Group decision making in best-worst method when the best and worst are not unique: case study of scholar selection

Abstract


Introduction
Scholarship for a student is a type of financial aid, and it is in the form of a grant that does not have to be repaid by the student.Scholarships may be in different shapes and sizes.They are offered to students by government, university, a private company or an organization for financial need or achievement in academics [1], [2].Although there are many reasons that a student has to apply the scholarship, the main reason is that scholarship helps to reduce the cost burden of the education life of the students [3].Sometimes the number of applicants who have the necessary skills and knowledge to successfully fulfill the needs of the scholarship may exceed the number of the scholars designed by a scholarship provider.Under this circumstance, selecting eligible students from competing applicants requires a detailed analysis to determine the eligibility of applicants [2].The selection process involves tasks of interviewing with the applicants, evaluating their applications, and making final decision.These tasks are to be completed by a selection committee.During the selection process, many multiple criteria have to be taken into consideration simultaneously to specify the students who deserve scholarship [4].
In the literature, the scholarship student selection has been assumed as Multi Criteria Decision Making (MCDM) problem, and there are few studies about it.Yeh [5], [6] formed this selection as Multi Attribute Decision Making (MADM) problem, and presented a solution based on total sum, Simple Additive Weighting (SAW), Weighted Product (WP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) methods.Uyun and Riadi [7] proposed to apply fuzzy MADM including TOPSIS and WP for scholarship student selection at Universitas Islam Negeri Sunan Kalijaga.Similarly, Wimatsari et al. [8] demonstrated the scholarship selection with a case by using fuzzy MADM including TOPSIS method.Saptarini et al. [9] used Analytic Hierarchy Process (AHP) for finding scholarship criteria weights and TOPSIS method for ranking the students who apply the scholarship.Purba and Sembiring [10] proposed a decision support system that uses Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE) method for the scholarship recipients selection at Polytechnic Unggul LP3M.Mesran et al. [11] performed Preference Selection Index (PSI) method for scholarship student selection process.They argued that PSI method facilitates the selection process in terms of not assigning criteria weights.Marfuah and Widiantoro [4] used AHP method for scholarship selection process at Universal University.Similarly, Puspitasari et al. [12] developed a decision support system based on AHP for scholarship determination.Rizana and Soesanto [13] integrated AHP and factor rating to find the selection criteria weights, and rank the scholarship applicants.Mahmud et al. [14] determined the criteria and sub-criteria for student selection, and applied fuzzy AHP method to select the students.Anamisa et al. [15] proposed to use SAW and TOPSIS methods for scholarship grantee selection.Mufizar et al. [16] combined Multifactor Evaluation Process (MFEP) and Distance to the Ideal Alternative (DIA) to specify the majors and the scholarship recipient.Utami and Ruskan [17] developed a decision support system applying Multi-Objective Optimization method on the basis of Ratio Analysis (MOORA) to determine the eligible students among the participants for Yayasan Alumni scholarship.MOORA method was also used as a selection method by Mardhiyyah et al. [2] for the scholarship selection process.Oktaviani et al. [18] established a decision support system that uses WP and SAW methods for the scholarship recipient selection process.Rafida et al. [19] proposed to apply Simple Multi-Attribute Rating Technique (SMART) method for scholarship selection, and demonstrated a case study in STMIK Widya Cipta Dharma.
In this study, it is aimed to solve scholarship student selection problem with two MCDM methods, Best-Worst Method (BWM) and COmbined COmpromise SOlution (CoCoSo).BWM is performed for weighting the scholarship student selection criteria while CoCoSo is performed for ranking of the applicants.The educational foundation that provides the scholarship wants to determine the scholarship criteria and the criteria weights together with the selection committee.From this point of view, the use of BWM is appropriate for the subjective determination of criteria weights.BWM was developed by Rezaei [20], and it is an improved version of AHP that was developed by Saaty [21].Although AHP is widely preferred for solving MCDM problems, there are some disadvantages.The main disadvantage of AHP is the number of pairwise comparisons that are based on the number of the levels in the hierarchy.Namely, the hierarchy expansion requires more pairwise comparisons, computation, more time and effort [22], [23].BWM overcomes this disadvantage of AHP in terms of determining the best and the worst criteria, and forming pairwise comparison between each of two criteria (best and worst) and the other criteria [20].On the other hand, CoCoSo was proposed by Yazdani et al. [24].It is the integration of two well-known MCDM methods, SAW and Exponentially Weighted Product (EWP).It uses different aggregation procedure for ranking the alternatives [25].With this feature, the CoCoSo method offers a solution that uses different methods as well as different ways of aggregation.This solution also includes the advantages of all the methods that CoCoSo method contains.In this sense, as Yazdani [24] stated, the CoCoSo method provides a compromise solution, and the compromise solution is very important in that it is a solution that meets all the requirements simultaneously for problems that contain generally unmeasurable and conflicting criteria.Considering all these situations, it is appropriate to use the CoCoSo method in scholarship selection.Also, application results show that the usage of two methods is suitable and efficient for the application of scholarship selection in terms of being simple, and requiring fewer and understandable formulations.
In the line with the brief explanations mentioned above regarding scope of the study, the contributions of the study can be stated as follows:


The scholarship selection problem is designed as a group decision-making problem,


BWM is performed to determine the scholarship selection criteria weights,


The main characteristic of the handled problem is that the best and worst criteria are not unique for each selection committee member.BWM method is applied to address this situation,


Selection of the best scholar among scholars who fulfil all criteria is performed by CoCoSo method.With this selection, it is thought that an easy-to-understand and practical methodology has been developed for the decision makers on this subject,


To the best of our knowledge, the integration of BWM and CoCoSo methods in scholarship selection is new in MCDM literature.
The rest of this study is organized as follows.The proposed methodology, the methodological backgrounds of BWM and CoCoSo methods are provided in Section 2. The application of the scholarship selection based on BWM and CoCoSo methods is demonstrated with a real case in Section 3.Then, application results are given and discussed.Lastly, recommendations for further studies are presented.

Proposed methodology
An integrated method based on BWM and CoCoSo methods is proposed in this study.In the first phase of the proposed method, criteria weights are determined by BWM.It is conducted in a group decision making environment.Also, the best and worst criteria of the decision makers are not common, and they are more than one.In the second phase of the proposed method, the decision alternatives are listed with CoCoSo.The criteria weights obtained from the BWM in the first phase are transferred to the second phase.The flowchart for the proposed methodology in this study is shown in Figure 1.In this sense, first of all, BWM and CoCoSo methods are explained in detail.Also, the integrated method proposed in this study is explained below.

BWM when the best and worst are not unique
BWM is one of the MCDM methods.It was firstly proposed by Rezaei [20].It is a comparison-oriented method, and assumed as an enhancement of AHP method [26], [27].Namely, it can be used to derive the subjective criteria weights.For this, first of all, the best and worst criteria are determined by the decision maker.Then, pairwise comparisons are formed between each of these two criteria and the other criteria.A maxi-min mathematical model based on these comparisons is solved, and the reliability of the comparisons is checked by a consistency ratio [26].In contrast to AHP, BWM requires fewer data, and provides more consistent results [28].There are many studies that have applied BWM or combined it with other methods and theories in the literature.Some studies from the BWM literature are provided in Table 1.
In this section, the steps of BWM method, in which the best and worst criteria are not unique, are explained.In addition, the situation where the decision makers cannot reach a common decision on the best and worst criteria is also taken into account.The application steps are described as follows [20], [29], [30]: Step 1.First of all, criteria of the problem (Cj ; j = 1,2, …,n) are defined.
Step 2. It is assumed that the decision will be made by more than one decision maker.So, this part is designed as the group decision making.Considering the criteria defined in Step 1, each decision maker identifies his/her best (CB) and worst (CW) criteria from his/her own perspective.The most desired or important criterion is accepted as the best criterion, while the least desired or important criterion is considered as the worst criterion.
Step 3.Each decision maker determines the pairwise comparisons of the best criterion over all other criteria by using Saaty's 1 to 9 scale shown in Table 2. Best-to-Others vector for the kth decision maker (k=1,2,…,K) is expressed as: a Bj k is the preference of the best criterion over jth criterion for kth decision maker, and aBB = 1.
Step 4. Each decision maker determines the pairwise comparisons of all criteria over the worst criterion by using Saaty's 1 to 9 scale shown in Table 2. Others-to-Worst vector for the kth decision maker (k=1,2,…,K) is expressed as: a jW k is the preference of the jth criterion over the worst criterion for kth decision maker, and aWW = 1.
Step 5.The aggregated Best-to-Others and Others-to-Worst vectors are computed.In this step, one of the following two situations occurs: a) If the decision makers at the decision committee reach a common decision about the best and the worst criteria, then aggregated Best-to-Others and Othersto-Worst vectors are computed by performing various aggregation ways such as arithmetic mean, geometric mean, etc. Aggregated pairwise comparison matrix is developed as: Table 1.Some studies from the BWM literature.

BWM
Bakker [33] Strategy selection for trolley supply chain Rezaei [20] Mobile phone selection Rezaei et al. [29] Supplier segmentation and development Sadaghiani et al. [34] External forces evaluation for oil and gas industry Rezaei et al. [31] Supplier selection Rezaei et al. [35] Determination of optimal freight bundling configuration for transport Gupta and Barua [36] Evaluation of enablers for technological innovation Ren et al. [32] Sustainability assessment of the technologies for the treatment of urban sewage sludge Bezerianos [28] Country selection for oil and gas industry Ghaffari et al. [27] Evaluation of key success factors for remotely-piloted helicopters industry Abadi et al. [37] Strategy evaluation for medical tourism development Ecer [38] Sustainability assessment of onshore wind plants Ulutaş [39] Evaluation of vehicle selection criteria Bilgiç et al. [40] Evaluation of renewable energy sources Arsu and Uğuz Arsu [41] Personnel selection Öz [42] Launch site location selection BWM and TOPSIS Evaluation of skyline web services Abdulkareem et al. [46] Intelligent algorithms selection for image dehazing BWM and SWOT analysis Chitsaz and Azarnivand [47] Evaluation of water shortage alleviation strategies Fuzzy BWM Guo and Zhao [48] Three case studies including transportation mode selection, car selection and supplier performance evaluation Ketabchi and Ghaeli [49] Risk assessment in oil projects Xu et al. [50] Allocation of water rights Z-BWM Aboutorab et al. [51] Supplier development Fuzzy BWM and COPRAS Amoozad Mahdiraji et al. [52] Evaluation of key factors of sustainable architecture BWM, VIKOR and GRA Parhizgarsharif et al. [53] Facility location selection BWM and WASPAS Kolagar [54] Evaluation of cities for urban agriculture Arslanhan and Tosun [55] Selection of transportation mode BWM and ARAS Bahrami et al. [56] Determination of Cu potential zones Çakır and Can [57] Evaluation of outsourcing companies for an accommodation company Çalık [58] Target market selection Fuzzy BWM and DEA From here, the aggregated Best-to-Others and Others-to-Worst vectors can be calculated using the aggregated pairwise comparison matrix.For this, first of all, row sums of the aggregated pairwise comparison matrix are calculated.The criterion with the highest row total is considered as the best criterion, while the criterion with the lowest row total is considered as the worst criterion.Concordantly, in the following steps of the method, the row with the best criterion is used as the Best-to-Others vector, and the row with the worst criterion is used as Others-to-Worst vector.
Step 7. Consistency level of comparison is calculated in this step.A comparison is fully consistent when aBj × ajW = aBW (j=1,2,…,n).The consistency ratio of Linear Chebyshev BWM is calculated by Eq. ( 7): Consistency Ratio =  *    (7) Consistency index depends on the value of aBW.The value of aBW may not be integer due to aggregation process in Step 5.In this case, the value of aBW can be rounded up.The corresponding consistency index is presented in Table 3.The consistency ratio takes its value from the interval [0,1].If the value of consistency ratio is close to zero, it means that comparison is consistent [32].

CoCoSo method
CoCoSo method is one of the MCDM methods, and was developed by Yazdani et al. [24].CoCoSo consists of two commonly used methods, SAW and EWP [72].From this point of view, the ranking results of alternatives obtained from the method can be considered as a summary or presentation of compromise solutions [25].Also, compromise solutions are computed by using different aggregation strategies in CoCoSo [73].Since the day CoCoSo was proposed, it has many successful applications in the literature.Some of these studies are presented in Table 4. CoCoSo method requires the following steps [24],[72],[74]: Step 1.The initial step is the generation of decision matrix (X).

𝑋 = [𝑥
where xij is the performance value of ith alternative with respect to jth criterion.
Step 2: The decision matrix is normalized by performing the linear normalization procedure.Eq. (9a) and Eq.(9b) are used for the beneficial and non-beneficial criteria, respectively: where rij is the normalized value of xij.
Step 3. The weighted comparability sequences, the sum and power (Si and Pi), for each alternative are computed.
where wj is the weight of jth criterion.In this study, the criteria weights used in Eq.( 10) are obtained from BWM given in Section 2.1.
Step 4. Relative performance scores of the alternatives are specified with three ways as shown in Eq.( 11)-( 13).
Eq.( 11) is the mean of scores obtained from WSM (Weighted Sum Method) and WPM (Weighted Product Method).Eq.( 12) is the sum of relative scores obtained from WSM and WPM compared to the best alternative.Finally, Eq.( 13) is the comprise solution for WSM and WPM.In the current study,  value is set to 0.5 as in many studies in the literature.However, in the application section, the effect of change of this value on the rankings is investigated.
Step 5.The overall value of alternative (ki) is determined by using Eq. ( 14): Alternatives are ranked in ascending order.The alternative with the highest value of ki is ranked as the best.

Application
In this section, the mentioned methodologies are applied to a real case, which refers to Pamukkale University in Denizli.One of educational foundations operated in Turkey gives scholarships to successful undergraduate students around Turkey every year.Educational foundation is determined the quota of the Faculty of Economics and Administrative Sciences of Pamukkale University as one student.In other words, the number of the students who will benefit from scholarship program is only one for one academic year.Scholar is chosen by a committee.For this purpose, firstly the scholarship committee is created including one representative from educational foundation (DM1) and three academicians from the faculty (DM2, DM3 and DM4).
The criteria used in this study are determined by the educational foundation that provides the scholarship.The criteria that are used in evaluating the applicants are defined as follows: Grade Point Average (GPA) of the student (C1): The student is required to obtain a minimum GPA of 2.0 to get his/her scholarship.
The payment amount of the student for accommodation (C2): It includes rent of their house or dormitory fee.It is expressed as TRY.If the student lives with his/her family in Denizli, this value will be 0.
The amount of payment, salary or scholarship received by the student from any private or public institution (C3): It is expressed as TRY.If the student does not get any payment or have any scholarship, this value will be 0.
The marital status and living arrangements of a student's parents (C4): For this criterion, 1 shows that the student's parents are both living and married to each other whereas 0 shows that the student's parents are divorced or separated.
The number of family member that must be liable to look after (C5): It shows the number of family member that must be liable to look after by their parents.
The number of the siblings in the family who are still students (C6): It shows the number of siblings in the family who are still students.
Annual family net income (C7): It shows the annually financial position of the family, and expressed as TRY.
The amount of rent received by the family from their immovable property (C8): It is expressed as TRY.If the family does not have any flat, building etc., this value will be 0.
The market value of the automobile that the family owns (C9): It is expressed as TRY.If family does not own the automobile, this value will be 0.
The selection process begins with finding the weight of each criterion.Firstly, each decision maker identifies his/her best and worst criteria among all criteria shown in Table 5.
Then, each decision maker states pairwise comparisons between the best criterion over all the other criteria, and all the criteria over the worst one, respectively.Saaty's 1 to 9 scale shown in Table 2 is used while making the pairwise comparisons.Table 6 and Table 7 show these pairwise comparisons.As shown in Table 5, the decision makers do not have any common decision about the best and the worst criteria.Also, the numbers of best or worst criteria are not unique for some decision makers.In this situation, 9x9 pairwise comparison matrix is completed for every decision maker by the help of secondary comparisons.As an example, 9x9 pairwise comparison matrix of DM1 is given in Table 8.The best and worst criteria of DM1 are C1 and C9, respectively.The pairwise comparisons of C1 and C9 over criteria, primary comparisons, are highlighted in Table 8.The others are secondary comparisons.For instance, C2 and C3 are neither best nor worst criteria.We know that a23 = w2/w3, and w2/w3 can be found as (wB/w3)/(wB/w2) in the following: The similar secondary comparisons are calculated, and 9x9 pairwise comparison matrix of DM1 is completed as seen in Table 8.The same procedure is applied for other decision makers.Afterwards, by using Eq.( 4) pairwise comparison matrices of four decision makers are aggregated, and the result is shown in Table 9.In the aggregation process, the geometric mean of the pairwise comparison matrices of 4 decision makers is computed.Clearly, an aggregated pairwise comparison matrix is formed by taking the geometric mean of the elements in the same position of 4 comparison matrices in 9x9 size.
The row totals of Table 9 are computed to determine the best and the worst criteria of the aggregated pairwise comparison matrix.The criterion with the highest row total value is considered as the best criterion whereas the criterion with the smallest value among the row totals is also considered as the worst criterion.In this manner, C1 and C9 are the best and worst criteria of aggregated pairwise comparison matrix, respectively.The row elements of C1 and C9 are used as the Best-to-Others vector and Others-to-Worst vector, respectively.In other words, these values are used as inputs for the model shown in Eq.( 6).Then, optimal criteria weights are computed with the model shown in Eq.( 6).The results are shown in Table 10.The optimal value of model ( *  ) is 0.052.In this study, aBW value is computed as 8.74 therefore, this value is rounded up, and it is assumed as aBW=9.While calculating the consistency ratio, the consistency index value corresponding to 9 in Table 3 is used.Consistency ratio proposed by Rezaei [20] is computed as 0.009, and this value is in the acceptable range.After making necessary announcement about the scholarship program in the faculty, 35 students (S1, S2, …, S35) apply for this scholarship.Then, these students fill out the application form including their personal information and the data related with criteria.The data collected from the applicants are tabulated, and utilized as decision matrix of the selection problem.Each student's data with respect to each criterion is shown in Table 11.
The types of criteria used in the study are different from each other.The types of C1, C2, C5 and C6 are beneficial whereas C3, C4, C7, C8 and C9 are non-beneficial criteria.The different types of data in the decision matrix are normalized separately by Eq.( 9a)-(9b), and normalized decision matrix is obtained.From now on, the necessary operations for CoCoSo method are applied.Firstly, Si and Pi values are computed using Eq. ( 10).Then, kia, kib, and kic values are calculated using Eq. ( 11)- (13).Finally, ranking score of each student is computed by Eq.( 14).All of these values are shown in Table 12.According to Table 12, the highest value belongs to the twelfth student (S12) so it can be said that twelfth student (S12) is the best in terms of fulfilling the needs of the scholarship successfully.In the calculation of the values given in Table 12, the threshold value () is taken as 0.5.In order to check if  has an effect on the ranking, new rankings are obtained by using different  values.These rankings and the change in rankings can be observed in Figure 2. When Figure 2 is examined, it is seen that when  changes, there are slight changes in the rankings.However, S12 is in the first place in all rankings.

Discussion
In this study, scholarship offered by an educational foundation is allocated to the best-suited student among multiple candidates effectively.In this manner identifying, weighting and evaluating the competing applicants are assumed as an MCDM problem.For solving this problem, a selection process based on two MCDM methods, BWM and CoCoSo, is developed.
Firstly, scholarship selection criteria are determined by the committee.In this study, nine criteria are discussed for the problem.Then, each decision maker separately identifies his/her best and worst criteria.In this situation, they are not unique for a decision maker and also, the decision makers do not reach any consensus about the best or worst criterion.These situations are the main characteristics of this study.To overcome this situation and solve the problem, secondary comparisons between criteria are considered.After computing the secondary comparisons and completing the each decision maker's pairwise comparison matrix, they are aggregated with geometric mean.The necessary operations are performed with aggregated pairwise comparison matrix.The best and worst criteria are determined by taking the row total of aggregated pairwise comparison matrix.In our study, grade point average of the student and the market value of the automobile that the family owns are derived from the aggregated pairwise comparison matrix as the best and worst criteria, respectively.From now on, the necessary mathematical models of BWM are developed, and criteria weights are computed.The results of BWM indicate that the most important criterion in selecting an appropriate scholar is grade point average of the student with an importance level of 0.264.This criterion is followed by annual family net income with an importance level of 0.243.We can say that they are very close in terms of their importance levels.Actually, this result is not too surprising for us, because annual family net income is among the best criteria of three decision makers.On the other hand, the market value of the automobile that the family owns is the least important criterion with an importance level of 0.022.Consistency ratio proposed by Rezaei [20] is also computed in the study, and this value (0.009) is assumed as acceptable.Performing BWM for scholarship selection problem is thought as appropriate method.As in our case study, the scholarship is provided by an educational foundation.This educational foundation also wants to determine the importance levels of criteria while selecting the best student among the scholarship applicants.In this sense, it seems quite reasonable to use a subjective weighting method.Simplicity of the method performed is also important for decision makers, and it comes from the decision maker's role on the selection process.In this method, decision makers are only asked to compare the best and the worst criteria with the others.So, the numbers of pairwise comparisons between the criteria are decreased to 2n-3, if there are n criteria in the problem [20].The time requiring for solving the problem is also decreased.To understand and fill the pairwise comparison matrix is easy for the decision makers.In such problems, each decision maker's pairwise comparisons can be aggregated with different ways.In further studies, criteria such as the disability status of students, students coming to university for education from low-income regions, and the scores of students in the university entrance exam can be added.In this case, the selection problem will become more complex as the number of criteria increases.When BWM is compared to AHP, BWM produces more consistent results for large and complex decision problem like our case study including nine criteria and four decision makers.
Determining the importance levels of the scholarship selection criteria is not sufficient to solve the scholarship selection problem.It is absolutely necessary to select among the students who apply the scholarship.In this study, the CoCoSo method is used as the selection method.An important feature that makes the CoCoSo method superior to other available MCDM methods is its simplicity and user friendly.It only requires the criteria weights and performances of the alternatives with respect to criteria.Types of criteria as beneficial and non-beneficial are considered.Students are ranked efficiently with CoCoSo method which considers criteria weights derived from BWM and performances of students with respect to criteria.The results of the method suggest that twelfth student (S12) is the best in terms of performing the needs of the scholarship successfully.In this sense, the result includes both the importance level of the criteria determined by the scholarship provider and the information of the student applying for the scholarship, as stated before.From this point of view, the integration of BWM and CoCoSo methods is very effective for the decision committee.It should also be noted that the results of this study are valid for only this faculty and the specified academic term.If the decision makers participating in the study, the students applying for the scholarship, data of the applicants, and the criteria change, the results of this study will also change.As Akıllı and İpekçi Çetin [97] stated, increasing the degree of expertise of the decision makers who are effective in determining the selection criteria and the number of decision makers participating in the decision process will increase the reliability of the results.

Conclusion
In this study, BWM is integrated with CoCoSo method for scholarship selection.To present the applicability of integrated methodology, a real case study is solved, and results are given above sections.The contributions of this study to the literature can be summarized in the following:


To the best of our knowledge, the scholarship selection problem is solved with the integration of BWM and CoCoSo methods for the first time in the literature,


The scholarship selection problem designed as a group decision-making problem,


The weights of selection criteria are computed subjectively by using pairwise comparisons.In this manner, BWM is performed to eliminate the drawbacks of AHP method,


The main characteristic of the handled problem is that the best and worst criteria are not unique for each selection committee member.This situation is overcome by using secondary comparisons.Although the use of secondary comparisons seems to complicate the method, secondary comparisons are calculated using simple mathematical operations, not pairwise comparisons of decision makers.The basis for these calculations is the best and worst criteria that are identified by the decision makers,  Another characteristic of handled problem is that the decision makers do not reach any consensus about the best or worst criterion.This situation is addressed by using aggregation concept.The common best and worst criteria are specified after getting aggregated pairwise comparison matrix,


The selection of best scholar among scholars who fulfil all criteria is performed by CoCoSo method.With this selection, it is thought that an easy-to-understand and practical methodology has been developed for decision makers on this subject.
BWM and CoCoSo methods are thought as appropriate methods for weighting and ranking the candidates or alternatives because of satisfactory results.In future academic terms, the same problem may be updated by changing the numbers of criteria, students and decision makers in the committee.Different aggregation ways may be tried.Fuzzy extensions of these methods may be applied.Finally, these methods are applicable to other management problems.

Author contribution statements
In the scope of this study, Esra AYTAÇ ADALI, in the formation of the idea, the literature review, the construction of the theoretical background, performing analyzes and the writing and checking the article; Atalay ÇAĞLAR, in the formation of the idea, the literature review, the construction of the theoretical background, performing analyzes and the writing and checking the article, were contributed.

Ethics committee approval and conflict of interest statement
Ethics committee permission is not required for the article prepared.
There is no conflict of interest with any person/institution in the article prepared.

Figure 1 .
Figure 1.The flowchart for the proposed methodology.

Figure 2 .
Figure 2. The effects of  on the rankings.
|  −     | ≤   for all j |  −     | ≤   for all j 6. Rezai et al. [31] developed a new Linear Programming (LP) model for finding optimal criteria weights namely, they proposed to use Linear Chebyshev BWM.In this study, the model for Linear Chebyshev BWM is performed for finding the optimal criteria weights.Maximum absolute differences |  −     | and |  −     | for all j are minimized, and this model is shown in Eq. (5): min max  {|  −     | , |  −     |} s.t.∑   = 1    ≥ 0 for all j (5) Model shown in Eq. (5) is transferred to the following LP model: min   s.t.

Table 4 .
Some studies about CoCoSo method from the literature.

Table 5 .
The best and the worst criterion/criteria for each decision maker.

Table 6 .
The preference of the best criterion over all the other criteria.

Table 7 .
The preferences of all the other criteria over the worst criterion.

Table 10 .
Optimal weight for each criterion found by Eq. (6).

Table 12 .
Overall value of each student based on Table11.