EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1959-1981
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Multi-Criteria Decision Making for Determining
Critical Success Factors in Achieving Top University

Graduation

Peter Nga Assi1,∗, Edet Effiong Bassey2, Jeremiah Ugeh Atsu1,3,
Ekpenyong Effiong Ibok4, Emeka Samuel Nnaji5, Bernard Azinu Ugbe1,
Samuel Chukwudike Amadigwe2

1 Department of Mathematics, Faculty of Physical Sciences, University of Calabar, Calabar,
Cross River State, Nigeria
2 Department of Statistics, Faculty of Physical Sciences, University of Calabar, Calabar,
Cross River State, Nigeria
3 Department of Mathematics, Faculty of Physical Sciences, Cross River State University of
Calabar, Calabar, Cross River State, Nigeria
4 Department of Science Education, Faculty of Vocational and Science Education,
University of Calabar, Calabar, Cross River State, Nigeria
5 Department of Educational Management, Faculty of Educational Foundation Studies,
University of Calabar, Calabar, Cross River State, Nigeria

Abstract. Within the context of students’ academic achievements, the aspect of “peak-performance”
among students in HEIs, still remain underexplored. Graduating with/at maximum achievement
is a desired goal for any student into HEI, and can never be less important. Admittedly, what
students do during school periods truly can help create the high grades border difference in respect
to maintaining and achieving their initial educational goal. This study seeks to identify and distill
from an academic perspective, key independent variables/factors called CSFs that are decisive and
significantly impact on the ability of students in HEIs to graduate with/at maximum achievement.
We gleaned a set of CSFs from the literature, then refined the list to 32 so that the final list reflects
broadly 6 main CSFs that relate to HEI studentship: - Behaviour towards academics, Hard work,
Organization, Peers and Socials, Skills, and Studying Style or Pattern. We then applied a Multi
Attributes Decision making tool, AHP on the set of CSFs. The AHP prioritizes a subset of these
CSFs as being most likely contributive variables to finishing top in university. The approach in this
study is unique and provides a strategic framework for decision-making. Prospective and current
students, parents, educators, academic institutions and the society at large who depend on these
graduates will find the study results cardinal.

2020 Mathematics Subject Classifications: 97M40, 90C29,90B50, 91B06. 97M70 97B40

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5149

Email addresses: orassipee@gmail.com (P. N. Assi), crayzstatistics93@gmail.com (E.E. Bassey),
atsujeremiah@gmail.com (J.U. Atsu), ibokekpenyong@yahoo.com (E.E. Ibok),
nnajiemeka88@yahoo.com (E.S. Nnaji), bernardugbe405@gmail.com (B. A. Ugbe),
samuel.c.amadigwe@gmail.com (S. C. Amadigwe)

https://www.ejpam.com 1959 © 2024 EJPAM All rights reserved.



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1960

Key Words and Phrases: Analytic Hierarchy Process, Critical Success Factors, Graduating at
maximum achievement, Pair-wise comparison matrix, Priority Vector

1. Introduction

Over the last decades, there has been a dramatic change in the higher education
environment globally. Educational privatization, online learning, and the rising cost of
education are some of the key elements that characterized this change [18]. Yet globally,
in today’s society, 8 out of every 10 (about 80 percent) high school graduates yearn to
attain post-secondary and higher levels of education [29]. Once in university, students’
likelihoods for academic ability to succeed top-performing can be different dynamically
and to a greater degree. Many of these students do not achieve their post-secondary ed-
ucational goals or least benefit at optimal levels from the actual university experience in
terms of their performance. Some students more than others are better prepared academ-
ically. That some students come out on top while others do not is indisputable.
According to [[33], [13]], only about a few students who begin their studies end up having
high grades and for postgraduate masters could also earn them qualifications for Ph.D.
admissions. There exist high grades achievement disparities among students in higher
education institution (HEIs). For every student into HEIs, graduating with/at maximum
achievement is a desired goal, and can never be less important or overemphasized. While
the emphasis in this study is not particular to a university, department or discipline, we
recognized that almost every HEI globally uses a slightly different grading system/policy.
In the light of this, we adopt a synchronized version that appears to cut-across all univer-
sities globally, as in Table 1. However, while really in school, many malleable factors could

Table 1: Basic grade system, [10]

Score Grouping Letter Grade Implication in this study

80− 100 A Desired decision

70− 79 B Undesired decision

50− 69 C Undesired decision

0− 49 F No student desire

be responsible for deviating from or not achieving this goal. Admittedly, what students do
during school hours- the activities they engage in and the company of friends they keep,
and their overall mindset towards academic practices and behaviors such as attendance,
studying and applying effort in class works and on homework and assignments truly can
help create the high grades border distinction in respect to maintaining and realizing their
initial educational goal [[20], [1], [8]]. So why do some students consistently top-perform
and improve faster than others? Graduating with/at maximum achievement in university
has never been more important. There is need to understand the variables that play out
critically in obtaining higher grades in universities. While the focus on students’ academic
success and performance keeps growing both in scope and dimensions, limited attention is



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1961

given to students in universities. Even more neglected is the grades nature of this unique
class of students. To find out why some students performed at maximum achievement
where others do not, we examined what these high-performing students have in common
and what others can do to maintain or improve their outcomes or grades. To this end, this
study seeks to unearth the critical success factors (CSFs) that are pivotal for a student to
have high grades. These CSFs reflect complete behavioral change and alignment that must
be fully demonstrated by students. They are niche areas for the already top-performing
students, and will serve as key focus areas for those who which to perform better or come
out on top in universities if put into practice effectively. There is need for improvement
planning for non-top-performing students,and will find these key elements useful for im-
plementing their improvement efforts, as [20] CSFs are planning tools. Every student
performance can be increased, as there are many different ways to improve. Therefore, we
take a cumulative and longitudinal view of what matters most to a student to earn a high
grade.
Till this moment, all similar publications are centered on the qualitative determination
of CSFs important for institutional success, and students’ academic success and perfor-
mance. However, our paper is different, both in terms of the problem definition and the
decision-making approach as it focuses quantitatively/mathematically on the CSFs crucial
for students getting high grades. This paper contributes to the literature by proposal of a
Multiple-criteria decision analysis (MCDA) or Multiple-criteria decision making (MCDM)
tool, the analytic hierarchy process (AHP) with which the attainment of “A” grade is
guaranteed for students in HEIs. This study seeks to unearth the CSFs that are pivotal
for a student to have high grades. The rest of the paper is organized as follows: After es-
tablishing the introductory points, section 2 describes a concise literature review of CSFs
necessary for institutions and students’ academic success and performance and highlights
the research gaps. In Section 3, we provide an overview of CSFs for graduating top of the
scale, while Section 4 demonstrates the use of AHP for graduating top in HEIs. Result,
analysis and discussion on findings are presented in Section 5, and Section 6 ends with
conclusions.

2. Literature Review

The concept of critical success factor (CSFs) was propounded by Daniel in 1961, and
made understandable by Rockart, who coined the term “Critical Success Factor” [1].
Daniel asserted that organizational planning information should focus on “success fac-
tors,” which he described as “key jobs that must be done exceedingly well for a company
to be successful” [[8], [21]]. In the literature there are several definitions of CSFs. For
example, Rockart, in presenting one of the most frequently cited definitions, uses ideas
from Daniel in defining CSFs as “the limited number of areas in which results, if they are
satisfactory, will ensure successful competitive performance for the organization”. Con-
sequently, he stresses, that these particular areas of activity should be constantly and
carefully managed by an enterprise if the business is to survive and flourish. Conversely,
underperformance in these core functions would prohibit the organization from achiev-



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1962

ing its strategic imperatives. Each factor requires multiple measures focused on results
[[24], [4], [7], [6]]. According to [22], CSFs are the few key areas where ‘things must go
right’ for the business to flourish and for the manager’s goals to be attained. A CSFs
analysis tries to identify a small number of goals or factors that, if they are reached or
successfully completed, will tend to predict or indicate success. Therefore, CSFs are the
activity areas that should receive constant and careful attention. The concept of CSFs
has been widely applied. CSFs differ among industries and for individual business within
a particular industry. This assertion came from a study by [21] in an attempt to define
CSFs that are crucial for implementing business intelligence (BI) systems in small and
medium enterprises (SMEs). [22] carried out a study on CSFs and they identified those
CSFs that are really significant in explaining the success of Project Management. In a
study to explore the CSFs of a New Product Development (NPD), [12] discovered that the
role of top Management is the most critical factor that leads to NPD success. [6] reviews
and proposes five CSFs for organizations managing high-rise (non-low cost) residential
buildings. Their results revealed that the CSFs can help to improve the organization’s
performance to understand the critical areas for successful management to meet the resi-
dents’ expectations for a harmonious communal living.
While CSFs analysis has often been used to elicit information requirements in systems
analysis and design, they could also be used to identify factors for institutional effective-
ness and improvement, and academic success factors necessary for graduating top of the
scale. Educationally, CSFs have been used as measures of institutional enhancement and
effectiveness. For example, [17] identifies through explorative investigations the critical
success factors (CSF) factors pertinent to school improvement under two commonly ob-
served existing models based on their missing links. [18] explored the CSFs involved in
creating a constituent relationship management (CRM) strategy that combines people,
processes, and technology in a higher education institution aim at creating a lifelong rela-
tionship across the multiplexity of the university constituents namely applicants, students,
alumni, employers, and companies. The study by [32] proposes a conceptual model rep-
resentation of the CSFs of total quality management (TQM) for implementation in HEIs
and their impacts on institutions performance. [4] reviewed CSFs responsible for gradu-
ate student’s success to determine the quality of the graduate program. They used CSFs
to develop strategies to create and maintain a community of connectedness for students,
and initiatives to halt attrition and sustain graduate student success. In academic perfor-
mance, CSFs for graduating top of the Scale are the same, though the degree, or extent of
satisfying them may vary from one individual to another. [33] provided an integrated view
of the causal effect of academic success by combining CSFs related to students’ academic
success with those factors identified in previous research studies. [13] determined that;
knowledge of IT, self-confidence, the particular university that the student graduated from,
and willingness to learn are the pivotal CSFs for employers when recruiting new employee
from among university graduates. Based on meta-analysis techniques of previous multiple
studies to determine CSFs that affect m-learning platforms in higher education settings, [2]
concludes that: - collaborations during studies, prospects of ubiquitous learning in space
and time, and user-friendly applications design are the most important perceived advan-



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1963

tages of m-learning. A study to understand the variables surrounding graduate faculty
perspectives of graduate student success, [7] discovers that faculty perceptions indicated;
variables of academics and self-responsibility to be critical to the success of graduate stu-
dents. An exploration of students’ descriptions of the personal resources that they use to
succeed while attempting to reach their goals as well as those personal characteristics or
obstacles that keep them from reaching their goals at universities is found in [29]. The
study discovered prominent themes supportive of student success including having a future
orientation, persistence, and strong executive functioning skills such as time management
and organization, were listed as CSFs for undergraduate students to reach their goals while
attending university.
From the aforementioned literature, Students’ academic achievements has remained pas-
sive, with no study being reported in particular on their top performance. While graduat-
ing top of the scale is a goal often cherished by many students pursuing higher education, it
is also commendable goal that demand strategic approach. However, this accomplishment
is often multifaceted and influenced by both qualitative and quantitative factors that may
not be immediately apparent. Graduating top of the scale is a decision-making problem
that demands making a trade-off between multiple tangible and intangible, and often con-
flicting criteria. It is not dependent on only one criterion, but involves taking a definite
number of alternative actions. Therefore, it is a classic example of a Multiple Attribute
Decision Making (MADM), a class of MCDA/MCDM. Since the past decades, MCDM
has emerged as a subdiscipline or branch of operations research that focuses on creating
mathematical and computational tools for supporting decision-making in subjective eval-
uation of a finite number of decision alternatives under a finitenumber of performance
criteria [[16], [19], [14]]. To effectively navigate and evaluate/prioritize these criteria, the
Analytical Hierarchy Process (AHP) which is one of the MADM tool is employed in this
study to assists individuals in evaluating and comparing different criteria and alternatives
in a hierarchical fashion. According to [31], the AHP approach simplifies a complex multi
decision making process, makes it more systematic, and introduces transparency while
saving cost and resources. The AHP can be applied to a multitude of decision-making
problems involving a finite number of alternatives [5]. In the context of academic per-
formance and graduating with/at maximum achievement, the AHP frame work can be
valuable approach to identify, assess and prioritize the CSFs (criteria) that contribute to
this goal. By using AHP, individuals can assign relative weights to these factors, enabling
a more systematic and data-driven approach to decision making and goal setting.

3. CSFs for Graduating with Maximum Achievement

A number of factors have been identified in some research literature as being impor-
tant for students to achieve success and high performance in HEIs. To make informed
decisions, a set of diverse CSFs were gleaned from the literature. For example, factors
responsible for students’ academic success are found in [[4], [29], [7], [3], [23]], and those
CSFs predictive for
students’ high academic performance can be found in [ [3], [33], [10], [13], [15]].



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1964

The identified CSFs were refined and synthesized to thirty-two (32) factors and then
grouped into six different CFSs categories so that the final list reflects broadly six main
CSFs that relate to a HEI studentship: - Behaviour towards academics, Hard work, Orga-
nization, Peers and Socials, Skills, and Studying Style or Pattern. These CSFs which are
products from the above research studies intersect with different academic departments
and disciplines as our focus is students-based or oriented. In other words, we are concern
with identifying what factors can help a student in any department or discipline to attain
Grade ‘A’ in table 1, in any university the world over. The results of this study is not
restricted to departments or disciplines in universities or countries, but purports a global
academic domain.
In the jargons of MADM, the category names represent the main-attributes (CSFs), and
the factors grouped under particular category are called sub-attributes (CSFs). Figure 2
depicts the identified self-explanatory main-CSFs and sub-CSFs considered for decision
making for graduating with/at maximum achievement, as summarized due to space lim-
itations. As attributes play very key role in the MCDM decision-making process, their
determination is very crucial [14]. The next section
presents the basics of the AHP.

4. Basics of Analytic Hierarchy Process (AHP)

The AHP approach was developed and introduced by Dr. Thomas L. Saaty to sup-
port decision-making problems with multiple criteria, [16]. [[25], [28]] defined AHP as
a structured approach for dealing with complex decisions by breaking them down into a
hierarchical structure of criteria and alternatives. Since then, the AHP has been a decision-
making methodology used to prioritize and make choices among various alternatives by
structuring complex problems into a hierarchical model. When it comes to graduating
with maximum achievement in an educational context, identifying the critical success fac-
tors using AHP can be highly beneficial. The primary goal is to excel academically and
achieve the highest possible grades. According to [[26], [27]], the AHP methodology com-
prises of three fundamental developmental stages, namely: structuring the problem in a
hierarchy; performing comparative judgments between elements and decision alternatives,
and priority analysis.
In a typical graduating top-of-the-scale situation, AHP starts by creating a hierarchical
structure that breakdown the goal into its components thereby presenting the most im-
portant elements and their relations as presented in Figure 2. It showcases the functional
interactions of the components and their impact towards realizing the goal, [9]. In the
judgment phase, peer-to-peer comparison is carryout on all the criteria (CSFs) in a given
level with the connected criteria (CSFs) in the level just above to which it is linked directly
using Saaty’s 9-point judgment scale that represents the intensity of a particular element
over another [26].
On this scale of relative importance: 1 signifies equal degree of importance while 9 is
complete importance. The nitty-gritty of the AHP algorithm for graduating with/at max-
imum achievement is fully explained in a stepwise fashion below.



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1965

In order to investigate the relative priorities of the CSFs (Criteria) for graduating top of
the scale in HEI using AHP, we adopt the AHP methodology shown in Figure 1 below.

4.1. Development of AHP methodology for graduating with Maximum
Achievement

In graduating with/at maximum achievement decision process, there is one goal (grad-
uate with “A” grade) and a finite set of alternatives, X = {x1, · · · , xn} from which the de-
cision maker (student), is expected to select the best combination. Since real-life problems
come with various complexities, by AHP’s strategy, the original problem is decomposed
into many smaller subproblems, [27]. Figure 2 shows a four-level decision hierarchy. The
basic elements of the Saaty’s hierarchical model are the main objective or goal to achieve,
the criteria that affect the overall goal, the sub-criteria that influence the main criteria
and finally the alternatives available to the problem. By structuring the problem in this
way, it is possible to better understand the decision to be achieved, the criteria to be used
and the alternatives to be evaluated [30].
To obtain the degree of relative importance of factors at each level, a pair-wise compar-
ison matrix is developed using Saaty 1-9 suggested preference scale [26] of numbers that
indicates how many times more important or dominant one factor is over another factor
with respect to the criterion to which they are compared. The eigenvector and the max-
imum eigenvalue (λmax) are then derived from the pair-wise comparison matrices. The
importance of the eigenvalue is to determine the degree of the consistency ratio, CR, [27]
of the comparative matrix in order to confirm whether the pair-wise comparison matrix
provides a completely consistent assessment. Next step is to derive the consistency index
(CI) Table 2 and consistency ratio (CR), and finally is to determine the global weights of
each CSF, sub-CSF and Alternatives,[11].
To graduate with/at maximum achievement using AHP model we proceed according to
the following seven major steps:
Step 1: Defining the goal/objectives
Step 2: Developing the AHP hierarchy by decomposing the goal into lower-level CSFs or
sub-CSFs etc.
Step 3: Construct a set of pair-wise comparison matrices for each of the levels.
Step 4: Construct the weighted, normalized decision matrix.
Step 5: Check for consistency in the pair-wise comparison matrix to obtain, λMax.
Step 6: Compute the Consistency Index (CI) and Consistency Ratio (CR).
Step 7: Calculate the global weights of each CSF, sub-CSF and Alternatives.
Steps 3-6 are performed for purpose of ascertaining the relative importance of each CSF
for all levels in the hierarchy. The detailed descriptions of each step follow:

Step 1. The goal or objective in our case is identified as: “To graduate with/at
maximum achievement”. The major-CSFs and sub-CSFs (or elements) considered for
decision making to graduate with/at maximum achievement as well as the alternatives
considered - Grade A, B, and C respectively are as presented in Figure 2.



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1966

Step 2. The goal is decomposed into 6 main-CSFs, 32 sub-CSFs and 3 Alternatives so as
to form a hierarchical abstraction of the decision problem.as in figure 2. Here, the goal or
objective is the first level, main-CSFs are in the second level, sub-CSFs are in third level,
while the alternatives are at the last level. Here, Grade A, Grade B, and Grade C are
considered Alternatives as they are usually conceivable in any HEI and since no student
would potentially desire Grade ‘F’. Most importantly, Grade ‘B’ and Grade ‘C’ were added
to represent the worst-case scenarios of the goal. The AHP model is an optimization
procedure that converges to the best alternative. In the language of optimization, they
represent sub-optimal solutions as against the optimal, Grade ‘A’. See Table 1.
Step 3. Construct a set of pair-wise comparison matrices for each of the levels.
From the hierarchical structure, any CSF in the higher level is a controlling CSF for
those in the lower level, as it either influences it or contributes to it. The CSFs in the
lower level are then compared to each other based on their influence on the controlling
CSFs above. To determine the relative importance among the CSFs and sub-CSFs at
each level we compare carefully CSFs of each hierarchy level by assigning Saaty’s scale of
relative importance (1-9), [27] in a pair-wise fashion with respect to the goal or objective
of the model. The result of this collective or group decision of the authors gave birth to
matrices of comparisons according to equation (1). These group’s decision judgements
were synthesized using Saaty’s fundamental scale of absolute numbers, [27], not outlined

here due to lack of space. Thus, with n CSFs, there will be n(n−1)
2 judgements required

to develop a single pair-wise comparison matrix subject to equation (2) below, and the
pairwise comparison of CSF j with CSF k produces a square matrix An×n of
judgement, such that:

An×n =

Attributes
1
2
3
...
n


a11 a12 a13 · · · a1n
a21 a22 a23 · · · a2n
a31 a32 a33 · · · a3n
...

...
...

...
...

an1 an2 an3 · · · ann

 (1)

Subject to the constraints

ajj = 1, ajk > 0, ajk =
1

akj
,∀j, k = 1, 2, 3, · · · , n (2)

So, if the entries exactly represent ratios between weights and by the condition of multi-
plicative reciprocity in equation (2), then matrix A can be express as:

A = (
wj

wk
)n×n =


1 a12 a13 · · · a1n
1

a21
1 a23 · · · a2n

1
a31

1
a32

1 · · · a3n
...

...
...

...
...

1
an1

1
an2

1
an3

· · · 1

 (3)



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1967

Figure 1: A Flow chart of AHP model for graduating with Maximum Achievement

where
ajk denotes the comparative importance of attribute j with respect to attribute k. From
Figure 2, there are 39 pairwise comparison matrices in all. Seven, i.e. One for the major
CSFs; C1 , C2, C3, C4, C5, and C6 with respect to the goal (Level 1), which is shown in
Table 3, ( One 6 x 6 ), and six for the sub-CSFs with respect to the major-CSFs (Level
2, presented in Tables 4 - 9):- the first for the sub-CSF under C1: C11, C12, C13, C14,
C15, C16 and C17 shown in Table 4, (One 7 x 7 ); for the sub-CSF under C2: C21, C22,
C23, C24, C25 and C26 shown in Table 5, (One 6 x 6 ); for the sub-CSF under C3: C31,
C32, C33, C34, and C35 as shown in Table 6, (One 5 x 5 ); for the sub-CSF under C4:
C41, C42, and C43,that is shown in Table 7, (One 3 x 3 ); for the sub-CSF under C5: C51,
C52, C53, C54, C55 and C56 shown in Table 8 (One 6 x 6 ) , and for the sub-CSF under



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1968

C6: C61, C62, C63, C64 and C65 as shown in Table 9 (One 5 x 5 ). Next, there are 32

Figure 2: AHP model of CSFs for graduating with Maximum Achievement

comparison matrices for the three alternatives with respect to all the ‘covering CSFs’- the
lowest level CSFs or sub-CSFs connected to the alternatives. These 32 covering CSFs are:
C11, C12, C13, C14, C15, C16 and C17, (meaning Seven 3 x 3 ); C21, C22, C23, C24, C25,
and C26, (Six 3 x 3 ); C31, C32, C33, C34 and C35, (Five 3 x 3 ); C41, C42, and C43, (Three
3 x 3 ); C51, C52, C53, C54, C55, and C56, (Six 3 x 3 ), and C61, C62, C63, C64 and C65,
(Five 3 x 3 ). All are sub-CSFs in Level 3 and are shown in Table 10. Tables 3 – 10
show the respective pair-wise comparison matrices along with their corresponding vector
of priorities. A vector of priorities is the principal eigenvector of the matrix that gives the
relative priority of the criteria measured on a ratio scale.
Step 4: Construct the weighted, normalized decision matrix
To obtain the Eigen values of the pair-wise comparison matrices, the matrix is first nor-
malized by dividing each entry in column j by the sum of the entries in column j . This
generates the normalized matrix in which the sum of the entries in each column is ‘1’,
see equation (4). Next, obtain the priority weights, also called the principal vector (PV)
or Eigen value, by computing the average of the entries in row of the normalized matrix.
Priority means the relative importance or strength of influence of a CSF in relation to
other CSFs that are placed above it in the hierarchy, see equation (5).

Cjk =
ajk
n∑

j=1
ajk

,∀j, k = 1, 2, 3, · · · , n (4)



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1969

Table 2: Average Randomly Generated Consistency Indexes (RGCI), [11]

N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
RI 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.48 1.56 1.57 1.59

wj =

n∑
j=1

cjk

n
,∀j, k = 1, 2, 3, · · · , n,W =


w1

w2
...
wn

 (5)

Step 5: Check for consistency in the pair-wise comparison matrix to obtain, λMax

Since pair-wise comparison matrices are products of subjective judgements it is possible
that the decision maker may be inconsistent in the judgements. To ensure that the judg-
ments of decision makers are consistent, the AHP procedure incorporates this step into
the model to examines whether the created pairs of matrices are consistent or not, and
also provides the decision makers with a measure of the inconsistencies. To check for
the consistency of judgements as presented in Tables 3 – 10, let the considered pair-wise
comparison matrix be denoted as δ1 and the corresponding column vector, PV be denoted
as δ2 , then compute the column δ and the column eigenvector λ as:

δ = δ1 ∗ δ2 λMax = Max
∀j,k

(
δ

δ2

)
= Max

∀j,k

(
jth entry in δ

jth entry in δ2

)
(6)

Step 6: Compute the Consistency Index (CI) and Consistency Ratio (CR)
The CR is used to confirm whether a criterion (CSF) can be used for decision-making. It is
obtained from the ratio of the consistency of the results being examined to the consistency
of the same problem evaluated with a random number as:

CI =
(λMax − n)

(n− 1)
CR =

CI

RGCI
(7)

where n denotes the order of matrix and the Randomly Generated Consistency Index.
If CI is sufficiently small, the decision maker’s comparisons are significantly consistent
enough to give useful estimates of the priority weights for the goal or objective. If CR <
0.10 (10%), the degree of consistency in the judgments is significantly acceptable, so
the obtained weights can be used for decision making. However, if CR > 0.10 , serious
inconsistencies may exist, and the AHP may not yield meaningful results, so the assessment
can be revised as indicated in Figureb1 above. Table 2 shows average Randomly Generated
Consistency indexes (RGCI) of the matrices of order 1–15, [11].

Step 7: Calculate the global weights of each CSF, sub-CSF and Alternatives The
priority weights (or Principal vectors (PVs)) are divided into ‘local weights (LWs)’ –
the Eigen value or priority weight with respect to the preceding hierarchical level, and
‘global weights (GWs)’– the Eigen value or priority weight with respect to the highest
hierarchical level – the objective or goal. To carry out an overall ranking for sub-CSFs,
AHP consolidates the priority weights of CSFs with the comparison rating for sub-CSFs
and alternatives. That is, each Priority weight (PW) of the main CSF in level 2 (L2-PW)



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1970

is multiplied by the respective PW value of sub-CSF in level 3 (L3-PW), which in turn
is again multiplied by the PW value for each respective alternative (L4-PW) to get the
desirability index (DI) of the alternative for each sub-CSF, using equation 8. Finally,
the sum of the desirability indexes (DIs) obtained for each of the sub-criterion gives the
overall DI for each alternative (Figure 3). The alternative with the highest DI is chosen
or preferred [30].

[H]GWs =

n∑
j,k=1

(LW for criterion j× LW for sub-criterion k w.r.t. criterion j) (8)

4.2. Comparison Matrices and their Local priorities

Table 3: Pair-wise Comparison Matrix for the Goal Level 1

Goal C1 C2 C3 C4 C5 C6 Priority

C1 1 0.2 0.5 0.25 2 4 0.088

C2 5 1 4 2 7 9 0.418

C3 2 0.25 1 0.5 4 5 0.15

C4 4 0.2 2 1 5 7 0.256

C5 0.5 0.14 0.25 0.2 1 2 0.05

C6 0.25 0.11 0.2 0.14 0.5 1 0.032

Sum 12.75 2.2 7.95 4.09 19.5 28 1

λMax = 6.231, C.I. = 0.046, C.R. = 0.040

Table 4: Pair-wise Comparison Matrix for the Main CSF (C1) Level 2

Sub-CSFs C11 C12 C13 C14 C15 C16 C17 Priority

C11 1 0.25 0.1667 2 0.333 0.5 0.25 0.05

C12 4 1 0.333 4 3 3 2 0.205

C13 6 3 1 8 4 5 3 0.387

C14 o.50 0.25 0.125 1 0.333 0.333 0.25 0.094

C15 3 0.33 0.25 3 1 2 0.5 0.105

C16 2 0.33 0.2 3 0.5 1 0.33 0.075

C17 4 0.5 0.333 4 2 3 1 0.159

Sum 20.5 5.66 2.4077 22 11.166 14.833 7.33 1

λMax = 8.400, C.I. = 0.123, C.R. = 0.090



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1971

Table 5: Pair-wise Comparison Matrix for the Main CSFs (C2) Level 2

Sub-CSFs C21 C22 C23 C24 C25 C26 Priority

C21 1 0.2 0.5 0.25 2 4 0.088

C22 5 1 4 2 7 9 0.418

C23 2 0.25 1 0.5 4 5 0.15

C24 4 0.2 2 1 5 7 0.256

C25 0.5 0.14 0.25 0.2 1 2 0.05

C26 0.25 0.11 0.2 0.14 0.5 1 0.032

Sum 12.75 2 7.95 4.09 19.5 28 1

λMax = 6.231, C.I. = 0.046, C.R. = 0.030

Table 6: Pair-wise Comparison Matrix for the Main CSFs (C3) Level 2

Sub-CSFs C31 C32 C33 C34 C35 Priority

C31 1 2 3 4 5 0.404

C32 0.5 1 2 4 5 0.28

C33 0.33 0.5 1 3 2 0.157

C34 0.25 0.25 0.33 1 3 0.098

C35 0.2 0.2 0.5 0.33 1 0.06

Sum 2.28 3.95 6.83 12.33 16 1

λMax = 5.374, C.I. = 0.094, C.R. = 0.080

Table 7: Pair-wise Comparison Matrix for the Main CSFs (C4) Level 2

Sub-CSFs C41 C42 C43 Priority

C41 1 5 4 0.667

C42 0.2 1 0.33 0.103

C43 0.25 3 1 0.23

Sum 1.45 9 5.33 1

λMax = 3.014, C.I. = 0.007, C.R. = 0.010

Table 8: Pair-wise Comparison Matrix for the Main CSFs (C5) Level 2

Sub-CSFs C51 C52 C53 C54 C55 Priority

C51 1 2 3 4 5 0.404

C52 0.5 1 2 4 5 0.28

C53 0.33 0.5 1 3 2 0.157

C54 0.25 0.25 0.33 1 3 0.098

C55 0.2 0.2 0.5 0.33 1 0.06

Sum 2.28 3.95 6.83 12.33 16 1

λMax = 6.230, C.I. = 0.046, C.R. = 0.040



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1972

Table 9: Pair-wise Comparison Matrix for the Main CSFs (C6) Level 2

Sub-CSFs C61 C62 C63 C64 C65 Priority

C61 1 2 3 4 5 0.404

C62 0.5 1 2 4 5 0.28

C63 0.33 0.5 1 3 2 0.157

C64 0.25 0.25 0.33 1 3 0.098

C65 0.2 0.2 0.5 0.33 1 0.06

Sum 2.28 3.95 6.83 12.33 16 1

λMax = 5.374, C.I. = 0.094, C.R. = 0.090

Table 10: Pair-wise comparison matrix for Level 3

C11 A B C Priority C12 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 0.2 1 0.068 C 0.125 0.25 1 0.069

Sum 1.36 5.2 15 1 Sum 1.292 7.25 13 1

λMax = 3.100, C.I. = 0.050, C.R. = 0.086 λMax = 3.083, C.I. = 0.042, C.R. = 0.070

C13 A B C Priority C14 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 3 1 0.068 C 0.125 0.25 1 0.069

Sum 1.36 5.2 15 1 Sum 1.45 5.33 9 1

λMax = 3.101, C.I. = 0.051, C.R. = 0.090 λMax = 3.083, C.I. = 0.042, C.R. = 0.072

C15 A B C Priority C16 A B C Priority

A 1 5 7 0.724 A 1 4 9 0.701

B 0.2 1 3 0.193 B 0.205 1 5 0.231

C 0.143 0.33 1 0.083 C 0.11 0.2 1 0.068

Sum 1.343 6.33 11 1 Sum 1.36 5.2 15 1

λMax = 3.036, C.I. = 0.018, C.R. = 0.030 λMax = 3.100, C.I. = 0.050, C.R. = 0.090

C17 A B C Priority C21 A B C Priority

A 1 6 8 0.739 A 1 5 7 0.724

B 0.167 1 4 0.192 B 0.2 1 3 0.193

C 0.125 0.25 1 0.069 C 0.143 0.33 1 0.083

Sum 1.292 7.259 13 1 Sum 1.343 6.33 11 1

λMax = 3.083, C.I. = 0.040, C.R. = 0.070 λMax = 3.036, C.I. = 0.020, C.R. = 0.034

C22 A B C Priority C23 A B C Priority

A 1 5 7 0.724 A 1 4 9 0.726

B 0.2 1 3 0.193 B 0.25 1 5 0.209

C 0.143 0.333 1 0.083 C 0.11 0.2 1 0.065

Sum 1.343 6.333 11 1 Sum 1.36 5.2 15 1

λMax = 3.036, C.I. = 0.018, C.R. = 0.030 λMax = 3.100, C.I. = 0.050, C.R. = 0.090



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1973

Table 11: Continuation of Table 10 Pair-wise comparison matrix for Level 3

C24 A B C Priority C25 A B C Priority

A 1 3 7 0.657 A 1 6 8 0.739

B 0.33 1 5 0.274 B 0.167 1 4 0.192

C 0.143 0.2 1 0.069 C 0.125 0.25 1 0.069

Sum 1.473 4.2 13 1 Sum 1.292 7.25 13 1

λMax = 3.013, C.I. = 0.006, C.R. = 0.010 λMax = 3.083, C.I. = 0.041, C.R. = 0.072

C26 A B C Priority C + 31 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 0.2 1 0.068 C 0.125 0.25 1 0.069

Sum 1.361 5.2 9 1 Sum 1.292 7.25 13 1

λMax = 3.092, C.I. = 0.046, C.R. = 0.070 λMax = 3.100, C.I. = 0.050, C.R. = 0.090

C32 A B C Priority C33 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 0.2 1 0.068 C 0.125 0.25 1 0.069

Sum 1.361 5.2 15 1 Sum 1.292 7.25 13..000 1

λ+Max = 3.010, C.I. = 0.050, C.R. = 0.090 λMax = 3.083, C.I. = 0.042, C.R. = 0.070

C34 A B C Priority C35 A B C Priority

A 1 6 8 0.739 A 1 5 7 0.724

B 0.167 1 4 0.192 B 0.2 1 3 0.193

C 0.125 0.25 1 0.069 C 0.143 0.333 1 0.083

Sum 1.292 7.25 13 1 Sum 1.343 6.333 11 1

λMax = 3.083, C.I. = 0.042, C.R. = 0.070 λMax = 3.014, C.I. = 0.069, C.R. = 0.012

C41 A B C Priority C42 A B C Priority

A 1 6 8 0.739 A 1 4 9 0.535

B 0.167 1 4 0.192 B 0.25 1 5 0.285

C 0.125 0.25 1 0.069 C 0.11 0.2 1 0.18

Sum 1.292 7.25 13 1 Sum 1.36 5.2 15 1

λMax = 3.083, C.I. = 0.042, C.R. = 0.072 λMax = 3.100, C.I. = 0.050, C.R. = 0.090

C43 A B C Priority C51 A B C Priority

A 1 5 7 0.724 A 1 4 9 0.701

B 0.2 1 3 0.193 B 0.25 1 5 0.231

C 0.143 0.33 1 0.083 C 0.11 0.2 1 0.068

Sum 1.343 6.33 11 1 Sum 1.7 3.33 9 1

λMax = 3.014, C.I. = 0.068, C.R. = 0.012 λMax = 3.100, C.I. = 0.051, C.R. = 0.086



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1974

Table 12: Continuation of Table 10 Pair-wise comparison matrix for Level 3

C52 A B C Priority C53 A B C Priority
A 1 6 8 0.739 A 1 6 8 0.739

B 0.167 1 4 0.192 B 0.167 1 4 0.192

C 0.125 0.25 1 0.069 C 0.125 0.25 1 0.069

Sum 1.292 7.25 13 1 Sum 1.292 7.25 13 1

λMax = 3.100, C.I. = 0.050, C.R. = 0.090 λMax = 3.100, C.I. = 0.050, C.R. = 0.086

C54 A B C Priority C55 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 0.2 1 0.068 C 0.125 0.25 1 0.069

Sum 1.36 5.2 15 1 Sum 1.292 7.25 13 1

λMax = 3.100, C.I. = 0.050, C.R. = 0.086 λMax = 3.083, C.I. = 0.042, C.R. = 0.072

C56 A B C Priority C61 A B C Priority

A 1 6 8 0.571 A 0.167 6 8 0.739

B 0.167 1 4 0.286 B 0.125 1 4 0.192

C 0.125 0.25 1 0.143 C 0.25 0.25 1 0.069

Sum 1.292 7.25 7.1 1 Sum 1.292 7.25 13 1

λMax = 3.080, C.I. = 0.042, C.R. = 0.072 λMax = 3.083, C.I. = 0.042, C.R. = 0.072

C62 A B C Priority C63 A B C Priority

A 1 5 7 0.724 A 1 4 9 0.701

B 0.2 1 3 0.193 B 0.25 1 5 0.231

C 0.143 0.33 1 0.083 C 0.11 0.2 1 0.068

Sum 1.343 6.33 11 1 Sum 1.36 5.2 15 1

λMax = 3.036, C.I. = 0.020, C.R. = 0.030 λMax = 3.100, C.I. = 0.050, C.R. = 0.090

C64 A B C Priority C65 A B C Priority

A 1 4 9 0.701 A 1 6 8 0.739

B 0.25 1 5 0.231 B 0.167 1 4 0.192

C 0.11 0.2 1 0.068 C 0.125 0.25 1 0.069

Sum 1.36 5.2 15 1 Sum 1.292 7.25 13 1

λMax = 3.100, C.I. = 0.050, C.R. = 0.090 λMax = 3.082, C.I. = 0.041, C.R. = 0.072

5. Results, Analysis and discussions

5.1. Level 2 towards achieving the Goal (Level 1)

Analysis of the pair-wise comparison matrix (Table 3) identified C2 (Hard work) with
the highest PV = 0.418 as the major CSF to achieve the goal of graduating at the top in



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1975

Figure 3: Data summary of the complete CSFs analysis

school.

5.2. Level 3 towards achieving the Main CSFs (Level 2)

The following Sub-CSFs were identified as being crucial to achieving the goal of grad-
uating top of the scale: C13 (Taking studies serious at all times), having the highest
PV = 0.387 confirms that students must take their academic studies serious at all times.
C22 (Reviewing lecture notes with lecturer), having the highest PV = 0.418 advices stu-
dents to always review their lecture notes on each course with their course lecturer to
prepare well for their test and exam to graduate at the top. C31 (Keeping to study plan)
having the highest PV = 0.404 as one of the key sub-CSFs for graduating top. C41

(low in-campus social involvement) having the highest PV = 0.667 advices to keep a
low in-campus social involvement. C51 (Ability to work independently) with the highest
PV = 0.404 calls for self-reliance in achieving a goal. C61 (Prefer reading at Night or
Day) with the highest PV = 0.404 advices students to choose comfortable time to read as



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1976

Figure 4: Continuation of Figure 3:Data summary of the complete CSFs analysis

their choice to perform well in their academics.

5.3. Level 4 towards achieving the Desired Alternative

To obtain Grade A (highest PV = 0.701), pay attention during lectures (C11). To
obtain Grade A (highest PV = 0.739), you must possess a Passionate Desire/interest
to learn (C12). To obtain Grade A (highest PV = 0.701), take studies serious at all
times (C13). Attending lectures early and regularly (C14) will lead to achieving Grade A
(PV = 0.739). Being goal – Oriented (C15) will lead to achieving Grade A (PV = 0.722).



P. N. Assi et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1959-1981 1977

Taking every lecture note (C16) will lead to achieving Grade A (PV = 0.701). Being
serious about CA tests (C17)) will lead to achieving Grade A (PV = 0.7039). Studying
every day (C21) will lead to achieving Grade A (PV = 0.724). Reviewing lecture note with
lecturer (C22) will lead to achieving Grade A (PV = 0.724). Studying difficult courses
regularly (C23) will lead to achieving Grade A (PV = 0.726). Studying past question
before exam (C24) will lead to achieving Grade A (PV = 0.607). Reading before every
exam (C25) will lead to achieving Grade A (PV = 0.739). Doing all Assignments/home
works by oneself (C26) will lead to achieving Grade A (PV = 0.726). Keeping to study
plan (C31) will lead to achieving Grade A (PV = 0.739). Efficient time management
(C32) will lead to achieving Grade A (PV = 0.701). Scheduling proper time for studying
(C33) will lead to achieving Grade A (PV = 0.739). Cutting back on socials (C34) will
lead to achieving Grade A (PV = 0.739). Being always prepared (C35) will lead to
achieving Grade A (PV = 0.724). Low in - campus social environment (C41) will lead to
achieving Grade A (PV = 0.739). Low off - campus social environment (C42) will lead
to achieving Grade A (PV = 0.701). Giving little attention to friends (C43) will lead
to achieving Grade A (PV = 0.724). Ability to work independently (C51) will lead to
achieving Grade A (PV = 0.701). Having Perseverance (C52) will lead to achieving Grade
A (PV = 0.739). Ability to think creatively and critically (C53) will lead to achieving
Grade A (PV = 0.739). Having self-motivation (C54) will lead to achieving Grade A
(PV = 0.701). Having a driving Curiosity to understand (C55) will lead to achieving
Grade A (PV = 0.739). A personal responsibility to learn (C56) will lead to achieving
Grade A (PV = 0.739). Preferring reading at Night or Day (C61) will lead to achieving
Grade A (PV = 0.739). Reading for long or short hours (C62) will lead to achieving
Grade A (PV = 0.724). Dependence on lecture notes or course materials (C63) will lead
to achieving Grade A (PV = 0.701). Study in quiet or noisy setting (C64) will lead to
achieving Grade A (PV = 0.701). Like reading alone or group always (C65) will lead to
achieving Grade A (PV = 0.739).
Tables 3 − 10 will help the students (readers) understand the importance of each major
CSF and sub-CSF. For instance, in Table 5, the PV/Eigen vector depicts the importance
(Ranking) of each major CSF with respect to the goal. In this case, concern is about
students’ Hard work (C2) as a key factor to graduating top with an A grade having the
highest PV value. Similarly, Table 10 establishes the relative importance (Rankings) of the
sub-CSFs and alternatives (Grade A, Grade B and Grade C) within the main-CSFs and
are interpreted same accordingly with their respective PV values. The results of the AHP
in Table 11 reveal that Grade A is achievable, as the overall desirability index (0.9268)
is higher when compared to the rest of the alternatives.

6. Conclusions

Graduating with maximum achievement(GradeA = 80%−100%) can never be less im-
portant or overemphasized. This work has made a serious and solemn attempt to show the
validity of the application of an MCDM tool, the AHP for modelling one of the strategic
decisions in an important function of the educational system – namely, academic achieve-



REFERENCES 1978

ment and performance towards graduating with/at maximum achievement. A literature
review on academic achievement and performance was carried out to provide an overview
of the CSFs pivotal for higher grades achievements apart from highlighting the differences
between them. Later, various research gaps were also identified. The application of AHP
serves to re-awaken the awareness about the importance of high grades by distilling a sub-
set of key independent variables called CSFs from an academic perspective, for students to
graduate with/at maximum achievement in HEIs. By systematically evaluating and pri-
oritizing CSFs such as behaviour towards academics, hard work, organization, peers and
socials, skills and studying style or pattern, individuals can optimize their academic per-
formance. Ultimately, the results of integrating AHP into educational strategies provide a
structured framework for decision-making, enabling students to focus on the most impact-
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their chances of graduating with/at maximum achievement. This paper provides a com-
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which could form the basis for future generalization. It is hoped that the results from
this study will assist researchers and post-secondary personnel in better understanding
the important factors for student success and provide the framework for further in-depth
exploration. By employing AHP, we aim to provide a comprehensive understanding of the
key factors that will enable students to excel academically and graduate with distinction.

Acknowledgements

The authors thank the readers of European Journal of Pure and Applied Mathematics,
for making our journal successful.

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