Although the inputs and outputs in this case are measured by exact values, we believe that the proposed method can be adapted to deal with fuzzy data.