Abstract
In this paper, we propose soft decision making methods based on fuzzy and soft set theory. We also use matrix representation of the soft sets that is very useful for computations of the method. We finally present an example which shows that the method can be successfully applied to many problems that contain uncertainties.
Introduction
There are theories that can be considered as mathematical tools for dealing with uncertain, fuzzy, not clearly defined objects. Some of them are probability,fuzzy sets [49], intuitionistic fuzzy sets [4], vague sets [15], rough sets [39] and soft sets [33]. Their role is significant when applied to complex problem not easily described by classical mathematical methods.
The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets. It has been successfully applied to the problems in many branches of science that contain uncertainties. In order to use the fuzzy set, its membership functions must be determined first since its performance is more dependent on membership function design. The membership function gives a degree to each object with all possible grades of truth between 0 and 1.Since the gradation is extremely individual and dependent on the situation, it is not easy to determine the membership function correctly in some cases. In this paper, we introduce a method that gives us an optimum determination of membership functions for a fuzzy set with the help of binary information.
To deal with uncertainty there is another theory called soft set theory that classifies the objects with help of binary information. The problem of setting the membership function does not arise in this theory. Therefore, in this paper, we define soft evaluation operator that transforms soft sets into fuzzy sets. On one hand, the operator gives an optimum determination of membership functions for a fuzzy set based on the soft set theory, on the other hand, the operator gives a decision making method on the soft sets.
The concept of soft sets was firstly introduced by Molodtsov [33] in 1999 as a general mathematical apparatus for dealing with uncertain. In [33–36], Molodtsov successfully applied the soft set theory into several directions, such as smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, probability, theory of measurement and so on.
The properties and applications on the soft set theory have been studied increasingly in recent years [2, 52]. The algebraic structure of soft set theory has also been studied in more detail, some of which are soft group [3], soft ring [1], soft semiring [12], soft WS-algebras [38], BCK/BCI-algebras [17], soft ideals [18], soft ordered semigroups [20], soft equality [40], vague soft sets [46], soft lattices [22], soft BL-algebras [50], exclusive disjunctive soft sets [43], soft p-ideals [19] and soft topology [9].
In recent years, many interesting applications of soft set theory have been expanded by embedding the ideas of fuzzy sets. The fuzzy soft sets [8, 41], fuzzy soft relation [42], fuzzy soft groups [5] and intuitionistic fuzzy soft sets [32] are studied. The soft set theory is compared with the related concepts of fuzzy sets and rough sets [3]. The fuzzy soft set theory is also used for decision making [14, 44]. The concept of soft fuzzy sets is first given by Yao et al. [48]. The fuzzy soft sets and soft fuzzy sets are also compared in [48].
In this work, we first defined weighted parameters set of a soft set, and then defined a soft aggregation operator by which several approximate functions of the soft set are combined to produce a single fuzzy set that is called fuzzy soft set. They can be used to evaluate each element of the universe. This evaluation gives a decision making method on the soft set theory. To make easy computation, matrix representations of the soft set are used. Finally, an example is given to show the method can be successfully applied to many problems that contain uncertainties.
Preliminary
In this section, we present the basic definitions of soft sets [33] and fuzzy sets [49] that are useful for subsequent discussions.
Here, f A is called approximate function of the soft set F A , and the value f A (x) is a set called x-element of the soft set for all x ∈ E. It is worth noting that the set f A (x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus, a soft set over U can be represented by the set of ordered pairs
From now on, the set of all soft sets over U will be denoted by .
More detailed explanations related to the soft set theory can be found in [6, 33].
We shall assume the reader has met fuzzy set theory [49] before, and it is not given here except the definition of fuzzy sets; a fuller introduction can be found in [10, 51].
It should be noted that the set of all the fuzzy sets on a set U will be denoted by .
Soft decision making
In this section, we define soft decision making methods based on fuzzy and soft set theory.
We first define weighted parameters set of a soft set which is a fuzzy set over the parameters set.
By using the soft sets and weighted parameters sets, now we can define a soft fuzzy set which is a kind of fuzzy set. It is produced by combining the approximate functions of a soft set as in the following way.
From now on, we use instead of (χf A (e1) (u) , χf A (e2) (u) , . . . , χf A (e m ) (u)) for each u ∈ U. It is clear that
This soft aggregation operator is an operation by which several e-elements of a soft set are combined to produce a fuzzy set that can be used to evaluate the alternative.
Theorem 3.4 is very important for application because membership degrees are obtained from the binary inputs. We can use a soft fuzzy set to make a decision by the following algorithm.
Choose a subset A of E to approximate the alternatives. Construct the soft set F
A
over U, Find the weighted parameters set of F
A
, Compute the soft fuzzy set of F
A
, Use the set to make decisions.
Now, we can present an example for the the method. It is possible that the method can be applied to many complicated problems containing uncertainties. But, a very easy example is given here to understand it clearly.
Then the soft set F
A
can be written by
By the similar way, |f
A
(e2) |=5, |f
A
(e3) |=7 and |f
A
(e4) |=4 are obtained. Therefore the weighted parameters set is written by,
By the similar way we can obtain the other degrees; , , , , , , , , . Hence, the soft fuzzy set of is constructed by
Matrix representation of the method
Soft matrix theory is a matrix representation of the soft sets is defined by Çağman et al. [7]. It is presented an almost analogous representation in the form of a binary table. In this section, we use the soft matrices that will be useful for computations of the method.
From now on, I n = {1, 2, . . . , n} is a index set for each positif integer n, and then U = {u j : j ∈ I n }, A = {e i : i ∈ I m } and A ⊆ E.
Let . Then the F
A
can be presented by a table as in the following form
If a
ij
= χf
A
(e
i
) (u
j
) for every i ∈ I
m
and j ∈ i
n
, then the soft set F
A
is uniquely characterized by a matrix,
If for every i ∈ I
m
and j = 1, then the weighted parameters set is uniquely characterized by a matrix,
If for every i ∈ I
n
and j = 1, then the soft fuzzy set is uniquely characterized by a matrix,
If ai1 = χ
U
(u
i
) =1 for every i ∈ I
n
and j = 1, then the set U is uniquely characterized by a matrix,
wherem = |A|, n = |U|, and the superscriptTis used to indicate transposition of the matrices.
Choose a subset A of E to approximate the alternatives. Construct the matrix Mm×n [A, U] of F
A
, Find the matrix of , Compute the matrix of , Use the set to make decisions.
Now, we can present an example for the the method.
Hence, matrix of the soft set is written as
[0.375 0.425 0.3 0.275 0.325 0 0.45 0.4 0.325 0.275]
Group decision making
If there are more then one decision makers, they transform their soft sets into the fuzzy sets. To make group decision makings, we now can use the fuzzy aggregation operations on the obtained fuzzy sets. There are several publications available on fuzzy aggregation operators, of which a few notable ones are [11, 47]. Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set. Aggregation operation on k fuzzy set (2 ≤ k) is defined by a function
which may be t-norms are also used to construct the intersection of fuzzy sets or as a basis for aggregation operators. In this case group decision-making goes down to constructing the membership function resulting from all given fuzzy sets. The method should be more comprehensive in the future to solve the related problems.
Conclusion
In this work, we first defined weighted parameters set of a soft set, and then defined soft evaluation operator by which several approximate functions of the soft set are combined to produce a single fuzzy set. An other way to sat that a decision maker construct a soft set and transforms own soft set into a fuzzy set by using soft evaluation operator. The fuzzy set can be used to evaluate each element of the universe. This evaluation give a decision making method on the soft set theory.
