Intuitionistic fuzzy parametrized soft set theory and its decision making

0.13 MB·English

by  İrfan Deli

Intuitionistic fuzzy parametrized soft set theory and its
decision making
Irfan Delia,∗, Naim C¸a˘gmanb
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1 aDepartment of Mathematics, Faculty of Arts and Sciences, 7 Aralık University , 79000
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Kilis, Turkey
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bDepartment of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa¸sa University
n
60250 Tokat, Turkey
a
J

3

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Abstract
L
. In this work, we first define intuitionistic fuzzy parametrized soft sets (intu-
h
itionistic FP-softsets) andstudy some of their properties. We then introduce
t
a
anadjustableapproachestointuitionisticFP-softsetsbaseddecisionmaking.
m
We also give an example which shows that they can be successfully applied
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to problems that contain uncertainties.
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v
Keywords: Soft sets, fuzzy sets, FP-soft sets, intuitionistic FP-soft sets,
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5 soft level sets, decision making.
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0
.
1 1. Introduction
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3
Many fields deal with the uncertain data which may not be successfully
1
: modeled by the classical mathematics, probability theory, fuzzy sets [33],
v
i rough sets [30], and other mathematical tools. In 1999, Molodtsov [28] pro-
X
posed a completely new approach so-called soft set theory that is more uni-
r
a versal for modeling vagueness and uncertainty.
Afterdefinitions oftheoperationsofsoftsets [3, 11, 26,32], theproperties
and applications on the soft set theory have been studied increasingly (e.g.
[3, 7, 9]). The algebraic structure of soft set theory has also been studied
increasingly (e.g. [1, 2, 5, 13, 17, 18, 19, 20, 31]). In recent years, many
interesting applications of soft set theory have been expanded by embedding
∗
Corresponding author
Email addresses: [email protected](Irfan Deli), [email protected]
(Naim C¸ag˘man)
Preprint submitted to Elsevier December 11, 2013
the ideas of fuzzy sets (e.g. [4, 6, 10, 14, 20, 22, 25, 27]), rough sets (e.g.
[4, 15, 16]) and intuitionistic fuzzy sets (e.g. [23, 24, 29])
C¸a˘gman et al.[8] defined FP-soft sets and constructed an FP-soft set
decision making method. In this paper, we first define intuitionistic fuzzy
parametrized soft sets (intuitionistic FP-soft sets), and study their opera-
tions and properties. We then introduce a decision making method based
on intuitionistic FP-soft sets. This method is more practical and can be
successfully applied to many problems that contain uncertainties.
2. Preliminary
Inthissection, wepresentthebasicdefinitionsofsoftsettheory[28],fuzzy
set theory [33], intuitionistic fuzzy set theory [12] and FP-soft set theory [8]
that are useful for subsequent discussions.
Definition 1. Let U be an initial universe, P(U) be the power set of U, E
is a set of parameters and A ⊆ E. Then, a soft set F over U is defined as
A
follows:
F = {(x,f (x)) : x ∈ E}
A A
where f : E → P(U) such that f (x) = ∅ if x ∈/ A.
A A
Here, f is called approximate function of the soft set F , and the value
A A
f (x) is a set called x-element of the soft set for all x ∈ E. It is worth noting
A
that the sets f (x) may be arbitrary.
A
Definition 2. Let U be a universe. Then a fuzzy set X over U is a function
defined as follows:
X = {(µ (u)/u) : u ∈ U}
X
where µ : U → [0.1]
X
Here, µ called membership function of X, and the value µ (u) is called
X X
the grade of membership of u ∈ U. The value represents the degree of u
belonging to the fuzzy set X.
Definition 3. Let E be a universe. An intuitionistic fuzzy set A on E can
be defined as follows:
A = {< x,µ (x),γ (x) >: x ∈ E}
A A
where, µ : E → [0,1] and γ : E → [0,1] such that 0 ≤ µ (x)+γ (x) ≤ 1
A A A A
for any x ∈ E.
2
Here, µ (x) and γ (x) is the degree of membership and degree of non-
A A
membership of the element x, respectively.
If A and B are two intuitionistic fuzzy sets on E, then
i. A ⊂ B if and only if µ (x) ≤ µ (x) and γ (x) ≥ γ (x) for ∀x ∈ E
A B A B
ii. A = B if and only if µ (x) = µ (x) and γ (x) = γ (x) ∀x ∈ E
A B A B
iii. Ac = {< x,γ (x),µ (x) >: x ∈ E}
A A
iv. A∪B = {< x,max(µ (x),µ (x)),min(γ (x),γ (x) >: x ∈ E},
A B A B
v. A∩B = {< x,min(µ (x),µ (x)),max(γ (x),γ (x) >: x ∈ E},
A B A B
vi. A+B = {< x,µ (x)+µ (x)−µ (x)µ (x),γ (x)γ (x) >: x ∈ E},
X Y X Y X Y
vii. A·B = {< x,µ (x)µ (x),γ (x)+γ (x)−γ (x)γ (x) >: x ∈ E}.
A B A B A B
Definition 4. Let U be an initial universe, P(U) be the power set of U, E
be a set of all parameters and X be a fuzzy set over E. Then a FP-soft set
(f ,E) on the universe U is defined as follows:
X
(f ,E) = {(µ (x)/x,f (x)) : x ∈ E}
X X X
where µ : E → [0.1] and f : E → P(U) such that f (x) = ∅ if µ (x) = 0.
X X X X
Here f called approximate function and µ called membership function
X X
of FP-soft sets.
3. Intuitionistic FP-soft sets
In this section, we define intuitionistic fuzzy parametrized soft sets (in-
tuitionistic FP-soft sets) and their operations.
Definition 5. Let U be an initial universe, P(U) be the power set of U, E
be a set of all parameters and K be an intuitionistic fuzzy set over E. An
intuitionistic FP-soft sets ∐ over U is defined as follows:
K
∐ = (< x,α (x),β (x) >,f (x)) : x ∈ E
K K K K
n o
where α : E → [0.1], β : E → [0.1] and f : E → P(U) with the property
K K K
f (x) = ∅ if α (x) = 0 and β (x) = 1.
K K K
Here, the function α andβ calledmembershipfunction andnon-membership
K K
of intuitionistic FP-soft set, respectively. The value α (x) and β (x) is the
K K
degree of importance and unimportant of the parameter x.
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Obviously, each ordinary FP-soft set can be written as
∐ = (< x,α (x),1−α (x) >,f (x)) : x ∈ E
K K K K
n o
Note that the sets of all intuitionistic FP-soft sets over U will be denoted by
IFPS(U).
Definition 6. Let ∐ ∈ IFPS(U). If α (x) = 0 and β (x) = 1 for all
K K K
x ∈ E, then ∐ is called a empty intuitionistic FP-soft sets, denoted by ∐ .
K Φ
Definition 7. Let ∐ ∈ IFPS(U). If α (x) = 1, β (x) = 0 and f (x) = U
K K K K
for all x ∈ E, then ∐ is called universal intuitionistic FP-soft set, denoted
K
by ∐ .
E˜
Example 1. Assume that U = {u ,u ,u ,u ,u } is a universal set and
1 2 3 4 5
E = {x ,x ,x ,x } is a set of parameters. If
1 2 3 4
K = {< x ,0.2,0.5 >,< x ,0.5,0.5 >,< x ,0.6,0.3 >}
2 3 4
and
f (x ) = {u ,u },f (x ) = ∅,f (x ) = U
K 2 2 4 K 3 K 4
then an intuitionistic FP-soft set ∐ is written by
K
∐ = {(< x ,0.2,0.5 >,{u ,u }),(< x ,0.5,0.5 >,∅),(< x ,0.6,0.3 >,U)}
K 2 2 4 3 4
If L = {< x ,0,1 >,< x ,0,1 >,< x ,0,1 >,< x ,0,1 >}, then the
1 2 3 4
intuitionistic FP-soft set ∐ is an empty intuitionistic FP soft set.
L
If M = {< x ,1,0 >,< x ,1,0 >,< x ,1,0 >,< x ,1,0 >} and
1 2 3 4
f (x ) = U, f (x ) = U,f (x ) = U and f (x ) = U, then the intu-
M 1 M 2 M 3 M 4
itionistic FP-soft set ∐ is a universal intuitionistic FP-soft set.
M
Definition 8. ∐ ,∐ ∈ IFPS(U). Then ∐ is a intuitionistic FP-soft
K L K
subset of ∐ , denoted by ∐ ⊆∐ , if and only if α (x) ≤ α (x), β (x) ≥
L K L K L K
β (x) and f (x) ⊆ f (x) for all x ∈ E.
L K L
e
Remark 1. ∐ ⊆∐ does not imply that every element of ∐ is an ele-
K L K
ment of ∐ as in the definition of classical subset. For example, assume that
L
e
U = {u ,u ,u ,u } is a universal set of objects and E = {x ,x ,x } is a set
1 2 3 4 1 2 3
of all parameters. If K = {< x ,0.4,0.6 >} and L = {< x ,0.5,0.5 >,
1 1
< x ,0.4,0.5 >}, and ∐ = {(< x ,0.4,0.6 >,{u ,u })}, ∐ = {(<
3 K 1 2 4 L
4
x ,0.5,0.5 >,{u ,u ,u }),(< x ,0.4,0.5 >,{u ,u })}, then for all x ∈ E,
1 2 3 4 3 1 5
α (x) ≤ α (x), β (x) ≥ β (x) and ∐ (x) ⊆ ∐ (x) is valid. Hence
K L K L K L
∐ ⊆∐ . Itis clearthat (< x ,0.4,0.6 >,{u ,u }) ∈ ∐ but (< x ,0.4,0.6 >
K L 1 2 4 K 1
, {u ,u }) ∈/ ∐ .
2 4 L
e
Proposition 1. Let ∐ ,∐ ∈ IFPS(U). Then
K L
i. ∐ ⊆∐
K E˜
ii. ∐ ⊆e∐
Φ K
iii. ∐ e⊆∐
K K
e
Definition 9. ∐ ,∐ ∈ IFPS(U). Then ∐ and ∐ are intuitionistic FP-
K L K L
soft-equal, written by ∐ = ∐ , if and only if α (x) = α (x) , β (x) =
K L K L K
β (x) and f (x) = f (x) for all x ∈ E.
L K L
Proposition 2. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
i. ∐ = ∐ and ∐ = ∐ ⇔ ∐ = ∐
K L L M K M
ii. ∐ ⊆∐ and ∐ ⊆∐ ⇔ ∐ = ∐
K L L K K L
iii. ∐ ⊆e∐ and ∐ ⊆e∐ ⇒ ∐ ⊆∐
K L L M K M
Definitieon 10. ∐ ∈eIFPS(U).eThen complement of ∐ , denoted by ∐c ,
K K K
is a intuitionistic FP-soft set defined by
∐c = (< x,β (x),α (x) >,f (x)) : x ∈ K
K K K Kc
n o
where f (x) = U \f (x).
Kc K
Proposition 3. Let ∐ ∈ IFPS(U). Then
K
i. (∐c )c = ∐
K K
ii. ∐c = ∐
Φ E˜
iii. ∐c = ∐
E˜ Φ
5
Definition 11. ∐ ,∐ ∈ IFPS(U). Then union of ∐ and ∐ , denoted
K L K L
by ∐ ∪∐ , is defined by
K L
e
∐K∪∐L = (< x,max(αK(x),αL(x)),min(βK(x),βL(x)) >,fK∪eL(x)) : x ∈ E
n o
e
where fK∪eL(x) = fK(x)∪fL(x).
Proposition 4. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
i. ∐ ∪∐ = ∐
K K K
ii. ∐ ∪e∐ = ∐
K Φ K
iii. ∐ ∪e∐ = ∐
K E˜ E˜
iv. ∐ ∪e∐ = ∐ ∪∐
K L L K
v. (∐ e∪∐ )∪∐ e= ∐ ∪(∐ ∪∐ )
K L M K L M
Definitioen 12e. ∐ ,∐ e∈ IFePS(U). Then intersection of ∐ and ∐ ,
K L K L
denoted by ∐ ∩∐ , is a intuitionistic FP-soft sets defined by
K L
e
∐K∩∐L = < x,min(αK(x),αL(x)),max(βK(x),βL(x) >,fK∩eL(x)) : x ∈ E
n o
e
where fK∩eL(x) = fK(x)∩fL(x).
Proposition 5. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
i. ∐ ∩∐ = ∐
K K K
ii. ∐ ∩e∐ = ∐
K Φ Φ
iii. ∐ ∩e∐ = ∐
K E˜ K
iv. ∐ ∩e∐ = ∐ ∩∐
K L L K
v. (∐ e∩∐ )∩∐ e= ∐ ∩(∐ ∩∐ )
K L M K L M
Remarke2. Leet ∐ ∈ IFPeS(Ue). If ∐ 6= ∐ or ∐ 6= ∐ , then ∐ ∪∐c 6=
K K Φ K E˜ K K
∐ and ∐ ∩∐c 6= ∐ .
E˜ K K Φ
e
Propositione 6. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
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i. ∐ ∪(∐ ∩∐ ) = (∐ ∪∐ )∩(∐ ∪∐ )
K L M K L K M
ii. ∐ ∩e(∐ ∪e∐ ) = (∐ ∩e∐ )∪e(∐ ∩e∐ )
K L M K L K M
e e e e e
Proposition 7. Let ∐ ,∐ IFPS(U). Then following DeMorgans types of
K L
results are true.
i. (∐ ∪∐ )c = ∐c ∩∐c
K L K L
ii. (∐ ∩e∐ )c = ∐c ∪e∐c
K L K L
Definitioen 13. ∐ ,∐e ∈ IFPS(U). Then OR-sum of ∐ and ∐ , denoted
K L K L
by ∐ ∨+ ∐ , is defined by
K L
∐ ∨+ ∐ (x) =
K L
(< x,αK(x)+αL(x)−αK(x)αL(x),βK(x)βL(x) >,fK∪eL(x)) : x ∈ E
n o
where fK∪eL(x) = fK(x)∪fL(x).
Definition 14. ∐ ,∐ ∈ IFPS(U). Then AND-sum of ∐ and ∐ , de-
K L K L
noted by ∐ ∧+ ∐ , is defined by
K L
∐ ∧+ ∐ (x) =
K L
(< x,αK(x)+αL(x)−αK(x)αL(x),βK(x)βL(x) >,fK∩eL(x)) : x ∈ E
n o
where fK∩eL(x) = fK(x)∩fL(x).
Proposition 8. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
i. ∐ ∨+ ∐ = ∐
K Φ K
ii. ∐ ∨+ ∐ = ∐
K E˜ E˜
iii. ∐ ∨+ ∐ = ∐ ∨+ ∐
K L L K
iv. ∐ ∧+ ∐ = ∐ ∧+ ∐
K L L K
v. (∐ ∨+ ∐ )∨+ ∐ = ∐ ∨+ (∐ ∨+ ∐ )
K L M K L M
vi. (∐ ∧+ ∐ )∧+ ∐ = ∐ ∧+ (∐ ∧+ ∐ )
K L M K L M
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Definition 15. ∐ ,∐ ∈ IFPS(U). Then OR-product ∐ and ∐ , denoted
K L K L
by ∐ ∨× ∐ , is defined by
K L
∐ ∨× ∐ (x) =
K L
< x,αK(x)αL(x),βK(x)+βL(x)−βK(x)βL(x) >,fK∪eL(x)) : x ∈ E
n o
where fK∪eL(x) = fK(x)∪fL(x).
Definition 16. ∐ ,∐ ∈ IFPS(U). Then AND-product ∐ and ∐ , de-
K L K L
noted by ∐ ∧× ∐ , is defined by
K L
∐ ∧× ∐ (x) =
K L
< x,αK(x)αL(x),βK(x)+βL(x)−βK(x)βL(x) >,fK∩eL(x)) : x ∈ E
n o
where fK∩eL(x) = fK(x)∩fL(x).
Proposition 9. Let ∐ ,∐ ,∐ ∈ IFPS(U). Then
K L M
i. ∐ ∧× ∐ = ∐
K Φ Φ
ii. ∐ ∧× ∐ = ∐
K E˜ K
iii. ∐ ∧× ∐ = ∐ ∧× ∐
K L L K
iv. ∐ ∨× ∐ = ∐ ∨× ∐
K L L K
v. (∐ ∧× ∐ )∧× ∐ = ∐ ∧× (∐ ∧× ∐ )
K L M K L M
vi. (∐ ∨× ∐ )∨× ∐ = ∐ ∨× (∐ ∨× ∐ )
K L M K L M
4. Intuitionistic FP-soft decision making method
In this section, we have defined a reduced intuitionistic fuzzy set of an
intuitionistic FP-soft set, that produce an intuitionistic fuzzy set from an
intuitionistic FP-soft set. We then have defined a reduced fuzzy set of an
intuitionistic fuzzy set, that produce a fuzzy set from an intuitionistic fuzzy
set. These sets present an adjustable approach to intuitionistic FP-soft sets
based decision making problems.
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Definition 17. Let ∐ be an intuitionistic FP-soft set. Then, a reduced
K
intuitionistic fuzzy set of ∐ , denoted by K , defined as follows
K rif
K = < u,α (u),β (u) >: u ∈ U
rif K K
rif rif
n o
where
1
α : U → [0,1], α (u) = α (x)χ (u)
Krif Krif |U| K fK(x)
x∈XE,u∈U
1
β : U → [0,1], β (u) = β (x)χ (u)
Krif Krif |U| K fK(x)
x∈XE,u∈U
where
1, u ∈ f (x)
χ (u) = K
fK(x) (cid:26) 0, u ∈/ f (x)
K
Here, α and β are called reduced-set operators of K . It is clear that
K K rif
rif rif
K is an intuitionistic fuzzy set over U.
rif
Definition 18. ∐ ∈ IFPS(U) and K be reduced intuitionistic fuzzy set
K rif
of ∐ . Then, a reduced fuzzy set of K is a fuzzy set over U, denoted by
K rif
K , defined as follows
rf
K = µ (u)/u : u ∈ U
rf K
rf
n o
where
µ : U → [0,1], µ (u) = α (u)(1−β (u))
K K K K
rf rf rif rif
Now, we construct an intuitionistic FP-soft decision making method by
the following algorithm to produce a decision fuzzy set from a crisp set of
the alternatives.
According to the problem, decision maker;
i. constructs a feasible intuitionistic fuzzy subsets K over the parameters
set E,
ii. constructs an intuitionistic FP-soft set ∐ over the alternatives set U,
K
iii. computes the reduced intuitionistic fuzzy set K of ∐ ,
rif K
iv. computes the reduced fuzzy set K of K ,
rf rif
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v. chooses the element of K that has maximum membership degree.
rf
Now, we can give an example for the intuitionistic FP-soft decision making
method. Some of it is quoted from example in [11].
Example 2. Assume that a company wants to fill a position. There are 5
candidateswho fill in a form in orderto applyformallyforthe position. There
is a decision maker (DM), that is from the department of human resources.
He want to interview the candidates, but it is very difficult to make it all of
them. Therefore, by using the intuitionistic FP-soft decision making method,
the number of candidates are reduced to a suitable one. Assume that the set
of candidates U = {u ,u ,u ,u ,u } which may be characterized by a set of
1 2 3 4 5
parameters E = {a ,a ,a ,a }. For i = 1,2,3,4 the parameters a stand for
1 2 3 4 i
experience, computer knowledge, training and young age, respectively. Now,
we can apply the method as follows:
Step i. Assume that DM constructs a feasible intuitionistic fuzzy subsets
K over the parameters set E as follows;
K = {< x ,0.7,0.3 >,< x ,0.2,0.5 >,< x ,0.5,0.5 >,< x ,0.6,0.3 >}
1 2 3 4
Step ii. DM constructs an intuitionistic FP-soft set ∐ over the alterna-
K
tives set U as follows;
∐ = (< x ,0.7,0.3 >,{u ,u ,u }),(< x ,0.2,0.5 >,U),(< x ,
K 1 1 2 4 2 3
n
0.5,0.5 >,{u ,u ,u }),(< x ,0.6,0.3 >,{u ,u })
1 2 4 4 2 3
o
Step iii. DM computes the reduced intuitionistic fuzzy set K of ∐ as
rif K
follows;
K = (< u ,0.28,0.26 >,< u ,0.40,0.32 >,< u ,0.16,0.16 >,
rif 1 2 3
n
< u ,0.28,0.32 >,< u ,0.04,0.10 >
4 5
o
Step iv. DM computes the reduced fuzzy set K of K as follows;
rf rif
K = 0.2072/u ,0.2720/u ,0.1344/u ,0.1904/u ,0.0360/u
rf 1 2 3 4 5
n o
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