Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

397 
https://internationalpubls.com 

Ranking Fuzzy Numbers by Defuzzification using Volumes 

 

P. N. V. L. Sasikala, P. Phani Bushan Rao
1
 

Department of Mathematics, School of Science, GITAM Deemed to be University, Visakhapatnam, India-530045 

 

Article History: 

Received: 01-06-2024 

Revised: 03-07-2024 

Accepted: 29-07-2024 

Abstract:  

Fuzzy numbers (FNs) are used to represent uncertain and ambiguous data and are mainly 

useful in decision-making. Ranking fuzzy numbers is an important concept in fuzzy set 

theory and has various applications in decision-making, data analysis, artificial 

intelligence, and optimization problems. To overcome the shortcomings in some of the 

existing fuzzy ranking methods, in this paper, we introduce a new ranking method for 

ranking generalized trapezoidal fuzzy numbers (GTrFNs) by a defuzzification technique 

using a score function defined using the volume of the solid obtained by revolving the left 

and right inverse membership functions of GTrFN about a vertical line. This score 

represents the defuzzified value of the GTrFN and is used to rank FNs. The proposed 

ranking method can rank different types of FNs in an effective manner and can be 

implemented in real-time applications like multicriteria decision-making and risk 

analysis. 

Keywords: Generalized Trapezoidal fuzzy numbers; Defuzzification; Volume of the 

solid; Score Function 

 

1 Introduction 

Zadeh (1965) introduced the fuzzy set theory, which determines the impreciseness and ambiguity in 

decision-making problems. Fuzzy number (FN) ranking is an important aspect of decision-making, 

giving the best alternative among options. Ranking FNs is crucial in various fields like decision-

making, data analysis, linear programming, risk analysis, and supply chain management. Several 

methods have been proposed over the years to rank FNs, each with its own benefits and constraints. 

Jain proposed the concept of ranking FNs (1976). Ranking FNs using the centroid concept was 

initiated by Yager (1978). Cheng (1998) proposed a ranking approach using the distance method. 

Later, Yao & Wu (2000) ranked FNs based on the decomposition principle and signed distance. Chen 

& Lu (2001) introduced an approximate approach for ranking FNs considering the left & right 

spreads at each α-level of FN. Later, Wang & Lee (2008) suggested an updated approach to Chu & 

Tsao's (2002) ranking method by considering the FNs based on the area between the centroid and the 

original points of an FN. Using distance minimization, Asady & Zendehnam (2007) ranked FNs. 

Abbasbandy & Hajjari (2009) proposed to rank FNs based on their left and right spreads. 

Additionally, Chen & Chen (2009) proposed to rank FNs according to their heights and spreads, the 

improved distance minimization approach for ranking FNs was proposed by Asady (2011), Nejad 

and Maschinchi (2011) presented a novel FN ranking approach on regions of the left and right sides 

                                                           
1
Corresponding author. Phani Bushan Rao Peddi, Department of Mathematics, GITAM Deemed to be University, Visakhapatnam, Andhra Pradesh, 

India. Email: ppeddi@gitam.edu. First author: P.N.V.L. Sasikala, Email: spalepu@gitam.in 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

398 
https://internationalpubls.com 

utilizing the deviation degree method, which can successfully rank various FNs and their images. 

Later, Chen et al. (2012) proposed a novel ranking algorithm for ranking generalized fuzzy numbers 

(GFNs) with varying left and right heights, Yu et al. (2013) suggested an epsilon deviation-based 

ranking function, Eslamipoor et al. (2015) suggested a novel ranking algorithm for GFNs based on 

Euclidean distance, Rezvani (2015) proposed ranking generalized exponential fuzzy numbers based 

on variance. Chutia (2017) developed a modified epsilon deviation approach for ranking FNs. Dombi 

& Jónás (2020) proposed a new ranking algorithm to rank FNs using a probability-based preference 

intensity index method. Using radius, midpoint, and left & right spread values of TrFNs, Ponnialagan 

et al. (2018) proposed a complete ranking method. Patra (2022) introduced the ranking of 

generalized trapezoidal fuzzy numbers (GTrFNs), considering FNs' mean position, area, and 

perimeter as major factors. Hop (2022) proposed a new ranking method using relative relationships 

and shape characteristics of FNs. Prasad &Sinha (2022) introduced a ranking index using the left & 

right limits of the α-cut integral and mode area integral of the FN. Additionally, Jeevaraj (2022) 

proposed an improved ranking principle on GTrFNs and discussed the drawback of the Marimuthu 

and Mahapatra (2021) ranking approach. A setback in ranking FNs was introduced by Sotoudeh-

Anvari & Sotoudeh-Anvari (2022), which indicates the most confusing state. They addressed some 

of the drawbacks of the existing ranking methods by giving counter-examples and studied the 

setback in fuzzy risk assessment in diabetes prediction. Later, Bihari et al. (2023a) ranked GTrFNs 

based on diagonal distance and mean also applied to supplier selection problem. Bihari et al. (2023b) 

introduced a new geometric approach using centroids to rank GTrFNs and applied it to a Multi-

criteria decision-making (MCDM) problem in selecting the best security guard. Also, Bihari et al. 

(2024c) introduced a complete ranking function based on diagonal distance scores to rank GrTrFNs. 

It also addressed the drawbacks of Marimuthu & Mahapatra's (2021) approach and introduced the 

cocoso approach to solving MCDM problems. 

In this paper, we introduce a new ranking method for ranking GTrFNs by a defuzzification technique 

using a score function defined using the volume of the solid obtained by revolving the left and right 

inverse membership functions of GTrFN about a vertical line. This score represents the defuzzified 

value of the GTrFN and is used to rank FNs. The rest of the paper is divided into six sections: The 

definitions related to the study are given in Section 2, and the proposed method is introduced in 

Section 3. Section 4 presents some properties, and some reasonable properties defined by Wang & 

Kerre (2001) are given in Section 5. Section 6 presents some numerical examples, Section 7 presents 

the comparative study, and Section 8 concludes the paper. 

2 Preliminaries 

The definitions of GFNs in this section are drawn from (Zimmermann, 2013). 

Definition 2.1 If 𝑆 is a universe of discourse and 𝑠 be any particular element of 𝑆. The fuzzy set �̅� 

defined on 𝑆 is a collection of ordered pairs, 

�̅� = {(𝑠, 𝜇𝑍(𝑠))|𝑠 ∈ 𝑆+                                                                                                                          

(1) 

where 𝜇𝑍: 𝑆 → ,0,1- 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

399 
https://internationalpubls.com 

Definition 2.2 A FN �̅�   shown in Fig.1 is a fuzzy subset of real line R with a membership function 

(MF) 𝑓�̅� satisfying the below properties: 

1. 𝑓�̅� is a continuous from R to ,0, 𝑡-, 

2. 𝑓�̅� is strictly increasing on ,𝑟1, 𝑟2-, 

3. 𝑓�̅�(𝑥) = 𝑡, for all 𝑥 ∈ ,𝑟2, 𝑟3-, 

4.𝑓�̅� is strictly decreasing on ,𝑟3, 𝑟4-, 

5.𝑓�̅�(𝑥) = 0, otherwise 

The MF of 𝑓�̅� can be expressed as: 

𝑓�̅� =

{
 

 
𝑓�̅�
𝐿(𝑥);   𝑟1 ≤ 𝑥 ≤ 𝑟2,
𝑡;           𝑟2 ≤ 𝑥 ≤ 𝑟3,

𝑓�̅�
𝑅(𝑥);  𝑟3 ≤ 𝑥 ≤ 𝑟4,

0;              𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒.

                        

(2) 

where 𝑓�̅�
𝐿 ∶ ,𝑟1, 𝑟2- → ,0, 𝑡-, and 𝑓�̅�

𝑅 ∶ ,𝑟3, 𝑟4- → ,0, 𝑡-. 

 

Fig .1. GFN representation 

Definition 2.3 A GTrFN �̅� = (𝑟1, 𝑟2, 𝑟3, 𝑟4; 𝑡), shown in Fig. 2, is a fuzzy subset of the real line R 

with MF defined as follows: 

𝑓�̅�(𝑥) =

{
 
 

 
 𝑡 .

𝑥−𝑟1

𝑟2−𝑟1
/ ,                  𝑖𝑓 𝑟1 ≤ 𝑥 ≤ 𝑟2,

     𝑡,                             𝑖𝑓 𝑟2 ≤ 𝑥 ≤ 𝑟3,

𝑡 .
𝑟4−𝑥

𝑟4−𝑟3
/ ,                 𝑖𝑓 𝑟3 ≤ 𝑥 ≤ 𝑟4,

0,                          𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒.

                 (3) 

here 𝑟1, 𝑟2, 𝑟3, 𝑟4 are real numbers, and 0 ≤ 𝑡 ≤ 1. If 𝑡 = 1, then �̅� is called a trapezoidal fuzzy 

number (TrFN), and if  𝑟2 = 𝑟3, then �̅� = (𝑟1, 𝑟2, 𝑟3; 𝑡)  is called a generalized triangular fuzzy 

number (GTFN). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

400 
https://internationalpubls.com 

 

Fig. 2. MF of GTrFN �̅�. 

Definition 2.4 Thomas et al. (2014) The volume of the solid generated by revolving the region 

between the 𝑦-axis and the graph of a continuous function 𝑥 = 𝑓−1(𝑦) ≥ 0, 𝐿 ≤ 0 ≤ 𝑦 ≤ 𝑤, about a 

vertical line 𝑦 = 𝐿 is 

𝑉 = ∫ 2𝜋(𝑦 − 𝐿)𝑓−1(𝑦)𝑑𝑦
𝑤

0
                                               (4) 

3 Proposed Method 

This section presents the new method to rank FNs by defuzzification using the volume. A score 

function that represents the defuzzified value of a TrFN is defined by using the volume of the solid 

obtained by revolving the images of left and right membership functions (MFs) of the TrFN about a 

vertical line.  

The volumes of the solid obtained by revolving the image of the left membership function (MF) of 

the GTrFN �̅� = (𝑟1, 𝑟2, 𝑟3, 𝑟4; 𝑡) about the vertical line 𝑦 = 𝐿 is given by: 

     𝐿𝑉 = ∫ 2𝜋𝑦𝑓𝐿
−1(𝑦)𝑑𝑦

𝑡

0
.      (5) 

The volumes of the solid obtained by revolving the image of the right membership function (MF) 

about the vertical line 𝑦 = 𝐿 is given by: 

     𝑅𝑉 = ∫ 2𝜋𝑦𝑓𝑅
−1(𝑦)𝑑𝑦

𝑡

0
.         (6) 

  The score function represents the defuzzified value of the GTrFN �̅� = (𝑟1, 𝑟2, 𝑟3, 𝑟4; 𝑡), is defined 

as:       

                 𝑠𝑐𝑜𝑟𝑒(�̅�) = 𝐿𝑉 + 𝑅𝑉 = ∫ 2𝜋𝑦𝑓𝐿
−1(𝑦)𝑑𝑦

𝑡

0
+ ∫ 2𝜋𝑦𝑓𝑅

−1(𝑦)𝑑𝑦
𝑡

0
   (7) 

                                = 2𝜋 0
𝑟1𝑡

2

2
+
(𝑟2−𝑟1)𝑡

2

3
1 + 2𝜋 0

𝑟4𝑡
2

2
−
(𝑟4−𝑟3)𝑡

2

3
1                 (8) 

               𝑠𝑐𝑜𝑟𝑒(�̅�) =
2𝜋𝑡2

3
,(𝑟1 + 𝑟4) + 2(𝑟2 + 𝑟3)-       (9) 

Ranking order 

If �̅�1 and �̅�2 are two GTrFNs, then by using the above score, the ranking order is defined as follows: 

i) �̅�1 is less preferred to �̅�2, expressed as �̅�1 ≺ �̅�2, if 𝑠𝑐𝑜𝑟𝑒(�̅�1) < 𝑠𝑐𝑜𝑟𝑒(�̅�2). 

ii) �̅�1 is more preferred to �̅�2, expressed as �̅�1 ≻ �̅�2, if 𝑠𝑐𝑜𝑟𝑒(�̅�1) > 𝑠𝑐𝑜𝑟𝑒(�̅�2). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

401 
https://internationalpubls.com 

iii) �̅�1 is equal to �̅�2, expressed as �̅�1 ≈ �̅�2, if 𝑠𝑐𝑜𝑟𝑒(�̅�1) = 𝑠𝑐𝑜𝑟𝑒(�̅�2). 

4 Properties 

1) Let 𝐼 ̅ = (𝑖1, 𝑖2, 𝑖3, 𝑖4; 𝑡) & 𝐽 ̅ = (𝑗1, 𝑗2, 𝑗3, 𝑗4; 𝑡) be two GTrFNs, then 

i) 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ + 𝐽)̅ = 𝑠𝑐𝑜𝑟𝑒(𝐼)̅ + 𝑠𝑐𝑜𝑟𝑒(𝐽)̅ 

ii) 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ − 𝐽)̅ = 𝑠𝑐𝑜𝑟𝑒(𝐼)̅ − 𝑠𝑐𝑜𝑟𝑒(𝐽)̅ 

Proof: i) Given 𝐼 ̅ = (𝑖1, 𝑖2, 𝑖3, 𝑖4; 𝑡) & 𝐽 ̅ = (𝑗1, 𝑗2, 𝑗3, 𝑗4; 𝑡) are two GTrFNs. From Eq. (9) we have  

𝑠𝑐𝑜𝑟𝑒(�̅�) =
𝜋

3
𝑡2((𝑟1 + 𝑟4) + 2(𝑟2 + 𝑟3)) 

Now, 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ + 𝐽)̅ =
𝜋

3
𝑡2((𝑖1 + 𝑗1 + 𝑖4 + 𝑗4) + 2(𝑖2 + 𝑗2 + 𝑖3 + 𝑗3)) 

         ⇒ 
𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) +

𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) 

∴ 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ + 𝐽)̅ = 𝑠𝑐𝑜𝑟𝑒(𝐼)̅ + 𝑠𝑐𝑜𝑟𝑒(𝐽)̅ 

Similarly 

ii) 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ − 𝐽)̅ =
𝜋

3
𝑡2((𝑖1 − 𝑗1 + 𝑖4 − 𝑗4) + 2(𝑖2 − 𝑗2 + 𝑖3 − 𝑗3)) 

         ⇒ 
𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) −

𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) 

∴ 𝑠𝑐𝑜𝑟𝑒(𝐼 ̅ + 𝐽)̅ = 𝑠𝑐𝑜𝑟𝑒(𝐼)̅ − 𝑠𝑐𝑜𝑟𝑒(𝐽)̅ 

2) Let 𝐼 ̅ = (𝑖1, 𝑖2, 𝑖3, 𝑖4; 𝑡) be a GTrFN. Then  

i) 𝑠𝑐𝑜𝑟𝑒(𝑘𝐼)̅ = 𝑘𝑠𝑐𝑜𝑟𝑒(𝐼)̅ 

ii) 𝑠𝑐𝑜𝑟𝑒(−𝐼)̅ = −𝑠𝑐𝑜𝑟𝑒(𝐼)̅ 

Proof: i) Given 𝐼 ̅ = (𝑖1, 𝑖2, 𝑖3, 𝑖4; 𝑡) be a GTrFN. From the above Eq. (9) we have 

𝑠𝑐𝑜𝑟𝑒(�̅�) =
𝜋

3
𝑡2((𝑟1 + 𝑟4) + 2(𝑟2 + 𝑟3)) 

Now,                𝑠𝑐𝑜𝑟𝑒(𝑘𝐼)̅ =
𝜋

3
𝑡2((𝑘𝑖1 + 𝑘𝑖4) + 2(𝑘𝑖2 + 𝑘𝑖3)) 

⇒ 𝑠𝑐𝑜𝑟𝑒(𝑘𝐼)̅ = 𝑘
𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) 

∴ 𝑠𝑐𝑜𝑟𝑒(𝑘𝐼)̅ = 𝑘𝑠𝑐𝑜𝑟𝑒(𝐼)̅ 

ii) Given 𝐼 ̅ = (𝑖1, 𝑖2, 𝑖3, 𝑖4; 𝑡) be a GTrFN. From the above Eq. (9) we have 

𝑠𝑐𝑜𝑟𝑒(�̅�) =
𝜋

3
𝑡2((𝑟1 + 𝑟4) + 2(𝑟2 + 𝑟3)) 

Now,       𝑠𝑐𝑜𝑟𝑒(−𝐼)̅ =
𝜋

3
𝑡2((−𝑖1 − 𝑖4) + 2(−𝑖2 − 𝑖3)) 

⇒ 𝑠𝑐𝑜𝑟𝑒(−𝐼)̅ =
𝜋

3
𝑡2(−(𝑖1 + 𝑖4) − 2(𝑖2 + 𝑖3)) 

⇒ 𝑠𝑐𝑜𝑟𝑒(−𝐼)̅ = −
𝜋

3
𝑡2((𝑖1 + 𝑖4) + 2(𝑖2 + 𝑖3)) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

402 
https://internationalpubls.com 

∴ 𝑠𝑐𝑜𝑟𝑒(−𝐼)̅ = −𝑠𝑐𝑜𝑟𝑒(𝐼)̅ 

3) Let 𝐼 = (0,0,0,0; 𝑡) be a GTrFN, then 𝑠𝑐𝑜𝑟𝑒(𝐼) = 0. 

Proof: Given 𝐼 ̅ = (0,0,0,0; 𝑡) be a GTrFN. From the above Eq. (9) we have 

𝑠𝑐𝑜𝑟𝑒(�̅�) =
𝜋

3
𝑡2((𝑟1 + 𝑟4) + 2(𝑟2 + 𝑟3)) 

Now,           𝑠𝑐𝑜𝑟𝑒(𝐼) =
𝜋

3
𝑡2((0 + 0) + 2(0 + 0)) = 0 

∴ 𝑠𝑐𝑜𝑟𝑒(𝐼) = 0 

5 Reasonable properties 

In this section, we present some reasonable properties Wang &Kerre (2001) 

Let 𝑨 be the ordering approach and 𝑩 be the set of fuzzy quantities for which the method 𝑨 can be 

applied. 𝑴 is a finite subset of 𝑩 and �̅�, �̅�, �̅� are elements of 𝑴. 

𝑷𝟏) For an arbitrary finite subset 𝑴 of 𝑩 and �̅� ∈ 𝑩 and �̅� ∈ 𝑴, �̅� ≽ �̅� by 𝑨 on 𝑴. 

𝑷𝟐) For an arbitrary finite subset 𝑴 of 𝑩 and (�̅�, �̅�) ∈ 𝑴𝟐, �̅� ≽ �̅� and �̅� ≽ �̅� by 𝑨 on 𝑴, we should 

have �̅� ∽ �̅� by 𝑨 on 𝑴. 

𝑷𝟑) For an arbitrary finite subset 𝑴 of 𝑩 and (�̅�, �̅�, �̅�) ∈ 𝑴𝟑, �̅� ≽ �̅� and �̅� ≽ �̅� by 𝑨 on 𝑴, we 

should have �̅� ≿ �̅� by 𝑨 on 𝑴. 

𝑷𝟒) For an arbitrary finite subset 𝑴 of 𝑩 and (�̅�, �̅�) ∈ 𝑴𝟐, inf 𝑠𝑢𝑝𝑝(�̅�) > sup 𝑠𝑢𝑝𝑝(�̅�), we should 

have �̅� ≿ �̅� by 𝑨 on 𝑴. 

𝑷𝟒
′ ) For an arbitrary finite subset 𝑴 of 𝑩 and (�̅�, �̅�) ∈ 𝑴𝟐, inf 𝑠𝑢𝑝𝑝(�̅�) > sup 𝑠𝑢𝑝𝑝(�̅�), we should 

have �̅� ≻ �̅� by 𝑨 on 𝑴. 

𝑷𝟓) Let 𝑩 and 𝑩′ be two arbitrary finite sets of fuzzy quantities in which 𝑨 can be applied and  

�̅� & �̅� are in 𝑩 ∩ 𝑩′.  We obtain the ranking order �̅� ≻ �̅�, by 𝑨 on 𝑩′ iff �̅� ≻ �̅� by 𝑨 on 𝑩. 

𝑷𝟔) Let �̅�, �̅�, �̅� + �̅� & �̅� + �̅� be the elements of 𝑩. If �̅� ≿ �̅� by 𝑨 on *�̅�, �̅�+, then �̅� + �̅� ≿ �̅� + �̅� by 

𝑨 on *�̅� + �̅�, �̅� + �̅�+. 

𝑷𝟔
′ ) If �̅� ≻ �̅� by 𝑨 on *�̅�, �̅�+, then �̅� + �̅� ≻ �̅� + �̅� by 𝑨 on *�̅� + �̅�, �̅� + �̅�+ when �̅� ≠ 0. 

𝑷𝟕) Let �̅�, �̅�, �̅��̅� & �̅��̅� be the elements of 𝑩 and �̅� ≥ 0. �̅� ≿ �̅� by 𝑨 on *�̅�, �̅�+, then �̅��̅� ≿ �̅��̅� by 𝑨 

on *�̅��̅�, �̅��̅�+. 

6 Numerical examples 

1) Consider two GTrFNs 𝑄1 = (0.1,0.2,0.3,0.5; 1) & 𝑄2 = (0.1,0.3,0.4,0.6; 1) taken from Le & 

Chu (2023), shown in Fig. 3  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

403 
https://internationalpubls.com 

 

Fig.3 𝑄1 = (0.1,0.2,0.3,0.5; 1.0), 𝑄2 = (0.1,0.3,0.4,0.5; 1.0) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 1.6761, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 2.2, so, the 

ranking order is 𝑄1 ≺ 𝑄2 and our result matches with Le & Chu's (2023) method. 

2) Consider two GTrFNs 𝑄1 = (0.1,0.2,0.3,0.5; 1) & 𝑄2 = (−0.5, −0.3, −0.2, −0.1; 1) taken 

from Le & Chu (2023), shown in Fig. 4 

 

Fig.4 𝑄1 = (0.1,0.2,0.3,0.5; 1.0), 𝑄2 = (−0.5, −0.3, −0.2, −0.1; 1.0) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 1.6761, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = −1.6761, so, the 

ranking order is 𝑄2 ≺ 𝑄1 and our result matches with Le & Chu's (2023) method. 

3) Consider three GTrFNs 𝑄1 = (1,2,3,5; 0.7) & 𝑄2 = (2,4,6,7; 0.6), 𝑄3 = (4,6,6,8; 0.8) taken 

from Bihari et al. (2023), shown in Fig. 5 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

404 
https://internationalpubls.com 

 

Fig.5 𝑄1 = (1,2,3,5; 0.7), 𝑄2 = (2,4,6,7; 0.6), 𝑄3 = (4,6,6,8; 0.8) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 8.2133, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 10.9371, 

𝑠𝑐𝑜𝑟𝑒(𝑄3) = 24.1371, so the ranking order is 𝑄1 ≺ 𝑄2 ≺ 𝑄3 and our result matches with Bihari 

et al. (2023) method. 

4) Consider two FNs 𝑄1 = (2,3,8; 0.8) & 𝑄2 = (2,4,6,8; 0.7) taken from Bihari et al. (2023), 

shown in Fig. 6 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 14.7504, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 15.4, so the 

ranking order is 𝑄1 ≺ 𝑄2 and our result matches with Bihari et al. (2023) method. 

 

Fig.6 𝑄1 = (2,3,3,8; 0.8), 𝑄2 = (2,4,6,8; 0.7) 

5) Consider the following fuzzy sets taken from Haji et al. (2014) shown in Fig. 7. 

𝑔𝑄1 = {

𝑥 − 2, 2 ≤ 𝑥 ≤ 4
6 − 𝑥

2
, 4 ≤ 𝑥 ≤ 6

0,                   𝑜𝑡𝑕𝑒𝑤𝑖𝑠𝑒

 

𝑔𝑄2 = {
𝑥 − 3, 3 ≤ 𝑥 ≤ 5
6 − 𝑥, 5 ≤ 𝑥 ≤ 6
0,                   𝑜𝑡𝑕𝑒𝑤𝑖𝑠𝑒

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

405 
https://internationalpubls.com 

𝑔𝑄3 =

{
 
 

 
 
𝑥 − 3, 3 ≤ 𝑥 ≤ 4
1,                  4 ≤ 𝑥 ≤ 5
7 − 𝑥

2
, 5 ≤ 𝑥 ≤ 7

0,                   𝑜𝑡𝑕𝑒𝑤𝑖𝑠𝑒

 

 

Fig.7 𝑔𝑄1 = (2,4,4,6; 1.0), 𝑔𝑄2 = (3,5,5,6; 1.0), 𝑔𝑄3 = (3,4,5,7; 1.0) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 47.1, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 56.52,       

𝑠𝑐𝑜𝑟𝑒(𝑄3) = 59.66, so the ranking order is 𝑄1 ≺ 𝑄2 ≺ 𝑄3 and our result matches with Haji et al. 

(2014) method. 

6) Consider four FNs 𝑄1 = (0.1,0.2,0.2,0.3; 1) & 𝑄2 = (0.3,0.4,0.4,0.5; 1),                                    

𝑄3 = (0.6,0.7,0.8; 1), 𝑄4 = (0.8,0.9,0.9,1.0; 1) taken from Ponnialagan et al. (2017) shown 

in Fig. 8. 

 

Fig.8 𝑄1 = (0.1,0.2,0.2,0.3; 1.0), 𝑄2 = (0.3,0.4,0.4,0.5; 1.0), 

𝑄3 = (0.6,0.7,0.7,0.8; 1.0), 𝑄4 = (0.8,0.9,0.9,1.0; 1.0) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 1.2571, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 2.5142, 

𝑠𝑐𝑜𝑟𝑒(𝑄3) = 4.4, 𝑄4 = 5.6571, so the ranking order is 𝑄1 ≺ 𝑄2 ≺ 𝑄3 ≺ 𝑄4 and our result 

matches with Ponnialagan et al. (2017) method. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

406 
https://internationalpubls.com 

7) Consider three FNs 𝑄1 = (0.1,0.3,0.3,0.8; 1) & 𝑄2 = (0.4,0.5,0.5,0.6; 1), 𝑄3 =

(1,1,1,1; 1),taken from Ponnialagan et al. (2017) shown in Fig. 9. 

 

Fig.9 𝑄1 = (0.1,0.3,0.3,0.8; 1.0), 𝑄2 = (0.4,0.5,0.5,0.6; 1.0), 𝑄3 = (1,1,1,1; 1.0) 

By applying the proposed method, we get 𝑠𝑐𝑜𝑟𝑒(𝑄1) = 2.2, 𝑠𝑐𝑜𝑟𝑒(𝑄2) = 3.1428, 𝑠𝑐𝑜𝑟𝑒(𝑄3) =

6.2857, so, the ranking order is 𝑄1 ≺ 𝑄2 ≺ 𝑄3 and our result matches with Ponnialagan et al. 

(2017) method. 

7 Comparative study 

7.1) Consider the following fuzzy sets taken from Patra (2022), shown in Fig. 10. 

a) Set I 𝑍1 = (0.1,0.2,0.2,0.3; 1.0), 𝑍2 = (0.1,0.2,0.2,0.3; 0.8) 

b) Set II 𝑍1 = (1,1,1,1; 1.0), 𝑍2 = (1,1,1,1; 0.8), 𝑍3 = (1,1,1,1; 0.5) 

c) Set III 𝑍1 = (5,6,6,7; 1.0), 𝑍2 = (5.9,6,6,7; 1.0), 𝑍3 = (6,6,6,7; 1.0) 

d) Set IV 𝑍1 = (0.4,0.5,0.5,1.0; 1.0), 𝑍2 = (0.4,0.7,0.7,1.0; 1.0), 𝑍3 = (0.4,0.9,0.9,1.0; 1.0) 

e) Set V 𝑍1 = (0.2,0.5,0.5,0.8; 1.0), 𝑍2 = (0.4,0.5,0.5,0.8; 1.0) 

Table 1 Comparative study taken from Patra (2022) 

Methods Set I   Set II   Set 

III 

  Set IV  Set V  

 
   

 
         

Yager (1978) 0.2 0.2  -   -  - 6 6.3  0.6333 0.7 0.7666 0.5 0.5 

Wang et al. (2006) 0.3887 0.3333  -  -  - 6.0092 6.3088 6.3421 0.7157 0.775 0.8359 0.35 0.45 

Chen & 

Sanguansat (2011) 

0.2 0.1882 1 0.9412 0.8 0.8571 0.8892 0.8928 0.6 0.7 0.8 0.5 0.5 

Chen & Chen 

(2009) 

0.1849 0.1479 1 0.8 0.5 0.7676 0.8279 0.8333 0.4721 0.562 0.6295 0.4016 0.462 

Nasseri et al. 

(2013) 

0.89 0.7118 2.5 2.32 2.125 12 12.766 12.853 1.6227 1.817 2.0227 1.4174 1.49 

Rezvani (2015) 0.0116 0.0115  -  -  - 8.952 10.179 10.327 0.1366 0.137 0.1366 0.0782 0.063 

Asady (2010) 0.1666 0.1666 0 0 0 0.6666 0.7101 0.7142 0.1818 0.375 0.8 0.3745 0.375 

𝑍2 𝑍1 𝑍3 𝑍1 

𝑍2 

𝑍1 𝑍2 𝑍3 𝑍1 𝑍2 𝑍3 𝑍1 𝑍2 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

407 
https://internationalpubls.com 

Yu et al. (2013) 1 1 1 1 1 0.0343 12.972 57.601 0.0467       1 
 

21.396 0.1176 0.074 

Abbasbandy & 

Hajjari (2009) 

0.2 0.2 1 1 1 6 6.075 6.0833 0.5333 0.7 0.8666 0.5 0.5 

Chutia (2017) 5.5597 0.213 60.523 1.0805 0.0017 0.0291 13.101 67.996 0.0426 0.934 23.431 0.1263 9.748 

Patra (2022) 0.2 0.131 1 0.8 0.5 6 2.4977 2.212 0.6 0.691 0.8 0.5 0.455 

Proposed method 1.2571 0.8045 6.2857 4.0228 1.5714 37.714 38.657 38.7619 3.5619 4.4 5.238 3.1428 3.352 

 

1) For Set I from Table 1, we can see that Yager (1978), Asady (2010), Yu et al. (2013), and 

Abbasbandy & Hajjari (2009) couldn't give the correct ranking order. From Fig.10, we can 

see that 𝑍1 & 𝑍2 has the same support, but the core is different due to different heights, so the 

ranking order should be 𝑍1 > 𝑍2. Our results match with all the other methods. 

2) For Set II from Table 1, we can see that Yager (1978), Rezvani (2015), and Wang et al. (2006) 

failed to rank the FNs. Asady (2010), Yu et al. (2013). Abbasbandy & Hajjari (2009), gave 

incorrect ranking order. Our ranking order is 𝑍1 > 𝑍2 > 𝑍3 which matches with the other 

methods. 

3) For Set III from Table 1, we can see that Patra (2022) couldn’t give the correct ranking order 

i.e. 𝑍1 < 𝑍2 < 𝑍3, and all other methods' results match with the proposed method's result as 

the x-coordinate centroid values of the FNs are the same as the order of the proposed method. 

4) For Set IV from Table 1, we can see that, the proposed method results match with all the 

other methods i.e. 𝑍1 < 𝑍2 < 𝑍3. 

5) For Set V from Table 1, we can see that Yager (1978), Chen & Sanguansat (2011), Rezvani 

(2015), Asady (2010), Yu et al. (2013), Abbasdandy & Hajjari (2009), Patra (2022) couldn’t 

give the correct ranking order i.e. 𝑍1 < 𝑍2, and all other methods' results match with the 

proposed method's result as the x-coordinate centroid values of the FNs are the same as the 

order of the proposed method. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

408 
https://internationalpubls.com 

 

Fig. 10 Fuzzy Sets taken from Patra (2022) 

7.2) Consider the following fuzzy sets taken from Cheng et al.  (2022), shown in Fig. 11.  

a) Set I 𝑍1 = (0.1,0.3,0.3,0.5; 1.0), 𝑍2 = (−0.5,−0.3, −0.3, −0.1; 1.0) 

b) Set II 𝑍1 = (0.1,0.2,0.4,0.5; 1.0), 𝑍2 = (1,1,1,1; 1.0) 

c) Set III 𝑍1 = (0.1,0.3,0.3,0.5; 0.8), 𝑍2 = (0.1,0.3,0.3,0.5; 1.0)  

d) Set IV 𝑍1 = (0.1,0.3,0.3,0.5; 1), 𝑍2 = (0.3,0.5,0.5,0.7; 1)  

e) Set V 𝑍1 = (0,0.4,0.6,0.8; 1), 𝑍2 = (0.2,0.5,0.5,0.9; 1), 𝑍3 = (0.1,0.6,0.7,0.8; 1.0) 

 

Set I 𝑍1 = (0.1,0.2,0.2,0.3; 1.0), 𝑍2 = (0.1,0.2,0.2,0.3; 0.8) 

 
𝑍3 = (1.0,1.0,1.0,1.0; 0.5) 

Set II 𝑍1 = (1.0,1.0,1.0,1.0; 1.0), 𝑍2 = (1.0,1.0,1.0,1.0; 0.8),                 

Set III 𝑍1 = (5,6,6,7; 1.0), 𝑍2 = (5.9,6,6,7; 1.0), 𝑍3 = (6,6,6,7; 1.0) 

 𝑍3 = (0.4,0.9,0.9,1.0; 1.0) 

Set IV 𝑍1 = (0.4,0.5,0.5,1.0; 1.0), 𝑍2 = (0.4,0.7,0.7,1.0; 1.0),                   

Set V 𝑍1 = (0.2,0.5,0.5,0.8; 1.0), 𝑍2 = (0.4,0.5,0.5,0.8; 1.0)  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

409 
https://internationalpubls.com 

Table 2 Comparative study table taken from Cheng et al. (2022) 

Methods Set I  Set II  Set III  Set IV   Set V  

 
           

Chen et al. (2012) 0.2553 -0.2533 0.2533 1 0.2462 0.2553 0.2553 0.4444 0.4 0.4667 0.5057 

Baker & Gegoy (2014) 0.0867 -0.0867 0.1096 0.3333 0.0715 0.0867 0.0867 0.1444 0.1197 0.1363 0.1452 

Madhuri et al. (2014) 0.5774 0.5774 0.5885  - 0.4934 0.5774 0.5774 0.7024 0.6794 0.7052 0.7684 

Wang (2015) 0.25 0 0.5 1  -  0.5 0.25 0.75 0.4615 0.5119 0.5275 

Jiang (2015) 0.2882 -0.2882 0.2869 1 0.2306 0.2882 0.2882 0.4804 0.4146 0.4898 0.5103 

Wu et al. (2018) 0.5906 -0.5906 0.5884 1 0.5332 0.5906 0.5906 0.7014 0.6506 0.7071 0.7003 

Barazandeh & Ghazanfari (2021) 0.25 -0.25 0.2833 1 0.23 0.25 0.25 0.4167 0.396 0.4444 0.4594 

Cheng et al. (2022) 0.74 0.26 0.36 1 0.332 0.34 0.34 0.5 0.47 0.5214 0.5348 

Proposed method 1.8857 -1.8857 1.8857 6.2857 1.2068 1.8857 1.8857 3.1428 2.934 3.247 3.66 

 

1) For Set I from Table 2, we can see that Baker & Gegoy (2014) couldn’t give the correct 

ranking order i.e. 𝑍2 < 𝑍1 and the proposed method's result matches with all the other 

methods. 

2) For Set II from Table 2, we can see that Madhuri et al. (2014) couldn’t rank the FNs. The 

ranking order is 𝑍1 < 𝑍2 and the proposed method's result matches with all the other 

methods.  

3) For Set III from Table 2, we can see that the proposed method results match with all other 

methods i.e. 𝑍1 < 𝑍2. 

4) For Set IV from Table 2, we can see that the proposed method results match with all other 

methods i.e. 𝑍1 < 𝑍2. 

5) For Set V from Table 2, we can see that Wu et al. (2018) couldn’t give the correct ranking 

order i.e. 𝑍1 < 𝑍2 < 𝑍3 and the proposed method's result matches with all the other methods. 

 

 

 

  

𝑍1 𝑍2 𝑍2 𝑍2 𝑍1 𝑍1 𝑍1 𝑍2 𝑍3 𝑍1 𝑍2 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

410 
https://internationalpubls.com 

 

Fig. 11 Fuzzy Sets taken from Cheng et al.  (2022) 

 

 

Set I 𝑍1 = (0.1,0.3,0.3,0.5; 1.0), 𝑍2 = (−0.5, −0.3, −0.3, −0.1; 1.0)  Set II  𝑍1 = (0.1,0.2,0.4,0.5; 1.0), 𝑍2 = (1.0,1.0,1.0,1.0; 1.0)  

Set III  𝑍1 = (0.1,0.3,0.3,0.5; 0.8), 𝑍2 = (0.1,0.3,0.3,0.5; 1.0)  Set IV  𝑍1 = (0.1,0.3,0.3,0.5; 1.0), 𝑍2 = (0.3,0.5,0.5,0.7; 1.0)  

Set V 𝑍1 = (0,0.4,0.6,0.8; 1.0),  𝑍2 = (0.2,0.5,0.5,0.9; 1.0),                   

                               𝑍3 = (0.1,0.6,0.7,0.8; 1.0)  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

411 
https://internationalpubls.com 

8 Conclusion 

In this paper, we proposed a new method to rank GTrFNs using the concept of defuzzification by a 

score function using the volume of solid by revolving the images of left and right membership 

functions about a vertical line. The ranking score obtained is the defuzzified value of GTrFN and is 

used to select the best alternative from the available alternatives. The proposed method overcomes 

the limitations of some of the existing methods, and it can rank different types of FNs along with 

their images and crisp numbers. The proposed method can be applied to many applications, such as 

risk assessment, decision-making, and optimization problems.  

References 

[1] Abbasbandy, S., & Hajjari, T. (2009). A new approach for ranking of trapezoidal fuzzy numbers. Computers & 

mathematics with applications, 57(3), 413-419. 

[2] Asady, B. (2010). The revised method of ranking LR fuzzy number based on deviation degree. Expert Systems 

with Applications, 37(7), 5056-5060. 

[3] Asady, B. (2011). Revision of distance minimization method for ranking of fuzzy numbers. Applied 

Mathematical Modelling, 35(3), 1306-1313. 

[4] Asady, B. (2011). Revision of distance minimization method for ranking of fuzzy numbers. Applied 

Mathematical Modelling, 35(3), 1306-1313. 

[5] Asady, B., & Zendehnam, A. (2007). Ranking fuzzy numbers by distance minimization. Applied Mathematical 

Modelling, 31(11), 2589-2598. 

[6] Bakar, A. S. A., & Gegov, A. (2014). Ranking of fuzzy numbers based on centroid point and spread. Journal of 

Intelligent & Fuzzy Systems, 27(3), 1179-1186. 

[7] Barazandeh, Y., & Ghazanfari, B. (2021). A novel method for ranking generalized fuzzy numbers with two 

different heights and its application in fuzzy risk analysis. Iranian Journal of Fuzzy Systems, 18(2), 81-91. 

[8] Bihari, R., Jeevaraj, S., & Kumar, A. (2023a). A new ranking principle for ordering generalized trapezoidal 

fuzzy numbers based on diagonal distance, mean and its applications to supplier selection. Soft Computing, 1-

26. 

[9] Bihari, R., Jeevaraj, S., & Kumar, A. (2023b). Geometric approach for ranking generalized trapezoidal fuzzy 

numbers and its application in selecting security guard service company. Expert Systems with Applications, 234, 

121052. 

[10] Bihari, R., Jeevaraj, S., & Kumar, A. (2024c). Complete ranking for generalized trapezoidal fuzzy numbers and 

its application in supplier selection using the GTrF-CoCoSo approach. Expert Systems with Applications, 255, 

124612. 

[11] Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy sets and 

systems, 95(3), 307-317. 

[12] Chen, L. H., & Lu, H. W. (2001). An approximate approach for ranking fuzzy numbers based on left and right 

dominance. Computers & Mathematics with Applications, 41(12), 1589-1602. 

[13] Chen, S. M. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy 

numbers. Expert Systems with Applications, 38(3), 2163-2171. 

[14] Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with 

different heights and different spreads. Expert systems with applications, 36(3), 6833-6842. 

[15] Chen, S. M., Munif, A., Chen, G. S., Liu, H. C., & Kuo, B. C. (2012). Fuzzy risk analysis based on ranking 

generalized fuzzy numbers with different left heights and right heights. Expert Systems with Applications, 39(7), 

6320-6334. 

[16] Cheng, R., Kang, B., & Zhang, J. (2022). A novel method to rank fuzzy numbers using the developed golden 

rule representative value. Applied Intelligence, 52(9), 9751-9767. 

[17] Chu, T. C., & Tsao, C. T. (2002). Ranking fuzzy numbers with an area between the centroid point and original 

point. Computers & mathematics with applications, 43(1-2), 111-117. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

412 
https://internationalpubls.com 

[18] Chutia, R. (2017). Ranking of fuzzy numbers by using value and angle in the epsilon-deviation degree 

method. Applied Soft Computing, 60, 706-721. 

[19] Dombi, J., & Jónás, T. (2020). Ranking trapezoidal fuzzy numbers using a parametric relation pair. Fuzzy sets 

and systems, 399, 20-43. 

[20] Eslamipoor, R., Hosseini-nasab, H., & Sepehriar, A. (2015). An improved ranking method for generalized fuzzy 

numbers based on Euclidian distance concept. Afrika Matematika, 26, 1291-1297. 

[21] Jain, R. (1976). Decision-making in the presence of fuzzy variables. 

[22] Janizade-Haji, M., Zare, H. K., Eslamipoor, R., & Sepehriar, A. (2014). A developed distance method for 

ranking generalized fuzzy numbers. Neural Computing and Applications, 25, 727-731. 

[23] Jeevaraj, S. (2022). A note on multi-criteria decision-making using a complete ranking of generalized 

trapezoidal fuzzy numbers. Soft Computing, 26(21), 11225-11230. 

[24] Jiang, W., Luo, Y., Qin, X. Y., & Zhan, J. (2015). An improved method to rank generalized fuzzy numbers with 

different left heights and right heights. Journal of Intelligent & Fuzzy Systems, 28(5), 2343-2355. 

[25] Wu, D., Liu, X., Xue, F., Zheng, H., Shou, Y., & Jiang W (2018). Fuzzy risk analysis based on a new method for 

ranking generalized fuzzy numbers. Iranian journal of fuzzy systems, 15(3), 117-139. 

[26] Le, T. H. P., & Chu, T. C. (2023). Novel Method for Ranking Generalized Fuzzy Numbers Based on Normalized 

Height Coefficient and Benefit and Cost Areas. Axioms, 12(11), 1049. 

[27] Madhuri, K. U., Babu, S. S., & Shankar, N. R. (2014). Fuzzy risk analysis based on the novel fuzzy ranking with 

new arithmetic operations of linguistic fuzzy numbers. Journal of Intelligent & Fuzzy Systems, 26(5), 2391-

2401. 

[28] Marimuthu, D., & Mahapatra, G. (2021). Multi-criteria decision-making using a complete ranking of 

generalized trapezoidal fuzzy numbers. Soft Computing, 25(15), 9859-9871. 

[29] Nasseri, S. H., Zadeh, M. M., Kardoost, M., & Behmanesh, E. (2013). Ranking fuzzy quantities based on the 

angle of the reference functions. Applied Mathematical Modelling, 37(22), 9230-9241. 

[30] Nejad, A. M., & Mashinchi, M. (2011). Ranking fuzzy numbers based on the areas on the left and the right sides 

of fuzzy number. Computers & Mathematics with Applications, 61(2), 431-442. 

[31] Patra, K. (2022). Fuzzy risk analysis using a new technique of ranking of generalized trapezoidal fuzzy 

numbers. Granular Computing, 7(1), 127-140. 

[32] Prasad, S., & Sinha, A. (2022). Ranking fuzzy numbers with unified integral value and comparative 

reviews. Journal of Scientific Research, 14(1), 131-151. 

[33] Ponnialagan, D., Selvaraj, J., & Velu, L. G. N. (2018). A complete ranking of trapezoidal fuzzy numbers and its 

applications to multi-criteria decision making. Neural Computing and Applications, 30, 3303-3315. 

[34] Rezvani, S. (2015). Ranking generalized exponential trapezoidal fuzzy numbers based on variance. Applied 

Mathematics and Computation, 262, 191-198. 

[35] Sotoudeh-Anvari, M., & Sotoudeh-Anvari, A. (2023). Setback in ranking fuzzy numbers: a study in fuzzy risk 

analysis in diabetes prediction. Artificial Intelligence Review, 56(5), 4591-4639. 

[36] Thomas Jr, G. B., Weir, M. D., Hass, J., Heil, C., (2014). Thomas’ Calculus Early Transcendentals (ed.13). 

Pearson  

[37] Van Hop, N. (2022). Ranking fuzzy numbers based on relative positions and shape characteristics. Expert 

Systems with Applications, 191, 116312.  

[38] Wang, Y. J. (2015). Ranking triangle and trapezoidal fuzzy numbers based on the relative preference 

relation. Applied mathematical modelling, 39(2), 586-599. 

[39] Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy sets and 

systems, 118(3), 387-405. 

[40] Wang, Y. M., Yang, J. B., Xu, D. L., & Chin, K. S. (2006). On the centroids of fuzzy numbers. Fuzzy sets and 

systems, 157(7), 919-926.  

[41] Wang, Y. J., & Lee, H. S. (2008). The revised method of ranking fuzzy numbers with an area between the 

centroid and original points. Computers & Mathematics with Applications, 55(9), 2033-2042. 

[42] Yager RR (1978) Ranking fuzzy subsets over the unit interval. In: 1978 IEEE conference on decision and 

control including the 17th symposium on adaptive processes, pp 1435–1437 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 7s (2024) 

 

413 
https://internationalpubls.com 

[43] Yao, J. S., & Wu, K. (2000). Ranking fuzzy numbers based on decomposition principle and signed 

distance. Fuzzy sets and Systems, 116(2), 275-288. 

[44] Yu VF, Chi HTX, Shen CW (2013) Ranking fuzzy numbers based on epsilon-deviation degree. Applied Soft 

Computing, 13(8), 3621-3627. 

[45] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353. 

[46] Zimmermann, H. (2013). Fuzzy Set Theory—and Its Applications. Springer Science & Business Media.