Fuzzy hypersoft sets and its weightage operator for decision making

Corresponding Author: somen008@rediffmail.com
10.22105/jfea.2021.275132.1083
E-ISSN: 2717-3453 | P-ISSN: 2783-1442
|
Abstract
1 | Introduction
Uncertainty is a part and parcel of our daily life activities and it exists in various forms. The
classical set-theoretic approach could not find any way to deal with incomplete information i.e
the information which is blurred. For the scientific computation of vague data, we need a
powerful tool that gives us a precise idea about objects so that we get insight into the objects and
classify them into different groups. Finally, fuzzy set theory was introduced by Zadeh [24] in 1965
for dealing with uncertain, incomplete, indeterministic information in a systematic way. After the
introduction of fuzzy set theory, it has been used successfully in various fields such as engineering,
social science, computer science, control theory, game theory, pattern recognition, logic, etc. In
the fuzzy set, every object has some membership value and it is called the degree of membership
and each membership value belongs to the unit closed interval [0, 1]. So, a fuzzy set is an extension
of a crisp set where there are only two choices (0 or 1) to denote the membership of an object i.e
Journal of Fuzzy Extension and Applications
www.journal-fea.com
J. Fuzzy. Ext. Appl. Vol. 2, No. 2 (2021) 163–170.
Paper Type: Research Paper
Fuzzy Hypersoft Sets and Its Weightage Operator for
Decision Making
Somen Debnath *
Department of Mathematics, Umakanta Academy, Agartala-799001, Tripura, India; somen008@rediffmail.com.
Citation:
Debnath, S. (2021). Fuzzy hypersoft sets and its weightage operator for decision making. Journal of
fuzzy extension and application, 2 (2), 163-170.
Accepted: 17/05/2021
Revised: 29/04/2021
Reviewed: 11/03/2021
Received: 06/02/2021
Hypersoft set is an extension of the soft set where there is more than one set of attributes occur and it is very much
helpful in multi-criteria group decision making problem. In a hypersoft set, the function F is a multi-argument function.
In this paper, we have used the notion of Fuzzy Hypersoft Set (FHSS), which is a combination of fuzzy set and hypersoft
set. In earlier research works the concept of Fuzzy Soft Set (FSS) was introduced and it was applied successfully in various
fields. The FHSS theory gives more flexibility as compared to FSS to tackle the parameterized problems of uncertainty.
To overcome the issue where FSS failed to explain uncertainty and incompleteness there is a dire need for another
environment which is known as FHSS. It works well when there is more complexity involved in the parametric data i.e
the data that involves vague concepts. This work includes some basic set-theoretic operations on FHSSs and for the
reliability and the authenticity of these operations, we have shown its application with the help of a suitable example.
This example shows that how FHSS theory plays its role to solve real decision-making problems.
Keywords: Fuzzy set, Soft set, Hypersoft set, Decision making.
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http://dx.doi.org/10.22105/jfea.2021.275132.1083

164
Debnath
|J.
Fuzzy.
Ext.
Appl.
2(2)
(2021)
163-170
we choose 1 for belongingness and 0 for non-belongingness of an object. Compared to classical set, by
using fuzzy set theory we are enabled to extend the range of domain under the fuzzy environment where
each object is a fuzzy word or fuzzy sentence or fuzzy axiom, etc. It gives a general formula to model
vague or uncertain or indeterministic or incomplete concepts lucidly. But we know that every theory has
its limitations so does fuzzy theory. Using fuzzy theory, we only determine the degree of membership
of an object but, there is no scope of non-membership degree. We have experienced the co-existence
of two opposite concepts like agreement-disagreement, truth-falsity, success-failure, yes-no, attraction-
repulsion in a real-life scenario to make a balance. Like this, there is a demand for the coexistence of
membership value and non-membership value. To realize the importance of non-membership value
along with membership value another set-theoretical notion known as Intuitionistic Fuzzy Set (IFS) was
introduced by Atanassov [5] in 1986. In IFS, every object has two values i.e membership value and non-
membership value, and their sum range from 0 to 1. For the need of the hour, fuzzy set theory has been
applied successfully to develop new theories, propositions, axioms, etc. Some of them are given in [6],
[10], [11], [12], [25], [26].
Due to the more complex like uncertainty in data the fuzzy set and its variants are not sufficient for
mathematical modeling. It is due to the parameters involved in an attribute. To handle parametric data
comprehensively we need another tool to solve the issue. This creates the invention of the Soft Set (SS).
Soft set theory was introduced by Molodtsov [16] in 1999. It gives more rigidity to model the vague
concept in a parametric way. It is the more general framework as compared to the fuzzy set and its
variants. The soft set has been progressed more rapidly and applied in different fields with great success.
Some of the novel’s work on soft set theory given in [3], [4], [7], [8], [14], [15]. In 2001, a new concept
known as Fuzzy Soft Set (FSS) was introduced by Maji et al. [13]. FSS is a combination of Fuzzy Set
(FS) and SS. Some applications and extensions of FSS are given in [9], [17], [21], [22].
Recently, Smarandache [20] generalize the concept of the soft set to the hypersoft set, where the function
F is transformed into a multi-attribute function. The main motivation of using Hypersoft Set (HSS) is
that when the attributes are more than one and further bisected, the SS environment cannot be applied
to handle such types of cases. So, there is a worth need to define a new approach to solve these.
Afterward, Saeed et al. [19] studied the fundamentals of hypersoft set theory, Abbas et al. [1] defined
the basic operations on hypersoft sets, Yolcu and Ozturk [23] introduced fuzzy hypersoft set and its
application in decision-making, Ajay and Charisma [2] defined neutrosophic hypersoft topological
spaces, aggregate operators of neutrosophic hypersoft set studied in [18], Extension of TOPSIS method
under intuitionistic fuzzy hypersoft set environment is discussed in [27], some fundamental operations
on interval-valued neutrosophic hypersoft set are discussed in [28].
In this work, we have used the notion of the Fuzzy Hypersoft Set (FHSS), which is an amalgamation of
the FS and HSS. Afterward, we define different set-theoretic operations on them, and then there is an
attempt to use this concept effectively in multi-criteria decision-making problems using weightage
aggregate operator.
2 | Preliminaries
This section includes some basic definitions with examples that are relevant for subsequent discussions.
Definition 1. [24] and [26]. Let X be a non-empty set. Then a fuzzy set A , defined on X , is a set
having the form  
 
 
A
A x, μ x : x X
  , where the function  
A
μ : X 0,1
 is called the membership
function and  
A
μ x is called the degree of membership of each element x X
 .

165
Fuzzy
hypersoft
sets
and
its
weightage
operator
for
decision
making
Definition 2. [16]. Let U be an initial universe and E be a set of parameters. Let  
P U denotes the power
set of U , and A E
 . Then the pair  
F, A is called a soft set over U , where F is a mapping given by
 
F : A P U
 .
Definition 3. [13]. Let U be an initial universe and E be a set of parameters. Let U
I be the set of all
fuzzy subsets of U , and A E
 . Then the pair  
F, A is called a fuzzy soft set over U , where F is a
mapping given by U
F : A I
 .
Definition 4. [20]. Let ξ be the set of the universe and  
P ξ denotes the power set of ξ . Consider
1 2 n
l ,l ,.......,l , for n 1
 , be n well-defined attributes, whose corresponding values are respectively the set
1 2 n
L ,L ,.......,L with i j
L L
   , for i j
 and  
i, j 1,2,......,n
 , then the pair  
1 2 n
F,L L ....... L
   is said
to be hypersoft set over ξ , where  
1 2 n
F : L L ....... L P ξ
    .
Example 1. Let  
1 2 3 4
U c ,c ,c ,c
 be the set of the universe of cars under consideration and  
1 3
A c ,c U
 
. We consider the attributes to be 1
x  size, 2
x  color, 3
x  cost price (in a dollar), 4
x  mileage,and 5
x 
model, and their respective values are given by Size= 1
X {small, medium, big}; color = 2
X 
{white, black, red}; cost price (in dollar)= 3
X  {1000,1050,1080}; model= 4
X  {honda amaze, tata
tigor, ford figo}.
Let the function be:  
1 2 3 4
F : X X X X P U
    . In respect of A , one has assumed that F  ({small,
white, 1000, Honda amaze}, {small, white, 1000, tata tigor}, {small, white, 1000, ford figo}, {small, white,
1050, Honda amaze}, {small, white, 1050, tata tigor}, {small, white, 1050, ford figo}, {small, white, 1080,
Honda amaze}, {small, white, 1080, tata tigor}, {small, white, 1080, ford figo}, {small, blue, 1000,
Honda amaze}, {small, blue, 1000, tata tigor}, {small, blue, 1000, ford figo}, {small, blue, 1050,
Honda amaze}, {small, blue, 1050, tata tigor}, {small, blue, 1050, ford figo}, {small, blue, 1080,
Honda amaze}, {small, blue, 1080, tata tigor}, {small, blue, 1080, ford figo}, {small, red, 1000,
Honda amaze}, {small, red, 1000, tata tigor}, {small, red, 1000, ford figo}, {small, red, 1050,
Honda amaze}, {small, red, 1050, tata tigor}, {small, red, 1050, ford figo}, {small, red, 1080,
Honda amaze}, {small, red, 1080, tata tigor}, {small, red, 1080, ford figo}, {medium, white, 1000,
Honda amaze}, {medium, white, 1000, tata tigor}, {medium, white, 1000, ford figo}, {medium, white,
1050, Honda amaze}, {medium, white, 1050, tata tigor}, {medium, white, 1050, ford figo}, {medium,
white, 1080, Honda amaze}, {medium, white, 1080, tata tigor}, {medium, white, 1080, ford figo},
{medium, blue, 1000, Honda amaze}, {mediuml, blue, 1000, tata tigor}, {medium, blue, 1000, ford figo},
{medium, blue, 1050, Honda amaze}, {medium, blue, 1050, tata tigor}, {medium, blue, 1050, ford figo},
{medium, blue, 1080, Honda amaze}, {medium, blue, 1080, tata tigor}, {medium, blue, 1080, ford figo},
{medium, red, 1000, Honda amaze}, {medium, red, 1000, tata tigor}, {medium, red, 1000, ford figo},
{big,white, 1000, Honda amaze}, {big, white, 1000, tata tigor}, {big, white, 1000, ford figo}, {big, white,
1050, Honda amaze}, {big, white, 1050, tata tigor}, {big,white, 1050, ford figo}, {big,white, 1080,
Honda amaze}, {big, white, 1080, tata tigor}, {big, white, 1080, ford figo}, {big, blue, 1000,
Honda amaze}, {big, blue, 1000, tata tigor}, {big, blue, 1000, ford figo}, {big, blue, 1050, Honda amaze},
{big, blue, 1050, tata tigor}, {big, blue, 1050, ford figo}, {big, blue, 1080, Honda amaze}, {big, blue, 1080,
tata tigor}, {big, blue, 1080, ford figo}, {big, red, 1000, Honda amaze}, {big, red, 1000, tata tigor}, {big,
red, 1000, ford figo}, {big, red, 1050, Honda amaze}, {big, red, 1050, tata tigor}, {big, red, 1050,
ford figo}, {big, red, 1080, Honda amaze}, {big, red, 1080, tata tigor}, {big, red, 1080, ford figo})= 
1 3
c ,c .

166
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(2021)
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Thus, there are 81 possible hypersoft sets to describe  
1 3
c ,c .
Definition 5. [23]. Let ς be the universe of discourse and  
P ς be the power set of ς . Suppose
1 2 n
l ,l ,.......,l , for n 1
 , be n well-defined attributes, whose corresponding values are respectively the set
1 2 n
L ,L ,.......,L with i j
L L
   , for i j
 and  
i, j 1,2,......,n
 , and 1 2 n
L L ....... L S
    , then the
pair  
F,S is said to be the FHSS over ς , where  
1 2 n
F : L L ....... L P ς
    and
   
 
 
1 2 n
S
Γ F L L ....... L x, μ F S : x ς
      , where μ is the membership function which
determines the value of the degree of belongingness and  

μ : ς 0,1 .
Example 2. Let us consider an example where we have proposed a data set that is suitable for selecting
a plot of land by the decision-makers. Suppose ς be the set of decision-makers to decide the best plot.
We consider  
 1 2 3 4 5
ς d ,d ,d ,d ,d and  
 
1 3 5
A d ,d ,d ς .
Now we consider the sets of attributes as 1
P = plot size (in sq. ft), 2
P = plot location, 3
P = cost of the
plot (in dollars), and 4
P = landmark surrounding of a plot. Their corresponding values are given as
 

1
P 2000,1745,1100,900,1245 , 
2
P {Agartala, Lucknow, Amritsar, Greater Noida, Hooghly},
 

3
P 4135,3812,3907,2547 and 
4
P {shopping mall, Railway Station, Airport, Multi specialist
Hospital, Highway}.
Therefore,  
   
1 2 3 4
F : P P P P P ς .
We consider the following tables to assign their membership values:
Table 1. Decision making fuzzy values for the size of the plot.
Table 2. Decision making fuzzy values for the location of the plot.
Table 3. Decision making fuzzy values for the cost of the plot.
(Plot size in sq. ft) 1
d 5
d 2
d 3
d 4
d
2000 0.6 0.5 0.7 0.4 0.8
1745 0.6 0.9 0.1 0.5 0.5
1100 0.6 0.7 0.4 0.8 0.5
900 0.3 0.2 0.5 0.6 0.2
1245 0.5 0.7 0.8 0.6 0.8
(Plot location) 1
d 2
d 3
d 4
d 5
d
Agartala 0.8 0.9 0.7 0.8 0.9
Lucknow 0.7 0.8 0.6 0.7 0.6
Amritsar 0.6 0.8 0.5 0.6 0.8
Greater Noida 0.4 0.3 0.2 0.3 0.2
Hooghly 0.8 0.6 0.5 0.7 0.9
(Plot cost in sq. ft) 1
d 2
d 3
d 4
d 5
d
4135 0.5 0.4 0.8 0.5 0.7
3812 0.8 0.6 0.8 0.6 0.8
3907 0.8 0.4 0.8 0.6 0.9
2547 0.6 0.6 0.5 0.7 0.6

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making
Table 4. Decision making fuzzy values for the landmark surrounding of a plot.
Then for the set  
1 3 5
A d ,d ,d
 , we define the fuzzy hypersoft in the following way:
F = {1100, Agartala, 3812, shopping mall} =        
       
       
1
d , 1100,0.6 , Agartala,0.8 , 3812,0.8 , Shopping Mall,0.7 ,
3
d , 1100,0.4 , Agartala,0.7 , 3812,0.8 , Shopping Mall,0.8 ,
5
d , 1100,0.5 , Agartala,0.9 , 3812,0.8 , Shopping Mall,0.8
 
 
 
 
 
 
 
 
 
Similarly, we can construct    
5 5 4 5 500 such fuzzy hypersoft sets for the set A as per the attribute
values are concerned. To find these 500 sets manually is a very tedious job but with the blessings of Data
Science the computing and the storage process are very easy to practice and we use it for various practical
purposes. So, with an aid of hypersoft set, there is a lot of choices made by the decision-makers among
which we fix with one choice that is suitable for use in all perspective. Such type of facility is not available
if we use the fuzzy soft set. So, the concept of a fuzzy hypersoft set gives us a scope to enhance our critical
thinking systematically. Also, by using it we redefine or extend the earlier concepts by introducing various
properties on fuzzy hypersoft sets and all these properties are more generalized as compared to the other
existing set theories.
It is to be noted that the set of all fuzzy hypersoft sets over  is denoted by  
ς
FHSS .
3 | Different Types of Fuzzy Hypersoft Sets and Their Properties
Definition 6. Let  
ς

S
Γ FHSS , where    
1 2 n
L L ....... L S . If  
x ς ,  
S then S
Γ is called an FHS-
null set and it is denoted by 
Γ .
ς

S
Γ FHSS , where    
1 2 n
L L ....... L S . If S is a crisp set and  
x ς , 
S
Γ ς then
S
Γ is called the FHS-universal set and it is denoted by U
Γ .
ς

S T
Γ ,Γ FHSS . Then S
Γ is said to be an FHS-subset of T
Γ i.e 
S T
Γ Γ iff 
S T ,
and  
   
 

μ F S μ F T .
Proposition 1. For  
ς

S T R
Γ ,Γ ,Γ FHSS , we have the following results:
1) 
S T
Γ Γ and   
T S S T
Γ Γ Γ Γ .
2) 
S T
Γ Γ and   
T R S R
Γ Γ Γ Γ .
3)   S
Γ Γ and 
S U
Γ Γ
ς

S
Γ FHSS . Then the complement of S
Γ is denoted by  
c
S
Γ and it is defined as
   
 
 
  
c
S
Γ x,1 μ F S : x ς .
(Landmark surrounding of a plot) 1
d 2
d 3
d 4
d 5
d
Shopping Mall 0.7 0.6 0.8 0.5 0.8
Railway Station 0.6 0.4 0.5 0.8 0.7
Airport 0.5 0.7 0.6 0.8 0.7
Multispecialist Hospital 0.3 0.7 0.5 0.4 0.7
Highway 0.4 0.7 0.6 0.6 0.8

168
Debnath
|J.
Fuzzy.
Ext.
Appl.
2(2)
(2021)
163-170
ς

S T
Γ ,Γ FHSS . Then their union is denoted by 
S T
Γ Γ and is defined by
ς

S T
Γ ,Γ FHSS . Then their union is denoted by 
S T
Γ Γ and is defined by
.
Proposition 2. We have the following propositions that are based on FHS-complementary set:
1)   
c
c
S S
Γ Γ .
2)  
 
c
Γ ς (in case of crisp set).
3) (iii)  
  
c c c
S T S T
Γ Γ Γ Γ and  
  
c c c
S T S T
Γ Γ Γ Γ (De Morgan Laws).
4) (iv)  
  
S S T S
  
S S T S
Γ Γ Γ Γ (Absorption Laws).
Proposition 3. For  
ς

S T R
Γ ,Γ ,Γ FHSS , we have the following results:
1) 
S T
Γ Γ = 
T S
Γ Γ and 
S T
Γ Γ = 
T S
Γ Γ (Idempotent Laws).
2)  
 
S T R
Γ Γ Γ = 
 
S T R
Γ Γ Γ and  
 
S T R
Γ Γ Γ = 
 
S T R
Γ Γ Γ (Associative Laws).
3)  
 
S T R
Γ Γ Γ =   
  
S T S R
 
S T R
Γ Γ Γ =   
  
S T S R
Γ Γ Γ Γ
(Distributive Laws).
4 | Weightage of Fuzzy Hypersoft Set in Decision Making
In example 2, we have determined only one fuzzy hypersoft set though there are 500 different choices
available. As we know that computer science is an integral part of Mathematics, then by using suitable
software or programming we can easily enumerate all these 500 different sets within few minutes as it is
quite difficult to do manually. That's why here we avoid such lengthy calculations. But we need an
operator by which we can select the best alternative for the customer among these 500 cases. Without
any proper decision-making of all the items we do not say, this or that is the best choice or there may
be multiple choices for the customer. For that purpose, we need to find the weightage of each fuzzy
hypersoft set by using the following operator.
 
   
 
 
 
 
 
 

   


 
  
 
 

 
 
S T
x,max μ F S ,μ F T : x S T
Γ Γ x,μ F S , if x S
x,μ F T ,if x T
 
   
 
 
 
 
 
 
 
  
 
 
  
 
 

 
 
S T
x,min μ F S ,μ F T : x S T
Γ Γ x,μ F S if x S
x,μ F T if x T
 
 
 

 
n i
FV
i 1
1
Ω max μ d Ξ .
FHSS ς

169
Fuzzy
hypersoft
sets
and
its
weightage
operator
for
decision
making
Where  
FHSS ς =number of fuzzy hypersoft sets over ς ,  
 
i
max μ d =maximum fuzzy membership
value concerning for to the decision-makers, and FV
Ξ =number of fuzzy attribute values corresponding
to the set of attributes.
We have to determine Ω for every each possible fuzzy hypersoft set of the given problem. Afterward, we
choose the best choice for the client which has the maximum weightage. In case of a tie, there may be
multiple choices for a client. In such a case, he or she may choose any one of the suitable choices.
5 | Application
There is a huge scope of the use of hypersoft set in different areas such as forecasting, business
management, traffic control, similarity measures, neural networking, data science, sociology, etc.
6 | Conclusions
Here we have used the novel concept known as a hypersoft set which was introduced by F. Smarandache
in 2018. By combining the fuzzy set and hypersoft set a new theory called fuzzy hypersoft set is introduced
in [23]. The Fuzzy hypersoft set is a more generalized form of fuzzy set, soft set, fuzzy soft set, etc. In this
article, an attempt has been made to study fuzzy hypersoft set and their kinds and discuss some set-
theoretic operations and propositions on them. With the help of a valid and concrete example, it also
shown that how this concept is more effective in multi-criteria decision-making problems compared to
other existing theories. In future, it has a wide range of application in the fields of topology, game theory,
computer science, neural network, decision-making etc.
Acknowledgments
Author is grateful to the anonymous referees for their valuable suggestions.
Conflict of interest
Author declared no conflict of interest regarding the publication of the paper.
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