#### Abstract

Soft set theory is a newly emerging mathematical tool to deal with uncertain problems. In this paper, by introducing a generalization parameter, which itself is trapezoidal fuzzy, we define generalized trapezoidal fuzzy soft sets and then study some of their properties. Finally, applications of generalized trapezoidal fuzzy soft sets in a decision making problem and medical diagnosis problem are shown.

#### 1. Introduction

Many complicated problems in economics, engineering, social sciences, medical sciences, and many other fields involve uncertain data. These problems, which one comes face to face with in life, cannot be solved using classical mathematic methods. There are several well-known theories to describe uncertainty, for instance, fuzzy set theory [1], rough set theory [2], and other mathematical tools. But all of these theories have their inherit difficulties as pointed out by Molodtsov [3]. To overcome these difficulties, in 1999 Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set theory seems to be free from the difficulties affecting the existing methods. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts [4]. Soft set theory has potential applications in many different fields including the smoothness of functions, game theory, operational research, Perron integration, probability theory, and measurement theory [3, 5]. Research works on soft sets are very active and progressing rapidly in these years. Maji et al. [6] defined several operations on soft sets and made a theoretical study on the theory of soft sets. Jun [7] introduced the notion of soft BCK/BCI-algebras. Jun and Park [8] discussed the applications of soft sets in ideal theory of BCK/BCI-algebras. Feng et al. [9] applied soft set theory to the study of semirings and initiated the notion of soft semirings. Furthermore, based on [6], Ali et al. [10] introduced some new operations on soft sets and improved the notion of complement of soft set. They proved that certain De Morgan’s laws hold in soft set theory. Qin and Hong [11] introduced the notion of soft equality and established lattice structures and soft quotient algebras of soft sets. Park et al. [12] discussed some properties of equivalence soft set relations.

The study of hybrid models combining soft sets with other mathematical structures is emerging as an active research topic of soft set theory. Maji et al. [13] initiated the study on hybrid structures involving fuzzy sets and soft sets. They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of soft sets. Furthermore, based on [13], Majumdar and Samanta [14] modified the definition of fuzzy soft sets and presented the notion of generalized fuzzy soft sets theory. Yang et al. [15] presented the concept of the interval-valued fuzzy soft sets by combining interval-valued fuzzy set [16, 17] and soft set models. By combining the multifuzzy set and soft set models, Yang et al. [18] presented the concept of the multifuzzy soft set and provided its application in decision making under an imprecise environment. Feng et al. [19] provided a framework to combine fuzzy sets, rough sets, and soft sets all together, which gave rise to several interesting new concepts such as rough soft sets, soft rough sets, and soft rough fuzzy sets. The combination of soft set and rough set models was also discussed by some researchers [20, 21].

The trapezoidal fuzzy number, as a vital concept of fuzzy set, is increasingly applied in many references [22, 23]. The membership function of a trapezoidal fuzzy number is piecewise linear and trapezoidal, which can express vagueness information caused by linguistic assessments through transforming them into numerical variables objectively. By combining the concepts of trapezoidal fuzzy sets and soft set models, Xiao et al. [24] presented the concept of the trapezoidal fuzzy soft sets which can deal with certain linguistic assessments. However, we can note that the attribute of the parameters in the soft set may be imprecise and vague such as the attribute “beautiful.” In order to further capture the vagueness of the attribute with linguistic assessments information, it is natural for us to generalize the concept of the trapezoidal fuzzy soft sets as introduced by Xiao et al. [24]. In this paper, by introducing a generalization parameter, which itself is trapezoidal fuzzy, we define generalized trapezoidal fuzzy soft sets. In our generalization of trapezoidal fuzzy soft sets, a degree is attached to the parametrization of trapezoidal fuzzy sets while defining a trapezoidal fuzzy soft set. This definition is more realistic as it involves uncertainty in the selection of a trapezoidal fuzzy set corresponding to each value of the parameter. Some operations on the generalized trapezoidal fuzzy soft sets are also investigated. Then we present an example which shows that the decision making method of generalized trapezoidal fuzzy soft sets can be successfully applied to many problems that contain uncertainties. Finally, application of generalized trapezoidal fuzzy soft sets in a medical diagnosis problem is shown.

The rest of this paper is organized as follows. The following section briefly reviews some backgrounds on soft sets, trapezoidal fuzzy sets, and trapezoidal fuzzy soft sets. In Section 3, the concept of generalized trapezoidal fuzzy soft sets is presented. Some operations on the generalized trapezoidal fuzzy soft sets are then defined. Also some of their interesting properties are investigated. An application of generalized trapezoidal fuzzy soft sets in a medical diagnosis problem is shown in Section 4. Section 5 concludes the paper.

#### 2. Preliminaries

The following definitions and preliminaries are required in the sequel of our work and hence presented in brief.

Definition 1 (see [3]). Let be an initial universe set and let be a universe set of parameters. A pair is called a soft set over if and , where is the set of all subsets of  .

Definition 2 (see [13]). Let be an initial universe set and let be a universe set of parameters. A pair is called a fuzzy soft set over if and , where is the set of all fuzzy subsets of .

Definition 3 (see [25]). A trapezoidal fuzzy number can be defined as shown in Figure 1 which has the membership function as follows:

A set which is consisted by a trapezoidal fuzzy number or several trapezoidal fuzzy numbers is called trapezoidal fuzzy set, denoted by .

The membership function of a trapezoidal fuzzy number is piecewise linear and trapezoidal which can capture the vagueness of those linguistic assessments as in Figure 2. For example, the linguistic variable “medium poor” can be represented as (0.2, 0.3, 0.4, 0.5), the membership function of which is

Let and be two trapezoidal fuzzy numbers; then we have the following operations.

We can say [25] if and only if The complement of which is denoted by can be defined as [25] The union of and which is denoted by can be defined as [25] The intersection of and which is denoted by can be defined as [25] The multiplication of and which is denoted by can be defined as [26]

Next the defuzzification method of a trapezoidal fuzzy number will be introduced (see [27]). Take a trapezoidal fuzzy number parameterized by a quadruplet as shown in Figure 1.

Then the defuzzification value of the trapezoidal fuzzy number is calculated from the figure as follows:

By using the concepts of trapezoidal fuzzy sets and soft sets, Xiao et al. [24] presented the concept of trapezoidal fuzzy soft sets.

Definition 4 (see [24]). Let be the set of all trapezoidal fuzzy subsets of . A pair is called a trapezoidal fuzzy soft set over , where is a mapping given by .

Example 5 (see [24]). Let be a set of five houses under consideration of a decision maker to purchase, which is denoted by . Let be a parameter set, where = {cheap; beautiful; size; location; in the green surroundings}. Someone describes the optional five houses under various attributes with linguistic variables intuitively as in Table 1.
Then, we can have a corresponding trapezoidal fuzzy soft set over the universe through the rule of conversion between linguistic variables and numerical variables showed in Figure 2.
Consider

Remark 6. In Example 5, we can note that the attribute of the parameters in the trapezoidal fuzzy soft sets is imprecise and vague such as the attribute “beautiful.” However, Xiao et al. do not point out how to depict the vague attribute of the parameters at all. So the model has its disadvantages which do not solve some problems that contain uncertainties when the attribute of the parameters in the trapezoidal fuzzy soft sets is imprecise and vague. In order to overcome the difficulty, we will introduce a generalization parameter, which itself is trapezoidal fuzzy, and present the concept of generalized trapezoidal fuzzy soft sets in the following section.

#### 3. Generalized Trapezoidal Fuzzy Soft Sets

##### 3.1. Concept of Generalized Trapezoidal Fuzzy Soft Sets

In this subsection, we generalize the concept of trapezoidal fuzzy soft sets as introduced by Xiao et al. [24]. In our generalization of trapezoidal fuzzy soft sets, a degree is attached to the parametrization of trapezoidal fuzzy sets while defining a trapezoidal fuzzy soft set.

Definition 7. Let be an initial universe and let be a set of parameters. The pair is called a soft universe. Suppose that and is a trapezoidal fuzzy subset of ; that is, . We say that is a generalized trapezoidal fuzzy soft set (GTFSS, in short) over the soft universe if and only if is a mapping given by where , such that for all , and .

Here for each parameter , indicates not only the trapezoidal fuzzy membership degree of belongingness of the elements of in but also the trapezoidal fuzzy membership degree of possibility of such belongingness of the parameters of which is represented by .

We can note that a generalized trapezoidal fuzzy soft set is actually a soft set because it is still a mapping from parameters to , and it can be written as where

Sometimes we write as . If , we can also have a GTFSS .

Example 8. Suppose that a married couple, Mr. X and Mrs. X, come to the real estate agent to buy a house. Assume that a real estate agent has a set of different types of houses which may be characterized by a set of parameters . For the parameters stand for “location,” “cheap,” and “size,” respectively. Considering his own needs, Mr. X describes the optional five houses under various attributes with linguistic variables intuitively as in Table 2.
Then, we can have a corresponding generalized trapezoidal fuzzy soft set over the universe through the rule of conversion between linguistic variables and numerical variables showed in Figure 2.
Consider

Remark 9. In Example 8, we consider the vagueness of the attribute of the parameters. For example, the attribute “cheap” is imprecise and is characterized by a trapezoidal fuzzy number . But as we have pointed out before, in Example 5 the vagueness of the attribute of the parameters is not considered. So the difference between the trapezoidal fuzzy soft set and the generalized trapezoidal fuzzy soft set is whether the vagueness of the attribute of the parameters has been depicted or not. Compared with the trapezoidal fuzzy soft set, the generalized trapezoidal fuzzy soft set can further capture the vagueness of the attribute with linguistic assessments information.

Definition 10. Let and be two GTFSSs over . Now is said to be a generalized trapezoidal fuzzy soft subset of if and only if(1)for all , trapezoidal fuzzy numbers ; that is, , , , and ;(2)for all , is a trapezoidal fuzzy subset of ; that is, , , , and in and over .In this case, we write .

Example 11. Consider the GTFSS over given in Example 8. Let be another GTFSS over defined as follows:
Clearly, we have .

Definition 12. Let and be two GTFSSs over . Now and are said to be a generalized trapezoidal fuzzy soft equal if and only if(1) is a generalized trapezoidal fuzzy soft subset of ;(2) is a generalized trapezoidal fuzzy soft subset of ,which can be denoted by .

##### 3.2. Operations on Generalized Trapezoidal Fuzzy Soft Sets

Definition 13. Let be a GTFSS over . Then the complement of , denoted by , is defined by , where and .

From the above definition, we can see that .

Example 14. Consider the GTFSS over defined in Example 11. Thus, by Definition 13, we have

Definition 15. The union operation on the two GTFSSs and , denoted by , is defined by a mapping given by , such that for all , where and .

Definition 16. The intersection operation on the two GTFSSs and , denoted by , is defined by a mapping given by , such that for all , where and .

Example 17. Let us consider the GTFSS in Example 8. Let be another GTFSS over defined as follows:
Then by Definition 15, we have
By Definition 16, then

Definition 18. A GTFSS is said to be a generalized empty trapezoidal fuzzy soft set, denoted by , if , such that , where and ,  for all .

Definition 19. A GTFSS is said to be a generalized universal trapezoidal fuzzy soft set, denoted by , if , such that , where and ,  for all .

From Definitions 18 and 19, obviously we have(1), (2),(3).

Theorem 20. Let be a GTFSS over ; then the following holds:(1), ,(2), .

Proof. Straightforward.

Remark 21. Let be a GTFSS over ; if or , then and .

Theorem 22. Let , , and be any three GTFSSs over ; then the following holds:(1), (2), (3), (4).

Proof. The properties follow from the properties of trapezoidal fuzzy sets.

Theorem 23. Let and be two GTFSSs over . Then De-Morgan’s laws are valid:(1), (2).

Proof. For all , Likewise, the proof of (2) can be made similarly.

Theorem 24. Let , , and be any three GTFSSs over . Then,(1), (2).

Proof. The proof follows from definition and distributive property of trapezoidal fuzzy sets.

Definition 25. Let and be two GTFSSs over . The “,” denoted by , is defined by , where , for all , such that and .

Definition 26. Let and be two GTFSSs over . The “,” denoted by , is defined by , where , for all , such that and .

Remark 27. Let and be two GTFSSs over . For all , if , then and .

Theorem 28. Let and be two GTFSSs over . Then(1), (2).

Proof. Suppose that .
Here , for all . By Definition 25, for all , we have and .
Again suppose that , where , for all , such that and . Hence .
Likewise, the proof of (2) can be made similarly.

Theorem 29. Let , , and be any three GTFSSs over . Then we have(1), (2), (3), (4).

Proof. The proof follows from Definitions 25 and 26 and distributive property of trapezoidal fuzzy sets.

In group decision problems, when the attribute of the parameters is imprecise and vague, the generalized trapezoidal fuzzy soft set is more realistic than the trapezoidal fuzzy soft set as each person has various opinions on the vague attributes of the same parameter. In order to further illustrate the point, the following example is given by us.

Example 30. Reconsider Example 8. Each person has different opinions on the vague attributes of the same house. Now Mrs. X also describes the optional five houses under various attributes with linguistic variables intuitively as in Table 3.
Similarly to Example 8, we can obtain a corresponding generalized trapezoidal fuzzy soft set as follows:
Here, we must use AND operation since the different opinions of the married couple have to be considered.
By Definition 25, we can obtain the result of “AND” operation on the generalized trapezoidal fuzzy soft sets and as follows:
Then, defuzzifying each element of , by (8), we get a fuzzy matrix showed in Table 4.
Now to determine the best house which will satisfy the needs of the married couple, we mark the highest numerical grade in each column excluding the last row which is the possibility defuzzification grade of such belongingness of a house against each pair of parameters (see Table 5). Now, the score of each such house is calculated by taking the sum of the products of these numerical grades with the corresponding possibility defuzzification grade . The house with the highest score is the desired one. We do not consider the numerical grades of the house against the pairs , as both the parameters are the same: Score ,Score ,Score .
The married couple will select the house with the highest score. Hence, they will buy the house .

Remark 31. In Example 30, we can note that the generalized trapezoidal fuzzy soft set in the application of group decision problems is more realistic and better than the trapezoidal fuzzy soft set as each person has various opinions on the vague attributes of the same house. For example, Mr. X thinks that the better size of a house is “medium poor,” but Mrs. X may not think that. She thinks that the better size of a house is “good.” Since it may be impossible or unnecessary to obtain more accurate values, for the subjective judgment of the decision maker and the vagueness of parameters information, piecewise linear trapezoidal membership functions are good enough to capture the vagueness of the attribute of the parameters. Therefore, the generalized trapezoidal fuzzy soft set is effective to express these decision making problems when the attribute of the parameters is imprecise and vague. This makes the decision making based on generalized trapezoidal fuzzy soft sets to be more preferable to reflect the reality.

#### 4. Application of Generalized Trapezoidal Fuzzy Soft Sets in Medical Diagnosis

In this section, inspired by Çelik’s method to diagnose which patient is suffering from what disease in [27], we also present a method of medical diagnosis based on generalized trapezoidal fuzzy soft sets theory.

Since soft set was introduced by Molodtsov in 1999, soft set and its various extensions have been applied in dealing with practical problems. Especially, applications of fuzzy soft set theory in medical diagnosis problems have been studied by many researchers. De et al. [28] have studied Sanchez's [29, 30] method of medical diagnosis using an intuitionistic fuzzy set. Saikia et al. [31] have extended the method in [28] using intuitionistic fuzzy soft set theory. In [32], Chetia and Das have studied Sanchez's approach of medical diagnosis through interval-valued fuzzy soft set obtaining an improvement of the same set presented in De et al. [28]. However, the above existing medical diagnosis approaches could inevitably have their limitations. For example, in the existing approaches we cannot present the precise membership degree how painful a patient's head is. But in the generalized trapezoidal fuzzy soft sets we can solve it. Suppose that a patient tells the doctor that his headache is “very good” with his linguistic assessments. So the doctor knows that the degree of his patient's headache is characterized by a trapezoidal fuzzy number . Hence, as we know that the membership function of a trapezoidal fuzzy number can express vagueness information caused by linguistic assessments through transforming them into numerical variables objectively, it is natural for us to apply generalized trapezoidal fuzzy soft sets to medical diagnosis in order to capture the vagueness of linguistic assessments information in medical diagnosis.

In the following subsections, we will present the steps and the algorithm for the new approach, respectively, in detail.

##### 4.1. Algorithm of Medical Diagnosis Problem Based on GTFSS

Assume that there is a set of patients with a set of symptoms related to a set of diseases .

Thus, we first construct a trapezoidal fuzzy soft set over where is a mapping . This trapezoidal fuzzy soft set gives a relation matrix called patient-symptom matrix, where the entries are trapezoidal fuzzy numbers , for and . The patient-symptom matrix is given as follows:

Secondly, we construct a generalized trapezoidal fuzzy soft set over where is a mapping . This generalized trapezoidal fuzzy soft set also gives a relation matrix called symptom-disease matrix, where each element is also taken as trapezoidal fuzzy numbers , for and . The symptom-disease matrix is given as follows: where the th row vector represents , the th column vector represents , and the last column represents the values of which is a trapezoidal fuzzy set on .

Thirdly, performing the transformation operation , we get the patient-diagnosis matrix as follows: where