Fuzzy Analytic Hierarchy Process (FAHP) in Analyzing Mathematical E-Learning Success Factors

First Author\[1\]<sup>\*</sup>, Second Author<sup>2</sup>  
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*<sup>1</sup>First affiliation, City and Country (Please do not type or edit anything here, until final camera-ready paper submission)*

*<sup>2</sup>Second affiliation, City and Country (Please do not type or edit anything here, until final camera-ready paper submission)*

<table>
<tbody>
<tr class="odd">
<td>ARTICLE INFO</td>
<td></td>
<td>ABSTRACT</td>
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<tr class="even">
<td><p><em>Article history:</em></p>
<p>Received</p>
<p>Revised</p>
<p>Accepted</p>
<p>Online first</p>
<p>Published 1 March 2024</p></td>
<td></td>
<td>E-learning approaches have tremendously become a prominent method in educational institution in Malaysia since the covid-19 outbreak. Moreover, the rapid growth in educational technologies with new software and high-end hardware has provided better accessibility for e-learning. For conducting an efficient and a conducive e-learning environment, the factors and sub-factors that leads towards e-learning success must be recognized and identified. Therefore, this research aims to rank the important factors and sub-factors influencing the success of e-learning in Mathematics from lecturers’ perspectives. Fuzzy Analytic Hierarchy Process (FAHP) is used in analyzing the data collected. The result of this study shows that among the four chosen success factors, the Quality of the Infrastructure and System is the most important factor (0.3876), followed by Characteristics of Students towards e-learning (0.2428), Quality of Design and Courses (0.1942) and lastly the Characteristics of Lecturers Towards e-learning (0.1753). The top four ranking out of twelve e-learning success sub-factors are Design and User Interface System (0.1447), Students’ Attitude Towards e-Learning (0.1232), Understanding the Use of Infrastructure (0.1218) and Level of Product Reliability (0.1211). This finding may help the e-learning process to be more effective not only for schools and universities but also for corporate and business sector especially in global training programs. For future study, the students’ perspective towards e-learning success should also be considered to have a complete view of success factors towards e-leaning.</td>
</tr>
<tr class="odd">
<td><p><em>Keywords:</em></p>
<p>E-Learning</p>
<p>Fuzzy Analytic Hierarchy Process</p>
<p>Success factors</p>
<p>Success sub-factors</p>
<p><em>DOI:</em></p>
<p>10.24191/jcrinn.v9i1</p></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

# 1\. Introduction

E-learning is a way of delivering lessons solely online without having physical interactions between educators and students. Nowadays, the learning and teaching approach used by students and teachers has changed significantly as accrued from newly advanced technologies in educational institutions (Hooshyar et al., 2020). The online platform’s dominance, on the other hand, creates a distinctive learning environment. The most exciting feature of e-learning is the capability to vary the time and location of educational engagement. Moreover, e-learning also allows students to get information from various sources in various formats to easily access their learning contents, which takes advantage of all media attributes (Anderson, 2000). According to Nguyen (2015), e-learning improves student learning while utilising limited resources, particularly in higher education. E-learning allows students to enrol in a course without having to attend a classroom. Students can also take a course from the convenience of their own home or somewhere else. Furthermore, because e-learning appears to have the ability to provide more flexible access to content and teaching material, it has gained in popularity. It is seen to gather and deliver learning content in an organised way, in addition to enhance the student-instructor ratio while achieving the same degree of learning outcomes as face-to-face learning (Bakia et al., 2012).

The Novel Coronavirus started in 2020 has dawned on new ways of teaching and learning process. Educational institutions around the world have been using e-learning platforms to continue the teaching and learning process. The new educational paradigm is based on an altered educational model, with e-learning at its core. It is no longer uncommon for educators and students around the world to rely on digital learning. Due to this new technology, many educational institutions have had to adapt to an unorthodox way of teaching and learning. Hence, it is very important to analyse the many aspects affecting the implementation of e-learning (Anggrainingsih et al., 2018). Some educational institutions have already returned to normal face-to-face learning system by the middle of 2022, but some still using e-learning and blended e-learning. Hence, it is very important to analyse the many aspects affecting the implementation of e-learning.

This study aims to analyse the mathematical e-learning success factors and to rank the sub-factors from lectures’ perspectives as he or she has a great experience an extensive involvement in implementing e-learning classes. Everybody knows that mathematics is a subjects that requires students to intensively understand and do practices on mathematical problems regularly. It is very challenging for teachers as well as for students to go through the process of teaching and learning successfully. Some students will be excel in mathematics while some will not. Therefore, it is necessary to have a study on the factors contributing to success in mathematics.

# 2\. LITERATURE REVIEW

Many studies have been conducted on evaluating factors affecting e-learning success. Mehregan (2011) introduced a new method to the e-learning system evaluation by defining and ranking the initial e-learning key success factors (CSFs), or enablers, which universities and educational institutions should focus on. The outcome of such performance appraisal then serves as an instructive resource for the development of an e-learning systems plan. It proposed a comprehensive strategy to evaluate e-learning programs using the CSF methodology and the Fuzzy AHP method. The AHP is made up of seven main CSFs: namely, characteristics of the instructors, characteristics of the students, quality of the content, quality of the information technology, the interaction between participants, support from educational institutions, and knowledge management. Each CSF is followed by its own sub-categories, respectively. The result indicates that student characteristics and IT quality are more important than instructor characteristics, content quality, support from educational institutions, participation, interaction, and knowledge management. Besides, as for the sub-categories, the most important things to focus on are students’ computer skills, motivation, and financial support from educational institutions.

In addition, an experiment was also conducted at Sebelas Maret University using the Fuzzy AHP approach to determine the rank of priority of many aspects that influence the effectiveness of e-learning. The important success factors for e-learning at that university, based on the lecturers’ and students’ point of views, are university involvement, quality of infrastructure and system, quality of design and courses, student’s characteristics and lecturer’s characteristics are the most critical success criteria for e-learning at that university (Anggrainingsih et al., 2018). Each of the aforementioned factors has four sub-factors, respectively. The study indicates that university involvement is the most critical factor of e-learning success, and the most important sub-factor is university policy (financial and regulatory policy). Thus, the e-learning management at Sebelas Maret University can use this finding and come up with a plan for a successful implementation of e-learning by taking these factors into account.

Furthermore, the success of e-learning also depends on the website’s level of quality. Consequently, a study has been done to identify the most important influences on online learning website success. Two types of questionnaires were employed in this investigation. There is an Analytical Hierarchy Process questionnaire included in the mix. The literature uncovered these factors, and the Fuzzy AHP approach was used to prioritise them (Nilashi & Janahmadi, 2012). The findings of this study show that the quality of website, the content of website, as well as the design of the website contributed positively to the success of e-learning. Furthermore, this study also demonstrates that website quality has the most favourable effect on online learners’ perception of an e-learning website.

# 3\. methodology

**This section explains in detail the data collection method and the essential step in applying FAHP in analysing the data.**

**3.1 Data Collection**

Data was collected by distributing questionnaires to a senior Mathematical lecturer in UiTM Perlis Branch Arau Campus. Several prior research based on the aspects that contributed to efficacy of e-learning and success have been analysed in order to determine the critical factors that influence the success of e-learning. Four criteria chosen namely quality of infrastructure and system, quality of design and courses, characteristics of students toward e-learning, as well as characteristics of lecturers toward e-learning were used in the study. Each factor was later defined by the three sub-factors that correspond to it. The sub-factors for the quality of infrastructure and system are identified as level of product reliability, understanding the use of infrastructure, as well as design and user interface system. Next, the sub-factors under the quality of design and courses are quality, relevance, and completeness of the content. From the students’ characteristics factor, the sub-factors that were considered are expertise in using computers and the internet as well as their attitudes toward e-learning. Lastly, as for the lecturers’ characteristics, the sub-factors examined are attitudes toward students and e-learning and their timely responses.

**3.2 FAHP Model**

The followings are the details of implementing FAHP:

**Step 1**: Selecting an expert group for the decision-making process. The lecturers of Calculus subject at UiTM Perlis Branch Arau Campus are selected as the experts with substantial knowledge and experience in organising e-learning programmes.

**Step 2**: Compute the fuzzy triangular number. A pairwise comparison between the factors and sub-factors is conducted by the experts to determine the relative score. The Fuzzy AHP is a range of values that considers the decision-makers’ uncertainty in place of a crisp value. The pairwise comparison matrix, denoted by \(\text{C}_{\text{ij}}\ \)is represented by the following matrix:

|  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |     |
|  | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- |
|  | \[{\text{\ \ \ \ \ \ \ \ \ \ }\text{C}}_{\text{ij}}\text{\ }\text{=}\text{\ }\begin{matrix}
\text{C}_{\text{1}} \\
\text{C}_{\text{2}} \\
\text{⫶} \\
\text{C}_{\text{k}} \\
\end{matrix}\text{\ }\begin{bmatrix}
\text{f}_{\text{11}} & \text{f}_{\text{12}} & \text{⋯} & \text{f}_{\text{1k}} \\
\text{f}_{\text{21}} & \text{f}_{\text{22}} & \text{⋯} & \text{f}_{\text{2k}} \\
\text{⫶} & \text{⫶} & \text{⋯} & \text{⫶} \\
\text{f}_{\text{k1}} & \text{f}_{\text{k2}} & \text{⋯} & \text{f}_{\text{kk}} \\
\end{bmatrix}\] | (1) |

for i = 1, 2 …, k and j = 1, 2 …, k, where f<sub>11</sub>, f<sub>12</sub>, … f<sub>1k</sub> is a triangular fuzzy number. Then, Table 1 will be used as a reference to compare the two factors for the decision maker to consider.

Table 1. Linguistic Variable of Pairwise Comparison Matrix for Factors and Sub-Factors

| Saaty Scale | Linguistic Variable            | Fuzzy Triangular Scale |
| ----------- | ------------------------------ | ---------------------- |
| 1           | Equally important (EI)         | (1,1,1)                |
| 3           | Weakly More Important (WI)     | (2,3,4)                |
| 5           | Strongly More Important (SI)   | (4,5,6)                |
| 7           | Very Strongly Important (VSI)  | (6,7,8)                |
| 9           | Absolutely More Important (AI) | (9,9,9)                |
| 2           |                                | (1,2,3)                |
| 4           | Intermediate Values            | (3,4,5)                |
| 6           |                                | (5,6,7)                |
| 8           |                                | (7,8,9)                |

**Step 3** : Calculate the consistency ratio (CR) to assure the consistency judgement of expert(s) by using the formula:

|  |  |     |
|  |  | --- |
|  |  | (2) |
|  |  | (3) |

where

: the largest eigenvalue of the comparison matrix

N: the dimension of matrix or number of criteria

CI: consistency index

RI: Random inconsistency index

CR: consistency Ratio

If CR is equal or less than 0.1, then the comparison is acceptable. When CR is greater 0.1, the value is indicative of inconsistent judgment.

**Step 4**: Calculate the geometry mean of fuzzy comparison value by using Eq. (2). \({\widetilde{\text{r}}}_{\text{i}}\) in Eq. (2) indicates as triangular values.

|  |                                                                                                                                                                |     |
|  | -------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- |
|  | \({\widetilde{\text{r}}}_{\text{i}}\text{=}\left( \prod_{}^{}{\widetilde{\text{d}}}_{\text{ij}} \right)^{\frac{\text{1}}{\text{n}}}\)*,*\(\text{i=}\)*1,2,…,n* | (4) |

**Step 5**: Calculate fuzzy weights. The vector sums of each geometric mean will be computed to determine the fuzzy weights, \({\widetilde{\text{w}}}_{\text{i}}\). Following that, the summation vector's reciprocal is computed, and the fuzzy triangular number is sorted in ascending order. To obtain the fuzzy weight, geometric averages must be multiplied by this reverse vector.

|  |                                                                                                                                                                                                                                                                                                                                                                                                          |     |
|  | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- |
|  | \({\widetilde{\text{w}}}_{\text{i}}\text{\ }{\text{=\ }{\widetilde{\text{r}}}_{\text{i}}\text{⊗}\left( {\widetilde{\text{r}}}_{\text{i}}\text{⊕}{\widetilde{\text{r}}}_{\text{i}}\text{⊕⋯⊕}{\widetilde{\text{r}}}_{\text{i}} \right)}^{\text{-1}}\) \(\text{\ \ \ \ \ \ \ \ \ \ \ \ }\text{=}\text{\ }\text{(}\text{lw}_{\text{i}}\text{\ ,\ m}\text{w}_{\text{i}}\text{,u}\text{w}_{\text{i}}\text{)}\) | (5) |

**Step 6**: Defuzzification comes next. The fuzzy weights must be defuzzified using the centre of area method because they are still in a fuzzy triangular number. This method was proposed by Chou and Chang (2008) using the equation stated below:

|  |                                                                                                                                                        |     |
|  | ------------------------------------------------------------------------------------------------------------------------------------------------------ | --- |
|  | \(\text{M}_{\text{i}}\) \(\text{=}\frac{\text{lw}_{\text{I}}\text{\ +\ m}\text{w}_{\text{i}}\text{+u}\text{w}_{\text{I}}\text{\ \ \ \ \ }}{\text{3}}\) | (6) |

**Step 7**: The final step is to normalise the defuzzification result using the following equation:

|  |                                                                                                                                            |     |
|  | ------------------------------------------------------------------------------------------------------------------------------------------ | --- |
|  | \(\text{N}_{\text{i}}\text{=}\frac{\text{\ \ \ }\text{M}_{\text{i}}\text{\ \ \ \ \ \ }}{\sum_{\text{i=1}}^{\text{n}}\text{M}_{\text{i}}}\) | (7) |

where n represents as the total number of \(\text{M}_{\text{i}}\)*.**The criterion with the highest score will be considered as the most important factor and sub-factor based on the results.***

# 4\. RESULT AND DISCUSSION

The lecturer’s evaluation (expert’s evaluation) on the four criteria used in the study is summarized in Table 2.

Table 2. Pairwise Comparison Matrix for Decision Criteria of Factors Affecting to E-Learning Success from Lecturer’s Perspective

> ![](66308dafe6c47_media/media/image4.emf)

In this study,gives the consistency ratio of 0.02246 (\<0.1) and it indicates that the evaluation of the expert is consistent.

Table 3 indicates the triangular fuzzy matrix and normalised weights for each criterion of factors influencing e-learning success from lecturers’ perspectives. Based on the table, **with the highest score of 0.3876, the quality of infrastructure and system is the most significant factor determining the success of e-learning, followed by characteristics of students toward e-learning with a score of 0.2428. Next is the quality of design and courses with a score of 0.1942, followed by characteristics of lecturers toward e-learning with a score of 0.1753.**

**Table 3. Triangular Fuzzy Matrix and Normalised Weight of E-Learning Success Factors from Lecturers’ Perspectives**

![](66308dafe6c47_media/media/image6.emf)

**Table 4 displays the triangular fuzzy matrix and normalised weight for the sub-factors that affect the e-learning success. The normalised weights were then used to compute the overall weightage based on the factors’ weights.**

**Table 4. Triangular Fuzzy Matrix and Normalised Weight of E-Learning Success Sub-Factors from Lecturers’ Perspectives**

![](66308dafe6c47_media/media/image7.emf)

**The overall weightage of sub-factors per each criterion are shown in Table 5, Table 6, Table 7 and Table 8.**

**Table 5. Overall Weightage of Sub-Factors under Quality of Infrastructure and System Criteria**

![](66308dafe6c47_media/media/image8.emf)

**Table 6. Overall Weightage of Sub-Factors under Characteristics of Students toward E-Learning Criteria**

![](66308dafe6c47_media/media/image9.emf)

**Table 7. Overall Weightage of Sub-Factors under Quality of Design and Courses Criteria**

![](66308dafe6c47_media/media/image10.emf)

**Table 8. Overall Weightage of Sub-Factors under Characteristics of Lecturers toward E-Learning Criteria**

![](66308dafe6c47_media/media/image11.emf)

**From the overall weightage per criteria, the sub-factors affecting to e-learning success were then ranked. From the results, it indicates that design and user interface system is the most important sub-factor in determining the success of e-learning with a score of 0.1447. With a score of 0.1232, the second most important sub-factor is students’ attitudes toward e-learning. Next, understanding the use of infrastructure and system was ranked third with a score of 0.1218. Level of product reliability was ranked fourth with a score of 0.1211followed by the fifth ranking, which is completeness of content with a score of 0.0773. Expertise is using computer was ranked sixth with a score of 0.0673, and lecturers’ timely response was ranked seventh with a score of 0.0664. Next, the eighth and ninth most important sub-factors affecting to e-learning are relevance of content as well as lecturers’ attitudes toward e-learning with each scored 0.0620 and 0.0577, respectively. Quality of content was ranked tenth with a score of 0.0549. Finally, expertise in using internet was ranked eleventh with a score of 0.0523 and lecturers’ attitudes toward students were placed in the twelfth ranking with a score of 0.0512. Table 8 illustrates the ranking of e-learning success sub-factors from the lecturers’ perspectives.**

**Table 9. Sub-Factors’ Ranking**

![](66308dafe6c47_media/media/image12.emf)

# 5\. conclusion

E-learning platforms are widely used by educational institutions around the world for teaching and learning activities. As a result, it is important to consider the factors that contribute to the success of e-learning. There is no reason to be apprehensive about deploying e-learning if all of the success factors and sub-factors can be implemented well. As a result of examining the factors and sub-factors affecting e-learning success, the e-learning process will be successful and yield such outstanding results. It can help students and instructors comprehend technology more deeply while also saving time and energy, in keeping with the current state of IT.

It is recommended to successfully manage the important success aspects of e-learning by utilising existing resources such as technology and software. Furthermore, suitable training of the e-learning system should be offered to students and lecturers so that they can gain appropriate learning and teaching skills. A lack of training might impede proper utilisation of technology and limit the potential benefits that can be obtained. Professional training can help lecturers become more capable of utilising virtual classroom tools.

In addition, professional development aids in the expansion of multimedia in e-learning settings. The lecturer will be able to manage a range of technical applications, such as quizzes and tests, the discussion board, and grading, for effective course management. The full benefits of the system can be realised by students with adequate instruction. Lecturers can also foster a favourable learning atmosphere by utilising instructional notes, films, and other types of multimedia. While users may share their e-learning system experiences on social media sites like Facebook, Twitter, and YouTube, lecturers can use social media to improve their interactions with students.

For future research, it is recommended to analyse the success factors of e-learning from the perspective of students as well because the students are also involved in the e-learning system and have experience in attending e-learning classes.

**REFERENCES**

Anderson, T. (2000). Teaching in an Online context. *Theory and Practice of Online Learning*, 273–294. http://www.atl.ualberta.ca/cmcAnggrainingsih, R., Umam, M. Z., & Setiadi, H. (2018). Determining e-learning success factor in higher education based on user perspective using Fuzzy AHP. *MATEC Web of Conferences*, *154*, 0–5. https://doi.org/10.1051/matecconf/201815403011Bakia, M., Shear, L., Toyama, Y., & Lasseter, A. (2012). Understanding the Implications of Online Learning for Educational Productivity. *Educational Technology*, 1–75. http://ctl.sri.com/publications/displayPublication.jsp?ID=913Chou, S. W., & Chang, Y. C. (2008). The implementation factors that influence the ERP (enterprise resource planning) benefits. *Decision Support Systems*, *46*(1), 149–157. https://doi.org/10.1016/j.dss.2008.06.003Hooshyar, D., Pedaste, M., & Yang, Y. (2020). Mining educational data to predict students’ performance through procrastination behavior. *Entropy*, *22*(1), 12. https://doi.org/10.3390/e22010012Mehregan, M. (2011). Application of Fuzzy Analytic Hierarchy Process in Ranking Modern Educational Systems’ Success Criteria. *International Journal of E-Education, e-Business, e-Management and e-Learning*, *1*(4). https://doi.org/10.7763/ijeeee.2011.v1.49Nguyen, T. (2015). The Effectiveness of Online Learning: Beyond No Significant Difference and Future Horizons. *MERLOT Journal of Online Learning and Teaching*, *11*(2), 309–319.

1.  <sup>\*</sup> Corresponding author. *E-mail address*: <donottypehere@email.com> (Please do not type or edit anything here, our editors will do the work for you)