https://doi.org/10.31449/inf.v45i1.3199 Informatica 45 (2021) 143–148 143 
 
Study of Fuzzy Distance Measure and Its Application to Medical 
Diagnosis 
Taruna
 
and H. D. Arora  
Department of Mathematics, Amity University, Noida, India 
E-mail: hdarora@amity.edu  
 
Vijay Kumar 
Manav Rachna International Institute of Research & Studies, Faridabad, India 
Keywords: fuzzy sets, directed divergence, fuzzy relative information measures, multi criteria decision making 
Received: June 12, 2020 
Ambiguity has an important part in the contrary observations around peripheral world. Entropy is 
imperative for measuring uncertain information which was first introduced by Shannon (1948) to measure 
the uncertain degree of randomness in a probability distribution. Fuzzy information measures have been 
applied widely in the area of decision making. Jensen–Shannon divergence is a useful distance measure 
in the probability distribution space. The present communication we propose a way of measuring the 
difference between two fuzzy sets by means of a function, called divergence. In addition, study of their 
detailed properties for its validity is also discussed. The applications of these newly developed fuzzy 
divergence measure have been provided to the optimal decision making based on the weights of 
alternatives. Numerical verification has been illustrated to demonstrate the proposed method for solving 
optimal decision-making problem under fuzzy environment. 
Povzetek: Za probleme medicinske diagnostike je narejena študija nedorečenosti in entropije. 
 
1 I ntr o duct i o n 
Information theory advanced out of mathematical studies 
of the problems linked with communication, storage, and 
transmission of massages. It originated from the 
fundamental paper “The Mathematical Theory of 
Communication” published by Shannon [1]. Shannon 
developed mathematical schemes for quantitatively 
defining the ideas of facts and proved several very trendy 
outcomes with deeper effects. Various generalizations of 
Shannon entropy were studied by Renyi [2], Arimoto [3], 
Sharma and Taneja [4], De Luca and Termini [5], 
Kaufmann [6] and Peerzada et al. [7]. Uncertainty and 
fuzziness are the primary nature of human wondering and 
of many real-world objectives. Fuzziness is found in our 
decision, in our language and inside the way we process 
information. The fundamental use of information is to get 
rid of uncertainty and fuzziness. 
In reality, we degree data furnished by using the 
quantity of probabilistic uncertainty eliminated in an 
experiment and the measure of uncertainty eliminated is 
also called as a measure of information while degree of 
fuzziness is the measure of vagueness and ambiguity of 
uncertainties. The theory of fuzzy sets (FSs) developed by 
Zadeh [8], as a generalization of classical set theory, for 
representing vague and indistinct phenomena. This idea 
serves as an effective tool for know-how of the behaviour 
of humanistic systems in which human judgment, 
perceptions and feelings play a critical role. In fuzzy set 
concept, the entropy is described as a degree of fuzziness 
which expresses the quantity of ambiguity or problem in 
we decide whether an element belongs to a set or not. 
Bhandari and Pal [9] extended the probabilistic 
exponential entropy idea of Pal and Pal [10] to the fuzzy 
phenomenon. Kapur [11] discussed fuzzy measures 
uncertainty due to fuzziness of information. 
In fuzzy context, several measures have been 
proposed to measure the degree of difference between two 
fuzzy sets. Measure of fuzzy divergence between two 
fuzzy sets gives the difference between two fuzzy sets and 
this measure of difference between two fuzzy sets is called 
the fuzzy divergence measure. 
The similarity measure is important tools that can be 
used in decision-making problem to deal with uncertainty 
through IFS theory. Various distance measures have been 
proposed by different researchers. It has been observed 
that different distance measure produces different values 
while measuring the distance degree between two IFSs. 
Also, sometimes existing distance measures are not able 
to give an appropriate and convenient result for a pair of 
IFSs. For this reason, it is always necessary to derive 
advanced measures for better decision making. 
To explain the distinction among fuzzy sets, the 
distance measure was set up and was regarded as dual 
model of correspondence measure. Many researchers, 
such as Yager [12], Kosko [13] and Kaufmann [6] had 
used distance measure to define fuzzy entropy. Several 
recent methods of fuzzy entropy generated by distance 
measure and properties of distance measure were extended 
by Fan et al. [14]. The distances among two fuzzy subsets 

144 Informatica 45 (2021) 143–148 H.D. Arora et. al  
 
on a fuzzy subset of R+ were characterized by Dubois and 
Prade [15]. Thus, the set of distances between two sets was 
simplified whereas the shortest distance between two crisp 
sets was not simplified. The shortest distance among two 
fuzzy sets as a density function on non-negative reals was 
described by Rosenfeld [16]. Thus, related to Kullback 
and Leibler [17] probabilistic measure of divergence, the 
subsequent measure of fuzzy directed divergence was 
initiated by Bhandari and Pal [9]. Montes et al. [18] 
proposed an axiomatic form to measure the difference 
between fuzzy sets and we study in detail the case of local 
divergence. 
Luo and Zhao [19] gave the algorithms for pattern 
recognition and use it to solve medical diagnosis 
problems. Gupta and Tiwari [20] and Datta and Goala [21] 
proposed cosine similarity measure for intuitionistic and 
interval-valued intuitionistic fuzzy sets using an advanced 
distance measure on intuitionistic fuzzy sets.  
2 Preliminaries 
The model of entropy was initiated to arrange numerical 
quantity of ambiguity.  
Shannon [1] originated a quantity  
𝐻 ( 𝑃 ) = − ∑ 𝑝 𝑖 𝑙𝑜𝑔 𝑝 𝑖 𝑛 𝑖 =1
     (1) 
for the uncertainty of a probability distribution 
( 𝑝 1
, 𝑝 2
, 𝑝 3
… 𝑝 𝑛 ) and called it entropy. 
A fuzzy set 𝐴 ̃
 in a finite Universe of discourse X = 
( 𝑥 1
, 𝑥 2
, 𝑥 3
… 𝑥 𝑛 ) is given by  
𝐴 ̃
 = {〈𝑥 , 𝜇 𝐴 ̃
( 𝑥 ) 〉|𝑥 ∈ 𝑋 }    (2) 
where 𝜇 𝐴 ̃
: X → [0,1]  𝑖𝑠 𝑡 ℎ𝑒 membership function of 
𝐴 ̃
.The number 
𝜇 𝐴 ̃
( 𝑥 ) 𝑑𝑒𝑠𝑐𝑟𝑖𝑏𝑒𝑠 𝑡 ℎ𝑒 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑏𝑒𝑙𝑜𝑛𝑔𝑖𝑛𝑔𝑛𝑒𝑠𝑠 𝑜𝑓 𝑥 ∈ 𝑋 𝑖𝑛 𝐴 ̃
. 
De Luca and Termini [5] defined fuzzy entropy for a fuzzy 
set A corresponding to Shannon Entropy (1948) as 
H( 𝐴 ̃
) = −
1
𝑛 ∑ [𝜇 𝐴 ̃
( 𝑥 𝑖 ) 𝑙𝑜𝑔 ( 𝜇 𝐴 ̃
( 𝑥 𝑖 ) )+  ( 1 −
𝑛 𝑖 =1
𝜇 𝐴 ̃
( 𝑥 𝑖 ) ) 𝑙𝑜𝑔 ( 1 − 𝜇 𝐴 ̃
( 𝑥 𝑖 ) ) ]   (3) 
Motivated by the fundamental properties of directed 
divergence, Kapur [11] explained the concept of fuzzy 
directed divergence as follows: The directed divergence of 
fuzzy set A from the fuzzy set B is a function D (A; B) 
that should comply with the subsequent requirements 
which satisfies the following conditions: 
1. 𝐷 ( 𝐴 ; 𝐵 ) ≥ 0 
2. 𝐷 ( 𝐴 ; 𝐵 ) = 0 𝑖𝑓𝑓 𝐴 = 𝐵 
3. 𝐷 ( 𝐴 ; 𝐵 ) ≥ 0 is a convex function in (0,1) 
4. 𝐷 ( 𝐴 ; 𝐵 ) ≥ 0 should not change, when 𝜇 𝐴 ( 𝑥 𝑖 ) is 
changed to1 − 𝜇 𝐴 ( 𝑥 𝑖 )  and 𝜇 𝐵 ( 𝑥 𝑖 ) is changed to 1 −
𝜇 𝐵 ( 𝑥 𝑖 ) . 
Now, corresponding to Kullback – Leibler’s [17] 
measure of divergence, Bhandari and Pal [9] proposed a 
fuzzy divergence measure A and B given by 
𝐷 ( 𝐴 ; 𝐵 ) =
1
𝑛 ∑ [𝜇 𝐴 ( 𝑥 𝑖 ) 𝑙𝑜𝑔
𝜇 𝐴 ( 𝑥 𝑖 )
𝜇 𝐵 ( 𝑥 𝑖 )
+ ( 1 −
𝑛 𝑖 =1
𝜇 𝐴 ( 𝑥 𝑖 ) ) 𝑙𝑜𝑔
( 1−𝜇 𝐴 ( 𝑥 𝑖 ) )
( 1−𝜇 𝐵 ( 𝑥 𝑖 ) )
]     (4) 
Later, Shang and Jiang [22] was pointed out that the 
expression (4) has some limitations, i.e., if 𝜇 𝐴 ( 𝑥 𝑖 ) 
approaches to 0 or 1, then its value tends to ∞. Therefore 
they proposed a modified version of fuzzy divergence 
measure (4), given as  
𝐽 ( 𝐴 ; 𝐵 ) = ∑ [𝜇 𝐴 ( 𝑥 𝑖 ) 𝑙𝑜𝑔
𝜇 𝐴 ( 𝑥 𝑖 )
𝜇 𝐴 ( 𝑥 𝑖 ) +𝜇 𝐵 ( 𝑥 𝑖 )
2
+ ( 1 −
𝑛 𝑖 =1
𝜇 𝐴 ( 𝑥 𝑖 ) ) 𝑙𝑜𝑔
( 1−𝜇 𝐴 ( 𝑥 𝑖 ) )
1−(
𝜇 𝐴 ( 𝑥 𝑖 ) +𝜇 𝐵 ( 𝑥 𝑖 )
2
)
]   (5) 
Corresponding to Kerridge [23] inaccuracy measure, 
Verma and Shrama [24] define a measure of inaccuracy of 
fuzzy set B with respect to fuzzy set A, as 
𝐼 ( 𝐴 ; 𝐵 ) = −
1
𝑛 ∑ [𝜇 𝐴 ( 𝑥 𝑖 ) 𝑙𝑜𝑔 𝜇 𝐵 ( 𝑥 𝑖 )+ ( 1 −
𝑛 𝑖 =1
𝜇 𝐴 ( 𝑥 𝑖 ) ) 𝑙𝑜𝑔 ( 1 − 𝜇 𝐵 ( 𝑥 𝑖 ) ) ]    (6) 
Ohlan [25] proposed a parametric generalized 
measure of divergence between two fuzzy sets A and B 
corresponding to Taneja [26] as 
𝐿 𝑡 ( 𝐴 , 𝐵 ) = ∑
( 𝜇 𝐴 ( 𝑥 𝑖 ) +𝜇 𝐵 ( 𝑥 𝑖 ) )
2
2
𝑡 𝑛 𝑖 =1
×
[
( 𝜇 𝐴 ( 𝑥 𝑖 ) +𝜇 𝐵 ( 𝑥 𝑖 ) )
𝑡 √𝜇 𝐴 ( 𝑥 𝑖 ) 𝜇 𝐵 ( 𝑥 𝑖 )
𝑡 +1
+
( 2−𝜇 𝐴 ( 𝑥 𝑖 ) +𝜇 𝐵 ( 𝑥 𝑖 ) )
𝑡 √( 1−𝜇 𝐴 ( 𝑥 𝑖 ) ) .( 1−𝜇 𝐵 ( 𝑥 𝑖 ) )
𝑡 +1
]  (7) 
    𝑡 = 0,1,2, … 
The generalized measure of fuzzy directed divergence 
of order 𝛼 and type 𝛽 is given by Arora and Dhiman [27] 
as 
𝐷 𝛼 𝛽 ( 𝐴 : 𝐵 ) =
1
( 1 − 𝛼 ) 𝛽 ∑ [{
𝜇 𝐴 ( 𝑥 𝑖 )
𝛼 𝜇 𝐴 ( 𝑥 𝑖 )
𝜇 𝐵 ( 𝑥 𝑖 )
𝛼 𝜇 𝐵 ( 𝑥 𝑖 )
𝑛 𝑖 =1
+
( 1 − 𝜇 𝐴 ( 𝑥 𝑖 ) )
𝛼 ( 1−𝜇 𝐴 ( 𝑥 𝑖 ) )
( 1 − 𝜇 𝐵 ( 𝑥 𝑖 ) )
𝛼 ( 1−𝜇 𝐵 ( 𝑥 𝑖 ) )
}
𝛽 − 2
𝛽 ] 
  where 𝛼 > 0, 𝛼 ≠ 1, 𝛽 ≠ 0. (8) 
Prakash and Kumar [28] proposed a new fuzzy 
divergence measure of fuzzy set B with respect to fuzzy 
set A, as follows: 
𝐾 ( 𝐴 , 𝐵 ) =
−𝑙𝑜𝑔 (
1+
1
𝑛 ∑ [√𝜇 𝐴 ( 𝑥 𝑖 ) 𝜇 𝐵 ( 𝑥 𝑖 ) +√( 1−𝜇 𝐴 ( 𝑥 𝑖 ) ) .( 1−𝜇 𝐵 ( 𝑥 𝑖 ) ) ]
𝑛 𝑖 =1
2
)
      (9) 
Kumari et al. [29] proposed Weighted Fuzzy 
Exponential J-Divergence as 
𝐻 ( 𝐴 ; 𝑊 ) =
1
𝑛 ( √ 𝑒 −1)
∑ ∑ 𝑤 𝑖𝑗
[( 𝜇 𝐴 𝑓 𝑖𝑗
) 𝑒 1−𝜇 𝐴 𝑓 𝑖𝑗
+ ( 1 −
𝑀 −1
𝑗 =0
𝑀 −1
𝑖 =0
( 𝜇 𝐴 𝑓 𝑖𝑗
) 𝑒 1−𝜇 𝐴 𝑓 𝑖𝑗
− 1]                (10) 

Study of Fuzzy Distance Measure and Its Application to... Informatica 45 (2021) 143–148 145 
 
where 𝜇 𝐴 𝑓 𝑖𝑗
 is the membership values of the pixels in the 
image and 𝑓 𝑖𝑗
 is the (i, j) th pixel of the image A. 
Tiwari and Gupta [30] proposed entropy measures 
and erived relation between distance, entropy, and 
similarity measures for IvIFSs. 
3 Proposed Fuzzy D i s t a nce Measure 
Let X = ( 𝑥 1
, 𝑥 2
, 𝑥 3
… 𝑥 𝑛 ) be the universe of discourse. 
Let 𝐴 = {〈𝑥 𝑖 , 𝜇 𝐴 ( 𝑥 𝑖 ) 〉|𝑥 𝑖 ∈ 𝑋 } and B = 
{〈𝑥 𝑖 , 𝜇 𝐵 ( 𝑥 𝑖 ) 〉|𝑥 𝑖 ∈ 𝑋 }  be two fuzzy sets. Then we 
propose new distance measure as follows: 
𝐷 ( 𝐴 ; 𝐵 ) =
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
1+ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
 (11) 
Theorem 3.1. The fuzzy distance measure 𝐷 ( 𝐴 ; 𝐵 ) 
defined in equation (11) is a valid measure of fuzzy 
divergence. 
Proof. All the necessary four conditions to be a 
distance measure are satisfied by the new distance 
measure which are as follows: 
(P1) 0 ≤ 𝐷 ( 𝐴 ; 𝐵 ) ≤ 1 
(P2) 𝐷 ( 𝐴 ; 𝐵 ) = 0 if and only if 𝜇 𝐴 ( 𝑥 𝑖 ) = 𝜇 𝐵 ( 𝑥 𝑖 ) . 
(P3) 𝐷 ( 𝐴 ; 𝐵 ) = 𝐷 ( 𝐵 ; 𝐴 ) 
(P4) If A, B and C be three fuzzy sets, then the 
distance measure satisfies the triangular inequality, i.e., 
𝐷 ( 𝐴 ; 𝐶 ) ≤ 𝐷 ( 𝐴 ; 𝐵 ) + 𝐷 ( 𝐵 ; 𝐶 ) . 
Proof: We will now prove these conditions one by 
one: 
(P1)  As we know that, 𝐴 = {〈𝑥 𝑖 , 𝜇 𝐴 ( 𝑥 𝑖 ) 〉|𝑥 𝑖 ∈ 𝑋 } for 
degree of membership 0 ≤ 𝜇 𝐴 ( 𝑥 𝑖 ) ≤ 1. 
That is, for 𝐴 = { 〈𝑥 𝑖 , 𝜇 𝐴 ( 𝑥 𝑖 ) 〉|𝑥 𝑖 ∈ 𝑋 } and B= 
{ 〈𝑥 𝑖 , 𝜇 𝐵 ( 𝑥 𝑖 ) 〉|𝑥 𝑖 ∈ 𝑋 } 
0 ≤ |𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) | ≤ 1 ⇒ 0 ≤
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )−
𝜇 𝐵 ( 𝑥 𝑖 ) | ≤
𝜋 2
  
 ⇒ 0 ≤ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |} ≤ 1 
                  (12) 
 ⇒ 0 ≤ 1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |} ≤ 2
                  (13) 
From (12) and (13), we have  
0 ≤ 2.
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
≤ 1 
      ⇒  0 ≤
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
1+ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
≤ 1 ⇒
0 ≤ 𝐷 ( 𝐴 ; 𝐵 ) ≤ 1. 
(P2)  𝐷 ( 𝐴 ; 𝐵 ) = 0 
⇔     
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
= 0 
⇔     
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
= 0 
⇔ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |} = 0
⇔  |𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) | = 0 
⇔ 𝜇 𝐴 ( 𝑥 𝑖 ) = 𝜇 𝐵 ( 𝑥 𝑖 ) ⇔  𝐴 = 𝐵 
Therefore, 𝐷 ( 𝐴 ; 𝐵 ) = 0 if and only if 𝜇 𝐴 ( 𝑥 𝑖 ) = 𝜇 𝐵 ( 𝑥 𝑖 ) .  
(P3)  As  
𝐷 ( 𝐴 ; 𝐵 ) = 
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
 
            =
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 )− 𝜇 𝐴 ( 𝑥 𝑖 ) |}
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 )− 𝜇 𝐴 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
 
= 𝐷 ( 𝐵 ; 𝐴 ) 
To prove the triangular inequality, i.e., 𝐷 ( 𝐴 ; 𝐶 ) ≤ 
𝐷 ( 𝐴 ; 𝐵 ) + 𝐷 ( 𝐵 ; 𝐶 ) , we have to prove that   
(P4) In order to prove fourth necessary condition, we 
must first prove the identity 
sin( 𝐴 + 𝐵 ) ≤ sin( 𝐴 )+ sin ( 𝐵 ) , where A and B 
are acute angles. 
or in other words, we have to show that 
sin( 𝐴 )+ sin( 𝐵 )− sin ( 𝐴 + 𝐵 ) ≥ 0 
⇒ sin 𝐴 + sin 𝐵 − sin𝐴 . 𝑐𝑜𝑠𝐵 − 𝑐𝑜𝑠𝐴 . 𝑠𝑖𝑛𝐵 ≥ 0 
⇒ sin 𝐴 ( 1 − 𝑐𝑜𝑠𝐵 )+ sin 𝐵 ( 1 − 𝑐𝑜𝑠𝐴 ) ≥ 0 
Since A and B are acute angles, therefore,  
sin 𝐴 , ( 1 − 𝑐𝑜𝑠𝐵 ), sin 𝐵 , ( 1 − 𝑐𝑜𝑠𝐴 ) are all positive 
and hence the identity holds good. 
Now, consider 𝐴 = {〈𝑥 , 𝜇 𝐴 ( 𝑥 ) 〉|𝑥 ∈ 𝑋 } , 𝐵 = 
{〈𝑥 , 𝜇 𝐵 ( 𝑥 ) 〉|𝑥 ∈ 𝑋 } and 𝐶 = {〈𝑥 , 𝜇 𝐶 ( 𝑥 ) 〉|𝑥 ∈ 𝑋 } be 
three fuzzy sets. 
As,  
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) | ≤ |𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) | +
|𝜇 𝐵 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |  
 ( ∵ 𝐼𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑜𝑓 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 ) 
⇒ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |} ≤ 𝑠𝑖𝑛 𝜋 2
[{|𝜇 𝐴 ( 𝑥 𝑖 )−
𝜇 𝐵 ( 𝑥 𝑖 ) | + |𝜇 𝐵 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |}]  
⇒ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |} ≤ 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )−
𝜇 𝐵 ( 𝑥 𝑖 ) |} + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |}  
Also. 

146 Informatica 45 (2021) 143–148 H.D. Arora et. al  
 
1 + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐶 ( 𝑥 𝑖 ) |} ≤ 1 +
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 )− 𝜇 𝐵 ( 𝑥 𝑖 ) |} + 𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 )−
𝜇 𝐶 ( 𝑥 𝑖 ) |}  
⇒ 
1
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
≥
1
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
  
⇒ 1 −
1
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
≤ 1 −
1
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
  
⇒ 
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
 ≤
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
  
⇒ 
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
     ≤
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
+
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
  
⇒
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
     ≤
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
+
2
𝑛 ∑
𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
1+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐴 ( 𝑥 𝑖 ) −𝜇 𝐵 ( 𝑥 𝑖 ) |}+𝑠𝑖𝑛 {
𝜋 2
|𝜇 𝐵 ( 𝑥 𝑖 ) −𝜇 𝐶 ( 𝑥 𝑖 ) |}
𝑛 𝑖 =1
  
⇒ 𝐷 ( 𝐴 ; 𝐶 ) ≤ 𝐷 ( 𝐴 ; 𝐵 ) + 𝐷 ( 𝐵 ; 𝐶 ) . 
Hence, the proposed distance measure satisfies all the 
necessary properties. 
4 Application of Proposed Fuzzy 
Measure to Medical Diagnosis 
In a classical problem of medical diagnosis, assume that if 
a doctor needs to diagnose some of patients "𝑃 =
{ 𝐴𝑙𝑒𝑥 , 𝐶 ℎ𝑟𝑖𝑠 , 𝐽𝑎𝑚𝑒𝑠 , 𝑀𝑖𝑘𝑒 𝑎𝑛𝑑 𝑆 ℎ𝑎𝑤𝑛 } " under 
some defined diagnosis "𝐷 = { 𝑉𝑖𝑟𝑎𝑙 𝑓𝑒𝑣𝑒𝑟 ( 𝑉𝐹 ) ,
𝑀𝑎𝑙𝑎𝑟𝑖𝑎 ( 𝑀 ) , 𝑇𝑦 𝑝 ℎ𝑜𝑖𝑑 ( 𝑇 ) ,
𝑆𝑡𝑜𝑚𝑎𝑐 ℎ 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 ( 𝑆𝑃 ) 𝑎𝑛𝑑 𝐶 ℎ𝑒𝑠𝑡 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 ( 𝐶𝑃 ) } " 
and a set of symptom "𝑆 = { 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 ( 𝑇𝑒𝑚𝑝 . ) ,
𝐻𝑒𝑎𝑑𝑎𝑐 ℎ𝑒 ( 𝐻 ) , 𝑆𝑡𝑜𝑚𝑎𝑐 ℎ 𝑝𝑎𝑖𝑛 ( 𝑆 . 𝑃𝑎𝑖𝑛 ) ,
𝐶𝑜𝑢𝑔 ℎ( 𝐶 ) 𝑎𝑛𝑑 𝐶 ℎ𝑒𝑠𝑡 𝑝𝑎𝑖𝑛 ( 𝐶𝑃 ) } ". The following 
tables (table 1 and table 2) serve the purpose of the 
proposed computational application: 
In view of the table 3, it is being concluded that 
"𝐴𝑙𝑒𝑥 " is suffering from  "𝑀𝑎𝑙𝑎𝑟𝑖𝑎 "; "𝐶 ℎ𝑟𝑖𝑠 ", 
"𝐽𝑎𝑚𝑒𝑠 " and "𝑀𝑖𝑘𝑒 " are suffering from 
 "𝐶 ℎ𝑒𝑠𝑡 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 " and "𝑆 ℎ𝑎𝑤𝑛 " is suffering from 
"𝑉𝑖𝑟𝑎𝑙 𝑓𝑒𝑣𝑒𝑟 ". 
This is because smaller value of the patient 
against each distance measure indicates the more 
probability of having the disease. 
5 Comparative Study 
Jain and Kumar [31] proposed the intuitionistic fuzzy 
based trigonometric entropy as: 
 Temp. H S. Pain C CP 
VF 0.4 0.4 0.3 0.1 0.6 
M 0.4 0.2 0.5 0.6. 0.7 
T 0.3 0.6 0.1 0.5 0.7 
SP 0.2 0.3 0.7 0.4 0.3 
CP 0.4 0.6 0.5 0.4 0.6 
Table 1: Fuzzy membership values for diseases and their 
symptoms. 
 Temp. H S. Pain C CP 
Alex 0.8 0.1 0.7 0.6 0.4 
Chris 0.3 0.6 0.8 0.4 0.7 
James 0.7 0.7 0.6 0.6 0.4 
Mike 0.5 0.4 0.5 0.4 0.6 
Shawn 0.5 0.6 0.3 0.2 0.9 
Table 2: Fuzzy membership values for patients and related 
symptoms. 
 Alex. Chris James  Mike Shawn 
VF 0.680965 0.493065 0.634603 0.273332 0.327418 
M 0.421368 0.421368 0.469442 0.296955 0.538941 
T 0.689085 0.24253 0.546709 0.444991 0.40803 
SP 0.42172 0.381062 0.516246 0.398187 0.670988 
CP 0.596858 0.233026 0.42181 0.148478 0.367724 
Table 3: Values of Fuzzy relative entropy measure for the 
patients and the likely diseases. 

Study of Fuzzy Distance Measure and Its Application to... Informatica 45 (2021) 143–148 147 
 
𝐸 𝐼𝐹
( 𝐴 )  =
1
𝑛 ∑ [𝑐𝑜𝑠
𝜋 2
( |𝜇 𝐴 2
( 𝑥 𝑖 )− 𝜈 𝐴 2
( 𝑥 𝑖 ) | ) ]
𝑛 𝑖 =1
 
The fuzzy version of the entropy is: 
𝐸 ( 𝐴 )  =
1
𝑛 ∑ [𝑐𝑜𝑠 𝜋 2
( |𝜇 𝐴 2
( 𝑥 𝑖 )− 𝜇 𝐵 2
( 𝑥 𝑖 ) | ) ]
𝑛 𝑖 =1
 
From the table, it is concluded that the larger value in 
the column is the decision value. 
Wei et al. [32] proposed the generalized fuzzy entropy 
as: 
𝐻 ( 𝐴 )  =
1
𝑛 ∑ [{ 𝑐𝑜𝑠 𝜋 (
𝜇 𝐴 ( 𝑥 𝑖 )− 𝜈 𝐴 ( 𝑥 𝑖 )
4
) − 1}
𝑛 𝑖 =1
× 
1
√2 − 1
]  
From the table, it is concluded that the smaller value 
in the column is the decision value. 
6 Conclusions 
In this paper, we have proposed a relative distance 
measure for fuzzy sets. Proof of its validity is also 
considered through numerical computations. Some of the 
essential properties of the measure are also studied. It has 
been observed that this measure is more flexible in terms 
of their previous derived measures. Application of this 
measure is also studied in medical diagnosis to check its 
legitimacy. Also, from the table 4 and 5, it is concluded 
that the result obtained from the proposed entropy is 
similar with the results of the existing entropies (shown in 
table nos.), which validates the fact that the proposed 
entropy is valid and have applications across disciplines. 
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