## 3 T-S Fuzzy Model  
Previous studies indicate that the trust value between two nodes influences the degree of information sharing between them. A higher degree of sharing correlates with increased frequency of information interaction [6; 30; 31]. For instance, ref. [30] categorizes the degree of information sharing into four levels based on trust value and provides a detailed description of the interaction status between nodes at these levels. Consequently, the trust value informs us about the strength of information interaction between nodes and the duration of any communication lag. Additionally, as trust value increases, the frequency of interaction between nodes rises, leading to reduced inventory loss from backlogs; thus, a higher trust value corresponds to a lower inventory loss rate. For simplicity, consider the supply chain system scenario consisting of two sub-chains, with \(\theta_{j}(t)(j=1,2)\) being the premise variable and \(W_{j}^{i}(i=1,2;j=1,2)\) being the fuzzy set. In this paper, \(\theta_{j}(t)=GT_{j}\) is each sub-chain's global trust value level. Nine scenarios are considered based on the high and low global trust levels of the two sub-chains, as shown in Table 1. Here, the superscript \(i=(1,2)\) in \(W_{j}^{i}\) denotes the number of sub-chains, while the subscript \(j=(1,2,3)\) represents the three trust scenarios: high, medium, and low, respectively. This framework can fully approximate the dynamic relationships among the nine scenarios in the actual product and supply process. Moreover, it enables the construction of T-S fuzzy set divisions and fuzzy rules for the nine subsystems. According to the subsystem division rule, let \(GT_{i}>1\) denote complete trust, and \(GT_{i}<0\) denote complete distrust, while \(0\leq GT_{i}\leq 1\) is the fuzzy range that lies between the two cases. A fuzzy set is established based on the above, and the affiliation function of \(GT_{i}(i=1,2)\) is \(W_{j}^{i}\), whose expression is shown in (11)-(13). \[W_{1}^{i}(GT_{i})= \begin{cases}0&0\leq\text{GT}_{i}\leq 0.5\\ 2*GT_{i}-1&0.5\leq GT_{i}\leq 1\end{cases}  \tag{11}\]  
\[W_{2}^{i}(GT_{i})= \begin{cases}2*GT_{i}&0\leq\text{GT}_{i}\leq 0.5\\ 2-2*GT_{i}&0.5\leq GT_{i}\leq 1\end{cases}  \tag{12}\]  
\[W_{3}^{i}(GT_{i})= \begin{cases}1-2*GT_{i}&0\leq\text{GT}_{i}\leq 0.5\\ 0&0.5\leq GT_{i}\leq 1\end{cases}  \tag{13}\]  
The system dynamics equations for the production process of the supply chain under different rules are as follows. Rule 1: If \(\theta_{1}(t)\) is \(W_{1}^{1}\), and \(\theta_{2}(t)\) is \(W_{2}^{1}\), then  
\[x(k+1)=(I_{N}\otimes A_{r1})x(k)+(I_{N}\otimes B\hat{K})\big{(}x(k-\tau_{1})- x(0)\big{)}-(L_{r1}\otimes Bh_{r1}\bar{K})x(k-\tau_{2})+\omega(k) \tag{14}\]  
\[\text{where }A_{r1}= \begin{bmatrix}1-\rho_{1}&0&0&0\\ 0&1-\rho_{2}&0&0\\ 0&0&1-\rho_{3}&0\\ 0&0&0&1-\rho_{4}\end{bmatrix} ,L_{r1}= \begin{bmatrix}1&-\bar{\varepsilon}\\ -\bar{\varepsilon}&1\end{bmatrix} ,\tau_{r1}(k)= \begin{bmatrix}\tau_{1}\\ \bar{\tau}_{2}\end{bmatrix} \]  
Rule 4: If \(\theta_{1}(t)\) is \(W_{1}^{2}\), and \(\theta_{2}(t)\) is \(W_{2}^{2}\), then  
\[x(k+1)=(I_{N}\otimes A_{r4})x(k)+(I_{N}\otimes B\hat{K})\big{(}x(k-\tau_{1})- x(0)\big{)} \tag{15}\]  
\[-(L_{r4}\otimes Bh_{r4}\bar{K})x(k-\tau_{2})+\omega(k)\]  
\[\text{where }A_{r4}= \begin{bmatrix}1-\rho_{1}&0&0&0\\ 0&1-\rho_{2}&0&0\\ 0&0&1-\rho_{3}&0\\ 0&0&0&1-\rho_{4}\end{bmatrix} ,L_{r4}= \begin{bmatrix}1&-\varepsilon\\ -\varepsilon&1\end{bmatrix} ,\tau_{r4}(k)= \begin{bmatrix}\tau_{1}\\ \tau_{2}\end{bmatrix} \]  
Rule 9: If \(\theta_{1}(t)\) is \(W_{1}^{3}\), and \(\theta_{2}(t)\) is \(W_{2}^{3}\), then  
\[x(k+1)=(I_{N}\otimes A_{r9})x(k)+(I_{N}\otimes B\hat{K})\big{(}x(k-\tau_{1})-x( 0)\big{)} \tag{16}\]  
\[-(L_{r9}\otimes Bh_{r9}\bar{K})x(k-\tau_{2})+\omega(k)\]  
\[\text{where }A_{r9}= \begin{bmatrix}1-\overline{\rho_{1}}&0&0&0\\ 0&1-\overline{\rho_{2}}&0&0\\ 0&0&1-\overline{\rho_{3}}&0\\ 0&0&0&1-\overline{\rho_{4}}\end{bmatrix} ,L_{r9}= \begin{bmatrix}1&-\underline {\varepsilon}\\ -\underline{\varepsilon}&1\end{bmatrix} ,\tau_{r9}(k)=