Hybrid cross-efficiency approach based on Ideal and Anti-Ideal points and the CRITIC method for ranking decision-making units: A case study on ranking the methods of rice weevil disinfestation

. by the authors; licensee Growing Science, Canada 1 2 20 ©


Introduction
Agriculture in Thailand, with more than 114,880 km 2 for rice cultivation and an output of about 32.63 million tons per year, is a diverse industry. One of the most important crops in Thailand is rice. In 2019, the production value of rice in Thailand reached almost 300 billion Thai baht. At the same time, the production volume of rice was forecast to be 28.36 million tons (Statista, 2019). However, there are insect pests, such as the Red flour beetle, Corn weevil and Rice weevil etc. that damage milled rice while it is in the warehouse awaiting export. Hence, entrepreneurs have discovered ways to protect rice from these insect pests. One of the most common methods for rice pest control is fumigation, which is easy and cheap. However, it uses substances toxic to the environment and humans. Furthermore, rice treated this way reduces consumer confidence, while rice exports do not comply with agricultural standards. A second method to protect rice is through vacuum seal packaging. This is a non-fumigant approach which complies with the agricultural standards appropriate for organic rice. However, it is expensive and has lower elimination efficiency. A new method to eliminate insect pests is using infrared. This method is environmentally friendly and it preserves the quality of the rice. There are various other methods for rice insect pest control. Each method has its own advantages and disadvantages. It is difficult to choose a best rice insect pest control method, because there are various criteria (both advantages and disadvantages) that we should consider. Therefore, it is not easy to rank the alternatives/decision making units (DMUs) for rice insect pest control, because there are several conflicting criteria (both inputs and outputs) such as cost, rice quality and production capacity. These factors should be taken into consideration simultaneously. Therefore, one important challenge to tackle in this complicated problem is to select an appropriate method to rank multiple DMUs with multiple factors (inputs and outputs).
valuable and capable approach for measuring performance of DMUs. It is well known that the main disadvantage of the classic DEA model is that efficient DMUs (Efficiency score of one) cannot be compared with each other on the basis of this criterion anymore. Therefore, it seems necessary to provide other models for further discrimination among these DMUs. Hence, many studies in the literature have proposed methods for ranking efficient DMUs. For example, Sexton et al. (1986) proposed the cross-efficiency evaluation to provide a full ranking for all DMUs. Later, applications of the cross-efficiency evaluation method have been widely applied in many fields. However, the main drawback of the cross-efficiency evaluation method is that the weights may be not unique, which clearly cannot provide results to help decision makers to improve their performance (Si & Ma, 2019;J. Wu et al., 2011). To overcome this problem, Sexton et al. (1986) and Doyle & Green (1994) have proposed benevolent and aggressive models for ranking all DMUs. Wang & Chin (2010) proposed a neutral DEA model for overcoming the difficulty of the choice between the aggressive and benevolent models, and also providing a full ranking for all the DMUs. All of the above models are based on the concept of the cross-efficiency method. Besides the above ranking methods, another way to solve the ranking problem is to combine the results of multiple ranking methods based on methodologies for determining criteria weights in calculating the rating of all the DMUs. Determining the weight of each criterion is one of the key factors of the decision-making process. Generally, most weighting methods can be divided into subjective and objective approaches. The subjective approach is based on determining the weight of each criterion using information from experts included in the decision-making process. On the other hand, the objective approach disregards the opinion of decision makers and is based on determining the weight of each criterion using data that is present in the initial decision matrix. The best known objective approaches include: Entropy (Shannon, 1948), CRITIC (Diakoulaki et al., 1995) and FANMA (Diakoulaki et al., 1995). The CRITIC method is one of the best known and most widely used in the literature (Abdel-Basset & Mohamed, 2020;H.-W. Wu, Zhen, & Zhang, 2020). For example, Rostamzadeh et al. (2018) proposed a framework for sustainable supply chain risk management using fuzzy TOPSIS based on the CRITIC method. Kazan and Ozdemir (2014) proposed the TOPSIS -CRITIC method for financial performance assessment of large scale conglomerates. Wang and Zhao (2016) proposed the CRITIC method and AHP for designing optimization of mechanical properties of ceramic tool materials. Wei et al. (2020) proposed the GRA based on the CRITIC method for location planning of electric vehicle charging stations. Zhao et al. (2020) proposed a combined prospect theory, the Copula-CRITIC method, to evaluate the construction schedule robustness. As shown in the above literature, the CRITIC method has been accepted as a powerful technique to generate criteria weights in decision-making problems. These are therefore the major reasons for choosing the CRITIC method to determine the weight of each criterion in this paper.

Methodology
There are many different ranking methods that have been proposed for solving the ranking problem in DEA. However, the results obtained for each model may produce different ranked decision-making units for similar ranking problems. Hence, it is wise to try effective methods that provide more reliable results in solving the ranking problems effectively. In this section, a new hybrid method is offered for solving the ranking problems. The framework for the proposed ranking method is shown in Fig. 1.

Generate the Ideal CEM and Anti-Ideal CEM based on Ideal and Anti-Ideal models
Assume that there be a set of n DMUs to be measured, where each DMU has m inputs to produces s outputs. We denote by xij (i = 1,. . .,m) and yrj (r = 1,. . ., s) the values of inputs and outputs of DMUj (j = 1,. . .,n), which are all known and positive. An IDMU and an ADMU can be defined as follows: Definition 1. An IDMU is a virtual DMU, which can use the least inputs to generate the most outputs. While an ADMU is a DMU, which consumes the most inputs only to produce the least outputs.

Start
3.1 Generate the Ideal CEM and Anti-Ideal CEM based on Ideal and Anti-Ideal models 3.2 Combine the Ideal CEM and Anti-Ideal CEM using the new formula 3.3 Calculate the weights of DMUs based on CRITIC method and rank all DMUs

End
We denote by xand y + the Ideal input and Ideal output of the IDMU, and by x + and ythe Anti-Ideal input and Anti-Ideal output of the ADMU, respectively. They are defined by the following formulae: Then the average cross-efficiency (ACE) of each decision making unit is defined as follows.
According to the efficiency concept (Hou et al., 2018), the efficiency of Ideal and Anti-Ideal models can be defined as Eq.

Combine the Ideal CEM and Anti-Ideal CEM using the new formula
In calculating the weight of each criterion using the CRITIC method, there are three calculation steps as follows. The combined Cross-Efficiency Matrix (combined CEM) will be generated using the results of the Ideal CEM and Anti-Ideal CEM in Section 3.1. Details are shown in Table 1.

Calculate the weights of each criterion
While generating the weights of criterion j, the standard deviation of criterion j ( j  ) and correlation between the criterion i and criterion j (rij) can be calculated using Excel 2010. In this regard, the weight of the criterion j (wj) is obtained as where Cj is the quantity of information contained in criterion j determined as

Calculate the weights of DMUs and rank all DMUs
The weight of each DMUi is obtained by multiplying the CRITIC weight value by the corresponding decision matrix using Eq. (14).
where i  is the integrated weight of each DMUi.
After calculating i  using Eq. (14), all DMUs can be ranked so that a higher value of i  means that the DMU's ranking is higher.

Numerical examples
This section uses the proposed ranking method to evaluate three numerical examples. The first is six nursing homes (Sexton et al., 1986), the second fourteen international passenger airlines (Tofallis, 1997a), and the third is a case study on choosing a suitable rice weevil disinfestation. Details of the calculation steps of the proposed methodology are shown in Sections 4.1, 4.2 and 4.3 respectively.

Efficiency evaluation of six nursing homes
In Table 2, the six nursing homes, proposed by Sexton et al. (1986), has two inputs (X1 and X2) and two outputs (Y1 and Y2).
X1: staff hours per day, including nurses, physicians, etc. X2: supplies per day, measured in thousands of dollars. Y1: total Medicare-plus-Medicaid reimbursed patient days. Y2: total privately paid patient days. Step 1: Generate the Ideal CEM and Anti-Ideal CEM based on Ideal and Anti-Ideal models for six nursing homes Consider a data set of six nursing homes, each DMU with two inputs and two outputs as in Table 2. The efficiency scores based on the CCR model (Equation (1)) must be evaluated first. After that, the Ideal model (Equation (4)) and Anti-Ideal model (Equation (5)) were coded using LINGO software. The results (using LINGO) of all models are shown in Appendix A. As a result, the Ideal CEM and Anti-Ideal CEM can be obtained as listed in Table 3 and Table 4 respectively. Step 2: Combine the Ideal CEM and Anti-Ideal CEM using the new formula for six nursing homes After obtaining the Ideal and Anti-Ideal CEMs, the combined CEM can be generated using new formula (Eqs. (8-10) as listed in Table 5. Step 3: Calculate the weights of DMUs based on CRITIC method and rank all DMUs for six nursing homes Consider the combined CEM in Table 5, where each DMU is viewed as an alternative, and the target DMU is viewed as a criterion. After that, the combined CEM was normalized using Eq. (11). Then, j  was computed with Excel 2010. As a result, the normalized CEM can be obtained as listed in Table 6. After obtaining the normalized CEM, the next step is to compute the correlation between target DMUi and target DMUj (rij) using Excel 2010. As a result, the correlation matrix can be obtained as listed in Table 7. After obtaining the correlation matrix for six nursing homes, the weight of the target DMUj (wj) was obtained using Eq.
(12) and Eq. (13). Cj was evaluated using Eq. (14). For example, Likewise, the values of C2 to C6 were obtained from the same calculation as the C1 value. Finally, w1, w2,...,w6 are shown in Table 8. After obtaining the wj of each criterion, each DMU weight ( i  ) can be obtained using Equation (14). As a result, DMUs were ranked as listed in Table 9. 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 10.  As seen in Table 9, the rating and ranking of all DMUs were obtained. The proposed method and Hou's model (Hou et al., 2018) assess that DMU1> DMU4 > DMU5 > DMU2 > DMU3 > DMU6. The aggressive model, Hou's model (Hou et al., 2018) and proposed method agree that the best DMU and the worst DMU are DMU1 and DMU6 respectively, but the benevolent model cannot discriminate between DMU1 and DMU4. As seen in Table 10, the correlation coefficients for the proposed method and benevolent efficiency and aggressive efficiency and Hou's efficiency values are evaluated as rs = 0.928, 0.943, 1.000 respectively. This is a guarantee that the proposed method is more reliable.

Efficiency evaluation of fourteen international passenger airlines
In Table 9, 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). X1: aircraft capacity in ton kilometers, X2: operating cost, X3: non-flight assets such as reservation systems, facilities and current assets, Y1: passenger kilometers, Y2: non-passenger revenue. Step 1: Generate the Ideal CEM and Anti-Ideal CEM based on Ideal and Anti-Ideal models for fourteen international passenger airlines Consider a data set of fourteen international passenger airlines; each DMU has three inputs and two outputs as shown in Table 11. The efficiency scores based on the CCR model (Eq. (1)) must be evaluated first. After that, the Ideal model (Eq. (4)) and Anti-Ideal model (Eq. (5)) were coded using LINGO software. The results (using LINGO) of all models are shown in Appendix B. As a result, the Ideal CEM and Anti-Ideal CEM can be obtained as listed in Table 12 and Table 13 respectively.  Step 2: Combine the Ideal CEM and Anti-Ideal CEM using the new formula for fourteen international passenger airlines After obtaining the Ideal and Anti-Ideal CEMs, using Equations 8 to 10, the combined CEM can be obtained as listed in Table 14.  Step 3: Calculate the weights of DMUs based on CRITIC method and rank all DMUs Considering the combined CEM of fourteen international passenger airlines in Table 14, the evaluation steps are the same as Step 2 of Section 4.1. As a result, the normalized decision matrix was generated as shown in Table 15. After obtaining the normalized CEM, the next step is to compute the correlation between target DMUi and target DMUj (rij) using Excel 2010. As a result, the correlation matrix can be obtained as listed in Table 16. After obtaining the correlation matrix for the fourteen international airlines, the weight of the target DMUj (wj) was obtained using Eq. (12) and Eq. (13). Cj was evaluated using Eq. (14). Likewise, the values of C1 to C14 were obtained from the same calculation as Step 2 of Section 4.1. Finally, w1, w2,...,w14 are shown in Table 17.

Table 17
Criteria weights for fourteen international passenger airlines using the CRITIC method  After obtaining the wj of each criterion, each DMU weight ( i  ) can be obtained using Equation (14). As a result, DMUs were ranked as listed in Table 18. 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 19.  As seen in Table 18, we use the proposed approach to calculate the efficiency rating and ranking of all DMUs. The proposed approach and aggressive model agree that DMU5 is the best DMU, but the benevolent model and Hou's method (Hou et al., 2018) indicate that DMU11 is the best DMU. All of the methods agree that DMU2 is the worst DMU. As seen in Table 19, after the Spearman correlation test, the Spearman's rank correlation coefficients for the proposed method and the CCR efficiency value, benevolent efficiency value, aggressive efficiency and Hou's efficiency values are calculated as rs = 0.960, 0.991 and 0.952 respectively. This is a guarantee that the proposed method is highly reliable.

Application to rank the methods of rice weevil disinfestations
Thailand is an agricultural country in Southeast Asia having a large amount of rice, which is an important economic crop for Thailand. However, in the harvesting season, there are many problems with various rice insect pests during warehouse storage while awaiting export that reduce the quality of milled rice, such as the Red flour beetle, Corn weevil and Rice weevil etc. Hence, various machines have been developed to prevent rice weevil disinfestations in Thailand. These machines should be evaluated and ranked in order to guide the development of more appropriate machines for rice weevil disinfestation. However, the selection of suitable machines for rice weevil disinfestation must consider various factors (inputs or outputs) and many alternatives (DMUs) at the same time. This is one multi-criteria decision-making problem that is difficult to evaluate. There are many tools for solving multi-criteria decision-making problems. However, the DEA crossefficiency evaluation is one approach that is effective for evaluating and ranking DMUs. Hence, this paper has applied this tool for solving this problem. Fig.1 shows an example of the machine developed by the research team.

Fig.1
An example of the machine developed by the research team In this case, six machines for rice weevil disinfestations are evaluated and ranked using the DEA approach. As seen in Table  20, the six machines selected have three inputs (x1, x2 and x3) and two outputs (y1 and y2). X1 : Cost of equipment (Million baht). X2 : Production cost (Baht/kg). X3 : Environmental risk (The risk scale is between 1 and 5). Y1 : Capacity (Ton/day). Y2 : Rice quality (The rice quality is between 1 and 5). Step 1: Generate the Ideal CEM and Anti-Ideal CEM based on Ideal and Anti-Ideal models for the methods of rice weevil disinfestation Consider a data set of the method of rice weevil disinfestations; each DMU with three inputs and two outputs as in Table  20. The efficiency scores based on the CCR model (Equation (1)) must be evaluated first. After that, the Ideal model (Equation (4)) and Anti-Ideal model (Equation (5)) were coded using LINGO software. The results (using LINGO) of all models are shown in Appendix C. As a result, the Ideal CEM and Anti-Ideal CEM can be obtained as listed in Table 21 and  Table 22 respectively.  Step 2: Combine the Ideal CEM and Anti-Ideal CEM using the new formula After obtaining Ideal and Anti-Ideal CEMs, using Equations 8 to 10, the combined CEM can be obtained as listed in Table  23. Step 3: Calculate the weights of DMUs based on CRITIC method and rank all DMUs Consider the combined CEM in Table 23 in which each DMU is viewed as an alternative, and the target DMU is viewed as a criterion. After that, the combined CEM was normalized using Equation (11). Then, j  was computed by Excel 2010. As a result, the normalized CEM can be obtained as listed in Table 24. After obtaining the normalized CEM, the next step is to compute the correlation between target DMUi and target DMUj (rij) using Excel 2010. As a result, the correlation matrix can be obtained as listed in Table 25. After obtaining the correlation matrix for the methods of rice weevil disinfestations, the weight of the target DMUj (wj) was obtained using Equation (12) and Equation (13). Cj was evaluated using Equation (14), For example,  After obtaining the wj of each criterion, each DMU weight ( i  ) can be obtained using Equation (14). As a result, DMUs were ranked as listed in Table 27. 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 28.

Table 27
The ranking of each DMU for the methods of rice weevil disinfestations  As seen in Table 27, the rating and ranking of all DMUs were obtained. The proposed method and Hou's method (Hou et al., 2018) assess that DMU3> DMU4 > DMU1 > DMU2 > DMU5 > DMU6. The aggressive model and proposed approach agree that the best DMU and the worst DMU are DMU3 and DMU6 respectively, but the benevolent model cannot discriminate between DMU3 and DMU4.
As seen in Table 28, the correlation coefficients for the proposed method and benevolent efficiency, aggressive efficiency and Hou's efficiency values are evaluated as rs = 0.986, 1.000 and 1.000 respectively. This is a guarantee that the proposed method is more reliable.

Conclusions
This paper presents a novel hybrid approach to tackle ranking problems with multiple inputs, multiple outputs and multiple DMUs. The proposed approach was tested with three numerical examples. We first utilized Ideal and Anti-Ideal models to calculate the efficiency rating of decision making units. The results of both models were used to generate an Ideal Cross-Efficiency Matrix (Ideal CEM) and Anti-Ideal Cross-Efficiency Matrix (Anti-Ideal CEM). In each Ideal Cross-Efficiency Matrix, the target DMUs of Ideal and Anti-Ideal models were viewed as criteria and DMUs were viewed as alternatives. Secondly, the combined CEM was generated using the new formula for combining Ideal and Anti-Ideal CEMs. After that, the criteria weights were generated by the CRITIC method. Finally, decision making units were ranked. The proposed hybrid approach showed potential in ranking decision making units, which differ from other models in the literature. We believe that the proposed ranking method can be employed to solve other real-world ranking problems.
For future research, in order to enhance the validity of the research output further, application of the proposed hybrid approach should be tested with more cases.