## 3 Model Design
### Making Complete Use of the Fuzzy Decision Support System
#### 3.3.1 The Fuzzifier  
In the domain \(U\), the fuzzy subset \(\widetilde{A}\) is essentially a function of \(U\rightarrow[0,1]\). Consequently, the primary task of dealing with a fuzzy situation in practice is to determine membership functions \(\widetilde{A}\left(u\right)\). There are three main methods for determining the membership function: the fuzzy statistical method, the tripartite method, and the Delphi method. The tripartite method is an experimental method that uses the idea of random intervals to study fuzziness, which is consistent with the analysis of specific application scenarios in this paper. Therefore, the tripartite method is used to determine the membership function. In working time, the factor set is \(U=\left\{u_{1},\,u_{2}\right\}\), where \(u_{1},u_{2}\) respectively represent the DI and the ST. The judgment set is \(V=\left\{v_{1},\,v_{2},v_{3}\right\}\), where \(v_{1},v_{2},v_{3}\) respectively represent low, medium, and high. In non-working hours, the factor set is \(U=\left\{u_{1},\,u_{2}\right\}\), where \(u_{1},u_{2}\) respectively represent IS and AS; the evaluation set is \(V=\left\{v_{1},\,v_{2},v_{3}\right\}\), where \(v_{1},v_{2},v_{3}\) respectively mean three different levels of membership subsets (low, medium, and high). Obviously, we can define the same membership function to calculate the degree of membership between nodes in the above two scenarios. A common function is defined as a parent function for fuzzification of working time and non-working time. We define a generic function as:  
\[\widetilde{CF}(i,j,t)=\left\{ \begin{array}{ll}DI(i,j,t)\times W_{DI}+ST(i,j, t)\times W_{ST}&t=1;\\ IS(i,j,t)\times W_{IS}+AS(i,j,t)\times W_{AS}&t=0;\end{array} \right. \tag{7}\]  
In this general function, \(t\) is set as the label to identify whether the node is in working time or non-working time. When \(t\) is equal to one, nodes \(i\) and \(j\) live in the working time, Otherwise, when \(t\) is equal to zero, nodes \(i\) and \(j\) live in the non-working time. In the above formula, \(W\) can be adjusted according to the experiment. Finally, the weight of the working time is \(W_{DI}=W_{ST}=0.5\), and the weight of the non-working time is \(W_{IS}=W_{AS}=0.5\). After standardizing \(\widetilde{CF}(i,j,t)\), the input value of the fuzzy system is obtained. \[CF(i,j,t)=\tanh(\widetilde{CF}(i,j,t))=\frac{e^{\widetilde{CF}(i,j,t)}-e^{ \widetilde{CF}(i,j,t)}}{e^{\widetilde{CF}(i,j,t)}+e^{\widetilde{CF}(i,j,t)}} \tag{8}\]  
According to the tripartite method and the attribute setting in this article, we define \(A_{low},A_{medium},A_{high}\in F(R)\), and \((U,V)  
On the basis of the theory of \(\forall CF(i,j,t)\in R,(u,v)\), the equation could be expanded as:  
\[f(u,v)(CF(i,j,t))=\left\{ \begin{array}{l}A_{low}(CF(i,j,t)),CF(i,j,t)\leq u\\ A_{medium}(CF(i,j,t)),u<\!CF(i,j,t)\leq v\\ A_{high}(CF(i,j,t)),CF(i,j,t)>v\end{array} \right. \tag{10}\]  
As demonstrated in Figure 2A-C, when the Common Function changes over time, low, medium and high membership functions present different curves and the codomain is always between 0 and 1. According to the actual application scenarios and a large number of literature references, obeying the normal distribution is set, and \(u(\xi_{1},\eta_{1}^{2})\), \(v(\xi_{2},\eta_{2}^{2})\). \[\left\{ \begin{array}{l}A_{low}(CF(i,j,t))=\int_{CF(i,j,t)}^{+\infty}P_{u}(x) dx=1-\int_{-\infty}^{CF(i,j,t)}P_{v}(x)dx=1-\varphi(\frac{CF(i,j,t)-\xi_{1}}{\eta_{1}}) \\ A_{medium}(CF(i,j,t))=1-\int_{CF(i,j,t)}^{+\infty}P_{u}(x)dx-\int_{CF(i,j,t)}^ {+\infty}P_{v}(x)dx=\varphi(\frac{CF(i,j,t)-\xi_{1}}{\eta_{1}})-\varphi(\frac {CF(i,j,t)-\xi_{2}}{\eta_{2}})\\ A_{high}(CF(i,j,t))=\int_{-\infty}^{CF(i,j,t)}P_{v}(x)dx=\varphi(\frac{CF(i,j, t)-\xi_{2}}{\eta_{2}})\end{array} \right. \tag{11}\]  
In the above formula, \(\varphi(x)\) is described as \(\varphi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}t^{2})dt\), which is the standard normal distribution function.