=Paper=
{{Paper
|id=Vol-3900/Paper6
|storemode=property
|title=Evaluation of Faculty Teaching Effectiveness through Student Feedback Utilizing Fuzzy TOPSIS Method
|pdfUrl=https://ceur-ws.org/Vol-3900/Paper6.pdf
|volume=Vol-3900
|authors=Dhrubajyoti Ghosh,Sandip Dey,Anita Pal
|dblpUrl=https://dblp.org/rec/conf/dosier/GhoshDP24
}}
==Evaluation of Faculty Teaching Effectiveness through Student Feedback Utilizing Fuzzy TOPSIS Method==
https://ceur-ws.org/Vol-3900/Paper6.pdf
Evaluation of Faculty Teaching Effectiveness through
Student Feedback Utilizing Fuzzy TOPSIS Method
Dhrubajyoti Ghosh1,*,† , Sandip Dey2,† and Anita Pal3,†
1
OmDayal Group of Institutions, Uluberia, Howrah, 711316, West Bengal, India
2
Sukanta Mahavidyalaya, Dhupguri, Jalpaiguri, 735210, West Bengal, India
3
National Institute of Technology, Durgapur, 713209, West Bengal, India
Abstract
Today, many colleges and universities have implemented a paper or web based process of how to obtain feedback
from students on faculty instruction. Recent studies have confirmed what many teachers already know: that
feedback in a constructive and actionable manner can dramatically enhance students’ performance on a daily
basis. So, students write their feedback forms on such criteria because they are by default concerned for their
teachers who affect their life even more as both an educator and personal guide. The ultimate purpose of faculty
performance evaluation is to discover the strengths and weaknesses in their professional growth. This specific
study uses the Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) which is a special
method of Multi-criteria Decision Making Problem that will improve effectiveness in ranking alternative(s).
Keywords
Multi-criteria decision making (MCDM), Fuzzy TOPSIS Method, Feedback system, Ranking, Fuzzy Positive Ideal
Solution, Fuzzy Negative Ideal Solution,
1. Introduction
Over the last few decades, Multiple Criteria Decision Making (MCDM) that is multiple objective decision
making has gain more popularity in terms of use and practice worldwide. The fundamental concept
behind this method is relatively straightforward. It uses two points of reference, namely the positive
ideal solution (PIS) and negative ideal solution (NIS) for assessment. For choose, the one closest to the
PIS and farthest from the NIS. The PIS is opposite of NIS which gives all benefit criteria their maximum
value and all cost criteria their minimum value.
In traditional MCMD approaches, such as the conventional TOPSIS method, the weights and ratings
assigned to criteria are predetermined. However, crisp statistics often fall short in accurately representing
the complexities of the real world, as human judgments, including preferences, are frequently ambiguous
and difficult to quantify without some level of uncertainty. Conversely, employing linguistic evaluations
in place of numerical values—by utilizing linguistic variables to assess the ratings and weights of the
criteria relevant to the problem at hand—may offer a more authentic representation. The classic TOPSIS
method depends on the information supplied by the expert or decision maker (DM) as precise numerical
values. Nevertheless, in certain real-world scenarios, the DM may struggle to articulate the significance
of the ratings of alternatives concerning specific criteria, or the expert may resort to using linguistic
expressions [1].
In circumstances where the evaluations made by decision makers depend on information that is either
unquantifiable, incomplete, or not readily available, alternative measurement approaches can be utilized.
These approaches encompass interval numbers [2] and [3], fuzzy numbers [4], ordered fuzzy numbers
[5], group decision making [6], ordered fuzzy numbers [7]. The utilization of these methodologies
has been thoroughly examined across a variety of fields, with numerous instances of fuzzy TOPSIS
The 2024 Sixth Doctoral Symposium on Intelligence Enabled Research (DoSIER 2024), November 28–29, 2024, Jalpaiguri, India
*
Corresponding author.
†
These authors contributed equally.
$ krizz27@gmail.com (D. Ghosh); dr.ssandip.dey@gmail.com (S. Dey); anita.buie@gmail.com (A. Pal)
0000-0001-5739-1901 (D. Ghosh); 0000-0002-1005-3304 (S. Dey); 0000-0002-2514-5463 (A. Pal)
© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
applications documented in the literature. Significant areas of application include the assessment of
service quality [8], comparisons [9] between companies, aggregate production planning [10], selection
of facility locations [11], and large-scale nonlinear programming [12], mobile health (mHealth)
applications [13], Supplier selection among others [14], Project Risk Variable Ranking [15], analysis
of Multi criteria decision making [16], weights detection of multi-criteria [17] and many others
The aim of this paper is to introduce a more efficient methodology for TOPSIS-based fuzzy group
decision-making (GDM) to enhance the ranking of fuzzy alternatives. The data utilized are presented in
linguistic form within a feedback system. By employing the Fuzzy TOPSIS Method, we represent the
feedback system and identify the optimal solution among the various options available for the given
problem, as well as evaluate the performance rankings of faculties. This paper is structured into five
sections: Section two outlines the fundamental components of fuzzy set theory. The TOPSIS method
is detailed in Section three. The application of fuzzy TOPSIS to the problems discussed is elaborated
in Section four, while Section five provides an illustrative example. The sixth section compares Fuzzy
TOPSIS with other multi-criteria decision-making (MCDM) methods. The seventh section concludes
our research, and the eighth section discusses future research directions.
2. Preliminaries
Here we give some of the notations which will be used frequently in the material of the present paper
and some equations related to them.
2.1. Definition 1
A fuzzy set 𝐴 within a given universe of discourse 𝑋 includes a membership function, 𝜇𝐴 (𝑥), that
assigns a real number between zero and one to each element of the universal set 𝑋. This value represents
the degree of membership of 𝑥 in 𝐴, as defined by Zadeh [18].The membership function is presented
as Equation (1)
𝜇𝐴 (𝑥) : 𝑋 → {0, 1} (1)
2.2. Definition 2
A fuzzy set 𝐴 is consider convex if and only if for all 𝑥1 and 𝑥2 in 𝑋. The convex fuzzy set is presented
as Equation (2)
𝜇𝐴 (𝜆𝑥1 + (1 − 𝜆)𝑥2 ) ≥ 𝑚𝑖𝑛(𝜇𝐴 (𝑥1 ), 𝜇𝐴 (𝑥2 )) (2)
Where 𝜆 ∈ [0, 1]
2.3. Definition 3
A fuzzy set A consider normal fuzzy set is presented as Equation (3)
∃𝑥 ∈ 𝑋 : 𝜇𝐴 (𝑥) = 1 (3)
2.4. Definition 4
The 𝜆 cut of fuzzy number 𝑛
˜ is defined as Equation (4)
˜ 𝛼 = {𝑥𝑖 : 𝜇𝑛˜ (𝑥𝑖 ) ≥ 𝛼, 𝑥𝑖 ∈ 𝑋}
𝑛 (4)
˜ 𝛼 , as defined in Equation (4), represents a non-empty, bounded closed interval that lies within X. This
𝑛
interval can be denoted as 𝑛 ˜ 𝛼 = [𝑛˜1 𝛼 , 𝑛˜2 𝛼 ], where 𝑛˜1 𝛼 and 𝑛˜2 𝛼 represent the lower and upper bounds
of the closed interval, respectively, as defined by Zimmermann [19].
2.5. Definition 5
A triangular fuzzy number 𝑛
˜ consider by a triplet(𝑛1 , 𝑛2 , 𝑛3 ), then the membership function 𝜇𝑛˜ (𝑥) is
defined as Equation (5):
{︃ 𝑥−𝑛1
𝑛2 −𝑛1 , if𝑛1 ≤ 𝑥 ≤ 𝑛2 ,
𝑌 𝜇𝑋˜ = 𝑥−𝑛3
𝑛2 −𝑛3 , if𝑛2 ≤ 𝑥 ≤ 𝑛3 , (5)
0, otherwise.
2.6. Definition 6
IF 𝑛
˜ is a triangular fuzzy number and 𝑛
˜ 𝛼𝑙 and 𝑛
˜ 𝛼2 , then 𝑛
˜ is called a normalized positive fuzzy number.
2.7. Definition 7
Zadeh [18] defined linguistic variables whose values are words or sentences in a natural language. In
general sense, when the problems encountered are so sophisticated or have such vague/fuzzy definitions
that they cannot be expressed in more precise languages such as for example mathematized forms, the
idea of linguistic variable is quite handy. One of the many linguistic variables is weight, which can have
fuzzy numbers in very low, low, medium, high and very high value.
3. TOPSIS Method
Hwang and Yoon (1981) developed the TOPSIS method, which assigns a ranking to the alternatives
based on their separations from both the ideal and negative ideal solutions. The best alternative has not
only the least Euclidean distance, but also the greatest distance in the negative ideal solution. Hence
in a positive solution, one strived at arranging a solution that has maximum gain at lesser cost. On
the other hand, negative solution is centered on the pains and tries to minimize gains [20]. Six stages
systems are designed to provide level of the TOPSIS approach:
• Compute the normalized decision matrix as Equation (6).
⎯
⎸𝑚 2
⎸∑︁
𝑟𝑖𝑗 = 𝑥𝑖𝑗 ⎷ 𝑥𝑖𝑗 , 𝑖 = 1, 2, . . . , 𝑚 and 𝑗 = 1, 2, . . . , 𝑛 (6)
𝑖−1
• Establish the weighted normalized decision matrix as Equation (7).
𝜐𝑖𝑗 = 𝑟𝑖𝑗 × 𝜔𝑗 𝑖 = 1, 2, . . . , 𝑚 and 𝑗 = 1, 2, . . . , 𝑛 (7)
Where, 𝜔𝑗 is the weight of the 𝑗𝑡ℎ criterion or attribute and 𝑛𝑗=1 𝜔𝑗 = 1.
∑︀
• Compute the positive ideal (𝐴* ) as Equation (8)and negative (𝐴− ) ideal solutions as Equation (9)
𝜐 𝜐
𝐴* = {(𝑚𝑎𝑥𝑖 𝑖𝑗 |𝑗 ∈ 𝐶𝑏 ), (𝑚𝑖𝑛𝑖 𝑖𝑗 |𝑗 ∈ 𝐶𝑐 )} = {𝜐𝑗* |𝑗 = 1, 2, · · · , 𝑚} (8)
𝜐 𝜐
𝐴− = {(𝑚𝑖𝑛𝑖 𝑖𝑗 |𝑗 ∈ 𝐶𝑏 ), (𝑚𝑎𝑥𝑖 𝑖𝑗 |𝑗 ∈ 𝐶𝑐 )} = {𝜐𝑗− |𝑗 = 1, 2, · · · , 𝑚} (9)
• The distance are computed by calculating the Euclidean distance across multiple dimensions. The
specific separation measures for each alternative, in relation to both the 𝐴* as Equation (10) and
the 𝐴− as Equation (11), are outlined as following way:
⎯
⎸𝑚
⎸∑︁
𝑆𝑖* = ⎷ (𝜐𝑖𝑗 − 𝜐 * ), 𝑗 = 1, 2, · · · , 𝑚
𝑗 (10)
𝑗=1
⎯
⎸𝑚
⎸∑︁
𝑆 = ⎷ (𝜐𝑖𝑗 − 𝜐 − ), 𝑗 = 1, 2, · · · , 𝑚
−
𝑖 𝑗 (11)
𝑗=1
• Determine the relative proximity to the ideal solution 𝑆.
• The proximity of the alternative 𝐴𝑖 with respect to 𝐴* is characterized in the following way and
ranking the preference order as Equation (12).
√︃
𝑆𝑖−
𝑅𝐶 *𝑖 = , 𝑖 = 1, 2, · · · , 𝑚 (12)
𝑆𝑖* + 𝑆𝑖−
4. Fuzzy TOPSIS Method
This analysis of the various alternatives utilizes the Technique for Order Preference by Similarity to
Ideal Situation (TOPSIS), which includes methods such as the fuzzy TOPSIS technique. This approach
employs criteria weights and ratings for alternatives expressed in linguistic variables that are converted
into Triangular Fuzzy Numbers [21]. According to this methodology, the most favored alternatives
should be positioned as close as possible to the Fuzzy Positive Ideal Solution (FPIS) while being situated
as far away as possible from the Fuzzy Negative Ideal Solution (FNIS).
Fuzzy TOPSIS method steps are provided below and the techniques of weights of criteria and ranking
of alternatives [22]:
Step 1: Create the classified diagram.
Step 2: Implement the data scaling process according to the defined criteria and alternatives. The
linguistic variables utilized by the decision-maker(s) for evaluating the alternatives are outlined in
Table 1, while the allocation of weights assigned to the criteria is presented in Table 2.
Table 1
Linguistic variables and Triangular Fuzzy Numbers for the criteria
Linguistic variables TFNs
Indeed-superior(A1) (8,9,10)
Superior(A2) (7,8,9)
Above-average(A3) (4,5,6)
Average(A4) (3,4,5)
Below-average(A5) (2,3,4)
Poor(A6) (1,2,3)
Very-poor(A7) (1,1,1)
Table 2
Linguistic variables and TFNs for alternative weights
Linguistic variables TFNs
Indeed-critical (W1) (8,9,10)
Rather-critical(W2) (7,8,9)
Very-important(W3) (4,5,6)
Important(W4) (3,4,5)
Rather-important(W5) (2,3,4)
Step 3: Calculate the total fuzzy weight of each of the criterion, 𝑤
˜𝑗 of the 𝑘𝑡ℎ decision maker described
as Equation (13) and as Equation(14):
𝑤
˜𝑗 = (𝑤𝑗1 , 𝑤𝑗2 , 𝑤𝑗3 ) (13)
Where,
∑︀𝐾
𝑘
𝑊𝑗1 = 𝑚𝑖𝑛𝑘 {𝑊𝑗1 } 𝑊𝑗2 = 𝐾1 𝑘
𝑘=1 𝑊𝑗2
𝑘
𝑊𝑗3 = 𝑚𝑎𝑥𝑘 {𝑊𝑗3 } (14)
where, 𝑗 = 1, 2, · · · , 𝑛𝑡ℎ criteria.
Step 4: To construct the fuzzy decision matrix as Equation (15), represented as 𝐷, ˜ which in-
cludes the alternatives, 𝑖, and the criteria, 𝑗, the subsequent elements will be necessary .
⎡ ⎤
𝐴1 𝑥 ˜11 𝑥
˜12 ··· 𝑥
˜1𝑛
𝐴2 ⎢ 𝑥
˜21 𝑥
˜22 ··· 𝑥
˜2𝑛 ⎥
˜ = . ⎢ (15)
.. ⎣ ... .. .. .. ⎥
𝐷
⎥
. . . ⎦
⎢
𝐴𝑚 𝑥 ˜𝑚2 · · · 𝑥
˜𝑚1 𝑥 ˜𝑚𝑛
where, 𝑖 = 1, 2, . . . , 𝑚 (alternatives), 𝑗 = 1, 2, . . . , 𝑛 (criteria).
Let 𝑥
˜𝑖𝑗 represent the associated fuzzy evaluations of alternative 𝑖 with respect to each criterion 𝑗,
as determined by a group of 𝐾 decision-makers. The calculation of 𝑥 ˜𝑖𝑗 is performed in the following
manner as Equations (16)-(18):
𝑥
˜𝑗 = (𝑎𝑖𝑗 , 𝑏𝑖𝑗 , 𝑐𝑖𝑗 ) (16)
where,
𝑎𝑖𝑗 = 𝑚𝑖𝑛𝑘 {𝑎𝑘𝑖𝑗 } (17)
𝐾
1 ∑︁ 𝑘
𝑏𝑖𝑗 = 𝑏𝑖𝑗
𝐾
𝑘=1
𝑐𝑖𝑗 = 𝑚𝑎𝑥𝑘 {𝑐𝑘𝑖𝑗 }
˜𝑘𝑖𝑗 = (𝑎𝑘𝑖𝑗 , 𝑏𝑘𝑖𝑗 , 𝑐𝑘𝑖𝑗 )
𝑥 (18)
Step 5: Normalize the fuzzy decision matrix using Equations (19)- (20):
(︁ )︁
𝑎𝑖𝑗 𝑏𝑖𝑗 𝑐𝑖𝑗
𝑟˜𝑖𝑗 = 𝑐*𝑗 , 𝑏*𝑗 , 𝑐*𝑗 (19)
and 𝑐*𝑗 = 𝑚𝑎𝑥𝑖 {𝑐𝑖𝑗 } (benefit criteria).
(︁ −
𝑎𝑗 𝑎− 𝑎−
)︁
𝑟˜𝑖𝑗 = 𝑗 𝑗
𝑐𝑖𝑗 , 𝑏𝑖𝑗 , 𝑎𝑖𝑗
(20)
and 𝑎−
𝑗 = 𝑚𝑖𝑛𝑖 {𝑎𝑖𝑗 } (cost criteria).
Step 6: Compute the weighted normalized fuzzy decision matrix, 𝑉˜ as Equation (21), as presented
below, by multiplying 𝑟˜𝑖𝑗 , with 𝑤𝑗 , in the following manner:
𝑉˜ = [˜
𝜐𝑖𝑗 ]𝑚𝑛 (21)
where, 𝑖 = 1, 2 · · · , 𝑚 alternatives , 𝑗 = 1, 2, . . . , 𝑛 criteria, 𝑣˜𝑖𝑗 = 𝑟˜𝑖𝑗 × 𝑤
˜𝑖𝑗 .
Step 7: Compute the FPIS, 𝐴* and FNIS, 𝐴− using Equations (22)-(23):
𝜐1* , 𝜐˜2* , 𝜐˜3* ), where 𝜐˜1* = 𝑚𝑎𝑥𝑖 {𝜐𝑖𝑗3 }
𝐴˜ = (˜ (22)
𝜐1− , 𝜐˜2− , 𝜐˜3− ), where 𝜐˜1− = 𝑚𝑖𝑛𝑖 {𝜐𝑖𝑗3 }
𝐴˜ = (˜ (23)
Step 8: Compute the distance of each alternative, (𝑑*𝑖 , 𝑑−
𝑖 ) where 𝑖 = 1, 2, 3, · · · , 𝑚 from the FPIS
and FNIS by applying the Equations (24)-(26) below:
𝑛
∑︁
𝑑*𝑖 = 𝜐𝑖𝑗 , 𝜐˜𝑗* )
𝑑(˜ (24)
𝑗=1
𝑛
∑︁
𝑑−
𝑖 = 𝜐𝑖𝑗 , 𝜐˜𝑗− )
𝑑(˜ (25)
𝑗=1
√︂
1
𝑎, ˜𝑏) =
𝑑(˜ [(𝑎1 − 𝑎2 )2 + (𝑏1 − 𝑏2 )2 + (𝑐1 − 𝑐2 )2 ] (26)
3
Step 9: Compute the closeness coefficient, 𝐶𝐶𝑖 using Equation (27):
𝑑−
𝐶𝐶𝑖 = 𝑖
(27)
𝑑−
𝑖 + 𝑑*𝑖
where,𝑖 = 1, 2, 3, · · · , 𝑚.
Step 10: Arrange the alternatives based on the calculated values of the closeness coefficient to the
ideal solution in descending order. It can be understood that the optimal alternative, characterized by
the highest 𝐶𝐶𝑖 value, is nearer to the FPIS and more distant from the FNIS. Figure 1 shows the visual
representation of the Fuzzy TOPSIS process.
Figure 1: Fuzzy TOPSIS Steps
5. Illustrative Example
Today, feedback system is very important for many institutions. This system can help improve a
teacher’s performance, behavior during the class, methods of teaching as well as style of teaching
among other things. The feedback system has many items such as “timeliness,” “communication skill,”
“control of the class,” and many others. Importance of performance is always an influential factor on
a venture example if a factor has extremely high performance. Now we define secondary linguistic
values for performance factors which are described in Table 1 and for secondary linguistic values of
significance factor described in T. This paper will employ the fuzzy TOPSIS approach to present an
empirical application for the evaluation of the teaching effect of college. Hence in this paper, three
expert decision makers, namely 𝐷𝑖 (𝑖 = 1, 2, 3, 4, 5) have been selected to complete the feedback form
to assess the faculties’ teaching performance where five faculties, 𝐹𝑖 (𝑖 = 1, 2, 3, 4, 5) are involved and
some criteria are used. Twelve benefit criteria are considered: Twelve benefit criteria are considered:
C1. Timeliness
C2. Attempt to complete syllabus
C3. Whether well performed and enough Knowledgeable about the topic
C4. Communication skill
C5. Control of the class
C6. Involvement in doubt clearance
C7. When attending the class, present the material in a clear manner
C8. Daily reminder of class assignments and adherence to lecture plan
C9. Proper experimental guidance
C10. Responsibility
C11. Teacher is approachable outside the class
C12. assessment is a guide and a well-wisher
In this paper, the assessments conducted by the decision makers concerning the criteria are presented
in Table 3for Decision Maker (D1), Table 4 for Decision Maker (D2) and Table 5 for Decision Maker
(D1) so on, which are further clarified by Figures 2- 4 through the use of line charts. Furthermore, the
linguistic variables related to the criteria are converted into Triangular Fuzzy Numbers (TFNs) utilizing
the scales, as illustrated in Equation (16).
Table 3
Decision Maker (D1)
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
F1 (7 8 9) (7 8 9) (7 8 9) (8 9 10) (7 8 9) (4 5 6) (3 4 5) (7 8 9) (4 5 6) (7 8 9) (7 8 9) (8 9 10)
F2 (3 4 5) (2 3 4) (3 4 5) (3 4 5) (2 3 4) (3 4 5) (4 5 6) (2 3 4) (3 4 5) (3 4 5) (2 3 4) (3 4 5)
F3 (8 9 10) (7 8 9) (8 9 10) (3 4 5) (4 5 6) (3 4 5) (7 8 9) (8 9 10) (7 8 9) (7 8 9) (8 9 10) (7 8 9)
F4 (3 4 5) (3 4 5) (1 2 3) (1 2 3) (3 4 5) (3 4 5) (4 5 6) (2 3 4) (4 5 6) (3 4 5) (1 2 3) (3 4 5)
F5 (2 3 4) (3 4 5) (2 3 4) (1 2 3) (4 5 6) (1 2 3) (1 2 3) (1 1 1) (1 2 3) (3 4 5) (4 5 6) (4 5 6)
Table 4
Decision Maker (D2)
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
F1 (7 8 9) (7 8 9) (8 9 10) (8 9 10) (7 8 9) (8 9 10) (7 8 9) (7 8 9) (7 8 9) (3 4 5) (7 8 9) (3 4 5)
F2 (2 3 4) (2 3 4) (3 4 5) (3 4 5) (3 4 5) (3 4 5) (2 3 4) (3 4 5) (2 3 4) (4 5 6) (3 4 5) (4 5 6)
F3 (7 8 9) (4 5 6) (3 4 5) (7 8 9) (8 9 10) (7 8 9) (8 9 10) (7 8 9) (7 8 9) (7 8 9) (7 8 9) (7 8 9)
F4 (3 4 5) (3 4 5) (1 2 3) (4 5 6) (3 4 5) (3 4 5) (1 2 3) (3 4 5) (3 4 5) (4 5 6) (3 4 5) (4 5 6)
F5 (3 4 5) (4 5 6) (1 2 3) (1 2 3) (2 3 4) (4 5 6) (4 5 6) (3 4 5) (3 4 5) (1 2 3) (3 4 5) (1 2 3)
Table 5
Decision Maker (D3)
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
F1 (3 4 5) (3 4 5) (7 8 9) (8 9 10) (4 5 6) (7 8 9) (7 8 9) (7 8 9) (7 8 9) (7 8 9) (8 9 10) (7 8 9)
F2 (4 5 6) (4 5 6) (3 4 5) (3 4 5) (3 4 5) (2 3 4) (3 4 5) (2 3 4) (2 3 4) (3 4 5) (3 4 5) (2 3 4)
F3 (7 8 9) (7 8 9) (7 8 9) (7 8 9) (7 8 9) (4 5 6) (7 8 9) (7 8 9) (8 9 10) (8 9 10) (3 4 5) (7 8 9)
F4 (4 5 6) (4 5 6) (3 4 5) (3 4 5) (4 5 6) (3 4 5) (3 4 5) (3 4 5) (1 2 3) (3 4 5) (1 2 3) (3 4 5)
F5 (1 2 3) (1 2 3) (3 4 5) (4 5 6) (1 2 3) (4 5 6) (3 4 5) (3 4 5) (4 5 6) (2 3 4) (1 2 3) (3 4 5)
Figure 2: Line chart of Decision Maker 1
Figure 3: Line chart of Decision Maker 2
Figure 4: Line chart of Decision Maker 3
5.1. Ranking the Alternatives
Using combined decision matrix with aggregated fuzzy weights of importance criteria 𝑤𝑗 is calculated
by Equation (18).
Table 6
Combined Decision Matrix
W3 W1 W3 W3 W4 W5 W4 W1 W2 W4 W1 W2
F1 (3,6.7,9) (3,6.7,9) (7,8.3,10) (8,9,10) (4,7,9) (4,7.3,10) (3,6.7,9) (7,8,9) (4,7,9) (3,6.7,9) (7,8.3,10) (3,7,10)
F2 (2,4,6) (2,3.6,6) (3,4,5) (3,4,5) (2,3.6,5) (2,3.7,5) (2,4,6) (2,3.3,5) (2,3.3,5) (3,4.3,6) (2,3.7,5) (2,4,6)
F3 (7,8.3,10) (4,7,9) (3,7,10) (3,6.6,9) (4,7.3,10) (3,5.6,9) (7,8.3,10) (7,8.3,10) (7,8.3,10) (7,8.3,10) (3,7,10) (7,8,9)
F4 (3,4.3,6) (3,4.3,6) (1,2.7,5) (1,3.7,6) (3,4.3,6) (3,4,5) (1,3.7,6) (2,3.7,5) (1,3.7, 6) (3,4.3,6) (1, 2.6,5) (3,4.3,6)
F5 (1,3,5) (1,3.7,6) (1,3,5) (1,3,6) (1,3.3,6) (1,4,6) (1,3.7,6) (1,3,5) (1,3.7,6) (1,3,5) (1,3.7,6) (1,3.7,6)
Equation(19)-(20), calculated the normalized decision matrix which are shown in Table 7.
Table 7
Normalized Fuzzy Decision Matrix
W3 W1 W3 W3 W4 W5 W4 W1 W2 W4 W1 W2
F1 (0.3,0.7,0.9) (0.3,0.7,1) (0.7,0.8,1) (0.8,0.9,1) (0.4,0.7,0.9) (0.4,0.7,1) (0.3,0.6,0.9) (0.7,0.8,0.9) (0.4, 0.7,0.9) (0.3,0.6,0.9) (0.7,0.8,1) (0.3,0.7,1)
F2 (0.2,0.4,0.6) (0.2,0.4,0.6) (0.3,0.4,0.5) (0.3,0.4,0.5) (0.2,0.3,0.5) (0.2,0.3,0.5) (0.2,0.4,0.6) (0.2,0.3,0.5) (0.2,0.3,0.5) (0.3,0.4,0.6) (0.2,0.3,0.5) (0.2,0.4,0.6)
F3 (0.7,0.8,1) (0.4,0.7,1) (0.3,0.7,1) (0.3,0.6,0.9) (0.4,0.7,1) (0.3,0.5,0.9) (0.7,0.8,1) (0.7,0.8,1) (0.7,0.83,1) (0.7,0.8,1) (0.3,0.7,1) (0.7,0.8,0.9)
F4 (0.3,0.4,0.6) (0.3,0.4,0.6) (0.1,0.27,0.5) (0.1,0.3,0.6) (0.3,0.4,0.6) (0.3,0.4,0.5) (0.1,0.37,0.6) (0.2,0.37,0.5) (0.1,0.37,0.6) (0.3, 0.4,0.6) (0.1,0.27,0.5) (0.3,0.4,0.6)
F5 (0.1,0.3,0.5) (0.1,0.4,0.67) (0.1,0.3,0.5) (0.1,0.3,0.6) (0.1,0.3,0.6) (0.1,0.4,0.6) (0.1,0.3,0.6) (0.1,0.3,0.5) (0.1,0.3,0.6) (0.1,0.3,0.5) (0.1,0.3,0.6) (0.1, 0.3,0.6)
Equation (21)-(23) compute 𝑉 matrix which are shown in Table 8.
Table 8
Weighted Normalized Fuzzy Decision Matrix
W3 W1 W3 W3 W4 W5 W4 W1 W2 W4 W1 W2
F1 (0.12,3.3,5.4) (2.6,6.6,10) (2.8,4.1,6) (3.2,4.5,6) (1.2,2.8,4.5) (0.8,2.2,4) (0.9,2.6,4.5) (5.6,7.2,9) (2.8,5.6, 8.1) (0.9,2.67,4.5) (5.6,7.5,10) (2.1,5.6,9)
F2 (0.08,2,3.6) (1.7,3.6,6.6) (1.2,2,3) (1.2,2,3) (0.6,1.4,2.5) (0.4,1.1,2) (0.6,1.6,3) (1.6,3,5) (1.4,0.03,4.5) (0.9,1.7,3) (1.6,3.3,5) (0.02,3.2,5.4)
F3 (0.2,4.1,6) (3.5,7,10) (1.2,3.5,6) (1.2,3.3,5.4) (1.2,2.9,5) (0.6,1.7,3.6) (2.1,3.3,5) (5.6,7.5,10) (4.9,0.08,9) (2.1,3.3,5) (2.4,6.3,10) (0.07,6.4,8.1)
F4 (0.1,2.1,3.6) (2.6,4.3,6.6) (0.4,1.3,3) (0.4,1.8,3.6) (0.9,1.7,3) (0.6,1.2,2) (0.3,1.4,3) (1.6,3.3,5) (0.7,0.03,5.4) (0.9,1.7,3) (0.8,2.4,5) (0.03,3.4,5.4)
F5 (0.04,1.5,3) (0.8,3.6,6.6) (0.4,1.5,3) (0.4,1.5,3.6) (0.3,1.3,3) (0.2,1.2,2.4) (0.3,1.4,3) (0.8,2.7,5) (0.7,0.03, 5.4) (0.3,1.2,2.5) (0.8,3.3,6) (0.01,2.9,5.4)
A* (0.2,4.1,6) (3.5,7,10) (2.8,4.1,6) (3.2,4.5,6) (1.2,2.9,5) (0.8,2.2,4) (2.1,3.3,5) (5.6,7.5,10) (4.9,5.6,9) (2.1, 3.3,5) (5.6,7.5,10) (2.1, 6.4, 9)
A- (0.04,1.5,3) (0.8,3.6,6.6) (0.4,1.3,3) (0.4,1.5,3) (0.3,1.3,2.5) (0.2,1.1,2) (0.3,1.4,3) (0.8,2.7,5) (0.7,0.03,4.5) (0.3,1.2,2.5) (0.8,2.4,5) (0.01,2.9,5.4)
Equation (24)- (26) show the process of measuring distance of each of the alternative of FPIS and
FNIS. In Table 8, the alternatives are organized according to their closeness coefficients in relation to
the ideal solution. The most favorable alternative is determined as the one that is the farthest from the
FNIS and nearest to the FPIS, distinguished by a higher 𝐶𝐶𝑖 score derived from Equation (27), as shown
in Table 9. Furthermore, Figure 5 depicts the ranking of the alternatives using a 3D pie chart.
Table 9
Rankings of the alternatives
RANK 𝐶𝐶𝑖 FACULTY
1 0.853677381 F1
3 0.241583701 F2
2 0.756295088 F3
4 0.235655207 F4
5 0.197371895 F5
6. Comparison with other Methods
The Analytic Hierarchy Process (AHP) [23], Data Envelopment Analysis (DEA) [24], and the Technique
for Order of Preference by Similarity to Ideal Solution (TOPSIS) [25] are well-established multi-criteria
decision-making (MCDM) techniques that are extensively applied in diverse domains, including supply
chain management, healthcare, and environmental sustainability.
AHP is a methodical approach that enables decision-makers to create a hierarchical model of complex
Figure 5: Ranking of the Alternative
issues. It entails breaking down an issue into its component elements so that criteria and options
can be compared pairwise. By translating subjective assessments into numerical values that are then
combined to create priority weights, AHP makes it easier to quantify subjective assessments. This
approach is appropriate for supplier selection and project prioritization.
On the other hand, DEA is a non-parametric technique that compares the input-output ratios of
decision-making units (DMUs) to assess their efficiency. In order to identify best practices and
benchmark, DEA operates under the premise that the DMUs are operating in a similar environment.
It is especially useful in situations when there are several inputs and outputs, such measuring the
effectiveness of production processes or the performance of healthcare facilities.
The idea of the geometric distance from an ideal solution, however, is the foundation of TOPSIS. It
takes into account the worst-case situation and evaluates options according to how close they are to the
ideal solution. Because TOPSIS clearly ranks solutions according to their performance, it is especially
helpful when decision-makers need to compare several options to a set of criteria.
By incorporating fuzzy logic into the conventional TOPSIS framework, fuzzy TOPSIS overcomes these
drawbacks. Decision-makers can now communicate their preferences in a more nuanced way thanks
to this adaption, which permits the portrayal of ambiguity and uncertainty in the decision-making
process. A ranking of options can be obtained by processing fuzzy integers that reflect criteria weights
and performance ratings in fuzzy TOPSIS.
While comparing them, DEA is excellent at assessing efficiency across a variety of inputs and outputs,
AHP is useful for combining subjective assessments and qualitative criteria. TOPSIS is appropriate for
situations needing prompt selections based on a number of factors since it provides a simple method for
evaluating possibilities. Fuzzy TOPSIS has a major edge in managing ambiguity and uncertainty, which
makes it especially helpful in complicated decision-making situations where qualitative considerations
are important. Table 10 shows the ranking comparison of the alternatives of the discussed methods for
the feedback system.
Table 10
Comparison of the results with other methods
RANK FACULTTY
Fuzzy TOPSIS F1 > F3 > F2 > F4 > F5
AHP F1 > F3 > F2 > F5> F4
Traditional TOPSIS F1 > F3 > F2 > F5> F4
DEA F1 > F3 > F2 > F5> F4
7. Conclusion
In this study, the implementation of Fuzzy TOPSIS utilized real-world data pertaining to the evaluation
of faculty selection in an academic institution. The feedback regarding faculty teaching effectiveness
tends to be general and ambiguous, often lacking quantifiable data from students, which leads to a
reliance on the opinions of decision-makers. The said method effectively transforms the linguistic
variables representing the preferences of decision-makers into Triangular Fuzzy Numbers (TFNs). This
approach proves advantageous in addressing these issues, as it allows for the measurement of criterion
weights and the assessment of all options relative to each criterion. To test the hypothesis that F1 >
F3 > F2 > F4 > F5, the results indicated a discernible difference in the evaluation of the alternatives.
Consequently, F1 emerged as the most preferred option, while F5 was identified as the least preferred
candidate. Therefore, the study’s objective of evaluating criterion weights and decision-making options
for conducting Fuzzy TOPSIS analysis in faculty selection has been clearly articulated
8. Future Scope
In our upcoming initiatives, the established model and algorithm will play a crucial role in tackling a
range of real-world multi-attribute group decision-making (MAGDM) issues, employing supplementary
aggregation operators including the Einstein geometric aggregation operator, Bonferroni mean
aggregation operator, and Yager ordered weighted average (OWA) aggregation operators.
9. Declaration on Generative AI
The author(s) have not employed any Generative AI tools.
References
[1] G. K. Koulinas, O. E. Demesouka, P. K. Marhavilas, A. P. Vavatsikos, D. E. Koulouriotis, Risk
assessment using fuzzy topsis and prat for sustainable engineering projects, Sustainability 11
(2019) 615. doi:https://doi.org/10.3390/su11030615.
[2] M. I. G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, An algorithmic method to extend topsis for
decision-making problems with interval data, Applied Mathematics and Computation 175 (2006)
1375–1384. doi:https://doi.org/:10.1016/j.amc.2005.08.048.
[3] Z. Yue, An extended topsis for determining weights of decision makers with interval numbers,
Applied Mathematics and Computation 24 (2011) 146–153. doi:https://doi.org/10.1016/j.
knosys.2010.07.014.
[4] C.T.Chen, Extensions of the topsis for group decision-making under fuzzy environment,
Applied Mathematics and Computation 114 (2000) 1–9. doi:https://doi.org/10.1016/
S0165-0114(97)00377-1.
[5] P. Prokopowicz, The use of ordered fuzzy numbers for modelling changes in dynamic processes,
Information Sciences 470 (2019) 1–14. doi:https://doi.org/10.1016/j.ins.2018.08.045.
[6] D. Kacprzak, Fuzzy topsis method for group decision making, Multiple Criteria Decision Making
13 (2018) 116–132. doi:https://doi.org/10.22367/mcdm.2018.13.07.
[7] D. Kacprzak, A doubly extended topsis method for group decision making based on ordered fuzzy
numbers, Expert Systems with Applications 116 (2019) 243–254.
[8] S.-H. Tsaur, T.-Y. Chang, C.-H. Yen, The evaluation of airline service quality by fuzzy mcdm,
Tourism Management 23 (2002) 107–115. doi:https://doi.org/10.1016/S0261-5177(01)
00050-4.
[9] H. Deng, C. Yeh, R.J.Willis, Inter-company comparison using modified topsis with objective
weights, Computers and Operations Research 27 (2000) 963–973.
[10] R.C.Wang, T.F.Liang, Application of fuzzy multi-objective linear programming to aggregate
production planning, Computers and Industrial Engineering 46 (2004) 17–41. doi:https://doi.
org/10.1016/j.cie.2003.09.009.
[11] T. Chu, Facility location selection using fuzzy topsis under group decisions, International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10 (2002) 687–701. doi:https:
//doi.org/10.1016/j.cie.2003.09.009.
[12] M.A.Abo-Sinna, , A.H.Amer, Extensions of topsis for multi-objective large-scale nonlinear pro-
gramming problems, Applied Mathematics and Computation 162 (2005) 243–256. doi:https:
//doi.org/10.1016/j.amc.2003.12.087.
[13] M. Rajak, K. Shaw, Technology in society evaluation and selection of mobile health (mhealth)
applications using ahp and fuzzy topsis, Technology in Society 59 (2019). doi:https://doi.org/
10.1016/j.techsoc.2019.101186.
[14] S. Kumar, S. Kumar, A. Barman, Supplier selection using fuzzy topsis multi criteria model for a
small scale steel manufacturing unit, Procedia Computer Science 133 (2018) 905–912. doi:https:
//doi.org/10.1016/j.procs.2018.07.097.
[15] S. Husin, F. Fachrurrazi, M. Rizalihadi, M. Mubarak, Implementing fuzzy topsis on project risk
variable ranking, Hindawi Advances in Civil Engineering (2019) 1–11. doi:https://doi.org/
10.1155/2019/9283409.
[16] D. Ghosh, A. Pal, Analysis of faculty teaching using multi-criteria decision making approach,
Hindawi Advances in Civil Engineering 7 (2018) 74–78.
[17] Fachrurrazi, Y. Away, S.Husin, The weights detection of multi-criteria by using solver, International
Journal of Electrical and Computer Engineering (IJECE) 7 (2017) 858–868. doi:10.11591/ijece.
v7i2.pp858-868.
[18] L. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. doi:https://doi.org/10.
1016/S0019-9958(65)90241-X.
[19] H. J. Zimmermann, Fuzzy Set Theory and its Applications, 4 ed., Kluwer Academic Publishers,
Boston/Dordredcht/London, 1991.
[20] C.L.Hwang, K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer-
Verlag, New York, 1981.
[21] A. N. Gani, S. N. M. Assarudeen, A new operation on triangular fuzzy number for solving
fuzzy linear programming problem, Applied Mathematical Sciences 6 (2012) 525–532. doi:https:
//doi.org/10.1016/S0019-9958(65)90241-X.
[22] A.Awasthi, S. Chauhan, H. Omrani, Application of fuzzy topsis in evaluating sustainable
transportation systems, Expert Systems with Applications 38 (2011) 12270–12280. doi:http:
//dx.doi.org/10.1016/j.eswa.2011.04.005.
[23] J. Tian, Z. Yan, Fuzzy analytic hierarchy process for risk assessment to general-assembling of
satellite, Journal of Applied Research and Technology 11 (2013) 568–577. doi:https://doi.org/
10.1016/S1665-6423(13)71564-5.
[24] Y. Goksen, O.Dogan, B.Ozkarabacak, A data envelopment analysis application for measur-
ing efficiency of university departments, Procedia Economics and Finance 19 (2015) 226–237.
doi:https://doi.org/10.1016/S2212-5671(15)00024-6.
[25] M. M. M, H. Taherdoost, A comprehensive guide to the topsis method for multi-criteria decision
making, Sustainable Social Development 1 (2023) 1–6. doi:https://doi.org/10.54517/ssd.
v1i1.2220.