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.
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 [
Soft set theory has received much attention and indepth study since its introduction. Maji et al. [
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 [
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
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 [
A pair
Let
Let us denote
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 [
An intervalvalued fuzzy set
Let
Let us denote
Let
A trianglevalued fuzzy soft set is a parameterized family of trianglevalued fuzzy subsets of
If
If
Suppose
Suppose
Suppose
Suppose
Let
Suppose the following:
The tabular representation of an intervalvalued triangular fuzzy soft set
An intervalvalued triangular fuzzy soft set




































Assuming
Given two intervalvalued triangular fuzzy soft sets
Let
Suppose
Obviously, we have
Considering another intervalvalued triangular fuzzy soft set
An intervalvalued triangular fuzzy soft set





























Assuming
Assuming
Assuming
Therefore,
Assuming
Because
For
Therefore,
Assuming
Assuming
Here,
Assuming
For
Therefore,
Assuming
From the law of operation among sets, we have
Therefore,
Assuming
Only prove
Obviously,
Then,
Therefore,
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 [
Theoretically, time
(1) If
The weight
(2) If
The weight
Attenuation coefficient
The integrated thought in [
Let
Suppose
Suppose
Suppose
Use mathematical induction to prove that, when
Then
Similarly,
Suppose the formula holds up when
From the above we can know that formula (
For simplicity,
Both [
Let
Determine the attenuation coefficient
Use formula (
Use formulas (
Make decision making analysis based on the decision values; the corresponding element of the biggest decision value is the best one.
A venture capital firm considers choosing one of the 4 alternative enterprises
Evaluation value of the first year.





















Evaluation value of the second year.





















Evaluation value of the third year.





















Let the attenuation coefficient be
Comprehensive intervalvalued triangle fuzzy soft set.





















Selection values of each enterprise calculated based on formula (
Decision values of each enterprise calculated based on formula (
Because
In order to improve fuzzy soft set theory, the concept of intervalvalued triangle fuzzy soft set is presented; in addition, some relevant operational properties are given and proven. Because the dynamic decision model based on intervalvalued triangle fuzzy soft considers the influence of time variation, the decision making process is more realistic, and the decision making result got by integrated operation is more reliable. At last, the practical example analysis shows that the decision making method presented in this paper is effective and that programming is easy to do with it. Besides, it is necessary to point out that the research method by intervalvalued triangle fuzzy soft set can be extended to other methods, like the intervalvalued trapezoidal fuzzy soft set theory.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Natural Science Foundation of China (no. 51105135) and young talents funded projects of Heilongjiang University of Science and Technology (no. 20120501).