Integrated fuzzy AHP and TOPSIS as innovative student selection methodology at institutions of higher learning

Abstract

BACKGROUND:

The selection of students at academic institutions has been a challenging affair given multiple criteria that need to be considered by the institution. Additionally, multiple evaluators and decision makers are involved in the student selection process, rendering it inconsistent. The complexity and subjectiveness in such decisions making requires new and innovative approach in order to be more systematic and transparent.

OBJECTIVE:

This paper presents an innovative methodology for student selection for admission into an Institute of Higher Learning (IHL) using Fuzzy Analytical Hierarchy Process (FAHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Drawing on the success of using these methods in other fields, this study applies the technique and principles on student selection process.

METHOD:

Fuzzy Analytical Hierarchy Process (FAHP) is used in determining the weights of the criteria by the decision makers which avoids the vagueness and inconsistencies in decision making process and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method ranks finds out the best alternative solution for student selection by calculating the relative closeness from the positive ideal solution.

RESULTS AND CONCLUSION:

This research finds using the hybrid method is effective in student selection for IHL and makes the process efficient and bias-free. This method can be applied to various fields and uses where multi-criteria decision making is involved.

Keywords

Multi-criteria decision-making student selection bias-free education technology

[]Dr Nisa James is an Assistant Professor in the Department of Management Studies, Kannur University and is the Innovation Ambassador of the University. A trained engineer, with over 10 years of experience in Postgraduate education at reputed B-Schools of South India, she has completed a PhD programme at Bharathiar University, Coimbatore. Conferred Accredited Management Teacher by AIMA, she is a resource person for various workshops and seminars at various colleges. An ardent researcher, with publications in reputed journals, she has presented research papers in national and international conferences and is the reviewer of MDPI and Inderscience journals. She is currently involved in the Unnat Bharat Abhiyan project of MHRD, India and an international research program, funded by The Sumitomo Foundation, Japan

[]Swetha Loganathan is a PhD Scholar at the Department of Economics, Christ (Deemed to be University, Bengaluru. She is presently undertaking her PhD under the Junior Research Fellowship Scheme of the University Grants Commission, Government of India. She has published papers in areas related to macroeconomics and international trade, focusing on India’s trade relations with Europe. Her research interests include international trade and the impacts of trade liberalisation on Indian agriculture

[]Assoc Prof Dr Robert Jeyakumar heads the department for Academic Innovation and Product Intelligence at Multimedia University, Malaysia. He works on program innovation & data analytics at the university and teaches Management, Marketing, Innovation, Research Methodology and Leadership. He is a certified Design Thinking Trainer, by Stanford University Design School. Prior to joining academia, he worked for Siemens Semiconductor, a semiconductor manufacturing firm based in Munich, Germany. Robert participates in the Standing Committee for Copyright and Related Rights (SCCR) with the World Intellectual Property Organization (WIPO).

Dr. Vijay Victor is an assistant professor of Economics, T A Pai Management Institute, Manipal, India. He is also attached to the College of Economics and Business, University of Johannesburg, as a research associate. He obtained his PhD from the Szent Istvan University, Hungary under the Stipendium Hungaricum framework, a fully-funded PhD fellowship granted by the Hungarian Government. He was also awarded the prestigious Erasmus Mundus Fellowship instituted by the European Commission in 2014. His areas of interest include behavioural economics, macroeconomic policies, and econometricss

Assoc Prof Ir Ts Dr Poh Kiat Ng is an Associate Professor and Deputy Dean of the Faculty of Engineering and Technology, Multimedia University, Malaysia. He is a Professional Engineer of the Board of Engineers Malaysia (PEng), a Chartered Engineer of the Engineering Council and Institution of Mechanical Engineers UK (CEng MIMechE), an ASEAN Chartered Professional Engineer (ACPE), and a Professional Technologist from the Malaysian Board of Technologists (PTech). He is also formally trained on a problem-solving technique known as the theory of inventive problem-solving (TRIZ), and has published a number of papers on design using TRIZ. He is an International TRIZ Level 3 Professional (MA TRIZ Level 3) and has been actively delivering workshops on TRIZ as an art of innovation. His research interests include design, human factors and ergonomics, biomechanics, TRIZ, innovation, and teaching and learning

1 Introduction

With higher education becoming increasingly important in the labor market, there is an increasing demand in student selection for various roles and professions. The Ministry of Human Resources, Government of India reported that gross enrollment ratio increased from 25.8 in academic year 2017–18 to 26.3 in the academic year 2018–19, rising Indian student population in higher education to 37.4 million. In this context, the process of student selection from a large pool of applicants has become more complex for educational institutions worldwide. There is a pertinent need for the process to be designed in an objective manner, taking into consideration the multifaceted nature of applicants and evolving demands of tertiary education. There are several significant aspects in relation to this problem, including the impact of higher education on labor market outcomes for individuals and the economy, the evolving concept of meritocracy, and changing contours of academic performance. Therefore, there is a need to develop a more nuanced and intricate mode of student selection to accommodate the emerging trends in education and industry, while addressing inherent biases identified in existing systems.

Choosing the right student for the college ensures success and adds value to the institution in the current global environment. Several criteria such as marks obtained at various levels and scores of different tests tend to be taken into consideration. The specific criteria selected is often affected by various factors such as the platform used and the opinion of experts. In addition, some criteria tend to be given prominence over others by individual decision-makers. Therefore, there is a need for systematic alternatives to decision-making in the context of student selection.

The fuzzy Analytical Hierarchy Process (AHP) is a tool designed to address the subjectiveness and uncertainty in the student selection process that could cause fuzziness and inconsistencies in ranking alternatives. An extension of the classical AHP, fuzzy AHP incorporates fuzzy logic based on the concept of degrees of truth, in contrast to the true or false logic, to accommodate subjectivity and fuzziness. This hybrid method is particularly effective as it takes into account the subjectivity in decision making by incorporating both abstract and target related aspects [2]. Further, it has been effectively applied in a variety of decision making contexts such as material science, logistics, management and engineering [3].

In light of the above challenges faced in student selection, this study seeks to develop an integrated approach to student selection by addressing the subjectiveness and uncertainty, by deploying Fuzzy AHP, followed by TOPSIS, which helps select the best alternative from different alternatives. It takes into consideration seven important factors for decision making, namely tenth-grade marks, twelfth-grade marks, degree marks, entrance test score, institute test, group discussion marks, and interview marks. The weightage for each criterion is obtained using a fuzzy AHP multi-decision tool to help grasp the importance of the criterion in selecting a student for the institution. On the basis of the weightage, student alternatives are ranked using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), a decision-making method involving multiple criteria. The findings of this study contribute towards the development of an objective method for student selection in circumstances involving subjective selection criteria. Additionally, it can also be extended to decision-making contexts involving multiple decision makers.

2 Literature review

This section seeks to provide an overview of existing literature relevant to the aim of this study. Firstly, previous work based on decision-making approaches applied to student selection are examined. Secondly, the application of fuzzy AHP and TOPSIS in various areas such as industrial production and in consumer behavior is reviewed.

2.1 Multi-criteria decision-making: Theoretical advances

Multi-criteria decision-making (MDCM) is concerned with decision-making based on theory, process, and analytical methods in situations where there exists a degree of uncertainty and dynamism along with the existence of multiple evaluating criteria [4]. As one of the models for decision making, MCDM tries to evaluate various alternatives using the selected criteria to arrive at the best alternative or decision. It becomes specifically useful in circumstances where multiple non-commensurable and competing criteria are taken into account such that no single alternative simultaneously meets all criteria [5]. Particularly in such contexts, decision support models need to be able to handle subjectivity and this has led to the evolution of fuzzy methods [6].

MCDM can be categorized as multi-objective deci-sion-making (MODM) and multi-attribute decision-making (MADM) [5]. In MODM, the decision variable of interest tends to be an integer on a continuous scale and can have an infinite or very large number of alternatives. On the other hand, MADM tends to have a discrete number of alternatives and the decision is made through preference based on multiple attributes associated with the alternatives. The uncertainty and dynamism involved in these decisions make “fuzzy” methods of MCDM increasingly popular and relevant in making decisions [7], creating the need to develop decision-making methods that focus on both abstract and target attributes. MADM in particular tends to apply scoring and compromising methods for the purpose of decision-making [8]. Scoring methods look at choosing an alternative that tends to have the highest score with regard to the multiple criteria applied while compromising methods examine the ideal best and ideal worst alternatives.

In the context of the present study, fuzzy AHP is used as a scoring method while TOPSIS is used as a compromising method. This hybrid method is particularly effective as it takes into account the subjectivity in decision making by incorporating both abstract and target related aspects [2]. Further, it has been effectively applied in a variety of decision making contexts such as material science, logistics, management and engineering [3].

2.2 Fuzzy AHP and TOPSIS

The first study of fuzzy AHP was proposed by Van Laarhoven and Pedrycz [9], which compared fuzzy ratios described by triangular fuzzy numbers. Buckley and Hayashi [10] initiated trapezoidal fuzzy numbers to express the decision-maker’s evaluation on alternatives with respect to each criterion, while Chang [11] introduced a new approach for handling fuzzy AHP, with the use of triangular fuzzy numbers for pair-wise comparison scale of fuzzy AHP. The fuzzy AHP method is a popular approach for multiple criteria decision-making and has been widely used in the literature. Fuzzy AHP is carried out mainly in five steps. Firstly, it converts the linguistic terms into the membership function via fuzzification in comparison with the scale of relative importance and the reciprocal conversion using, A-1 = (l, m, u) –1 = (1/u, 1/m, 1/l) where l, m, and u are the lower, middle, and upper value in a fuzzy set. After obtaining the membership function the geometric mean value of the fuzzy numbers are calculated using, $\begin{matrix} \bar{A} 1 \otimes \bar{A} 2 & = (l_{1}, m_{1}, u_{1}) \otimes (l_{2}, m_{2}, u_{2}) \\ = (l_{1} * l_{2}, m_{1} * m_{2}, u_{1} * u_{2}) \end{matrix}$ (1)

Geometric mean value is used for computing fuzzy weight: ${\hat{w}}_{i} = {\underset{˙}{\bar{r}}}_{i} \otimes ({\underset{˙}{\bar{r}}}_{1} \oplus {\underset{˙}{\bar{r}}}_{2} \oplus \dots \dots \dots {\underset{˙}{\bar{r}}}_{n-1}$ (2)

The last step involved in fuzzy AHP is de-fuzzification i.e., converting back to the crisp logic. The weight obtained from de-fuzzification can be used as the weight of each criterion used in the TOPSIS method for finding the student alternatives.

TOPSIS identifies the best alternative from different alternatives where the decision-maker of a college scores the alternatives before selecting the students. The main difference between fuzzy AHP and TOPSIS is in doing the pairwise comparison. TOPSIS ranks the student alternatives based on the decision-maker’s perception while in fuzzy AHP the pairwise comparison of each criterion is made and the weight is obtained using the de-fuzzification method. TOPSIS method considers different types of attributes or criteria which include qualitative benefit attributes/ criteria and quantitative benefit attributes. In this method, the artificial alternatives are hypothesized: ideal and negative ideal alternatives. The one which has the best level for all attributes is the ideal alternative. On the other hand, the one with the worst attribute values is the negative ideal alternative.

The main objective of TOPSIS method is to select the alternative that is closest to the ideal solution and farthest from the negative extreme. It is a method of compensatory aggregation that compares a set of alternatives by identifying weights for each criterion, normalizing scores for each criterion, and calculating the geometric distance between each alternative and the ideal alternative, which is the best score in each criterion. An assumption of TOPSIS is that the criteria are monotonically increasing or decreasing. Normalization is usually required as the parameters or criteria are often of incongruous dimensions in multi-criteria problems. Compensatory methods such as TOPSIS allow trade-offs between criteria, where a poor result in one criterion can be negated by a good result in another criterion. This provides a more realistic form of modelling than non-compensatory methods, which include or exclude alternative solutions based on hard cut-offs. These advantages associated with TOPSIS, makes it an ideal and effective application for students selection into an institution.

2.3 Applications of MCDM in education

The value of multi-criteria decision-making in student selection has been studied to explore the potential of various measures. In the context of India, one of the earliest studies by Kousalya, & Reddy [12] mentioned the use of fuzzy AHP and TOPSIS methods for the selection of a student from an Engineering College for All Round Excellence Award. Using a small sample of students who performed well, to the study supported the idea that the methods can be extended to a larger sample of students such as student applications. Similarly, to determine and solve the problem in the selection of the best student at a college and generate more efficient decisions Fadlina and colleagues [13] proposed an extended multi-criteria decision-making tool known as PROMETHEE II in which the effectiveness of the resulting decisions can motivate students to be more active in learning. Furthermore, Uyun and Riadi [14] applied TOPSIS and weighted product (WP) methods to select the candidates for academic and non-academic scholarships at Universitas Islam Negeri Sunan Kalijaga. The results of this study showed that TOPSIS and WP FMADM methods can be used to select the most suitable candidates to receive the scholarships since the preference values applied in these methods can identify applicants with the highest eligibility.

Student decision-making can also involve multi-criteria decision-making (MCDM) methods in making decisions pertaining to the selection of supervisors and colleges. The MCDM approach using COPRAS-G was suggested by Datta et al. [15] to compromise for the ranking method in supervisor selection. This approach presents the highlights of evaluating the means by which the quality and acceptability of a supervisor can be estimated quantitatively, which is a helpful way in selecting the best one prior to starting a research career under the guidance of a supervisor. MCDM methods such as Analytical Network Process and PROMETHEE multi-criteria decision-making tools are used to assess the sustainability of students’ preferences for university selection. Kabak and Dagdeviren [14] indicated that the priorities of the alternatives and the proposed decision-making approach can help decision-makers choose and analyze factors and attributes easily.

2.4 Applying fuzzy AHP and TOPSIS in education

For decision-making in the field of education, there is growing interest among researchers to adopt integrated MCDM methods, particularly owing to the specific advantages associated with them [17]. Several studies have validated the advantages involved using an integrated method, specifically that of AHP and TOPSIS [18, 19]. In this regard, the present study combines fuzzy AHP (a scoring method) with TOPSIS (a compromising method). The review of the above studies provided sufficient evidence to argue that the use of fuzzy AHP in weighing alternatives and TOPSIS in selecting alternatives provides a superior method of student selection. Fuzzy AHP provides a method of overcoming inconsistencies and supports decision-making through the determination of a consensus. These factors give fuzzy AHP an advantage over other methods of MCDM [20]. These findings have been validated in the context of student selection by the works of Kusumawardani and Agintiara [21] and by Altunok et al. [22]. In addition, TOPSIS involves the selection of an alternative in which there is less deviation from the ideal and more deviation from the most negative extreme [20]. Studies by Agrawal et al. [23] also shows the effectiveness of this hybrid methodology is the selection of B-School by students when deciding which institution to select to do their graduate studies. In more recent studies, Fuzzy-AHP TOPSIS hybrid method has been applied for the study of distance learning models [24, 25].

3 Methodology

The current study is applied in the education industry for the identification of the best students for admission to a university using MCDM. The manual research admission process is tedious and time-consuming. However, dilution in the rigor may negatively affect the quality of students admitted.

3.1 Tools and techniques used

3.1.1 Fuzzy sets

Fuzzy set theory was first introduced by Zadeh in 1965 [26]. It is a class of objects with a continuum of grade of membership ranges from zero to one. A fuzzy set is an extension of crisp set which allows only full membership or non-membership functions. Fuzzy sets and fuzzy logic are powerful mathematical tools for modelling. Modelling using fuzzy sets has proven to be an effective way of formulating decision problems where the information available is subjective and imprecise.

3.1.2 Linguistic variable

A linguistic variable is a variable which values are words or sentences in a natural or artificial language. The concept of a linguistic variable provides a means of approximate characterization of phenomena which are too complex or too ill-defined to be amenable to the description in conventional quantitative terms.

3.1.3 Triangular fuzzy numbers

Triangular fuzzy numbers (TFN) can be defined as a triplet (l, m, u). The parameters l, m, and u are the smallest, most promising, and the largest value that describe a fuzzy event.

Two methods proposed in the study are fuzzy FAHP and TOPSIS. It is possible to use different fuzzy numbers according to the situation. In applications, it is often convenient to work with TFNs because of their computational simplicity. TFNs are useful in promoting representation and information processing in a fuzzy environment. In this study, TFNs are adopted in the fuzzy AHP. A careful and intelligent selection of students enhances the reputation of the college The criteria for student selection were finalized on the basis of consultation with experts and by referring journal articles. decision criteria for student selection were selected. The proposed methodology comprises two steps. In step 1, AHP is improved by the fuzzy set theory. By using fuzzy set theory in AHP, the uncertainty of human judgments can be addressed. In step 2, the obtained results are used as input weights in the TOPSIS algorithm. TOPSIS, by considering ideal and non-ideal solution, helps the decision-maker evaluate ranking alternatives and select the best one.

3.2 Data analysis

As per suggestions of experts in the academic field, the seven eligibility criteria included for academic screening are tenth grade marks, twelfth grade marks, degree marks, entrance marks, institute marks, group discussion marks, and interview marks.

3.2.1 Scale of relative importance

Pairwise comparison using AHP

The stakeholders of a college were asked to do a pairwise comparison of the seven criteria. Table 1 describes the linguistic variables and the corresponding numeric values, which are then transformed into fuzzy numbers [27].

Table 1
Scale of relative importance

Title 1 Title 2 Title 3

1 (1,1,1) Equal Importance

3 (2,3,4) Moderate Importance

5 (4,5,6) Strong Importance

7 (6,7,8) Very Strong Importance

9 (9,9,9) Extreme Importance

2 (1,2,3) Intermediate Values

4 (3,4,5) Intermediate Values

6 (5,6,7) Intermediate Values

8 (7,8,9) Intermediate Values

1/3,1/5,1/7,1/9 Value for Inverse Comparison

Title 1	Title 2	Title 3
1	(1,1,1)	Equal Importance
3	(2,3,4)	Moderate Importance
5	(4,5,6)	Strong Importance
7	(6,7,8)	Very Strong Importance
9	(9,9,9)	Extreme Importance
2	(1,2,3)	Intermediate Values
4	(3,4,5)	Intermediate Values
6	(5,6,7)	Intermediate Values
8	(7,8,9)	Intermediate Values
1/3,1/5,1/7,1/9		Value for Inverse Comparison

3.2.1.2.Pairwise comparison using fuzzy AHP.

1. Fuzzification

Fuzzification is the process of converting linguistic terms into membership functions. Reciprocal Conversion: A-1 = (l, m, u) -1 = (1/u, 1/m, 1/l), where l, m, and u are the upper, lower, and middle value. Fuzzy AHP is a widely used MCDM method, and one of its advantages is that it is based on pairwise comparison and evaluates the inconsistency index. Another distinguishing feature of fuzzy AHP is that it is a powerful tool for solving MCDM problems that are difficult to interpret.

The fuzzy geometric mean value can be identified using the following equation. $\begin{matrix} \bar{A} 1 \otimes \bar{A} 2 & = (l_{1}, m, u_{1}) \otimes (l_{2}, m_{2}, u_{2}) \\ = (l_{1} * l_{2}, m_{1} * m_{2}, u_{1} * u_{2}) \end{matrix}$ (3)

2. Combined geometric means for stakeholders’ pairwise comparison

Table 2 gives the Fuzzy Geometric Mean and Combined Geometric Mean for each Criterion, computed from all the four stakeholders.

Table 2

Fuzzy geometric mean and combined geometric mean for each criterion¹

Criterion	Geometric Mean for Pair-wise Comparison				Combined Geometric Mean
	S I	S II	S III	S IV
Tenth Grade Marks	(3.63,4.36,5.05)	(0.80,1.08,1.57)	(1.23,1.59,2.04)	(3.30,3.74,4.15)	(1.85,2.30,2.86)
Twelfth Grade Marks	(2.11,2.43,2.73)	(0.98,1.23,1.73)	(0.33,0.44,0.54)	(1.05,1.42,1.88)	(0.92,1.16,1.47)
Degree Marks	(1.30,1.43,1.53)	(1.29,1.77,2.15)	(0.87,1.05,1.31)	(0.89,1.16,1.51)	(1.06,1.32,1.59)
Entrance Marks	(0.77,0.95,1.15)	(1.73,2.37,2.90)	(1.31,1.49,1.70)	(0.85,1.05,1.42)	(1.10,1.36,1.60)
Institute Marks	(0.29,0.38,0.47)	(0.46,0.59,0.86)	(0.49,0.58,0.70)	(0.27,0.31,0.39)	(0.36,0.44,0.57)
Group Discussion	(0.41,0.54,0.77)	(0.28,0.32,0.40)	(1.25,1.69,2.11)	(0.30,0.36,0.30)	(0.45,0.56,0.66)
Interview	(0.26,0.32,0.44)	(0.60,0.90,1.16)	(1.07,1.32,1.54)	(0.70,1.04,1.45)	(0.58,0.79,0.77)

¹The primary data used for this estimated has been included in Table A1, A2, A3 and A4 in the appendix.

3. Combined fuzzy weight of the criterion $Fuzzy weight {\hat{w}}_{i} = {\underset{˙}{\bar{r}}}_{i} \otimes ({\underset{˙}{\bar{r}}}_{1} \oplus {\underset{˙}{\bar{r}}}_{2} \oplus \dots \dots \dots {\underset{˙}{\bar{r}}}_{n-1}$ (4)

Equation for adding two fuzzy numbers: $\begin{matrix} \bar{A} 1 \oplus \bar{A} 2 & = (l_{1}, m_{1}, u_{1}) \oplus (l_{2}, m_{2}, u_{2}) \\ = (l_{1} + l_{2}, m_{1} + m_{2}, u_{1} + u_{2}) \\ = (1 / u, 1 / m, 1 / 1)] \end{matrix}$ (5)

Table 3 gives the Combined Fuzzy Weight for each Criterion by multiplying Fuzzy Geometric Mean by Reciprocal of Summation.

Table 3

Combined fuzzy weight for each criterion

Criterion	Fuzzy Geometric Mean multiplied by Reciprocal of Summation	Combined Fuzzy Weight for Criterion
Tenth Grade Marks	(1.85,2.30,2.86) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.19,0.29,0.45)
Twelfth Grade Marks	(0.92,1.16,1.47) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.09, 0.14,0.23)
Degree Marks	(1.06,1.32,1.59) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.11,0.16,0.25)
Entrance Marks	(1.10,1.36,1.68) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.11,0.17,0.26)
Institute Marks	(0.36,0.44,0.57) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.03,0.05,0.09)
Group Discussion	(0.45,0.56,0.66) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.04,0.07,0.10)
Interview	(0.58,0.79,0.77) ⊗ (1/9.60, 1/7.93, 1/6.32)	(0.06,0.09,0.12)

4. De-fuzzification and normalization of weights

De-fuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets, and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. The following expression describes the Center of gravity (COG) / Centroid of Area (COA) Method $Center of Area, wi = ((l + m + u) / 3)$ (6)

The total weight of the criteria is obtained as 1.65 which is greater than 1. To get a value of less than 1, the weight of the criteria obtained is normalized by dividing the total weight obtained with each of the criterion.

Table 4 shows the outcome of fuzzy AHP where the priorities of each criterion is obtained. Analyzing the results, priority is more for the tenth-grade marks, followed by the entrance test score, degree marks, twelfth grade marks, interview marks, group discussion marks, and institute test.

Table 4

Normalized weight for each criterion

Criterion	Centre of Area, wi	Weight	Normali-zed Weight
Tenth Grade Marks	((0.19+0.29+0.45)/3)	0.31	0.18
Twelfth Grade Marks	((0.09+0.14+0.23)/3)	0.15	0.09
Degree Marks	((0.11+0.16+0.25)/3)	0.17	0.10
Entrance Marks	((0.11+0.17+0.26)/3)	0.18	0.11
Institute Marks	((0.03+0.05+0.09)/3)	0.05	0.03
Group Discussion	((0.04+0.07+0.10)/3)	0.07	0.04
Interview	((0.06+0.09+0.12)/3)	0.09	0.05
Total		1.65	0.6 1

5. Consistency checking

Consistency ratio (CR) checking is required to inspect the correctness of the weights assigned based on expert reasoning. Usually, the CR value is less than 0.1, indicating that the weights are consistent.

The formula for calculating CR is as follows, $CR = CI / RI$ (7) where, CI is the consistency index and RI is the random consistency index. The following equation describes the Consistency index $CI = (λ max - n) / n - 1$ (8) where, λ max refers to the average normalized weight of the criteria and n refers to the number of comparisons.

Table 5 gives the number of total responses (N) and the corresponding Random Index (RI). $\begin{matrix} Consistency Ratio, CR & = CI / RI = 0.0 85 / 1.32 \\ = 0.0 64 < 0.16 \end{matrix}$

Table 5

Random consistency index

N	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.9	1.12	1.24	1.32	1.41	1.45	1.49

Based on the RI data in Table 5, the CI and RI are found to be 0.085 and 1.32 respectively. Thus, the CR can be calculated as 0.064. Since the CR value obtained is less than 1 the criteria weights are considered consistent.

3.2.2 TOPSIS calculation

TOPSIS is a multi-criteria decision-making tool that ranks the best student for the admission process. In this circumstance, the decision-makers score the student alternatives before choosing the eligible students in the admission stage for the college. The output weightage of each criterion obtained from the de-fuzzification stage of fuzzy AHP is given as the input to the TOPSIS calculation. Table 5 depicts the actual score given by the decision-makers (stakeholders) of the educational institution to the students based on different criteria. Criterion values are normalized for further analysis, across Tables 6 to 9.

Table 6
Decision matrix based on weightage¹

Criterion 10th 12th Degree Entrance Institute GD Interview

(0.18) (0.09) (0.10) (0.11) (0.03) (0.04) (0.05)

Student 1 95 9.5 84 8.4 89 8.9 42 4.2 80 8 78 7.8 62 6.2

Student 2 81 8.1 68 6.8 80 8 24 2.4 78 7.8 80 8 60 6

Student 3 90 9 87 8.7 70 7 48 4.8 85 8.5 54 5.4 64 6.4

Student 4 74 7.4 84 8.4 74 7.4 27 2.7 11 1.1 58 5.8 58 5.8

Student 5 90 9 81 8.1 78 7.8 52 5.2 70 7 67 6.7 60 6

Student 6 85 8.5 92 9.2 63 6.3 20 2 88 8.8 69 6.9 55 5.5

Student 7 74 7.4 74 7.4 68 6.8 21 2.1 80 8 74 7.4 57 5.7

Criterion	10th	12th	Degree	Entrance	Institute	GD	Interview
Student 1	95	9.5	84	8.4	89	8.9	42	4.2	80	8	78	7.8	62	6.2
Student 2	81	8.1	68	6.8	80	8	24	2.4	78	7.8	80	8	60	6
Student 3	90	9	87	8.7	70	7	48	4.8	85	8.5	54	5.4	64	6.4
Student 4	74	7.4	84	8.4	74	7.4	27	2.7	11	1.1	58	5.8	58	5.8
Student 5	90	9	81	8.1	78	7.8	52	5.2	70	7	67	6.7	60	6
Student 6	85	8.5	92	9.2	63	6.3	20	2	88	8.8	69	6.9	55	5.5
Student 7	74	7.4	74	7.4	68	6.8	21	2.1	80	8	74	7.4	57	5.7

¹The values have been scaled down by 10 for the purpose of analysis.

Table 7

Calculation of ∑ (x²) ^1/2

Student	10th Marks	12th Marks	Degree	Entrance	Institute	GD	Interview
Student 1	90.25	70.56	79.21	17.64	64	60.84	38.44
Student 2	65.61	46.24	64	5.76	60.84	64	36
Student 3	81	75.69	49	23.04	72.25	29.16	40.96
Student 4	54.76	70.56	54.76	7.29	121	33.64	33.64
Student 5	81	65.61	60.84	27.04	49	44.89	36
Student 6	72.25	84.64	39.69	4	77.44	47.61	30.25
Student 7	54.76	54.76	46.24	4.41	64	54.76	32.49
∑ (x²ij)	499.63	468.06	393.74	89.18	508.53	334.9	247.78
∑ (x²) ^1/2	22.35	21.63	19.84	9.44	22.55	18.30	15.74

Table 8

Normalized data

Student	10th Marks	12th Marks	Degree	Entrance	Institute	GD	Interview
Student 1	0.42	0.38	0.44	0.44	0.35	0.42	0.39
Student 2	0.36	0.31	0.40	0.25	0.34	0.43	0.38
Student 3	0.40	0.40	0.35	0.50	0.37	0.29	0.40
Student 4	0.33	0.38	0.37	0.28	0.48	0.31	0.36
Student 5	0.40	0.37	0.39	0.55	0.31	0.36	0.38
Student 6	0.38	0.42	0.31	0.21	0.39	0.37	0.34
Student 7	0.33	0.34	0.34	0.22	0.35	0.40	0.36

Table 9

New decision matrix

Student	10th Marks	12th Marks	Degree	Entrance	Institute	GD	Interview
Student 1	0.075	0.034	0.044	0.048	0.011	0.016	0.020
Student 2	0.064	0.027	0.04	0.027	0.010	0.017	0.019
Student 3	0.072	0.036	0.035	0.055	0.011	0.011	0.020
Student 4	0.059	0.034	0.037	0.030	0.014	0.012	0.018
Student 5	0.072	0.033	0.039	0.060	0.009	0.014	0.019
Student 6	0.068	0.037	0.031	0.023	0.011	0.015	0.017
Student 7	0.059	0.030	0.034	0.024	0.010	0.016	0.018

ii. Ideal Solution $\begin{matrix} A^{*} & = {0.0 75, 0.0 37, 0.0 44, 0.0 60, \\ 0.0 14, 0.0 17, 0.0 20} \end{matrix}$ (9) Where, A^* is the ideal solution

Separation from ideal solution, $Si * = [Σ (vj * - {vij}^{2})] \frac{1}{2} for each row$ (10)

Separation distance is applied by Hwang and Yoon [28] in the TOPSIS method to obtain the closeness coefficient of each alternative [28]. The distance Si^* of each alternative, from A^* can be currently calculated by the area compensation method. Chu (2002) applied distance in the study of fuzzy TOPSIS approach.

Table 10 is obtained after applying area compensation method.

Table 10

Ideal solution A^* and separation from ideal solution (Si^*)

Student	10th Marks	12th Marks	Degree	Entrance	Institute	GD	Interview	Si^*
Student 1	0	0.000009	0	0.0004	0.000049	0.000001	0	0.021
Student 2	0.00012	0.0001	0.000036	0.0029	0.0029	0	0.000001	0.021
Student 3	0.000009	0.000001	0.000016	0.000081	0.000081	0.000036	0	0.058
Student 4	0.00025	0.000009	0.000081	0.0024	0.0024	0.000025	0.000004	0.059
Student 5	0.000009	0.000016	0.000049	0	0	0.000009	0.000001	0.016
Student 6	0.000049	0	0.000025	0.0038	0.0038	0.000004	0.000009	0.069
Student 7	0.00025	0.00004	0.0001	0.0012	0.0036	0.000001	0.000004	0.067

iii. Negative ideal solution $\begin{matrix} A & = {0 . 059, 0 . 027, 0 . 031, 0 . 023, 0 . 009, \\ 0.010, 0 . 011, 0 . 017} \end{matrix}$ (11) Where, A’ is the negative ideal solution.

Separation from negative ideal solution, ${Si}^{'} = [Σ (vj * - {vij}^{2})] \frac{1}{2}$ (12)

Separation distance is applied by Hwang and Yoon [28] in the TOPSIS method to obtain the closeness coefficient of each alternative [28]. The distance Si’ of each alternative, from A’ can be currently calculated by the area compensation method. Chu (2002) applied distance in the study of fuzzy TOPSIS approach.

Table 11 is obtained after applying area compensation method.

Table 11

Negative ideal solution and separation from negative ideal solution Si’

Student	10th Marks	12th Marks	Degree	Entrance	Institute	GD	Interview	Si’
Student 1	0.00078	0.00012	0.00048	0.00176	0.000004	0.000081	0.000025	0.057
Student 2	0.000081	0	0.00025	0.00006	0.000004	0.0001	0.00001	0.021
Student 3	0.00048	0.00019	0.000049	0.0028	0.000009	0	0.000036	0.065
Student 4	0	0.00012	0.0001	0.00016	0.000081	0.000001	0.000004	0.059
Student 5	0.00048	0.000081	0.0001	0.0038	0.000004	0.000025	0.000016	0.057
Student 6	0.00022	0.00028	0	0	0.000016	0.000025	0	0.012
Student 7	0	0.000025	0.000025	0.000004	0.000004	0.00006	0.000014	0.023

iv. Relative closeness to the ideal solution $C i^{*} = {Si}^{'} / (S i^{*} + {Si}^{'}) where 0 < C i^{*} < 1$

Relative closeness of Separation distance by Hwang & Yoon [24].

The relative closeness, Ci^* is the value that lies between 0 and 1. The student’s alternative that is closest to 1 will be given more priority compared to the alternatives which are farthest from 1. The results from Table 12 show that the relative closeness for Student 5 is 0.78 which is much closer to 1 compared to the other students. Hence, this student can be chosen as the best student alternative for the college, followed by Student 1, Student 3, Student 2, Student 7, Student 6, and Student 4.

Table 12

Relative closeness to ideal solution

Student	Ci^*
Student 1	0.73
Student 2	0.5
Student 3	0.52
Student 4	0.14
Student 5	0.78
Student 6	0.15
Student 7	0.25

4 Discussion

This study provides valuable insights on managing the strenuous process of admitting potential candidates to institutions of higher learning. There are multiple criteria involved in deciding whether a candidate should be selected or not. The integration of fuzzy AHP and TOPSIS methods offers a reliable and bias-free framework for student selection. Based on the weight vector obtained from the analysis, the criteria for student selection can be redefined. The results of this study provides the following selection framework.

The tenth-grade marks criterion (0.18) which has the highest priority weight is selected followed by the entrance test score (0.11), degree marks (0.10), twelfth grade marks (0.09), interview marks (0.05), group discussion marks (0.04), and institute test marks (0.03) using fuzzy AHP. The output weightage of each criterion obtained from the de-fuzzification stage of fuzzy AHP is converted to the normalized weight and given as the input to the TOPSIS calculation. The fuzzy TOPSIS method is easy to understand and can be used to rank the alternatives effectively. The TOPSIS method identifies the best alternative solution for the student selection by calculating the relative closeness from the positive ideal solution. The results for this study further show the Student 5 (0.78) is the best alternative for the selection in the admission process followed by Student 1, Student 3, Student 2, Student 7, Student 6, and Student 4.

Fuzzy AHP and fuzzy TOPSIS methods are both appropriate for the selection of students or other multi-criteria decision-making problems of the organization. Other methods in comparison with the combined use of these methods have weaknesses such as difficulty in properly reflecting the qualitative judgements of the experts, issues with identifying and weighting selection criteria [4]. Fuzzy AHP and TOPSIS methods have been proven to be effective in multi criteria decision making [29 –31]. The results of this study further bolsters it by showing that the integrated Fuzzy AHP and TOPSIS is a very efficient and reliable method for the selection of students. The AHP method reports sophisticated multi criteria problems into a hierarchy. Subjective perceptions and statements can be handled by integrating the Fuzzy numbers with the AHP method which account for the linguistic evaluation. Using the TOPSIS method, the best solution can be identified.

In TOPSIS, there is no pairwise comparison. It is based on distance which assumes that there is an ideal and non-ideal solution. The method aims to find the shortest distance to the positive ideal solution. It is based on linear programming and works with criteria weights. Weights are normally obtained using the de-fuzzification stage in fuzzy AHP that determines subjective criteria. In fuzzy AHP, decision-makers make pairwise comparisons for the criteria and alternatives under each criterion. Subsequently, these comparisons are integrated and the decision-makers’ pairwise comparison values are transformed into TFNs. The best alternative is determined based on the combination of priority weights of criteria and alternatives. According to the fuzzy AHP, the best criterion for student selection is tenth grade marks, and the ranking order of the criteria is tenth grade marks >entrance test score >degree >twelfth grade marks >interview marks >group discussion marks >institute test.

Integration of fuzzy AHP and TOPSIS methods can be successfully used in many areas. The results of this study further extend its applications to resolving the admission criteria-related issues in universities. This method offers an efficient, bias free and transparent framework to the universities that will also cut down on their time and resources used in the decision making related to admission criteria.

5 Conclusion, limitations and suggestion for future research

Multi-criteria decision making involves many challenges. Methods like Fuzzy AHP and TOPSIS offer solutions that are effective and sensible. This paper applies the Fuzzy AHP and TOPSIS method in improving the student selection process in universities. The student selection criteria obtained were tenth grade marks, twelfth grade marks, degree marks, entrance test score, institute test, group discussion marks, and interview marks. These criteria were evaluated and ranked based on the priority for selecting the most appropriate students in the admission process of a college. Both fuzzy AHP and TOPSIS were used with the objective of selecting students, however, integrating these methods offers a better solution.

Despite the advantages, these two methods do have some limitations. The most appropriate method has to be chosen based on the problem. When comparing the two methods with respect to the number of computations, fuzzy AHP requires more complex computations than TOPSIS. The pairwise comparisons for criteria and alternatives are made in fuzzy AHP, while no pairwise comparison is done in TOPSIS.

Future research could further study the use of Fuzzy AHP and TOPSIS beyond the education sector, such as advanced logistics for IR4.0 [32], online retailing [33], financial planning and insurance [34], textile and fashion [35], SME and entrepreneurship [36], and recruitment industries [37]; where multi-criteria decision making are becoming increasing prevalent and complex. A software program, if designed based on the fuzzy AHP and TOPSIS logic, would help automate the decision-making process and can be effectively applied in various fields of multi-criteria decision-making.

Footnotes

Acknowledgments

We thank Kannur University, India for providing the necessary resources to carry out this research.

Author contributions

CONCEPTION: Nisa James and Vijay Victor

METHODOLOGY: Nisa James, Robert Jeyakumar Nathan and Swetha Loganathan

DATA COLLECTION: Nisa James

INTERPRETATION OR ANALYSIS OF DATA: Nisa James, Vijay Victor and Robert Jeyakumar Nathan

PREPARATION OF MANUSCRIPT: Nisa James, Swetha Loganathan and Vijay Victor

REVISION OF IMPORTANT INTELLECTUAL CONTENT: Nisa James, Robert Jeyakumar Nathan, Vijay Victor, Swetha Loganathan and Poh Kiat Ng

SUPERVISION: Robert Jeyakumar Nathan and Poh Kiat Ng

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