Plane Thermoelastic Waves in Infinite Half-Space Caused


Decision Making: Applications in Management and Engineering  
Vol. 5, Issue 1, 2022, pp. 246-263. 
ISSN: 2560-6018 
eISSN: 2620-0104  
DOI: https://doi.org/10.31181/dmame211221090a 

* Corresponding author. 
E-mail addresses: hdarora@amity.edu (H. D. Arora), anathani@amity.edu (A. 

Naithani) 

SIGNIFICANCE OF TOPSIS APPROACH TO MADM IN 
COMPUTING EXPONENTIAL DIVERGENCE MEASURES 

FOR PYTHAGOREAN FUZZY SETS  

H. D. Arora1 and Anjali Naithani1* 

1Department of Mathematics, Amity Institute of Applied Sciences, 
Amity University Uttar Pradesh, Noida, India 

 

Received: 29 March 2022;  
Accepted: 17 December 2021;  
Available online: 21 December 2021. 

 
Original scientific paper 

Abstract: Managers nowadays face challenging decisions on daily basis and 
must weigh a growing number of factors while making such decisions. One of 
the most common and popular research domains in decision theory is the 
Multiple Attribute Decision-Making (MADM) problem which allows us to 
consider several factors into consideration. In this paper, the primary goal is 
to uncover the important aspect of divergence measures based on exponential 
function under Pythagorean Fuzzy Sets (PFSs) and study its application to 
multi attribute decision making. PFSs is a more tensile and powerful approach 
than intuitionistic fuzzy set (IFS) to depict uncertainty. Numerical example has 
been illustrated and sensitivity analysis has been carried out to validate our 
proposed measures. Moreover, a comparative study of the results for the 
proposed measures demonstrates the efficacy of the proposed distance 
measures.    

Key words: Intuitionistic fuzzy set, Pythagorean fuzzy sets, similarity 
measure, exponential measure, TOPSIS method, multi attribute decision 
making 

1. Introduction 

Evaluations of alternative measures are a challenging and complex task due to 
several variables which relate to specific decisions in many decision-making 
problems, such as environmental, social, physical, organizational, and social criteria. 
Researchers have developed various decision-making methods over the last few years 
to help policymakers to analyze the strategic planning in the industry and to resolve 
them. Furthermore, because of the increasing uncertainty and complexity of attributes 
and the vagueness of human thinking, the study of MADM in an uncertain environment 
has received much emphasis. For decision-makers, therefore, it is important to 



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

247 

understand the nature and significance of insecurity to improve their capabilities to 
take the best decision to decrease risk in their final decisions. Multi-attribute decision-
making, a systematic method, can be a useful tool for fuzzy set has its participation and 
non-participation values totaled to 1. Atanassov (1986, 1989) created the notion of 
intuitionistic fuzzy set (IFS) to better express uncertainty by easing this constraint. 
The participation degree and non-participation degree of an IFSs are both real 
numbers in the range [0, 1], and their sum is less than 1. Another IFSs parameter, the 
hesitation degree, is derived from the difference between 1 and their summation. IFSs 
theory has been successfully applied to a variety of real-world challenges such as 
decision-making. Remarkable outcomes on IFSs have been carried out by many 
researchers (Peng et al., 2017; Thao, et al., 2019). 

PFSs is a generalization of IFS that has the prerequisite that the sum of square of 
perception and non-perception grade ≤ 1. The space of all intuitionistic membership 
values is also Pythagorean membership values, but not the other way around. Garg 
(2017) introduced an improved ranking order interval valued PFSs using TOPSIS 
technique. Indeed, the hypothesis of PFSs has been widely considered, as 
demonstrated by various researchers (Garg, 2018; Liang and Xu, 2017). In association 
with the uses of PFSs, Rahman et al., (2017, 2018) proposed a few ways to deal with 
aggregation operators (AO) and MCDM problems. PFSs have drawn the attention of 
researchers and are being applied in decision making (Liu et al. 2020; Mahanta and 
Panda, 2021; Fei and Deng, 2020, Farhadinia, 2021), medical diagnosis (Ejegwa, 2020; 
Zhou et al. 2020), stock portfolio problem (Kalifa, 2020), belief function (Xiao, 2020).  
The characteristics and applicability of the measures in pattern recognition, medical 
diagnosis, multi-criteria decision-making, and clustering analysis were reviewed by 
Singh and Ganie (2020). Overall, the possibility of PFSs has pulled in incredible 
considerations of numerous researchers, and the idea has been functional to a few 
applied regions viz., aggregation operators (Khan et al., 2019), social network analysis 
(Wang et al. 2020), MCDM (Gao and Wei, 2018; Rahman and Abdullah, 2019), 
information measures and many more (Yager, 2014; Ejegwa, 2019). Pamucar et al. 
(2017) investigated the sensitivity of MADM approaches to changes in criteria weight, 
as well as the methods' consistency in response to changes in measurement scale and 
created criteria. 

A divergence metric for PFSs is a tool that reflects how analogous two or more PFSs 
are to each other. Indeed, there is a second concept of similarity measurement for 
PFSs. PFS similarity measures have been investigated from several angles in recent 
times (Ejegwa, 2018). To address the shortcomings of existing measures, Peng (2019) 
proposed new Pythagorean distance and similarity measurements (Adabitabar et al. 
2020). Modification of Zhang and Xu’s (2014) distance measure for PFSs and its 
application to pattern recognition was carried out (Ejegwa, 2020). Some formulae of 
Pythagorean fuzzy information measures on similarity measures and corresponding 
transformation relationships were also developed (Peng et al., 2017, 2019). Similarity 
measures for trigonometric function for FSs, IFSs and PFSs were also proposed 
(Taruna et al., 2021), IFSs and PFSs (Wei and Wei, 2018; Mohd and Abdullah, 2018) 
were also proposed (Maoying, 2013). Some Complex PFSs distance measures have 
been established, and their features have been investigated (Ullah et al. 2020). The 
similarity measures of the IFSs and PFSs are broadly used in various disciplines, 
comparable to the pattern identification (Peng and Garg, 2019), the clinical finding 
(Son and Phong, 2016), decision-making (Zhang et al., 2019). However, Lu and Ye 
(2018) offered similarity measure of IVFSs on log function. Agheli et al. 2021 recently 
proposed a method for calculating Pythagorean similar measure for two Pythagorean 
fuzzy value by making use of T-norm and S-norm. 



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

248 

For supplier evaluation and selection, Pamucar et al. (2020) suggested a fuzzy 
neutrosophic decision-making approach. Many researchers analysed MADM approach 
using TOPSIS method (Hwang and Yoon, 1981). Many Researchers like Adeel et al., 
2019; Akram and Adeel, 2019; Akram et al., 2018; Balioti et al., 2018; Biswas and 
Kumar, 2018;  Askarifar et al., 2018; Wang and Chen, 2017; Gupta et al., 2018; Kumar 
and Garg, 2018 and many more have applied TOPSIS method in various problems of 
decision making like supplier selection, selection of land, robotics, medical diagnosis, 
ranking of water quality, human resource selection personnel problem, and many 
other real life situations flavoured with FSs and generalized FSs. 

In this article, we are exploring the resourcefulness of exponential divergence 
measures of PFSs in the application to pattern recognition and multi attribute decision 
making. This paper is organized as follows: Section 2 introduces preliminaries of FSs, 
IFSs and the PFSs. Section 3 comprises of the concept of proposed exponential 
similarity measures of PFSs. We introduce exponential similarity measures and 
weighted similarity measures of the PFSs and its numerical computations to validate 
our measures. Application is also provided in Section 4. Section 5 deliberates 
discussion about the methodology discussed and sensitivity analysis of the proposed 
measures. Section 6 compares the new exponential similarity measures with the 
existing similarity measure by an example. Finally, Section 7 summarizes the 
document and delivers directions for future experiments. 

2. Preliminaries 

In this segment, we bring in some basic theories related to FSs, IFSs and PFSs 
applied in the article.  
Definition 2.1. [Zadeh, 1965]. Let X be a nonempty set. A fuzzy set 𝑃 in 𝐸 =
{𝑥1, 𝑥2 … , 𝑥𝑛 } is characterized by a membership function: 

𝑃 = {〈𝑥, 𝛿𝑃(𝑥)〉|𝑥 ∈ 𝐸}                        (1) 

where 𝛿𝑃(𝑥): 𝐸 → [0,1] is a measure of belongingness of degree of membership of an 
element 𝑥 ∈ 𝐸 in 𝑃.  
Definition 2.2. [Atanassov, 1986]. An IFS P in X is given by 

𝑃 = {〈𝑥, 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥)〉|𝑥 ∈ 𝐸}                        (2) 

where 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥): 𝐸 → [0,1], 0 ≤ 𝛿𝑃(𝑥) + 𝜁𝑃 (𝑥) ≤ 1, ∀ 𝑥 ∈ 𝐸. The number 𝛿𝑃(𝑥) and 
𝜁𝑃 (𝑥) represents, respectively, the membership degree and non-membership degree 
of the element 𝑥 to the set P.       
For each IFS 𝑃 in 𝐸, if  

𝜂𝑃(𝑥) = 1 − 𝛿𝑃(𝑥) − 𝜁𝑃 (𝑥), ∀ 𝑥 ∈ 𝐸.                (3)  

Then 𝜂𝑃(𝑥) is called the degree of indeterminacy of 𝑥 to Ã.  
Definition 2.3. [Yager, 2013]. An IFS 𝑃 in 𝐸 is given by 

𝑃 = {〈𝑥, 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥)〉|𝑥 ∈ 𝐸}       

where 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥): 𝐸 → [0,1], with the condition that 

0 ≤ 𝛿𝑃
2(𝑥) + 𝜁𝑃

2(𝑥) ≤ 1, ∀ 𝑥 ∈ 𝐸                          (4) 

and the degree of indeterminacy for any PFS 𝑃 and 𝑥 ∈ 𝐸 is given by  

𝜂𝑃
2 (𝑥) = √1 − 𝛿𝑃

2(𝑥) − 𝜁𝑃
2(𝑥)                           (5) 



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

249 

3. Exponential Divergence Measures 

In this segment, new exponential divergence measures of the PFSs are proposed. 
Preposition 1. Let 𝛸 be nonempty set and P, Q, R ∈ PFS (𝛸). The divergence measure 
between P and Q is a function 𝐷𝑖𝑣: 𝑃𝐹𝑆 × 𝑃𝐹𝑆 → [0,1] satisfies 
(P1) Boundedness: 0 ≤ 𝐷𝑖𝑣(𝑃, 𝑄) ≤ 1  
(P2) Separability: 𝐷𝑖𝑣(𝑃, 𝑄) = 0 ⇔ 𝑃 = 𝑄. 
(P3) Symmetric: 𝐷𝑖𝑣(𝑃, 𝑄) =  𝐷𝑖𝑣(𝑄, 𝑃)  
(P4) Inequality: If R is a PFS in 𝛸 and 𝑃 ⊆ 𝑄 ⊆ 𝑅, then 𝐷𝑖𝑣(𝑃, 𝑄) ≤ 𝐷𝑖𝑣(𝑃, 𝑅) and 
                                 𝐷𝑖𝑣(𝑄, 𝑅) ≤ 𝐷𝑖𝑣(𝑃, 𝑅). 

In several circumstances, the weight of the elements 𝑥𝑖  ∈ 𝑋 must be considered. For 
instance, in decision making, the attributes usually have distinct significance, and thus 
ought to be designated unique weights. As a result, we propose two weighted 
logarithmic divergence measures between P and Q, as follows:  
Let 𝑃, 𝑄 ∈ PFS (𝛸) such that 𝑋 = {𝑥1, 𝑥2 … , 𝑥𝑛 } then   

𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) =
2

𝑛 
∑ [|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑄
2(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2 (𝑥𝑖)|−1 + |𝜁𝑃
2(𝑥𝑖 ) −

𝑛
𝑖=1

𝜁𝑄
2 (𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑄

2 (𝑥𝑖)|−1]                        (6)  

𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) =
2

𝑛 
∑ 𝜔𝑖 [|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑄
2(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2 (𝑥𝑖)|−1 + |𝜁𝑃
2 (𝑥𝑖 ) −

𝑛
𝑖=1

𝜁𝑄
2 (𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑄

2 (𝑥𝑖)|−1]                                        (7)  

𝜔 = (𝜔1, 𝜔2, … , 𝜔𝑛 )
𝑇  is the weight vector of 𝑥𝑖 (𝑖 = 1,2, … , 𝑛), with 𝜔𝑘 ∈ [0,1], 𝑘 =

1,2, … , 𝑛,   ∑ 𝜔𝑘 = 1
𝑛
𝑘=1 . If 𝜔 = (

1

𝑛
,

1

𝑛
, …

1

𝑛
)

𝑇

, then the weighted exponential divergence 

measure reduces to proposed measure. If we take 𝜔𝑘 = 1, 𝑘 = 1,2, … , 𝑛, then then 
𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄).  
 
Theorem 3.1. The Pythagorean fuzzy divergence measures defined in equation (6) - 
(7) are valid measures of Pythagorean fuzzy divergence. 
Proof. All the necessary four conditions to be a divergence measure are satisfied by 
the new divergence measures as follows: 
 
(P1) Boundedness: 0 ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 1 
Proof. Since the values 0 ≤ 𝛿𝑃(𝑥𝑖 ) ≤ 𝛿𝑄(𝑥𝑖 ) ≤ 1 and 0 ≤ 𝜁𝑃 (𝑥𝑖 ) ≤ 𝜁𝑄 (𝑥𝑖 ) ≤ 1, 

therefore, 0 ≤ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2(𝑥𝑖 )| ≤ 1 and  0 ≤ |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑄

2(𝑥𝑖 )| ≤ 1. Since minimum 

values of all the expression is 0, then the measure 𝐷𝑃𝐹𝑆𝐿 (𝑃, 𝑄) will have value as 

𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) =
2

1
(0 . 2−1 + 0 . 2−1 ) = 0 . Also, if the maximum value of the above 

expressions is 1, then 𝐷𝑃𝐹𝑆𝐿 (𝑃, 𝑄) =
2

1
(1 . 2−2 + 1 . 2−2 ) = 1.  

Thus, 0 ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 1.  
Measure 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) can be proved similarly.  
 
(P2) Separability: 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0 ⇔ 𝑃 = 𝑄.  
Proof.  For two PFSs P and Q in 𝑋 = {𝑥1, 𝑥2 … , 𝑥𝑛 }, if 𝑃 = 𝑄, then 𝛿𝑃

2(𝑥𝑖 ) = 𝛿𝑄
2(𝑥𝑖 ) and 

𝜁𝑃
2(𝑥𝑖 ) = 𝜁𝑄

2 (𝑥𝑖 ). Thus, |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2(𝑥𝑖 )| = 0 and |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑄

2(𝑥𝑖 )| = 0. 

 Therefore, 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0.   
If 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0, this implies 

|𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2(𝑥𝑖 )|. 2
−|𝛿𝑃

2 (𝑥𝑖)−𝛿𝑄
2 (𝑥𝑖)|−1 + |𝜁𝑃

2(𝑥𝑖 ) − 𝜁𝑄
2(𝑥𝑖 )|. 2

−|𝜁𝑃
2 (𝑥𝑖)−𝜁𝑄

2 (𝑥𝑖)|−1 = 0 

⇒ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2 (𝑥𝑖 )| = 0 and |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑄

2(𝑥𝑖 )| = 0.  



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

250 

Therefore 𝛿𝑃
2(𝑥𝑖 ) = 𝛿𝑄

2(𝑥𝑖 ) and 𝜁𝑃
2(𝑥𝑖 ) = 𝜁𝑄

2 (𝑥𝑖 ). Hence P = Q.  

Measure 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) can be proved similarly. 
 
(P3) Symmetric: 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) =  𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑃)   
Proofs are self-explanatory and straight forward. 
 
(P4) Inequality: If R is a PFS in 𝛸 and 𝑃 ⊆ 𝑄 ⊆ 𝑅, then 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 
𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). 
 Proof.  If 𝑃 ⊆ 𝑄 ⊆ 𝑅, then for 𝑥𝑖  ∈ 𝛸, we have 0 ≤ 𝛿𝑃(𝑥𝑖 ) ≤ 𝛿𝑄(𝑥𝑖 ) ≤ 𝛿𝑅 (𝑥𝑖 ) ≤ 1 and 

1 ≥ 𝜁𝑃 (𝑥𝑖 ) ≥ 𝜁𝑄 (𝑥𝑖 ) ≥ 𝜁𝑅 (𝑥𝑖 ) ≥ 0. 

This implies that 0 ≤ 𝛿𝑃
2(𝑥𝑖 ) ≤ 𝛿𝑄

2(𝑥𝑖 ) ≤ 𝛿𝑅
2(𝑥𝑖 ) ≤ 1 and 1 ≥ 𝜁𝑃

2(𝑥𝑖 ) ≥ 𝜁𝑄
2(𝑥𝑖 ) ≥

𝜁𝑅
2(𝑥𝑖 ) ≥ 0. Thus, we have, 

|𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2(𝑥𝑖 )| ≤ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑅

2(𝑥𝑖 )| ; |𝛿𝑄
2(𝑥𝑖 ) − 𝛿𝑅

2(𝑥𝑖 )| ≤ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑅

2(𝑥𝑖 )| and  

|𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑄

2 (𝑥𝑖 )| ≤ |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑅

2(𝑥𝑖 )|  ; |𝜁𝑄
2 (𝑥𝑖 ) − 𝜁𝑅

2(𝑥𝑖 )| ≤ |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑅

2(𝑥𝑖 )|  

From the above we can write, 2−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2 (𝑥𝑖)|−1 ≤ 2−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑅

2 (𝑥𝑖)|−1 

⇒ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2 (𝑥𝑖 )|. 2
−|𝛿𝑃

2 (𝑥𝑖)−𝛿𝑄
2 (𝑥𝑖)|−1 ≤ |𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑅
2(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑅

2 (𝑥𝑖)|−1 

⇒ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑄

2 (𝑥𝑖 )|. 2
−|𝛿𝑃

2 (𝑥𝑖)−𝛿𝑄
2 (𝑥𝑖)|−1 + |𝜁𝑃

2(𝑥𝑖 ) − 𝜁𝑄
2 (𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑄

2 (𝑥𝑖)|−1 

≤ |𝛿𝑃
2(𝑥𝑖 ) − 𝛿𝑅

2(𝑥𝑖 )|. 2
−|𝛿𝑃

2 (𝑥𝑖)−𝛿𝑅
2 (𝑥𝑖)|−1 + |𝜁𝑃

2(𝑥𝑖 ) − 𝜁𝑅
2(𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑅

2 (𝑥𝑖)|−1   

⇒  
2

𝑛 
[∑ {|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑄
2(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2 (𝑥𝑖)|−1 + |𝜁𝑃
2(𝑥𝑖 ) − 𝜁𝑄

2 (𝑥𝑖 )|. 2
−|𝜁𝑃

2(𝑥𝑖)−𝜁𝑄
2 (𝑥𝑖)|−1}

𝑛

𝑖=1

] 

      ≤   
2

𝑛 
[∑ {|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑅
2(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑅

2 (𝑥𝑖)|−1 + |𝜁𝑃
2(𝑥𝑖 ) −

𝑛
𝑖=1

𝜁𝑅
2(𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑅

2 (𝑥𝑖)|−1}]   

⇒  𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). Similarly, 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). 
Similar proofs can be made for 𝐷𝑊𝑃𝐹𝑆𝐿 (𝑃, 𝑄) ≤ 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤
𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅). 

3.1. Numerical Verification of the Distance Measures 

Based on the parameters suggested by Wei and Wei (2018), we verify whether 
proposed divergence measures satisfy above four properties: 
Example 1. Let 𝑃, 𝑄, 𝑅 ∈ 𝑃𝐹𝑆(𝑋) for 𝑋 = {𝑥1, 𝑥2, 𝑥3}. Suppose  
𝑃 = {⟨𝑥1, 0.6, 0.2⟩, ⟨𝑥2, 0.4, 0.6⟩, ⟨𝑥3, 0.5, 0.3⟩},  
𝑄 = {⟨𝑥1, 0.8, 0.2⟩, ⟨𝑥2, 0.7, 0.3⟩, ⟨𝑥3, 0.6, 0.3⟩} and 
 𝑅 = {⟨𝑥1, 0.9, 0.1⟩, ⟨𝑥2, 0.8, 0.2⟩, ⟨𝑥3, 0.7, 0.1⟩}  

Calculating the distance using proposed distance measures are as follows: 

𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) =
2

3
[

{|0.62 − 0.82|. 2−|0.6
2−0.82|−1 + |0.22 − 0.22|. 2−|0.2

2−0.22|−1} +

{|0.42 − 0.72|. 2−|0.4
2−0.72|−1 + |0.62 − 0.32|. 2−|0.6

2−0.32|−1} +

{|0.52 − 0.62|. 2−|0.5
2−0.62|−1 + |0.32 − 0.32|. 2−|0.3

2−0.32|−1}

]  

=  
2

3
[0.115302742 + 0 + 0.131263519 + 0.111958138 + 0.050962343 + 0] 

= 0.272991161.     

𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) =
2

3
[

{|0.62 − 0.92|. 2−|0.6
2−0.92|−1 + |0.22 − 0.12|. 2−|0.2

2−0.12|−1} +

{|0.42 − 0.82|. 2−|0.4
2−0.82|−1 + |0.62 − 0.22|. 2−|0.6

2−0.22|−1} +

{|0.52 − 0.72|. 2−|0.5
2−0.72|−1 + |0.32 − 0.12|. 2−|0.3

2−0.12|−1}

]  



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

251 

=  
2

3
[0.16470964 + 0.014691304 + 0.172074629 + 0.12817118 + 0.101609437 +

0.037842305] = 0.41273233. 

𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) =
2

3
[

{|0.82 − 0.92|. 2−|0.8
2−0.92|−1 + |0.22 − 0.12|. 2−|0.2

2−0.12|−1} +

{|0.72 − 0.82|. 2−|0.7
2−0.82|−1 + |0.32 − 0.22|. 2−|0.3

2−0.22|−1} +

{|0.62 − 0.72|. 2−|0.6
2−0.72|−1 + |0.32 − 0.12|. 2−|0.3

2−0.12|−1}

]  

=  
2

3
[0.075551627 + 0.014691304 + 0.067593784 + 0.024148408 + 0.05939904 +

0.037842305] = 0.186150981. 
The detailed computation for the proposed measures can be summarized in the 

table 1: 
Table 1. Numerical illustration to validate proposed measures 

Proposed 
Measure 1 

Numerical 
Values 

Proposed 
Measure 2 

Numerical 
Values 

 
𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) 0.272991 

 
𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) 0.093874 

 
𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) 0.412732 

 
𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) 0.138443 

 
𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) 0.186151 

 
𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) 0.061395 

 
From the above computations, it supports that 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 

𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). Also, 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤
𝑆𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅).   

4. Applications of Exponential Divergence Measures 

One of the most crucial aspects of building model is deciding on criteria. As a result, 
criteria are critical components that allow options to be compared from a particular 
perspective. Users are often satisfied with a product when its characteristics fit their 
tastes and expectations. The most essential product selection criteria for consumers 
must be determined to design an effective decision model.   

PFSs has been frequently used to handle multi attribute decision making (MADM) 
problems in Pythagorean fuzzy environments due to its excellent skill in representing 
uncertain information. Many PFSs based MADM algorithms have been proposed. We 
are presenting a novel Pythagorean fuzzy MADM approach based on the TOPSIS 
method in this part. Under the Pythagorean fuzzy condition, MADM problem can be 
depicted by a flowchart as shown in Figure 1. 

 



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

252 

Figure 1. TOPSIS Algorithm Approach 

4.1. A Case Study 

A multinational company wishes to buy smartphones for their white-collar 
workers. Because of the large number of smart phones to be acquired, the process is 
crucial. With members as Procurement Manager (Ɗ𝟏), Human Resource Manager (Ɗ𝟐) 
and Quality Manager (Ɗ𝟑), a decision committee of three experts is formed by the 
company with the goal of determining the most appropriate smartphone among five 
possible options as Model 1 (𝒜1), Model 2 (𝒜2), Model 3 (𝒜3), Model 4 (𝒜4) and 
Model 5 (𝒜5). Experts assist in the decision-making process and employ smartphone 
choosing criteria. There are five options available as storage capacity in gigabytes(ℂ𝟏), 
weight in grams(ℂ𝟐), camera specifications in pixels(ℂ𝟑), screen size in inches(ℂ𝟒), 
battery life in hours(ℂ𝟓). These models were chosen due to their similar costs in the 
Indian market. Organization assigns weights to these criteria as 𝜔 =0.25, 0.35, 0.20, 
0.12, and 0.08, respectively.  

 
Step 1: Establish the decision matrix (X) 

For each criterion, the options are first examined by the decision makers Ɗ𝟏, Ɗ𝟐 
and Ɗ𝟑 using pair-wise comparisons. The assessment of the given alternatives in the 
form of Pythagorean fuzzy sets by these three decision makers are examined in table 
2-4. 
  



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

253 

Table 2. Data set in the form of a decision matrix (X) of decision maker Ɗ1 
Decision 

Maker 
Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟏 

𝒜1 <0.8,0.1> <0.1,0.6> <0.2,0.8> <0.6,0.1> <0.1,0.6> 
𝒜2 <0.0,0.8> <0.4,0.4> <0.6,0.3> <0.1,0.7> <0.1,0.8> 
𝒜3 <0.6,0.1> <0.4,0.5> <0.3,0.0> <0.7,0.2> <0.3,0.4> 
𝒜4 <0.7,0.3> <0.3,0.4> <0.7,0.2> <0.8,0.1> <0.2,0.5> 
𝒜5 <0.5,0.3> <0.5,0.4> <0.7,0.2> <0.6,0.1> <0.4,0.7> 

 
Table 3. Data set in the form of a decision matrix (X) of decision maker Ɗ2 

Decision 
Maker 

Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟐 

𝒜1 <0.4,0.0> <0.0,0.7> <0.3,0.3> <0.1,0.8> <0.4,0.0> 
𝒜2 <0.3,0.5> <0.6,0.2> <0.6,0.1> <0.2,0.4> <0.3,0.5> 
𝒜3 <0.1,0.7> <0.9,0.0> <0.2,0.7> <0.8,0.0> <0.1,0.7> 
𝒜4 <0.4,0.3> <0.8,0.1> <0.2,0.6> <0.2,0.7> <0.4,0.3> 
𝒜5 <0.4,0.5> <0.5,0.3> <0.8,0.2> <0.7,0.3> <0.3,0.6> 

 
Table 4. Data set in the form of a decision matrix (X) of decision maker Ɗ3 

Decision 
Maker 

Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟑 

𝒜1 <0.2,0.6> <0.8,0.3> <0.4,0.5> <0.1,0.7> <0.6,0.5> 
𝒜2 <0.6,0.3> <0.5,0.3> <0.6,0.3> <0.5,0.2> <0.2,0.6> 
𝒜3 <0.7,0.2> <0.6,0.2> <0.7,0.2> <0.6,0.3> <0.3,0.4> 
𝒜4 <0.3,0.8> <0.2,0.6> <0.3,0.6> <0.6,0.3> <0.4,0.8> 
𝒜5 <0.2,0.6> <0.8,0.3> <0.4,0.5> <0.1,0.7> <0.6,0.5> 

 
Step 2: Calculation of Normalized decision matrix (X) 

In the crisp environment, to avoid the complicated normalization formula used 
in classical TOPSIS, simpler formulas are used to transform the various criteria scales 
into a comparable scale. ℂ𝟏, ℂ𝟑, and ℂ𝟒 are benefit criteria, while ℂ𝟐 is cost qualities, 
according to these experts. However, in case of Pythagorean fuzzy environment, 
normalized matrix can be constructed by replacing membership and non-membership 
values in cost attributes, whereas there will not be any change in case of benefit 
attributes. The results are shown in table 5-8. 

 
Table 5. Normalized values of decision maker Ɗ1 in terms of PFSs (X) 

Decision 
Maker 

Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟏 

𝒜1 <0.8,0.1> <0.6,0.1> <0.2,0.8> <0.6,0.1> <0.1,0.6> 
𝒜2 <0.0,0.8> <0.4,0.4> <0.6,0.3> <0.1,0.7> <0.1,0.8> 
𝒜3 <0.6,0.1> <0.5,0.4> <0.3,0.0> <0.7,0.2> <0.3,0.4> 
𝒜4 <0.7,0.3> <0.4,0.3> <0.7,0.2> <0.8,0.1> <0.2,0.5> 
𝒜5 <0.5,0.3> <0.4,0.5> <0.7,0.2> <0.6,0.1> <0.4,0.7> 

 
 
 
 
 
 



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

254 

Table 6. Normalized values of decision maker Ɗ2 in terms of PFSs (X) 

Decision 
Maker 

Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟐 

𝒜1 <0.4,0.0> <0.7,0.0> <0.3,0.3> <0.1,0.8> <0.4,0.0> 
𝒜2 <0.3,0.5> <0.2,0.6> <0.6,0.1> <0.2,0.4> <0.3,0.5> 
𝒜3 <0.1,0.7> <0.0,0.9> <0.2,0.7> <0.8,0.0> <0.1,0.7> 
𝒜4 <0.4,0.3> <0.1,0.8> <0.2,0.6> <0.2,0.7> <0.4,0.3> 
𝒜5 <0.4,0.5> <0.3,0.5> <0.8,0.2> <0.7,0.3> <0.3,0.6> 

 
Table 7. Normalized values of decision maker Ɗ3 in terms of PFSs (X) 

Decision 
Maker 

Alternatives 
vs Criteria 

ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 

 
 

Ɗ𝟑 

𝒜1 <0.2,0.6> <0.3,0.8> <0.4,0.5> <0.1,0.7> <0.6,0.5> 
𝒜2 <0.6,0.3> <0.3,0.5> <0.6,0.3> <0.5,0.2> <0.2,0.6> 
𝒜3 <0.7,0.2> <0.2,0.6> <0.7,0.2> <0.6,0.3> <0.3,0.4> 
𝒜4 <0.3,0.8> <0.6,0.2> <0.3,0.6> <0.6,0.3> <0.4,0.8> 
𝒜5 <0.2,0.6> <0.3,0.8> <0.4,0.5> <0.1,0.7> <0.6,0.5> 

 
Step 3: Identify the Fuzzy Positive Ideal Solution (FPIS) and Negative Ideal Solution 
(FNIS)  

FPIS maximizes the benefit and minimizes the cost, whereas the FNIS maximizes 
the cost and minimizes the benefit. For each decision maker, we compute FPIS and 
FNIS for the PFSs using 

𝐴𝑘+ = {𝑟1
𝑘+, 𝑟2

𝑘+, … , 𝑟𝑛
𝑘+} = {(max

𝑖
(𝑟𝑖𝑗

𝑘 ) ∕ 𝑗 ∈ 𝐼) , ((min
𝑖

(𝑟𝑖𝑗
𝑘 ) ∕ 𝑗 ∈ 𝐽))};            (8) 

𝐴𝑘− = {𝑟1
𝑘−, 𝑟2

𝑘−, … , 𝑟𝑛
𝑘−} = {(min

𝑖
(𝑟𝑖𝑗

𝑘 ) ∕ 𝑗 ∈ 𝐼) , ((max
𝑖

(𝑟𝑖𝑗
𝑘 ) ∕ 𝑗 ∈ 𝐽))}             (9) 

where I refer to the benefit criteria and J, the cost criteria. The subsequent values 
are presented in table 8. 

 
Table 8. Fuzzy Positive and Negative Ideals for each decision makers  

Decision 
Maker 

FPIS 
and 
FNIS 

ℂ1 ℂ2 ℂ3 ℂ4 ℂ5 

 
Ɗ1 

𝐴+ 
𝐴− 

<0.8,0.1> 
<0.0,0.8> 

<0.6,0.1> 
<0.4,0.5> 

<0.7,0.0> 
<0.2,0.8> 

<0.8, 0.1 
<0.1,0.7> 

<0.4,0.4> 
<0.1,0.8> 

 
Ɗ2 

𝐴+ 
𝐴− 

<0.4,0.0> 
<0.1,0.7> 

<0.7,0.0> 
<0.0,0.9> 

<0.8,0.1> 
<0.2,0.7> 

<0.8,0.0> 
<0.1,0.8> 

<0.4,0.0> 
<0.1,0.7> 

 
Ɗ3 

𝐴+ 
𝐴− 

<0.7,0.2> 
<0.2,0.8> 

<0.6, 02> 
<0.2,0.8> 

<0.7,0.2> 
<0.3,0.6> 

<0.7,0.2> 
<0.1,0.7> 

<0.6,0.0> 
<0.2,0.8> 

 
Step 4: Calculate the separation distance of each competitive alternative from the ideal 
and non- ideal solution 

Separation measures 𝐷(𝒜𝑖 , 𝐴
+), 𝐷(𝒜𝑖 , 𝐴

−) and the weighted exponential 
divergence measure proposed in equation (7) of each alternative from FPIS and FNIS 
have been calculated using formulae (10) and (11) and presented in table 9.  



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

255 

𝐷(𝒜𝑖 , 𝐴
+) =

2

𝑛 
∑ 𝜔𝑖 [|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑄
2+(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2+(𝑥𝑖)|−1 + |𝜁𝑃
2(𝑥𝑖 ) −

𝑛
𝑖=1

𝜁𝑄
2+(𝑥𝑖 )|. 2

−|𝜁𝑃
2(𝑥𝑖)−𝜁𝑄

2+(𝑥𝑖)|−1]                      (10) 

𝐷(𝒜𝑖 , 𝐴
−) =

2

𝑛 
∑ 𝜔𝑖 [|𝛿𝑃

2(𝑥𝑖 ) − 𝛿𝑄
2−(𝑥𝑖 )|. 2

−|𝛿𝑃
2 (𝑥𝑖)−𝛿𝑄

2−(𝑥𝑖)|−1 + |𝜁𝑃
2(𝑥𝑖 ) −

𝑛
𝑖=1

𝜁𝑄
2−(𝑥𝑖 )|. 2

−|𝜁𝑃
2 (𝑥𝑖)−𝜁𝑄

2−(𝑥𝑖)|−1]               (11) 

Table 9. Separation measures for ideal solutions w.r.t. each decision maker  
Alternativ

es  
Ɗ1 Ɗ2  Ɗ3 

𝐷1 (𝐴𝑖 , 𝐴
+) 𝐷1 (𝐴𝑖 , 𝐴

−) 𝐷2(𝐴𝑖 , 𝐴
+) 𝐷1 (𝐴𝑖 , 𝐴

+) 𝐷1 (𝐴𝑖 , 𝐴
−) 𝐷2(𝐴𝑖 , 𝐴

+) 

𝒜1 0.03861 0.08584 0.03659 0.10277 0.06359 0.05815 
𝒜2 0.09158 0.03120 0.08431 0.07717 0.10660 0.03200 
𝒜3 0.05103 0.08746 0.11772 0.01963 0.05403 0.06523 
𝒜4 0.03305 0.09852 0.10238 0.04890 0.04483 0.06909 
𝒜5 0.05061 0.07748 0.05728 0.09883 0.06889 0.06916 

 
Step 5: Measure the relative closeness of each location to the ideal solution and rank the 
preference order 

For each competitive alternative the relative closeness of the potential model 
with respect to the ideal solution is computed. Relative closeness coefficient with 
respect to each decision maker can be found using the formula 

𝑅𝑖 =  
𝐷(𝐴𝑖,𝐴

−)

𝐷(𝐴𝑖,𝐴
+)+𝐷(𝐴𝑖,𝐴

−) 
                        (12)  

where 0 ≤ 𝑅i ≤ 1, 𝑖 = 1,2, … , 𝑚 
The value of 𝑅𝑖  signifies that higher the value of the relative closeness, the higher 

the ranking order and hence the better the performance of the alternative. Ranking of 
the preference in descending order thus allows relatively better performances to be 
compared. The ranking results obtained by Pythagorean fuzzy TOPSIS approach is 
demonstrated in table 10. 

 
Table 10. Ranking Results Obtained from TOPSIS Approach  

Alternatives  Ɗ1 Ɗ2 Ɗ3 

𝑅1 Ranking 𝑅2 Ranking 𝑅3 Ranking 

𝒜1 0.6897 2 0.7374 1 0.4776 4 
𝒜2 0.2541 5 0.4779 3 0.2308 5 
𝒜3 0.6315 3 0.1428 5 0.5469 2 
𝒜4 0.74882 1 0.3232 4 0.6064 1 
𝒜5 0.6048 4 0.6330 2 0.5009 3 

5. Discussion 

Table 10 shows the ranking of the 5 considered smart phones according to the 
three decision makers. For decision maker 1, 𝐷1 , Model 4 (𝒜4) is the best smartphone. 
Same is the case for decision makers three, 𝐷3.  On the other hand, Model 1 (𝒜1) is the 
best choice for decision maker 𝐷2. Further, to validate this result, Sensitivity analysis 
will be carried out in the next section, 

 



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

256 

5.1. Sensitivity Analysis 

If decision makers find distinct ranking for the alternatives, the overall findings of 
the best alternatives remain unclear. To overcome the ambiguity about the best 
alternatives with respect to the decision-makers, we aggregate the ideal distance 
measurement values of every decision-maker different values of the experts are 
aggregated by assigning a priority value value 𝜌 = (𝜌1, 𝜌2, … , 𝜌𝑠)

𝑇  to each expert such 
that 𝜌𝑠 > 0 and ∑ 𝜌𝑘 = 1

𝑠
𝑘=1 .  

The distance measure of each expert is aggregated by using these weight vectors 
and the overall measurement values of the alternatives are obtained which can be 
depicted in table 11 as    
 𝜗𝑖

+ = ∑ 𝜌𝑘 𝐶𝑖𝑗
+𝑠

𝑘=1                                  (13) 

 𝜗𝑖
− = ∑ 𝜌𝑘 𝐶𝑖𝑗

−𝑠
𝑘=1                                                                (14) 

Also, ℜ𝑖 =  
𝜗𝑖

−

𝜗𝑖
++𝜗𝑖

−                 (15) 

where 0 ≤ ℜ𝑖 ≤ 1, 𝑖 = 1, 2, … , 5 

Table 11. Aggregated closeness coefficient and ranking for each smartphone 
𝐶𝑎𝑠𝑒 1: 𝜌1 = 0.45, 𝜌2 = 0.35, 𝜌3 = 0.20  

Alternatives ℜ𝑖  Ranking Selected smartphone 
𝒜1 0.6677 1  
𝒜2 0.3401 5  
𝒜3 0.4415 4 𝒜1 
𝒜4 0.5577 3  
𝒜5 0.5953 2  

𝐶𝑎𝑠𝑒 2: 𝜌1 = 0.35, 𝜌2 = 0.27, 𝜌3 = 0.37 
Alternatives ℜ𝑖  Ranking Selected smartphone 

𝒜1 0.6268 1  
𝒜2 0.3154 5  
𝒜3 0.4637 4 𝒜1 
𝒜4 0.5679 3  
𝒜5 0.5743 2  

𝐶𝑎𝑠𝑒 3: 𝜌1 = 0.38, 𝜌2 = 0.33, 𝜌3 = 0.29 
Alternatives ℜ𝑖  Ranking Selected smartphone 

𝒜1 0.6485 1  
𝒜2 0.3325 5  
𝒜3 0.4423 4 𝒜1 
𝒜4 0.5536 3  
𝒜5 0.5855 2  

𝐶𝑎𝑠𝑒 4: 𝜌1 = 0.40, 𝜌2 = 0.30, 𝜌3 = 0.30 
Alternatives ℜ𝑖  Ranking Selected smartphone 

𝒜1 0.6448 1  
𝒜2 0.3250 5  
𝒜3 0.4565 4 𝒜1 
𝒜4 0.5659 3  
𝒜5 0.5834 2  

 
  



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

257 

𝐶𝑎𝑠𝑒 5: 𝜌1 = 0.29, 𝜌2 = 0.25, 𝜌3 = 0.46 
Alternatives ℜ𝑖  Ranking Selected smartphone 

𝒜1 0.6092 1  
𝒜2 0.3081 5  
𝒜3 0.4659 4 𝒜1 
𝒜4 0.5653 3  
𝒜5 0.5655 2  

𝐶𝑎𝑠𝑒 6: 𝜌1 = 0.29, 𝜌2 = 0.25, 𝜌3 = 0.46 
Alternatives ℜ𝑖  Ranking Selected smartphone 

𝒜1 0.6626 1  
𝒜2 0.3347 5  
𝒜3 0.4495 4 𝒜1 
𝒜4 0.5639 3  
𝒜5 0.5926 2  

 
 Sensitivity Analysis concluded that by assigning different priorities to the 
opinions of decision makers, the result of the proposed method remains the same, as  
𝒜1 came out to be the Best-Model in all the cases, thereby substantiating the validity 
and reliability of the proposed method. 

6. Comparative Study 

To demonstrate the dominance of the proposed divergence measure, a comparison 
between the proposed weighted exponential divergence measure and the existing 
measures is conducted based on the numerical cases suggested. Table 12 represents a 
comprehensive evaluation of the divergence measures for PFSs.  

The numerical data in table 12 have been analysed, and it has been discovered that 
the results produced using our suggested divergence measure given in equation 8 are 
like those obtained using existing measures. As a result, the accompanying table 
demonstrates that the proposed divergence measure is consistent across all 
approaches, as the best alternative remains the same.   
 

Table 12. Comparison of existing with the proposed divergence measures 
Measure Ranking 
Peng et al. (2017) 𝒜 1 › 𝒜 4 › 𝒜 5 › 𝒜 3 › 𝒜 2 
Ejegwa, 2018 Measure I 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Ejegwa, 2018 Measure II 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Ejegwa, 2018 Measure III 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Zhang et al., 2019 Measure I 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Zhang et al., 2019 Measure II 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Zhang et al., 2019 Measure III 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 
Zhang et al., 2019 Measure IV 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 

It can be determined by studying the above findings that there are differences in 
ranks when different scenarios are applied, indicating that the model is sensitive to 
changes. It is observed that 𝒜1 is the best alternative. In accordance with the findings 
of Bohar et al., (2020), Spearman’s rank correlation is used to find the correlation of 
attributes.  

When written in mathematical form, Spearman’s Coefficient of correlation is 
denoted by ℛ and is defined by   



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

258 

 ℛ = 1 −
6 ∑ Ď𝑖

2𝑛
𝑖=1

𝑛(𝑛2−1)
                                                                                                                            (16)                                 

where Ď𝑖 = difference in ranks of the “ith” element: 𝑛 = number of observations 
ℛ = value of correlation coefficient 

The value of ℛ  will always lies between -1 and 1. If ℛ = 1, there is a perfectly 
positive correlation; If ℛ = −1, then the ranks are exactly opposite. However, if ℛ =
0, then the ranks are uncorrelated. Spearman’s rank correlation among existing and 
proposed measures is shown in table 13 as 

Table 13. Spearman’s rank correlation for various existing and proposed measures 
  

P
e

n
g

 e
t 

a
l.

 (
2

0
1

7
) 

E
je

g
w

a
, 2

0
1

8
 M

e
a

su
re

 I
 

E
je

g
w

a
, 2

0
1

8
 M

e
a

su
re

 I
I 

E
je

g
w

a
, 2

0
1

8
 M

e
a

su
re

 I
II

 

Z
h

a
n

g
 e

t 
a

l.
, 2

0
1

9
 M

e
a

su
re

 I
 

Z
h

a
n

g
 e

t 
a

l.
, 2

0
1

9
 M

e
a

su
re

 I
I 

Z
h

a
n

g
 e

t 
a

l.
, 2

0
1

9
 M

e
a

su
re

 
II

I 
Z

h
a

n
g

 e
t 

a
l.

, 2
0

1
9

 M
e

a
su

re
 

II
I 

Z
h

a
n

g
 e

t 
a

l.
, 2

0
1

9
 M

e
a

su
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 I
V

 

P
ro

p
o

se
d

 M
e

a
su

re
 

Peng et al. (2017) 1                   

Ejegwa, 2018 Measure I 0.9 1                 

Ejegwa, 2018 Measure II 0.9 1 1               

Ejegwa, 2018 Measure III 0.9 1 1 1             

Zhang et al., 2019 Measure I 0.9 1 1 1 1           

Zhang et al., 2019 Measure II 0.9 1 1 1 1 1         

Zhang et al., 2019 Measure III 0.9 1 1 1 1 1 1       

Zhang et al., 2019 Measure III 0.9 1 1 1 1 1 1 1     

Zhang et al., 2019 Measure IV 0.9 1 1 1 1 1 1 1 1   

Proposed Measure 0.9 1 1 1 1 1 1 1 1 1 
 
The table of Spearman's coefficient of correlation values shows that there is 

perfectly positive correlation in almost all the cases. 
The graphical representation (Figure 2) is also shown for better understanding of the 
selection procedure. 



Significance of TOPSIS approach to MADM in computing exponential divergence measures… 

259 

 

Figure 2. Comparison of existing with the projected divergence measure 

7. Conclusion 

The paper offers new exponential divergence measures which comply with the 
conventional parameters of PFSs. The credibility of the proposed divergence measures 
through numerical computations has been confirmed as well. Further, these 
divergence measures have been employed to the application of MADM problem for the 
selection of smartphones. This analysis depicts an extension of TOPSIS methodology 
under Pythagorean fuzzy sets (PFSs) environment. The technique for order preference 
by similarity to ideal solutions (TOPSIS) notable and powerful technique for multi 
attribute decision making (MADM) issues. The goal of this investigation is to broaden 
TOPSIS to handle MADM problems under PFSs. However, in this TOPSIS approach, 
sometimes there could be severe loss of data and misleading results in the fuzzy 
environment. To overcome this, a sensitivity analysis has been done for better 
reliability and accuracy of the decision. A case study is taken to rank five leading 
smartphones based on five criteria using the proposed divergence measures. These 
weighted divergence measures can be applied to complex decision making and risk 
analysis in the future. Furthermore, criteria weights can be chosen using Entropy and 
sensitivity analysis of the obtained results with the results obtained by the basic 
classical methods TOPSIS, fuzzy TOPSIS methods, and Intuitionistic fuzzy TOPSIS can 
be done.    

Author Contributions: H. D. Arora made a significant contribution to the article's 
concept or design. The author is also responsible for methodology, drafting, and 
investigations. Anjali Naithani contributed to software validation, formal analysis, 
data processing, and critical supervision. She also assisted in critically edited the 
article for essential intellectual content. Both authors agreed to be responsible for all 
aspects of the work, including ensuring that any questions about the work's accuracy 
or integrity are thoroughly examined and resolved. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

D1(P,Q) D2(P,Q) D3(P,Q) D4(P,Q) D5(P,Q) D6(P,Q) D7(P,Q) D8(P,Q) Proposed
Measure

Comparison of existing with the proposed 
divergence measures

R1 R2 R3 R4 R5



Arora/Decis. Mak. Appl. Manag. Eng.  5 (1) (2022) 246-263 

260 

Funding: This research received no external funding. 

Data Availability Statement: The authors certify that the data supporting the 
findings of this study are available within the article. 

Acknowledgments: Authors thank the anonymous reviewers and editors for their 
careful reading of our manuscript and their insightful comments and suggestions. 

Conflicts of Interest: The authors declare that they have no known competing 
financial interests or personal relationships that could have appeared to influence the 
work reported in this paper. 

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