EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1727-1736
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On The Picture Fuzzy-TOPSIS

Raden Sulaiman1,∗, Yuliani P. Astuti1, Dwi Nur Yunianti1, N Azzah Awang2

1 Mathematics Department, Faculty of Mathematics and Natural Sciences,
State University of Surabaya, Surabaya, East Java, Indonesia
2 Faculty of Computer and Mathematical Sciences, UiTM Shah Alam, Shah Alam, Malaysia

Abstract. Research related to the Technique for Order Preference by Similarity to Ideal Solotion
(TOPSIS) method have been developed using fuzzy or intuitionistic fuzzy sets. In this article,
we provide the development of the topsis method based on the generalization of fuzzy sets. We
present a new method of topsis based on picture fuzzy sets. In this article, we propose steps of
TOPSIS picture fuzzy sets method. Finally, we presented the illustrative example of this method.

2020 Mathematics Subject Classifications: 03E72, 08A72, 94D05

Key Words and Phrases: TOPSIS, Picture Fuzzy Sets

1. Introduction

In the development of set theory, classical sets were expanded into fuzzy sets by Zadeh
[18]. He has generalized classical sets theory to fuzzy sets by allowing intermediate situ-
ations between the whole and nothing. In fuzzy sets, the membership function replaced
the characteristic function in classical sets. Then Atanassov [3, 4] expanded again concept
of fuzzy sets into intuitionistic fuzzy sets. Due to the limitations of membership values in
intuitionistic fuzzy sets, Yunianti [16] developed the concept of intuitionistic fuzzy set into
collection of intuitionistic fuzzy set, Yager [15] introduced a general class of intuitionistic
fuzzy sets. Then the researchers developed the concept of picture-fuzzy sets. Cuong and
Kreinovich [6] introduced the concept of picture-fuzzy sets. Then Dinh and Thao [7], and
Dutta [8] did further research. There has been a lot of research on the application of the
fuzzy concept of fuzzy TOPSIS. In previous studies, the intuitive fuzzy topsis method was
introduced by Boran [5], Rouyendegh [12, 13], Tlig and Rebai [14], and was used by Astuti
et al [2] who used Intuitionistic Fuzzy Topsis with Euclidean distance to determine the
dominant factors that influence the resilience of COVID-19 patients. Meanwhile, Ashraf
et al. [1] developed the method for multiple criteria decision making (MCDM) problem

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5225

Email addresses: radensulaiman@unesa.ac.id (R. Sulaiman), yulianipuji@unesa.ac.id (Y.P. Astuti),
dwiyunianti@unesa.ac.id (D N. Yunianti), azzahawang@uitm.edu.my (N Azzah Awang)

https://www.ejpam.com 1727 © 2024 EJPAM All rights reserved.



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1728

for picture fuzzy environment. Some researchers have written the application of pic-
ture fuzzy and TOPSIS. Jiulin et.al [10] developed Picture Fuzzy-TOPSIS method. This
method based on covering-based picture fuzzy rough set (CPRS). Turdan and Kahraman
[11] proposed method that combined concepts of the picture fuzzy, Z-AHP, and TOPSIS.
While Zeng et.al [19] developed application of extended version of linguistic picture fuzzy-
TOPSIS method in Enterprise Resource Planning Systems. Somewhat different from all
the methods above, we developed a method which is a combination of picture fuzzy and
TOPSIS. This article proposed the new method in MCDM, namely picture fuzzy-TOPSIS
method. This method simpler and gives good result in making decisions. Therefore, the
construction of the TOPSIS method on the Picture-Fuzzy set is very important to do.

2. Preliminaries

2.1. Picture Fuzzy Set (PFS)

In this part, we gave some basic definitions and results which will be used later on. We
wrote again preliminary the concepts of intuitionistic fuzzy sets (IFS), and picture fuzzy
sets (PFS).

Every crisp set X can be represented as fuzzy set A = {(x, 1), x ∈X}. The definition of
fuzzy set the first introduces by Zadeh [17] in 1965. Then in 1986 Atanassov [3] generalized
the concept of fuzzy set by introducing the concept of intuitonistic fuzzy set.

Definition 1. Let X be a crisp set. Intuitionistic Fuzzy Set (IFS) A of X is defined as
A = {(x, µA(x), vA(x)) : x ∈ X},where:
0 ≤ µA(x) ≤ 1, 0 ≤ vA(x) ≤ 1, and 0 ≤ µA(x) + vA(x) ≤ 1, for all x ∈ X.

The function µA is called the membership function and the the function vA is called
non-membership function. The number µA(x) is called the degree of membership of x to
the set A, while the vA(x) is called the degree of non-membership of x to the set A. The
amount πA(x) = 1 − µA(x) − vA(x) is called the degree of indeterminacy or hesitation
part, which may cater to either membership value or non-membership value or both.

Every fuzzy set A = {(x, µA(x)) : x ∈ X} can be viewed as intuitionistic fuzzy set
A = {(x, µA(x), 1 − µA(x)) : x ∈ X}. In 2017, Yanger [15] introduced the picture fuzzy
set (PFS) as an extension of FS and IFS.

Definition 2. (see [6])Let X be a crisp set. Picture fuzzy set (PFS) A of X is defined as
A = {(x, µA(x), vA(x), γA(x)) : x ∈ X},where:
0 ≤ µA(x) ≤ 1, 0 ≤ vA(x) ≤ 1, 0 ≤ γA(x) ≤ 1 and 0 ≤ µA(x) + vA(x) + γA(x) ≤ 1, for all
x ∈ X.

Example 1. Let X = {x1, x2, x3}. The following is an example of PFS of X.
A = {(x1, 0.6, 0.2, 0.1), (x2, 0.5, 0.3, 0.2), (x3, 0.6, 0.2, 0.1)},

Every intuitionistic fuzzy set A = {(x, µA(x), vA(x)) : x ∈ X} can be viewed as PFS
A = {(x, µA(x), vA(x), 1−µA(x))−vA(x) : x ∈ X}. The number µA(x) is called the degree



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1729

of membership of x to the set A, the number vA(x) is called the degree of non-membership
of x to the PFS A, while γA(x) is called the degree of neutral membership of x in the PFS
A. The amount 1 − µA(x)− vA(x) − γA(x) is called the degree of refusal membership of
x in the PFS A.

It can be seen that FS is an extension of classical sets, IFS is an extension of FS, and PFS
is an extension of IFS.

2.2. Technique for Order Preference by Similarity to Ideal Solution (TOP-
SIS) method

The developing method that provides a solution to a given multiple criteria decision
making (MCDM) problem is always a challenging endeavor. The one method that has been
developed is the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS).
In general, MCDM problem has the objective of assessing and ranking alternative Ai based
on certain attributes or criteria Cj . Every alternative Ai represents the available options
for the decision maker which requires to be ranked. While every criteria Cj represents
the factors influencing the decision maker’s choice while ranking the alternative Ai. The
weights indicating the relative significance of the criteria Cj were represented by Cj .

In this section, we present a literature review of existing the classical TOPSIS method.
This method proposed by Hwang and Yoon [9] as a simple and useful MCDM method, is
a distance-based method aiming to choose the best alternative with the farthest distance
from the negative ideal solution and the shortest distance from the positive ideal solution.
Decision-makers express their opinions by assigning crisp values in the classical TOPSIS
method.

Below, is shown the five steps of the TOPSIS method.

Step 1: Normalize the decision matrix as follows

Y =


y11 y12 . . . y1j
y21 y22 . . . y2j
. . . . . . . . . . . .
yi1 yi2 . . . yij

 (1)

where
yij =

xij√∑I
i=1 x

2
ij

(2)

and xij is the performance of every alternative Ai for every criteria Cj .

Step 2: Aggregate the criteria weights to the normalized matrix as follows

V =


.v11 v12 . . . v1j
v21 v22 . . . v2j
. . . . . . . . . . . .
vi1 vi2 . . . vij

 (3)



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1730

where vij = W jyij .

Step 3: Find positive ideal solution A+and negative ideal solution A−as follows

A+ =
[
v+1 v+2 . . . v+J

]
,

A− =
[
v−1 v−2 . . . v−J

]
,

(4)

where

v+j =

{
max vij , if Cj is beneficial criteria,

min vij , if Cj is non beneficial criteria,
(5)

and

v−j =

{
min vij , if Cj is beneificial criteria,

max vij , if Cj is non beneficial criteria,
. (6)

Step 4: Compute the separation measure for each alternative as follows

S+
i =

√√√√ J∑
j=1

(
vij − v+j

)2
, (7)

and

S−
i =

√√√√ J∑
j=1

(
vij − v−j

)2
. (8)

Step 5: Compute the closeness coefficient of each alternative to the ideal solution as
follows

Vi =
S−
i

S−
i + S+

i

, (9)

where alternatives are descending ordered by value of V i.

3. Proposed Method

3.1. Picture Fuzzy-TOPSIS method

In this section, we provide the picture fuzzy-TOPSIS method. It is a modification
of TOPSIS method that consist of seven step. The seven setps of picture fuzzy-TOPSIS
method is shown below.

Step 1: Compute the weight of every decision makers

λk =

(
µk + vk

(
µk

µk+γk

))
∑l

k=1

(
µk + vk

(
µk

µk+γk

)) (10)



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1731

where
λk = weight of k-th decision maker,
µk = membership degree of k-th decision maker,
vk = nonmembership degree of k-th decision maker,
γk = neutral degree of k-th decision maker,
and

l∑
k=1

λk = 1. (11)

Step 2: Compute the weight of each criterion as follows

W = (W 1,W 2, . . . ,Wm) (12)

where

Wj = IFWArλ

(
W

(1)
j ,W

(2)
j ,W

(3)
j , . . . ,W

(l)
j

)
= λ1W

(1)
j ⊕ λ2W

(2)
j ⊕ λ3W

(3)
j ⊕ · · · ⊕ λlW

(l)
j

=

(
1−

l∏
k=1

(
1− µ

(k)
j

)λk

,

l∏
k=1

(
v
(k)
j

)λk

,

l∏
k=1

(
γ
(k)
j

)λk

)
.

(13)

Step 3 : Construct the picture fuzzy decision matrix R as follows

R =

 r11 r12 r13 r14 · · · r1m
...

...
...

...
. . .

...
rn1 rn2 rn3 rn4 · · · rmn

 , (14)

where
rij = IFWArλ

(
rij

(1), rij
(2), rij

(3), . . . , rij
(l)
)

= λ1rij
(1) ⊕ λ2rij

(2) ⊕ λ3rij
(3) ⊕ · · · ⊕ λlrij

(l)

=

(
1−

l∏
k=1

(
1− µij

(k)
)λk

,
l∏

k=1

(
vij

(k)
)λk

,
l∏

k=1

(
γij

(k)
)λk

) (15)

Step 4 : Construct the weighted fuzzy picture decision aggregate matrix

The aggregate weighted fuzzy picture decision matrix (R
′
) is obtained from multiplying

the picture fuzzy decision matrix (R) in Step 3 and the weight matrix (W) in Step 2 as
follows.

R′ = R⊗W =
[
r′ij
]

(16)



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1732

where
r′ij =

(
µ′
ij , v

′
ij , γ

′
ij

)
µ′
ij = µij ∗ µj

v′ij = vij + vj − vij · vj
γ′ij = γij + γj − γij ∗ γj

(17)

Step 5 : Find positive ideal solution A+and negative ideal solution A−as follows

A∗ =
(
r′+1 , r′+2 , . . . , r′+n

)
A− =

(
r′−1 , r′−2 , . . . , r′−n

) (18)

where
r′+j =

(
µ′+
j , v′+j , γ′+j

)
, j = 1, 2, . . . , n,

r′−j =
(
µ′−
j , v′−j , γ′−j

)
, j = 1, 2, . . . , n,

µ′+
j =

((
max

i
µ′
ij | j ∈ J1

)
,

(
min
i

µ′
ij | j ∈ J2

))
,

v′+j =

((
min
i

v′ij | j ∈ J1

)
,

(
max

i
v′ij | j ∈ J2

))
,

γ′+j
++

=

((
min
i

γ′ij | j ∈ J1

)
,

(
max

i
γ′ij | j ∈ J2

))
,

µ′−
j =

((
min
i

µ′
ij | j ∈ J1

)
,

(
max

i
µ′
ij | j ∈ J2

))
,

v′−j =

((
max

i
v′ij | j ∈ J1

)
,

(
min
i

v′ij | j ∈ J2

))
,

γ′−j =

((
max

i
γ′ij | j ∈ J1

)
,

(
min
i

γ′ij | j ∈ J2

))
,

(19)

J1 is beneficial criteria, J2 is non beneficial criteria.

Step 6: Compute the separation measure for each alternative as follows

S∗ =

√√√√ 1

2n

n∑
j=1

(
µ′

ij − µ′+
j

)2
+
(
v′ij − v′+j

)2
+
(
γ′ij − γ′+j

)2
,

S− =

√√√√ 1

2n

n∑
j=1

(
µ′
ij − µ′−

j

)2
+
(
v′ij − v′−j

)2
+
(
γ′ij − γ′j

−
)2

.
(20)

Step 7: Compute the closeness coefficient of each alternative to the ideal solution as
follows

Ci∗ =
si−

si+ + si−
, 0 ≤ Ci∗ ≤ 1. (21)



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1733

3.2. Illustrative Example

In this example (for the simulation), we will determine the type of educational YouTube
content that viewers are most interested in. There are 3 alternatives, namely: A1 =school/
college subject matter. A2 =discussion/solving questions, and A3 =enrichment materials.
Meanwhile, the criteria used are: C1 =accuracy of the content, C2 =suitability of the title
to the content, C3 = attractiveness of the introductory video, C4 = existence of related
video links, and C5 =attractiveness of the material presented.

Step 1: Compute the weight of every decision makers.
To obtain the data, we conducted interviews or distributed questionnaires to several re-
spondents. In this example, the respondent are the decision makers and the number of
decision makers are 5: R1, R2, R3, R41, and R5. The importance levels/rating of the de-
cision makers are considered based on linguistic term, that are: Very Very Important
(VVI), Very Important (VI), Important (I), Medium (M), and Unimportant (UI). The
linguistic terms were assigned Picture Fuzzy Number (PFN) and we write them as Dk
= (µA(x), vA(x), γA(x)). The importance levels of the decision-makers in the PFN as
shown in in Table 1 below.

Table 1: Criteria Important level of the decision-makers and PFN’s.

Criteria/rating Important Level PFN’s (Dk)

V V I (0.9, 0.1, 0)
V I (0.7, 0.2, 0.1)
I (0.6, 0.3, 0.1)
M (0.5, 0.4, 0.1)
UI (0.3, 0.5, 0.2)

By using the formula (10) we get the weight of every decision-makers as Table 2 below.

Table 2: Criteria and weight decision makers.

Decision maker weight

R1 0.25773195
R2 0.18556701
R3 0.18556701
R4 0.18556701
R5 0.18556701

Step 2: Compute the weight of each criterion.
The results of data from respondents obtained the data of criteria as in Table 3 below.

By using the formula (12) we get the weight of each criterion as Table 4 below.
Step 3: Based on data from respondents regarding alternative ratings, and by using

formula (15) the following results of matrix R were obtained.



R. Sulaiman et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1727-1736 1734

Table 3: The rating of each criterion.

Decision Maker C1 C2 C3 C4 C5

R1 V V I V V I M Ui V V I
R2 V V I V I V V I V V I V V I
R3 V V I V I V V I V V I V V I
R4 V I V I UI M V I
R5 M V I V I UI V V I

Table 4: The weight each criterion.

Weight of the jth criteria (Wj) PFN ′s

W1 (0.01566, 0.98247, 0)
W2 (0.01566, 0.98247, 0)
W3 (0.011006, 0.98635, 0)
W4 (0.008299, 0.989428, 0)
W5 (0.018566, 0.979975, 0)

R=


(0.65, 0.14, 0.00077) (0.48, 0.23, 0.00219) (0, 62, 0.16, 0.00178)
(0.65, 0.14, 0.000776) (0.48, 0.23, 0.00219) (0, 62, 0.16, 0.00178)
(0.65, 0.14, 0.000776) (0.48, 0.23, 0.00219) (0, 62, 0.16, 0.00178)
(0.65, 0.14, 0.00077) (0.48, 0.23, 0.00219) (0, 62, 0.16, 0.00178)
(0.65, 0.14, 0.00077) (0.48, 0.23, 0.00219) (0, 62, 0.16, 0.00178)


Step 4: By using the formula (17) the following results of matrix R′ were obtained.

R′=


(0.0102, 0.98, 0.0007) (0.0076, 0.98, 0.0021) (0.0098, 0.98, 0.0017)
(0.0102, 0.98, 0.0007) (0.0076, 0.98, 0.0021) (0.0098, 0.98, 0.0017)

(0, 0.14, 0.0117) (0, 0.239, 0.013) (0, 0.16, 0.012)
(0.64, 0.14, 0.0007) (0.48, 0.23, 0.002) (0.62, 0.16, 0.001)
(0.005, 0.99, 0.0007) (0.004, 0.99, 0.002) (0.005, 0.99, 0.001)


Step 5 & 6, and 7: By using the formula (19) we have the positive ideal solution

and negative ideal solution as Table 5 follow.

Table 5: Ideal solution picture fuzzy.

Criteria A+ A−

C1 (0.0102, 0.985, 0.0007) (0.0076,0.98,0.0021)
C2 (0.0102, 0.985, 0.0007) (0.0076,0.98,0.0021)
C3 (0, 0.1429, 0.0117) (0,0.239,0.0131)
C4 (0.64, 0.14, 0.0007) (0.482,0.239,0.0021)
C5 (0.004, 0.99, 0.002) (0.0054,0.99,0.00077)

Then, by using the formula (20) we have:
S∗ for A1 = 0.000702, S∗ for A2 = 0.067083, S∗ for A3 = 0.013221.



REFERENCES 1735

S− for A1 = 0.9962,S− for A2 = 1.0000, S− for A3 = 0.9975.
Finally, by applying the formula (21) we have:
Ci∗ for A1 = 0.999296, Ci∗ for A2 = 0.937135, Ci∗ for A3 = 0.986919. It means, we rank
according to the descending order of Ci∗ to show the measure of relative closeness of each
alternative that is A1, A3, A2.

4. Conclusion

In the classical TOPSIS method, the decision-makers express their opinions by assign-
ing crisp values. However, these crisp values are often insufficient and inadequate for the
solution of real decision-making problems when uncertain and vague information is taken
into account in decision-making. The picture-fuzzy TOPSIS method involves the member-
ship degree, the non-membership degree, and the neutral degree. Therefore, this method
that we provide is better to capture the uncertainty in the evaluations of decision-makers.

Acknowledgement

We sincerely acknowledge the anonymous reviewers for their suggestions which improve
the quality of this article.

References

[1] S. Ashraf, T. Mahmood, S. Abdullah, and Q. Khan. Different approaches to multi-
criteria group decision making problems for picture fuzzy environment. Bulletin of
the Brazilian Mathematical Society, 50(2):373–397, 2019.

[2] Y.P. Astuti. Application of intuitionistic fuzzy topsis in determining online learning
platform during the covid-19 pandemic. Advanced in Social Science, Education an
Humanities Research, 529:–, 2021.

[3] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20:87–96, 1986.

[4] K. Atanassov. Intuitionistic Fuzzy Sets Theory and Applications. Springer-Verlag,
Berlin, Heidelberg, 1999.

[5] F.E. Boran, S. Genç, M. Kurt, and D. Akay. A multi-criteria intuitionistic fuzzy
group decision making for supplier selection with topsis method. Expert Systems with
Applications, 36(8):11363–11368, 2009.

[6] B. C. Cuong and V. Kreinovich. Picture fuzzy sets - a new concept for computational
intelligence problems. In Third World Congress on Information and Communication
Technologies 2013., pages 1–6, -, 2013. IEEE.

[7] N. V. Dinh and N. X. Thao. Some measures of picture fuzzy sets and their application
in multi-attribute decision making. International Journal of Mathematical Science
and Computing, 4:23–41, 2018.



REFERENCES 1736

[8] P. Dutta. Medical diagnosis based on distance measures between picture fuzzy sets.
International Journal of Fuzzy Systems, -:15–36, 2018.

[9] C. Hwang and K. Yoon. Multiple Attribute Decision Making-Method. Springer,
Newyork, 1981.

[10] J. Jiulin, P. Zhao, and Taijie You. Picture fuzzy topsis method based on cpfrs model:
An application to risk management problems. Scientific Programming, Maret:1–15,
2021.

[11] T. Nurdan and C. Kahraman. An integrated picture fuzzy z-ahp topsis methodology:
Application to solar panel selection. Applied Soft Computing, 149:Part A, 2023.

[12] B.D. Rouyendegh. Developing an integrated anp and intuitionistic fuzzy topsis model
for supplier selection. Journal of Testing and Evaluation, 43(3):664–672, 2015.

[13] B.D. Rouyendegh, A. Yildizbasi, and U.Z.B. Arikan. Using intuitionistic fuzzy topsis
in site selection of wind power plants in turkey. Advances in Fuzzy Systems, -):1–14,
2018.

[14] H. Tlig and A. Rebai. A topsis method based on intuitionistic fazzy value: A case
study of north african airport. Management Science Letters, -):351–358, 2017.

[15] R. Yager. Generalized orthopair fuzzy sets. IEEE Transactions on Fuzzy Sets and
Systems, -:1222–1230, 2017.

[16] D.N. Yunianti and Raden Sulaiman et.all. Some properties of operations in the
collection of intuitionistic fuzzy sets : A novel approach. European Journal Of Pure
and Applied Mathematics, 16(4):2198–2207, 2023.

[17] L.A. Zadeh. Fuzzy sets. Fuzzy Inf. Eng, 6:338–353, 1965.

[18] L.A. Zadeh. Fuzzy logic. IEEE Computer, 21:83–93, 1988.

[19] S. Zeng, M. Qiyas, M. Arif, and T. Mahmood. Extended version of linguistic picture
fuzzy topsis method and its applications in enterprise resource planning systems.
Mathematical Problems in Engineering, Jan:1–8, 2019.