Research Article  Open Access
Xiaoguo Chen, Hong Du, Yue Yang, "The IntervalValued Triangular Fuzzy Soft Set and Its Method of Dynamic Decision Making", Journal of Applied Mathematics, vol. 2014, Article ID 132806, 12 pages, 2014. https://doi.org/10.1155/2014/132806
The IntervalValued Triangular Fuzzy Soft Set and Its Method of Dynamic Decision Making
Abstract
A concept of intervalvalued triangular fuzzy soft set is presented, and some operations of “AND,” “OR,” intersection, union and complement, and so forth are defined. Then some relative properties are discussed and several conclusions are drawn. A dynamic decision making model is built based on the definition of intervalvalued triangular fuzzy soft set, in which period weight is determined by the exponential decay method. The arithmetic weighted average operator of intervalvalued triangular fuzzy soft set is given by the aggregating thought, thereby aggregating intervalvalued triangular fuzzy soft sets of different timeseries into a collective intervalvalued triangular fuzzy soft set. The formulas of selection and decision values of different objects are given; therefore the optimal decision making is achieved according to the decision values. Finally, the steps of this method are concluded, and one example is given to explain the application of the method.
1. Introduction
Complex problems involving vagueness and uncertainty are pervasive in many areas of modern technology. These practical problems arise in diverse areas such as economics, engineering, environmental science, and social science [1]. While a wide variety of mathematical disciplines like probability theory, fuzzy set theory [2], rough set theory [3], and interval mathematics [4] are well known and frequently employed as useful mathematical approaches to modeling, each of them has its advantages as well as inherent limitations. One major weakness of these theories may be the inadequacy of the parameterization tool in them [5]. Consequently, Molodtsov initiated the concept of soft theory as a mathematical tool, which is free from the above difficulties for dealing with uncertainties [5].
Soft set theory has received much attention and indepth study since its introduction. Maji et al. [6] presented the concept of the fuzzy soft set which is based on a combination of the fuzzy set and soft set models. In the same year, Maji et al. [7] presented the notion of the intuitionistic fuzzy soft set theory which is based on a combination of the intuitionistic fuzzy set and soft set models. Yang et al. [8] proposed the concept of the intervalvalued fuzzy soft set by combining the intervalvalued fuzzy set and soft set models. Jiang et al. [9] presented the theory of intervalvalued intuitionistic fuzzy soft set by combining the intervalvalued intuitionistic fuzzy set and soft set. Feng et al. [10] made a primary discussion of soft set through combining the rough set and fuzzy set. Ali [11] raised the concept of rough soft set by combining rough set and soft set. Xu et al. [12] used vague set [13] to make an extension of soft set and proposed vague soft set theory. Majumdar and Samanta [14] proposed the generalized concept of fuzzy soft set theory. On the basis of previous studies, many researchers have carried out particular exploration on the operations of intersection, union, complement, and subset among various soft sets [15–21]. Xiao et al. [22] employed fuzzy soft set into the application of economic forecasting. Maji and Roy firstly introduced the soft set and fuzzy soft set into decision making problem and employed the rough set theory in attribute reduction [23, 24], but the decision making method in [24] pretended some errors which had been discussed by Kong et al. [25]. Çağman and Enginoğlu [26] redefined the concept of soft set proposed by Molodtsov and discussed its relevant algorithm. In the meantime, they also defined decision function and applied it into the decision making problem. Çağman and Enginoğ [27] defined soft matrix on the basis of soft set, discussed its relevant properties and operation, and constructed the maximum and minimum soft functions to solve practical decision making problems. Feng et al. [28] and Jiang et al. [29] carried out further study of soft set, respectively, and proposed the threshold vector for constructing the level soft set in decision making, converted different types of soft sets into classic soft sets, and then obtained the optimal decision by the opportunity value of different objects. Based on [28, 29], Mao et al. [30] proposed a decision making method on the basis of timing fuzzy soft set. Kuang and Xiao [31, 32] raised the definition of introduced triangular fuzzy soft set and trapezoidal fuzzy soft set, introduced the relevant operating properties of them, and built the corresponding decision making models.
Despite the fact that the studies above have widely discussed the extension of the soft set, there is relatively little literature on the study of the intervalvalued triangle fuzzy soft set. Besides, the studies on decision above are almost static, while practical problems always need dynamic analysis, so it is essential to introduce the time variate into the decision making of the fuzzy soft set. Based on this, definition of the intervalvalued triangle fuzzy soft set is presented and relevant operational properties are discussed in this paper. The paper establishes the decision making model of the dynamic intervalvalued triangle fuzzy soft set which considers the time variate and whose time weights can be determined by exponential decay method of [33]. Using the thought of integration [34], the paper defines the arithmetic weighted average operator of intervalvalued triangle fuzzy soft set and gives methods to integrate the dynamic intervalvalued triangle fuzzy soft set. In this paper, the decision methods of [8, 31] are used in combination to calculate a solution formula of the selection values and decision values based on the presented intervalvalued triangle fuzzy soft set. Then the drawn solution formula is applied to decision analysis.
The purpose of this paper is to combine the intervalvalued triangular fuzzy set and soft set, from which we can obtain a new soft set model: intervalvalued triangular fuzzy soft set. To facilitate our discussion, we first introduce the standard soft set, fuzzy soft set, intervalvalued fuzzy soft set, and triangular fuzzy soft set in Section 2. The concept of intervalvalued triangular fuzzy soft set is presented in Section 3. Then the “AND,” “OR,” intersection, union, and complement operations on the intervalvalued triangular fuzzy soft sets are defined. In Section 4, the intervalvalued triangular fuzzy soft set is used to analyze a decision making problem. Then a specific example is analyzed and solved with this technique in Section 5. At last, we draw conclusions in Section 6.
2. Preliminary
In this section, we briefly review the concepts of soft set, fuzzy soft set, intervalvalued fuzzy soft set, and triangular fuzzy soft set. Further details could be found in [5–9]. Molodtsov [5] defined the soft set in the following way. Let be an initial universe of objects and let be the set of parameters in relation to objects in . Parameters are usually attributes, characteristics, or properties of objects. Let denote the power set of and .
Definition 1 (see [5]). A pair is called a soft set over , where is a mapping given by . In other words, the soft set is not a kind of set but a parameterized family of subsets of the set . For any parameter , may be considered as the set of approximate elements of the soft set .
Definition 2 (see [6]). Let be the set of all fuzzy subsets of . Let be a set of parameters and . A pair is called a fuzzy soft set over , where is a mapping given by .
Let us denote by the membership degree that object holds parameter where and ; then can be written as a fuzzy set such that .
In the following, we will introduce the notion of intervalvalued fuzzy soft set. Firstly, let us briefly introduce the concept of the intervalvalued fuzzy set of [8].
An intervalvalued fuzzy set on a universe is a mapping such that , where stands for the set of all closed subintervals of ; the set of all intervalvalued fuzzy sets on is denoted by . Suppose that , , is called the degree of membership of an element to . and are referred to as the lower and upper degrees of membership of to , where .
Definition 3 (see [8]). Let be an initial universe and let be a set of parameters. denotes the set of all intervalvalued fuzzy sets of . Let . A pair is an intervalvalued fuzzy soft set over , where is a mapping given by .
Let us denote by the membership degree that object holds parameter where and ; then can be written as an intervalvalued fuzzy set as
Definition 4 (see [31]). Let be an initial universe and let be a set of parameters. denotes the set of all triangular fuzzy sets of . Let . A pair is called a trianglevalued fuzzy soft set over , where is a mapping given by .
A trianglevalued fuzzy soft set is a parameterized family of trianglevalued fuzzy subsets of ; thus its universe is the set of all trianglevalued fuzzy sets of , that is, . Obviously, a trianglevalued fuzzy soft set is a special case of a fuzzy soft set.
3. IntervalValued Triangular Fuzzy Soft Sets
Definition 5 (see [35]). If
where , then is called an intervalvalued triangular fuzzy number and also is denoted as .
If , then is called a standard intervalvalued triangular fuzzy number. The following intervalvalued triangular fuzzy numbers described in this paper stand for standard intervalvalued triangular fuzzy numbers.
Definition 6. Suppose and are intervalvalued triangular fuzzy numbers; if , , , , and , it is said .
Definition 7. Suppose and are intervalvalued triangular fuzzy numbers; then we have
Definition 8. Suppose is an intervalvalued triangular fuzzy number; then the complementary set of is denoted as
Definition 9. Suppose and are intervalvalued triangular fuzzy numbers; for the constant , then we have
Definition 10. Let be a universe and let be a set of parameters. Suppose that stands for all the intervalvalued triangular fuzzy sets over , ; then we say that is an intervalvalued triangular fuzzy soft subset of , where mapping . For , where is the intervalvalued triangular fuzzy number of in .
Example 11. Suppose the following: is the set of the houses under consideration and
is the set of parameters and .
The tabular representation of an intervalvalued triangular fuzzy soft set is shown in Table 1. Obviously, the precise evaluation for each object on each parameter is unknown, while the intervalvalued triangular fuzzy set of such an evaluation is given. The intervalvalued triangular fuzzy soft set is given as , where

Definition 12. Assuming and are the intervalvalued triangular fuzzy soft sets on the universe , then if and only if and , is the intervalvalued triangular fuzzy subset of . For , , and , the intervalvalued triangular fuzzy numbers and satisfy the condition ; then we say that is the intervalvalued triangular fuzzy soft subset of , and it is written as . If is the intervalvalued triangular fuzzy soft subset of , then is the intervalvalued triangular fuzzy soft superset of .
Example 13. Given two intervalvalued triangular fuzzy soft sets and and a set of houses under consideration , where then, apparently, we have .
Definition 14. Let and be two intervalvalued triangular fuzzy soft sets over universe ; if and , then we say that and are equal, and it can be written as .
Definition 15. Suppose is an intervalvalued triangular fuzzy soft set over universe ; is called the complement of , where . For ,
where is the intervalvalued triangular fuzzy number of in .
Obviously, we have according to Definition 15.
Example 16. Considering another intervalvalued triangular fuzzy soft set shown in Table 2, the universe is the same as the universe in Table 1, while the set of the parameters thus we have the following formulas according to Definition 15:

Definition 17. Assuming and are intervalvalued triangular fuzzy soft sets over universe , we say that is the “AND” operation on the two sets and . For , we have where and are the corresponding intervalvalued triangular fuzzy numbers of in and , respectively.
Definition 18. Assuming and are intervalvalued triangular fuzzy soft sets over universe , we say that is the “OR” operation on the two sets, and . For , we have where and are the corresponding intervalvalued triangular fuzzy numbers of in and , respectively.
Theorem 19. Assuming and are intervalvalued triangular fuzzy soft sets over universe , then we have
Proof. and have similar proof, so only is proven in detail. Suppose
, where
For , we have
Therefore, is proven.
Theorem 20. Assuming , , and are intervalvalued triangular fuzzy soft sets over universe , then we have
Proof. Because and have similar proof, only is proven in detail. Suppose
For , , we have
where , , and are the corresponding intervalvalued triangular fuzzy numbers of in , , and , respectively.
For , , we have
Therefore, is established.
Definition 21. Assuming and are intervalvalued triangular fuzzy soft sets over universe , we say that is the interaction operation on the two sets and , where . For , we have where and are the corresponding intervalvalued triangular fuzzy numbers of in and , respectively.
Definition 22. Assuming and are intervalvalued triangular fuzzy soft sets over universe , we say that is the union operation on the two sets and , where . For , we have
Here,
where and are the corresponding intervalvalued triangular fuzzy numbers of in and , respectively.
Theorem 23. Assuming and are intervalvalued triangular fuzzy soft sets over universe , then we have
Proof. and have similar proof, so only is proven in detail. Suppose , where , and .
For , we have
Therefore, is proven.
Theorem 24. Assuming , , and are intervalvalued triangular fuzzy soft sets over universe , then we have
Proof. and have similar proof, so only is proven in detail. Suppose
where . For , we have ,
For , we have ,
For , we have ,
For , we have ,
From the law of operation among sets, we have .
Therefore, is established.
Theorem 25. Assuming , , and are intervalvalued triangular fuzzy soft sets over universe , then we have
Proof. Only prove , because and have similar proof. Assume
Obviously, . For , we have ,
For , we have
For , we have
For , we have
For , we have
Then, .
Therefore, is established.
4. Decision Making Method of IntervalValued Triangle Fuzzy Soft Set
4.1. Determination of Time Weight
In practical decision making problems, the decisionmaker masters different information at different time. Normally, the nearer the final moment of the decision is, the more information the decisionmaker masters. Thus, they have greater influence on the final decision. While the farther the final moment of the decision is, the less information the decisionmaker masters. Thus they have less influence on the final decision. That is to say, the information the decisionmaker masters increases with time; accordingly, the effect on the final decision decays forward with time. The effect of different time information on the final decision can be measured in weight, so the problem is turned into how to determine the weight of different times. Reference [33] introduced the weighting methods of basic unitinterval monotonic (BUM) function model, normal distribution model, exponential distribution model, and so forth. Therefore, the weights determination method of exponential decay model can be built on the basis of previous studies.
Theoretically, time can be divided into continuous and discrete types described, respectively, as follows.
(1) If is discrete time, is denoted. The weight of moment is ; thus, , where is constant and is attenuation coefficient.
The weight should satisfy , and we can work out