#### Abstract

A concept of interval-valued 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 interval-valued triangular fuzzy soft set, in which period weight is determined by the exponential decay method. The arithmetic weighted average operator of interval-valued triangular fuzzy soft set is given by the aggregating thought, thereby aggregating interval-valued triangular fuzzy soft sets of different time-series into a collective interval-valued 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 . While a wide variety of mathematical disciplines like probability theory, fuzzy set theory , rough set theory , and interval mathematics  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 . Consequently, Molodtsov initiated the concept of soft theory as a mathematical tool, which is free from the above difficulties for dealing with uncertainties .

Soft set theory has received much attention and in-depth study since its introduction. Maji et al.  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.  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.  proposed the concept of the interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set models. Jiang et al.  presented the theory of interval-valued intuitionistic fuzzy soft set by combining the interval-valued intuitionistic fuzzy set and soft set. Feng et al.  made a primary discussion of soft set through combining the rough set and fuzzy set. Ali  raised the concept of rough soft set by combining rough set and soft set. Xu et al.  used vague set  to make an extension of soft set and proposed vague soft set theory. Majumdar and Samanta  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 . Xiao et al.  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  pretended some errors which had been discussed by Kong et al. . Çağman and Enginoğlu  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  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.  and Jiang et al.  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.  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 interval-valued 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 interval-valued 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 interval-valued triangle fuzzy soft set which considers the time variate and whose time weights can be determined by exponential decay method of . Using the thought of integration , the paper defines the arithmetic weighted average operator of interval-valued triangle fuzzy soft set and gives methods to integrate the dynamic interval-valued 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 interval-valued triangle fuzzy soft set. Then the drawn solution formula is applied to decision analysis.

The purpose of this paper is to combine the interval-valued triangular fuzzy set and soft set, from which we can obtain a new soft set model: interval-valued triangular fuzzy soft set. To facilitate our discussion, we first introduce the standard soft set, fuzzy soft set, interval-valued fuzzy soft set, and triangular fuzzy soft set in Section 2. The concept of interval-valued triangular fuzzy soft set is presented in Section 3. Then the “AND,” “OR,” intersection, union, and complement operations on the interval-valued triangular fuzzy soft sets are defined. In Section 4, the interval-valued 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, interval-valued fuzzy soft set, and triangular fuzzy soft set. Further details could be found in . Molodtsov  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 ). 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 ). 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 interval-valued fuzzy soft set. Firstly, let us briefly introduce the concept of the interval-valued fuzzy set of .

An interval-valued fuzzy set on a universe is a mapping such that , where stands for the set of all closed subintervals of ; the set of all interval-valued 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 ). Let be an initial universe and let be a set of parameters. denotes the set of all interval-valued fuzzy sets of . Let . A pair is an interval-valued 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 interval-valued fuzzy set as

Definition 4 (see ). 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 triangle-valued fuzzy soft set over , where is a mapping given by .
A triangle-valued fuzzy soft set is a parameterized family of triangle-valued fuzzy subsets of ; thus its universe is the set of all triangle-valued fuzzy sets of , that is, . Obviously, a triangle-valued fuzzy soft set is a special case of a fuzzy soft set.

#### 3. Interval-Valued Triangular Fuzzy Soft Sets

Definition 5 (see ). If where , then is called an interval-valued triangular fuzzy number and also is denoted as .
If , then is called a standard interval-valued triangular fuzzy number. The following interval-valued triangular fuzzy numbers described in this paper stand for standard interval-valued triangular fuzzy numbers.

Definition 6. Suppose and are interval-valued triangular fuzzy numbers; if , , , , and , it is said .

Definition 7. Suppose and are interval-valued triangular fuzzy numbers; then we have

Definition 8. Suppose is an interval-valued triangular fuzzy number; then the complementary set of is denoted as

Definition 9. Suppose and are interval-valued 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 interval-valued triangular fuzzy sets over , ; then we say that is an interval-valued triangular fuzzy soft subset of , where mapping . For , where is the interval-valued 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 interval-valued triangular fuzzy soft set is shown in Table 1. Obviously, the precise evaluation for each object on each parameter is unknown, while the interval-valued triangular fuzzy set of such an evaluation is given. The interval-valued triangular fuzzy soft set is given as , where

Definition 12. Assuming and are the interval-valued triangular fuzzy soft sets on the universe , then if and only if and , is the interval-valued triangular fuzzy subset of . For , , and , the interval-valued triangular fuzzy numbers and satisfy the condition ; then we say that is the interval-valued triangular fuzzy soft subset of , and it is written as . If is the interval-valued triangular fuzzy soft subset of , then is the interval-valued triangular fuzzy soft superset of .

Example 13. Given two interval-valued triangular fuzzy soft sets and and a set of houses under consideration , where then, apparently, we have .

Definition 14. Let and be two interval-valued 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 interval-valued triangular fuzzy soft set over universe ; is called the complement of , where . For , where is the interval-valued triangular fuzzy number of in .
Obviously, we have according to Definition 15.

Example 16. Considering another interval-valued 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 interval-valued 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 interval-valued triangular fuzzy numbers of in and , respectively.

Definition 18. Assuming and are interval-valued 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 interval-valued triangular fuzzy numbers of in and , respectively.

Theorem 19. Assuming and are interval-valued 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 interval-valued 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 interval-valued triangular fuzzy numbers of in , , and , respectively.
For , , we have
Therefore, is established.

Definition 21. Assuming and are interval-valued 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 interval-valued triangular fuzzy numbers of in and , respectively.

Definition 22. Assuming and are interval-valued 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 interval-valued triangular fuzzy numbers of in and , respectively.

Theorem 23. Assuming and are interval-valued 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 interval-valued 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 interval-valued 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 Interval-Valued Triangle Fuzzy Soft Set

##### 4.1. Determination of Time Weight

In practical decision making problems, the decision-maker masters different information at different time. Normally, the nearer the final moment of the decision is, the more information the decision-maker masters. Thus, they have greater influence on the final decision. While the farther the final moment of the decision is, the less information the decision-maker masters. Thus they have less influence on the final decision. That is to say, the information the decision-maker 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  introduced the weighting methods of basic unit-interval 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 . Thus, Particularly, , ; thus weight of time is

(2) If is continuous time, is denoted. The weight of time is ; thus, , where is constant and is attenuation coefficient.

The weight should satisfy ; we can work out . Thus we have

Attenuation coefficient in the model has important practical significance; it reflects accumulative rate of decision-maker’s amount of information over time. If accumulative rate at the final moment is higher, it indicates that information which is nearer to the final moment has more effect on the final decision, and the attenuation coefficient is larger; otherwise, the coefficient is smaller. So, decision-maker can determine the value of attenuation coefficient according to the change of information at different times. This weight determining method can be more flexibly applied to practice.

##### 4.2. Integrated Method of Interval-Valued Triangle Fuzzy Soft Set

The integrated thought in  is used to similarly define the arithmetic weighted average operator of interval-valued triangle fuzzy soft set. Then interval-valued triangle fuzzy soft sets of different time are integrated into comprehensive interval-valued triangle fuzzy soft set. According to , fuzzy soft set and fuzzy soft matrix are one to one correspondence, so the discussion of the integrated method of interval-valued triangle fuzzy soft set is turned into the research of the integrated interval-valued triangle fuzzy soft matrix.

Let denote the domain and let denote the parameter set; the matrix is corresponding with interval-valued triangle fuzzy set , where is the interval-valued triangle fuzzy number that is corresponding with in ; the operation rules of interval-valued triangle fuzzy soft matrix are defined as follows.

Definition 26. Suppose and are two interval-valued triangle fuzzy soft matrices, constant ; then

Definition 27. Suppose is an interval-valued triangle fuzzy soft matrix, and the weight satisfies and ; then the arithmetic weighted average operator of interval-valued triangle fuzzy soft set is

Theorem 28. Suppose is an interval-valued triangle fuzzy soft matrix, and the weight satisfies and . is the interval-valued triangle fuzzy number which is corresponding with in ; then where

Proof. Use mathematical induction to prove that, when , there are
Then , where
Similarly, satisfy formula (46).
Suppose the formula holds up when ; then when , we have Similarly