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

**ABSTRACT**

E-learning approaches have emerged as a prominent method in educational institutions in Malaysia since the Covid-19 outbreak. Moreover, rapid growth in educational technologies with the introduction of new software and high-end hardware has provided better accessibility for e-learning. Toensure efficient and 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, Quality of 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 Characteristics of Lecturers Towards e-learning (0.1753). The top four out of 12 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 to improve the effectiveness of e-learning process to not only for schools and universities, but also for corporate and business sectors especially for global training programs. For futurestudies, students’ perspective towards e-learning success should also be considered to have a complete view of e-learning success factors.

*Keywords:* E-learning, Fuzzy Analytic Hierarchy Process, Success factors, Success sub-factors

# 1\. Introduction

E-learning is a way of delivering lessons solely online without having and 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). Online platform’s dominancecreates a distinctive learning environment and the most exciting feature of e-learning is its 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 students learning process while utilising limited resources, particularly in higher education. E-learning allows students to enrol in a course without having to physically attend the subject. Students can also take courses from the convenience of their own home or somewhere else. Furthermore, e-learning has gained in popularity due to its ability to provide more flexible access to contents and teaching materials. It is ableto gather and deliver learning contents in an organised way, in addition to enhance student-instructor ratio while achieving the same degree of learning outcomes as face-to-face learning (Bakia et al., 2012).

The Novel Coronavirus that started in 2019 (Covid-19)has dawned on new ways of teaching and learning educational institutions around the world started to use e-learning platforms to conduct the said 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 to rely on digital learning. Due to this new technology, many educational institutions 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 use e-learning and blended e-learning methods. Hence, it is very important to analyse the many aspects affecting the implementation of e-learning.

This study aims to analyse the e-learning success factors for Mathematics subject and rank the relevant sub-factors from lecturers’ perspectives. It is well-known that this subject requires students to intensively understand and solve questions problems regularly. The subject’s teaching and learning process is very challenging for both teachers and only a handful ofstudents excel in Mathematics. Therefore, it is necessary to study the factors contributing to success in Mathematics.

# 2\. LITERATURE REVIEW

Many studies have been conducted to evaluate factors affecting e-learning success. Mehregan (2011) introduced a new method to evaluate the e-learning system 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 instructors, characteristics of students, quality of content, quality of information technology, interaction between participants, support from educational institutions, and knowledge management. Each CSF is followed by its own sub-categories. The result indicates that students’ characteristics and IT quality are more important than instructors’ characteristics, content quality, support from educational institutions, participants, interaction, and knowledge management. As for the sub-categories, the most important things to focus on are students’ computer skills, motivation, and financial support from educational institutions.

Another study conducted an experiment in a higher learning institution Sebelas Maret University using the Fuzzy AHP approach to determine the rank of 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. 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). Higher learning institutions can use this finding to come up with a plan for successful implementation of e-learning by taking these factors into account.

Furthermore, e-learning success also depends on a website’s level of quality. A study has been done to identify the most important factors influencing online learning website success. Two types of questionnaires were employed in this investigation with one of them being Analytical Hierarchy Process questionnaire. 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 quality of website, content of website, as well as design of 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** **on** **data collection method and** **essential steps used to** **apply FAHP in analysing the data.**

**3.1 Data Collection**

Data was collected by distributing questionnaires to a senior Mathematics lecturer teaching in UiTM Perlis Branch Arau Campus. Based on literature review done, the study has determined four critical factors that influence e-learning success. namely quality of infrastructure and system, quality of design and courses, characteristics of students toward e-learning, and characteristics of lecturers toward e-learning. Each factor was later defined by three sub-factors that correspond to it. The sub-factors for quality of infrastructure and system are level of product reliability, understanding the use of infrastructure, as well as design and user interface system. Next, sub-factors for quality of design and courses are quality, relevance, and completeness of the content. For students’ characteristics factor, the sub-factors include expertise in using computers and the internet as well as their attitudes toward e-learning. Lastly, for lecturers’ characteristics, the sub-factors are attitudes toward students and e-learning and their timely responses.

**3.2 FAHP Model**

Provided are the steps used to of implement FAHP:

**Step 1**: Selecti an expert group for the decision-making process. The lecturers of Calculus subject at UiTM Perlis Branch Arau Campus are selected as the experts who possess 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. 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

: largest eigenvalue of the comparison matrix

N: 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.

|  |                                                                                                                                                                       |     |
|  | --------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --- |
|  | \({\widetilde{\text{r}}}_{\text{i}}\text{=}\left( \prod_{}^{}{\widetilde{\text{d}}}_{\text{ij}} \right)^{\frac{\text{1}}{\text{n}}}\)*,*\(\text{i}\text{=}\)*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**: Perform defuzzification. 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 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 lecturers’ evaluation (experts’ 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 E-Learning Success from Lecturers’ Perspective

> ![](6672974b818e1_media/media/image4.emf)

In this study,resulting to consistency ratio of 0.02246 (\<0.1) This indicates that the experts’ evaluation 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, quality of **infrastructure and system is the most significant factor** **followed by characteristics of students** **(0.2428), quality of design and courses** **( 0.1942),** **and characteristics of lecturers** **(0.1753).**

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

![](6672974b818e1_media/media/image6.emf)

**Table 4 displays the triangular fuzzy matrix and normalised weight for the sub-factors that affect** **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**

![](6672974b818e1_media/media/image7.emf)

**The overall weightage of sub-factors** **for** **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**

![](6672974b818e1_media/media/image8.emf)

**  
**

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

![](6672974b818e1_media/media/image9.emf)

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

![](6672974b818e1_media/media/image10.emf)

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

![](6672974b818e1_media/media/image11.emf)

**From the overall weightage** **for each** **criteria, the sub-factors affecting** **e-learning success were then ranked.** **The results indicate 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** **followed by** **Students’** **Attitudes** **toward** **E-learning** **(0.1232),Understanding the** **Use** **of** **Infrastructure** **and** **System** **( 0.1218),** **Level of** **Product** **Reliability** **( 0.1211),** **Completeness of Content** **( 0.0773),. Expertise** **in** **Using** **Computer** **(0.0673), and** **Lecturers’** **Timely** **Response** **(0.0664).** **The rest of the sub-factors are** **Relevance** **of** **Content (0.0620),** **Lecturers’** **Attitudes** **toward** **E-learning** **(0.0577). Quality of** **Content** **(0.0549),.** **Expertise in** **Using** **Internet** **(0.0523) and finally** **Lecturers’** **Attitudes** **toward** **Students** **(0.0512). Table 8 illustrates the ranking of e-learning success sub-factors from the lecturers’ perspectives.**

**  
**

**Table 9. Sub-Factors’ Ranking**

![](6672974b818e1_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 methodology 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 properly manage the important aspects of e-learning by utilising relevant resources such as technology and software. Furthermore, suitable e-learning system trainings should be offered to students and lecturers so that they can gain appropriate 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. Lecturer will be able to manage a range of technical applications for effective course management including online, quizzes and tests, discussion board, and grading,. 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 because they are directly involved in the e-learning system and have experience in attending e-learning classes.

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