## 3 Model
### Event-Based Supply Chain Partnership Integration Model
#### 3.2.6 Supplier Performance Updating and Partnership Integration  
The purpose of partnership integration is to select appropriate suppliers to collaborate strategically and provide operational benefits. Depending upon the nature of the supply chain partnership integration problems, we formulated a multi-objective model that contains four objectives. Theobjectives of selecting suitable suppliers are to maximize trustworthiness \(T\), supplier service \(\rho\), qualified products rate \(\theta\), and to minimize cost \(C\). The objective functions are shown as follows:  
\[\text{Max T}=\sum_{i=1}^{n}\sum_{j=1}^{m}T_{i,j}u_{ij} \tag{11}\]  
\[\text{Max }\rho=\sum_{i=1}^{n}\sum_{j=1}^{m}\rho_{i,j}u_{ij} \tag{12}\]  
\[\text{Max }\theta=\sum_{i=1}^{n}\sum_{j=1}^{m}\theta_{i,j}u_{ij} \tag{13}\]  
\[\text{Min C}=\sum_{i=1}^{n}\sum_{j=1}^{m}C_{i,j}u_{ij} \tag{14}\]  
Constraints:  
\[\sum_{j=1}^{m}u_{ij}=1,i=1,2,\ldots,n \tag{15}\]  
\[u_{ij}=\left\{ \begin{array}{ll}1,&\text{ if supplier $s_{i,j}$ is choosen}\\ 0&\text{ otherwise}\end{array} \right. \tag{16}\]  
Equation (11) defines the total trustworthiness of the supply chain, Equation (12) defines the total supplier service in the prescribed time, and Equations (13) and (14) give the objectives about qualified product rate and the total cost of the supply chain. Constraint (15) guarantees choosing only one supplier from each stage, and constraint (16) is a variable constraint. In order to select suppliers to meet all the objectives easily, we use the linear-weighted method to deal with multiple objective optimization. By means of the technique for order preference by similarity to an ideal solution (TOPSIS) method, the multi-objective optimization supply chain partnership integration problem is reformulated into a single objective nonlinear programming problem. TOPSIS finds out the positive ideal solution and the negative ideal solution of multiple targets after the normalization of data standardization matrix, and then calculates the distance between evaluation targets and ideal targets to find the best solutions which have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution [38]. For the four objectives of the model, their positive ideal solutions and negative ideal solutions are \((T_{\text{max}},\rho_{\text{max}},\theta_{\text{max}},C_{\text{max}})\) and \((T_{\text{min}},\rho_{\text{min}},\theta_{\text{min}},C_{\text{min}})\), respectively. The distance \(d_{ij}^{+}\) and \(d_{ij}^{-}\) of each evaluation target from the positive ideal solution and negative ideal solution are the following:  
\[d_{ij}^{+}=w_{1}\frac{T_{\text{max}}-T_{i,j}}{T_{\text{max}}+T_{\text{min}}}+ w_{2}\frac{\rho_{\text{max}}-\rho_{i,j}}{\rho_{\text{max}}+\rho_{\text{min}}}+w_{3} \frac{\theta_{\text{max}}-\theta_{i,j}}{\rho_{\text{max}}+\theta_{\text{min} }}+w_{4}\frac{C_{\text{max}}-C_{i,j}}{C_{\text{max}}+C_{\text{min}}} \tag{17}\]  
\[d_{ij}^{-}=w_{1}\frac{T_{i,j}-T_{\text{min}}}{T_{\text{max}}+T_{\text{min}}}+ w_{2}\frac{\rho_{i,j}-\rho_{\text{min}}}{\rho_{\text{max}}+\rho_{\text{min}}}+w_{3} \frac{\theta_{i,j}-\theta_{\text{min}}}{\rho_{\text{max}}+\theta_{\text{min}} }+w_{4}\frac{C_{i,j}-C_{\text{min}}}{C_{\text{max}}+C_{\text{min}}} \tag{18}\]  
We use the linear normalization instead of vector normalization to reduce the complexity of computing when the numbers of variables and indices are large. Evaluation of supplier \(s_{i,j}\) can be given as:  
\[z_{ij}=\frac{d_{ij}^{-}}{d_{ij}^{+}+d_{ij}^{-}} \tag{19}\]  
There are two categories of method to set the index weights. One is the subjective method, such as Delphi and AHP, which determines the relative importance of each index according to the experts'knowledge and experience.