#### Abstract

Molodtsov introduced the theory of soft sets, which can be used as a general mathematical tool for dealing with uncertainty. This paper aims to introduce the concept of the type-2 fuzzy soft set by integrating the type-2 fuzzy set theory and the soft set theory. Some operations on the type-2 fuzzy soft sets are given. Furthermore, we investigate the decision making based on type-2 fuzzy soft sets. By means of level soft sets, we propose an adjustable approach to type-2 fuzzy-soft-set based decision making and give some illustrative examples. Moreover, we also introduce the weighted type-2 fuzzy soft set and examine its application to decision making.

#### 1. Introduction

Soft set theory [1], firstly proposed by Molodtsov, is a general mathematical tool for dealing with uncertainty. Compared with some traditional mathematical tools for dealing with uncertainties, such as the theory of probability, the theory of fuzzy sets [2], and the theory of rough sets [3], the advantage of soft set theory is that it is free from the inadequacy of the parametrization tools of those theories. It has been demonstrated that soft set theory brings about a rich potential for applications in many fields like functions smoothness, Riemann integration, decision making, measurement theory, game theory, and so forth [1].

Soft set theory has received much attention since its introduction by Molodtsov. Maji et al. [4] first defined some operations on soft sets and introduced the soft set into the decision making problems [5]. Chen et al. [6] proposed a new definition of soft set parameterization reduction and compared it with attributes reduction in rough set theory [3]. Kong et al. [7] introduced the definition of normal parameter reduction into soft sets. Ali et al. [8] gave some new operations in soft set theory. Zou and Xiao [9] presented some data analysis approaches of soft sets under incomplete information. Çağman and Enginoğlu [10] defined soft matrices which were a matrix representation of the soft sets. Moreover, they [11] redefined the operations of soft sets and constructed a uni-int decision making method. Herawan and Deris [12] presented an alternative approach for mining regular association rules and maximal association rules from transactional datasets using soft set theory. Gong et al. [13] proposed the concept of bijective soft set and defined some operations on it. The algebraic structure of soft set theories has been investigated in recent years. In [14], Aktaş and Çağman gave a definition of soft groups and studied their basic properties. Jun [15] introduced the notion of soft BCK/BCI-algebras and soft subalgebras. Jun and Park [16] examined the algebraic structure of BCK/BCI-algebras. Feng et al. [17] initiated the study of soft semirings by using the soft set theory and investigated several related properties. Acar et al. [18] defined soft rings and introduced their initial basic properties such as soft ideals and soft homomorphisms. Yamak et al. [19] studied soft hypergroupoids. Anvariyeh et al. [20] investigated the algebraic hyperstructures of soft sets associated to semihypergroups.

It should be noted that all of above works are based on the classical soft set theory. The soft set model, however, can also be combined with other mathematical models. Maji et al. [21] first introduced the concept of fuzzy soft sets by combining the soft sets and fuzzy sets. Majumdar and Samanta [22] defined generalised fuzzy soft sets and discussed application of generalised fuzzy soft sets in decision making problem and medical diagnosis. By combining the vague set and soft set models, Xu et al. [23] introduced the notion of vague soft set. Yang et al. [24] introduced the concept of the interval-valued fuzzy soft set which is a combination of the soft set and the interval-valued fuzzy set. Feng et al. [25] focused on a tentative approach to soft sets combined with fuzzy sets and rough sets and proposed three different types of hybrid models, which are called rough soft sets, soft rough sets, and soft-rough fuzzy sets, respectively. Bhattacharya and Davvaz [26] introduced the concepts of intuitionistic fuzzy lower soft rough approximation and IF upper soft rough approximation space. Maji et al. [27, 28] proposed the notion of intuitionistic fuzzy soft sets by integrating the soft sets and intuitionistic fuzzy sets [29]. By combining the interval-valued intuitionistic fuzzy sets and soft sets, Jiang et al. [30] obtained a new soft set model: interval-valued intuitionistic fuzzy soft set theory. Aygünoğlu and Aygün [31] focused on fuzzy soft groups, homomorphism of fuzzy soft groups, and normal fuzzy soft groups.

According to Mendel [32], there exist at least four sources of uncertainties in type-1 fuzzy logic systems (T1 FLS), which are as follows: (1) meanings of the words that are used in the antecedents and consequents of rules can be uncertain (words mean different things to different people); (2) consequents may have a histogram of values associated with them, especially when knowledge is extracted from a group of experts, all of whom do not collectively agree; (3) measurements that activate a T1 FLS may be noisy and therefore uncertain; (4) the data that are used to tune the parameters of a T1 FLS may also be noisy. All these uncertainties lead to uncertain fuzzy-set membership functions. Ordinary type-1 fuzzy sets cannot model such uncertainties directly, because they are characterized by crisp membership functions. Type-2 fuzzy sets are capable of modeling the four uncertainties. The concept of type-2 fuzzy sets, first proposed by Zadeh [33], is an extension of a type-1 fuzzy set in which its membership function falls into a fuzzy set in the interval . Because type-2 fuzzy sets can improve certain kinds of inference better than do fuzzy sets with increasing imprecision, uncertainty, and fuzziness in information, type-2 fuzzy sets are gaining more and more in popularity. The basic concepts of type-2 fuzzy set theory and its extensions, as well as some practical applications, can be found in [34–40].

However, in the practical applications, we are often faced with the situation in which the evaluation of parameters is a fuzzy concept. For instance, when we are going to buy a car, we need to consider the safety of car. We can provide some linguistic terms, such as good, medium, and bad, as the evaluation about the safety of car. Here, good, medium, and bad are fuzzy concepts and they can be represented by fuzzy sets rather than exact numerical values, interval numbers, intuitionistic fuzzy numbers, and interval-valued intuitionistic fuzzy numbers. Obviously, it is very difficult for the classical soft set and its existing extensions to deal with the above case because the evaluation of parameters of the object is a fuzzy concept rather than an exact numerical value, an interval number, an intuitionistic fuzzy number, and an interval-valued intuitionistic fuzzy number. Hence, it is necessary to extend soft set theory to accommodate the situations in which the evaluation of parameters is a fuzzy concept. As mentioned above, type-2 fuzzy set can be used to represent the fuzziness of the above evaluation of parameters directly. Thus, it is very necessary to extend soft set theory using type-2 fuzzy set. The purpose of this paper is to further extend the concept of soft set theory by combining type-2 fuzzy set and soft set, from which we can obtain a new soft set model: type-2 fuzzy soft sets. We present the concept of type-2 fuzzy soft sets and define some operations on type-2 fuzzy soft sets. Moreover, we also investigate the applications of type-2 fuzzy soft sets and weighted type-2 fuzzy soft sets in decision making problems.

The remainder of this paper is organized as follows. After recalling some preliminaries, Section 3 presents the concept of the type-2 fuzzy soft set and some operations on type-2 fuzzy soft sets. In the sequel, applications of type-2 fuzzy soft sets and weighted type-2 fuzzy soft sets in decision making problems are, respectively, shown in Sections 4 and 5. Finally, conclusions are given in Section 6.

#### 2. Preliminaries

In this section, we will review the concepts of soft set, type-2 fuzzy set, and interval type-2 fuzzy set.

##### 2.1. Soft Sets

Let be an initial universe of objects and (, for short) the set of parameters in relation to objects in . Parameters are often attributes, characteristics, or properties of objects. Let denote the power set of and . Molodtsov [1] defined a soft set as follows.

*Definition 2.1 (see [1]). *A pair is called a soft set over , where is a mapping given by

In other words, a soft set over is a parameterized family of subsets of the universe . For , is regarded as the set of -approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [1].

*Definition 2.2 (see [5]). *Let be a soft set over . The choice value of an object is , given by
where if then , otherwise .

As an illustration, let us consider the following example originally introduced by Molodtsov [1].

*Example 2.3 (A house purchase problem). *Suppose the following. The universe is the set of six houses under consideration. is the set of parameters that Mr. X is interested in buying a house. , where () stands for the parameters in a word of “expensive,” “beautiful”, “wooden”, “in the green surroundings”, and “convenient traffic”, respectively. That means, out of available houses in , Mr. X is to select that house which qualifies with all (or with maximum number of) parameters of the set . In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. The soft set describes the “attractiveness of the houses” which Mr. X (say) is going to buy.

Suppose that

The soft set is a parametrized family of subsets of the set and gives us a collection of approximate descriptions of an object. Consider the mapping which is “houses ” where dot is to be filled up by a parameter . For instance, means “houses (expensive)” whose functional value is the set . Thus, we can view the soft set as a collection of approximations as below:

In order to store a soft set in a computer, we could represent a soft set in the form of a 0-1 two-dimensional table. Table 1 is the tabular representation of the soft set . If , then , otherwise , where are the entries in Table 1.

##### 2.2. Type-2 Fuzzy Sets and Interval Type-2 Fuzzy Sets

In the current subsection, we recall the notions of type-2 fuzzy sets and interval type-2 fuzzy sets from [32, 35, 39].

*Definition 2.4. *Let be a finite and nonempty set, which is referred to as the universe. A type-2 fuzzy set, denoted by , is characterized by a type-2 membership function , where , and , that is,
where . can also be expressed as
where .

The class of all type-2 fuzzy sets of the universe is denoted by .

*Definition 2.5. *At each value of , say , the 2D plane whose axes are and is called the vertical slice of . A secondary membership function is a vertical slice of . It is for and , that is,
where . The amplitude of a secondary membership function is called a secondary grade. In Definition 2.4, and are all secondary grades.

*Definition 2.6. *The domain of a secondary membership function is called the primary membership of . In Definition 2.5, is the primary membership of .

*Definition 2.7. *If all the secondary grades of a type-2 fuzzy set are equal to , that is, , and , then is defined as an interval type-2 fuzzy set.

##### 2.3. Operations of Type-2 Fuzzy Sets

Let be a nonempty universe, :

The union, intersection, and complement for type-2 fuzzy sets are defined as follows.(1) Union of two type-2 fuzzy sets : where is the minimum operation, is the maximum operation, is called the join operation, , , and are the secondary membership functions, and all are type-1 fuzzy sets.(2) Intersection of two type-2 fuzzy sets : where is the minimum operation, is called the meet operation, , , and are the secondary membership functions, and all are type-1 fuzzy sets.(3) Complement of a type-2 fuzzy set :

*Example 2.8. * Let a nonempty universe, and let and be two type-2 fuzzy sets over the same universe .

Suppose that Then, we have that

##### 2.4. Cut Sets of Type-2 Fuzzy Sets

*Definition 2.9 (see [41, 42]). *Let be a type-2 fuzzy set on the universe , the secondary membership function of , and , , . Then the secondary -cut set of is defined by
where and .

By Definitions 2.7 and 2.9, we can see that the secondary -cut set of is an interval type-2 fuzzy set.

*Definition 2.10 (see [41, 42]). *Let be an interval type-2 fuzzy set on the universe , the secondary membership function of , and , , . Then the primary -cut set of is defined by
where and .

By Definition 2.10, the primary -cut set of the secondary -cut set of the type-2 fuzzy set is defined as follows.

*Definition 2.11 (see [41, 42]). *Let be the secondary -cut set of the type-2 fuzzy set . Then the primary -cut set of is defined by
where , , and .

The cut set of the type-2 fuzzy set contains the secondary cut set and the primary cut set, and it is the primary -cut set of the secondary -cut set of the type-2 fuzzy set. Therefore, when we compute the cut set of the type-2 fuzzy set , the first step is to compute the secondary -cut set of , and the second step is to compute the primary -cut set of .

#### 3. Type-2 Fuzzy Soft Sets

In this section, we will initiate the study on hybrid structures involving both type-2 fuzzy sets and soft sets. In Section 3.1, we introduce the concept of the type-2 fuzzy soft set which is an extension of the soft set [32]. Next, in Section 3.2, we discuss some operations on type-2 fuzzy soft sets.

##### 3.1. Concept of Type-2 Fuzzy Soft Sets

*Definition 3.1. *Let be an initial universe and a set of parameters; a pair is called a type-2 fuzzy soft set over , where is a mapping given by

In other words, a type-2 fuzzy soft set is a parameterized family of type-2 fuzzy subsets of . For any , is referred as the set of -approximate elements of the type-2 fuzzy soft set , it is actually a type-2 fuzzy set on , and it can be written as.

Here, , are, respectively, the primary membership degree and secondary membership degree that object holds on parameter .

To illustrate the idea, let us consider the following example (adapted from Mendel and Wu [43]).

*Example 3.2 (see [43]). *Consider a type-2 fuzzy soft set over , where is a set of six houses under the consideration of a decision maker to purchase, which is denoted by , and is a parameter set, where . The type-2 fuzzy soft set describes the “attractiveness of the houses” to this decision maker.

Suppose that

We can convert the above linguistic terms into the corresponding fuzzy sets and obtain the following results:

The type-2 fuzzy soft set is a parameterized family of type-2 fuzzy sets on , and

Table 2 gives the tabular representation of the type-2 fuzzy soft set . We can see that the precise evaluation for each object on each parameter is unknown. For example, we cannot present the precise degree of how expensive house is; however, we can say that house is inexpensive.

*Definition 3.3. *For two type-2 fuzzy soft sets and over , one says that is a type-2 fuzzy soft subset of if and only if and , . One denotes this relationship by . is said to be a type-2 fuzzy soft super set of , if is a type-2 fuzzy soft subset of . One denotes it by .

*Example 3.4. *Let and be two type-2 fuzzy soft sets over the same universe as follows:
where is the set of houses, , and .

Clearly, by Definition 3.3, we have .

*Definition 3.5. *Two type-2 fuzzy soft sets and over a common universe are said to be type-2 fuzzy soft equal if is a type-2 fuzzy soft subset of and is a type-2 fuzzy soft subset of , which can be denoted by .

##### 3.2. Operations on Type-2 Fuzzy Soft Sets

*Definition 3.6 (see [4]). *Let be a parameter set. The not set of , denoted by , is defined by , where .

*Definition 3.7. *The complement of a type-2 fuzzy soft set is denoted by , and it is defined by
where is a mapping given by , .

*Example 3.8. *Following Example 3.2, the complement of the type-2 fuzzy soft set is given below:

*Definition 3.9. * Let and be two type-2 fuzzy soft sets over . Then “ AND ” is defined by , where , .

*Definition 3.10. * Let and be two type-2 fuzzy soft sets over . Then “ OR ” is defined by , where , .

*Example 3.11. *Let and be two type-2 fuzzy soft sets over the same universe . Here is the set of houses, , and .

Suppose that

Compute the results of the “AND” operation and “OR” operation on and , respectively. Let . Then, by Definition 3.9, we have that

That is,

Let . Then, by Definition 3.10, we have that

That is,