Dynamic Decision Making of Decision-Makers’ Psychological Expectations Based on Interval Triangular Fuzzy Soft Sets

Abstract

Dynamic decision-making is the process of seeking optimal choice with multiple related attributes under the multi-time-point situation. Considering that the time-varying nature of decision information can have a specific impact on the psychology of decision makers, in this paper, a dynamic decision-making method based on the cumulative prospect theory is proposed. Combining this with infinite parameterization of fuzzy soft sets, a time series interval triangular fuzzy soft set is presented, which has characteristics of boundedness, monotonicity, and symmetry. Then, based on the new information priority principle, the exponential decay model is used to determine the time weight coefficient, and a static fuzzy soft matrix is obtained. Furthermore, a method of calculating psychological utility values is proposed, in which the threshold-reference point set is introduced to analyze the psychological “profit and loss” values. Simultaneously, the time probability of the decision-making scenario is transformed into the corresponding weight function. On the basis of prospect maximization theory and maximum entropy theory, an optimization model for determining the weight of decision parameters is established. The cumulative prospect values of the alternatives are given to achieve the best choice for the alternatives. Finally, an example showed the feasibility and effectiveness of this method.

1. Introduction

Decision making is to evaluate and rank the merits and disadvantages of an alternative proposal in a fair and reasonable manner, and it has been extensively employed in practical decision-making problems, such as supplier selection [1], investment selection [2], smart medical device selection [3], signal processing [4], etc. How to effectively deal with the inherent uncertainty and fuzziness of decision-making problems and decision makers’ (DMs) evaluation information is a vital issue before deter mining the optimal alternative.

In the decision-making process, due to the inherent complexity and ambiguity of objective phenomena, it is challenging to measure the relevant decision-making information precisely using mathematical tools. Although the development of probability theory [5], rough set theory [6], and fuzzy analysis theory [7] have provided a theoretical foundation for addressing such problems, these theories still suffer from the insufficiently comprehensive nature of parameterized tools. In view of this, Molodtso [8] proposed a new theory—soft set theory. This can make full use of parametric tools, so it has been effectively promoted and used. On this basis, Santos-Buitrago, Murat, et al., gradually try to apply the soft set maximum and minimum method to solve the uncertainty and decision-making problems in various fields [9,10]. Given the unique superiority of this theory, scholars have integrated soft sets with fuzzy set theory, proposing new concepts of fuzzy soft set and intuitionistic fuzzy soft sets, highlighting their outstanding role in solving practical problems [11,12,13]. Among them, Qian Yu developed some novel Maclaurin symmetric mean operators for complex cubic q-Rung orthopair fuzzy sets [14]. Liu et al. improved the existing decision methods for fuzzy soft sets and rough soft sets. For fuzzy soft sets, a calculation tool called the D-score table was introduced to improve the decision process of the classical method. In terms of rough soft sets, several new decision algorithms were introduced to meet the needs of different decision makers, and a multi-criterion group decision-making method was proposed [15]. Yang et al. proposed a new decision-making method with entropy weight based on direct fuzzy soft sets theory, using the direct fuzzy new entropy measurement method instead of standard weight [16]. In some existing literature, linguistic variables are also converted to triangular fuzzy numbers in the decision-making process [17]. Xiaoguo Chen et al. [18] combined the interval-valued triangular fuzzy set with the soft set, and a concept of interval-valued triangular fuzzy soft set was presented. Then, they proposed the related properties of the interval triangular fuzzy soft set, and integrated the soft set of different schemes. They calculated the selection value and decision value, and obtained the optimal decision scheme from the maximum decision value [19]. At the same time, the application of the fuzzy number information fusion operator in the integration of fuzzy information is also discussed. These studies provide new ideas and methods for multi-attribute decision making [15]. Rish et al. pointed out that classical fuzzy soft sets have relative limitations in solving uncertain parameter problems. Combining the advantages of interval intuitionistic fuzzy set theory and soft set theory, they defined the interval intuitionistic fuzzy soft set and proposed intersection, union, and/or related rules and operational qualities [20].

Meanwhile, in the actual decision-making process, because of the complexity of decision problems and the limitation of decision makers’ cognition, decision makers usually have the psychological characteristics of limited rationality [21]. Considering the influence of decision makers’ psychological behavior, Peng et al. put forward two new interval-valued fuzzy soft set methods, which combine regret theory and take decision makers’ subjective preferences and objective information into consideration in decision making. These two methods can better deal with uncertainty and risk in decision making and have high practicability and feasibility [22]. Chen et al. proposed a method based on generalized fuzzy soft sets to reduce errors in the decision-making process [23]. Krohling and Souza proposed a multi-attribute decision-making method based on prospect theory [24].

However, the research mentioned above mainly focuses on ensemble algorithms and their preliminary applications in static decision-making, with less research on complex decision-making processes under dynamic risk-based decision making. In fact, the decision making is usually a dynamic and evolving process, such as the multi-stage dynamic investment decision problem [25], decision problem of emergency [26], etc. Therefore, scholars further explored the decision-making problem under the multi-time-point situation [27,28]. Usually, researchers will dynamically integrate the evaluation of each stage [29], but less research has focused on the complex psychological changes of decision makers at different stages in the dynamic decision-making process.

The time-varying nature of decision information can have a specific impact on the psychology of decision-makers. Therefore, in this paper, we propose a dynamic decision-making method based on the cumulative prospect theory, in view of the psychological behavior of decision makers regarding the dynamic evolution of expected value under time-varying uncertain information. Because of the intuition, usability, and computational simplicity of interval triangular fuzzy numbers, as well as the infinite constraints of soft set parameter settings, the interval triangular fuzzy numbers are combined with soft sets to represent decision information, and a time series interval triangular fuzzy soft set is defined. On this basis, a threshold reference point set considering the evolution of the decision maker’s psychological expected value is introduced. Furthermore, a dynamic decision-making method based on the evolution of psychological expected value is constructed, making the decision-making results more objective, reasonable, and effective. To do that, the rest of this paper is organized as follows. In Section 2, we briefly introduce the interval triangular fuzzy soft sets and their related properties. Section 3 describes the proposed dynamic fuzzy decision-making method. Section 4 investigates a numerical example, including an application to select a company with development prospects to invest in to illustrate the applicability of the proposed method. Finally, the conclusion is given in Section 5.

2. Time Series Interval Triangular Fuzzy Soft Sets and Their Related Properties

The interval triangular fuzzy number is intuitive, easy to use, and simple in calculation. This paper combines the interval triangular fuzzy numbers with soft sets to represent decision information. In this section, the definition of the interval triangular fuzzy soft sets and their related properties are introduced. Considering the time-varying characteristics of dynamic decision information, a time series interval triangular fuzzy soft set is presented.

2.1. Interval Triangular Fuzzy Soft Sets

Definition 1. 

Let $\tilde{a}=\left[{\tilde{a}}^L,{\tilde{a}}^U\right]$, the lower interval ${\tilde{a}}^L=\left(a_1^L,a_2,a_3^L\right)$, upper interval ${\tilde{a}}^U=\left(a_1^U,a_2,a_3^U\right)$ and $0<a_1^L\le a_1^U\le a_2\le a_3^L\le a_3^U\le 1$, then $\tilde{a}=\left[\left(a_1^L,a_1^U\right);a_2;\left(a_3^L,a_3^U\right)\right]$, $\tilde{a}$ is called interval triangular fuzzy number. If $a_1^L=a_1^U;a_3^L=a_3^U$, then the interval triangular fuzzy number degenerates into a regular triangular fuzzy number [17].

Definition 2. 

Let $\tilde{a}=\left[\left(a_1^L,a_1^U\right);a_2;\left(a_3^L,a_3^U\right)\right]$, $\tilde{b}=\left[\left(b_1^L,b_1^U\right);b_2;\left(b_3^L,b_3^U\right)\right]$ be interval triangular fuzzy numbers, and $\lambda$ be any positive real number [30]. Define the following operations:

$$\begin{array}{cc}\hfill \tilde{a}\oplus \tilde{b} =& [\left(a_1^L+b_1^L-a_1^L\times b_1^L,a_1^U+b_1^U-a_1^U\times b_1^U\right);\left(a_2+b_2-a_2+b_2\right);\hfill \\ \hfill & \left(a_3^L+b_3^L-a_3^L\times b_3^L,a_3^U+b_3^U-a_3^U\times b_3^U\right)]\hfill \end{array}$$

(1)

$$\lambda \tilde{a}=\left[\left(1-{\left(1-a_1^L\right)}^{\lambda },1-{\left(1-a_1^U\right)}^{\lambda }\right);1-{\left(1-a_2\right)}^{\lambda };\left(1-{\left(1-a_3^L\right)}^{\lambda },1-{\left(1-a_3^U\right)}^{\lambda }\right)\right]$$

(2)

$${\tilde{a}}^c=\left[\left(1-a_3^U,1-a_3^L\right),1-a_2,\left(1-a_1^U,1-a_1^L\right)\right]$$

(3)

Definition 2 ensures the closure of operations on normalized interval triangular fuzzy numbers.

Definition 3. 

Let $U$ be the domain of a non-empty set, $E$ be the parameter set concerning objects in $U$, and $\Gamma (U)$ represent the collection of all interval triangular fuzzy sets on the domain $U$, $A\subseteq E$. The pair $\left(F,A\right)$ is referred to as an interval triangular fuzzy soft set on the domain $U$, where $F:A$$\to$$\Gamma (U)$ is a mapping from $A$ to the collection of all interval triangular fuzzy sets on $\Gamma (U)$, i.e., $\forall e\in A$, there exists:

$$F\left(e\right)=\left\{〈x,{\tilde{a}}_{F\left(e\right)}\left(x\right)〉\left|x\in U\right.\right\}$$

(4)

In the formula, ${\tilde{a}}_{F\left(e\right)}\left(x\right)$ is the interval triangular fuzzy number corresponding to $e$ of $x$ in $F\left(e\right)$ [30].

2.2. Time Series Interval Triangular Fuzzy Soft Sets

In the dynamic decision-making process, the information of the decision system fluctuates over time. To effectively address this issue, we introduce a time series $T$, and construct a type of interval triangular fuzzy soft set that takes time sequences into account. Thus, we have:

Definition 4. 

Let $U$ be the domain of a non-empty set, $E$ be the parameter set concerning objects in $U$, $T$ be the time set, and $\Gamma (U)$ represent the entire set of interval triangular fuzzy sets on the domain $U$, $A\times T\subseteq E\times T$. The ordered pair $\left(F,A\times T\right)$ is a time series interval triangular fuzzy soft set on the domain $U$, where $F: A\times T$$\to$$\Gamma (U)$ is the mapping from $A\times T$ to $\Gamma (U)$, i.e., $\forall \left(e,t\right)\in A\times T$, there exists:

$$F\left(e,t\right)=\left\{〈x,{\tilde{a}}_{F\left(e,t\right)}\left(x\right)〉\left|x\in U\right.\right\}$$

(5)

In the formula, ${\tilde{a}}_{F\left(e,t\right)}\left(x\right)$ belongs to the interval triangular fuzzy number corresponding to the parameter $e$ of $x$ at a certain time $t$ in $F\left(e,t\right)$.

According to Definitions 3 and 4, the interval triangular fuzzy soft set is the mapping from the parameter set to the interval triangular fuzzy set $\Gamma (U)$, while the temporal interval triangular fuzzy soft set is composed of all interval triangular fuzzy sets $\Gamma (U)$ at a certain time; i.e., $\forall \left(e,t\right)\in A\times T$, $F\left(e,t\right)$ is the interval triangular fuzzy set of parameter $e$ at a certain time $t$, which constitutes the temporal interval triangular fuzzy soft set.

For convenience, the fuzzy soft set is expressed in the form of fuzzy soft matrix [18], then the domain $U=\left(x_1,x_2,\cdots ,x_m\right)$, parameter set $A=\left(e_1,e_2,\cdots ,e_n\right)$, time set $T=\left(t_1,t_2,\cdots ,t_k\right)$, and the fuzzy soft matrix of time interval triangular fuzzy soft sets $\left(F,A\times T\right)$ at time $t_p$ can be represented as ${\tilde{F}}_{t_p}={\left[{\tilde{a}}_{F(e_j,t_p)}(x_i)\right]}_{m\times n}$$, i=1,2,\cdots ,m;$$j=1,2,\cdots ,n;$$p=1,2,\cdots ,k$.

Definition 5. 

Let ${\tilde{a}}_{F(e,t)}\left(x\right)=\left[\left(a_1^L,a_1^U\right);a_2;\left(a_3^L,a_3^U\right)\right]$, ${\tilde{b}}_{G(e,t)}\left(x\right)=\left[\left(b_1^L,b_1^U\right);b_2;\left(b_3^L,b_3^U\right)\right]$ be the interval triangular fuzzy numbers corresponding to $e$ of $x$ at a certain time $t$ in $F\left(e,t\right)$ and $G\left(e,t\right)$, respectively; $l_{\overline{a}}=a_2-\overline{a_1}$, $l_{\overline{b}}=b_2-\overline{b_1}$ be the upper half value length of ${\tilde{a}}_{F(e,t)}\left(x\right)$ and ${\tilde{b}}_{G(e,t)}\left(x\right)$, respectively; and $l_{\underset{¯}{a}}=\overline{a_3}-a_2$, $l_{\underset{¯}{b}}=\overline{b_3}-b_2$ be the lower half value length of ${\tilde{a}}_{F(e,t)}\left(x\right)$ and ${\tilde{b}}_{G(e,t)}\left(x\right)$, respectively.

$$p\left({\tilde{a}}_{F(e,t)}\left(x\right)\ge {\tilde{b}}_{G(e,t)}\left(x\right)\right)=\frac{\min \left\{l_{\overline{a}}+l_{\overline{b}},\max \left(a_2-{\overline{b}}_1,0\right)\right\}}{2\left(l_{\overline{a}}+l_{\overline{b}}\right)}+\frac{\min \left\{l_{\underset{¯}{a}}+l_{\underset{¯}{b}},\max \left({\overline{a}}_3-b_2,0\right)\right\}}{2\left(l_{\underset{¯}{a}}+l_{\underset{¯}{b}}\right)}$$

(6)

$p$ is the probability of ${\tilde{a}}_{F(e,t)}\left(x\right)\ge {\tilde{b}}_{G(e,t)}\left(x\right)$, where ${\overline{a}}_i=\left(a_i^L+a_i^U\right)/2$, ${\overline{b}}_i=\left(b_i^L+b_i^U\right)/2$, (i = 1, 2, 3).

When $p\ge 0.5$, it means that two interval triangular fuzzy numbers ${\tilde{a}}_{F(e,t)}\left(x\right)\ge {\tilde{b}}_{G(e,t)}\left(x\right)$, otherwise ${\tilde{a}}_{F(e,t)}\left(x\right)<{\tilde{b}}_{G(e,t)}\left(x\right)$.

Definition 6. 

${\tilde{a}}_{F(e,t_1)}\left(x\right)=\left[\left(a_{(1,1)}^L,a_{(1,1)}^U\right);a_{(1,2)};\left(a_{(1,3)}^L,a_{(1,3)}^U\right)\right],$${\tilde{a}}_{F(e,t_2)}\left(x\right)=\left[\left(a_{(2,1)}^L,a_{(2,1)}^U\right);a_{(2,2)};\left(a_{(2,3)}^L,a_{(2,3)}^U\right)\right]$ are interval triangular fuzzy numbers corresponding to parameter $e$ under time $t_1$ and $t_2$ in $F\left(e,t\right)$, where $t_1,t_2\in T$, let

$$\begin{array}{c}d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)=\\ \frac{\sqrt{6}}{6}\sqrt{{\left(a_{(1,2)}-a_{(2,2)}\right)}^2+\left(\zeta \left({\left(a_{(1,1)}^L-a_{(2,1)}^L\right)}^2+{\left(a_{(1,1)}^U-a_{(2,1)}^U\right)}^2\right)+\left(1-\zeta \right)\left({\left(a_{(1,3)}^L-a_{(2,3)}^L\right)}^2+{\left(a_{(1,3)}^U-a_{(2,3)}^U\right)}^2\right)\right)}\end{array}$$

(7)

$d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)$ is defined as the distance between ${\tilde{a}}_{F\left(e,t_1\right)}$ and ${\tilde{a}}_{F\left(e,t_2\right)}$.

In Equation (5) $\zeta \in [0,1]$, if $\zeta$ < 0.5, it indicates risk preference; if $\zeta$ > 0.5, it indicates risk aversion; if $\zeta$ = 0.5, it indicates risk neutrality.

Theorem 1. 

${\tilde{a}}_{F(e,t_1)}\left(x\right)=\left[\left(a_{(1,1)}^L,a_{(1,1)}^U\right);a_{(1,2)};\left(a_{(1,3)}^L,a_{(1,3)}^U\right)\right],$${\tilde{a}}_{F(e,t_2)}\left(x\right)=\left[\left(a_{(2,1)}^L,a_{(2,1)}^U\right);a_{(2,2)};\left(a_{(2,3)}^L,a_{(2,3)}^U\right)\right],$${\tilde{a}}_{F(e,t_3)}\left(x\right)=\left[\left(a_{(3,1)}^L,a_{(3,1)}^U\right) ;a_{(3,2)};\left(a_{(3,3)}^L,a_{(3,3)}^U\right)\right],$ are the interval triangular fuzzy numbers corresponding to the parameter $e$ of $x$ at times $t_1$, $t_2$, and $t_3$ in $F\left(e,t\right)$, respectively. $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)$ and $d\left({\tilde{a}}_{F\left(e,t_2\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)$ are the distances between ${\tilde{a}}_{F\left(e,t_1\right)}$ and ${\tilde{a}}_{F\left(e,t_2\right)}$, and ${\tilde{a}}_{F\left(e,t_2\right)}$ and ${\tilde{a}}_{F\left(e,t_3\right)}$; then

(1)

$d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)\ge 0$, if and only if $a_{(1,1)}^L=a_{(\mathrm{2,1})}^L,a_{(\mathrm{1,2})}=a_{(\mathrm{2,2})},a_{(\mathrm{1,3})}^L=a_{(\mathrm{2,3})}^L$, $a_{(\mathrm{1,1})}^U=a_{(\mathrm{2,1})}^U,a_{(\mathrm{1,3})}^U=a_{(\mathrm{2,3})}^U$, then $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)=0$;

(2)

$d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)=d\left({\tilde{a}}_{F\left(e,t_2\right)},{\tilde{a}}_{F\left(e,t_1\right)}\right)$;

(3)

If ${\tilde{a}}_{F\left(e,t_1\right)}\le {\tilde{a}}_{F\left(e,t_2\right)}\le {\tilde{a}}_{F\left(e,t_3\right)}$, then $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)\le d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)$; $d\left({\tilde{a}}_{F\left(e,t_2\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)\le d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)$.

Proof. 

From Definitions 2 and 6, we can see that for conclusion (1),

$$\sqrt{{\left(a_{(1,2)}-a_{(2,2)}\right)}^2+\left(\zeta \left({\left(a_{(1,1)}^L-a_{(2,1)}^L\right)}^2+{\left(a_{(1,1)}^U-a_{(2,1)}^U\right)}^2\right)+\left(1-\zeta \right)\left({\left(a_{(1,3)}^L-a_{(2,3)}^L\right)}^2+{\left(a_{(1,3)}^U-a_{(2,3)}^U\right)}^2\right)\right)}\ge 0$$

(8)

then $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)>0$. At the same time, it is obvious that if and only if $a_{(1,1)}^L=a_{(2,1)}^L,a_{(1,2)}=a_{(2,2)},a_{(1,3)}^L=a_{(2,3)}^L$, $a_{(1,1)}^U=a_{(2,1)}^U,a_{(1,3)}^U=a_{(2,3)}^U$, then $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)=0$.

For conclusion (2),

$$\begin{gathered} \sqrt{{\left(a_{(1,2)}-a_{(2,2)}\right)}^2+\left(\zeta \left({\left(a_{(1,1)}^L-a_{(2,1)}^L\right)}^2+{\left(a_{(1,1)}^U-a_{(2,1)}^U\right)}^2\right)+\left(1-\zeta \right)\left({\left(a_{(1,3)}^L-a_{(2,3)}^L\right)}^2+{\left(a_{(1,3)}^U-a_{(2,3)}^U\right)}^2\right)\right)}= \\ \sqrt{{\left(a_{(2,2)}-a_{(1,2)}\right)}^2+\left(\zeta \left({\left(a_{(2,1)}^L-a_{(1,1)}^L\right)}^2+{\left(a_{(2,1)}^U-a_{(1,1)}^U\right)}^2\right)+\left(1-\zeta \right)\left({\left(a_{(2,3)}^L-a_{(1,3)}^L\right)}^2+{\left(a_{(2,3)}^U-a_{(1,3)}^U\right)}^2\right)\right)} \end{gathered}$$

(9)

then $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)=d\left({\tilde{a}}_{F\left(e,t_2\right)},{\tilde{a}}_{F\left(e,t_1\right)}\right)$.

For conclusion (3), if ${\tilde{a}}_{F\left(e,t_1\right)}\le {\tilde{a}}_{F\left(e,t_2\right)}\le {\tilde{a}}_{F\left(e,t_3\right)}$, then $a_{(1,1)}^L\le a_{(2,1)}^L$, $a_{(2,1)}^L\le a_{(3,1)}^L$, $a_{(1,1)}^U\le a_{(2,1)}^U,a_{(2,1)}^U\le a_{(3,1)}^U$, $a_{(1,2)}\le a_{(2,2)},a_{(2,2)}\le a_{(3,2)},$$a_{(1,3)}^L\le a_{(2,3)}^L,a_{(2,3)}^L\le a_{(3,3)}^L,$$a_{(1,3)}^U\le a_{(2,3)}^U,$$a_{(2,3)}^U\le a_{(3,3)}^U,$ from this we can obtain $a_{(1,1)}^L\le a_{(3,1)}^L$, $a_{(1,1)}^U\le a_{(3,1)}^U$, $a_{(1,2)}\le a_{(3,2)},$$a_{(1,3)}^L\le a_{(3,3)}^L,$$a_{(1,3)}^U\le a_{(3,3)}^U$.

Next ${\left(a_{(1,1)}^L-a_{(2,1)}^L\right)}^2\le {\left(a_{(1,1)}^L-a_{(3,1)}^L\right)}^2,$${\left(a_{(1,1)}^U-a_{(2,1)}^U\right)}^2\le {\left(a_{(1,1)}^U-a_{(3,1)}^U\right)}^2,$${\left(a_{(1,2)}-a_{(2,2)}\right)}^2\le {\left(a_{(1,2)}-a_{(3,2)}\right)}^2,$${\left(a_{(1,3)}^L-a_{(2,3)}^L\right)}^2\le {\left(a_{(1,3)}^L-a_{(3,3)}^L\right)}^2,$${\left(a_{(1,3)}^U-a_{(2,3)}^U\right)}^2\le {\left(a_{(1,3)}^U-a_{(3,3)}^U\right)}^2$.

Then, $d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_2\right)}\right)\le d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)$.

Similarly, $d\left({\tilde{a}}_{F\left(e,t_2\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)\le d\left({\tilde{a}}_{F\left(e,t_1\right)},{\tilde{a}}_{F\left(e,t_3\right)}\right)$.

Complete. □

3. Considering the Dynamic Fuzzy Decision-Making Method for the Evolution of Decision-Makers’ Psychological Expected Values

The time-varying nature of decision information can have a specific impact on the psychology of decision-makers. In this section, a dynamic fuzzy decision-making method based on the cumulative prospect theory is proposed.

3.1. Problem Description

The multi-attribute decision-making problem represented by interval triangular fuzzy soft sets is considered. Let $U=\left(x_1,x_2,\cdots ,x_m\right)$ represent the set of $m$ alternatives, and $A=\left(e_1,e_2,\cdots ,e_n\right)$ represent the set of $n$ evaluation index parameters. $\omega =\left({\omega }_1,{\omega }_2,\cdots {\omega }_n\right)$ represents the weight vector of evaluation index parameters, $T=\left(t_1,t_2,\cdots ,t_k\right)$ represents the $k$ time sets of time series T, ${\gamma }_{t_p}=\left({\gamma }_{t_1},{\gamma }_{t_2},\cdots ,{\gamma }_{t_k}\right)$ represents the time weight vector, and meets ${\gamma }_{t_p}\in \left[0,1\right],{\displaystyle \sum _{p=1}^k{\gamma }_{t_p}}=1;p=1,2,\cdots ,k$. $R_b$ and $R_c$ represent the set of benefit type attributes and cost type attributes, respectively, and $R_b\cup R_c=A,R_b\cap R_c=0$. For the evaluation index $e_j\left(e_j\in A\right)$, the decision maker gives the corresponding interval triangular fuzzy number of scheme $x_i\left(x_i\in U\right)$ at the time of $t_p$$\left(t_p\in T\right)$, as ${\tilde{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)=\left[\left(a_{(i,j,p,1)}^L,a_{(i,j,p,1)}^U\right) ;a_{(i,j,p,2)};\left(a_{(i,j,p,3)}^L,a_{(i,j,p,3)}^U\right)\right]$. Then, we can obtain the soft set matrix: ${\tilde{F}}_{t_p}={\left[{\tilde{a}}_{F(e_j,t_p)}(x_i)\right]}_{m\times n}$$,i=1,2,\cdots ,m;$$j=1,2,\cdots ,n;$$p=1,2,\cdots ,k$.

In order to eliminate the difference in decision making caused by dimensions, the cost $R_c$ and benefit $R_b$ data types are treated as follows:

$${\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)=\left\{\begin{array}{c}{\tilde{a}}_{F\left(e_j,t_p\right)}^c\left(x_i\right)e_j\in R_c;x_i\in U;t_p\in T\\ {\tilde{a}}_{F\left(e_j,t_p\right)},\left(x_i\right)e_j\in R_c;x_i\in U;t_p\in T\end{array}\right.$$

(10)

The normalized fuzzy soft matrix can be obtained from Equation (10): ${\overline{F}}_{t_p}={\left[{\overline{a}}_{F(e_j,t_p)}(x_i)\right]}_{m\times n}$, $i=1,2,\cdots ,m;$ $j=1,2,\cdots ,n;$$p=1,2,\cdots ,k$.

The problem of dynamic decision making to be solved in this paper is as follows: how to make dynamic decisions obtain the ranking and selection of alternatives according to the interval triangular fuzzy decision-making soft matrix, considering the evolution of psychological expected value, reference dependence, and other behavior problems of decision makers at different time nodes.

3.2. Determination of Time Weight in Dynamic Decision Making

The dynamic decision-making process involves multi-period decision making, and the decision makers in each period make the optimal decision at that time based on the currently known information. Obviously, the latest information is based on the existing information. Therefore, as the decision-making time in the time series increases, the time weight given to the decision-making time should be greater, and this time weight also reflects the different attention paid to the time series $T=\left(t_1,t_2,\cdots ,t_k\right)$. In order to reflect the principle of new information priority, the exponential decay method is used to determine the time weight.

Definition 7 

[29]. Let the time weight vector of time series $T=\left(t_1,t_2,\cdots ,t_k\right)$ as ${\gamma }_{t_p}=\left({\gamma }_{t_1},{\gamma }_{t_2},\cdots ,{\gamma }_{t_k}\right),p=1,2,\cdots ,k$

$${\gamma }_{t_p}=\frac{e^{\lambda t_p}\left(1-e^{\lambda }\right)}{e^{\lambda }\left(1-e^{\lambda K}\right)},p=1,2,\cdots ,k$$

(11)

 where $\lambda$ is the attenuation coefficient, and $\lambda \in \left(0,1\right)$.

3.3. Reference Point Setting Considering the Evolution of Decision Makers’ Expected Value

The decision makers make dynamic decisions at different time nodes according to the current information, which is a process of continuous updating and adjustment of decisions. At the same time, at different decision-making times, the decision makers’ attitudes are different for the two situations that exceed or are lower than the expected value, that is, “gain” and “loss”. Considering the decision makers’ psychological behavior under uncertain or incomplete information, the dynamic psychological expectation value of the decision maker for each scheme at each decision-making moment is introduced as a reference point, the response function of the decision maker to the income loss is given, and then the prospect value of each scheme is obtained as the basis of the optimal decision-making result.

The soft set represents the decision information by analyzing the evaluated object from the perspective of parameters. In this process, the decision maker will naturally produce horizontal and vertical comparisons for different time and different evaluation objects. Therefore, this paper proposes a threshold reference point set: the reference points in each stage are based on the information obtained in the previous stage. The decision maker’s expectation value not only considers the evaluation values of all schemes in the current evaluation stage, it will also generate the expected threshold for the current stage of the scheme according to the development and connection of the initial stage of the self-evaluation of the scheme.

Let $Q\left(e,t\right)=\left({\overline{q}}_{Q(e_j,t_1)}(x_i),{\overline{q}}_{Q(e_j,t_2)}(x_i),\cdots ,{\overline{q}}_{Q(e_j,t_p)}(x_i)\right)$ denote the dynamic psychological expected threshold–reference point set generated by the decision maker at each stage of the evaluation index $e_j$, where ${\overline{q}}_{Q(e_j,t_p)}(x_i)$ denotes the threshold–reference point for the evaluation index $e_j$ scheme $x_i$ at the time $t_p$. The calculation formula of the reference point is:

$$\begin{array}{c}{\overline{q}}_{Q\left(e_j,t\right)}\left(x\right)=\\ \left\{\begin{array}{c}\varphi \max \left\{\underset{p}{\max }{\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right),\left(\underset{i}{\max {\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)}\right)\right\}+\left(1-\varphi \right)\min \left\{\underset{p}{\min }{\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right),\left(\underset{i}{\min {\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)}\right)\right\}\\ e_j\in R_b;x_i\in U;t_p\in T\\ \varphi \min \left\{\underset{p}{\min }{\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right),\left(\underset{i}{\min {\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)}\right)\right\}+\left(1-\varphi \right)\max \left\{\underset{p}{\max }{\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right),\left(\underset{i}{\max {\overline{a}}_{F\left(e_j,t_p\right)}\left(x_i\right)}\right)\right\}\\ e_j\in R_c;x_i\in U;t_p\in T\end{array}\right.\end{array}$$

(12)

where $\varphi$ is the adjustment coefficient of positive and negative ideal points. When $\varphi =1$, the best value of the scheme since the evaluation period and the current stage of each scheme is selected as the reference point. When $\varphi =0$, the worst value of the scheme since the initial stage of the evaluation period and the current stage of each scheme is selected as the reference point.

The threshold reference point set for parameter set A is thus obtained:

$${\overline{Q}}_E(x)=\left\{\left.{\overline{q}}_{Q\left(e_j,t\right)}\left(x\right)\right|e_j\in A\right\}$$

(13)

3.4. Determination of Dynamic Decision Scheme

Interval triangular fuzzy soft sets of time series are the unity of interval triangular fuzzy soft sets sorted by time. Under fixed $t_p$, $\left(F,E\times \left\{t_p\right\}\right)=\left(F_p,E\right)$. At the same time, reference [14] points out that each fuzzy soft set can be expressed as a corresponding fuzzy soft matrix, so the decision-making problem on interval triangular fuzzy soft set can be transformed into an aggregation problem of the interval triangular fuzzy soft matrix, and the optimal dynamic decision-making scheme can be determined by this method.

Suppose that at time $t_p$, the decision maker obtains the psychological expectation ${\overline{q}}_{Q(e_j,t_p)}(x_i)$ according to the current situation and the development trend of the situation. Suppose that the “profit and loss value” of the evaluation index $e_j$ for ${\overline{q}}_{Q(e_j,t_p)}(x_i)$ is ${\overline{v}}_{ij}^p$. From Definitions 1, 3, 4, and Theorem 1, it can be obtained that the profit and loss value of the evaluation index $e_j$ for ${\overline{q}}_{Q(e_j,t_p)}(x_i)$ at time $t_p$ is:

$${\overline{v}}_{ij}^p=d\left({\overline{a}}_{Q(e_j,t_p)}\left(x_i\right),{\overline{q}}_{Q(e_j,t_p)}(x_i)\right),i=1,2,\cdots ,m;j=1,2,\cdots ,n;p=1,2,\cdots ,k$$

(14)

${\overline{v}}_{ij}^p$ can be used to measure the psychological benefit and loss utility of decision makers for the expected value. When ${\overline{a}}_{Q(e_j,t_p)}\left(x_i\right)\ge {\overline{q}}_{Q(e_j,t_p)}(x_i)$, it means that at time $t_p$, scheme $x_i$ has a positive “gain” in index $e_j$ relative to reference point ${\overline{q}}_{Q(e_j,t_p)}(x_i)$. When ${\overline{a}}_{Q(e_j,t_p)}\left(x_i\right)<{\overline{q}}_{Q(e_j,t_p)}(x_i)$, it means that under the time $t_p$, the index $e_j$ of scheme $x_i$ has a negative “loss” relative to the reference point ${\overline{q}}_{Q(e_j,t_p)}(x_i)$. The fuzzy utility value matrix $V^p={\left[{\overline{v}}_{ij}^p\right]}_{m\times n}$ is obtained through the ${\overline{v}}_{ij}^p$ value. According to Definition 6 and cumulative prospect theory [31], the expressions of value functions ${\overline{v}}_{ij}^{\left(+\right)p}$ and ${\overline{v}}_{ij}^{\left(-\right)p}$ are obtained:

$${\overline{v}}_{ij}^{\left(+\right)p}={\left({\overline{v}}_{ij}^p\right)}^{\alpha },{\overline{a}}_{Q(e_j,t_p)}\left(x_i\right)\ge {\overline{q}}_{Q(e_j,t_p)}(x_i);i=1,2,\cdots ,m;j=1,2,\cdots ,n;p=1,2,\cdots ,k$$

(15)

$${\overline{v}}_{ij}^{(-)p}=-\theta {\left({\overline{v}}_{ij}^p\right)}^{\beta },{\overline{a}}_{Q\left(e_j,t_p\right)}\left(x_i\right)<{\overline{q}}_{Q(e_j,t_p}(x_i);i=\mathrm{1,2},\cdots ,m;j=\mathrm{1,2},\cdots ,n;p=\mathrm{1,2},\cdots ,k$$

(16)

where ${\overline{v}}_{ij}^{\left(+\right)p}$ and ${\overline{v}}_{ij}^{\left(-\right)p}$ represent the fuzzy utility value of scheme $x_i$ when it obtains positive “gain” and negative “loss” relative to the psychological expectation value ${\overline{q}}_{Q(e_j,t_p)}(x_i)$ under index $e_j$ at time $t_P$, respectively. $\alpha ,\beta$ represent the decision maker’s psychological perception sensitivity to profit and loss, where $0\le \alpha ,\beta \le 1$. The greater the value of $\alpha ,\beta$, the more the decision-maker tends to risk preference, and the lower the sensitivity to loss. $\theta$ is a positive number greater than 0, indicating the degree of decision makers’ avoidance of loss itself. The greater the value of $\theta$, the greater the loss aversion [31].

From Equations (15) and (16), it can be seen that the psychological utility value ${\overline{v}}_{ij}^{\left(-\right)p}\le 0$, ${\overline{v}}_{ij}^{\left(+\right)p}\ge 0$ is limited to 0, and the value of ${\overline{v}}_{ij}^p$ is arranged: ${\overline{v}}_{ij1}^p\le \cdots \le {\overline{v}}_{ij\varphi }^p\le 0\le {\overline{v}}_{ij\varphi +1}^p\le \cdots \le {\overline{v}}_{ijk}^p$. Let ${\overline{v}}_{ijh}^p$ denote the psychological utility value at position $h$. When $h\in \left\{1,2,\cdots ,\phi \right\}$, then ${\overline{v}}_{ijh}^{\left(-\right)p}\le 0$. When $h\in \left\{\phi +1,\phi +2,\cdots ,k\right\}$, then ${\overline{v}}_{ijh}^{\left(+\right)p}\ge 0$. Note that ${\overline{v}}_{ijh}^p$ is the psychological utility value of $t_p$ at the evaluation time, and the time probability corresponding to $t_p$ is ${\gamma }_{t_p}$. The weight function is used to convert the time probability into weight, so as to evaluate the effectiveness of the aggregation probability of decision samples. According to the cumulative prospect theory [32], the corresponding weight coefficients formula of ${\pi }_{ijh}^{\left(-\right)}$ and ${\pi }_{ijh}^{\left(+\right)}$ is:

$${\pi }_{ijh}^{\left(-\right)}={\omega }^{-}\left({\displaystyle \sum _{p=1}^h{\gamma }_{t_p}}\right)-{\omega }^{-}\left({\displaystyle \sum _{p=1}^{h-1}{\gamma }_{t_p}}\right),h=1,2,\cdots ,\phi ;p=1,2,\cdots k$$

(17)

$${\pi }_{ijh}^{\left(+\right)}={\omega }^{+}\left({\displaystyle \sum _{p=h}^k{\gamma }_{t_p}}\right)-{\omega }^{+}\left({\displaystyle \sum _{p=h+1}^k{\gamma }_{t_p}}\right),h=\phi +1,\phi +2,\cdots ,k;p=1,2,\cdots k$$

(18)

Here,

$${\omega }^{-}\left({\displaystyle \sum _{p=1}^h{\gamma }_{t_p}}\right)=\exp \left(-{\lambda }^{-}{\left(-\ln \left({\displaystyle \sum _{p=1}^h{\gamma }_{t_p}}\right)\right)}^{\delta }\right)$$

(19)

$${\omega }^{+}\left({\displaystyle \sum _{p=h}^k{\gamma }_{t_p}}\right)=\exp \left(-{\lambda }^{+}{\left(-\ln \left({\displaystyle \sum _{p=h}^k{\gamma }_{t_p}}\right)\right)}^{\delta }\right)$$

(20)

The value of parameters in Equations (14)–(19) refers to references [21] and [31], and it is appropriate to set $\alpha =\beta =0.88$, $\theta =2.25$, ${\lambda }^{-}={\lambda }^{+}=0.8$, and $\delta =1.0$. Expression of the psychological utility prospect value $V_i$ for scheme $x_i$ at time $t_p$ is:

$$V_{ij}\left(f^{-}\right)={\displaystyle \sum _{h=1}^{\phi }{\pi }_{ijh}^{\left(-\right)}{\overline{v}}_{ijh}^{\left(-\right)P}},i=1,2,\cdots m;j=1,2,\cdots ,n;p=1,2,\cdots ,k;h=1,2,\cdots ,\phi$$

(21)

$$V_{ij}\left(f^{+}\right)={\displaystyle \sum _{h=\phi +1}^k{\pi }_{ijh}^{\left(+\right)}{\overline{v}}_{ijh}^{\left(+\right)P}},i=1,2,\cdots m;j=1,2,\cdots ,n;p=1,2,\cdots ,k;h=\phi +1,\phi +2,\cdots ,k$$

(22)

Here, $V_{ij}\left(f^{+}\right)$ and $V_{ij}\left(f^{-}\right)$ represent the positive “gain” psychological utility prospect value and the negative “loss” psychological utility prospect value of scheme $x_i$ with respect to decision makers’ psychological expectations ${\overline{q}}_{Q\left(e_j,t_p\right)}(x_i)$ under index $e_j$. The cumulative prospect value $V_{ij}\left(f\right)$ is obtained by combining these positive and negative psychological utility prospect values:

$$V_{ij}\left(f\right)=V_{ij}\left(f^{+}\right)+V_{ij}\left(f^{-}\right),i=1,2,\cdots ,m;j=1,2,\cdots ,n$$

(23)

On this basis, the final comprehensive cumulative prospect value of each scheme is aggregated based on the weights of each evaluation indicator ${\overline{V}}_i$:

$${\overline{V}}_i={\displaystyle \sum _{j=1}^n{w}_j{V}_{ij}\left(f\right),i=1,2,\cdots ,m;j=1,2,\cdots ,n}$$

(24)

In order to determine the weight of evaluation indicators $\omega =\left({\omega }_1,{\omega }_2,\cdots {\omega }_n\right)$, the utility prospect values of each evaluation data are first normalized:

$$u_{ij}=\frac{V_{ij}-\underset{i,j}{\min }\left\{V_{ij}\right\}}{\underset{i,j}{\max }\left\{V_{ij}\right\}-\underset{i,j}{\min }\left\{V_{ij}\right\}},i=1,2,\cdots ,m,j=1,2,\cdots ,n$$

(25)

Using the entropy weight method based on the value function to obtain the weights of each evaluation indicator,

$$w_j=\frac{1-\left(-\frac{1}{\ln m}{\displaystyle \sum _{i=1}^m{l}_{ij}\ln l_{ij}}\right)}{n-\left(-\frac{1}{\ln m}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^m{l}_{ij}\ln l_{ij}}}\right)}$$

(26)

Here, $l_{ij}=\frac{u_{ij}}{{\displaystyle \sum _{i=1}^m{u}_{ij}}}$, and $\underset{l_{ij}\to 0}{\lim }{l}_{ij}\ln l_{ij}=0$.

Finally, the optimal selection of alternative decision-making solutions is achieved by using the comprehensive cumulative prospect value vector ${\overline{V}}_i=\left({\overline{V}}_1,{\overline{V}}_2,\cdots ,{\overline{V}}_m\right)$.

The specific steps for dynamic decision-making considering the psychological behavior of decision-makers are as follows:

Step 1: According to Equation (11), calculate the time weight vector ${\gamma }_{t_p}=\left({\gamma }_{t_1},{\gamma }_{t_2},\cdots ,{\gamma }_{t_k}\right),p=1,2,\cdots ,k$.

Step 2: According to Equation (11), calculate the threshold reference point set for the decision maker’s psychological expectations $Q\left(e,t\right)$.

Step 3: According to Equations (12) and (14), calculate the psychological utility ‘benefit loss value ${\overline{v}}_{ij}^p$ and fuzzy utility value matrix generated by each evaluation value relative to the decision maker’s psychological expectations at the corresponding time $V_i={\left[{\overline{v}}_{ij}^p\right]}_{m\times n},$ $i=\mathrm{1,2},\cdots ,m;j=\mathrm{1,2},\cdots ,n,p=\mathrm{1,2},\cdots ,k.$

Step 4: According to Equations (15)–(20), calculate the corresponding value functions ${\overline{v}}_{ij}^{\left(+\right)p}$ and ${\overline{v}}_{ij}^{\left(-\right)p}$, decision weight functions ${\pi }_{ijh}^{\left(+\right)}$ and ${\pi }_{ijh}^{\left(-\right)}$, $i=1,2,\cdots ,m;j=1,2,\cdots ,n$.

Step 5: According to Equations (21)–(23), the cumulative prospect value of each scheme is composed of positive and negative utility prospect values $V_{ij}\left(f\right),$ $i=1,2,\cdots ,m;j=1,2,\cdots ,n$.

Step 6: According to Equation (15) and Equation (22), calculate the weight of corresponding evaluation indicators $\omega =\left({\omega }_1,{\omega }_2,\cdots {\omega }_n\right)$.

Step 7: According to Equation (24), calculate the comprehensive cumulative prospect value ${\overline{V}}_i$ for each option, sort the decision options based on the ${\overline{V}}_i$ value, and take $X_i=\max \left\{x_i\left|{\overline{V}}_i,i=1,2,\cdots ,m\right.\right\}$.

Figure 1 describes the dynamic decision-making process that takes into account the psychological expectations of decision makers:

4. Example Analysis

In this section, a numerical example is presented to show the application in the dynamic investment decision-making problem. Comparison analysis and discussion are exhibited to illustrate the applicability and effectiveness of the proposed method.

4.1. Numerical Example

In China, venture capital has made important contributions to innovative high-tech enterprises and their business development. If the venture capital industry runs smoothly, it will be conducive to the technological research and development of high-tech enterprises and the healthy and rapid development of the high-tech industry. In essence, the process of venture capital selecting projects is the process of effectively guiding social capital to enterprises with the best return. Venture capital institutions should establish a scientific and effective investment decision-making mechanism instead of relying on feelings and intuitive experience. Among them, when selecting a specific investment project, the most important link is to evaluate the project according to the evaluation index and decide whether to invest or not.

An investment company plans to choose a company with development prospects for investment. After the background industry market survey, three candidate companies with equivalent comprehensive strength are selected $U=\left(x_1,x_2,x_3\right)$ and submitted to the internal project discussion, mainly focusing on the company scale, company reputation, social benefits, and innovation ability. The company is required to have a certain scale, good reputation, and good social benefits in recent years. At the same time, due to the strong pressure of the external competitive market, the company is required to have a certain scientific and technological innovation ability. The decision makers made a comprehensive investigation on the specific situation of the four companies in four years $\left(t_1,t_2,t_3,t_4\right)$ in four aspects: company size ($e_1$, unit: 1000 people), company reputation $\left(e_2\right)$, social benefits ($e_3$, unit: 100 million yuan), and scientific and technological innovation ability $\left(e_4\right)$. $e_1$ is represented by clear numbers; $e_2$, $e_3$, and $e_4$ are represented by interval triangular fuzzy numbers; and the specific data are shown in

Table 1. Parameters of investment company $X_1$ from $t_1$–$t_4$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.01	[(0.71, 0.83); 0.86; (0.89, 0.94)]	[(0.25, 0.36); 0.39; (0.45, 0.56)]	[(0.43, 0.55); 0.61; (0.67, 0.78)]
$t_2$	0.48	[(0.67, 0.76); 0.81; (0.84, 0.88)]	[(0.54, 0.63); 0.67; (0.69, 0.78)]	[(0.43, 0.54); 0.58; (0.64, 0.75)]
$t_3$	0.26	[(0.81, 0.92); 0.93; (0.95, 1.0)]	[(0.42, 0.53); 0.59; (0.64, 0.73)]	[(0.32, 0.43); 0.51; (0.56, 0.64)]
$t_4$	0.21	[(0.51, 0.60); 0.65; (0.68, 0.73)]	[(0.34, 0.43); 0.47; (0.51, 0.64)]	[(0.51, 0.63); 0.67; (0.72, 0.82)]

,

Table 2. Parameters of investment company $X_2$ from $t_1$–$t_4$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.39	[(0.42,0.51); 0.60; (0.65,0.71)]	[(0.61,0.71); 0.79; (0.85,0.94)]	[(0.70,0.81); 0.87; (0.90,0.98)]
$t_2$	0.61	[(0.62,0.71); 0.81; (0.85,0.91)]	[(0.56,0.68); 0.74; (0.78,0.86)]	[(0.74,0.85); 0.90; (0.92,1.0)]
$t_3$	0.65	[(0.51,0.60); 0.67; (0.69,0.74)]	[(0.53,0.60); 0.65; (0.70,0.81)]	[(0.53,0.67); 0.75; (0.79,0.87)]
$t_4$	0.52	[(0.56,0.65); 0.72; (0.75,0.80)]	[(0.72,0.81); 0.86; (0.89,0.96)]	[(0.62,0.72); 0.78; (0.81,0.92)]

and

Table 3. Parameters of investment company $X_3$ from $t_1$–$t_4$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.78	[(0.52, 0.65); 0.68; (0.72, 0.84)]	[(0.43, 0.54); 0.63; (0.65, 0.73)]	[(0.69, 0.81); 0.89; (0.92, 0.99)]
$t_2$	0.91	[(0.62, 0.73); 0.76; (0.80, 0.91)]	[(0.52, 0.62); 0.67; (0.71, 0.79)]	[(0.72, 0.84); 0.92; (0.94, 1.0)]
$t_3$	0.69	[(0.43, 0.55); 0.58; (0.62, 0.75)]	[(0.62, 0.73); 0.77; (0.79, 0.86)]	[(0.53, 0.63); 0.74; (0.78, 0.88)]
$t_4$	0.99	[(0.55, 0.69); 0.72; (0.76, 0.86)]	[(0.52, 0.65); 0.71; (0.76, 0.85)]	[(0.61, 0.72); 0.79; (0.83, 0.91)]

.

The following uses the proposed decision-making method to make decisions on investing in companies with the most development prospects.

Firstly, calculate the time weight vector ${\gamma }_{t_p}$ according to Equation (11) and obtain ${\gamma }_{t_p}=\left(0.15,0.20,0.28,0.37\right)$, which the decision-maker based on his own experience for the industry environment, $\lambda$ = 0.3.

Second, according to Equation (12), calculate the psychological expectation threshold reference point of decision makers for each scheme in different periods. The results are shown in

Table 4. Reference set of psychological expectations for investment company $X_1$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.78	[(0.71, 0.83); 0.86; (0.89, 0.94)]	[(0.61, 0.71); 0.79; (0.85, 0.94)]	[(0.70, 0.81); 0.87; (0.90, 0.98)]
$t_2$	0.91	[(0.67, 0.76); 0.81; (0.84, 0.88)]	[(0.56, 0.68); 0.74; (0.78, 0.86)]	[(0.74, 0.85); 0.90; (0.92, 1.0)]
$t_3$	0.69	[(0.81, 0.92); 0.93; (0.95, 1.0)]	[(0.62, 0.73); 0.77; (0.79, 0.86)]	[(0.72, 0.84); 0.92; (0.94, 1.0)]
$t_4$	0.99	[(0.62, 0.73); 0.76; (0.80, 0.91)]	[(0.72, 0.81); 0.86; (0.89, 0.96)]	[(0.72, 0.84); 0.92; (0.94, 1.0)]

,

Table 5. Reference set of psychological expectations for investment company $X_2$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.78	[(0.71, 0.83); 0.86; (0.89, 0.94)]	[(0.61, 0.71); 0.79; (0.85, 0.94)]	[(0.70, 0.81); 0.87; (0.90, 0.98)]
$t_2$	0.91	[(0.71, 0.83); 0.86; (0.89, 0.94)]	[(0.56, 0.68); 0.74; (0.78, 0.86)]	[(0.74, 0.85); 0.90; (0.92, 1.0)]
$t_3$	0.69	[(0.81, 0.92); 0.93; (0.95, 1.0)]	[(0.62, 0.73); 0.77; (0.79, 0.86)]	[(0.53, 0.67); 0.75; (0.79, 0.87)]
$t_4$	0.99	[(0.81, 0.92); 0.93; (0.95, 1.0)]	[(0.72, 0.81); 0.86; (0.89, 0.96)]	[(0.62, 0.72); 0.78; (0.81, 0.92)]

and

Table 6. Reference set of psychological expectations for investment company $X_3$.

$\mathit{t}$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.78	[(0.71, 0.83); 0.86; (0.89, 0.94)]	[(0.61, 0.71); 0.79; (0.85, 0.94)]	[(0.70, 0.81); 0.87; (0.90, 0.98)]
$t_2$	0.91	[(0.67, 0.76); 0.81; (0.84, 0.88)]	[(0.61, 0.71); 0.79; (0.85, 0.94)]	[(0.74, 0.85); 0.90; (0.92, 1.0)]
$t_3$	0.91	[(0.81, 0.92); 0.93; (0.95, 1.0)]	[(0.62, 0.73); 0.77; (0.79, 0.86)]	[(0.74, 0.85); 0.90; (0.92, 1.0)]
$t_4$	0.99	[(0.62, 0.71); 0.81; (0.85, 0.91)]	[(0.72, 0.81); 0.86; (0.89, 0.96)]	[(0.74, 0.85); 0.90; (0.92, 1.0)]

, in which the positive expectation of investment decision on the development of the company is considered; $\varphi$ = 1.

Again, assuming that the decision maker is risk averse, $\zeta =0.6$. According to Equations (12) and (14), we obtain the psychological utility of each evaluation value relative to the psychological expectation of the decision maker at different times; “Benefit utility value matrix” $V_i={\left[{\overline{v}}_{ij}^p\right]}_{m\times n}$:

$$\begin{array}{l}{V}_1=\left[\begin{array}{cccc}-0.25& -0.04& -0.04& 0\\ -0.41& 0& 0& 0\\ -0.08& -0.15& 0& -0.04\\ -0.25& 0& -0.03& -0.02\end{array}\right], V_2=\left[\begin{array}{cccc}-0.06& 0& -0.18& -0.08\\ -0.04& 0& 0& -0.11\\ 0& 0& -0.04& -0.07\\ -0.09& -0.11& -0.17& -0.01\end{array}\right],\\ V_3=\left[\begin{array}{cccc}0& -0.09& 0& 0\\ 0& 0& 0& 0\\ -0.02& -0.10& -0.01& -0.03\\ 0& 0& 0& -0.02\end{array}\right].\end{array}$$

Furthermore, the decision weight functions ${\pi }_{ijh}^{\left(+\right)}$ and ${\pi }_{ijh}^{\left(-\right)}$ are calculated according to Equations (15)–(20). The results are shown in

Table 7. Decision weight function value ${\pi }_{ijh}^{\left(+\right)}$ and ${\pi }_{ijh}^{\left(-\right)}$.

$\mathit{t}$	${\mathit{X}}_1$				${\mathit{X}}_2$				${\mathit{X}}_3$
	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.367	0.055	0.273	0.299	0.116	0.682	0.578	0.571	0.578	0.059	0.059	0.571
$t_2$	0.682	0.062	0.461	0.578	0.367	0.267	0.267	0.116	0.394	0.340	0.340	0.116
$t_3$	0.507	0.340	0.394	0.394	0.578	0.461	0.571	0.578	0.571	0.116	0.116	0.578
$t_4$	0.116	0.116	0.267	0.507	0.394	0.394	0.116	0.267	0.507	0.394	0.394	0.267

.

Again, we can obtain the “profit and loss” utility value ${\overline{v}}_{ij}^{\left(+\right)p}$ and ${\overline{v}}_{ij}^{\left(-\right)p}$, as well as the corresponding weight functions ${\pi }_{ijh}^{\left(+\right)}$ and ${\pi }_{ijh}^{\left(-\right)}$. The cumulative prospect value $V_{ij}$ under each index of different decision-making schemes can be obtained from Formulas (21) and (22), as shown in

Table 8. Cumulative prospect value of the scheme under each evaluation index $V_{ij}$.

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$X_1$	−0.104	−0.055	−0.104	−0.046
$X_2$	−0.154	−0.029	−0.061	−0.103
$X_3$	−0.005	−0.002	−0.043	−0.008

.

Finally, the comprehensive cumulative prospect value of each decision-making scheme is calculated by Equations (23)–(26) ${\overline{V}}_i$:

$${\overline{V}}_1=0.30\times \left(-0.104\right)+0.36\times \left(-0.055\right)+0.06\times \left(-0.104\right)+0.28\times \left(-0.046\right)=-0.070$$

$${\overline{V}}_2=0.30\times \left(-0.154\right)+0.36\times \left(-0.029\right)+0.06\times \left(-0.061\right)+0.28\times \left(-0.103\right)=-0.089$$

$${\overline{V}}_3=0.30\times \left(-0.005\right)+0.36\times \left(-0.002\right)+0.06\times \left(-0.043\right)+0.28\times \left(-0.008\right)=-0.007$$

Therefore, the ranking results of all schemes are ${\overline{V}}_3$ > ${\overline{V}}_1$ > ${\overline{V}}_2$. Aiming at the investment decision-making problem of the investment company in this paper, company $X_3$ is finally selected for investment.

4.2. Comparison Analysis and Discussion

In order to verify the effectiveness and practicability of the model in this paper, the model in this paper is compared with the extended GRA method [17], the arithmetic weighted average (AWA) operator [18], and the stochastic multi-criteria decision making (SMDM) approach [22]. Among them, the extended GRA method used interval-valued triangular fuzzy numbers to express criterion values, and developed an extended grey relational analysis method for solving decision-making problems. The AWA operator used in [18] made decisions through aggregating interval-valued triangular fuzzy soft sets of different time series into a collective interval-valued triangular fuzzy soft set. The SMDM approach put forward interval-valued fuzzy soft set methods, which combine decision makers’ subjective preferences.

The extended GRA method and the AWA operator do not consider the psychological and behavioral factors of decision makers, so such decision-making problems will be transformed into traditional decision-making problems. The SMDM approach only considers the prospect preference, and the parameters of interval triangular fuzzy numbers are processed to interval-valued fuzzy numbers, which are shown in

Table 9. Parameters of investment companies from $t_1$–$t_4$.

$\mathit{t}$	${\mathit{X}}_1$				${\mathit{X}}_2$				${\mathit{X}}_3$
	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$
$t_1$	0.01	(0.71, 0.94)	(0.25, 0.56)	(0.43, 0.78)	0.39	(0.42, 0.71)	(0.61, 0.94)	(0.70, 0.98)	0.78	(0.52, 0.84)	(0.43, 0.73)	(0.69, 0.99)
$t_2$	0.48	(0.67, 0.88)	(0.54, 0.78)	(0.43, 0.75)	0.61	(0.62, 0.91)	(0.56, 0.86)	(0.74, 1.00)	0.91	(0.62, 0.91)	(0.52, 0.79)	(0.72, 1.00)
$t_3$	0.26	(0.81, 1.0)	(0.42, 0.73)	(0.32, 0.64)	0.65	(0.51, 0.74)	(0.53, 0.81)	(0.53, 0.87)	0.69	(0.43, 0.75)	(0.62, 0.86)	(0.53, 0.88)
$t_4$	0.21	(0.51, 0.73)	(0.34, 0.64)	(0.51, 0.82)	0.52	(0.56, 0.80)	(0.72, 0.96)	(0.62, 0.92)	0.99	(0.55, 0.86)	(0.52, 0.85)	(0.61, 0.91)

. The time weight is still determined by the exponential decay model, $\lambda$ = 0.3.

The decision results of the above model are compared with those of this paper, and the results are shown in

Table 10. Comparison of decision results of each model.

Method	Rank Results	Optimization
the extended GRA method [17]	$X_1>X_3>X_2$	$X_1$
the AWA operator [18]	$X_3>X_2>X_1$	$X_3$
The SMDM approach [22]	$X_3>X_1>X_2$	$X_3$
model in this article	$X_3>X_1>X_2$	$X_3$

.

It can be seen from Table 9 that the decision-making results of the model in this paper are consistent with those of the SMDM approach for the optimal scheme, which verifies the effectiveness of the model in this paper, while the decision-making results of the extended GRA method and the AWA operator are different from those in this paper because the psychological and behavioral factors of the decision-maker are considered. The results obtained by the two kinds of decision-making methods are different, which shows that whether the decision-maker’s psychological behavior is considered will affect the selection results of the scheme. The cumulative prospect theory considers the decision maker’s bounded rationality, making the results more comprehensive and objective. Compared with the SMDM approach, the advantages of this method are as follows: (1) the multi-parametric interval triangular fuzzy soft set of time series is used to represent the fuzzy decision information in different periods, which makes the dynamic decision making more flexible and comprehensive. (2) Considering the evolution characteristics of the decision maker’s psychological expected value, the selection of threshold reference point uses the parameterized decision-making idea of soft set, which is more suitable for the decision maker’s psychological behavior.

5. Conclusions

In this paper, a dynamic decision-making method based on the interval triangular fuzzy soft set under the evolution of decision makers’ psychological expected value is proposed. This method combines the infinite parameterization advantage of fuzzy soft set processing decision information, and uses the cumulative prospect theory to realize the dynamic adjustment of the decision-making scheme based on the decision-maker’s psychological utility value and the time probability of the decision-making scenario, comprehensively considering the decision-maker’s psychological behavior factors according to both the utility uncertainty of the event at different times and the dynamic evolution of the decision-maker’s psychological expectation value.

The results show that (1) using interval triangular fuzzy soft sets of time series to represent decision information is more convenient for decision makers to make comprehensive and flexible dynamic decisions. (2) The threshold reference point of dynamic psychological expectation is set under different decision-making scenarios, so that each scheme generates corresponding profit and loss values relative to the threshold reference point. At the same time, combined with the time probability of the decision-making scheme under different decision-making scenarios, the prospect value obtained from the interaction of the two can better reflect the benefits and losses of the scheme. This treatment not only takes into account the decision-maker’s limited rational psychology, but also effectively reduces the impact of uncertain information on decision making. It can be applied to multi-stage dynamic investment decision problems, decision problems of emergency, supplier selection problems and other practical problems.

However, the related properties, operational laws and functions of time series interval triangular fuzzy soft set need further discussion. Therefore, future research will continue to enrich the soft set representation under time-varying information and dynamic decision-making. In addition, we will further study the psychological evolution mechanism of decision makers in the process of dynamic decision making.