Soft decision making methods based on fuzzy sets and soft sets

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.

Keywords

Soft sets fuzzy sets weighted parameters sets soft aggregation operator soft decision making

1 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.

2 Preliminary

In this section, we present the basic definitions of soft sets [33] and fuzzy sets [49] that are useful for subsequent discussions.

Definition 2.1. [6] Let U be an initial universe, P (U) be the power set of U, E be the set of all parameters and A ⊆ E. Then a soft set F_A over U is a set defined by a function representing a mapping $f_{A} : E \to P (U) such that f_{A} (x) = \emptyset if x \notin A$

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

$F_{A} = {\frac{f_{A} (e)}{e} : e \in E, f_{A} (e) \in P (U)}$

From now on, the set of all soft sets over U will be denoted by $S (U)$ .

More detailed explanations related to the soft set theory can be found in [6 , 33].

Example 2.2. Let U = {u_l, u₂, u₃, u₄, u₅, u₆} be a universal set and E = {x₁, x₂, x₃, x₄, x₅} be a set of parameters. If A = {x₁, x₃, x₄} ⊆ E and f_A (x₁) = {u₂, u₄}, f_A (x₃) = U, and f_A (x₄) = {u₁, u₃, u₅}, then the soft-set F_A is written by $F_{A} = {(x_{1}, {u_{2}, u_{4}}), (x_{3}, U), (x_{4}, {u_{1}, u_{3}, u_{5}})}$

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].

Definition 2.3. [49] Let U be a nonempty set. A fuzzy set X in U is characterized by its membership function $μ_{X} : U \to [0, 1]$ and μ_X (x) is interpreted as the degree of membership of element x in fuzzy set X for each x ∈ U. It is clear that X is completely determined by the set oftuples $X = {\frac{μ_{X} (x)}{x} : x \in U}$

It should be noted that the set of all the fuzzy sets on a set U will be denoted by $F (U)$ .

3 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.

Definition 3.1. Let $F_{A} \in S (U)$ . Then weighted parameters set of F_A, denoted $F_{A}^{w}$ , is defined by $F_{A}^{w} = {\frac{μ_{F_{A}^{w}} (e)}{e} : e \in E}$ which is a fuzzy set over E. Here, the membership function $μ_{F_{A}^{w}}$ is defined by $μ_{F_{A}^{w}} : E \to [0, 1], μ_{F_{A}^{w}} (e) = \frac{| f_{A} (e) |}{| U |}$ where |f_A (e) | and |U| are cardinality of e-element of F_A and universe U, respectively. $μ_{F_{A}^{w}} (e)$ is interpreted as the degree of weighted of parameter e in fuzzy set $F_{A}^{w}$ for each e ∈ E.

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.

Definition 3.2. Let $F_{A} \in S (U)$ and $F_{A}^{w}$ be its weighted parameters set. Then a soft fuzzy set, denoted $F_{A}^{s}$ , is defined by $F_{A}^{s} = {\frac{μ_{F_{A}^{s}} (u)}{u} : u \in U}$ which is a fuzzy set over U. Here, the membership degree $μ_{F_{A}^{s}} (u)$ of u is defined as follows

$\begin{matrix} μ_{F_{A}^{s}} : U \to [0, 1], \\ μ_{F_{A}^{s}} (u) = \frac{1}{| A |} \sum_{e \in E} μ_{F_{A}^{w}} (e) χ_{f_{A} (e)} (u) \end{matrix}$ where $χ_{f_{A} (e)} (u) = {\begin{matrix} 1, & if u \in χ_{f_{A} (e)} \\ 0, & if u \notin χ_{f_{A} (e)} \end{matrix}$ is the characteristic function of f_A (e) for every e ∈ A, and |A| is the cardinality of A. The value $μ_{F_{A}^{s}} (u)$ is interpreted as the degree of membership of element u in fuzzy set $F_{A}^{s}$ for each u ∈ U.

From now on, we use $χ_{f_{A} (\vec{e})} (u)$ instead of (χ_{f_A(e₁)} (u) , χ_{f_A(e₂)} (u) , . . . , χ_{f_A(e_m)} (u)) for each u ∈ U. It is clear that $χ_{f_{A} (\vec{e})} (u) \in {0, 1}^{m}$

Definition 3.3. Let $F_{A} \in S (U)$ , and f_A (e) be its e-element for each e ∈ E. Then soft aggregation operation on a soft set F_A, denoted s_agg, is defined by $\begin{matrix} S_{agg} : {0, 1}^{m} \to [0, 1], \\ S_{agg} χ_{f_{A} (\vec{e})} (u_{k}) = \frac{1}{mn} \sum_{i \in I_{m}} \\ (χ_{f_{A} (e_{i})} (u_{k}) \sum_{j \in I_{n}} χ_{f_{A} (e_{i})} (u_{j})) \end{matrix}$ for each k, 1 ≤ k ≤ n. Where m = |A|, n = |U|, and χ_{f_A(e)} is characteristic function of f_A (e) for eache ∈ E.

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.Let $F_{A} \in S (U)$ , f_A (e) be itse-element for eache ∈ Eand $F_{A}^{s}$ be its soft fuzzy set. Then $μ_{F_{A}^{s}} (u) = S_{agg} χ_{f_{A} (\vec{e})} (u)$ for eachu ∈ U.

Proof 3.5. It is easy from Definition 3.2 and Definition 3.3.

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.

Algorithm: Assume that we have a set of alternatives U, and have a set of all possible parameters E. Then,

Choose a subset A of E to approximate the alternatives.

Construct the soft set F_A over U,

Find the weighted parameters set $F_{A}^{w}$ of F_A,

Compute the soft fuzzy set $F_{A}^{s}$ of F_A,

Use the set $F_{A}^{s}$ 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.

Example 3.6. Let U = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀} be a set of ten students in a classroom. We want to give study grant to some good students. There are a lats of parameters to make them. The parameters may be “hardworking”, “honest”, “a good speaker”, “clever”, “well-organizer”, “ability to work with others”, “adaptability”, “emotional stability”, “leadership potential”, and so on. Let E be a set of all possible parameters. Then a decision maker can be made it as follows;

Step 1: It is very natural that each person considers a different subset of E for this work. Assume that the decision maker considers only four parameter; “hardworking”, “honest”, “clever”, and “well-organizer”. If they are denoted by e₁, e₂, e₃ and e₄, respectively, then A = {e₁, e₂, e₃, e₄} ⊆ E.

Step 2: Suppose that the decision maker approximate the alternatives according to own parameters as $\begin{matrix} f_{A} (e_{1}) & = & {u_{1}, u_{2}, u_{4}, u_{5}, u_{7}, u_{9}} \\ f_{A} (e_{2}) & = & {u_{1}, u_{3}, u_{4}, u_{7}, u_{8}} \\ f_{A} (e_{3}) & = & {u_{2}, u_{3}, u_{5}, u_{7}, u_{8}, u_{9}, u_{10}} \\ f_{A} (e_{4}) & = & {u_{1}, u_{2}, u_{8}, u_{10}} \end{matrix}$

Then the soft set F_A can be written by $\begin{matrix} F_{A} = & {\frac{{u_{1}, u_{2}, u_{4}, u_{5}, u_{7}, u_{9}}}{e_{1}}, \frac{{u_{1}, u_{3}, u_{4}, u_{7}, u_{8}}}{e_{2}}, \\ \frac{{u_{2}, u_{3}, u_{5}, u_{7}, u_{8}, u_{9}, u_{10}}}{e_{3}}, \frac{{u_{1}, u_{2}, u_{8}, u_{10}}}{e_{4}}} \end{matrix}$

Step 3: To construct the weighted parameters set $F_{A}^{w}$ , the cardinality of approximate functions f_A (e) of F_A can be found for each e ∈ E, $\begin{matrix} | f_{A} (e_{1}) | = & \sum_{u \in U} χ_{f_{A} (e_{1})} (u)) \\ = & 1 + 1 + 0 + 1 + 1 + 0 + 1 + 0 + 1 + 0 \\ = & 6 \end{matrix}$

By the similar way, |f_A (e₂) |=5, |f_A (e₃) |=7 and |f_A (e₄) |=4 are obtained. Therefore the weighted parameters set $F_{A}^{w}$ is written by, $F_{A}^{w} = {\frac{0.6}{e_{1}}, \frac{0.5}{e_{2}}, \frac{0.7}{e_{3}}, \frac{0.5}{e_{4}}}$

Step 4: Let us use Theorem 3.4 to get the membership degrees $μ_{F_{A}^{s}} (u)$ of soft fuzzy set $F_{A}^{s}$ for each u ∈ U, $\begin{matrix} μ_{F_{A}^{s}} (u_{1}) = & S_{agg} χ_{f_{A} (\vec{e})} (u_{1}) \\ = & S_{agg} (1, 1, 0, 1) \\ = & \frac{1}{4.10} (1 \cdot 6 + 1 \cdot 5 + 0 \cdot 7 + 1 \cdot 4) \\ = & 0.375 \end{matrix}$

By the similar way we can obtain the other degrees; $μ_{F_{A}^{s}} (u_{2}) = 0.425$ , $μ_{F_{A}^{s}} (u_{3}) = 0.3$ , $μ_{F_{A}^{s}} (u_{4}) = 0.275$ , $μ_{F_{A}^{s}} (u_{5}) = 0.325$ , $μ_{F_{A}^{s}} (u_{6}) = 0$ , $μ_{F_{A}^{s}} (u_{7}) = 0.45$ , $μ_{F_{A}^{s}} (u_{8}) = 0.4$ , $μ_{F_{A}^{s}} (u_{9}) = 0.325$ , $μ_{F_{A}^{s}} (u_{10}) = 0.275$ . Hence, the soft fuzzy set of $F_{A}^{s}$ is constructed by $\begin{matrix} F_{A}^{s} = & {\frac{0.375}{u_{1}}, \frac{0.425}{u_{2}}, \frac{0.3}{u_{3}}, \frac{0.275}{u_{4}}, \frac{0.325}{u_{5}}, \frac{0.45}{u_{7}}, \frac{0.4}{u_{8}}, \frac{0.325}{u_{9}}, \frac{0.275}{u_{10}}} \end{matrix}$

Step 5: Now, $F_{A}^{s}$ can be used to make decisions. We can arrange the students from weak to strong as follows, $u_{6}, u_{4} = u_{10}, u_{3}, u_{5} = u_{9}, u_{1}, u_{8}, u_{2}, u_{7} .$ where u₆ is the weakest student since it has smallest membership grade, and u₇ is the best student since it has largest membership grade.

4 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 $F_{A} \in S (U)$ . Then the F_A can be presented by a table as in the following form $\begin{array}{c} A \ U & u_{1} & u_{2} & \dots & u_{n} \\ e_{1} & χ_{f_{A} (e_{1})} (u_{1}) & χ_{f_{A} (e_{1})} (u_{2}) & \dots & χ_{f_{A} (e_{1})} (u_{n}) \\ e_{2} & χ_{f_{A} (e_{2})} (u_{1}) & χ_{f_{A} (e_{2})} (u_{2}) & \dots & χ_{f_{A} (e_{2})} (u_{n}) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ e_{m} & χ_{f_{A} (e_{m})} (u_{1}) & χ_{f_{A} (e_{m})} (u_{2}) & \dots & χ_{f_{A} (e_{m})} (u_{n}) \end{array}$ where χ_{f_A(e)} is the characteristic function of f_A (e) for every e ∈ A.

If a_ij = χ_{f_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, $[a_{ij}]_{m \times n} = [\begin{matrix} a_{11} & a_{12} & . . . & a_{1 n} \\ a_{21} & a_{22} & . . . & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & . . . & a_{mn} \end{matrix}]$ that is an m × n matrix of F_A denoted by M_m×n [A, U].

If $a_{ij} = μ_{F_{A}^{w}} (e_{i})$ for every i ∈ I_m and j = 1, then the weighted parameters set $F_{A}^{w}$ is uniquely characterized by a matrix, $[a_{i 1}]_{m \times 1} = [\begin{matrix} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{matrix}]$ that is an m × 1 matrix of $F_{A}^{w}$ denoted by $M_{m \times 1}^{w} [A]$ .

If $a_{i 1} = μ_{F_{A}^{s}} (u_{i})$ for every i ∈ I_n and j = 1, then the soft fuzzy set $F_{A}^{s}$ is uniquely characterized by a matrix, $[a_{i 1}]_{1 \times n} = [a_{11} a_{21} . . . a_{n 1}]$ that is an 1 × n matrix of $F_{A}^{s}$ denoted by $M_{1 \times n}^{s} [U]$ .

If a_i1 = χ_U (u_i) =1 for every i ∈ I_n and j = 1, then the set U is uniquely characterized by a matrix, $[a_{i 1}]_{1 \times n} = [1 1 . . . 1]$ that is an 1 × n matrix of U denoted by M_1×n [1].

Theorem 4.1.IfM_m×n [A, U], $M_{m \times 1}^{w} [A]$ and $M_{1 \times n}^{s} [U]$ are given, then

$M_{m \times 1}^{w} [A] = \frac{1}{n} M_{m \times n} [A, U] \times M_{1 \times n} [1]^{T}$

$M_{1 \times n}^{s} [U] = \frac{1}{m} M_{m \times 1}^{w} [A]^{T} \times M_{m \times n} [A, U]$

wherem = |A|, n = |U|, and the superscriptTis used to indicate transposition of the matrices.

Proof 4.2. The proofs are easy from the matricesmultiplication.

Theorem 4.3. If $M_{1 \times n}^{s} [U]$ is given and S _agg is the soft agreation operator, then $M_{1 \times n}^{s} [U] = [S_{agg} χ_{f_{A} (\vec{e})} (u_{1}) S_{agg} χ_{f_{A} (\vec{e})} (u_{2}) . . . S_{agg} χ_{f_{A} (\vec{e})} (u_{n})]$

Proof 4.4. It is easy from Theorem 3.4 and Theorem 4.1. The matrix operations give very easy way to compute the soft fuzzy set of a soft set easily. Especially, Theorem 4.1 and 4.3 are very applicable for the soft agreation operator. Therefore, we can make a decision by the following algorithm.

Algorithm: As before, assume that we have a set of alternatives U, and have a set of all possibleparameters E.

Choose a subset A of E to approximate the alternatives.

Construct the matrix M_m×n [A, U] of F_A,

Find the matrix $M_{m \times 1}^{w} [A]$ of $F_{A}^{w}$ ,

Compute the matrix $M_{1 \times n}^{s} [U]$ of $F_{A}^{s}$ ,

Use the set $M_{1 \times n}^{s} [U]$ to make decisions.

Now, we can present an example for the the method.

Example 4.5. Let us consider Example 3.6 that can be done by the matrix representation as it is.

Step 1: As before, A = {e₁, e₂, e₃, e₄} ⊆ E.

Step 2: Then the F_A can be presented by a table as in the following form $\begin{array}{c} A \ U & u_{1} & u_{2} & u_{3} & u_{4} & u_{5} & u_{6} & u_{7} & u_{8} & u_{9} & u_{10} \\ e_{1} & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ e_{2} & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ e_{3} & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ e_{4} & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{array}$

Hence, matrix of the soft set is written as $M_{4 \times 10} [A, U] = [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{matrix}]$

Step 3: By using Theorem 4.1-i matrix of the weighted parameters set can be computed by $\begin{matrix} M_{4 \times 1}^{w} [A] = \frac{1}{10} [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{matrix}] [\begin{matrix} 1 \\ ∥ 1 \\ ⋮ \\ 1 \end{matrix}] \\ = [\begin{matrix} 0.6 \\ 0.5 \\ 0.7 \\ 0.4 \end{matrix}] \end{matrix}$

Step 4: By using Theorem 4.1-ii matrix of the soft fuzzy set can be found by $\begin{matrix} M_{1 \times n}^{s} [U] = \frac{1}{4} [0.6 0.5 0.7 0.4] \\ [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{matrix}] \end{matrix}$ and after computation it is written as

$M_{1 \times n}^{s} [U] =$ [0.375 0.425 0.3 0.275 0.325 0 0.45 0.4 0.325 0.275]

Step 5: Now, $M_{1 \times n}^{s} [U]$ can be used to make decisions. We can arrange the students from weak to strong as before, $u_{6}, u_{4} = u_{10}, u_{3}, u_{5} = u_{9}, u_{1}, u_{8}, u_{2}, u_{7} .$ where u₆ is the weakest student since it has smallest membership grade, and u₇ is the best student since it has largest membership grade.

5 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 $f_{agg} : [0, 1]^{k} \to [0, 1]$

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.

6 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.

References

Acar

, Koyuncu

and Tanay

, Soft sets and soft rings, Comput Math Appl59 (2010), 3458–3463.

Ali

M.I.

, Feng

, Liu

, Min

W.K.

and Shabir

, On some new operations in soft set theory, Comput Math Appl57 (2009), 1547–1553.

Aktaş

and Çağman

, Soft sets and soft groups, Inf Sci177 (2007), 2726–2735.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Set Syst20 (1986), 87–96.

Aygünoglu

and Aygün

, Introduction to fuzzy soft groups, Comput Math Appl58 (2009), 1279–1286.

Çağman

and Enginoğlu

, Soft set theory and uni-int decision making, Eur J Oper Res207(2) (2010), 848–855.

Çağman

and Enginoğlu

, Soft matrix theory and its decision making, Comput Math Appl59 (2010), 3308–3314.

Çağman

, Citak

and Enginoğlu

, Fuzzy parameterized fuzzy soft set theory and its applications, Turk J Fuzzy Syst1/1 (2010), 21–35.

Çağman

, Karataş

and Enginoğlu

, Soft topology, Comput Math Appl62 (2011), 351–358.

10.

Dubois

and Prade

, Fuzzy Set and Systems: Theory and Applications, Academic Press, New York, 1980.

11.

Dubois

and Prade

, A review of fuzzy set aggregation connectives, Inf Sci36 (1985), 85–121.

12.

Feng

, Jun

Y.B.

and Zhao

, Soft semirings, Comput Math Appl56/10 (2008), 2621–2628.

13.

Feng

, Jun

Y.B.

, Liu

and Li

, An adjustable approach to fuzzy soft set based decision making, J Comput Appl Math234 (2010), 10–20.

14.

Feng

, Li

, Davvaz

and Ali

M.I.

, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Comput14 (2010), 899–911.

15.

Gau

W.L.

and Buehrer

D.J.

, Vague sets, IEEE Trans System Man Cybemet23(2) (1993), 610–614.

16.

Jiang

, Tang

, Chen

, Wang

and Tang

, Extending soft sets with description logics, Comput Math Appl59 (2010), 2087–2096.

17.

Jun

Y.B.

, Soft BCK/BCI-algebras, Comput Math Appl56 (2008), 1408–1413.

18.

Jun

Y.B.

and Park

C.H.

, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inf Sci178 (2008), 2466–2475.

19.

Jun

Y.B.

, Lee

K.J.

and Zhan

, Soft p-ideals of soft BCIalgebras, Comput Math Appl58 (2009), 2060–2068.

20.

Jun

Y.B.

, Lee

K.J.

and Khan

, Soft ordered semigroups, Math Log Quart56-1 (2010), 42–50.

21.

Jun

Y.B.

, Lee

K.J.

and Park

C.H.

, Fuzzy soft set theory applied to BCK/BCI-algebras, Comput Math Appl59 (2010), 3180–3192.

22.

Karaaslan

, Cagman

and Enginoglu

, Soft lattices, Journal of New Results in Science1 (2012), 5–17.

23.

Klir

G.J.

and Folger

T.A.

, Fuzzy Sets, Uncertainty and Information, Prentice-Hall, 1988.

24.

Klir

G.J.

and Yuan

, Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall, 1995.

25.

Kong

, Gao

, Wang

and Li

, The normal parameter reduction of soft sets and its algorithm, Comput Math Appl56 (2008), 3029–3037.

26.

Kovkov

D.V.

, Kolbanov

V.M.

and Molodtsov

D.A.

, Soft sets theory-based optimization, J Comput Sys Sc Int46/6 (2007), 872–880.

27.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy soft sets, J Fuzzy Math9/3 (2001), 589–602.

28.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Comput Math Appl45 (2003), 555–562.

29.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in a decision making problem, Comput Math Appl44 (2002), 1077–1083.

30.

Majumdar

and Samanta

S.K.

, Similarity measure of soft sets, New Math Nat Comput4/1 (2008), 1–12.

31.

Majumdar

and Samanta

S.K.

, Generalised fuzzy soft sets, Comput Math Appl59 (2010), 1425–1432.

32.

Mukherjee

and Chakraborty

S.B.

, On intuitionistic fuzzy soft relations, Bull Kerala Math Assoc5/1 (2008), 35–42.

33.

Molodtsov

D.A.

, Soft set theory-first results, Comput Math Appl37 (1999), 19–31.

34.

Molodtsov

D.A.

, The description of a dependence with the help of soft sets, J Comput Sys Sc Int40/6 (2001), 977–984.

35.

Molodtsov

D.A.

, The Theory of Soft Sets (in Russian), URSS Publishers, Moscow, 2004.

36.

Molodtsov

D.A.

, Leonov

V.Yu.

and Kovkov

D.V.

, Soft sets sechnique and its application, Nechetkie Sistemi i Myakie Vychisleniya1/1 (2006), 8–39.

37.

Mushrif

M.M.

, Sengupta

and Ray

A.K.

, Texture classification using a novel, soft-set theory based classification, algorithm, Lect Notes Comput Sci3851 (2006), 246–254.

38.

Park

C.H.

, Jun

Y.B.

and Öztürk

M.A.

, Soft WS-algebras, Commun Korean Math Soc23/3 (2008), 313–324.

39.

Pawlak

, Rough sets, Int J Comput Inform Sci11 (1982), 341–356.

40.

Qin

and Hongi

, On soft equality, J Comput Appl Math234 (2010), 1347–1355.

41.

Roy

A.R.

and Maji

P.K.

, A fuzzy soft set theoretic approach to decision making problems, J Comput Appl Math203 (2007), 412–418.

42.

Som

, On the theory of soft sets, soft relation and fuzzy soft relation, Proc of the National Conference on Uncertainty: A Mathematical Approach, UAMA-06, Burdwan, 2006, pp. 1–9.

43.

Xiao

, Gong

, Xia

and Zou

, Exclusive disjunctive soft sets, Comput Math Appl59 (2010), 2128–2137.

44.

Xiao

, Gong

and Zou

, Acombined forecasting approach based on fuzzy soft sets, J Comput Appl Math228 (2009), 326–333.

45.

Xiao

, Li

, Zhong

and Yang

, Research on synthetically evaluating method for business competitive capacity based on soft set, Stat Methods Med Res (2003), 52–54.

46.

, Ma

, Wang

and Hao

, Vague soft sets and their properties, Comput Math Appl59 (2010), 787–794.

47.

Yager

R.R.

, Multiple objective decision making using fuzzy sets, Int J Man Mach Stud9 (1977), 375–382.

48.

Yao

, Liu

and Yan

, Fuzzy soft sets and soft fuzzy sets, in Proceeding of Fourth International Conference on Natural Computation, ICNC-08, 2008, Volume 6, pp. 252–255.

49.

Zadeh

L.A.

, Fuzzy sets, Inform and Control8 (1965), 338–353.

50.

Zhan

and Jun

Y.B.

, Soft BL-algebras based on fuzzy sets, Comput Math Appl59 (2010), 2037–2046.

51.

Zimmermann

H.-J.

, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, 1993.

52.

Zou

and Xiao

, Data analysis approaches of soft sets under incomplete information, Knowl Base Syst21 (2008), 941–945.