1. Introduction
Many scientists wish to find appropriate solutions to some mathematical problems that cannot be solved by traditional methods. These problems lie in the fact that traditional methods cannot solve the problems of uncertainty in economy, engineering, medicine and the problems of decision-making and others. One of these solutions is fuzzy sets—the title of Zadeh’s first article about his new mathematical theory, which was published in a scientific journal in 1965. Since Zadeh published his new classic paper almost fifty years ago, fuzzy set theory has received more and more attention from researchers in wide range of scientific areas, especially in the past few years. The difference between a binary set and a fuzzy set is that in a “normal” set every element is either a member or a non-member of the set. Here, we see that it either has to be A or not A. In a fuzzy set, an element can be a member of a set to some degree and at the same time a non-member to some degree of the same set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the closed unit interval [0,1]. Fuzzy sets generalise classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Therefore, a fuzzy set A in a universe of discourse X is a function
, usually this function is referred to as the membership function and denoted by
. Some mathematicians use the notation 
to denote the membership function instead of
. A fuzzy set A is written symbolically in various
Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. After Molodtsov’s work, some operations and application of soft sets were studied by Chen et al. [2] , Maji et al. [3] and Maji et al. [4] . Also Maji et al. [5] have introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties and also Roy and Maji [6] used this theory to solve some decision making problems. Alkhazaleh et al. [7] introduced the concept of soft multisets as a generalization of soft set. They also defined the concepts of fuzzy parameterized interval-valued fuzzy soft set [8] and possibility fuzzy soft set [9] and gave their applications in decision making and medical diagnosis. Alkhazaleh and Salleh [10] introduced the concept of a soft expert set, where the user can know the opinion of all experts in one model without any operations. Even after any operation the user can know the opinion of all experts. So in this paper, we introduce the concept of a fuzzy soft expert set, which will be more effective and useful which is a combination of fuzzy set and soft expert set. We also define its basic operations, namely complement, union, intersection, AND and OR and study their properties. We give an application of this concept in decision making problem. Finally, we study a mapping on fuzzy soft expert classes and its properties.
2. Preliminaries
In this section, we recall some basic notions related to this work. Molodtsov defined soft set in the following way. Let U be a universe and E be a set of parameters. Let 
denote the power set of U and
.
Definition 1 [1] A pair 
is called a soft set over U where F is a mapping 
In other words, a soft set over U is a parameterized family of subsets of the universe U For 
may be considered as the set of 
-approximate elements of the soft set
.
Definition 2 [5] Let U be an initial universal set and let 
be a set of parameters. Let 
denote the power set of all fuzzy subsets of
. Let 
A pair 
is called a fuzzy soft set over U where F is a mapping given by
Let U be a universe, E a set of parameters, X a set of experts (agents), and O a set of opinions. Let Z = E × X × O and
.
Definition 3 [10] A pair 
is called a soft expert set over 
where 
is a mapping given by
where 
denotes the power set of 
Definition 4 [10] For two soft expert sets 
and 
over U, 
is called a soft expert subset of 
if 1. 
2. 
This relationship is denoted by
. In this case 
is called a soft expert superset of
.
Definition 5 [10] Two soft expert sets 
and 
over U are said to be equal if 
is a soft expert subset of 
and 
is a soft expert subset of 
Definition 6 [10] Let 
be a set of parameters and X a set of experts. The NOT set of
, denoted by
, is defined by 
Definition 7 [10] An agree-soft expert set 
over U is a soft expert subset of 
defined as follows:
Definition 8 [10] A disagree-soft expert set 
over 
is a soft expert subset of 
defined as follows:
Definition 9 [10] The complement of a soft expert set 
is denoted by 
and is defined by 
where 
is a mapping given by
Definition 10 [10] The union of two soft expert sets 
and 
over U, denoted by 
, is the soft expert set 
where 
and 
Definition 11 [10] The intersection of two soft expert sets 
and 
over U, denoted by 
, is the soft expert set 
where 
and 
Definition 12 [10] If 
and 
are two soft expert sets over U then “
AND
” denoted by
, is defined by
where
.
Definition 13 [10] If 
and 
are two soft expert sets over 
then “
OR
” denoted by
, is defined by
where
.
3. Fuzzy Soft Expert Set
In this section, we introduce the definition of a fuzzy soft expert set and give basic properties of this concept.
Let U be a universe, 
a set of parameters, 
a set of experts (agents), and 
a set of opinions. Let 
and
.
Definition 14 A pair 
is called a fuzzy soft expert set over 
where 
is a mapping given by
where 
denotes the set of all fuzzy subsets of 
Example 1 Suppose that a company produces new types of products and wants to take the opinion of some experts about these products. Let 
be a set of products, 
is a set of decision parameters where 
denotes the parameters “easy to use”, “quality” and “cheap”. Let 
be a set of experts. Suppose that
Then we can view the fuzzy soft expert set (F,Z) as consisting of the following collection of approximations:
Definition 15 For two fuzzy soft expert sets 
and 
over
, 
is called a fuzzy soft expert subset of 
if 1. 
2. 
is fuzzy subset of 
This relationship is denoted by
. In this case 
is called a fuzzy soft expert superset of
.
Definition 16 Two fuzzy soft expert sets 
and 
over U are said to be equal if 
is a fuzzy soft expert subset of 
and 
is a fuzzy soft expert subset of 
Example 2 Consider Example 1. Suppose that the company takes the opinion of the experts once again after a month of using the products. Let
and
Clearly
. Let 
and 
be defined as follows:
Therefore
.
Definition 17 An agree-fuzzy soft expert set 
over U is a fuzzy soft expert subset of 
defined as follows:
Definition 18 A disagree-fuzzy soft expert set 
over U is a fuzzy soft expert subset of 
defined as follows:
Example 3 Consider Example 1. Then the agree-fuzzy soft expert set 
over 
is
and the disagree-fuzzy soft expert set 
over 
is
Definition 19 The complement of a fuzzy soft expert set 
is denoted by 
and is defined by 
where 
is a mapping given by
where c is a fuzzy complement.
Example 4 Consider Example 1. By using the basic fuzzy complement, we have
Proposition 1 If 
is a fuzzy soft expert set over 
then 1. 
Proof From Definition 19 we have 
where
. Now,
where 
4. Union and Intersection
In this section, we introduce the definitions of union and intersection of fuzzy soft expert sets, derive their properties, and give some examples.
Definition 20 The union of two fuzzy soft expert sets 
and 
over U, denoted by 
, is the fuzzy soft expert set 
where 
and 
where s is an s-norm.
Example 5 Consider Example 1. Let
and
Suppose 
and 
are two fuzzy soft expert sets over 
such that
By using basic fuzzy union (maximum) we have 
where
Proposition 2 If
, 
and 
are three fuzzy soft expert sets over
, then 1. 
2. 
Proof a. We want to prove that 
By using definition 20 we have
We consider the case when 
as the other cases are trivial, then we have
.
We also consider her the case when 
as the other cases are trivial, then we have
.
b. The proof is straightforward.
Definition 21 The intersection of two fuzzy soft expert sets 
and 
over
, denoted by
, is the fuzzy soft expert set 
where 
and 
where t is a t-norm.
Example 6 Consider Example 5. By using basic fuzzy intersection (minimum) we have 
where
Proposition 3 If
, 
and 
are three fuzzy soft expert sets over
, then 1. 
2. 
Proof a. We want to prove that 
By using definition 21 we have
We consider the case when 
as the other cases are trivial, then we have
.
We also consider her the case when 
as the other cases are trivial, then we have
.
b. The proof is straightforward.
Proposition 4 If
, 
and 
are three fuzzy soft expert sets over
, then 1. 
2. 
Proof a. We want to prove that 
By using definitions 20 and 21 we have
We consider the case when 
as the other cases are trivial, then we have
.
We also consider her the case when 
as the other cases are trivial, then we have
.
b. We want to prove that 
By using definitions 20 and 21 we have
We consider the case when 
as the other cases are trivial, then we have
.
We also consider her the case when 
as the other cases are trivial, then we have
.
5. AND and OR Operations
In this section, we introduce the definitions of AND and OR operations for fuzzy soft expert sets, derive their properties, and give some examples.
Definition 22 If 
and 
are two fuzzy soft expert sets over 
then “
AND
” denoted by 
is defined by
such that
, where t is a t-norm.
Example 7 Consider Example 1. Let
and
.
Suppose 
and 
are two fuzzy soft expert sets over 
such that
By using basic fuzzy intersection (minimum) we have 
where
Definition 23 If 
and 
are two fuzzy soft expert sets over U then “
OR
” denoted by 
is defined by
such that
, where s is an s-norm.
Example 8 Consider Example 7. By using basic fuzzy union (maximum) we have 
where
Proposition 5 If 
and 
are two fuzzy soft expert sets over 
then [a.]
1. 
2. 
Proof a. Suppose that 
Therefore, 
Now,
where 
Now, take 
Therefore,
Then 
and 
are the same. Hence, proved.
b. Suppose that 
Therefore, 
Now,
where 
Now, take 
Therefore,
Then 
and 
are the same. Hence, proved.
Proposition 6 If
, 
and 
are three fuzzy soft expert sets over 
then 1. 
2. 
3. 
4. 
Proof We give the proofs of a and b. [a.]
1. Suppose that
,
.
Therefore
,
.
,
.
.
2. Suppose that
,
.
Therefore
,
.
,
.
.
Remark The commutativity do not hold in AND and OR operations since
.
6. An Application of Fuzzy Soft Expert Set in Decision Making
In this section, we present an application of fuzzy soft expert set theory in a decision making problem. Assume that a company wants to fill a position. There are four candidates who form the universe 
the hiring committee considers a set of parameters, 
the parameters 
stand for “experience”, “computer knowledge” and “good speaking” respectively. Let 
be a set of experts (Committee members). After a serious discussion the committee constructs the following fuzzy soft expert set
In Table 1 and Table 2 we present the agree-fuzzy soft expert set and disagree-fuzzy soft expert set respectively.
The following algorithm may be followed by the company to fill the position.
1. Input the fuzzy soft expert set
.
2. Find an agree-fuzzy soft expert set and a disagree-fuzzy soft expert set.
3. Find 
for agree-fuzzy soft expert set.
4. Find 
for disagree-fuzzy soft expert set.
5. Find 
6. Find m, for which 
Then 
is the optimal choice object. If 
has more than one value, then any one of them could be chosen by the company using its option.
Table 1. Agree-fuzzy soft expert set.
Table 2. Disagree-fuzzy soft expert set.
Now we use this algorithm to find the best choice for the company to fill the position. From Table 1 and Table 2, we have the following in Table 3:
Then 
so the committee will choose candidate 4 for the job.
7. Mapping on Fuzzy Soft Expert Classes
In this section, we introduce the notion of mapping on fuzzy soft expert classes. fuzzy soft expert classes are collections of fuzzy soft expert sets. We also define and study the properties of fuzzy soft expert images and fuzzy soft expert inverse images of fuzzy soft expert sets, and support them with example and theorems.
Definition 24 Let 
be a universe, 
a set of parameters, 
a set of experts (agents), and 
a set of opinions. Let
. Then the collection of all fuzzy soft expert sets over U with a parameters from 
is called a fuzzy soft expert class and is denoted as
.
Definition 25 Let 
and 
be fuzzy soft expert classes. Let 
and 
be mappings. Then a mapping 
is defined as follows:
For a fuzzy soft expert set 
in
, 
is a fuzzy soft expert in 
obtained as follows:
For
, 
and
. 
is called a fuzzy soft expert image of the fuzzy soft expert set
.
Definition 26 Let 
and 
be fuzzy soft expert classes. Let 
and 
be mappings. Then a mapping 
is defined as follows:
For a fuzzy soft expert set 
in
, 
is a fuzzy soft expert set in 
obtained as follows:
For 
and
. 
is called a fuzzy soft expert inverse image of the fuzzy soft expert set
.
Example 9 Let
, 
and let
and
Suppose that 
and 
are fuzzy soft expert classes. Define 
and 
as follows:
, 
, 
.
Let 
and 
be two fuzzy soft expert sets over U and Y respectively such that
Then we define a mapping 
as follows:
For a soft expert set 
in
, 
is a soft expert set in 
where
and is obtained as follows:
.
.
.
Then 
.
.
.
Then 
.
.
.
Then 
.
.
.
Then 
.
.
.
Then 
Hence
.
Next for the soft expert inverse images, the mapping 
is defined as follows:
For a soft expert set 
in
, 
is a soft expert set in 
where
and is obtained as follows:
.
.
.
Then 
.
.
.
Then 
By similar calculations, consequently, we get
Definition 27 Let 
be a mapping and 
and 
fuzzy soft expert sets in
. Then for
, 
, the fuzzy soft expert union and intersection of fuzzy soft expert images
and 
are defined as follows:
Definition 28 Let 
be a mapping and 
and 
fuzzy soft expert sets in
. Then for
, 
, the fuzzy soft expert union and intersection of fuzzy soft expert inverse images 
and 
are defined as follows:
Theorem 1 Let 
be a mapping. Then for fuzzy soft expert sets 
and 
in the fuzzy soft expert class
, [a.]
1.
.
2.
.
3.
.
4.
.
5. If
, then
Proof For (a), (b) and (e) the proof is trivial, so we just give the proof of (c) and (d).
c. For 
and
, we want to prove that
For left hand side, consider
. Then
(1.1)
such that 
where 
an interval-valued fuzzy union.
Considering only the non-trivial case, then Equation 0.1 becomes:
(1.2)
For right hand side and by using Definition 27, we have
(1.3)
From Equations (1.1) and (1.3), we get (c).
d. For 
and
, and using Definition 27, we have
.
This gives (d).
Theorem 2 Let 
be mapping. Then for fuzzy soft experts
, 
in the fuzzy soft expert class
, we have:
1.
.
2.
.
3.
.
4.
.
5. If
, then
.
Proof We use the same method as in the previous proof.
8. Conclusion
In this paper, we have introduced the concept of fuzzy soft expert set which is more effective and useful and studied some of its properties. Also the basic operations on fuzzy soft expert set namely complement, union, intersection, AND and OR have been defined. An application of this theory has been given to solve a decisionmaking problem. We also studied a mapping on fuzzy soft expert classes and its properties.
Acknowledgements
The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant AP-2013-009.
NOTES
*Corresponding author.